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(§t>o Reprint Editio.n^S) Analytical Mechanics A Comprehensive Treatise on the Dynamics of Constrained Systems John G Papastavridjs (Jbo Reprint Edition Analytical Mechanics A Comprehensive Treatise on the Dynamics of Constrained Systems WWW.EBOOK777.COM This page intentionally left blank WWW.EBOOK777.COM (£bo Reprint Edition Analytical Mechanics A Comprehensive Treatise on the Dynamics of Constrained Systems John G. Papastavridis, PhD. v> World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI WWW.EBOOK777.COM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Papastavridis, J. G. (John G.) Analytical mechanics : a comprehensive treatise on the dynamics of constrained systems / by John G Papastavridis. — Reprint edition, pages cm Reprint of 2002 edition. Includes bibliographical references and index. ISBN 978-981-4338-71-4 (hardcover : alk. paper) 1. Mechanics, Analytic. I. Title. QA805.P355 2014 531.01*515—dc22 2013017316 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. This edition is a corrected reprint of the work first published by Oxford University Press in 2002. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore by Mainland Press Pte Ltd. WWW.EBOOK777.COM To the living and loving memory of my father, GEORGE S. PAP AST A VRIDIS (rEOPriOY £. ITAITALTAYPIAH) A lawyer and fearless maverick, who, throughout his life, fought with fortitude, conviction, and class to better his world; and to whom I owe a critical part of my Weltanschauung. [The author’s father is shown here in the traditional Greek foustanella, in Athens, Greece (Themistokleous street, near Lqfos Strefi ), ca.1924] WWW.EBOOK777.COM PREFACE TO THE CORRECTED REPRINT This is a corrected reprint of a work first published in early 2002, by Oxford University Press, and which went out of print shortly thereafter. A few sign misprints and similar errors have been corrected; some notations have been, hopefully, improved (especially in Chs. 1, 2); a useful addition has been made on p. 336, and a couple of sections have been thoroughly revised (e.g. §3.12, §8.13). I am grateful to (a) the many reviewers, in some of the most prestigious professional journals and elsewhere (e.g. Bulletin of the American Mathematical Society, IEEE Control Systems Magazine, Zentralhlatt fiir Mathematit, amazon.com and amazon.co.uk, private communications, and references in advanced works of mechanics), for their enthusiastic comments; (b) the American Association of Publishers for selecting, in January 2003, the book for their “Annual Award for Outstanding Professional and Scholarly Titles of 2002, in Engineering”; and (c) last but not least, my WSPC editor, Dr. S. W. Lim, and his most capable and courteous staff, for their continuous and effective support. All these have been of essential moral and practical help to me in the making of this “new” edition! Here, I take the opportunity to restate that, in this book, I have: (a) Sought to combine the best of the old and new, i.e. no age discrimination; no knee-jerk disdain for “dusty old stuff” nor automatic following of “progress/modernity” — even in the exact sciences, and especially in mechanics, new is not necessarily or uniformly better; and (b) Avoided developments of considerable but nevertheless purely mathematical interest, especially those of the a-historical and intuition-deadening (“epsilonic”) type. May this treatise, as well as my other two works on mechanics [“Tensor Calculus and Analytical Dynamics ” (CRC, 1999) and “ Elementary Mechanics ” (WSPC, under production)], keep making many and loyal readers! John G. Papastavridis Atlanta, Georgia, Spring 2014 WWW.EBOOK777.COM PREFACE Many of the scientific treatises today are formulated in a half- mystical language, as though to impress the reader with the uncomfortable feeling that he is in the permanent presence of a superman. The present book is conceived in a humble spirit and is written for humble people. (Lanczos, 1970, pp. vii-viii) GENERAL DESCRIPTION This book is a classical and detailed introduction to advanced analytical mechanics (AM), with special emphasis on its basic principles and equations of motion, as they apply to the most general constrained mechanical systems with a finite number of degrees of freedom (this term is explained in Chapter 2). For the reasons detailed below, and in spite of the age of the subject, I think that no other single volume exists, in English and in print, that is comparable to the one at hand in breadth and depth of the material covered — and, in this nontrivial sense, this ca. 1400-page and 174-figure long work is unicpie. The book is addressed to graduate students, professors, and researchers, in the areas of applied mechanics, engineering science, and mechanical, aerospace, struc¬ tural, (even) electrical engineering, as well as physics and applied mathematics. Advanced undergraduates are also very welcome to browse, and thus get initiated into higher dynamics. The sole technical prerequisite here, a relatively modest one, is a solid working knowledge of “elementary/intermediate” (i.e., undergraduate) dynamics; roughly, equivalent to the (bulk of the) material covered in, say, Spiegel’s Theoretical Mechanics, part of the well-known Schaum’s outline series. Also, familiarity with the simplest aspects of Lagrange’s equations, that is, how to take the partial and total derivatives of scalar energetic functions, would be helpful; although, strictly, it is not necessary. [See also “Suggestions to the Reader” (Introduction, §3).] CONTENTS Specifically, the book covers in what I consider to be a most logical and pedagogical sequence, the following topics: Introduction: Introduction to analytical mechanics, brief summary of the history of theoretical mechanics; suggestions to the reader; and list of symbols/notations, abbre¬ viations, and basic formulae. WWW.EBOOK777.COM PREFACE Chapter 1: Background'. Algebra of vectors and Cartesian tensors, and basic concepts and equations of Newton-Euler (or momentum) mechanics of particles and rigid bodies; that is, a highly selective compendium of undergraduate dynamics, and (some of) its mathematics, from a mature viewpoint. Chapter 2: Kinematics of constrained systems (i.e., Lagrangean kinematics); including the general theory of up to linear velocity (i.e., Pfaffian) constraints, in both holonomic (or true) and nonholonomic (or quasi) coordinates', and a uniquely readable account of the fundamental theorem of Frobenius, for testing the nonholonomicity of such con¬ straints. Chapter 3: Kinetics of constrained systems (i.e., Lagrangean kinetics); including the fundamental principles of AM; that is, those of d’Alembert-Lagrange and of relaxation of the constraints, the central equation of Heun-Hamel; equations of motion with or without reactions, with or without multipliers, in true or quasi system variables; an introduction to servoconstraints (theories of Appell-Beghin, et al.); and rigid-body applications. This is the key chapter of the entire book, as far as engineering readers are concerned. Chapter 4: Impulsive motion, under ideal constraints; including the associated extremum theorems of Carnot, Kelvin, Bertrand, Robin, et al. Chapter 5: Nonlinear nonholonomic constraints; that is, kinematics and kinetics under nonlinear, and generally nonholonomic, velocity constraints. Chapter 6: Differential variational principles, of Jourdain, Gauss, Hertz, et al., and their derivative higher-order equations of motion of Nielsen, Tsenov, et al. Chapter 7: Time-integral theorems and variational principles, of Lagrange, Hamilton, Jacobi, O. Holder, Voss, Suslov, Voronets, Hamel, et ah; for linear and nonlinear velocity constraints in true and quasi variables, with or without multipliers; plus energy and virial theorems. Chapter 8: Introduction to Hamiltonian/Canonical methods; that is, equations of Hamilton and Routh-Helmholtz, cyclic systems, steady motion and its stability, varia¬ tion of constants, canonical transformations and Poisson’s brackets, Hamilton-Jacobi integration theory, integral invariants, Noether’s theorem, and action-angle variables and their applications to adiabatic invariants and perturbation theory. Chapters 2-8 each contain a large number of completely solved examples, and problems with their answers (and, occasionally, hints), to illustrate and extend the previous theories; short ones are integrated within each chapter section, and longer, more synthetic, ones are collected at each chapter’s end; and also, critical comments/ references for further study. The exposition ends with a relatively extensive, cumu¬ lative, and alphabetical list of References and Suggested Reading, including every¬ thing from standard textbooks all the way to epoch-making memoirs of the last (more than) two hundred years. This list complements those found in such well- known references as Neimark and Fufaev (1967/1972) and Roberson and Schwertassek (1988). Parts of the text have, unavoidably, state-of-the-art flavor. However, as far as fundamental ideas go, very little, if anything, of the topics covered is truly new — today, no one can claim much originality in classical mechanics! The newness here, a nontrivial one, I think, consists in restoring, clarifying, putting together, and pre¬ senting, in what I hope is a readable form, material most of which has appeared over the past one hundred fifty, or so, years; frequently in little known, and/or hard to find and decipher, sources. (In view of the thousands of books, lecture notes, articles, and so on, used in the writing of this work, failure to acknowledge an author’s WWW.EBOOK777.COM PREFACE particular contribution is not intentional, merely an oversight.) But, given the aston¬ ishing unfamiliarity, confusion, and intellectual provincialism so prevalent in many theoretical and applied mechanics circles today, even in the fundamental concepts and principles of analytical dynamics (like virtual displacements/work and principle of d’Alembert-Lagrange, which is, by far, the most misunderstood “principle” of physics!), I felt very strongly that this noble, beautiful, and powerful body of knowl¬ edge, that diamond of our cultural heritage, should be accurately preserved and represented, so as to benefit present and future workers in dynamics. No single volume can even pretend to cover satisfactorily all aspects of this vast and fascinating subject; in particular, both its theoretical and applied aspects, let alone the currently popular computational ones. Since this is not an encyclopedia of theoretical and applied dynamics, an inescapable and necessary selection has oper¬ ated, and so, the following important topics are not covered: applications of differ¬ ential forms/exterior calculus (of Cartan, Gallissot, et al.) and symplectic geometry to Lagrangean and Hamiltonian mechanics; group theoretic applications; nonlinear dynam¬ ics (inch regular and stochastic/chaotic motion) and stability of motion; theory of orbits; and computational/numerical techniques. For all these, there already exists an enormous and competent literature (see “Suggestions to the Reader”). However, with the help of this treatise, the conscientious reader will be able to move quickly and confidently into any par¬ ticular theoretical and/or computational area of modern dynamics. In this sense, the work at hand constitutes an optimal investment of the reader’s precious energies. RAISON D'ETRE, AND SOME PHILOSOPHY The customary words of explanation, or apology, for writing “another” book on advanced dynamics are now in order. The main theme of this work, like a Wagnerian leitmotiv, is deductive order, formal structure, and physical ideas, as they pertain to that particular energetic form of mechanics of constrained systems founded by Lagrange and known as analytical (= deductive) mechanics; to be differentiated from the also analytical but momentum, or “elementary,” form of system mechanics, founded by Euler. It is a book for people who place theory (theories), ideas, knowledge, and understanding above all else—and do not apologize for it. Here, AM is studied not as the “maid” of some other (allegedly) more important discipline, but as a sub¬ ject worth knowing in its own right; that is, as a “king or queen.” As such, it will attract those with a qualitative and theoretical bent of mind; while it may not be as agreeable to those with purely computational and/or intellectually local predilections. [In the words of the late Professor R.M. Rosenberg (University of California, Berkeley): “The held of dynamics is plowed by two classes of people: those who enjoy the inherent beauty, symmetry and consistency of this discipline, and those who are satisfied with having a machine that manufactures equations of motion of complex mechanisms” (private communication, 1986).] Generally, science is more than a collection of particular problems and special techniques, even involved ones — it is much more than mere information. However, practical people should be reminded that theory and application are mutually complementary rather than adversarial; in fact, contemporary important practical problems and the availability of powerful computational capabilities have made the thorough understanding of the fundamen¬ tal principles of mechanics more necessary today than before. Applications and computers have, among other things, helped resurrect, restore, and sharpen old academic curiosities (for engineers anyway), such as the differential variational WWW.EBOOK777.COM PREFACE principles of Jourdain and Gauss (which have found applications in such “un¬ related” areas as multibody dynamics, nonlinear oscillations, even the elasto-plastic buckling of shells); and Hamilton’s canonical equations in quasi variables (which have found applications in robotic manipulators). A more concrete reason for writing this book is that, outside of the truly monu¬ mental British treatise of Pars (1965) and the English translations of the beautiful (former) Soviet monographs of Neimark and Fufaev (1967/1972) and Gantmacher (1966/1970), there is no comprehensive exposition of advanced engineering-oriented dynamics in print, in the entire English language literature! True, the famous treatise of Whittaker (1904/1917/1927/1937), for many years out of print, has recently been reprinted (1988). However, even Whittaker, although undeniably a classic and in many respects the single most influential dynamics volume of the twentieth century (primarily, to celestial and quantum mechanics), nevertheless leaves a lot to be desired in matters of logic, fundamental principles, and their earthly applications; for example, there is no clear and general formulation of the principle of d’Alembert-Lagrange and its use, in connection with Hamel’s method of quasi variables, to uncouple the equations of motion and obtain constraint reactions; also, Whittaker would be totally unacceptable with the better of today’s educational philosophies. Such drawbacks have plagued most British texts of that era; for exam¬ ple, the otherwise excellent works of Thomson/Tait, Routh, Lamb, Ramsey, Smart, and many of their U.S.-made descendants. [In a way, Whittaker et al. have been pretty lucky in that most of the great continental European works on advanced dynamics — for example, those of Boltzmann, Heun, Maggi, Appell, Marcolongo, Suslov, Nordheim et al. (vol. 5 of Handhuch der Physik, 1927), Winkelmann (vol. 1 of Handbuch der Physikalischen und Technischen Mechanik , 1929), Prange (vol. 4 of Encyclopddie der Mathematischen Wissenschaften, 1935), Rose, Hamel, Peres, Lur’e, et al. were never translated into English.] Next, the comprehensive three-volume work of MacMillan (late 1920s to early 1930s) and the encyclopedic treatise of Webster (early 1900s), probably the two best U.S.-made mechanics texts, are, unfor¬ tunately, out of print. The very lively and deservedly popular monograph of Lanczos (1949-1970) does not go far enough in areas of engineering importance; for example, nonholonomic variables and constraints; and, also, lacks in examples and problems. Only the excellent encyclopedic article of Synge (1960) comes close to our objectives; but, that, too, has Lanczos’ drawbacks for engineering students and classroom use. The existing contemporary expositions on advanced dynamics, in English and in print, fall roughly into the following three groups: Formalistic I Abstract, of the by-and-for-mathematicians variety, and, as such, of next to zero relevance and/or usefulness to most nonmathematicians. Considering the high mental effort and time that must be expended toward their mastery vis-a-vis their meager results in understanding mechanics better and/or solving new and nontrivial problems, these works represent a pretty poor investment of ever scarce intellectual resources; that is, they are not worth their “money.” The effort should be commensurate to the returns. And, contrary to the impression given by authors of this group, even in the most exact sciences, books are written by and for concrete people; not by super- logical, detached, and cold machines. As Winner puts it: "The accepted form of ‘objec¬ tivity’ in scientific and technical reports (one can also include books and articles in social science) requires that the prose read as if there were no person in the room when the writing took place” (1986, p. 71). Also, I categorically reject soothing apologies of the type “oh, well, that is a book for mathematicians”; that is, the book has little or no consideration for ordinary folk. The distinguished physicist F. J. Dyson confirms our WWW.EBOOK777.COM PREFACE suspicions that “the marriage of mathematics and physics [about which we have been told so many nice things since our high school days] has ended in divorce” (quoted in M. Kline’s Mathematics, The Loss of Certainty, Oxford University Press, 1980, pp. 302-303). Applied, which either emphasize the numerical/computational aspects of mechanics, but, perhaps unavoidably, are soft and/or sketchy on its fundamental principles; or are so theoretically/conceptually impoverished and unmotivated that the reader is soon led to a narrow and dead-end view of mechanics. [Notable and refreshing exceptions to this style are the recent compact but rich-in-ideas works by Bremer et al. [1988(a), (b), 1992] in dynamics/control/flexible multibody systems.] Mainstream or traditionalist; for example, those by (alphabetically): Arya, Baruh, Calkin, Crandall et al., Corben et al., Desloge, Goldstein, Greenwood, Kilmister et al., Konopinski, Kuypers, Lanczos, Marion, McCauley, Meirovitch, Park, Rosenberg, Woodhouse. The problem with this group, however, is that its representatives either do not go far and deep enough (somehow, the more advanced topics seem to be mono¬ polized by the expositions of the first group); or they could use some improvements in the quality and/or quantity of their engineering^ relevant examples and problems. The book at hand belongs squarely and unabashedly to this last group, and aims to remedy its above-mentioned shortcomings by bridging the space between it and some of the earlier-mentioned classics, such as (chronologically): Heun (1906, 1914), Prange (1933-1935), Hamel (1927, 1949), Peres (1953), Lur’e (1961/1968), Gantmacher (1966/1970), Neimark and Fufaev (1967/1972), Dobronravov (1970, 1976), and Novoselov (1966, 1967, 1979). Hence, my earlier claim that this treatise is unique in the entire contemporary literature; and my strong belief that it does meet real and long overdue needs of students and teachers of advanced (engineering) dynamics of the international community. I have sought to combine the best of the old and new — no age discrimination here — and I hope that this work will help counter the very real and disturbing trend, brought about by the proponents of the first two groups, toward a dynamical tower of Babel. ON NOTATION To make the exposition accessible to as many willing and able readers as possible, and following the admirable and ever applicable example of Lanczos (1949-1970), I have chosen, wherever possible, an informal approach; and I have, thus, deliberately avoided all set-theoretic and functional-analytic formalisms, all unnecessary rigor (“epsilonics”) and similar ahistorical/unmotivating/intuition-deadening tools and methods. For the same reasons, 1 have also avoided the currently popular direct/ dyadic (coordinate-free) and matrix notations (except in a very small number of truly useful situations); and I have, instead, chosen good old-fashioned elementary/ geometrical (undergraduate) form, for vectors, and/or indicial Cartesian tensorial notation for vectors, tensors, etc. The ad nauseam advertised “advantages” of the coordinate-free (“direct”) nota¬ tion and matrices are vastly exaggerated and misguiding. To begin with, it is no accident that the solution of all concrete physical problems is intimately connected with a specific and convenient (or natural, or canonical) system of coordinates. Indicial tensorial notation seems to kill two birds with one stone: it combines both coordinate invariance (generality) and coordinate specificity; that is, one knows exactly what to do in a given set of coordinates/axes; see, for example, Korenev WWW.EBOOK777.COM PREFACE (1979), MaiBer (1988) for robotics applications. However, the systematic use of general tensors in dynamics has been kept out of this book. [That is carried out in my monograph. Tensor Calculus and Analytical Dynamics (CRC Press, 1999).] The only thing tensorial used here amounts to nothing more than the earlier-mentioned indicial Cartesian tensor notation; and for reasons that will become clear later, not even the well-known summation convention is employed. Indicial tensorial notation turns out to be the best tool in “unknown and rugged terrain”; and frequently it is the only available notation, for example, in dealing with nonvectorial/tensorial “geo¬ metrical objects,” such as the Christoffel symbols and the Ricci/Boltzmann/Hamel coefficients. Once the fundamental theory is thoroughly understood, and the numer¬ ical implementation of a (frequently large-scale) concrete problem is sought, then one can profitably use matrices, and so on. Heavy use of matrices, with their non¬ commutativity “straitjacket,” at an early stage [e.g., Haug, 1992(a)] is likely to restrict creativity and replace physical understanding with the local mechanical manipulation of symbols. FURTHER PHILOSOPHY: On Computerization, Applications, and Ultimate Coals of Research 1 do not think that the author of a book on analytical mechanics (AM) should be constantly defending it as simply a means to some other allegedly higher ends [e.g., a prerequisite to quantum mechanics, as Goldstein (1980) does], or in terms of its current “real life” applications in space or earth (e.g., artificial satellites, rocketry, robotics, etc.; i.e., in terms of dollars to be made); although, clearly, such connec¬ tions do exist and can be helpful. What should worry us is that these days, under what B. Schwartz calls “economic imperialism,” or what R. Bellah calls “market totalitarianism” (i.e., the penetration of purely monetary values into every aspect of social life; or, to regard all aspects of human relations as matters of economic self- interest, and model them after the market) every activity is fast becoming a means for something else, preferably quantifiable and monetary. In the process, daily work, craftsmanship, and the pleasure derived from the practice of that activity, have all been degraded. Unless we restore some internal, or intrinsic, goals and rewards to our subject and disseminate them to our young students, pretty soon such an activity will be no different from clerical or assembly-line work; that is, just a paycheck. As stated earlier, we view AM as a course worth pursuing in its own terms. We study it because it is worth learning, and because it is a grand and glorious part of our intellectual/ cultural heritage—those who do not care about the past cannot possibly care about the present, let alone the future. On a more practical level, a few years from now such applied areas as multibody dynamics, a subject with which so many dynamicists are preoccupied today, will be exhausted — some say that that has already happened. What are the practical mechanicians going to do then? Most of their expositions (second of the earlier groups) are too narrow and do not prepare the reader for the long haul. But there is a more fundamental reason for adopting “my” particular approach to mechanics: I strongly believe that every generation has to rediscover ( better, reinvent) AM, and most other areas of knowledge for that matter, anew and on its own terms; that is, replow the soil and not just be handed down from their predecessors, discontinuously, prepackaged and predigested “information” in a diskette (the electronic equivalent of ashes in an urn). To squeeze the “entire” mechanics into a huge master computer WWW.EBOOK777.COM PREFACE program, which (according to common but nevertheless vulgar advertisements) “does everything,” and makes it available to the reader (“user”) in the form of data inputs, is not only dangerous for the present (e.g., accidents, screw-ups, which are especially consequential in today’s large-scale systems — recall the omni¬ present Murphy’s laws), but also, being a degradation and dehumanization of knowledge, it guarantees the intellectual death of our society. If the job makes the person (mentally, psychologically, and physically), then how are we going to answer the question “What are people for?” Typical of such contemporary one-dimensional, or “digital,” approaches to dynamics are sweeping statements like: “pre-computer analytical methods for deriving the system equations must be replaced by systematic computer oriented formalisms, which can be translated conveniently into efficient computer codes for * generating the system equations based on simple user data describing the system model, * solving those complex equations yielding results ready for design evalua¬ tion” and “Emphasis is on computer based derivation of the system equations thus freeing the user from the time consuming and error-prone task of developing equa¬ tions of motion for various problems again and again.” [From advertisement of Roberson and Schwertassek (1988) in Ingenieur-Archiv, 59, p.A.3, 1989.] Here, the advertisers hide the well-known fact of how much error prone is the formulating and running of any complicated program; and how the combination of this with the absence of any physically simple and meaningful checks for finding errors — some¬ thing of a certainty for the structureless/formless mechanics of Newton-Euler, on which so much of multibody dynamics rests — is a recipe for chaos (=> arbitrariness)! Our reading of this ad is that the whole process will, eventually, “free” the user from thinking at all — first, we replace the human functions and then we replace humans altogether [first industrial revolution: mechanization of muscles, second (current) industrial revolution: automation of both muscles and brains]; and anyone who dares to criticize, or inquire about choices (i.e., politics), is summarily and arrogantly dismissed as a technophobe or, worse, a neo-Luddite! As the mathematicians Davis and Hersh put it accurately: By turning attention away from underlying physical mechanisms and towards the pos¬ sibility of once-for-all algorithmization, it encourages the feeling that the purpose of computation is to spare mankind of the necessity of thinking deeply.... Excessive computerization would lead to a life of formal actions devoid of meaning, for the computer lives by precise languages, precise recipes, abstract and general programs wherein the underlying significance of what is done becomes secondary. [Inimitably captured in M. McLuhan’s well-known dictum: The medium is the message.] It fosters a spirit-sapping formalism. The computer is often described as a neutral but willing slave. The danger is not that the computer is a robot but that humans will become robotized as they adapt to its abstractions and rigidities (1986, pp. 293, 16). And, in a similar vein, H. R. Post adds: “You understand a subject when you have grasped its structure, not when you are merely informed of specific numerical results” (quoted in Truesdell, 1984, p. 601). The issue is not whether the complete computer codification of (some version of ) dynamics can be achieved or not; it clearly can , somehow. The issue is the desirability of it; that is, the could versus the should , its scale compared with the other approaches, and the temporal order of such a presentation to the student (“user”). The only safe way for using such heavily computerized schemes is for the student to already possess a very thorough grounding in the fundamentals of mechanics—like WWW.EBOOK777.COM PREFACE vaccination against a virus! There is no painless and short way to bypass several centuries of hard thinking by a handful of great fellow humans — no royal road to mechanics! Otherwise, we are headed for more confusion, degradation, errors, and accidents, and eventual disengagement from our subject. [For iconoclastic, devastating, and sobering critiques of the contemporary mindless and rabid computeritis, see, for example, Truesdell (1984, pp. 594-631), Davis and Flersh (1986), and Mander (1985).] As for the applications of mechanics, there is nothing wrong with them; as long as they do not hurt or exploit people and nature — alas, several such contemporary applications do just that. Those preoccupied with them rarely, if ever, ask the natural question: What are the (most likely) applications of the applications; namely, their social/environmental consequences? In this light, common statements like “the com¬ puter is only a tool” are utterly naive and meaningless. I should also add that the current relentless emphasis, even in the academia, on applied research with quick tangible results — that is, dollars at the expense of every other nonmonetary aspect — is a relatively recent phenomenon imposed on us from outside ; it is neither intrinsic nor accidental to science, but instead is an intensely socio-economic activity — technology is neither autonomous nor neutral! [And as Truesdell concurs, with depressing accuracy (1987, p. 91): “It is not philosophers of science who will enforce one kind of research or another. No, it will be the national funding agencies, the sources of manna, nectar, and ambrosia for the corrupted scientists. The directors of funds are birds of a feather, chattering mainly to each other and at any one moment singing more or less the same cacophonous tune. There may come a time when even the scholarly foundations will give preference to those who claim to promote national ‘defense’ by research on the basic principles governing some new, as yet totally secret — that is, known only to the directors of war in the U.S. and Russia — allegedly secret idea for a broader and more effective death by torture in a world full of humanitarians and their -isms.”] If applications, even worthwhile ones, are but one motive for studying mechanics, and science in general, then what else is? Here are some plausible (existential?) reasons offered by Einstein, which I have found particularly inspiring, since my high school years: Man tries to make for himself in the fashion that suits him best a simplified and intelligible picture of the world; he then tries to some extent to substitute this cosmos of his for the world of experience, and thus to overcome it. This is what the painter, the poet, the speculative philosopher, and the natural philosopher do, each in his own fashion. Each makes this cosmos and its construction the pivot of his emotioned life, in order to find in this way the peace and security which he cannot find in the narrow whirlpool of personal experience (emphasis added; from "Principles of Research,” an address delivered in 1918, on the occasion of M. Planck’s sixtieth birthday). From a broader perspective, I am convinced that the quality of our lives depends not so much on specific gadgets/artifacts, no matter how technically advanced they may be (e.g., from artificial hearts to space stations), but on our collective abilities to formulate simple, clear, and unifying ideas that will allow us to understand (and then change gently and gracefully — sustainably) our increasingly complicated, unstable and fragile societies; and, in the process, understand ourselves. The resulting psycho¬ logical and intellectual peace of mind from such a liberal arts (= liberating) approach cannot be overstated. It is this kind of activity and attitude that gives human life meaning — we do not do science just to make money, merely to exchange and con¬ sume. This book is intended as a small but tangible contribution to this lofty goal. WWW.EBOOK777.COM PREFACE SOME PERSONAL HISTORY My interest in AM began during my undergraduate studies (mid-to-late 1960s) upon reading in Hamel (1949, pp. 233-236, 367) about the differences between the calculus of variations (mathematics) and Hamilton’s variational principle (mechanics) for nonholonomic systems. The need for a deeper understanding of the underlying kinematical concepts led me, about twenty years later, to the study of the original epoch-making memoirs of such mechanics masters as Appell, Boltzmann, Heun, and Hamel. Then, in the spring of 1986, in related studies on variational calculus, I had the good fortune to stumble upon the virtually unknown but excellent papers of Schaefer (1951) and Stiickler (1955), which, along with my earlier acquaintance with tensors, showed me the way toward the correct understanding of everything virtual: virtual displacements and virtual work/Lagrange’s principle; that is, 1 arrived at AM via the calculus of variations, just like Lagrange in the 1760s! Finally, the emphasis on the fundamental distinction between particle and system quantities I owe to the writings of Heun, the founder of theoretical engineering dynamics (early 20th century), and especially to those of his students: Winkelmann and the great Hamel. In closing, 1 would like to recommend the reading of the preface(s) of Lanczos (1949-1970); the present work has been conceived and driven by a similar overall philosophy. May this book make many and loyal friends! john.papas@me.gatech.edu Atlanta, Georgia J. G. P. Autumn 2001 WWW.EBOOK777.COM This page intentionally left blank WWW.EBOOK777.COM ACKNOWLEDGMENTS (Where the author recognizes, with gratitude and pleasure, the social dimension of his activity) Every book on analytical mechanics is better off the closer it comes to the simplicity, clarity, and thoroughness of Georg Hamel’s classic Theoretische Mechanik , arguably the best (broadest and deepest) single work on mechanics; and, secondarily, of Anatolii E Lur’e’s outstanding Analiticheskaya Mekhanika. I hope that this treatise follows closely and loyally the tradition created by these great masters. My indebt¬ edness to their monumental works is hereby permanently registered. Next, I express my deep appreciation and thanks to Ms. Katharine L. Calhoun, of the Georgia Tech library, for her most courteous and capable help in locating and obtaining for me, over the past several years (1986 to present), hundreds of rare and critically needed references, from all over the country. Katharine is an oasis of humanity and graciousness in an otherwise arid and grim campus. Drs. Feng Xiang Mei, professor at the Beijing Institute of Technology, China, and Zhen Wu, professor at the University of Tsiao Tong, China; and Drs. Sergei A. Zegzhda and Mikhail P. Yushkov, professors at the Mathematics and Mechanics faculty of St. Petersburg University, Russia [birthplace of the first treatise on theoretical/analytical mechanics (Euler’s, Mechanica Sive Motus Scientia ..., 2 vols, 1736)] for making available to me copies of their excellent textbooks and papers on advanced dynamics, which are virtually unavailable in the West (see the list of References and Suggested Reading). Dr. John L. Junkins, chaired (and distinguished) professor of aeronautical engineering at Texas A&M University, for early encouragement on a previous version (outline) of the manuscript (1986) — what John aptly dubbed “the zeroth approximation.” Dr. Donald T. Greenwood, professor of aerospace engineering at the University of Michigan (Ann Arbor) and a true Nestor among American dynamicists, also author of internationally popular and instructive graduate texts on dynamics, for his detailed and mature (and very time- and energy-consuming) comments on the entire technical part of the manuscript; and for sharing with me his (soon to appear in book form) notes on Special Advanced Topics in Dynamics. Dr. Wolfram Stadler, professor and scholar of mechanics at San Francisco State University and author of uniquely encyclopaedic work on robotics/mechatronics, for qualitative and constructive criticism on Chapter 1 (Background). Wolf remains a staunch and creative individualist, in an age of ruthless academic collectivization. WWW.EBOOK777.COM xviii ACKNOWLEDGMENTS A very special expression of indebtedness to my friend Dr. Hartmut Bremer (O. Univ.-Prof. Dr.-Ing. habil.), professor of mechatronics at the University of Linz, Austria (formerly, professor of mechanics at the Technical University of Munich, Germany) and author of two dense and comprehensive textbooks on dynamics/ control/multibody systems, for extensive, thoughtful and critical discussions/com¬ ments on several topics of theoretical and applied dynamics, including their histor¬ ical, cultural, and educational aspects; and for supplying me with rare and precious references on the subject. To Hartmut’s persistent efforts (1989-1993), 1 owe a very rare photograph of Karl Heun (to appear in a special gallery of photos of mechanics masters, which will be included in my forthcoming Elementary Mechanics). In addition, I consider myself very fortunate to have benefited from the “mechan¬ ical” knowledge and wisdom of my friend Dr. Leon Y. Bahar, professor of mechan¬ ical engineering and mechanics at Drexel University and another American Nestor. Leon, a veritable engineering science scholar, craftsman, and above all dedicated teacher (i.e., representative of an academic species that is somewhere between endan¬ gered and extinct, as professionalism goes up and scholarship goes down), has over the past several years selflessly provided me with critical and enlightening quanta of knowledge and insight (lecture notes, papers, and extensive letters), and much valued mentorship. These are not the best of times for writing “another” book on advanced theore¬ tical mechanics — to put it mildly. However, and this provides a certain consolation, even such all-time titans of mathematics and mechanics as Euler, Lagrange, and Gauss had considerable difficulties in publishing, respectively, their Theoria Motus ... (1765), Mechanique Analitique (1788), and Disquisitiones Arithmeticae (1801). [According to Truesdell (1984, p. 352), all that Euler received for his masterpiece on rigid-body dynamics (1765) was ... twelve free copies of it!] Aspiring academic writers in this area are forewarned that the contemporary “research” university is not particularly supportive to such scholarly activities; these latter, obviously “interfere” with the more lucrative business (to the university bureaucrats, but not necessarily to students and society at large) of contracts and grants from big business and big government. Such an inimical “academic” environment makes it much more natural than usual that I reserve the strongest expression of gratitude, by far, to my family, here in Atlanta, Georgia: my wife Kim Ann and daughter Julia Constantina; and in my native Athens, Greece: my mother Konstantina and brother Stavros (in my nonobjective but fair view, one of the brightest mathematicians of contemporary Greece, and an island of moral and intellectual nobility, in an archipelago of petty greed and irrationality) — for their continual and critical moral and material support throughout the several long and solitary years of writing of the book. Last, this volume (as well as my other two mechanics books) could not have been written without (i) the institution of academic tenure (much maligned and curtailed recently by reactionary ideologues, demagogues, and ignoramuses) and (ii) the (alas, fast disappearing) policy of most university libraries, of open , direct, and free access to books and journals. Regrettably, and in spite of high-tech millennarian promises, the next generation of scholarly authors will not be as lucky as I have been, in both these areas! WWW.EBOOK777.COM Contents INTRODUCTION 3 1 Introduction to Analytical Mechanics 4 2 History of Theoretical Mechanics: A Bird’s-Eye View 9 3 Suggestions to the Reader 13 4 Abbreviations, Symbols, Notations, Formulae 14 BACKGROUND: BASIC CONCEPTS AND EQUATIONS OF PARTICLE AND RIGID-BODY MECHANICS 71 1.1 Vector and (Cartesian) Tensor Algebra 72 1.2 Space-Time Axioms; Particle Kinematics 89 1.3 Bodies and their Masses 98 1.4 Force; Law of Newton-Euler 101 1.5 Space-Time and the Principle of Galilean Relativity 104 1.6 The Fundamental Principles (or Balance Laws) of General System Mechanics 106 1.7 Accelerated (Noninertial) Frames of Reference (or Relative Motion, or Moving Axes); Angular Velocity and Acceleration 113 1.8 The Rigid Body: Introduction 138 1.9 The Rigid Body: Geometry of Motion and Kinematics (Summary of Basic Theorems) 140 1.10 The Rigid Body: Geometry of Rotational Motion; Finite Rotation 155 1.11 The Rigid Body: Active and Passive Interpretations of a Proper Orthogonal Tensor; Successive Finite Rotations 178 1.12 The Rigid Body: Eulerian Angles 192 1.13 The Rigid Body: Transformation Matrices (Direction Cosines) Between Space-Fixed and Body-Fixed Triads; and Angular Velocity Components along Body-Fixed Axes, for All Sequences of Eulerian Angles 205 1.14 The Rigid Body: An Introduction to Quasi Coordinates 212 1.15 The Rigid Body: Tensor of Inertia, Kinetic Energy 214 1.16 The Rigid Body: Linear and Angular Momentum 222 1.17 The Rigid Body: Kinetic Energy and Kinetics of Translation and Rotation (Eulerian “Gyro Equations”) 225 1.18 The Rigid Body: Contact Forces, Friction 237 WWW.EBOOK777.COM xx CONTENTS 2 KINEMATICS OF CONSTRAINED SYSTEMS (i.e., LAGRANGEAN KINEMATICS) 242 2.1 Introduction 242 2.2 Introduction to Constraints and their Classifications 243 2.3 Quantitative Introduction to Nonholonomicity 257 2.4 System Positional Coordinates and System Forms of the Flolonomic Constraints 270 2.5 Velocity, Acceleration, Admissible and Virtual Displacements; in System Variables 278 2.6 System Forms of Linear Velocity (Pfaffian) Constraints 286 2.7 Geometrical Interpretation of Constraints 291 2.8 Noncommutativity versus Nonholonomicity; Introduction to the Theorem of Frobenius 296 2.9 Quasi Coordinates, and their Calculus 301 2.10 Transitivity, or Transpositional, Relations; Hamel Coefficients 312 2.11 Pfaffian (Velocity) Constraints via Quasi Variables, and their Geometrical Interpretation 323 2.12 Constrained Transitivity Equations, and Hamel’s Form of Frobenius’ Theorem 334 2.13 General Examples and Problems 345 3 KINETICS OF CONSTRAINED SYSTEMS (i.e., LAGRANGEAN KINETICS) 381 3.1 Introduction 381 3.2 The Principle of Lagrange (LP) 382 3.3 Virtual Work of Inertial Forces (61), and Related Kinematico-Inertial Identities 399 3.4 Virtual Work of Forces: Impressed (6'W) and Constraint Reactions (6 1 W R ) 405 3.5 Equations of Motion via Lagrange’s Principle: General Forms 409 3.6 The Central Equation (The Zentralgleichung of Heun and Hamel) 461 3.7 The Principle of Relaxation of the Constraints (The Lagrange- Hamel Befreiungsprinzip ) 469 3.8 Equations of Motion: Special Forms 486 3.9 Kinetic and Potential Energies; Energy Rate, or Power, Theorems 511 3.10 Lagrange’s Equations: Explicit Forms; and Linear Variational Equations (or Method of Small Oscillations) 537 3.11 Appell’s Equations: Explicit Forms 563 3.12 Equations of Motion: Integration and Conservation Theorems 566 3.13 The Rigid Body: Lagrangean-Eulerian Kinematico-Inertial Identities 581 3.14 The Rigid Body: Appellian Kinematico-Inertial Identities 594 3.15 The Rigid Body: Virtual Work of Forces 597 WWW.EBOOK777.COM CONTENTS XXI 3.16 Relative Motion (or Moving Axes/Frames) via Lagrange’s Method 606 3.17 Servo (or Control) Constraints 636 3.18 General Examples and Problems 650 APPENDIX 3.A1 Remarks on the History of the Hamel-type Equations of Analytical Mechanics 702 APPENDIX 3.A2 Critical Comments on Virtual Displacements/Work; and Lagrange’s Principle 708 4 IMPULSIVE MOTION 718 4.1 Introduction 718 4.2 Brief Overview of the Newton-Euler Impulsive Theory 718 4.3 The Lagrangean Impulsive Theory; Namely, Constrained Discontinuous Motion 721 4.4 The Appellian Classification of Impulsive Constraints, and Corresponding Equations of Impulsive Motion 724 4.5 Impulsive Motion via Quasi Variables 751 4.6 Extremum Theorems of Impulsive Motion (of Carnot, Kelvin, Bertrand, Robin, et al.) 784 5 NONLINEAR NONHOLONOMIC CONSTRAINTS 817 5.1 Introduction 818 5.2 Kinematics; The Nonlinear Transitivity Equations 819 5.3 Kinetics: Variational Equations/Principles; General and Special Equations of Motion (of Johnsen, Hamel, et al.) 831 5.4 Second- and Higher-Order Constraints 871 6 DIFFERENTIAL VARIATIONAL PRINCIPLES, AND ASSOCIATED GENERALIZED EQUATIONS OF MOTION OF NIELSEN, TSENOV, ET AL. 875 6.1 Introduction 875 6.2 The General Theory 876 6.3 Principle of Jourdain, and Equations of Nielsen 879 6.4 Introduction to the Principle of Gauss and the Equations of Tsenov 884 6.5 Additional Forms of the Equations of Nielsen and Tsenov 894 6.6 The Principle of Gauss (Extensive Treatment) 911 6.7 The Principle of Hertz 930 7 TIME-INTEGRAL THEOREMS AND VARIATIONAL PRINCIPLES 934 7.1 Introduction 935 TIME-INTEGRAL THEOREMS 936 7.2 Time-Integral Theorems: Pfaffian Constraints, Holonomic Variables 936 7.3 Time-Integral Theorems: Pfaffian Constraints, Linear Nonholonomic Variables 948 WWW.EBOOK777.COM xxii CONTENTS 7.4 Time-Integral Theorems: Nonlinear Velocity Constraints, Holonomic Variables 957 7.5 Time-Integral Theorems: Nonlinear Velocity Constraints, Nonlinear Nonholonomic Variables 958 TIME-INTEGRAL VARIATIONAL PRINCIPLES (IVP) 960 7.6 Hamilton’s Principle versus Calculus of Variations 960 7.7 Integral Variational Equations of Mechanics 966 7.8 Special Integral Variational Principles (of Suslov, Voronets, et al.) 974 7.9 Noncontemporaneous Variations; Additional IVP Forms 990 APPENDIX 7.A Extremal Properties of the Hamiltonian Action (Is the Action Really a Minimum; Namely, Least?) 1055 8 INTRODUCTION TO HAMILTONIAN/CANONICAL METHODS: EQUATIONS OF HAMILTON AND ROUTH; CANONICAL FORMALISM 1070 8.1 Introduction 1070 8.2 The Hamiltonian, or Canonical, Central Equation and Hamilton’s Canonical Equations of Motion 1073 8.3 The Routhian Central Equation and Routh’s Equations of Motion 1087 8.4 Cyclic Systems; Equations of Kelvin-Tait 1097 8.5 Steady Motion (of Cyclic Systems) 1115 8.6 Stability of Steady Motion (of Cyclic Systems) 1119 8.7 Variation of Constants (or Parameters) 1143 8.8 Canonical Transformations (CT) 1161 8.9 Canonicity Conditions via Poisson’s Brackets (PB) 1176 8.10 The Hamilton-Jacobi Theory 1192 8.11 Hamilton’s Principal and Characteristic Functions, and Associated Variational Principles/Differential Equations 1218 8.12 Integral Invariants 1230 8.13 Noether’s Theorem 1243 8.14 Periodic Motions; Action-Angle Variables 1250 8.15 Adiabatic Invariants 1290 8.16 Canonical Perturbation Theory in Action-Angle Variables 1305 References and Suggested Reading 1323 Index 1371 WWW.EBOOK777.COM Words of Wisdom and Beauty On Rigor It is not so much important to be rigorous as to be right. —A. N. KOLMOGOROV On Theory There is nothing more practical than a good theory. —L. BOLTZMANN We have no access to a theory-independent world — that is, a world unconditioned by our point of view_The world we see is ... theory-laden: it already bears the ineliminable mark of our involve¬ ment in it .... Knowledge is always a representation of reality from within a particular perspective_We cannot assume ... “the view from nowhere.” —T. W. CLARK I really do not at all like the now fashionable “positivistic” tendency of clinging to what is observable ... I think ... that theory cannot be fabricated out of the results of observation, but that it can only be invented. —A. EINSTEIN On Method In the sciences the subject is not only set by the method; at the same time it is set into the method and remains subordinate to the method .... In the method lies all the power of knowledge. The subject belongs to the method, (emphasis added) —M. HEIDEGGER The core of the practice of science — the thread that keeps it going as a coherent and developing activity—lies in the actions of those whose goals are internal to the practice. And these internal goals are all noneconomic, (emphasis added.) —B. SCHWARTZ On Beauty My own students, few they have been, I have tried to teach how to ask questions humbly and to see ways to some taste in a vulgar, obscene epoch. Taste is acquired by those who can face questions, especially insoluble questions. C. A. TRUESDELL It is by the steady elimination of everything which is ugly — thoughts and words no less than tangible objects — and by the sub¬ stitution of things of true and lasting beauty that the whole progress of humanity proceeds. —A. PAVLOVA WWW.EBOOK777.COM This page intentionally left blank WWW.EBOOK777.COM (Jbo Reprint Edition Analytical Mechanics A Comprehensive Treatise on the Dynamics of Constrained Systems WWW.EBOOK777.COM This page intentionally left blank WWW.EBOOK777.COM Introduction KOIMON TONAE, TON AYTON AI1ANTQN, OYTE Til 0EQN OYTE ANGPQnnN EIIOIHIEN, AAA'HN AEI KAI EZTIN KAI EITAI IlYP AEIZOON, AllTOMENON METPA KAI AIIOIBENNYMENON METPA. [HPAKAEITOZ (Herakleitos, Creek philosopher; Ephesos, Ionia, late 6th century b.c.)] [Translation: "This world [order], which is the same for all [beings], no one of gods or humans have created; but it was ever, is now, and ever shall be an ever-living Fire, that starts and goes out according to certain rules [laws]." This magnificent statement marks the beginning of science—one of the countless, fundamental, and original gifts of Greece to the world. (See, e.g., Burnet, 1930, p. 134; also Frankfort et al., 1946, chap. 8.)] Die Mechanik ist die Wissenschaft von der Bewegung; als ihre Aufgabe bezeichnen wir: die in der Natur vor sich gehenden Bewegungen vollstandig und auf die einfachste Weise zu beschreiben. (Translation: Mechanics is the science of motion; we define as its task the complete description and in the simplest possible manner of such motions as occur in nature.) (Kirchhoff, 1876, p. 1, author's emphasis) Dynamics or Mechanics is the science of motion .... The problem of dynamics according to Kirchhoff, is to describe all motions occurring in nature in an unambiguous and the simplest manner. In addition it is our object to classify them and to arrange them on the basis of the simplest possible laws. The success which has attended the efforts of physicists, mathematicians, and astronomers in achieving this object from the time of Calileo and Newton through that of Lagrange and Laplace to that of Helmhoitz and Kelvin, constitutes one of the greatest triumphs of the human intellect. (Webster, 1912, p. 3, emphasis added) Die Mechanik ist ein Teil der Physik. (Translation: Mechanics is a part of physics.) (Foppl, 1898, vol. 1, p. 1) WWW.EBOOK777.COM INTRODUCTION 1 INTRODUCTION TO ANALYTICAL MECHANICS What Is Analytical Mechanics? Classical mechanics (CM)—that is, the exact science of nonrelativistic and non¬ quantum motion (effects) and forces (causes)—was founded in the 17th century (Galileo, 1638; Newton, 1687), and was brought to fruition and generality during the next century, almost single-handedly, by Euler (1752: principle of linear momen¬ tum; 1775: principle of angular momentum). [D’Alembert too had formulated sepa¬ rate laws of linear and angular momentum (1743, 1758), but his approach came nowhere near that of Euler in generality and power.] That was the first complete dynamical theory in history. We shall call it, conveniently (even though not quite accurately), the Newton-Euler method of mechanics (NEM). The second such theory was also initiated in the (late) 17th century, this time by Huygens and Jakob Bernoulli; it was further developed during the 18th century by Johann Bernoulli (Jakob’s brother) and d’Alembert (early 1740s), and was finally brought to relative mathematical and physical completion by the other great math¬ ematician of that century, Lagrange (1760: principle of “least” action; 1764: princi¬ ple of d’Alembert in Lagrange’s form, or Lagrange's principle', 1780: central equation and Lagrange’s equations', 1788: Mechanique Analitique; 1811-1812: transitivity equa¬ tions). This second approach, what we shall call the method of d’Alembert Lagrange, or, simply and more accurately, the method of Lagrange, forms the basis of what has come to be known as analytical mechanics (AM); or, equivalently, Lagrangean mechanics (LM). Although both these methods are, roughly, theoretically equivalent, since there is only one classical mechanics, the second approach proved much more influential and fertile to the subsequent development not only of mechanics, but also of practically all areas of physics: from generalized coordinates and configuration space to Riemannian geometry and tensors, and from there to general relativity; and similarly for quantum mechanics. Analytical mechanics proved particularly significant and useful to engineers, although it took another century after Lagrange for this to be fully realized (see §2). The reason for this is that AM was specifically designed by its inventors to handle constrained (earthly) systems— the concept of constraint is central to AM. Not that NEM could not handle such systems, but AM proved incomparably more expedient both for formulating their simplest (or minimal) equations of motion, and also for offering numerous theoretical and practical insights and tools for their solution (e.g., theorems of conservation and invariance, variational “principles” and associated direct methods of approximation, etc.—detailed in chaps. 3-8). In NEM, the basic principles (or axioms) are those of linear and angular momen¬ tum, and, secondarily, that of action-reaction, for the internal (or mutual) forces (see chap. 1); that is, NEM is a mechanics of systems based on momentum principles. In LM, on the other hand, the primary axiom is the kinetic principle of virtual work for the constraint reactions [=Lagrange’s principle (LP)] and, secondarily, the principle of relaxation (or liberation, or freeing) of the constraints (see chap. 3); that is, LM is a mechanics of systems based on energetic principles. WWW.EBOOK777.COM §1 INTRODUCTION TO ANALYTICAL MECHANICS With the help of his LP, Lagrange and many others later (see §2) formulated the most general equations of motion of systems subject to general positional and/or motional constraints. [The former are called holonomic, while the latter, if they cannot be brought (integrated) to positional form are called nonholonomic (see chap. 2).] Last, from the viewpoint of applications, AM constitutes the theoretical founda¬ tion of advanced engineering dynamics ; which, in turn, is very useful to the following: structural dynamics (e.g., bridges, airport runways); machine dynamics (e.g., internal combustion engines); vehicle dynamics (e.g., automobiles, locomotives); rotor dynamics (e.g., turbines); robot dynamics (e.g., robotic manipulators); aero-jastro- dynamics (e.g., airplanes, artihcial satellites); control theoryjsystem dynamics (e.g., electromechanical systems, valves); celestial dynamics (e.g., astronomy), and so on. Comments on the Methodology of AM 1. From the otherwise physically complete (particle) mechanics of Newton two things were missing: rotation and constraints (and, of course, deformation, but we do not deal with continua here). The first was taken care of by Euler, Mozzi, Cauchy, Chasles, Rodrigues et al. (1750s to the mid 19th century), and the second by Lagrange (1760s 1780s) and later many others (1870-1910). Of course, special cases of both problems had been examined earlier: for example, Newton discussed motions on specified curves and the associated forces, and, as Heun points out, with the help of his third law of action/reaction, he could have built a constrained particle mechanics, had he pursued that possibility; d’Alembert worked with particles “con¬ strained in rigid body connections”; and even Huygens had such pendula involving several constrained particles. Much later (early 1810s), Lagrange brought rotation under his energetic plan (genesis of nonholonomic, or ywa,si-coordinates; special tran¬ sitivity equations—see bridge between Euler and Lagrange below). 2. Analytical versus synthetical, Euler versus Lagrange. To begin with, CM holds quite satisfactorily for sizes, or lengths, from 1CT 10 m (atom) to 10 20 m (galaxy), and for speeds up to c/10 (c = speed of light « 300,000 km/s). Outside of these broad ranges, CM is replaced by relativity (high speeds) and quanta (small sizes) (see, e.g., French, 1971, p. 8). Now, depending on the method adopted, CM can be classified as follows: Experimental MECHANICS Synthetical Theoretical Analytical kal Newton/Euler (momentum) D'Alembert/Lagrange (energy) This classification, a logically possible one out of many (see below; e.g., Hamel, 1917), stresses the following: 2(a). Contrary to popular declarations, and Lagrange himself is partly to blame for this, AM does not mean mechanics via mathematical analysis; that is, it does not WWW.EBOOK777.COM INTRODUCTION mean an ageometrical and figureless mechanics. [Even such 20th century mechanics authorities as Whittaker state that “The name Analytical Dynamics is given to that branch of knowledge in which the motions of material bodies,..., are discussed by the aid of mathematical analysis” (1937, p. 1).] Instead, and in the sense used in philosophy/logic, AM means a deductive mechanics: everything flowing from a few selected initial postulates/principles/axioms by logical (mathematical) reasoning; that is, from the general to the particular—as contrasted with inductive, or synthetic, mechanics; that is, from the particular to the general. As such, AM is by no means ageometrical (and, similarly, synthetic mechanics does not necessarily mean geome¬ trical and nonmathematical mechanics). Also, in the past (mainly 19th century) the terms theoretical, rational, and analytical have frequently been used synonymously. 2(b). In such a classification, the mechanics of Euler also deserves to be called analytical! The reason that we in this book, and most everybody else, do not have more to do with historical tradition and usage rather than with strict logic: today AM has come to mean, specifically, Lagrangean method = Principle of Lagrange (= Principle of d’Alembert + Johann Bernoulli’s principle of virtual work) + Principle of relaxation of the constraints (Hamel’s Befreiungsprinzip) After more than 200 years, AM is defined by its practice —that is, by its methods, tools, and range of problems dealt with by its practitioners—and because, contrary to the mechanics of Newton-Euler, it is capable of extending to other areas of physics: for example, statistical mechanics, electrodynamics. As the distinguished applied mathematician Gantmacher puts it [A]nalytical mechanics is characterized both by a specific system of presentation and also by a definite range of problems investigated. The characteristic feature ... is that general principles (differential or integral) serve as the foundation; then the basic differ¬ ential equations of motion are derived from these principles analytically. The basic content of analytical mechanics consists in describing the general principles of mechanics, deriving from them the fundamental differential equations of motion, and investigating the equations obtained and methods of integrating them (1970, p. 7). 2(c). Frequently, one is left with the impression that Eulerian mechanics is vec¬ torial, whereas Lagrangean mechanics is scalar. This, however, is only superficially true: LM can be quite geometrical and vectorial, but in generalized nonphysical/non- Euclidean (Riemannian and beyond) spaces [see, e.g., Papastavridis (1999), Synge (1926-1927, 1936), and references therein]. 2(d). Euler and Lagrange should be viewed as mutually complementary, not as adversarial—as some historians of mechanics do. And although it is undeniably true that, of the two, Euler was the greater “geometer” in both quantity and quality, yet it was the method of Lagrange that shaped and drove the subsequent epoch-making developments of theoretical physics and a fair part of applied mathematics (i.e., differential geometry/tensors —> relativity; Hamiltonian mechanics/phase space —> statistical mechanics, quantum mechanics). Lagrange himself, shortly before his death (in 1813), succeeded in building the bridge between his method and that of Euler (rigid-body equations) by obtaining a special case of “transitivity equations” WWW.EBOOK777.COM §1 INTRODUCTION TO ANALYTICAL MECHANICS [so named by Heun (early 20th century) because they allow the transition from Lagrangean to Eulerian], which appeared in the second volume of the second edition of his Mecanique Analytique (1815). And that is why the great mechanician Hamel, in 1903-1904, dubbed his own famous equations the “Lagrange-Euler equations”; and in his magnum opus Theoretische Mechanik (1949) he founded the entire mechanics on Lagrangean principles. [With the exception of Neimark and Fufaev (1972), the transitivity equations are completely and conspicuously absent from the entire English and French literature!] 3. Newtonian particles versus Eulerian continua. There is a certain viewpoint, particularly popular among celestial dynamicists/astronomers, (particle) physicists, and some applied mathematicians, according to which classical mechanics is the study of the motions of systems of particles under mutually attractive/repulsive forces, whose intensities depend only on the distances among these particles (mole¬ cules, etc.); and that, eventually, all physical phenomena are to be explained by such a “mechanistic” model. This Newtonian mindset dominated 19th century mechanics and physics almost completely, and obscured the fact that such a “central force + particle(s)” mechanics [launched, mainly, by P. S. de Laplace in his monumental five- volume Traite de Mecanique Celeste (1799-1825)] is but one possibility, even within the nonrelativistic and nonquantum confines of the 19th century. Under other, physically more realistic, possibilities the total interparticle force, generally, consists of a reaction to the geometrical and/or kinematical constraints imposed, and an impressed, or physical, part that can depend explicitly on time, position(s), and velocity(-ies) of some or all of the system particles. However, the introduction of such forces to mechanics creates effects that cannot be accounted by mechanics alone, such as thermal and/or electromagnetic phenomena; whereas, the conse¬ quences of Newtonian forces stay within mechanics. The “mechanistic theory of matter”—namely, to explain all nonmechanical phe¬ nomena via simple models of internal nonvisible (concealed) motions of the system’s molecules (second half of 19th century, proposed by physicists like W. Thomson, J. Thomson, Helmholtz, Hertz et al.)—was only partially successful, and eventually evolved to statistical mechanics and physics (Boltzmann, Gibbs) and quantum mechanics [Planck, Einstein, Bohr, Born, Heisenberg, Schrodinger, Dirac et al.; see also Stackel (1905, p. 453 ff)]. Finally, as Hamel (1917) points out, it should be remembered that AM is not restricted to particles: even though Lagrange himself starts with particles, that fact is totally unimportant to his method; he could have just as well spoken of “volume elements.” 4. Theory versus experiment. The logical consequences of the principles of AM should not contradict experience. This, however, does not mean that these conse¬ quences (theorems, corollaries, etc.) should be derived from experiments; the latter cannot supply missing mathematics, or be used to prove and/or verify something, but they can be used to disprove a hypothesis. As H. R. Post puts it: [There are] three items of religious worship inside present-day science, the third of which is experiment. [l]n the main the role of experiment constitutes a harmless myth in the philo¬ sophy of scientists. The myth considers experiment to be a generator of theories. In fact the role of experiment ... is solely to decide between two or more existing theories ... Experiment does not generate theories but rather is suggested by them. [As quoted in Truesdell (1987, p. 83). And, in a similar vein, Einstein declares: “Experiment never responds with a ‘yes’ to theory. At best, it says ‘maybe’ and, most frequently, simply ‘no.’ When it agrees with theory, this means ‘maybe’ and, if it does not, the verdict is ‘no.’ ”] WWW.EBOOK777.COM INTRODUCTION But if the axioms of mechanics do not flow simply (“mechanically”) and uniquely out of experiments, then where do they come from? Paraphrasing Hamel, Einstein et al., we may say that these axioms are erected from the facts of experience (the object) by the human mind (the subject) as cm equal and imaginative partner, from a little observation, a lot of thought and eventually a rather sudden (qualitative) under¬ standing and insight into nature. In other words, humans are not passive at all in the formation of scientific theories, but because of the enormous difficulty involved, the creation of a successful set of axioms is the rare act of genius [e.g. (chronologically): Euclid, Archimedes, Newton, Euler, Lagrange, Maxwell, Gibbs, Boltzmann, Planck, Einstein, Heisenberg, Schrodinger], In CM, although open and nontrivial problems still remain, yet they are to be solved by the adoped principles; namely, we do not risk much in stating that this science is essentially closed, and that is why here the analytical/deductive method is possible. Otherwise, we would have to adopt a synthetic/inductive approach and change it slowly, depending on the new empirical facts. 5. In addition to the Lagrangean (and Hamiltonian) analytical formulation of mechanics—namely, the classical tradition of Whittaker, Hamel, Lur’e, Pars, Gantmacher et al. followed here, and depending on the emphasis laid on their most significant aspects, the following complementary formulations of CM also exist: Variational (e.g., Lanczos, Rund). Vako-nomic (= Variational Axiomatic Kind; e.g., Arnold, Kozlov). Algebraic (= infinitesimal and finite canonical transformations. Lie algebras and groups, symmetries and conservation theorems; e.g., McCauley, Mittelstaedt, Saletan and Cromer, Sudarshan and Mukunda). Nonlinear dynamics (= regular and stochastic/chaotic motion; e.g., Gabor, Guggenheimer and Holmes, Lichtenberg and Lieberman, McCauley). Geometrical (= symplectic geometry, canonical structure; e.g., Arnold, Abraham and Marsden, MacLane). Statistical and thermodynamical (= Liouville’s theorem, equilibrium and nonequilibrium statistical mechanics, irreversible processes, entropy, etc.; e.g., Gibbs, Katz, Fiirth, Sommerfeld, Tolman). Many-body and celestial mechanics (= orbits and their stability, many-body problem; e.g., Charlier, Hagihara, Happel, Siegel and Moser, Szebehely, Wintner). All these, and other, formulations testify once more to the vitality and importance of CM for the entire natural science, even today. 6. For engineering purposes, the following (nonunique) partitioning of mechanics seems useful: MECHANICS Kinematics (motion) Dynamics (forces) Statics (forces and equilibrium) Kinetics (forces and motion) [We consider this preferable to the following partitioning, customary in the U.S. undergraduate engineering education: WWW.EBOOK777.COM §2 HISTORY OF THEORETICAL MECHANICS: A BIRD'S-EYE VIEW MECHANICS Statics Dynamics Kinematics Kinetics In addition, we will be using the following, not so common, terms: Stereomechanics: mechanics of rigid bodies (and, accordingly, stereostatics, stereo¬ kinetics, etc.—mainly, after Maggi, late 1800s to early 1900s); Kinetostatics: study of internal and external reactions in rigid bodies in motion (after Heun, early 1900s). 7. Finally, the problem of AM consists in the following: (a) . Formulation of the smallest, or minimal, number of equations of motion without (external and/or internal) constraint reactions; namely, the so-called kinetic equations; and also the ability to retrieve these reactions if needed; namely, the so- called kinetostatic equations. And then, (b) . The ability to solve these equations for the motion and unknown forces, respectively, either analytically (exactly or approximately) or numerically (computa¬ tionally or symbolically). This is aided by the possible existence of first integrals; for example, energy, momentum, and conservation/invariance principles; more on these in chapter 3. 2 HISTORY OF THEORETICAL MECHANICS: A BIRD'S EYE VIEW For us believing physicists the distinction between past, present, and future is only an illusion, even if a stubborn one. (A. Einstein, Aphorisms ) The past is intelligible to us only in the light of the present; and we can fully understand the present only in the light of the past .... Past, present, and future are linked together in the endless chain of history. (E. H. Carr, What is History?, 1961) But it is from the Greeks, and not from any other ancient society, that we derive our interest in history and our belief that events in the past have relevance for the present. (M. Lefkowitz, 1996, p. 6) For in a real sense, history isn't the past—it's a posture in the present toward the future. (L. Weschler, American author/journalist, 1986) Rootless men and women take no more interest in the future than they take in the past. (C. Lasch, The Minimal Self, 1984) WWW.EBOOK777.COM INTRODUCTION The devaluation of history is a prerequisite for the free exercise of pure power. (J. Rifkin, Time Wars, 1988) The complete history of analytical mechanics, including 20th century contributions, has not been written yet—in English, anyway—and lack of space prevents us from doing so here. However, we hope that the following brief, selective, subjective, and unavoidably incomplete (but essentially correct and fair) summary, and references, will inspire others to pursue such a worthwhile and long overdue task more fully. Most Important Milestones in the Evolution of Theoretical Engineering Dynamics (from the Viewpoint of Analytical Mechanics) Unconstrained System Mechanics (Momentum mechanics of Newton-Euler) 1638: Special particle motions (Galileo) 1687: Physical foundations of mechanics [Newton: incomplete principles, no method (no equations of motion in his Principia)] 1730s: Mechanics of a particle (Euler) 1740s: Mechanics of a system of particles (Euler, late 1740s: Newtonian equa¬ tions of motion!) 1750s: General principle of linear momentum (Euler); kinetics of rigid bodies (Euler) 1770s: General principle of angular momentum (Euler); kinematics and geo¬ metry of rigid body motion (Euler) Constrained System Mechanics (Energetic mechanics of Lagrange) 1743: Principle of d’Alembert (Jakob Bernoulli —> Jean Le Rond d'Alembert) 1760: Principle of least action (Maupertuis —> Euler —> Lagrange) 1764: Principle of Lagrange [Principle of virtual work (Johann Bernoulli, 1717, published 1725) -f- Principle of d’Alembert] 1780: Equations of Lagrange 1788: Mechanique Analitique (1st ed.; note old spelling) 1811: Special transitivity equations (Lagrangean derivation of rigid-body Eulerian equations) 1811-1815: Mecanique Analytique (2nd ed.; 3rd ed.: 1853-1855; 4th ed.: 1888— 1889; English translation, from 2nd ed.: 1997!) 1829: Gauss’ Principle of least constraint (or least deviation, or extreme compulsion) 1830s: Canonical formulation of mechanics (Hamilton) 1840s: Transformation/integration theory of dynamics (Jacobi) 1860s: Gyroscopic systems [Thomson (Lord Kelvin), Tait] 1870s: Cyclic coordinates, steady motion, and its stability (Routh) 1873: Earliest reactionless Lagrange-like equations for nonholonomic systems (Ferrers) WWW.EBOOK777.COM §2 HISTORY OF THEORETICAL MECHANICS: A BIRD'S-EYE VIEW 1879: 1870s-1910s: 1903-1904: 1910s 1930s: Gauss’ Principle for inequality constraints; Gibbs-Appell equations for unconstrained systems, but in general nonholonomic velocities (quasi velocities; Gibbs) Dynamics of nonholonomic systems , under linear (or Pfaffian) velocity constraints, possibly nonholonomic (Routh, Appell, Chaplygin, Voronets, Maggi, Heun, Hamel et al.—see below) Definitive and general study of nonholonomic systems (Pfaffian con¬ straints) in nonholonomic variables; Lagrange-Euler equations (Hamel) Dynamics of nonholonomic systems, under nonlinear velocity con¬ straints (Appell, Delassus, Chetaev, Johnsen, Hamel); Study of non¬ holonomic systems via general tensor calculus (Schouten, Synge, Vranceanu, Wundheiler, Horak, Vagner et al.) 1970s-present: Applications of the above to multi-body dynamics, computational dynamics [(alphabetically): Bremer, Haug, Huston, MaifSer, Roberson and Schwertassek, Schiehlen, Wittenburg et al.]; Nonlinear dynamics (regular and stochastic/chaotic motion) Let us elaborate a little on the dynamics of nonholonomic systems. The original Lagrangean equations (1780) were limited to holonomically constrained systems. At that time, and for several decades afterwards, velocity constraints (holonomic or not) were only a theoretical possibility; though one that could be easily handled by the Lagrangean method (i.e., principles of Lagrange and of the relaxation of the con¬ straints (detailed in chap. 3)). But it was not until about a century later that such constraints were studied systematically. However, that necessitated a thorough re¬ examination of the entire edifice of Lagrangean mechanics: roughly between 1870 and 1910, what may be accurately called the second golden age of analytical mechanics, a host of first-rate mathematicians (Ferrers, Lindelof, Hadamard, Appell , Volterra, Poincare, Klein, Jourdain, Stackel, Maurer), physicists (Gibbs, C. Neumann, Korteweg, Boltzmann), mechanicians (Routh, Maggi, Chaplygin, Voronets, Suslov, Heun , Hamel), and engineering scientists (Vierkandt, Beghin) developed the modern AM of constrained systems, including nonholonomic ones; and, also, the unified theory of differential variational principles of Lagrange, Jourdain, Gauss, Hertz et al. Up until then (ca. 1900), AM was used almost exclusively, by mathematicians and physicists, to study unconstrained systems: for example, celestial mechanics. The Promethean transition from heavens down to earth (i.e., constraints) was led by the great German mechanician Heun (1859 1929), who can be fairly considered as the founder of modern engineering dynamics; and, also, by his more famous student Hamel (1877-1954), arguably the greatest mechaniker of the 20th century. For instance, to these two we owe the correct formulation and interpretation of the d’Alembert-Lagrange principle (i.e., LP), and its successful application (along with additional geometrical and kinematical concepts, already in embryonic or special forms in Lagrange’s works) to systems under general holonomic and/or linear velocity (or Pfaffian, possibly nonholonomic) constraints. Therein lie the roots of all correct treatments of the subject. [Heun also made important contributions to applied mathematics. For example, the well-known Runge-Kutta method in ordinary differential equations should be called method of Range—Kutta—Heun; see, for example, Renteln (1995).] Between the two world wars, on the basis of the so-accumulated powerful insights into the mathematical structure of LM (especially from the differential variational principles), its methods were extended to nonlinear nonholonomic constraints; first WWW.EBOOK777.COM INTRODUCTION by Appell (1911-1925) and his student Delassus (1910s) [also by Prange and Muller (1923)] and then by Chetaev (1920s), Johnsen (1936-1941), and Hamel (1938). During the post World War II era, the entire field was summarized by Hamel himself in his magnum opus Theoretische Mechanik (1949); and then elaborated upon by a new generation of Soviet mechanicians [(alphabetically) Dobronravov, Fradlin, Fufaev, Lobas, Lur’e, Novoselov, Neimark, Poliahov, Rumyantsev (or Rumi¬ antsev), et al.], whose efforts culminated in the unique and classic monograph Dynamics of Nonholonomic Systems by Neimark and Fufaev (1967, transl. 1972). Both of these works are most highly recommended to all serious dynamicists. [(a) On the history of the nonholonomic equations of motion, see also chapter 3, appendix I. (b) Nonlinear (possibly nonholonomic) constraints are an area that, probably, constitutes the last theoretical frontier of LM and is of potential interest to nonlinear control theory. Also, the differential variational principles have rendered important services in the numerical treatment of problems of multibody dynamics, and promise to do more in the future.] Guide Through the Literature on the History of Mechanics 1. General (Mechanics and Physics): D’Abro (1939, 2nd ed.): Qualitative and quantitative tracing of the evolution of ideas from antiquity to modern quantum mechanics; excellent. Hoppe [1926(a), (b)]: Concise history of physics, with some quantitative detail; good place to start. Hund (1972): Panoramic, competent and compact history of physics, from antiquity to mod¬ ern quantum mechanics, cosmology, and so on; one of the best places to start. Simonyi (1990): Comprehensive and sufficiently quantitative history of physics from antiquity to modern; beautifully and richly illustrated; most highly recommended. 2. Mechanics — General: Dugas (1955): Comprehensive and quantitative history of classical and modern mechanics, from a French physicist’s viewpoint; quite useful. Diihring (1887): Comprehensive treatment of the history of mechanics from antiquity to the middle of the 19th century; difficult to read due to its complete absence of figures and almost complete absence of mathematics; for specialists/scholars. Haas (1914): Detailed and pedagogical treatment of the principles of classical mechanics, from antiquity to the early 19th century; very warmly recommended, especially for undergraduates in science/engineering. Mach (1883-1933): Leisurely and mostly qualitative history of the principles of classical mechanics, from antiquity to the end of the 19th century; interesting and influential, but in some respects incomplete and misleading. Papastavridis [.Elementary Mechanics (under production)] and references cited there. Szabo (1979): Selective history of entire mechanics, with lots of beautiful photographs and diagrams; combines features of Mach, Dugas, and Truesdell. Tiolina (1979) and Vesselovskii (1974): General histories of mechanics, with detailed accounts of Russian contributions; very highly recommended for both their contents and references. 3. Mechanics — Specialized: Cayley (1858, 1863): Excellent and authoritative summaries of theoretical devel¬ opments until the mid 19th century; by a very famous mathematician. Hankins (1970): Detailed account of the life and work of d’Alembert; highly recommended to mechanics historians/scholars. Hankins (1985): Physics during the 18th century (of enlightenment). Kochina (1985): Life and works of S. Kovalevskaya. Oravas and WWW.EBOOK777.COM §3 SUGGESTIONS TO THE READER McLean (1966): Detailed account of the development of energetic/variational prin¬ ciples, mainly of elastostatics. Polak (1959, 1960): Detailed and lively history of differential and integral variational principles of mechanics and classical/modern physics, from antiquity to the 20th century; most highly recommended. Stackel (1905): Excellent quantitative history of particle and rigid-body dynamics (elemen¬ tary to intermediate), from antiquity to the early 20th century; a must for mature dynamicists; complements Voss’s article. Truesdell (1968, 1984, 1987): Authoritative and lively detailing of the life and contributions of Euler; but invariably unfair/ misleading to Lagrange and to anything remotely connected to particles and physics; for mature mechanicians/physicists. Voss (1901): Detailed and quantitative history of the principles of theoretical mechanics; with extensive lists of original references, from antiquity to ca. 1900; very highly recommended to mechanics and physics specialists. Wheeler (1952): Life and works of J. W. Gibbs. Wintner (1941, pp. 410-443): Notes and references on the history of analytical mechanics, with special emphasis on the mathematical aspects of celestial mechanics—the book, in general, is not recommended to anyone but specialists in theoretical astronomy. Ziegler (1985): Detailed and quantitative history of geometrical approach to rigid-body mechanics; primarily for kinematicians, not dynamicists. 4. Histories of Mathematics: Bell (1937): Lively and enjoyable; concentrates on the lives and times of famous mathematicians. Bochner (1966): Informative, unconventional. Klein (1926(b), 1927): Detailed and authoritative. Kline (1972): Arguably, the best of its kind in English; encyclopedic, reliable, insightful, witty; a scholarly masterpiece. Kramer (1970): Like a more elementary version of Kline’s book; interesting account of the evolution of determinism in physics (pp. 204—245). Struik (1987): Compact, dependable; includes socioeconomic explanations of mathematical inventions. Also Dictionary of Scientific Biography (ed. Gillispie, 1970s). 3 SUGGESTIONS TO THE READER A cumulative and alphabetical bibliography is located at the end of the book (see References and Suggested Reading, pp. 1323-1370). The following grouping of textbooks and treatises aims to better orient the reader relative to (some of) the best available international mechanics/dynamics literature, and thus obtain max¬ imum benefit from this work. References in bold, below, happen to be our per¬ sonal favorites, and have influenced us the most in the writing of this book. 1. For Background (Elementary to Intermediate Level): Butenin et al. (1985), Coe (1938), Crandall et al. (1968), Easthope (1964), Fox (1967), Hamel (1912, 1st ed.; 1927), Loitsianskii and Lur’e (1983), Milne (1948), Nielsen (1935), Osgood (1937), Papastavridis (EM, in preparation), Parkus (1966), Rosenberg (1977), Smith (1982), Sommerfeld (1964), Spiegel (1967), Stackel (1905), Suslov (1946), Synge and Griffith (1959), Wells (1967). 2. For Concurrent Reading (Intermediate to Advanced Level): Boltzmann (1902, 1904), Butenin (1971), Dobronravov (1970, 1976), Gantmacher (1970), Gray (1918), Greenwood (1977, 2000), Hamel (1949), Heil and Kitzka (1984), Heun (1906), Lamb (1943), Lanczos (1970), Lur’e (1968), MacMillan (1927, 1936), Mei (1985, 1987), Neimark and Fufaev (1972), Nordheim (1927), Pars (1965), WWW.EBOOK777.COM INTRODUCTION Pasler (1968), Peres (1953), Poliahov et al. (1985), Prange (1935), Rose (1938), Synge (1960), Winkelman (1929, 1930). 3. For Further Reading: Theoretical Physics, Nonlinear Dynamics, and so on: Arnold (1989), Arnold et al. (1988), Bakay and Stepanovskii (1981), Birkhoff (1927), Born (1927), Corben and Stehle (1960), Dittrich and Reuter (1994), Dobronravov (1976), Fues (1927), Hagihara (1970), Liehtenberg and Lieberman (1983/1992), McCauley (1997), Mittelstaedt (1970), Nordheim and Fues (1927), Pars (1965), Prange (1935), Santilli (1978, 1980), Straumann (1987), Synge (1960), Tabor (1989), van Vleck (1926), Vujanovic and Jones (1989), Whittaker (1937). Special Topics (Analytical): Altmann (1986), Arhangelskii (1977), Chertkov (1960), Korenev (1967, 1979), Koshlyakov (1985), Leimanis (1965), Lobas (1986), Lur’e (1968), Merkin (1974, 1987), Neimark and Fufaev (1972), Novoselov (1969), Tinierding (1908). Applied (Multibody Dynamics/Computational/Numerical, etc.): Battin (1987), Bremer [1988(a)], Bremer and Pfeiffer (1992), Haug (1992), Hughes (1986), Huston (1990), Junkins and Turner (1986), Magnus (1971), McCarthy (1990), Roberson and Schwertassek (1988), Schiehlen (1986), Shabana (1989), Wittenburg (1977). 4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE These are the customary meanings; but, of course, some, hopefully easily under¬ stood, exceptions are possible. The reader is urged always to keep common sense handy! Numbering of Equations, Examples, and Problems Chapters are divided into sections; for example, §3.4 means chapter 3, section 4. Equations are numbered consecutively within each section. For example, reference to eq. (3.4.2) means equation (2) of chapter 3, section 4. Related equations are indicated, further, by letters; for example, eq. (3.4.2a) follows eq. (3.4.2) and some¬ how complements or explains it. In chapters 2-8, examples and problems are placed anywhere within a section, and are numbered consecutively within it; for example, ex. 5.7.2 means the second example of chapter 5, section 7 and prob. 5.7.3 means the third problem of the same section. Within examples/problems, equations are numbered consecutively alpha¬ betically; for example, reference to (ex. 5.7.3: b) means equation (b) of the third example of chapter 5, section 7. Related equations in examples/problems are followed by numbers; for example, (ex. 5.7.2: k2) is related to or explains (ex. 5.7.2: k). WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Abbreviations AD Analytical dynamics AM Analytical mechanics CM Classical mechanics GP Gauss’ principle (§6.4, §6.6) H Holonomic (coordinate/constraint/ system) HP Hamilton’s principle (ch. 7) HZP Hertz’s principle (§6.7) JP Jourdain’s principle (§6.3) LP Lagrange’s principle (or D’Alembert’s principle in Lagrange’s form, §3.2) NP Nonholonomic (coordinate/ constraint/system) VD Virtual displacement (§2.5) PVW Principle of virtual work (§3.2) Chapter 1: Background Scalars in italics: for example, a, A, to, Q Vectors in boldface italics: for example, a. A, to, Q Tensors/Dyads in boldface, uppercase, italics; Matrices in boldface (always), uppercase (usually), roman (usually, but sometimes in italics, like tensors; should be clear from context, or clarified locally) General symbols N h f k, l, p, r, ... /, /', I", ... D, D', D", ... B E (...)' £/crs Number of particles of a system Number of holonomic constraints (H=l,...,h) Number of Lagrangean (or global) coordinates (= 3 N - h) Number of Pfaffian (holonomic and/or non- holonomic) constraints Number of (local or global) degrees of freedom (= n — m ) General (system) indices (= 1,...,«) Independent variable indices (= m + 1,..., n) Dependent variable indices (= 1,..., m < n) A implies, or leads to, B (A 44- B, for both “directions”) Discrete summation; usually, over a pair of indices (one for each such pair) Summation over all the material points (par¬ ticles) of a system, for a fixed time; a three- dimensional material Stieltjes integral, equiva¬ lent to Lagrange’s famous integration sign S... Total/inertial time derivative The (...) have been subjected to some kind of transformation Evaluated at some special value: for example, initial or equilibrium; or with some constraints enforced in it Expressed as function of the variables t , q , u> (quasi velocities) Transpose of matrix (...) Inverse of matrix (...) Kronecker delta Permutation symbol (—► tensor, in rectangular Cartesian coordinates) WWW.EBOOK777.COM INTRODUCTION r/v/a r A/B/ v A/Bl a A/B dm m P df dfe df i dF dR O - XYZ ( O - IJK) or Q-x k , ( O - «k') ♦ — xyz (♦ 1 *5 0 ♦ :(♦ - «k) ♦ - XYZ (♦ - IJK) or ♦ - x k > (♦- - «*•') d(.. ■)/dt d(. ■ •)/dt da/dt = da/dt + wxal dT/dt = dT/dt + cjxT -Txto] A- = ( A k'k) A-' =A t R R(n, X ) R' =R-1 X y = tan(x72 )n y = ( 71 , 2 , 3 ) = ( 7 x,y,z) <t>,8, *P to/Q = A t • (dA/dt) a = dco/dt/A. = dQ/dt o' IQ’ = (dA/dt) ■ A t (Instantaneous, and usually inertial) position/ velocity I acceleration of a particle P Position/velocity/acceleration vector of particle A relative to particle B Mass of a particle P (Continuum approach) Mass of a particle P (Discrete approach) Total force on particle P (= df e + dfp Newton-Euler) (= dF + dR: D’Alembert-Lagrange) Total external force on particle P Total internal or mutual, force on particle P Total impressed, or physical, force on particle P Total constraint reaction, or geometrical/ kinematical force on particle P Space-fixed; namely, inertial, axes (basis) at O Body-fixed; namely, moving, axes (basis) at body point ♦ Comoving, translating but nonrotating (or intermediate, or accessory) axes at body point ♦ Rate of change of vector or tensor (...) relative to fixed (inertial) axes Rate of change of vector or tensor (...) relative to moving (noninertial) axes a arbitrary vector, T arbitrary (second-order) tensor, o angular velocity vector of moving axes relative to fixed ones Matrix of direction cosines between moving (e.g., body-fixed) axes and space-fixed ones; a (proper) orthogonal matrix ( passive interpreta¬ tion): A k'k = A kk' = cosine [(fixed)*,, (moving)*] = cosine [x*,, x k ] = cosine [x*,x*,j Determinant of A = Det A = +1 Rotation tensor ( active interpretation of A) Rotation tensor about a point O and axis through it specified by the unit vector n, by an angle x Rotator tensor Angle of finite rotation about a point O and axis specified by the unit vector n Gibbs vector of finite rotation Rodrigues parameters relative to O — XYZ Eulerian angles (sequence 3 —> 1 —> 3): precession (</>) —> nutation (0) —> proper (or eigen-) spin (ip) Angular velocity vector/tensor (moving axes components) Angular acceleration vector/tensor (moving axes components) Angular velocity vector/tensor (space-fixed axes components) WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE a ' / A! = di2'/dt E = A + ti ii G C ♦ • H ..absolute = H . = S( r ~ r ■) X dm v H -.relative = >> . = S ^ ~ '' ■ ) X dm (v - V . ) = $ r /. x dm v/, M... I... T = (1 /2) Q v ■ v dm Chapter 2: Kinematics q = (q t ,...,q„) r = r(t,q) r(t, q + Sq)- r(t, q) & Sr = E ( dr/dq k )8q k e k = dr/dq k , e 0 = e„ +1 = dr/dt v = {dqi/dt = q\ = v u ... ,dq„/dt = = v„) •’ = E e k q k + e 0 a = Y1 e kdk + No other ^ terms = dv/dq k = <9a/d<7/ c = • ■ ■ = e k ud = E a DlAk + a d = 0 d6o = E a Dkdq k + = 0 = E = 0 Angular acceleration vector/tensor (space- fixed axes components) Sometimes referred to as tensor of angular acceleration Center of mass of a rigid body Contact point between two bodies Generic/arbitrary body point Generic/arbitrary space point Absolute angular momentum about • Relative angular momentum about • Moment of a force (or couple) about ... Moment of inertia tensor about ... (Usually inertial) kinetic energy of a system Holonomic, or global, or Lagrangean, or system, coordinates; otherwise known as generalized coordinates Fundamental Lagrangean representation of position of typical system particle P (First-order) virtual displacement of P Fundamental holonomic particle and system vectors (Heun’s begleitvektoren) Holonomic, or global, or Lagrangean, or system, velocities; otherwise known as generalized velocities Particle velocity expressed in holonomic variables Particle acceleration expressed in holonomic variables Basic kinematical identity (holonomic variables) Pfaffian constraints in velocity form ( a Dk , a D : constraint coefficients, functions of t and q; ur. quasi velocities) Pfaffian constraints in kinematically admissible, or possible, form (6\ quasi coordinates) Pfaffian constraints in virtual form General, kinematically admissible, variations of (...): d{...) = ^2{d... / dq k ) dq k + (d... / dt) dt = ^2(d... / dO k ) dO k + {d... /dd„ +l ) dt Virtual variation of (...): &{■ ■ ■) = ... /dq k )8q k = JJ(d... /d9 k )69 k WWW.EBOOK777.COM INTRODUCTION Quasi chain rules d.../dd k = ^2(d... /dqf) (dq,/du k ) = ^ A, k (d... / dq,) d... /dq, = ^2(d... /de k ){doj k /dqi) = ^ a k ,(d... /d6 k ) d... /dd n+l =^2(d... /dq,)(dq,/duj n+l ) + d.../dt = M d ■ ■ ■ l dc li ) + d.../dt = d... /d(t) + d.../dt d.../dt = E a k {d. . . /dQ k ) + d... /dO n+l = - ^ A k (d... /dq k ) + d... /dO n+l GENERAL (LOCAL) QUASI-VELOCITY TRANSFORMATIONS Velocity form U> D = ^2, a DkQk + <*D — Oi UJi = ^2 a Ik4k + a I ^ 0, W„ +1 = q u+ \ = t = 1 Kinematically admissible (or possible) form dd D = ^2 a Dk d^k + a D dt = 0 , d6, = ^2 a ik dqk + «/ dt ^ 0 , dd n+i = dq n+ 1 = dt 0 Virtual form S6 d = E a Dk 6 dk = 0 , Mi = E a ' k6qk ^ °’ 60 n +' ~ Sq “+ l = 6t = 0 HOLONOMIC VELOCITIES EXPRESSED IN TERMS OF QUASI VELOCITIES [(«*/) and (A k ,) are inverse matrices] Velocity form <lk = v k — E Aki^i + A k — ^2 Aki^i + A k v/l 0 Kinematically admissible (or possible) form dq k = E A kl d0, + A k dt = ^2 A k i dd, + A k dt ^ 0 (under d0 D = 0) Virtual form 6q k = J2 Ak i 60 i = E AkI 601 ^ 0 (under 66 D = 0) PARTICLE KINEMATICS IN TERMS OF QUASI VARIABLES (9,u>, etc.) Virtual displacement Sr = E ek Sqk = E Ek 60k = E £ > 601 WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Velocity Acceleration v = a = W/8y + £„+l = 22 W/£/ + £o w^£ A - + No other w terms yy th/£/ + No other u> terms Basic kinematical identity [where / = f(t,q,q) = f*(t,q,u) = /*] dr/dQ k = dv*/duj k = da*/du k = ■ ■ ■ = s k Transformation relations between the holonomic and nonholonomic bases e , £ f-k = 22 (dch/duj^e, = 22 A Ik e, £ »+! — y^ £ ° = yz ^ /g/ e ' i+x = — yz a/ ' £,t e,i+i e i = 22 (duk/dq^Sk = 22 a k ,s k e n +1 = e 0 = a k E k + e H+l = — 22 ^l e l + £ «+l FROM PARTICLE TO SYSTEM VECTORS (i.e., vectors characterizing, or expressing, system variables) (particle vector) • e k = (system vector) /f (holonomic components) ^ (particle vector) • s k = (system vector) /f (nonholonomic components) SPECIAL FORMS OF PFAFFIAN CONSTRAINTS Chaplygin ^d = <1d- 22 b Dl Ch = 0; b D[ : functions of q I = (q m+1 q n ) ui = qi + 0 Voronets uj d = q D — 22 b DI qi — b D = 0; b D[l b D \ functions of t and all qs ^i = qi¥= 0 Qd = 22 boi^r + b D , qi = w/ Corresponding particle virtual displacement Sr=22 e k Sc ik = 22 i { ' s<ir WWW.EBOOK777.COM free ©b©©fe > www n ©b©©k777-©©mni 20 INTRODUCTION Corresponding particle velocity v = E <2lPl + Pi [ «+l = V o HAMEL COEFFICIENTS 7*, = -7*, = EE (da kh /dq c - da kc /dq b )A hr A cs = EE ®kb [-Acr(dAb s /dcjc) A cs (dAbr/dq c )\ = EE (A br A cs A cr Ab s ) (dcikb/dq c ) 7 r,rt+1 = ~7 «+l,r = 7 ^ EE (■ da kb /dq c - da kc /dq h )A hr A c + E ( da kb/ dt ~ da k /dq h )A hr TRANSITIVITY EQUATIONS (69 k ) - 8ui k — E a kl [(6qi) - <5(<//)] + E E 7 + E 7**^4 (%)' - = E - &>/] - EE 'y^^b - E 7*^*} (^«+t)' - <Wn = (^»+t)‘ - <5(?«+i) = (to)' - 6(t) = 0 or, equivalently. d{80 k ) — 8(d9 k ) — E a w [<5?(^/) - <5(J<7/)] + EE 'y k bs d0 s 60 h + Y / 1 k b dt60 b d(8q k ) - $(<%) = E {[<w - W] ^EE 7fc dB, 89 h - £ 7 i dt 89 h } </($0 n+1 ) - <$(</6>„ +1 ) = </(«*) - <5(J?) = d( 0) - fi(*) = 0-0 = 0; or, assuming (Hamel viewpoint) (■ 8qk)' = 8(q k ) or d(8q k ) = 8(dq k ) (89 k y -&;* = £ E 7 ^ + E 7 * ^ = E hk b S9 b d(89 k ) - 8(dd k ) = EE 1 k hs d6 s 89 h + Y, r l k b d t 80 h EE' ^ bs {d9 s 89 h - 89 s d0 h ) + J2 7 \ dt 89 h (where Y2 ' means that the summation extends over b and s only once; say, s <b) Generally [with o, • = 1,... ,n; 89 n+l = 8t = 0] d(60.) - 8(dB,) = EE 7 \ g d9 0 89' +Y^ 7*. dt 89. WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE FROBENIUS’ THEOREM (Necessary and sufficient conditions for holonomicity = complete integrability of a system of m Pfaffian constraints in the n + 1 variables q { ,..., </„; q n+ \) l D ii'= 0, l D i,n+\ = = 0 {D= l' = m+ Chapter 3: Kinetics BASIC QUANTITIES r = (1/2)5 v r dm (Usually inertial) Kinetic Energy of system S=(l/2)5«. a dm (Usually inertial) Gibbs—Appell function of system, or simply Appellian NOTATION = f[t,q,q D (t,q,u)] = f*(t,q,w) = f* [arbitrary function]; for example, T(t,q,q) = T[t,q,q D (t,q,w)] = T*(t,q,w ) = T* => T*{t, q, u> D = 0,w/) = T* 0 (t,q } uj) = T* 0 T{t, q, q) = T[t, q, q D (t, q, q,),q,} = T„(t, q, q,) = T 0 (and similarly for S) LAGRANGE’S PRINCIPLE S'W R > 0 =>• 6I>6'W (for unilateral constraints; for bilateral constraints, > is replaced by =) Particle (or raw) forms S'W R = gdR-6r, S'W= gdF-Sr, 61= gdma-Sr Holonomic variable forms S'W R = J2 R k6q k , 6'W=Y,QkSq k , R k = 5 dR • e k , Qk = S dF ' e *’ 61 = J2 [(dT/dq k y - dT/dq k \ 6q k = £ (dS/dq k ) 6q k = £ E k 6q k WWW.EBOOK777.COM INTRODUCTION Nonholonomic variable forms 6'W R = Y A- M k = Y A i 66,, A k = $ dR ■ e k , 6'W = Y J ®kMk, 0 k =SdF-s k , v = Y m*/du k y - dT*/ae k - r k ] 66 k = Y (ds*/du k ) 6o k = Y 4 w k INERTIAL “FORCES" IN HOLONOMIC VARIABLES E k = S dma-e k = ( dT/dq k )‘ — dT/dq k (Lagrangean form) = dS/dq k (Appellian form) = dT/dq k — 2(dT/dq k ) = N k (T) = N k (Nielsen form; see chap. 5) INERTIAL "FORCES" IN NONHOLONOMIC VARIABLES 4 = S dm “ ' Ek = (dT*/du) k )' — dT*/dd k — T k = E*(T*) — T k (Volterra—Hamel form) = dS*/du k (Gibbs—Appell form) = Y ( dq,/duj k )E, = Y A, k E, (Maggi form) Nonholonomic deviation r k = ^ dm v* • [(dv*/dui k y - dv*/dO k ] = £ dm v* • E k *(v*) = £ dm v* • y k (particle/raw form) = - E E l l ks{dT*/&j,)u s - Y 7 l k{dT*/^,) = - Y h'kidT*/^!) [ h 'k = Y 7 ‘ ks Us + 7 * TRANSFORMATION EQUATIONS El = Y, a k<d-k ^ A k = Yj A, k Ri Ql = Yj a k'®k ®k = Yj A 'kQl Ei = Y! ak]dk ^ 4 = Y^ A ikE, THE CENTRAL EQUATION (Lagrange-Heun-Hamel Zentralgleichung) First Form 6T + 6'W + 6D = ( 6Py WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Second Form Pk 69k +55 Pk[(6e k y - Suj k ] - 55 (dT*/de k ) se k = 55 & k se k where 8T = Q dm v • hv = E [(dT/dq k ) 6q k + (dT/dq k ) 8{q k )\ = E K dT */ dq k ) % + ( dT*/duj k ) &*;*] = E K dT */dOk) 89 k + (dT*/8u k ) 6co k ] = <5 7"* 6'W= S d F‘6r = Y,Qk % = E & k 66 k 8D = ^ dm v • [(<5r)' — £v] = 55 (dT*/aj k )[(se k y - &*] - 55 ri k (dT*/^ t ) se k 6 P= S dm v ■ 6 r = E Pk Sqk = E Pk S9k Pk = S dm v • e k = 97" /dv k (holonomic momentum) 7-*/^ = ^ dm v* • z k = dT*/dui k (nonholonomic momentum) Pi = E ^ ^ = E y 4/A-7’/ (transformation formulae) EQUATIONS OF MOTION COUPLED Routh-Voss (adjoining of constraints via multipliers) E k = Q k + R k (multipliers; holonomic variables) UNCOUPLED Maggi (projections) Kinetostatic: ^Dk^D = ^di<Qd + A d (multipliers; holonomic variables) Kinetic: ^ik^i = J2 9 ik Qi (no multipliers; holonomic variables) Hamel (embedding of constraints via quasi variables) Kinetostatic: E D *(T*) — r D = & D + A D (multipliers; nonholonomic variables) Kinetic: Ef*(T*) — 7~7 = 0j (no multipliers; nonholonomic variables) SPECIAL FORMS (constraints of form q D = Yh boiqi + b d: boi, bo functions of t, q ) Maggi —> Hadamard Ed = Qd + An (kinetostatic) E, = Q, - E b DI X D => 7t/ + 55 6 djEd = Qi + 55 ^ diQd = Qt,o = Qio (kinetic) WWW.EBOOK777.COM INTRODUCTION Hamel —> Voronets To = T„{t , q, q 7 ), q D = E b DI q, + b D (■ dT 0 /dq,)' - dT 0 /dq, - E b DI {dT 0 /dq D ) -EE ^ '{dT/dq D ) 0 q r - E w D fdT/dq D ) 0 = Q, + E A' db Dr /dq r + E V/(36n/'/dto')] ~ [ 9 &D// 9 ?/' + E h D'i'(db DI / dq D ') w°i — w D In+x = db D /dq! + E b D 'i{db D /dq D <) db DI /dt+^2 bry{db DI /dq r y) Voronets —> Chaplygin T 0 =T 0 (q I ,q I ), q D = E b D ,(q m+ y.. .,q n )q r : i.e., b D = 0 {dTg/dcfo)' — dTg/dqj -EE^ '(dT/dq n ) o q r = Q, + E h DiQ D t D ii' = db DI /dq r - db nv jdq, POWER (OR ENERGY RATE) THEOREMS Holonomic variables dh/dt dL/dt + ^ ^ C/c,nonpotential Qk ^ ^ ^D^Di h = E (aC/5^) q k — L = T 2 + ( V 0 — T 0 ), L = T — V; dE/dt = —dL/dt + d(T x + 2T 0 )/dt + Ea , nonpotential Qk ^ ^ - E=T+v 0 , L=r-F=r-(C 0 + C 1 ), A = C-(r 1 + 27’o) Nonholonomic variables dh*/dt = -dL*/de n+x + E ®I. nonpotential ^1 ~ R : h* = E (dL*/du) I )ui I - L* = T* 2 + (V 0 - T* 0 ) dL*/dd n+l = 3L*/9t + E A k (dL*/dq k ) R = E E 7 r i{dL*/dui r )u> I (Rheonomic nonholonomic power) EXPLICIT FORMS OF THE EQUATIONS OF MOTION Lagrangean equations: with T =T 2 +T x + T ( y. 2T 2 = E E M kr q r q k , T i = E ^rQn ~Tt = M 0 , M k i = M /k , M kn+X = M n+lk = M k o = M ok = M k , M, ;+1 „ +1 = Mqq = M 0 : Inertia coefficients , WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 2 rk.rs = 2 r ksr = dM kr /dq s + dM ks /dq r — dM rs /dq k : 1st kind Christoffels. G k = Y Skr k, = Y ( 9M r /dq k ~ dM k /dq r )q r : Gyroscopic “ force ”, Qk = Qk, nonpotential + (dV/dq k )' - dV/dq k , V = Y^ E k (t,q)q k + E 0 (t,q) = V\{t,q,q) + Vo(t,q): Generalized potential, the Lagrangean-type equations, say E k = Q k , assume the form E k (T 2 ) + E k (Ti) + E k (T 0 ) = Q k , E k (T 2 ) = Y M krkr + E E r k,rsQrks + Y ( dM kr/9t)kn E k {T\) = dM k /dt — G k , E k {T,) = -(1/2 ){dMjdq k ). Hamel equations (stationary case, no constraints), with 2 T* = 2T*2 = EE M kr ^r ^ki 2r\ rs = 2r\ sr = dM* kr /dO s + dM\ s /do r - dM*Jde k , Akjp — r kjp +E 7 r k iM*. p (“nonholonomic Christoffels”) Hamel-type equations, say I k = 0 k , assume the form E M *-/^+E E Ak,i P w / u p — 0 k . APPELLIAN FUNCTION Holonomic variables 25 = E M k,Mk + 2 E E E r k.ipkk ki k P + 4 EE Ek,l,n+\kk k, + 2Y E k ,n+\,n+\kk Nonholonomic variables (stationary case) 25* = E E M * kr Wfc + 2 E E E A k‘p u p LAGRANGEAN TREATMENT OF THE RIGID BODY Kinetic energy T — ^translation + T, 2 ^translation = V+ rotation 2 ' coupling (♦: arbitrary body point; nr. mass of body) 2 ^"rotation = m ' S dm ( r /+ X v /*) = a> ■ ^ dm [»■/*. x («i x i^,)] = co ■ h* = co ■ • co ^coupling = CO ■ S dm (iy* X v*) = m v«. • (<u x r G/4 ) = m r* • v G/ * WWW.EBOOK777.COM INTRODUCTION Momentum vectors 6P = ^ dmv • 5r = p • 5r. + H * • 56 (66: virtual rotation vector) p = ^ dm v = m v G \ linear momentum of body r/+ x (dmv) = h+ + r G /+ x (m v 4 ): absolute angular momentum of body about ♦ H 0 = ^ r x (dm v) = //* + r+/ 0 x p (O: fixed point) Kinetic energy in terms of the momentum vectors 2T = p-v.+H.-(o, p = dT/dv+, H+=dT/d(o Kinematico-inertial (KI) acceleration vectors 51 = ^ dm a - 5r = I • 6r. + A. • 56 I = ^ dm a = ma G : linear KI acceleration of body A. = ^r/+ x (dma): angular KI acceleration of body about ♦ Eulerian principles in Lagrangean form Linear momentum (£2: vector of angular velocity of moving axes) I = dp/dt = dp/dt + Qxp - d/dt(dT/dv .) + Q x ( dT/dv .) Angular momentum A. = dH./dt + v, xp = (dH./dt + Q x H.) + v, xp = d/dt(dT/dco) + Qx (dT/dco) + x (dT/dv.) (also A. = dH. /dt + v. xp\ •: any point) APPELLIAN FUNCTION (to within acceleration-proportional terms) 2S = m a. 2 + 2mr G j. • (a. x a) + 2m (to x v G /+) • (a* x to) + a • I. • a + 2(a x to) • /* • to = m a G 2 + a • I G ■ at + 2(a x a>) • I G - co (Appellian counterpart of Konig’s theorem) RELATIVE MOTION (/: inertial origin; O: moving origin) Positions i"j = r 0 (t ) + r(< 7 i,..., q„) (motion of O known, q\ noninertial coordinates) WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Velocities V = V 0 + V relative + S2 x r, v r eiative = dr/dt = ^ (dr / dq k ) q k Virtual displacements Srj = Srg + Sr = Srg + S TC ir + 60 x r (Cl = d&/dt: frame angular velocity) Kinetic energy transport = rn v Q + 2m v Q • (Q X r G ) + Q • I 0 • Q = 2 T 0 T — Ttransnnrt + Trplativp + T c 27) 2 Relative = dm V re p ve • V re p ve 2 / s ^"coupling 7*rel’ve ' V O.retvc * H — T\ Prel’ve = ^ V rel’ve = m(dr G /dt) = ( tS dm ( dr / d dk)) 4 (,noninertial linear momentum) #0,reEve = S X ( dr I dt ) = ^2(S ^" r X ( dr / d dkfj 4 (,noninertial absolute angular momentum) LAGRANGEAN TREATMENT OF RELATIVE MOTION [equations of carried body; say, E k {T) — Q k ] Ek (-^2) “1“ Q k, transporttransl’n “ 1 “ transport rotat'n “ 1 “ transport rotat’n centrifugal + C4 , Coriolis 7 transport transl’n = ^ ^translation /^Qki ^translation = W (Iq ' Qk ,transport rotat’n = -{da/dt) • (dH 0 , eVve /dq k ) = — ( dQ/dt ) • dm v x ( dr/dq k )j ; transport rotat’n centrifugal ^^centrifugal = ~S (12 X f) — f2* /$ • 12, Qk ,Coriolis = -2$ n x (dm v reFve ) • (dr/dq k ) = ^ gkl = gki • *2, Hk! = dm ^ dr / d( lk) X (df/dqi)\ Chapter 4: Impulsive Motion Fundamental impulsive variational equation (impulsive principle of Lagrange — LIP) : 6i = snv, WWW.EBOOK777.COM 28 INTRODUCTION where SI = ^ dm a - Sr = ^ A (dm v) • Sr: (first-order) virtual work of impulsive momenta, and tfw = $ dF-Sr = gdF -Sr: (first-order) virtual work of impulsive impressed “forces vw R = S dR ' br = Y.(S dR - e k) % = E R > SfV= SdF-6r=J^(SdF- e k) % = E & 6 SI = A(dmv) • Sr = dm Av • e k j Sq k = E A ( S dm v ■ e,i ) 5qk - E Apk Sq ^ and Pk = S ( dm V • e k ) = dT / dq k ^ Ap k = a(^£ dm v-ekj = g [A (dm v)- e k \: [holonomic (A)th component] impulsive system momentum change, Qk = S dF ' L ’ k = S dF ‘ ek ' [holonomic (A)th component] impulsive system impressed force; or, simply, impressed system impulse. R k =E dR ■ e k = gdR- e k : [holonomic (A)th component] impulsive system constraint reaction force, we finally obtain LIP in holonomic system variables: E Rk Sqk = °> E j ( dT l dq k) Sqk = E & k and similarly in quasi variables. Energetic theorem AT=T + ^T-= W_ /+ , 2T + = Q dm r + • v + , 2T = ^ dmv • v , WWW.EBOOK777.COM where §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE and W_ /+ = SdF-{^ + + m 2 In words: The sudden change of the kinetic energy of a moving system, due to arbitrary impressed impulses, equals the sum of the dot products of these impulses with the mean (average) velocities of their material points of application, immedi¬ ately before and after their action. APPELLIAN CLASSIFICATION OF IMPULSIVE CONSTRAINTS, AND CORRESPONDING EQUATIONS OF IMPULSIVE MOTION At a given initial instant t', new constraints are suddenly introduced into the system and/or some old constraints are removed, or suppressed. As a result, mutual percus¬ sions are generated, which, in the very short time interval r = t" — t' over which they are supposed to act and during which the shock lasts, produce finite velocity changes, but, according to our “first” approximation, produce negligible position changes; that is, for r —> 0: Aq = 0, A(dq/dt)f0. The constraints existing at the shock moment are either persistent or nonpersistent. By persistent we mean constraints that, existing at the shock “moment,” exist also after it, so that the actual postimpact displacements are compatible with them; whereas by nonpersistent we mean con¬ straints that, existing at the shock moment, do not exist after it, so that the actual postimpact displacements are incompatible with them. The constraints that exist at the shock instant can be classified into the following four distinct kinds or types: 1. Constraints that exist before , during , and after the shock ; that is, the latter neither introduces new constraints, nor does it change the old ones; the system, however, is acted on by impulsive forces. An example of such a constraint is the striking of a physical pendulum with a nonsticking (or nonplastic) hammer at one of its points, and the resulting communication to it of a specified impressed impulsive force. 2. Constraints that exist during and after the shock, but not before it; that is, the latter introduces suddenly new constraints on the system. Examples: (a) A rigid bar that falls freely, until the two inextensible slack strings that connect its endpoints to a fixed ceiling become taut (during) and do not break (after), (b) The inelastic central collision of two solid spheres (“coefficient of restitution” = e — 0—see below), (c) In a ballistic pendulum, the pendulum is constrained to rotate about a fixed axis, which is a constraint that exists before, during, and after the percussion of the pendulum with a projectile (i.e., first-type constraint). The projectile, however, originally inde¬ pendent of the pendulum, strikes it and becomes embedded into it, which is a case of a new constraint whose sudden realization produces the shock, and which exists during and after the shock but not before it (i.e., second-type constraint). 3. Constraints that exist before and during the shock , but not after it. For example, let us imagine a system that consists of two particles connected by a light and inextensible bar, or thread, thrown up into the air. Then, let us assume that one of these particles is suddenly seized (persistent constraint introduced abruptly; i.e., second type), and, at the same time, the bar breaks (constraint that exists before the shock but does not exist after it; i.e., third type). 4. Constraints that exist only during the shock , but neither before nor after it. For example, when two solids collide, since their bounding surfaces come into contact, a constraint is abruptly introduced into this two-body system. If these bodies are WWW.EBOOK777.COM INTRODUCTION elastic (e = 1—see coefficient of restitution, below), they separate after the collision, which is a case of a constraint that exists during the percussion but neither before nor after it (i.e., fourth type); while if they are plastic (e = 0), they do not separate (projectile and pendulum, above; i.e., second type). If 0 < e < 1, the bodies separate; that is, we have a fourth kind constraint. Clearly, the first two types contain the persistent constraints, while the last two contain the nonpersistent ones. Schematically, we have the classification shown in table 1. Table 1 Appellian Classification of Impulsive Constraints Preshock (before) Shock (during) Postshock (after) 1 (persistent) 2 (persistent) 3 (nonpersistent) 4 (nonpersistent) In impulsive problems: the excess of the number of unknowns (postimpact velo¬ cities and constraint reactions) over that of the available equations [those obtained front Lagrange’s impulsive principle; plus preimpact velocities, impressed impulsive forces, constraints, and, sometimes, knowledge of the postimpact state (second type; e.g., e = 0)]—namely, the degree of its indeterminancy—equals the number of its constraints, which, having existed before or during the shock, cease to do so at the end of it; that is: Degree of indeterminacy = Number of nonpersistent constraints ; that is, the persistent types 1 and 2 are determinate, while the nonpersistent ones 3 and 4 are indeterminate. COEFFICIENT OF RESTITUTION (e) (v 2 /t •n) + _ v 2/1 ,„ + Relative velocity of separation (r 2 /i • n )~ v 2 /i,n - Relative velocity of approach where 1 and 2 are the two points of bodies A and B that come into contact during the collision, and n is the unit vector along the common normal to their bounding surfaces there, say from A to B. This coefficient ranges from 0 {plastic impact, no separation) to 1 {elastic impact, no energy loss); that is, 0 < e < 1. ANALYTICAL EXPRESSION OF THE APPELLIAN CLASSIFICATION; PERSISTENCY VERSUS DETERMINACY 1. In terms of elementary dynamics: Consider a system that consists of N solids, in contact with each other at K points, out of which C are of the nonpersistent type, and/or with a number of foreign solid obstacles that are either fixed or have known motions. Assuming frictionless collisions, we shall have a total of 6TV + K unknowns (6 N postshock velocities, plus K percussions at the smooth contacts, along the common normals), and 6N + K C equations (6 N impulsive momentum equations, WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE plus K — C persistent-type constraints); and therefore the degree of indeterminacy equals the number of nonpersistent contacts C (i.e., the kind that disappear after the shock). Hence: (a) a free (i.e., unconstrained) solid subjected to given percussions or (b) a system subjected only to persistent constraints are impulsively determinate. 2. From the Lagrangean viewpoint: (a) A number of constraints, imposed on a system originally defined by n Lagrangean coordinates, can always be put in the equilibrium form: q x = 0 , <72 = 0 ,..., q m — 0 ( nr. number of such constraints < n). (b) Within our impulsive approximations, even Pfafhan constraints (including non- holonomic ones) can be brought to the holonomic form; that is, in impulsive motion, all constraints behave as holonomic; and to solve them, either we use impulsive multipliers, or we avoid them by choosing the above equilibrium coordinates; or we use quasi variables. Assuming, henceforth, such a choice of Lagrangean coordinates for all our impul¬ sive constraints (and, for convenience, re-denoting these new equilibrium coordi¬ nates by </i,..., < 7 ,„;..., q„), we can quantify the four Appellian types of impulsive constraints as follows: • First-type constraints (existing before , during , and after the shock). As a result of these constraints, let the system configurations depend on n, hitherto independent, Lagrangean parameters: q = {q x ,..., q„). During the shock interval ( t', t"), the cor¬ responding velocities q = (q x ,... ,q„) pass suddenly from the known values (q)~, at t ', to other values {q ) + , while the q s remain practically unchanged; that is, here we have {dk) before (gk) during 0 ) (<7fc)after Mk = (dkf - {dkY f 0 [fc) +: unknown, (q k )~: known], • Second-type constraints (additional constraints existing during and after the shock, but not before it). Here, with q D " = {q x ,... ,q m »), where m" < n, we have (to")before 7^ 0 , (to")during = 0 , (to")after = ^ {qD") f o, (qD") + = o => A{q D ") = -(to") f®- • Third-type constraints (additional constraints existing before and during the shock, but not after it). Here, with q D '« = {q m - ( ] ,..., q m '"), where m'" < n, we have (to'")before = (to")during = (to'")after f (<to'") - = 0 . (to"') + f~ h ~~r A{q D ") = (to'") + f °- • Fourth-type constraints (additional constraints existing only during the shock, but neither before nor after it). Here, with q D »» = {q m - + \,..., q,„""), where m"" < n, we have (to"") before ^ (to"") _ f 0: (to"")during ~ 0, (to"")after / ^ (to"") + f 0 => ^(to»") = (to"") + - (to'"') - f 0. WWW.EBOOK777.COM INTRODUCTION Hence, if no fourth-type constraints exist, m'" = m""; and if no third-type con¬ straints exist, m" = in'"; etc. Next, arguing as in the case of continuous motion (chap. 3), during the shock interval, we may view the constraints of the second, third, and fourth types as absent, provided that, in the spirit of the impulsive principle of relaxation (LIP), we add to the system the corresponding constraint reactions. All relevant equations of motion are contained in the LIP: ^2 A (dT/dq k ) Sq k = ^ g* % ( k = 1,...,«). If the virtual displacements 6q = (6q\,... ,6q„) are arbitrary, the right side of the above equation contains the impulsive virtual works of the reactions stemming from the second, third, and fourth type constraints, and operating during the shock interval (t',t"). Therefore, to eliminate these “forces,” and thus produce n — m"" reactionless, or kinetic, impulsive equations, we choose Sq’s that are com¬ patible with cdl constraints holding at the shock moment ; that is, we take 6cj \,..., Sq m , 6q m "+1 • ■ - - ? ^q m '"• m " ! +1 ■ • - -; ^q m ,m If 6q m "" + \,...,Sq„fO. Corresponding two (uncoupled) sets of equations: Impulsive kinetostatic: A{dT / dq D ) = Q D + (D = 1,... ,m""), Impulsive kinetic. A(dT/dq t ) = Q k (/ = m"" + 1,..., n). Further, since the velocity jumps Aq are produced only by the very large impulsive constraint reactions, operating during the very small interval t" — t' , within our approximations, the Qj [since they derive only from ordinary (i.e., finite, nonimpul- sive) forces, like gravity] vanish: Q r = 0; and so eq. (b) reduces to Appell’s rule: A{dT/dq,) = 0 => (dT/dq,) + = {dT/dqf-. In words: The partial derivatives of the kinetic energy relative to the velocities of those system coordinates cf s that are not forced to vanish at the shock instant (i.e., (/during 7 ^ 0) have the same values before and after the impact; or, these n — m"" unconstrained momenta, pi = dT/dqi , are conserved. To make the problem determinate, in the presence of nonpersistent-type con¬ straints, we must make particular constitutive {i.e., physical) hypotheses: for example, elasticity assumptions about the postshock state. EXTREMUM THEOREMS OF IMPULSIVE MOTION All based on the following master equation (impulsive Lagrange’s principle): ^ dm (v + — r ) ■ Sr = ^ dF • Sr Carnot (first part—collisions) Sr~v + , dF = 0^T + -T-<0 WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Carnot (second part—explosions) <5r~v”, dF = 0 T + - T > 0 Kelvin (prescribed velocities) 6r ~ v + , 5r~v + + 5 K v = v, v~ = 0 —> T(v) — C(v + ) > 0, 6 K T + = 0 Bertrand-Delaunay (prescribed impulses) <5r~v + , 6r ~ v + + S B/D v = v -► T(v) - T(v + ) < 0, 6 B/D T + = 0 [Taylor: r Kelvin (v) - T(v + ) > 7> + ) - C(v) Bertrand _ Delaunay ] Robin (prescribed impulses and constraints) 6r ~ v + , <5r ~ v + + 6 R v = v —> P = (dm/2)(v — v - ) 2 — ^dF • (v — v~): stationary and minimum Gauss (impulsive compulsion) Z = ^ (dm/2)(v — v — dF/dm) 1 = P+ (dF) 2 /2dm: stationary and minimum Chapter 5: Nonlinear Nonholonomic Constraints CONSTRAINTS fo{t,q,q) = o QUASI VARIABLES Velocity form Wfl = fo(t, q, q) = 0, u,= fj(t, q, q) ^ 0, w„ =1 = q ,,+1 = i = 1 Virtual form (by Maurer-Appell-Chetaev-Johnsen-Hamel) S0 D = Y ( 9f D /dq k ) Sq k = Y (9^o/dq k ) Sq k = 0, 69, = Y ( df,/dq k ) 6q k = Y (dui/dq k ) 6q k ± 0 , % = Y ( d 9k/duJi) 99, = Y ( d< ik/dui) 99, Compatibility Y ( df k /dq h )(dq h /duji) = Y (du, k /dq b )(dq b /du;,) = du> k /du>, = 6 kh Y (9F k /dui h )(duj h /dq,) = Y ( dq k /du b )(duj b /dqi) = dq k /dq, = 6 kl WWW.EBOOK777.COM INTRODUCTION PARTICLE KINEMATICS Virtual displacements Sr = 22 (dr*/89,) SO, = 22 s, 69, = Sr*, where £ i = 22 ( dr /dq k )(dq k /duj,) = 22 ( d 9 k/duJ,)e k , e k = 22 (dr*/99,)(du>,/9q k ) = 22 (dui/dq k )s,\ that is, d(.. .)/89, = *22 [#(• ■ -)/dq k \(dq k /duji), d(.. .)/dq k = 22 [^(- • ■)/d9,](du>,/dq k ) [nonlinear symbolic (nonvectorial/tensorial) quasi chain rules] Velocities v = 22 ?*(*, q, u)e k +e 0 [t = q n+l } = 22 V) £ k + £o = r*(t, q, uj) = v*, where a 0 = dr/89 n+l = 22 (dr/8q a )(dq a /duj n+l ) [a = 1,... ,n + 1] = 22 (dq k /duj n+l )e k + e 0 = 22 (Qk e k- ~ ^k £ k) + e 0 = Co + 22 (jlk - 22 (dqk/du,)u2)e k , and, inversely, e 0 = dr/dt = 22 (dr/d9 a )(9u; a /dq n+l ) [a = 1+ 1] = £o + 22 { UJk ~ H (du k /dq,)q^J s k . For any function /* = f*(t,q,u), 9f*/89 n+l = 22 (df*/dq k ){q k - 22 ( d q k /du,)u2) + df*/dt- which in the Pfaffian case reduces to the earlier df*/89 n+ 1 = 22 ( 5 /*/ dq k )A k + df*/dt = 9f*/d(t) + df*/dt. WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE In particular, for /* = q h we find dqb/dO s = dq h /duj s , dq h /dO n+l = dq h /du n+l = q h ~Y (dq b /du,)u>i [= q b — Yj A m uji = A b , in the Pfaffian case]; and, inversely, dO k /dt = du k /dq n+l = ui k — Y^ (a o k /dq,)q,. Accelerations a = dv/dt = Y, ( dv/dq k )q k + No other ij/d> terms = Y ( dv*/duj,)d>i H- = Y S A d-= E ( da */ ddj i)^i d- = a*(t,q,uj,u) = a*, where dv*/du>/ = E (dv/dq k )(dq k /duji) or £/ — ^ e k (dq k /du>i) (which is a vectorial transformation equation, and not some quasi chain rule). BASIC KINEMATIC IDENTITIES Holonomic variables dr/dq k = dv/dq k = da/dq k = • • • = e k Nonholonomic variables dr*/dO k = dv*/du k = da*/du k = • • • = s k System forms dq k /80 1 = dq k /du>i = dq k /dd>i = • • • dO,/dq k = dui,/ dq k = ddi,/dq k = ■■■ NONINTEGRABILITY RELATIONS Nonholonomic deviation (vector) 7k = E k *(v*) = Y E k *(q,)e, = Y V 'k e i = ~Y where Nonlinear Voronets-Chaplygin coefficients V l k = ( dq,/du k y - dq,/dO k = E,*^). WWW.EBOOK777.COM INTRODUCTION Nonlinear Hamel coefficients H k h = Y ( d 9l/^b)[(duk/dqiY - dw k /dq,} = Y (9q,/duj h )E,(uj k ) =* h k b = Y 7U = Y l k bs^ s + l\,n +1 (in the Pfaffian case) H\ = ~Y (&>b/dq,) V' k ^V l k = ~Y (dq,/^ b )H h k E t {u k ) = ~YY^ b / d ^ dUk i d ^ Eh *^^ w = -EE (dq,/du h )(dq s /duj k )Ei(u> k ) For a general function /* = f*(t,q, to), the following noncommutativity relations hold: d/de,(df*/de k ) - d/de k {df*/de,) =EEE [(^ q b /dq s dw k )(dq s /duji) ~ ( d 2 q b /dq s du,)(dqj du k )\(du p /dq h )(df* / d9 p ) THE NONLINEAR TRANSITIVITY EQUATIONS (69 k )' — 6u> k = E (du> k /dqi)[(6qi) — 6(q{)] + Y^ Ei(uj k ) 6q, = Y (a o k /dq,)[(6q,y - 6(qi)\ + Y ®b = E [(<$?/)' - (5(?,)] - E E E'bi^k/dqi) 69 h (6q,y - 6(q,) = £ (d?,/^)! W - &*] + £ v' k 69 k = Y ( d 9i/duJk)[(69 k y - 6u) k ] - E E ( dqi/du} b )H h k 69 k SPECIAL CHOICE OF QUASI VELOCITIES V D = fn(j,q,q) = 0, ujj = fiit, q, q) = q, ^0- => 9d = qD(t,q,qi) = Mt,q,qi) System virtual displacements 6q k : 6q D = Y ( d( Pn/dq,) 6q u 6qi = Y ( d 9t/ d 9i') b< h' = E (^') %' = St h Particle virtual displacements 6r = E 8c E = E B ' 8qi ~ where B b = dr/d(q,) = dr/dq 1 + ^ (dr/dq^idcfo/dc/j) =e, + Y (^>D/dqi)e D ] WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE and, in general, dBj/dqji ^ dBj'/dq! (i.e., the Bj are nongradient vectors) Particle velocities and accelerations v —► v 0 = E B l q l + No other q terms, a —+ a 0 = E Bjqj + No other q terms; => dr/d{qi) = dvjdq, = da 0 /dq I = = B, Special transitivity relations 6qD = '52(d(/) D /dq I )8q l and q D = q D {t,q,qi) = 4> D {t,q,qi) (8q D Y ~8q D = ^2 (^dI dq,)[{8qi)' - 8{qi)\ + E - d(j) D /d{q,)} 6q, 7 where d(bn/d{q,) = d<p n jdq ] + E (d<j) D /dq D '){d(j> D '/dqfi, d{- ■ -)/d{qi) = d{.. .)/dq, + E [d(.. .)ldq D ]{d<j> D /dq I ) Nonlinear Suslov transitivity relations (8q k y - 8(q k ): (6q D )' - Sq D = E W °i Sq, 0) («?/)■-«?/ = 0 [= 0; i.e., W 1 '/ = 0], where W D t = E f ((j) D ) - E {d(t) D /dq D ')(d(j) D :/dq,) = (d^o/dq,)' - dcj> D /d{q,) = E {I) {<j> D ) [special nonlinear Voronets coefficients] Nonlinear Chaplygin system qd = qD{qiAi) = <t>D{qi,qi) W D J -> T D j = {d^ D /dq,y - dci> D /dq, = E,{(j) D ) [special nonlinear Chaplygin coefficients] KINETIC PRINCIPLES (P k = 8T*/du k ) Central equation E ( dp k/ dt ) Mk - E (9T*/d0 k ) 86 k + E Pk[(S0 k )‘ - 8u k \ = E ®k ® k WWW.EBOOK777.COM INTRODUCTION Lagrange’s principle in NNH variables E (dPk/dt - 8T*/dO k + Y H‘ k P, - 0 A ) S0 k = 0 EQUATIONS OF MOTION (D = 1,... ,m; 1 = m + 1,...,«) Coupled Ek(T) = Qk + Y X D(df D /dq k ) ( Routh-Voss form) Uncoupled I D = 0 fl + /!/> (Kinetostatic) 7/ = 0 7 (Kinetic) where 4 = $ fl *' = E ( d <ii/ duj k)Ei = dS*/du k = ( dT*/du k y - 8T*/d6 k - r k {Raw form) {Maggi form) {Appell form) (Jolmsen—Hamel form) r k = $dm V* . E k *(v*) = J2 V kPl = -J2 H’ k P, [nonholonomic correction term] Transformation equations between holonomic and nonholonomic components 4 = E (dq//duJk)Ei <£> E, = Y ( duj k /dq,)I k , 0k = E ( dqi/du k )Q, &Q, = Y (■ duj k /dq,)0 k , Ek'* -r k ' = J2 (&>k/fa k ')(E k * - r k ) Transformation equations of E k *{T*) and r k between the quasi velocities u> <-> to' Ek ,*{T*') = E (a^/a^o E k *(T*) + Y ( dT*/av k )E k .*(u k ), Ek' = E {dT*/du k )E k ,*(u; k ) + Y {dto k /dw k ,)r k Johnsen-Hamel forms in extenso 4 = dP k /dt - dT*/dO k + Y H\P, = dP k /dt - dT*/dQ k — EE (dui/d q h ) V h k P, = dP k /dt - dT*/dO k - Y vb kPb* [ E = Pk = p k (t, q, q) = Pk*(t, q, <*>) = {dr y dq k )* WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE SPECIAL FORMS OF THE EQUATIONS OF MOTION FOR THE CHOICE UD = /d{U q, q)=qD- q, <li) =0, uj, = f,(t , q , q) = q, ± 0 and its inverse Qd =uj d + + ^o(t,q,Wi), qi = Wj, and with the notation E k = E k {T) = ( dT/dq k Y - dT/dq k = dS/dq k Maggi equations =>• nonlinear Hadamard equations Kinetostatic: Ed — Qd + -\d Kinetic. Ej + W> D /dqi)E D = Qi + 55 ( d <l>n/dq,)QD or dSJdqj = Qi + 55 ( d< P D /dqi)QD (= Qi.o = Qio ), where S = S(t, q,q,q) = • • • = S 0 (t, q , qi,qi) = S 0 , constrained Appellian V 0 Hamel equations =>■ nonlinear Voronets equations Ei(To) - 55 (' d<t> D /dqi)(dT 0 /dq D ) - E Io = £ (/) (r o ) - E Io = Q Io , H D ! - -£(/)(&,) = El - r /j0 = r /0 = 55 w D I {dT/dq D ) 0 = 55 Voronets equations => Chaplygin equations 4d = qo(qi,qi) = <t>D{qi,qi) and r 0 = r 0 (^,^) => tC 11 / = E^(c/) D ) —> Ej{(j) D ) = (d^D/dqj) — c^n/cty/ = T D , =► e Io -+ 55 (dT 0 /dq,y - dTjdqj - 55 T D I {dT/dq D ) 0 = Q Io Transformation of the nonlinear Hamel and Voronets coefficients V l k ,H ! k under uj b ’ = u> b '(t,q,q) <£> qi = qi(t,q,u'): r k = 55 ^ =55 = 55 (duk/dvk 1 ) V l k + 55 WWW.EBOOK777.COM INTRODUCTION ^ = EE {du) k /du> k ')(du>i'/du>i)H l k E [{duk/duv)' - du k /dO k ']{du v /du k ), H l ' k , = - Y, (&>v/dqi) V l k . V l k , = -J2 {dq l /du Jl ,)H , ' k , Chapter 6: Differential Variational Principles PRINCIPLE OF LAGRANGE ^ (dm a — dF) • Sr = 0, with St = 0 PRINCIPLE OF JOURDAIN (dm a — dF) • 6v = 0, with St = 0 and Sr = 0 PRINCIPLE OF GAUSS Q (dm a — dF) • Sa = 0, with St = 0, Sr = 0, and hv = 0 PRINCIPLE OF MANGERON-DELEANU (dma — dF) • $r = 0, (5=1,2,...) with <5? = 0, anh hr = 0, 6(f) = 0, h(r) = 0,..., < 5 ( ( s 7 1) ) =0 NIELSEN IDENTITY N k (T) = dT/ dq k - 2 (dT/dq k ) = (dT/dq k )' - ST/c^ = E k (T) TSENOV IDENTITIES Second kind E k (T) = C k {2 \T) = (1/2 )[df/dq k - 3(0r/0 ?ifc )] Third kind E k (T) = C k ®(T) = (1/3 )[dT/dq k - 4(dT/dq k )} WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE MANGERON-DELEANU IDENTITIES E k (T) = C k ®{T) = (l/s) « (s) dT/dq k -{s+l)(dT/dq k ) [C k {l) (T) = N k (T)} VARIOUS KINEMATICO-INERTIAL IDENTITIES (■s—1) W d T /dq k = dT/dq k = g dmr-(dr/dq k ) d T ] /d q k = s(dT/dq k ) =s $dmr- ( dr/dq k ) W ~ (-S+1) (s) (s+1) T = \ dm r • r + s \ dm r ■ r + no r terms, d/dt ■ (i-I) (s)' d T 0 /dq, - dT/dq, = d/dt(dT/dqi) — dT/dq { + ^ d/dt W M {dT/dq D )(dq D /dqi) (s-1) W (-s— i) («) / (s-i) W \ / W W\ 5 T„/dq I = d T jdq l + XI (j 9 T / d 9oJ [dq D /dq,j VIRTUAL DISPLACEMENTS NEEDED TO PRODUCE THE CORRECT EQUATIONS OF MOTION Constraints Lagrange Jourdain fit, q) = 0: df/dq Sf = (df/dq) Sq S'f = 0 6'f = (df/dq) Sq f(t,q,q) = 0: df/dq f(t,q,q,q) = 0: df/dq — S'f = (df/dq) Sq Gauss 6"f = 0, 6"f = 0 S"f= (df/dq) Sq 6"f = 0 6"f = (df/dq) Sq 6"f = (df/dq) Sq CORRECT EQUATIONS OF MOTION [Notation: M k = E k (T) - Q k = N k (T) - Q k = dS/dq k - Q k . Principle: £ M k S* k = 0, 6* k = 6q k , 6q k , 6q k ,...] Constraints Virtual Displacements f D (t, q) = 0 % = £ (df D /dq k ) Sq k /b(<, q, q) = 0 S'f D = £ (df D /dq k ) Sq k f D (t,q,q. q) = 0 S"f D = £ ( df D /dq k ) Sq k SPECIAL FORM OF CONSTRAINTS Equations of Motion M k = £ A D (df D /dq k ) M k = £ A D (df D /dq k ) M k = £ A D (df D /dq k ) 9d = Mfq,qi) ( D= = m+l,...,n). For an arbitrary differentiable function / = f(t,q,q ) = f[t,q,4>D{t,q,qi)Ai] = f 0 (t,q,qi) = L, WWW.EBOOK777.COM INTRODUCTION the following identity holds: Ni{fo) = E/ifo) + Y {.df 0 /dq D ){d(j) D /dq I ) =* Nj(T 0 ) = Ej{T 0 ) + Y (dTjdq^dfo/dq,), Ni{q D ) = E,(q D ) + Y ( d ^D/dq D ')(9^ D '/dqi). NIELSEN FORM OF SPECIAL NONLINEAR VORONETS EQUATIONS N,(T 0 ) - Y ( dT/dq D ) 0 Nj{q D ) -2Y(dT/dq D ) 0 {dt D /dq I ) = Q Io NIELSEN FORM OF SPECIAL NONLINEAR CHAPLYGIN EQUATIONS dtjdq, - 2 {dTjdqj) - Y ( d T/dq D ) 0 [ d <iD/dqi ~ 2 (dq D /dq,)\ = Q lo Special Pfaffian —> Voronets form <1d = Y bm + Mb Q) Then the above Voronets equations assume the special Nielsen form: dtjdq! - 2 {dTjdqi) ~ Y ( dT /dq D )o{Y ft 0 "' ~ 2 (dbnr/dq r )\cir + [b D [ - 2{db n /dq,)\} -2 Y^T/dq D ) 0 b DI = Q l0 , where b D a' = Yj [{dboi/dqD')bD'i' + ( db DI Jdq D ')b D 'j\ + (db DI /dqj' + db DI '/dqj), b D i = b D In+ 1 = Y^ [( db D /dq D i)b D ij + {db DI / dq D ')b D ') + ( db DI /dt + db D /dqj). Special Voronets —> Chaplygin form 4d = Y b Di(9i)9i, and dT/dq D = 0 Then the above Chaplygin equations assume the special Nielsen form: dtjdch - 2(dTjdq I ) ~YJ2^ dT / dq "t( db Di/dqi' - db nr Jdq,)q r = Q Io [b D = 0, b D i = 0, b D n i = dbojdqi' + db DI :/dqj\ WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE NIELSEN FORMS OF HIGHER-ORDER EQUATIONS Let ,, W (-5) N k ^\...)^d{...)/dq k -2 (,-1) (.5-1)' d(.. .)/d q k E k {s \...) = d/dt ’ (*-i) W d(- ■ ■)/dq k r (5-i) (5-i)i 3(...)/d q k Then, for any sufficiently differentiable function / = f(t,q,q), and any k= 1,25= 1,2,3,..., Let N k {s \f) = £*«(/). N k ^ s \...) = d(.:.)/de k - 2 E k * {s \...) = d/dt ■ (5-1) (5-1) d{...)/d e k ' (*-i) w' d(...)/de k ■ (5-1) (5-1) d(.. .)/d e k where (5-1) (5-1) d{...)/d e k = £ (s-l) (.5-1)' d{...)/8 q, ' M W dq,/d9 k [(j)th-ordcr quasi chain rule]. Then, for any sufficiently differentiable function /* = f*(t,q,w), and any k = 1,2,... ,n; s = 1,2,3,..., N k * [s) (D = £a-* w (./*), where (»-i) W f{t,q,q) => f => ■ ■ ■ f =>f, 5-1) (1) (5-1) (5)/ ' (i) (i-i) W\1 / t, q,q = q,- •> 9 ; t,q,q, ■ ■ , q , 0 ) (5-1)/ (1) (5-1) (5) \ = / [t,q,q,..., q ,6 1, w w ■ (i) (*-i) «/ (i) (5 1) (5) \ (S+ 1 )/ ' ( 1 ) (s 1 ) (5) (5+1) \ " /* = / t,q,q , ■ ■ • , q ; q[t,q,q,.. ., q ,0 > q ( t,q,q q ,0, e J ( 5 )/ (1) (5-1) ( 5 ) (5+1) = f[t,q,q,..., q ,9, 0 WWW.EBOOK777.COM INTRODUCTION Hamel-type equations (s = 1,2,3,...) ( 5 ) d/dt (■»—!) W d T*/d6j (,-i) (*-i) -d T*/d Oj E / w w\ ■ w (i-iy si dq k /dO, ) - dq k /d 0, „ ( W «\ = Y.[ d( ikl de i)Q k = & I (j-l) (j) \ * d T / dq k Nielsen-type equations / W W \ / I*” 1 ) (•»-!) (s) [dT*/d0 I j -(s+1) d T*/d 0, E (*+D w H d q k /dO l - (s + 1) dq k /d 0, (s-l) W \ * d T /dq k For s = 1, the above yield, respectively, d/dtid^/dO^-d^/dO, - E [( 5 4/<90/)' - dq k /dOj] ( dT/dq k )* = © h dT*/d0 I -2(dT*/d0 I ) ~ E [ d dk/dO, - 2(dq k /d0f)\(dT/dq k )* = 0,; and, for s = 2, 2(df*/d0,y -dt*/d0j ~ E [ 2 (d4/<94)' - d^k/dOj] (dt/dq k )* = 0 r , 2(df*/d0 I )-3(dt*/d0 I ) - £ [ 2 (dgyae,) - 3(^M)] ( df/dq k )* = GAUSS’ PRINCIPLE Compulsion Z = (1/2)^ dm [a — (dF/dm)] = (1/2) ^ (1 / dm) {dm a — dF) = (1/2) S(dR) 2 /dm = S (- dR) 2 /2dm = s (Lost force) 2 /2 dm > oj = S — ^ dF • a + terms not containing accelerations, where 5= (1/2) ^ dm a- a: Appellian. WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Gauss’ principle 8"Z = 0 , where 6"t = 0, 8" r = 0, <5% = 0, 8"(dF) = 0, but 8" a ± 0 [dF = dF(t, r, v) =► 8"(dF) = 0, 6"Q k = 0] a = 22 e k q k + no (/-terms = 22 e,u, + no w-terms, => = 22 Ck 8 ^ k = 22 £/ and so, explicitly, 8" Z = ( 1 / 2 )^ dm2 [a — ( dF/dm)\ ■ 8"a = ^ {dm a ~ dF) ' b" a = ^ ( dR/dm ) • 8"(dR) = ( dR/dm ) • 8"{dm a — dF) = Q (dR/dm) ■ dm 8"a = ^ dR ■ 8"a = 0 . COMPATIBILITY BETWEEN THE PRINCIPLES OF GAUSS AND LAGRANGE % = 22 ( d Vk/duj,) 86,, 86 1 ~ 22 ^djJ,/dq k ) 8q k = 22 (df,/dq k ) 8q k ± 0 Also, 8f D = 8 uj d = £ (df D /dv) ■Sr='22 ( df D /dq k ) 8q k = 0, instead of the formal (calculus of variations) definition Sf D = 22 ( d fn/ d dk) Sq k + 22 ( d fo/ d 9k) % = 0. The same conclusion can be reached by requiring compatibility between the principles of Lagrange and Jourdain. EQUATIONS OF MOTION 8"Z + 22^dS"/d = 0, where 8"Z = 22 \Ek(T) — Q k ] 8q k (Holonomic system variables) = 22 (dS*/dd) k — 0 k ) 8(u> k ) (Nonholonomic system variables) WWW.EBOOK777.COM INTRODUCTION 8"{/d) = ^"{dfn/dt + £ \(df D /dr) • v + ( df D /dv ) • a]| = ^ (df D /dv) ■ 6a (Particle form) = 6"{df D /dt + Y W„/dq k )q k + (df D /dq k )q k )} = (dfo/dqk) 6q k (Holonomic system variables) MINIMALITY OF THE COMPULSION A”Z = Z(a + 6"a) - Z(a) = (1/2 )^dm [(a + 6"a) — ( dF/dm )] 2 — (\/2) ^dm [a — (dF/dm)]" = 6"Z+(\/2)6 nl Z > 0, where 6"Z= g (dm a - dF) -6"a (= 0), 6" 2 Z = g(dm6"a-6"a) (> 0). Chapter 7: Time-Integral Theorems and Variational Principles GENERALIZED HOLONOMIC VIRIAL IDENTITY (dT/dq k )z k + (dT/dq k + Q k + Y^ ^D a Dk) z k} dt = ^Y(dT/dqk)zk\ x [z k = z k (t): arbitrary functions, but as well behaved as needed; and integral extends from t\ to ? 2 (arbitrary time limits). Specializations z k —+ 6q k [virtual displacement of q k ; and assuming 6q k = (6q k )']: (6T + 6'W)dt = {£ PkSq k ^, [Hamilton’s law of virtually/vertically varying action] z k —» Aq k = 6q k + q k At (noncontemporaneous, or skew, or oblique, variation of q k ): WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE (dT/dq k )(Aq k )' + Y ( dT/dq k + Q k ) Aq k - Y A D a D A D } dt = {T,PkMk}\ [Hamilton’s law of skew-varying action] (. Aq k )' — A(q k ) = q k (At)' [i.e., A{.. .) and (...)' do not commute] z k —> q k (actual system coordinate)'. J{S (dT/dq k )q k + Y^ [dT/dq k + Q k + Y^ A D a Dk)<lk} dt [Virial theorem (of Clausius, Szily et al.)] Zk —► q k ( actual system velocity)', power theorem in holonomic variables. GENERALIZED NONHOLONOMIC VIRIAL IDENTITY (E (dT*/du k )z k + Y {dT*/d9 k )z k - E E hh k( dT */^b)^ + Y ( &k + A k) Z k) dt = {E (dT*/du k )z k ^ Specializations z k —> 69 k (recalling that 59 D = 0, 69 n+l = St = 0, while 69 j ^ 0): (st*+Y 0 > 69 1) dt = {E p ‘ 6e t}\ [Hamilton’s law of virtual and nonholonomic action], z k —> 9 k = uj k (recalling that uj d = 0): power theorem in nonholonomic variables. z k —> 9 k : This case is meaningless because there is no such thing as 9 k . z k —> Lu k : This case does not seem to lead to any readily useful and identifiable result. z k -> A9 k \ = 69 h + 9 h At =69 h + u} h At (A0 h ) — Aio h = ( 69 h ) — 6{9 h ) + to b {At) = Y, h b k 69 k + to h (At ) (. 3T*/d9 k ) A9 k + Y (dT*/du k ) Aco k + El {dT*/duj k ) u> k (At) ~Y^l k b^bAt + E + A-k) A9 k | = {E {dT*/duj k ) A6 k } [Hamilton’s law of skew-varying action in nonholonomic variables]. dt 2 1 WWW.EBOOK777.COM INTRODUCTION NONLINEAR NONHOLONOMIC CONSTRAINTS; HOLONOMIC VARIABLES (dT / dq k )z k + Y [dT/dq k + Q k + Y X n(9fn/9q k )^z k } dt = {Y ( < dT i d ^ Zk } l Specializations z k —> q k (Virial theorem): J{£ ( dT/dq k )q k + Y \ dT / dc lk + Qk + Y X ni d f d! dq k )\ q k } dt = {Y^ dT / d ^ qk ] x z k —> q k (Nonlinear (nonpotential) generalized power equation): d/dt (E {dT/dq k )q k - T} = -dT/dt + Y Qkdk + Y E x o(df D /dq k )q k z k —> 8q k (Hamilton’s law of varying action); z* —> Aq k (Hamilton’s law of skew-varying action): J{£ (dT/dq k )(Aq k )' + Y ( dT/dq k + Q k ) Aq k +(EE ^D(df D /dq k )q^J At} dt = {Y^ dT l d ^ Zk } i NONLINEAR NONHOLONOMIC CONSTRAINTS; NONHOLONOMIC VARIABLES {8T* + 8'W)dt={YPkM k } 2 ^ where 8T* = => • = (E ( dT */^k) S0 k )' - Y (dT*/du k y 86 k -EE*** (dT*/du, k ) 86 h + ^ (dT*/dO k ) 86 k , 8T* dt Y [(dT*/du k y - + £ H h k {dT*/ du> h ) + {Y(dT* 1^)86^, WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE ( 89 b )' — 8u> b = £ E s {u b ) = £ £ E s {u b ){dqJdiJ k ) 89 k = J2 h\ se k = -££ E k {q,){duj h /dq,)89 k = - ££ V' k (duj h /dq,) 89 k , Ct = - £ H b k (dT*/ckv h ) = V h k (dT/dq h r [assuming again that (6q k )' = 6(q k )]. GENERAL INTEGRAL EQUATIONS {8T + ( dT/dq k )[(8q k y - 8(q k )] + 8'w) dt = {bT + 8'W + £ Pk[{50k) - t>u k \ + £ £ v \p k SOb } dt = {ST + S'W+Y, p k\W - 6u k ] - ££ n k bPk^b\dt = {£( a7 7 9 ?*) Sq k where (%)' - S(q k ) = ( dq k /du h )[(69 h y - Su b ] + £ V k h 69, b = £ (dq k /duj b )[{6e b y - 8u b ] - ^ £ ( dq k /duj,)H' h 69 h T = T[t,q,q(t,q,uj)\ = T*(t,q,u) = T*. The above yield the “equation of motion forms” [without the assumption (%)' = £(?*)]: £ [(dT*/du k y -dT*/de k ^{dT/dq^k-Qk £ [(dT*/fa k y-dT*/do k + Y J {dT*/du Jh )H h k -& k 89 k dt = 0, 69 k dt = 0. HOLDER-VORONETS-HAMEL VIEWPOINT (6q k )' = 5q k , whether the 6q k are further constrained or not. Then, with: 6'W* = : 89 k , the above yield (8T + 8' W) dt = I ( 8T * + 8 1 W*) dt = {£(9r*/au,)«5d,}j WWW.EBOOK777.COM INTRODUCTION CONSTRAINED INTEGRAL FORMS [i.e., in terms of T * -A T* a = T*(t, q , W;)] Generally: <5^* = 6T\ + J2 (dT*/du; D ) 0 6 u d , 6T* a = (dT* 0 /dOj) 69, + E (dT*J&,,) 6co, Under the Holder-Voronets-Hamel viewpoint: 6{qk) = (6q k ) > = 0, d(66 D ) = 0 => (#?/>) = 0; but 6(d9 D ) ^ 0, &ud = EE rVaWafc)»/ = -£ »,#0. and 6luj = (<5$/) — ^ ^ ^v 5 we obtain the constrained integral equation + E E ^T/dq k ) 0 V k , 68, + 6' W* 0 dt 6T* 0 — EE (dT*/du D ) o H D ,66, + 6'W* 0 dt (E(sr/%) %}j- Special form of the constraints: ?z> = 4>D(t,q,qi) => u n = Rd - 0x>(G q, qi) =0, U), = q,i- 0, Qd = + </>/)[?, q, q,(t , q, uj,)\ = ui D + <j) D (t , q,uj,), 69 n = 6q n - E {dfbn/dq,) 6q, = 0, <Sd 7 = % ^ 0. Suslov transitivity assumptions and integral equation: {6q D y l L 6{q D ), (6q,)'- 6(q,) = 0; but <W) = 0 — S(q D — <!>d) — 6{q D ) — 6<p D — 0 [and (50/)) — 0] =>■ = ^0/) [definition of <*)(#£,)]; => = (E ^o/dq,) 6q,^j - 6(f> D = ■ ■ ■ = E E^{(j> D ) 6q, = E 6qj ^ 0; 6T = 6T a . WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Suslov principle: {6T 0 + J2 0 dT/dq D ) 0 [(6q D y - S<f> D ] + 6'W 0 )dt = {&T 0 + EE (dr/dq D )W D ,6q 1 + 6'W 0 }dt = {Ew^)^}, Holder-Voronets-Hamel transitivity assumptions: S(qk) = {Sq k )', 66 d = 0, d(60 D ) = 0 => (60 D )' = 0; but «(</*/>) ^ 0 or — S(q£) — — <5(4o) — — (^n) — S(f>j) = E e (iMd) Sqi = Y W °I 5c li ± 0 [definition of d(?n)]; ^6T = 6T 0 + YY (' dT / d 9 d) W°, Sqj. Voronets principle: ST 0 + YH o dT / d 9D )«?/ + -5' In both cases: T —> T 0 (t,q,qi) —> (variation of constrained T), dT/dq D -> (dT/dq D ) a = p D [t, q, q,, (j) D (t, q, q,)\ = p Dfi {t,q,qi) = p Do , {Y( dT / d 4k) tq*} i = ••• = {E (■■•)/««/},. 6'W 0 = Y QioSqj- NONCONTEMPORANEOUS VARIATIONS AND RELATED THEOREMS Definition: zl(...) = 6 (...) + [rf(.. .)/<&] zlt: noncontemporaneous variation operator =Z + qk A t, zl? = + ( dt/dt) At = 0 + (1) zlt = zlt. WWW.EBOOK777.COM INTRODUCTION Basic identities: (...) dt — A(...)dt = 8(. ..)dt+ {(...) At} { {A(. ..) + (■■ .)[d(At)/dt]} dt [A(. ..)dt +(...) d(At)\, {8(...)-(...)[d(At)/dt]}dt+{(...)At}\ [8(...)dt-(..)d(At)] + {(...)At}\; (■••)*- A (...) dt = (...) d(At) (...) dt =••• = — {(.. .)[d(At)/dt]} dt; ^2 E k(---)Mkdt + | [dh(.. .)/dt + d(...)/dt] At dt + ( d - ■ ■ / d 9k) Mk ~ h(...) At - | Ek (- ■ ■) hqk dt + ■ • • / dq k) Ac lk ~ fl (- --) At } Y J E k (...)8q k dt + (d... /dq k ) 8q k + (...) At} where With />(•••) = ^ [<9(.. -)/dq k ]q k — (...): generalized energy operator. h = /?(L) = ^ p k q k — L = h(t, q, q): generalized energy, yl// = Ai I V) dt = Ldt: Hamiltonian action (functional), 22" Jt: Lagrangean action (functional), 2? = 7" + V: total energy of the system; we have the following mechanical integral theorems: A \ Tdt+ \ kw dt = + ( T - p - pk ) Al } WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE AA AA l - 6'W„ p dt= p k Aq k - h At^ = {^PkSqk + LAt} 1, (6E - S' W np ) dt = {55 Pk Mk ~ (55 PkVk - 2r) At} = {Y^P k8c lk + 2T At } l ’ SE dt + {E At}\, I (6E — S' W„ p ) dt + {£ PhAqk + (lT — 55 Pkdkj At \^ \{SE + S'W np )dt + {- 55 ^ Mk + ( 2v +55 Pk ^ k ) At ]p [.AT+ 2T(At)'+ t At\dt+ S'Wdt = | [ATdt + 2Td{At) + dT At + S'Wdt] = {^2PkS<lk + {2T)At} i = {51 Pk Ac lk - (55 Pkdk -2 r'j At^ E dt = 2 Tdt = 2 Vdt = SECOND (VIRTUAL) VARIATION OF A H Total (virtual) variation: S t A h = A H (q + Sq) — A H (q) — SA H + (1/2) 6 A H + ■ First (virtual) variation: SA H — SLdt = ••• = — E(q) Sqdt + {pSq}\ Second (virtual) variation (one Lagrangean coordinate): 6 A H = 8{8A H ) — 8 2 Ldt = ••• = — J J{8q) Sqdt + {8p8q}\, WWW.EBOOK777.COM INTRODUCTION where S 2 L = 6{6L) = [(d/dq) 6q + (d/dq) 8(q)] 2 L = ■ ■ ■ = ( d 2 L/dq 2 )(6q ) 2 + 2(d 2 L/dqdq) 6qSq + (d 2 L/dq 2 )(8q ) 2 Jacobi’s variational equation: J(8q) = {d/dt[d/d(6q)] - [d / d(8q)]}(\/2) S 2 L = (d 2 L/dq 2 ) 6q + (d 2 L/dq 2 )' 6q + [(d 2 L/dq dq)' — (d 2 L/dq 2 )] 8q = d/dt[(d 2 L/dq 2 ) 8q\ — \d 2 L/dq 2 — d/dt(d 2 L/dq dq )] 6q = 0 Equivalently: E[L(t, q + 8q,q + 8q)\ - E[L(t, q , q)} « 8E(q, 6q) (to first-order) = J{Sq;q) = J{8q) Chapter 8: Hamiltonian/Canonical Methods CONJUGATE (HAMILTONIAN) KINETIC ENERGY T' = (Y Pkik -T) =Y1 Pk9k{U CLP) - T (qp) = T\t , q,p) = Y ( dT/dq k )q k - T = (2T 2 + T x ) - (T 2 + 7) + T 0 ) = T 2 - T 0 : i.e., if T = T 2 (e.g., stationary constraints), then T 1 = T CANONICAL, OR HAMILTONIAN, CENTRAL EQUATION Y ( d Pk/dt + dT'/dq k - Q k ) 8q k + Y (' dq k /dt - dT'/dp k ) 6p k = 0 CANONICAL, OR HAMILTONIAN, EQUATIONS OF MOTION (for unconstrained variations) dpk/dt = ~(dT'/dq k ) + Q k (= dT/dq k + Q k => dT/dq k = -dT'/dq k ), dq k /dt = dT'/dp k If Q k = —dV(t,q)/dq k , the above assume the antisymmetrical form: dp k /dt = — dH/dq k , dq k /dt = dH jdp k , WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE where H=T'+V= (J2 PkQk -T+V) = 55 PkQk{t, <LP) ~ ( T ( qP ) ~ V) / q=q{t,qj>) ' = (51 PkQk ~ L ) q=q{t,q,p) = 55 Pk4k(t, <hP) ~ L (qp) q=q(t,q,p) = H(t,q,p): Hamiltonian of system (function of 2n + 1 arguments). If both potential and nonpotential forces (Q k ) are present, the above are replaced by dpk/dt = - dH/dq k + Q k , dq k /dt = dH/dp k ; also, dH/dq k = —dL/dq k and dL/dt = —dH/dt. For stationary (holonomic) constraints, H = T(t,q,p) + V Q {t,q) = E(t,q,p) = total energy , in FLamiltonian variables. In all cases, the following kinematico-inertial identities hold: dT'/dt = —dT/d t, dT'/dq k = —dT/dq k , dT'/dp k = dq k /dt ; dH/dt = —dL/dt, dH/dq k = —dL/dq k , dH / dp k = dq k /dt. LEGENDRE TRANSFORMATION (LT) An LT transforms a function Y(..., y ,...) into its conjugate function Z(... ,z ,...), where z = dY/dy, so that dZ/dz = y. Here in dynamics we have the following identifications: Y(...)->L, q,t, y^q, z = dY / dy-> p = dL/dq, Z(...) —> H, dZ/dz = y —> dH/dp = q. POWER THEOREM dH/dt = dH/dt + 55 QtAk If dH/dt = 0 (e.g., stationary constraints) and Q k = 0 (e.g., potential forces), then the Hamiltonian energy of the system is conserved: H = H(q,p) = constant. CANONICAL ROUTH-VOSS EQUATIONS Under the m Pfaffian constraints 55 a Dk fiqk = o, WWW.EBOOK777.COM INTRODUCTION the canonical equations are dpk/dt dT /dq k + Q k ^ ^ ^D®Dk 3H /dq k “h Qk ,nonpotential “h ^ ^ ^D^Dki dq k /dt = dT'/dp k (= dH/dp k ). ROUTHS EQUATIONS Ignorable (or cyclic) coordinates and momenta (q u ...,q M ) = (V’u-mV’m) = {ipi) = ip, (Pu---,Pm) = (Pi ,...,P M ) = ('Pi) = P Positional (or palpable) coordinates and velocities (qm+ i , • • •, q„) = (q P ) = q (qm +i ,---,q n ) = (q P ) = q Kinetic energy T = T(t; ip M \ q M+ i,... 7 q n \ Vh,..., V’m; Qm+ u • • • > 4n) = T(t,ij,q;ip 7 q) = T[t, ip,q;ip(t,'il;,q;'F,q), q] = T(t,ip,q ; ?',4) = Modified (Routhian) kinetic energy t " = (^ - E ^ 'ip='ip(t;ip,q;'F ,q) = T"(t, ip, q\ <P,q) Routhian central equation E (' d Pk/dt - dT"/dq k - Q k ) Sq k + E (/V - dT"/dq p ) 6q p - E (dipi/dt + dT"/dT/) ST, = 0 Routh’s equations (for unconstrained variations) dp k /dt = 3T"/dq k + Q k : d'Pi/dt = dT"/dpi + Q t (i = 1,..., M), dp p /dt = dT"/dq p + Q p (p = M + 1,..., n); d^/dt = -dT"/dV t (/ = 1,..., M), Pp = dT"/dq p (p = M+l,...,n) Hamilton-like Routh’s equations d'Pi/dt = -d(-T")/di/ji + Q h dipjdt = d(-T")/dPi Lagrange-like Routh’s equations dp p /dt = dT"/3q p + Q p , p p = dT"/dq p (= dT/dq p ) =► (dT"/dq p y-dT”/dq p = Q p WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Additional Routhian kinematico-inertial identities dT/dq k = dT"/dq k : dT/ty, = dT"/dtp t (i = 1,..., M), dT/dq p = dT"/dq p (p = M + 1,..., n) In sum, we have the following two groups of such kinematico-inertial identities: dT"/dip t = dT/chbi and dT"/dWi = -dip t /dP, dT"/dq p = dT/dq p and dT"/dq p = dT/dq p (= p p ). If p k = 8L/dq k , the above are replaced by the following: Hamilton-like Routh’s equations dTj/dt = dR/dipf + Q h dipjdt = —dR/d'Pj and Lagrange-like Routh’s equations dp p /dt = dR/dq p + Q p , p p = dR/dq p (= dL/dq p ) => E p (R) = ( dR/dq p Y - dR/dq p = Q p ; where R = (l - J2 ip=ip(t^,q-, *F,q) R{t]i>,q-, v,q) = Routhian function , or modified Lagrangean , L = L(t ; ip, q; T, q) = T^ w - V = L^ = Lagrangean expressed in Routhian variables; that is, the Routhian is a Hamiltonian [times (—1)] for the ip h and a Lagrangean for the q p . Relation between Routhian and Hamiltonian H = Y] PkVk -L, R = ^Z Ppdp - H = L - ' ¥ >^ i STRUCTURE OF THE ROUTHIAN Decomposition of T (scleronomic system): T=T qq+ T qi, + T w> = T M 9\ 9), where ^ Rqq ~ q p q q = homogeneous quadratic in the q’s (a pq = a qp : positive definite), r # E EE b p jq p ipj = homogeneous bilinear in the q’ s and ip’s (in general: b pi f b ip , sign indefinite), WWW.EBOOK777.COM INTRODUCTION — EE Cyipiipj = homogeneous quadratic in the ip's (cjj = Cji\ positive definite) [/. j = 1 q = M + 1and the coefficients are functions of all n ^ fc ’s]. Next, 'F, = dT-/dipt = Y Cji^j + E b pMp ^ E c i^i = l[, < ~ E dipj/dt = Y C ji( ,f/ i ~ Y b P‘dp) (since is positive definite =>■ Cy is nonsingular), where Cjj = [cofactor of element c /7 in Det(c / -,)]/Det(c /V ) = Cy (= known function of the cf s and ip' s). Then T = T 2 f) + Tq 2 = T(ip, q\ W, q), where 2r 2,o ^EE( a «-EE wjw. 2r o,2 = E EE that is, T = 7’('0, q\ T, 4) does not contain any bilinear terms in the </’s and T’s; and so T" = T-Y = r “E ^(E C ^/-E *„)) r . rrill rr r _ rr r // . rriH . rri / / 2,0 “r 1,1 — -*0,2 = -* 2,0 “t" J- 1,1 ~r 7 0,2 = T"{iP,q- W,q), where 2^"2,o = E E («* - E E c fi b pj b ^d P q q = Y E r P^)q P q q = 2T 2 o (= positive definite in the g’s), r "u = E E (E c a<) ^ = E »>(«> [No counterpart in T = T(ip,q\ 'F,q),i.e., T xl = 0; sign indefinite], it\ 2 = - E E ow = 2r V(?, y) = —27) ) 2 (= negative definite in the T’s). WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE Conversely, T=T" + J2 w iTpi = T " > Fi(dT"/d'Pj) = (T\ o + T" u + T\ 2 ) - (T\ x + 2T\ 2 ) = T" 2fi -T\ 2 = T{^,q- W,q). Hence, L=T-V= (T v0 + T 0i2 ) -V=T 2fi -(V- r 0i2 ) = (r" 2 , 0 - r" 0j2 ) — f = r" 2 , 0 - (f + rV) = ?; 'P,g) =>R = L~Y1 = L + Y, (' dT"/d'F i )'F i = (T \o - n, 2 - f) + (2r" 0j2 + r" M ) = iC + = ^(V'i?)> where ^2 = n,o = r 2 , 0l /?, = r" u , *0 = r " 0t2 -v = -t 0i2 - v. Additional results (i) With T=T tt+ «'> ?) ^t"=t-J 2 vdi = t -J 2 (dT/fyirit = T m - T u = T "^ ?; i>, ?); (ii) d^i/dt = —dT"/d'F i = ■■■ = dT Q2 /d'F i - dK 22 /dir n where 2 T 0i2 = - 2 r " 0]2 = ££) Cji'Pj'Pj, and 2K 22 = ^ ^ (y, b p j c/p'j V'4?) = Cji' K i 7 'i- Matrix form of these results: q T = (to+i,-•■,?„), = T t = (•?!,..., -Pm), ^ Ppq) i®qp) ® 5 ^ (Pip) P {bpi) ^ 3 Pij) ( ttji ) C , IT = q T aq + 2vj/'b T q + vj/ T c vj/, <977chj/ = b T q + cvj/ = T => \j/ = c(T - b T q) = COP - b T q) => ij/ T = (»P T - q T b)C WWW.EBOOK777.COM INTRODUCTION [since c is symmetric, so is its inverse C = (C /7 ): C = c 1 = (c *) T = C T ], T = ■ ■ ■ = (l/2)q T (a - bCb T )q+ (1/2)«F T C*P = T 2fi + T , 0 , 2 = T\ 0 - T\ 2 [since T t C b T q = q T b C4* ] Y T vj/ = = Y T C Y - Y T C b T q = -2 T\ 2 - Y t C b T q, R= (T — V) — *P T v|/ = ... = R 2 + R l +R 0 R 2 = T\ o = r 2 ,o = (1/2) q T (a - bCb T )q, R, = T \i = T t C b T q, Ro = T'\ 0,2 — V = —(V + T 02 ) = —(1/2)'P t C T - V. If b = 0 (i.e., q s and i/>’s uncoupled in the original T ), 7? reduces to R= (l/2)q T aq-(l/2)»P T CT- V. CYCLIC (OR GYROSTATIC) SYSTEMS (i) (qi,. ■ •, q M ) = {tpu---,ipM) = (V'i) = ^ do 770/ appear explicitly, neither in its kinetic energy nor in its nonvanishing impressed forces; only the corresponding Lagrangean velocities {q u ...,q M ) = = {'•Pi) = i> appear there, and, of course, time / and the remaining coordinates and/or velocities {Qm+u ■ ■ ■, c ln) = {q P ) = q and (q M+l ,..., q n ) = (,q p ) = q respectively; that is. dr/a^i = o but, in general, dT/dipi ± 0 =>• T = T(t;q,ip,q). (ii) The corresponding impressed forces vanish; that is, Qi = 0, but Q p = Q p (q) ± 0. If all impressed forces are wholly potential , the above requirements are replaced, respectively, by dL/dipj = 0 and dL/dipi ^ 0 => L = L(t\ q , ip, q). The coordinates ip, and corresponding velocities ip, are called cyclic (Helmholtz), or absent (Routh), or kinosthenic, or speed (J. J. Thomson), or ignorable (Whittaker). The remaining coordinates q, and corresponding velocities q , are called palpable, or WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE positional. Then the Lagrangean equations corresponding to the cyclic coordinates/ variables, become ( dT/d’tpi)' - dT/dip, = Qf. ( dT/dxpj )' = 0 =>■ dT/dip, = T,- = constant = C,-; that is, the momenta T, corresponding to the cylic coordinates ipi are constants of the motion. [Conversely, however, if dT/dipj = 0, then dT/dip t = 0, and, as a result, T = T(t;q,q); that is, the evolution of the ip’s does not affect that of the q's.] Hence, the Routhian of a cyclic system is a function of t, q 7 q and = (T,); that is, with C = (Q), R = (l - Y, Vii’. ip=ip(t;q;q,C) [after solving dT/dipj = T, = C, : for the ip in terms of t , q, q 7 C] = L[t 7 q 7 q 7 ip(p 7 q- 7 C 7 q)- 7 C]~^2 ' F pfit; q 7 q: C) = R{t-,q,q ; Q => L = <T T C) + R{t- 7 q 7 q ; C)]; that is, the system has been reduced to one with only n — M Lagrangean coordinates, new “reduced Lagrangean” R , and, therefore, Lagrange-type Routhian equations for the positional coordinates and the “palpable motion” q p (t): (dR/dqp) dR/dcjp C//anonpoLcntial impressed positional forces ■ Then, R = known function of time =>■ dR/dCj = known function of time = —/[■(?; C), fi = - ( dR/dT,) dt + constant = f(t: C) dt + constant = ipi(t 7 C) + constant. EQUATIONS OF KELVIN-TAIT Let T = T(q 7 q 7 ip) = homogeneous quadratic in the ip and q 7 =$• R = R 2 T R\ + R 07 where (i/2) r pq{q)q p (lq{= Tift) = R 2 (q 7 q) = homogeneous quadratic in the nonignorable velocities q 7 R\ = T" 1,1 = ^2 r p (q,C)q p = Rfq 7 q 7 C ) = homogeneous linear in the nonignorable velocities q 7 [apparent kinetic energy T'\{\\ WWW.EBOOK777.COM INTRODUCTION and fp ^ , PpiCj ppi — ^ ' Cjjbpj fpiiip ) J ) r 0 = t'\ B, 2 — v = —(v — t\ 2 ) = -(i/2) ^£ qqc, - V [— -(^+ To,2)] = R,,((/■ C) = homogeneous quadratic in the constant ignorable momenta = C [ 1 apparent potential energy T" 02 = —T Q 2 (< 0)]. Hence, the situation is mathematically identical to that of relative motion (§3.16) Lagrangean equations of palpable motion: (dR/dq p ) dR/dcjp Op.nonpolcntial impressed positional forces ■ From the above we obtain the following. Kelvin-Tait equations (with p, p' = M + 1 E p (R) = E p (R 2 + R[ + Rq) = E p {R 2 ) + £p(2?]) + E p (R 0 ) = Q p , or E P (R 2 ) — Qp — E p (R\) — E p (R 0 ), or, explicitly, (dR 2 /dq p y - dR 2 /dq p = Q p + dR 0 /dq p - [(dRi/dq p )' - dRfidq p ] = Qp-d(V- T" 02 )/dq p + Y {dr p ,/dq p - dr p /dq p ,)q p , = Qp-d(V-T\ 2 )/dq p + G p , where G p = -[(dRjdcjpY - dRjdq p \ = Y ( dr p'/ dc ip - dr p/ dc ip')<ip' = Y G pp'Qp' [Gyroscopic Routhian “force,” since G pp ' = — G p ' p = G pp \q ; C)]. These are the equations of motion of a fictitious scleronomic system (sometimes referred to as “conjugate” to the original, or reduced, system) with n — M positional coordinates q, and subject, in addition to the impressed forces Q p (nonpotential) and ~dV/dq p (potential), to two special constraint forces: a centrifugal-like dT" 02 /dq p , and a gyroscopic one G p . Ignorable motion, once the palpable motion has been determined: q p (t) => dfii/dt = —dR/dCj = -dRQdCj - dR^/dCt = -dT" op _/dC i - Y PpA Gyroscopic uncoupling G pp > = 0 => E p {Rf) = ( dR 2 /dq p Y - dR 2 /dq p = Q p + dR 0 /dq p WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE A system is gyroscopically uncoupled if, and only if, R x dt=^f r p (q; C) dq p is an exact differential. [A similar uncoupling occurs if all the C, vanish: r p = 0 => R t = 0; and Rq = — V(q ).] A cyclic power theorem dh R /dt = ^2 Q p d P , where h R = R 2 — Rq — T\ o + (V — T" 02 ) = T 2fi + (V + T 0j2 ) = T(q, q , C) + V{q) = E(q , q, C) = Modified (or cyclic) generalized energy; if E Q P dp = 0: h R = T" 20 + (V - T" oz ) = T(q,q, C ) + V(q) = constant. Alternatively, H = ^2 ( dL/dq k )q k — L (= constant, if Q p = 0 and dL/dt = dR/dt = 0) = - r +'52 ( dR / d d P )dp = — (R 2 + R\ + Rq) + ( 2R 2 + R\) = R 2 -R 0 = H(q,q,C) (= h R ). For rheonomic cyclic systems; that is, L = L(t, q , q, C) => R = L(t,q,q,C) - Cii>i(t,q,q,C) = R(t,q,q,C). STEADY MOTION (OR CYCLIC SYSTEMS) ipi = constant = c,- (in addition to 1 = constant = C,), and q p = constant = s P (=> q p = 0) (with i = 1,..., M; p = M + 1,..., n); that is, all velocities are constant (and, hence, all accelerations vanish); and, for scleronomic such systems, the Lagrangean has the form L = L(cj,s p ). Conditions for steady motion [necessary and sufficient conditions for the steady motion of an originally (scleronomic and holonomic) system; or, equivalently, for the equilibrium of the corresponding reduced ^-system]: Q P + dR 0 /dq p = Q P + (dT" 02 /dq p - dV/dq p ) = 0, or, if the forces are wholly potential: dR 0 /dq p = 0, or dT" ()2 /dq p = dV/dq p . WWW.EBOOK777.COM INTRODUCTION Equivalently, since R = R 2 (homogeneous quadratic in the q’s) + R\ (homogeneous bilinear in the Ws and q s) + Rq (homogeneous quadratic in the IP’s) and dR/dq p = dL/dq p , the above equations can be rewritten as {dR/dq p ) 0 = (,dL/dq p ) 0 = 0 [(.. .)„ = (• ■ •)I^= C , 9 =J> expressing q’s = s’s in terms of the arbitrarily chosen IP’s = C’s. The ip's can then be found from the second (Hamiltonian) group of Routh’s equations: dA/dt = -(dR/d'FA = ~(dR o /d'F i ) 0 = -(dT\ 1 /d'P i ) 0 = ^ CjjCj = constant = c t [with q p = 0] = Function of the s’ s and the (arbitrarily chosen) C’s, ^ Ipi(t') = Ci(t initial) T A, initial = Function of the s’s and the (now) arbitrarily chosen c,’s and VWiai’ s > i.e., in steady motion, the cyclic coordinates vary linearly with time. If we initially choose arbitrarily the IP’s, then the above equations relate them to the cf s. If, on the other hand, we choose the ip’s = c’s, then, to relate them directly to the cf s: first, we take T" 02 , and, using *P, = c jiA > change it to a homo¬ geneous quadratic function in the ip’s (with i, j, j', j": 1,..., M): 2T" 0 2 = 2 T"qnp = — EEw [recalling that ^ Cji c j'j — by = faj, = —2T^; or, since dT\ w /dq p = ~(dT"^/dq p ) = dT^/dq p , we can, finally, replace the steady motion conditions by ~( dT "^p/ d ^ = dV / d ^ or dT u/ d( lp = dv /dq p , relating the cf s to the ip’s; and, using *P, = c jiA > we can re l ate both to the IP’s. VARIATION OF CONSTANTS (OR PARAMETERS) Theorem of Lagrange-Poisson: Equations of motion: dp k ldt = f k (t, q.p) and dq k /dt = g k {t,q,p), [fk = —dH/dq k + Q k and g k = dH/dp k , for a Hamiltonian system]; WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE general solutions: Pk = Pk ( t\c) and q k = q k ( t;c), where c = (cj,..., c 2n ) = (c y ; v = 1,..., 2n\. constants of integration. Adjacent trajectory, II = I + 6(1), 6p k = Y (dp k /dc v ) 6c v and 6q k = Y {dq k /dc v ) 6c v . Linear variational, or perturbational, equations: (6p k y = 6(p k ) = Y K d fk/dp t ) 6p, + (df k /dq,) 6q,\, (%)' = S(q k ) = Y [(dgk/dpi) SPi + (■ dgk/dqj) 6q,}. Then, for a Hamiltonian system, d/dt (E (diPk 6 2 q k — 6 2 p k 6\q k )^ — Y^ (^i Qk & 2 Ik ~ & iQk )• Theorem of Lagrange-Poisson: In a holonomic and potential (i.e., Q k = 0, or dQk/dqi = dQi/dq k , for all k, 1= but possibly rheonomic, system, the bilinear expression I = Y Pk 6 2 Qk - S 2 p k 6 x q k ) is time-independent, that is, it is a constant of the motion. Lagrange’s brackets (LB): I ^ ^ ' [tfji Cv] Cfi where = Y [( d Pk/dCe){dqkldc v ) - (dp k /dc v )(dq k /dc ll )\ = Lagrange an bracket of c^, c v . Properties of LB: [tfn Cfi] [C/n t/y] [^1/: , d[c^ c v ]/dc x + d[c v , c a ]/<9c m + d[c x , c li ]/dc v = 0, M = d/dc v (Y q k (dp k /dc,Sj -d/dc^ (U <lk{dp k /dcS) ■ PERTURBATION EQUATIONS Unperturbed problem and its solution dpk/dt = —dH / dq k , dq k /dt = dH/dp k ; p k = p k (t; c), q k = q k (t ; c) WWW.EBOOK777.COM 66 INTRODUCTION Slightly perturbed problem dpk/dt = —dH/dq k + X k , dq k /dt = dH/dp k , where X k = X k (t,q,p) = given function of its arguments « X k ^\t-,c) [first-order approximation, upon substitution of unperturbed solution in it] 2 n first-order differential equations for the c^ = constant —> c M (t): X! {dPk/dc^dcJdt) = X k (x \ 22 (dqk/dcjidcp/dt) = 0. Lagrangean form of the perturbation equations: 22 [c v ,c^{dc v /dt) = 22 x k W {dq k /dc fJ ). If the perturbations are potential —that is, if X k = —dQ/dq k —then, since q k = q k {t\ c), the above specializes to 22 [ c "’ c v\( dc »/ dt ) = - dQ/dc Inverting, we obtain = hJt,q,p ) = first integral (constant) of the unperturbed problem, dc^/dt = E (dhJdp^Xk = ^ (dcJdp k )X k w . Poisson’s brackets. If the perturbations are potential —that is, if X k = -dQ/dq k = ~22 ( dQ l dc v)( dc vl d( lk), then dcjdt =~22 (dQ/dc v )(c^,c v ), where (c M ,c„) = 22 [( dc »/ d Pk)(dc v /dq k ) - {dc li /dq k ){dc v /dp k )\ = Poisson bracket of c^c,,. Compatibility with LB: 'y ^ [Cj/J C^] (Cy, C^) WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE First-order corrections. Setting in c M = c, M) + c„i, where c^ = unperturbed values and c„| = corresponding first-order corrections, we have dcpi/dt = - 55 ( dQ o/dCvJiC'to, c v0 ) [where Q a = Q{c 0 )]. Lagrange’s result. Let 1 Ik — c lkO + c lk\ t + fet 2 H-, Pk = PkO + Pkl t + Pk2 {2 H-■ Then, with Ck=qid> and c n+! =pio (k,l = 1,... ,n), the perturbation equations assume the canonical form: dc k /dt = dQ/dc n+k , dc n+k /dt = —dQ/dc k (k = 1,... ,n). CANONICAL TRANSFORMATIONS Transformations q = q(t,q',p') <->■ q' = q'(t,q,p)-, p=p(t,q',p') p' = p'{t,q,p), [with nonvanishing Jacobian \d(q' ,p')/d(q,p)\] that leave Hamilton’s equations form invariant. Requirements: L dt = L' dt + dF => 55 p k dq k — Ft dt = 55 p k ' dq k : — Ft' dt + dF, => 55 Pk d dk ~ Pk' dqk' = (H - H') dt + dF, where F is the generating function of the transformation (an arbitrary differentiable function of the coordinates, momenta, and time); and H' satisfies the Hamiltonian equations in the new variables. Alternatively, 55 Pk dqk -Hdt= df(t,q,p) and ^ p k , dq k , - H' dt = df\t,q',p'), => 55 Pk dqk ~ 55 Pk ' dqk ' ~( H ~ H ') dt = df(t, q,p) - df\t, q',p') = dF. Virtual form of a canonical transformation: 55 Pk s qk - 55 Pk' 6 qk' = SF - WWW.EBOOK777.COM INTRODUCTION Forms of F and their relations with the corresponding conjugate variables: F = Fft, q, q')\ Pk = dF x /dq k , Pk' = —dF x j dq k '\ H' = H+dFjdt; F = F 2 {t, q,p')\ Pk = 9F 2 /dq k , qk' = dF 2 /dp k r, H' =H + dF 2 /dt, F = F 3 {t,p,q'): q k = -dF 3 /dp h , Pk' = -dF 3 /dq k r, H' = H + dF 3 /dt; F = F 4 {t,p,p'y. q k = -dF 4 /dp k , qk' = dF 4 /dp k r, H' = H + dFjdt ; Fi = F\ + Y Pk'Qk', p 3 = Fi~Y^ PkQki F 4 = F\+^2 Pk'Qk' ~ Y P k q k = f 2 ~ Y PkQk = F i + Y Pk^k'- POISSON'S BRACKETS (PB) AND CANONICITY CONDITIONS The PB of f g (where f g. h are arbitrary differentiable dynamical quantities) is (/,#) = Y i( d f/ d Pk)(9g/dq k ) - {df/dq k ){dg/dp k )\ = Y d{f,g)/d{p k ,q k ). Then df /dt = df /dt + (H, f) + Y ( d f ! d Pk)Qk\ and so for / to be an integral of the motion, we must have df/dt + Y, ( df/d Pk )Q k +(H,f)=0 => {H, f) =0, if / = f(q,p) and Q k = 0, that is, its PB with the Hamiltonian of its variables must be zero. [Remarks on notation: A number of authors define PBs as the opposite of ours; that is, as {f,g) = Y [i d f/ dq k)(dg/dp k ) - {df /dp k ){dg/dq k )]. Therefore, a certain caution should be exercised when comparing references. Also, others denote our Lagrangean brackets, [...], by {...}; and our Poisson brackets, (■ ■ •). by [• • .]■] Properties/theorems of PBs (f,g) = -(g,f) = (-g,f) =>/,/) = o (f,c) = 0 (/i + fi,g) = {fug) = (fug) {flfug) = f\{fl,g) +fl{fug) =>• {cf,g) = c{f,g) => !r/ = Y C kfk - then {f,g) = Y C k{fk> g) (anti-symmetry) (c = a constant) (distributivity) (c = a constant) (c k = constants) WWW.EBOOK777.COM §4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 9/dt(f,g) = ( df/dt,g) + (f,dg/dt) (“Leibniz rule”) [Actually, d/dx{f,g) = (df /dx,g) + (/, dg/dx); x = any variable] (/, <lk) = df'/dp k , (f,Pk) = ~df / dq k , 0 <ik,qi ) = 0 , (Pk,Pi) = 0, ( Pk,cj /) = 8 k i (= Kronecker delta). [The last three types of brackets are called fundamental , or basic , PB] (/, (g,h)) + {g, ( h,f)) + ( h , (f,g)) = 0, {{f,g),h) + ((g, /?),/) + ((/?,/), g) = 0 (Poisson-Jacobi identity) Theorem of Poisson-Jacobi: If/and g are any two integrals of the motion, so is their PB; that is, iff = c x and g = c 2 , then (f,g) = c 3 {c X 2 , 3 = constants). Theorem: The PBs are invariant under CT\ that is, ( f,g) p = {f,g) q > p ' = • • •; where / and g keep their value, but not necessarily their form, in the various canonical coordinates involved. Canonicity conditions via PB L pv,Pk'] = 0, ( Pl',Pk 0 = 0 , Vli'iCik'} = 0 , fli’i c lk') = 0 , [Pk',<}r\ = Ski, (. Pr,qk') = 8, k , since both Poisson and Lagrange brackets are canonically invariant. Theorem of Jacobi (i) The integration of the canonical equations dq k /dt = dHldp k , dp k /dt = -dH/dq k , is reduced to the integration of the Hamilton-Jacobi equation (H — J): H(t , q, dA/dq) + dA/dt = 0, A = A(t,q,p'): generating function (Hamiltonian action). (ii) If we have a complete solution of H — J\ that is, a solution of the form A = A(t; q 1 ,...,q„;j3 1 ,...,0„)=A(t;q,0), where (3= ..., /3„) = n essential arbitrary constants, and \d 1 A/dqd/3\ f 0 (non¬ vanishing Jacobian), then the solution of the algebraic system: dA/dfk = a k [Finite equations of motion, a: new arbitrary constants => q k = q k (t,a,/3)\, WWW.EBOOK777.COM INTRODUCTION dA/dq k = p k [=>■ p k = p k (t,a,/3 ): canonically conjugate (finite) equations of motion], constitutes a complete solution of the canonical equations. Schematically, these are as follows. Hamilton: Differential equations of motion: dq/dt = dH/dp , dp/dt = —dH/dq (If these equations can be integrated, an action function can be obtained) Hamilton-J acobi: H(t,q , dA/dq ) + dA/dt = 0 =>■ A = A(t, q,/3) Jacobi: Finite equations of motion: dA/d/3 = a —> q = q(t , a, /?); dA/dq = p —> p = p{t, a, /?) (If an action function can be obtained, then Hamilton’s equations can be integrated.) No significant new notations are involved in the remaining sections §8.12—§8.16 (i.e. special topics on Hamiltonian mechanics). WWW.EBOOK777.COM Background Basic Concepts and Equations of Particle and Rigid-Body Mechanics Therefore it would seem right that any systematic treatment of classical dynamics should start with axioms carefully laid down, on which the whole structure would rest as a house rests on its foundations. The analogy to a house is, however, a false one. Theories are created in mid-air, so to speak, and develop both upward and downward. Neither process is ever completed. Upward, the ramifications can extend indefinitely, downward, the axiomatic base must be rebuilt continually as our views change as to what constitutes logical precision. Indeed, there is little promise of finality here, as we seem to be moving towards the idea that logic is a man-made thing, a game played according to rules to some extent arbitrary. (Synge, 1960, p. 5, emphasis added) In this chapter we summarize, without detailed proofs and/or elaborate discussions, in a handbook (not textbook) fashion, like a first-aid kit, but in a hopefully accurate and serviceable form, the basic concepts, definitions, axioms, and theorems of “elementary” (or momentum/Newton-Euler, or general) theoretical mechanics. This compact, highly selective, perhaps nonhomogeneous, and unavoidably incom¬ plete account should help to establish a common background with readers, and thus enhance their understanding of the rest of this relatively self-contained book. For complementary reading, we recommend (alphabetically): Fox (1967): one of the best, and most economically written, U.S. texts on elementary- intermediate general mechanics. Hamel (1909), (1912, 1st ed., 1922, 2nd ed.): arguably the best text on elementary- intermediate general mechanics written to date, (1927), (1949). Hund (1972): concise, insightful. Langner (1996-1997): dense, clear; “best buy.” Loitsianskii and Lur’e (1982, 1983): excellent. Marcolongo (1905, 1911/1912): rigorous, comprehensive. Milne (1948): interesting vectorial treatment of rigid dynamics. Papastavridis: Elementary Mechanics (EM for short), under production: encyclopedia/ handbook of Newton-Euler momentum mechanics, from an advanced and unified viewpoint; includes the elements of continuum mechanics. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Parkus (1966): an educational classic. Synge and Griffith (1959): clear, reliable. Synge (1960): comprehensive, encyclopedic, mature. Winkelmann (1929, 1930): concise, comprehensive. Additional references, at particular sections, and so on, will also be given, as deemed beneficial. 1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA Vectors: Basic Concepts/Definitions and Algebra Geometrically, vectors are straight line segments that, in the most general case, have the following five characteristics: (i) length, (ii) direction, (iii) sense, (iv) line of action (or carrier), and (v) origin (or point of application) on carrier; (iv) and (v) can be replaced with spatial origin. Also, vectors obey the well-known parallelogram law of addition (=> commutativity); that is, not all line segments with characteristics (i)-(v) are vectors (e.g., finite rotations, §1.10). Next, if only characteristics (i) (iii) matter, but (iv) and (v) do not, the vector is called free', if characteristics (i)—(iv) matter, but (v) does not, the vector is called line bound or sliding', and if all five characteristics matter, the vector is called point bound. As a rule, the vectors of continuum mechanics and the system vectors of analytical mechanics (chap. 2 ff.) are point bound', while those of rigid-body mechanics are line bound. Notation for vectors: a, b, ... (bold italic). Length, or magnitude, or modulus, or intensity, or norm, of a. \a\ = a > 0. If a = 0, the vector is called null', if a = 1 , the vector is called unit (or normalized). The physical space of classical mechanics is a three-dimensional Euclidean point space, denoted by E 3 or E', while the associated (also Euclidean) vector space is denoted by E 3 or E. An orthonormal basis (i.e., one whose vectors are unit and mutually orthogonal — see below) {«i,H 2 ,» 3 } = {" 1 , 2 , 3 } = {«*;£= 1,2,3} = {«/J = {u x , u y . u z } = \u xyz } = k}, (1-1-1) together with an “origin,” O, make up a (local) rectangular Cartesian frame: {O , u k .}. If the origin is not important, we simply write {«/{. [Since E is flat (noncurved), a single such frame, and associated rectilinear and mutually rectangular axes of coordinates <9—123 = O-xyz, can be extended to cover, or represent, the entire space: local frame —> global frame. For details, see, for exam¬ ple, Papastavridis (1999, pp. 84-91, 211-218), or Lur’e (1968, p. 807).] In such a basis, a vector a can be represented by its rectangular Cartesian com¬ ponents or {a\,a 2 ,afj = {a u , 3 } = {a k ',k = 1,2,3} = {a k } = {a x ,a y ,a z } = {a w }, (1.1.2a) a = a 1111 + a 2 u 2 + a 2 u 2 = a x u x + a y u v + a : ii z = ci k u k . WWW.EBOOK777.COM (1.1.2b) §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA In terms of the famous Einsteinian summation convention [= lone, or free, sub¬ scripts range over the integers 1, 2, 3, or x, y, z, while summation is implied over repeated (i.e., pairs) of subscripts ], we can simply write a = a k u k . In this book, how¬ ever, and for reasons that will gradually become clear (chap. 2), we shall NOT use this convention! Dotting (1.1.2b) with u k , and noting the six orthonormality (metric!) conditions or constraints: u k ■ u{. scalar , or dot , or inner, product of u k , Uj = &kl — £>lk (Kronecker delta) = 1 if k = l, =0 if kfl (k,l= 1,2,3, or x,y,z), (1.1.3) in extenso: i ■ j = j ■ j = k • k = 1 (normality), i -j = i • k = j • k = 0 (orthogonality). (1.1.3a,b) we obtain the following expression for the a-components: a k = a-u k . (1.1.2c) In such a basis, the dot product of two vectors a and b is expressed as a b=b a= a k u^j • b,u^j = ••• = ^ a k b k . (1.1.4) For a = b, the above yields the length, or norm, or magnitude, of a: N(a) = a = \a\ = (a ■ a) 1/2 = (E a k a k ^j > 0 (this book). (1.1.5) The basis {« 1A3 } is called O rtho N ormal D extral (i . e ., rig ht-handed) = OND, if, in addition to (1.1.3), it satisfies u k • (i/,. x «j) = (u k ,u r ,u s ) = e krs (permutation symbol, or alternator, of Levi—Civita) = +1/—1/0 according as k,r,s are an even/odd/no permutation of 1,2,3" [i.e., £123 = £23 1 = £312 = + 1 ; e 132 = £213 = £321 = — 1) £112 = £122 = £313 = £222 = • • • = 0 (two or more indices equal)], (1.1.6) or, equivalently, if u r xu s = Y^ £rVc«/c = e krs u k «/< = (1/2) ^ e krs {u r x uf: (1.1.6a) that is, («,. x u s ) k = e rsk , otherwise {iq 2 , 3 } is left-handed, or sinister, in which case (u k ,u r ,u s ) = —£ krs . Flenceforth, only OND bases will be used. • The symbols of Kronecker and Levi-Civita are connected by the following “ed iden¬ tity”: f ] Ckrs^hns ^ C skr £ s l m ^kfrm ^km fl: WWW.EBOOK777.COM (1.1.6b) CHAPTER 1: BACKGROUND which, for r = m (and then summation over repeated subscripts), produces E £ £krs£ > rs = 2SkI ; (1.1.6c) and this, for k = /, etc., yields £ E E £k " £k " = 2 (E 6kk ) = 2(3 >= 6 - (i 16d) • The dextrality of the orthonormal basis ( i,j,k ) (i.e., i x i =j xj = k x k = 0 ), is expressed by i x./ = ~{j x ') = k, ./ x k = -(k xj) = i, kxi = -(i x k) =j. (1.1.6e) With the help of the above, we express the vector , or cross, or outer, product of a and b as «xA = -(Axa) = EEE e klr a kblU r , that is, («x *), = ££ ZklrO-kbl — EE ZrklUkbl- It can be shown that |«1 X U 2 1 2 = |«2 X M 3 I 2 = |«3 X U\ | 2 = (» ll « 2 , l / 3) 2 = +1, where (a, b, c) = a ■ (b x c) = h • (c x a) = c • (a x b) = (a x b) • c = (b x c) • a = (c x a) • b ^ ^ ^ ^ ^ ^ ~'krs^l: b r C,■ (1.1.7a) (1.1.7b) (1.1.8a) (1.1.8b) [+, if (a, h , c) is ng7zt; —, if (a, A, c) is left', 0, if (a, b, c ) are coplanar or zero]: scalar triple product of a, b, c = signed volume of parallelepiped having a, b, c as sides; also [a, b, c\ = a x (b x c) = (a ■ c)b — (a • b)c If (a x b) x c = —c x (a x b) = (a • c)b — ( b • c)a\: (1.1.8c) vector triple product of a, b, c. The dyadic, or direct, or open, or tensor product of two vectors a and h, ab = a®b (fb®a, in general), (1.1.9a) is defined as (the tensor — see below): a h = a ® b = (E «*»*) ® (E b > u >) = E E a k b I (u k ®u,). (1.1.9b) • This product can also be defined as the tensor that assigns to each vector x the vector a (b ■ x): WWW.EBOOK777.COM §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA (a ® A) ■ x = a (A ■ x) = (A ■ x)a, (1.1.9c) and also x ■ (a® b) = (x ■ a)h = b [x ■ a). (1.1.9d) In components, these read, respectively, (a®b)-x = J2J2(a k b l Xi)u k , x-(a® b) = ^ j 'Y^(x l a 1 b k )u k . (1.1.9e) • It can be shown that [a, A, c] = [(A ® c) — (c® A)] • a. (1.1.8d) Tensors: Basic Concepts/Definitions and Algebra [For a detailed classical mostly indicial treatment of general tensors, see, for example, Papastavridis (1999), and owe Elementary Mechanics.} A second-order (or rank) tensor (or dyadic, from the Greek A YO = two ) or, here, simply tensor T (bold, in italics or roman) is defined as a linear transformation from V to F; or as a linear mapping assigning to each vector a another vector A: b=T-a , (1.1.10a) or in components ^2 b k u k = T ki a i u k => b k = ^2 T ki a h (1.1.10b) or as h =“- T = J2J2 a k T kl u, => b) — 'y ' T ki a k , where T k i = u k -{T-ui ) = ( T-u,)-u k = T ■ (u k ®u,), (1.1.10c) are the Cartesian components of T (see tensor products, below). Alternatively, a vector/tensor//(n)th order tensor associates a scalar/vector //(it — l)th order tensor with each spatial direction ii d = (u^)k- direction cosines of unit vector u d ), via a linear and homogeneous expression in the u^ k ; that is, for a (second-order) tensor: T —> v d = T ■ u d (direct notation), v^ k = ^ Tki u (d)i (component notation). Thus (and in addition to the well-known 3x3 matrix form), T has the following representations: T = ^ 2 ^2 Tkt u k ® u i ( Dyadic or nonion representation) = 11 k ® h, where t k = ^ T kl u h = '^2t,®ui, where r, = ^ T kl u k . WWW.EBOOK777.COM (1.1.lOd) (1.1.lOe) CHAPTER 1: BACKGROUND The nine tensors {u k ® «/} span the set of all (second-order) tensors; they form an orthonormal “tensor basis” there. If T l2 = T 1X {= —T 2 \), etc., then T is called sym¬ metric {anti-, or skew- symmetric). Generally [see definition of transpose, (.. .) T , below]: Symmetric tensor: T = T J , T u = (1.1.11a) Antisymmetric tensor: T = — J T , T k i = —T lk {^T kk = 0, no sum!) (1.1.11b) Algebra of Tensors: Basic Operations • Sum/difference of tensors T and S: T ± S = E E ( r « ± -Sjfc/)»jfc ® »/■ (1.1.12a) • Product of T with a scalar (number) A, XT: XT = YY1 ( A7 */W ® «/ (1.1.12b) • Tensor product of T and S, T ■ S, is defined by ™=EEEr* S ri u k ®i// S • T, in general); that is, (T- S) kl = Y T k ,S rI * {S ■ T) u = Y S kr T rl . (1.1.12c) • Inner, or dot, scalar product of T and S, T : S, is defined by (see trace below) = EE = Tr{S-T T ) = S:T, (1.1.12d) where Tr means “trace of.” If T = S, T = \ T\ = {T : T) l/2 : magnitude of T{> 0, unless J=0). (1.1.12e) If either of T, S is symmetric (as is almost always the case in mechanics), then, t -s = yy = EE TkiSik = Tr(T ■ S) = T•• S = EZ S, k T kl =Tr(S-T) = S-- T (1.1.12f) In sum, we have defined the following three tensorial products: ( r - s )ki = Y TkrSri ( Tens ° r ). T ■ s = E E T » s » ( Scalar )> T • • 5 = E E Tk > Sik ( Scalar ) • WWW.EBOOK777.COM (1.1.12g) §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA The reader should be warned that these notations are by no means uniform, and so caution should be exercised in comparing various references. • Transpose of T, T J , is defined uniquely by (T • a) ■ b = a • (T J ■ b) , for all a,b. (1.1.12h) • Trace of T is defined by Trace of T = Tr(T) = T U + T 22 + T 33 = £ T kk . (1.1.12i) • Determinant of T is defined by Determinant of T = Det(T) = Det(T ki ) = |TJy|. (1.1.12j) It can be shown that: (i) Tr(T)= Tr(T J ) : Det(T) = Det(T T ), (1.1.12k) (ii) For any two vectors a and b\ (a ® b) T = b ig) a, 7>(a 0 b) = a • b = a k b k , Det(a® h) = 0, (1.1.121) (iii) For any two tensors T and S. (T-S) t = S t -T t , Tr(T-S) = Tr(S ■ T) = T - S, (1.1.12m) Det(T ■ S) = Det(T) Det(S); (1.1.12n) also (in three dimensions): Det(tT) = P Det(T), for any real number t. (1.1.12o) • Inverse of T, T 1 , is defined uniquely by: T ■ T 1 = T 1 ■ T = 1 (unit tensor), [Det(T) f 0], (1.1.12p) From the above, we can easily deduce that (i) Det(T~ x ) = (Det T)~ x (1.1.12q) (ii) (T-S)~ l = S~ l -T~ l (T,S: invertible) (1.1.12r) (iii) d/dx(Det T) = (Det T)Tr[(dT/dx) • T~% (1.1.12s) where T = T(x) = invertible, x = real parameter, and dT/dx = ( dT kt /dx ). • A tensor can be built from two vectors; but, in general, it cannot be decom¬ posed into two vectors. • Every tensor can be decomposed uniquely into a sum of a symmetric part (T' ki ) and an antisymmetric part ( T" k /): T kl = T’ kl + T" kh 2T’ kI = T kl + T lk = 2T\ k , 2T" kl = T kl - T lk = -2T" lk - (1.1.13a) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND that is, T=T' + T", r' = (J') T , r" = -(J") T . (1.1.13b) • For any tensor T and any three vectors a, b, c, the following identities hold: (i) a- (T ■ b) = T: (a 0 b), ^ «*(E T kl b^j = ^ E Tki{a k bj) (in components). (1.1.14a) (ii) Since r—EE ( T k ia,)u k , « J = EE (■ a,Ti k )u k , we will have T • a = a ■ T, only if T is symmetric, from which we also conclude that (u k ® «/) : (« r 0 u s ) = 8 kr 8 h . (1.1.14b) (iii) (a x T) • b = a x (J • b), (T x a) • b = T • (a x h), (1.1.14c) where Jx «=EEEE ( T kr a s e rs i)u k ® </,; that is, (J x a) fc/ = EE £ irs T k,a s , (1.1.14d) and fl X J=EEEE ( T sl a r£rsk)» k ® «/, (« X 7)*/ = E E £ krs a r T sh (l-l-14e) (iv) (T ■ a, T • h, T ■ c) = (Det T)(a,b,c). (1.1.14f) (v) T t -(T- ax T-b) = (DetT)(ax b). (1.1.14g) Special Tensors Zero tensor O. 0-a = 0, for every vector a. (1.1.15a) Unit, or identity, tensor 1 : 1 • a - a, for every vector a, (1.1.15b) 1 = ^ ^2 8 k ju k ® «/ = ui ® a i + «2 0 a 2 + «3 0 a 3 (Dyadic form) = (8 k/ ) = diagonal^, 1,1) (Matrix form), =>Detl = +l. (1.1.15c) WWW.EBOOK777.COM §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA Diagonal tensor D. D = D n iii (g> ui + Z>22«2 ® «2 + ^ 33 m 3 ® u 3 (Dyadic form) = diagonal (£> n , D 22 ,D 33 ) (Matrix form). (1.1.15d) If _D n = £> 22 , D reduces to D — D\\l + (D 33 — £>ii)w 3 ® H 3 , (1.1.15e) a result that is useful in the representation of moments of inertia of bodies of revolu¬ tion. Alternator tensor E = E E £ klm U k ®u l ®u m . (1.1.15f) It can be shown that ( 1 ) Det J=|7’,H = EEEEEE( 1 / 6 ) e «^AW-- ( LU5 g) (ii) If S is symmetric, then T.S = J T : S = (l/2)( T + J T ): S', (1.1.15h) If S is antisymmetric , then T:S = —(J T :S) = (1/2)(T— J T ) : 5', (1.1.15i) If S is symmetric and T is antisymmetric , then T:S = 0. (1.1.15j) (iii) If T:S = 0 for every tensor S', then T = 0, (1.1.15k) If T:S = 0 for every symmetric tensor S, then T = antisymmetric , (1.1.151) If T : S = 0 for every antisymmetric tensor .S', then J = symmetric. (1.1.15m) Axial Vectors There exists a one-to-one correspondence between antisymmetric tensors and vectors: given a (any) antisymmetric tensor W —that is, W = —W 1 —there exists a unique vector w, its axial (or dual) vector or axis, such that for every vector a: W - a = w x a, (1.1.16a) that is, (recalling the earlier definitions of products, etc.) W •(...) = (w x 7) •(...) = w x (...) [=$■ a• (W• a) = 0]. (1.1.16b) And, conversely, given a vector yv, there exists a unique antisymmetric tensor W, such that (1.1.16a,b) hold. In components, the above read: w k = -(1/2) EE £ktmW/ m — (1/2) EE km Wlm t (1.1.16c) ^ ^ &Imk W k ^ ^ ^Ikm W k ? (1.1.16d) or, in matrix form: ( 0 W\ 2 — — W '3 IFi 3 — W2 W = {W lm ) = w 2 1 = w 3 0 w 2 3 = —H 7 ! ^ PT 31 = -w 2 ^32 = W’, 0 WWW.EBOOK777.COM CHAPTER 1: BACKGROUND [Sometimes (especially in general indicial tensorial treatments) w k is defined as the negative of the above; that is, Wk = (1 /2) £ £klm W[m ^ W ‘ m = £ £ > mkWk ’ (1.1.16f) or W • a = — w x a = a x w>, (1.1.16g) and so, here too, the reader should be careful when comparing references.] It can be shown that: (i) The axial vector of a general nonsymmetric tensor equals the axial vector of its antisymmetric part; that is, the axial vector of its symmetric part (and, generally, of any symmetric tensor) vanishes; and, conversely, the vanishing of that vector shows that that tensor is symmetric. (ii) The axial vector of T, t (or T x , or f x ), can be expressed as —2t = (Ty — 7 32 )tfi + (7 31 — 7 13 )h 2 + (T u — T 2 \ )w 3 = »i xf|+i/ 2 x( 2 +« 3 xt 3 . (1.1.16h) (iii) The axial vector of / 2 ) w ki( u k ®»/ -«/ ® «k ), w has the following dyadic representation (note k, l order): * = £££ [-(1/2 )e rkl W kl ]u r = ■■■ = ££ (1/2)^/(«/ x «,). (1.1.16i) (iv) Let w = wu\. Then, W = w(w 3 <8 « 2 — « 2 ® « 3 ), w = « 3 • (IF-« 2 ); (1.1.16j) and cyclically for w = vv « 2 , w = wk 3 . (v) The antisymmetric part of the tensor a 8 h equals (in matrix form) ( 0 — w 3 vv 2 \ vr 3 0 —w\ J, (1.1.16k) -w 2 wi 0 / where hi=(1/2)Ax« (note order). (1.1.161) (vi) The tensor a®b — b® a, where «, b are arbitrary vectors, is antisymmetric; and, by the preceding, its axial vector is b x a (note order). (vii) Let W \, w 2 be the axial vectors of the antisymmetric tensors W ,, W 2 , respec¬ tively. Then, W\ • W 2 = w 2 ® W\ - (wq • w 2 )l, Tr(W x • W 2 ) = — 2(wq • w 2 ). (1.1.16m) => W -W = w ® w - (w • w)l, or W 1 -w ® w - w 2 l. (1.1.16n) WWW.EBOOK777.COM §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA Spectral Theory of Tensors DEFINITION A scalar A is a principal , or characteristic, or proper value, or eigenvalue, of T if there exists a unit vector n = (n l ,n 2 ,n 2 ) such that T-n = Xn, or in components ^ T kl n/ = Xn k . (1.1.17a) Then n is called a principal, or characteristic, or proper, or eigen-direction of T corresponding to that value of A. DEFINITION The principal, or characteristic , or proper, or eigen-space of T corresponding to A is the subspace of V consisting of cdl vectors a satisfying (1.1.17a): T ■ a = A a, that is, the subspace of all the eigenvectors of T. If T is positive definite — that is, if a - (T - a) >0 for all a f 0 — then its eigen¬ values are strictly positive. THEOREM OF SPECTRAL DECOMPOSITION (of T ) If T = T t (i.e., symmetric ), there exists an orthonormal basis \n i,n 2 ,n 3 } for V and three real, but not necessarily distinct, eigenvalues A ls A 2 , A 3 of T such that T n k = X k n k (k = 1,2,3; no sum), (1.1.17b) and T = T ■ 1 = T • (^2 "k ® «a ) = ^2 ( T ' "*) 18 Hk = ^2 X k (n k <g> n k ) (Dyadic representation) = diagonal (Aj, A 2 , A 3 ) (Matrix representation); (1.1.17c) =>■ n k • (T • it/) = T • n k • ii t = X k 6 k i (= X k or 0, according as k = l or k fi l)] that is, with n k = («(£)/: components of n k ), T k i = X\n^ k n^i + X 2 n( 2 )k n ( 2 )i + ^ 3 w ( 3 )/t w ( 3 )/- ( 1 . 1 . 1 7d) Conversely, if J = A k (n k ® n k ), with {n k } = orthonormal, then T •n k = X k n k (no sum). Depending on the relative sizes of the three eigenvalues, we distinguish the follow¬ ing three cases: (i) If A[, X 2 , A 3 = distinct, then the eigendirections of T are the three mutually orthogonal lines, through the origin, spanned by ti\, n 2 , « 3 . (ii) If A| X 2 = A 3 (i.e., two distinct eigenvalues), then the spectral decomposition (1.1.17c) reduces to the following (with |/f 3 1 = 1): T = Ai(«i ® Hi) + A 2 (7 - Hi ® «i). (1.1.17e) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Conversely, if (1.1.17e) holds with A| f A 2 = A 3 , then Ai and A 2 are the sole distinct eigenvalues of J; which, in this case, has the two distinct eigenspaces: (a) the line spanned by n u and (b) the plane perpendicular to //,. (iii) If A[ = A 2 = A 3 = A, in which case T=X1 = A(«i ® n i + n 2 <g> n 2 + n 3 <g) n 3 ) (Dyadic representation) = diagonal( A, A, A) (Matrix representation), (1.1.17f) then the eigenspace of T is the entire space V. Conversely, if V is the eigenspace of J, then Thas the form (1.1.17f). [For extensions of the theorem to polynomial functions of T see books on linear algebra; also Bradbury (1968, pp. 113-116).] The require¬ ment of nontrivial solutions for n, in (1.1.17a), leads, in well-known ways, to the characteristic (polynomial) equation for T: - Det(T-XI) = Det(\l — T) = D( A) = A 3 - fX 2 + I 2 X - h = 0, (1.1.18a) where the coefficients, or principal invariants of T (i.e., quantities independent of the choice of the basis used for the representation of T), are given by I\(T) =h = Tr(T) = Y, T kk = Aj + A 2 + A 3 , h(T) = I 2 = (1/2)[(Tr T) 2 — Tr(T 2 )} = (1/2) Taj Tu'j — TkiTik'j = A[A 2 + AjA 3 + A 2 A 3 , / 3 (J) = I 2 = Det T = \T kl \ e kim T ki T, 2 T m3 = A[A 2 A 3 = (1/6) (TrTy - 3(TrT)(TrT 2 ) + 2Tr(T i ) also If - 21 2 = X x 2 + A 2 2 + A 3 2 = Tr(T 2 ). [(a) It is shown in linear algebra/matrix theory that: (1.1.18b) (1.1.18c) • In general, that is, T = nonsymmetric , eq. (1.1.18a) has either three real roots', or one real and two complex ( conjugate ) roots. • Every tensor T satisfies its own characteristic equation; that is, eq. (1.1.18a) with A replaced by T: T 2 - I\T 2 + I 2 T - I 3 1 = 0 ( Cayley-Hamilton theorem). And, more generally, if/(A) = real polynomial in an eigenvalue A of T , then/(A) is an eigenvalue of f(T ); and, an eigenvector of T corresponding to A is also an eigenvector of f(T) corresponding to /(A). (b) The above show that Tr T, Tr(T 2 ), Tr(T 3 ) may also be considered as princi¬ pal invariants of T.] Further, it can be shown, that: (i) If TV) 2 3 are the antisymmetric tensors whose axial vectors are, respectively, the three orthonormal eigenvectors of (the symmetric tensor) T. /t 123 , then T has, in addition to (1.1.17c), the following spectral decomposition: T= XfNi ■N 1 ) + X 2 (N 2 -N 2 ) + X 3 (N 3 -N 3 ) + Tr(T)l; (1.1.18d) WWW.EBOOK777.COM §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA and, therefore, for an arbitrary vector a, T-a = X l (N l ■N l )-a+X 2 (N 2 -N 2 )-a+X 3 (N 3 -N 3 )-a+Tr(T)a-, (1.1.18el) also, Tr(N\ • (V,) = Tr(N 2 -N 2 ) = Tr(N 3 -N 3 ) = -2. (1.1.18e2) (ii) If a = axial vector of A, then T • a = axial vector of [— ( T • A + A • T) + Tr(T)A], (1.1.18f) (iii) The principal invariants of T = Y.Y, T kl U k ® »/ = "k 0 t k , where t k = Y^ T kt u h (1.1.18g) can be expressed as h = ll \ • t\ + u 2 • h + m 3 • hi (1.1.18h) I 2 = M| • (t 2 X t 3 ) + «2 • (f 3 x fj) +83 • (#! x t 2 )j (1.1.18i) h = t\' (h x * 3 )- (1.1.18j) (iv) The principal invariants of an antisymmetric tensor W are /, = TrW = 0, (1.1.18k) I 2 = W 23 2 + W 3l 2 + W {2 2 = (—Hq) T (—vtq) + (—W 3 ) - = vv] T- nq T W 3 - , (1.1.181) / 3 = Z>e? IT = 0 |m’| 2 = w 2 = (axial vector of IF) 2 ; (1.1.18m) from which, and from (1.1.18a), we can deduce that W has a single real eigenvalue A = 0. (v) If T is a symmetric and positive definite tensor with (=> positive) eigenvalues, then Det T > 0 (i.e., T is invertible), T l = ^ X k ~ l (n k <g> «*). (1.1.18n) (vi) If T is an invertible tensor, and the characteristic equation of T~ l is Det(T~ l - /ii) =()=>• /z 3 -/' 1 /i 2 +/'2M-/'3 = 0, (1.1.18o) then /z = 1/A; i.e., the eigenvalues of T _1 are the inverse of those of T, (1.1.18p) I\=h/hi I'i = hlhi r ' 3 = 1//3, (1-l.lSq) T 1 = (T 2 -I x T + I 2 1)/I 3 . (1.1.18r) Orthogonal Transformations A tensor T is called orthogonal (or length-preserving ) if it satisfies T- T t = T t ■ T = 1 => T l =T t ; (1.1.19a) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND or, in components, E T ki( T \ = E T u T 'i = (1.1.19b) E ( T \l T ‘r = E T * T * = 4rl (1.1.19 C ) from which, since Det T = Det T t (always), and Det(T • T t ) = (Det T)(Det T J ) and Det 1 = 1, it follows that (DetT) 2 = 1 => Det T = ±1. (1.1.19d) THEOREM The set of all orthogonal tensors forms the (full) orthogonal group ; and the set of all orthogonal tensors with Det T = +1 forms the proper orthogonal (sub) group. THEOREM (transformation of bases and preservation of their dextrality) If A = (A k . k = A kk i) is a proper orthogonal tensor, or a rotation, and the basis {u k \ k= 1,2,3} is ortho-normal-dextral (OND), the new basis {u k e, k' = 1,2,3} defined by U k < = E A k’k«k <=> Hfc = E A kk'«k' (1.1.19e) is also OND. Conversely, if both {u k } and {u k f are OND, then there exists a unique proper orthogonal tensor such that (1.1.19e) holds. It is not hard to see that A k'k = COS (life-, Hfc) = COS (u k ,u k >) = A kk r, (1.1.19f) and in this commutativity of the indices lies one of the advantages of the non- accented/accented index notation: one does not have to worry about their order. [In a matrix representation: A = (A k ' k ), k'\ rows, k : columns; A T = (A kk >), k: rows, k'\ columns; where (in general): Ap 2 — A 2 g ^ A 2 p = A n > etc.] Also, in view of the earlier orthonormality conditions (or constraints): Uk' • u r = 6 kr and u k ■ u, = 6 kh (1.1.19g) [which, due to (1.1.19e) are none other than (1.1.19a): A • A T = A T • A = 1] only three of the nine elements (direction cosines) of A are independent. • For a vector a, we have the following component representations in {u k }, {u k '}\ a = E a k u k = E a k' u k'\ (1.1.19h) and from this, using the basis transformation equations (1.1.19e), we readily obtain the corresponding component transformation equations'. a k' = E A k'ka k = E A kk' a k & a k = E A kk>a k ' = E A k'kUk'- (1119i) • Polar versus axial vectors : In general tensor algebra, the word axial (vector, tensor) is frequently used in the following broader sense: (a) Vectors that transform as (1.1.19i) under any/all orthogonal transformations {u k } {u k i} proper or not, are called polar (or genuine ); whereas, WWW.EBOOK777.COM §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA (b) Vectors that, under such transformations, transform as a k ' = ( DetAy 1 A k' k a k = ( DetA ) ^ A k , k a k <=> a k = (Det ^ A kk m k , = ( DetA J y 1 ^ A kk ,a k , = (DetA) ^ A kk ,a k f, are called axial (or pseudo-) vectors. Hence, under a change from a right-hand system to a left-hand system (a reflection), in which case DetA = Det(A k ' k ) = — 1, the com¬ ponents of the axial vectors are unaffected; while those of polar vectors are multi¬ plied by — 1. Since only proper orthogonal transformations are used in this book, this difference disappears — all our vectors will be polar, in that sense. This polar/axial distinction is of importance in other areas of physics; for example, relativity, electro¬ dynamics (see, e.g., Bergmann, 1942, p. 56; Malvern, 1969, pp. 25-29). • Every orthogonal tensor is either a rotation, A —> R, or the product of a rotation with — 1; that is, R or — 1 • R (1: 3 x 3 unit tensor). • The eigenvectors of R — that is, the set of vectors satisfying R ■ x = x (R f 1) — build a one-dimensional subspace of V called the axis (of rotation) of R. • Under {u k } {h^} transformations, the components of a tensor T = (T k/ ) = ( T k n /) transform as follows: T'k'l' = EE A k'k A l'lT kl = A kk' A U'Tkl, (1.1.19j) T ki = E E A kk' A u' Tk'V = E E A k'k A viTk'V, (1 • 1 ■ 19k) or, in matrix form (also shown, frequently, in bold but roman), (l.l.lSj): (T k ' V ) = (A k > k )(T kl )(A lv ) or T' = A T A t , (1.1.191) (1.1.19k) :(T kl ) = (A kk ,)(T k , r )(A n ) or T = A 1 • T' • A. (1.1.19m) [(a) Here, T' should not be confused with the symmetrical part of T. (1.1.13a, b). The precise meaning should be clear from the context. (b) We do not see much advantage of (1.1.191,m) over (1.1.19j,k), especially as a working tool in new and nontrivial situations. However, (1.1.191,m) could be useful once the general theory has been thoroughly understood and is about to be applied to a concrete/numerical problem.] It can be shown that: (i) If W is antisymmetric, then (a) 1 + TV is nonsingular, that is, Det(l + IT) f 0; and (b) (1 — W) • (1 + W) “ 1 is orthogonal (a result useful in rigid-body rotations). (1.1.19n) (ii) If 0-u \ 2 3 and 0-u\: 2 'y originally coincide, then the rotation tensor of a counter¬ clockwise (positive) rotation of 0-u m through an angle <j> about » 3 = uy has the matrix form (with ctj> = cos cj>,stj> = sin f)\ / ctj) —s(j> 0\ A —* R = I c(j> 0 . V o 0 1 ) WWW.EBOOK777.COM (1.1.19o) CHAPTER 1: BACKGROUND Moving Axes Theorems for Vectors and Tensors Let us consider the following representation of a vector a and a tensor T , measured relative to inertial, or fixed, OND axes {u k i}, but expressed in terms of their com¬ ponents along (also OND) moving axes {u k } rotating with angular velocity m relative to {u k :}-. « = E a k»k, T = Tk ' Uk ( 1 ■ 1 - 20a ) Let us calculate their inertial rates of change [i.e., relative to the fixed axes, da/dt, dT/dt (t = t'\ time)], but in terms of their moving axes representations (1.1.20a) and their rates of change. (i) By d{. . .)/dt-differentiating the first of (1.1.20a) and invoking the fundamental kinematical result (most likely known from undergraduate dynamics)—a result which, along with the concept of angular velocity, is detailed in §1.7: du k /dt = a x u k , (1.1.20b) we obtain da/dt = ^ [(da k /dt)u k + afiio x u k )] = da/dt + to xa, (1.1.20c) where da/dt = ^ ( da k /dt)u k : rate of change of a relative to the moving axes. (1.1.20d) (ii) Repeating this process for the second of (1.1.20a) we obtain dT/dt = EE {( dT k j / dt)u k ® iij + T k /[(a> x u k ) ® U; + u k ® (<n x u/)\ } = dT/dt + coxT-Txa), (1.1.20e) where dT/dt = ^ ( dT kI /dt)u k ® uy. rate of change of T relative to the moving axes (or Jaumann. or corotational, derivative of T). (1.1.20f) Recalling the earlier results on the algebra of vectors/tensors and axial vectors [eqs (1.1.12), (1.1.14), (1.1.16)] we can rewrite (1.1.20c,e) in %-components as follows: (i) ( da/dt) k = da k /dt + (a) x a) k (fida k /dt) = da k /dt + E E £krs UJrCls = da kl dt + E Qk > a >- ( 1 • 1 - 2 °g) (ii) ( dT/dt) kl = ( dT/dt) kl + (to x T) k , - (T x (o) k , dT k i/dt -f 'y' y' s krs tu r T s/ y ( y' si rs u> s T kr = dT k i/dt + ^ Q ks T sl + ^ Q r T kr [after some index renaming in the last (third) group of terms, and noting that 17/, — 12 s /] = dT kt /dt + y ' Q ks T s i — y ^ T ks Q s i. WWW.EBOOK777.COM (1.1.20h) §1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA where ^=EE Qki u k <S> uf. moving axes representation of angular velocity tensor (of these axes relative to the fixed ones); i.e., antisymmetric tensor whose axial vector is a>: Q • a = coxa. in components: w* = -(l/2) EE ^krs^rs ^rs — E Ekrs^k • (1.1.20i) Thus, in dyadic/matrix notation (see table 1.1), eq. (1.1.20e) reads dT/ dt = 8T/ dt + £2 ■ T - T • £2 (1.1.20j) [ = dT/dt + £2 • T + (£2 • T) t , if T=T J ]. (1.1.20k) REMARKS (i) Overdots, like (...)', are unambiguous only when applied to well-defined com¬ ponents of vectors/tensors; that is, a k , a k ', T kh T k n',...\ not when applied to their direct or dyadic , and/or matrix representations; that is, does a mean da/dt or da/dtl This is a common source of confusion in rigid-body dynamics. (ii) We hope that this has convinced the reader of the superiority of the indicial notation over the (currently popular but nevertheless cumbersome and after-the- factish) dyadic/matrix notations. Coordinate Transformations versus Frame of Reference Transformations See also §1.2, §1.5. Let a' and a be the values of a vector as measured, respectively, in the fixed (inertial) and moving (noninertial) frames. Then [recalling (1.1.19e i)], we have Inertial: «' = ^ a' k u k = ^ a' k ai k t\ (1.1.201) => a'k’ = E A k'k a 'k ^ a'k = E A kk' a 'k' (definition of a' k , , a' k ) (1.1.20m) Noninertial: « = E a k»k = E °k' u k'\ (1 • 1 -On) => a k i = 'y ) A k ' k a k a k = E A kk ,a k : (definition of a k i,a k ). (1.1.20o) Tabled Common Tensor Notations Direct/Dyadic Matrix Indicial/Component a- h = h-a (Dot product) a T • b = b T • a E a kbk T = a® b (Outer product) T = a • b T II E-i h= Ta b = Ta b k — E T k i a i h = a-T b T = a T • T or b = T t • a b k = E a iT, k a - T ■ h (Bilinear form) a T • T ■ b E E T kl a k bi T ■S (Tensor product) TS E T kr S rl T ■ S T (Tensor product) T-S t E T kr S) r T:S = S:T (Dot product) Tr(T-S T ) = Tr(S • T t ) E E TuSki T ■ ■ S = S ■ ■ T (Dot product) Tr(T-S) = Tr(S-T) E E T kl s lk Note: In matrix notation, the product dot is, frequently, omitted. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND However, to relate the noninertial components a k >, a k to the inertial components a' k i, a' k , say, to be able to write something like a k = ci'k ^ a k' = a'k’, (1.1.20p) we need additional assumptions (postulates) or derivations—eqs. (1.1.20p) express frame of reference ( physical) transformations; that is, they do not follow from eqs. (1.1.20m,o), which are simply coordinate system (geometrical[projection) trans¬ formations; (1.1.20p) have to be either postulated or derived from these postulates! Mathematically, a frame of reference transformation is equivalent to an explicitly time-dependent transformation between coordinate systems representing the two frames: x k i = x k '(x k , t) <^> x k = x k (x k ', t ), while an ordinary coordinate transforma¬ tion is explicitly time-independent: x k i = x k t(x k ) x k = x k (x k i). For example, let us consider an inertial frame represented by the (fixed) axes 0-x k i and a noninertial one represented by the (moving) axes 0-x k , related by the homogeneous transformation (common origin!) x k> = X A k'kX k ^ x k = X A kk’*k', (1.1.20q) where A k'k = A kk' = A k'k(t)- Clearly, from geometry [i.e., (1.1.20p)-type postulates]: x' k ’=x k t, x' k = x k . (1.1.20r) By (...)'-differentiating the first of (1.1.20q), and since dx' k >/dt = dx k '/dt = v' k >: inertial velocity of particle (with inertial coordinates x k ') resolved along inertial axes, dx' k /dt = dx k /dt = v k : noninertial velocity of same particle (with noninertial coordinates x k ) resolved along noninertial axes, we get v'k’ = X A k'kVk + X (' dA k'k/dt)x k = v k ' + X ( dA k'k/dt)x k , (1.1.20s) [invoking (1.1.20o)], where dA k > k /dt = J2^k't' A i'k = E A k'i-hk (see §1.7); that is, v'k 1 7 ^ v k ', even if the x k and x k > are, instantaneously, aligned (i.e., A k i k = 6 k ' k —see §1.7); and, similarly, from the second of (1.1.20q), v' k ^ v k , where v' k = J2 A kk’ v 'k'- As eq. (1.1.20s) shows, v' k t depends on both the relative orientation between x k and x k ' (term ^ A k ' k v k = v k '\ noninertial particle velocity, but resolved along inertial axes — a geometrical effect) as well as on their relative motion [term J2(dA k ' k /dt)x k — a kinematical effect]. There is more on moving axes theorems/ applications in §1.7. Vectors transforming between frames as (1.1.20p) are called objective — namely, frame-independent; otherwise they are called nonobjective. Similarly for tensors: if T' kr = T kr , or T' kl = T kh where T kr = A k'k A nT' k i and T k 'i’ =E 22 A k'k A nTkh that tensor is called objective. These concepts are important in continuum mechanics: the constitutive (physical) equations — namely, those relating stresses with strains/deformations and their time rates of change — must be objective. They also constitute the fundamental, or guid¬ ing, philosophical principle of the “Theory of Relativity” [A. Einstein, 1905 (special theory); 1916 (general theory)]. Classical mechanics does not admit of a fully phy¬ sically invariant formulation (although its geometrically invariant formulation is easy via tensor calculus), and the reason is that it is based on Euclidean geometry WWW.EBOOK777.COM §1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS and on a sharp separation between space and ( absolute , or Newtonian) time. Hence, to obtain such a physically invariant mechanics, one had to change these concepts— and this was the great achievement of relativity: The latter replaced classical space and time with a more general non-Euclidean “space-time,” a fusion of both space and time (and gravity). In this new “space,” physical invariance is again expressed as geometrical invariance, via a “physical tensor calculus.” (See, e.g., Bergmann, 1942.) Table 1.1 summarizes, for the readers’ convenience, common vector and tensor operations in all three notations. [We are reminded that in matrix notation, vectors are displayed as 3 x 1 column matrices, so that, in order to save space, we write a —> a T = (a\,a 1 ,a- i ) T ] Differential Operators (Field Theory) The most important differential operators of scalar (/)/vector («)/tensor (T) field theory, needed not so much in analytical mechanics as in continuum mechanics/ physics, are (■ d/dr)f = grad f = df /dr = ^ (df '/dx k )u k - (1.1,21a) (d/dr) ® a = grad a = da/dr = E ( da l/ ax k) (u k ® U/) , (1.1.21b) (d/dr) • a = Tr(grada) = diva = (da k /dx k ), (1.1.21c) (d/dr) x a = curl a = £ krs(9 a s/dx r )u k ; (1.1.21d) (d/dr) 0 T = grad T = ^ ^ E (9T rs /dx k )[u k 0 (u r ®u s )\, (1.1.21 e) (d/dr) • T = Tr(grad T) = div T -EE (dT ks /dx k )u s - (1.1.21f ) where r = (x. y. z): position vector, from some origin O , on which/, a , T depend; and (a k ), (T kl ) are the respective components of «, T relative to an OND basis { O. u k }. 1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS Space, Time, Events Classical mechanics (CM), the only kind of mechanics studied here, and that of which analytical mechanics is the most illustrious exponent, studies the motions of material bodies, or systems, under the action of mechanical loads (forces, moments). Hence, bodies, forces, and motions are its fundamental ingredients. Before examining them, however, we must postulate a certain space-time, or stage, where these phe¬ nomena occur, so that we may describe them via numbers assigned to elements of length/area/volume/time interval. In CM: (i) space is assumed to be three-dimensional and Euclidean (C 3 ); that is, in good experimental agreement with the Pythagorean theorem, both locally and globally, and (ii) there is a definite method for assigning numbers to time intervals, which is based on the existence of perfect clocks’, that is, on completely periodic physical systems (i.e., such that a certain of their configurations is repeated indefinitely; e.g., an oscillating pendulum in vacuo, or our Earth in its daily rotation about its axis). Further, we assume that space and time are homoge¬ neous (i.e., no preferred positions), and that space is also isotropic (i.e., no preferred directions). A physical phenomenon that is more or less sharply localized spatially WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and temporally (i.e., one that is occurring in the immediate neighborhood of a space point at a definite time: e.g., the arrival of a train at a certain station at a certain time) is called an event. Geometrically, events can be viewed as points in space-time, or event space ; that is, in a four-dimensional mathematical space formed jointly by three-dimensional space and time. There, the four coordinates of an event, three for space and one for time, are measured by observers using geometrically invariant, or rigid, yardsticks (space) and the earlier postulated perfect clocks (time). [Fuller understanding of this measurement process requires elaboration of the concepts of immediate ( spatial) neighborhood and ( temporal) simultaneity. This is done in relativistic physics. Here, we take them with their intuitive meaning.] Frame of Reference A frame of reference is a rigid material framework, or rigid body, relative to which spatial and temporal measurements of events are made, by a team of (equivalent) observers, distributed over that body (at rest relative to it), equipped with rigid yardsticks and mutually synchronized perfect clocks. Clearly, some, if not all, of these measurements will depend on the state of motion of the frame (relative to some other frame!); that is, this “coordinate-ization of events” is, generally, nonunique. The relation between the measurements of the same event(s), as registered in two such frames, in relative motion to each other, is called a frame of reference transfor¬ mation-, and the latter is expressed, mathematically, by an explicitly time-dependent coordinate transformation — one coordinate system rigidly embedded to each frame and “representing” it. Inertial Frame of Reference This is a frame determined by the center of mass (“origin”) of our Sun and the so- called fixed stars (directions of axes of frame). This primary, or astronomical, frame is Newton’s absolute space', and, like a cosmic substratum, is assumed to exist (in Newton’s words) “in its own nature, and without reference to anything external, remains always similar and immovable.” Similarly, Newton assumes the existence of absolute time, which is measured by standard clocks, and flows uniformly and inde¬ pendently of any physical phenomena or processes —something that, today, is con¬ sidered physically absurd: “[I]t is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon” (Einstein, 1956, pp. 55-56). In spite of its logically/epistemologically crude and no longer tenable foundations, CM is astonishingly accurate in several areas. For example, the planet Mercury in its motion around our Sun sweeps out a total angle of 150,000°/century; which is only 43" more than the Newtonian predic¬ tion! In this sense, of Machean Denkokonomie (~ Principle of economy, in the for¬ mation of concepts), CM is an extremely economical intellectual and practical investment. As the mathematical structure of the Newton-Euler laws of motion shows (§1.4,5), any other frame moving with (vectorially) constant velocity, relative to the primary frame, is also inertial; so we have a family, or group, of secondary inertial frames. In inertial frames, the laws of motion have their simplest form [the familiar “force equals mass times acceleration (relative to that frame)”]. WWW.EBOOK777.COM §1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS Particle Kinematics The instantaneous position , or place , of a particle P relative to an origin, or reference point, O, fixed in a, say, inertial frame F in E 3 , is given by its position vector r = (x,y,z); where x, y. z are at least twice (piecewise) continuously differentiable functions of time t. Clearly, r depends on O while x, y, z depend on the kind of coordinates used in F. (Also, we are reminded that in kinematics, the frame does not really matter; any frame is as good as any other.) At time t, a collection of particles, or body B , occupies in E 3 a certain shape, or configuration, described by the single¬ valued and invertible mapping r = f(P,t ): Place of P, in F, at time t: (1.2.1a) from which, inverting (conceptually), we obtain p=r\v,t). (i. 2 .ib) A motion of B is a change of its configuration with time; that is, it is the locus of r of each and every P of B, for all time in a certain interval. Formally, this is a one- parameter family / of configurations with time as the (real) parameter. Often, especially in continuum mechanics, the motion of P is described as r=f(r 0 ,t)=r(r 0 ,t), (1-2.2) where r a is the position of P at some “initial or reference” time; that is, a reference configuration used as the name of P (see also §2.2 ff.). The above representation — in addition to being single-valued, continuous, and twice (piecewise) continuously differentiable in t — must also be single-valued and invertible in r 0 ; that is, one-to-one in both directions. (In mathematicians’ jargon: a configuration is a smooth homeomorphism of B onto a region ofE 3 .) The velocity and acceleration of P , relative to a frame F, are defined, respectively, by (assuming rectangular Cartesian coordinates) v = dr/dt = (dx/dt,dy / dt,dz / dt), a= dv/dt = d 2 r/dt 2 = ( d 2 x/dt 2 ,d 2 y/dt 2 ,d 2 z/dt 2 ). (1.2.3) Clearly, v and a depend on the frame, but not on its chosen fixed origin O. The representation of the velocity and acceleration of P, relative to F, moving on a general space, or skew, (F-fixed) curve C, along its natural, or intrinsic, ortho-normal-dextral moving trihedron/triad {u t , u n , uf\ = {t,n,b} (see fig. 1.1 for definitions, etc.) is v = dr/dt = {dr/ds)(ds/dt) = (ds/dt)t = v,t (= v t t + 0n + 0b), (1.2.3a) a = dv/dt = ( d 2 s/dt 2 )t + [(ds/dt) 2 /p]n = (dv,/dt)t + ( v, 2 / p)n = (dv,/dt)t + (v 2 / p)n (= a t t+ a„n + Ob, see below). (1.2.3b) • The speed of P is defined as the magnitude of its velocity: Speed = v = |v| = |v,| = \ds/dt\ = + [(dx/dt) 2 + ( dy/dt ) 2 + (dz/dt) 2 ~\ V ~ f 0; i.e., v t = v ■ t = s = ±v. (1.2.3c) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Center of Curvature Figure 1.1 Natural, or intrinsic, triad representation in particle kinematics, s: arc coordinate along C, measured (positive or negative) from some origin A on C\ p\ radius of curvature of C at P(0 < p < oo); orthonormal and dextral (OND) triad: {Uf, u n , u b } = {f, n, to); (oriented) tangent: ut = t = dr/ds) [always pointing toward (algebraically) increasing values of s (= positive C-sense)]; (first, or principal) normal: u n = n = p(dt/ds ) (always in sense of concavity, toward center of curvature); (second) normal, or binormal: u b = to = f x n; osculating plane: plane spanned by f and n (locus of tip of acceleration vector); rectifying plane: plane spanned by fand to; normal plane: plane spanned by n and to; f • (nx b) = (f, n, b) = +1 > 0. [More in §1.7: (1,7.18a)ff.[ Hence, in general, since v 2 = v t v t = (,sj 2 , a=\a\ = \dv/dt\ = [{dv/dt) 1 + (v 4 /p 2 )] 1/2 ^ \dv,/dt\ = \d 2 s/di 2 \ = \dv/dt\; i.e., a, = s (tangential accel’n), a„ = ( s) 2 /p (normal a.), a b = 0 (binormal a.) => v a = ss = (i 2 /2)‘ , vxa = (s 3 /p)b. (1.2.3.d) REMARKS (i) The difference between speed v = \dr/dt\ = |v| = \ds/dt\ > 0 and the ( tangential ) velocity component v, = v-t = ds/dt = ± v [i.e. by equation (1.2.3c): v, = +v > 0 if ds > 0, and v, = — v < 0 if ds < 0] results from the oriented-ness of the curve C [i.e. that it is equipped with (a) an origin A, and (b) a positive /negative sense of traverse =T ± ,s]; i.e. in any motion of P along it, the unit tangent vector t = dr/ds (f 0) points always towards the increasing arcs s (just like i = dr/dx always points towards the positive/ increasing x- see below). Fortunately, this v, versus v difference (almost never noticed in the literature) rarely results in fatal errors. (ii) Thus, it becomes clear that s [= (intrinsic) arc/path/trajectory curvilinear coordinate/abscissa, of P relative to a chosen C-origin A] is the “natural” curvilinear generalization of the rectilinear position (-al) coordinates x, y, z (and [t. n.b) are of [i. j, k), respectively). (iii) The equation s = s(t), resulting by integrating ds = ±.\v{t)\dt = ± v(t)dt [say, from t(A) to f(P)], is referred to as the equation / law of motion of P on C. (iv) Last, (a) the length of the arc AP is defined as the absolute value of ,v, |,v > 0, while (b) the (total) distance traveled by our particle P, along C, from an origin A to its current /final position (i.e. what a car odometer shows) is defined by: I \ds\ (> |s| >0). (1.2.3e) J origin —f current C-position For details on arc length, admissible curve parametrizations etc, see works on differential geometry. WWW.EBOOK777.COM §1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS It can be shown that [with the additional notation (...)' = d{- ■ •)/ du \: (i) t = r' /s' = r' / (r' • r') 1 ! 2 = ( dx/ds)i + ( dy/ds)j + ( dz/ds)k, (1.2.4a) (ii) n = p{dt/ds) = p{d 2 r/ds 2 ) = p[{d 2 x/ds 2 )i + (d 2 y/ds 2 )j + ( d 2 z/ds 2 )k\ = P{t'/s') = pt '/(/'' -r') 1/2 = [p/{y' -r'f 2 W •r') x ^ 2 v" — (r 1 • r')~ x ^ 2 {r' •r")r'] = [p/( r ' -r')r" -(/•/>']; (1.2.4b) (iii) k = \/p = \dt/ds\ = \d 2 r/ds 2 \ (^ 0 , 0 ^ p < + 00 ): (first) curvature of C, at P, (1.2.4c) (iv) (v) K 2 = l/p 2 = (/ X r") 2 /{r'-r'f = [{/■ r'){r" ■ r") - (/*' • r") 2 ]/[r' • r ') 3 [= {d 2 x/ds 2 ) 2 + {d~y / ds 2 ) 2 + {d 2 z/ds 2 ) 2 , if u = s]\ (1.2.4d) / /. r =st, r w = s"t + s V = s"t + s'[{dt / ds){ds / du)] [ = s"t + (s') 2 (dt/ds)] = s"t + k(s') 2 ii = s"t + [{s') 2 / p]n (1.2.4e) [= (dv,/dt)t + {v 2 / p)n, v,v, = vv = {ds/dt) 2 ; if u = t]; (vi) t = v/(ds/dt) = v/v „ (1.2.4f) n - p[r 2 a - (v • a)v]/v 4 = - • ( 1 - 2 .4g) b = t x « = p[(r/r/cfe) x (J 2 r/cf.r)] = p(r' x r")/{r' -r ') 3 ^ 2 = p(v x a)/v , 3 = p(v x a)/v t v 2 ; (1.2.4h) (vii) k 2 = 1/p 2 = (v x a) 2 /v 6 = [v 2 a 2 — (v-a) 2 ]/v 6 ; (1.2.4i) (viii) a, = a-t= [v x (dv x /dt) + v y (dv y /dt) + v z {dv z /dt)]/vp (1.2.4j) (ix) = |a x #| = {[v x (</v ; ,/t/t) — v J ,(t/v x /t/?)] 2 + [v y (</v z /t/t) — v,(iiv v /Jt)] 2 + [v.-f^M) - v x {dv z /dt)] 2 } l/2 / (v Y 2 + v y 2 + v z 2 ) 1/2 . (1.2.4k) • In plane polar coordinates, the position/velocity/acceleration of a particle P are (where a,, u,,;. unit vectors along OP and perpendicular to it, in the sense of increasing r, </> respectively): du r /dt = {dfi/dfju^ and du^/dt = —{d<j>/dt)up or du r = d<j)u ( p and du^ = —d<j>u r , r = r u r [= (r)u r + (0)ity], (1.2.5a) v = ( 1 dr/dt)u r + r{d</>/dt)u^ = v r u r + v^Ua, (1.2.5b) a = [d 2 r/dt 2 — r{d</>/dt) 2 ]u r + {r~ x d / dt[r 2 (d<f> / dt)]}u^, = [d 2 r/dt 2 — r{d(f>/dt) 2 ]u r + [2{dr / dt){d(/> / dt ) + r{d 2 (j)/dt 2 )\u$ = a (r) u r + a w « 0 . (1.2.5c) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND The vectors ( d(j>/dt)k and ( d 2 cpldt 2 )k are, respectively, the angular velocity and angular acceleration of the radius OP = r relative to O—xy. It can be shown that (i) a, = a-t= ±(v ,.a r + v^af) / ( v r 2 + v 0 2 ) 1/2 [+ if v, > 0, - if v, < 0], (1.2.5d) (ii) The rectangular Cartesian components of the velocity and acceleration are, respectively, dx/dt = ( dr/dt ) cos <f> — [r(d(/)/dt)\ sin p, dy/dt = ( dr/dt ) sin <p + [r(d(j)/dt)] cos </; (1.2.5e) d 2 x/dt 2 = [ d 2 r/dt 2 — r(dcf>/dt)~} cosp — [2 {dr / dt){d<p / dt) + r{d 2 <f>/dt 2 )\ simp, d 2 y/dt 2 = [ d 2 r/dt 2 — r{d<p/dt) 2 ] sin p + [2 (dr/dt)(d<f>/dt) + r(d 2 (/)/dt 2 )\ sin p, (1.2.5f) and, inversely, dr/dt = (xv x +yVy)/(x 1 + y 2 ) 1 ^ 2 , dp/dt = (xv y ~yv x )/{x 2 +y 2 ), etc. (1.2.5g) [A more precise notation of vector components along various bases of orthogonal curvilinear (i.e., nonrectangular Cartesian) coordinates is introduced below.] • In general (i.e., not necessarily plane) motion, the areal velocity dA/dt of a particle equals dA/dt = (1/2) |r x v| = (1/2) | angular momentum of particle about origin , per unit mass\. (1.2.6a) It can be shown that (assuming r f 0) d 2 A/dt 2 = (r x v) • (r x «)/2|r x v|. (1.2.6b) Velocity and Acceleration in Orthogonal Curvilinear Coordinates (A certain familiarity with the latter is assumed—otherwise, this topic can be omitted at this point.) In such coordinates, say q = (q \, q 2 -qfj = (< 71 , 2 , 3 ) [see fig. 1.2(a)] the position vector r, of a particle P , is expressed as: r = x{q)i + y{q)j + z(q)k = r(q), (1.2.7a) and so the corresponding unit tangent vectors along the coordinate lines < 71 , 2,35 w i, 2 , 3 > are M, = (\/h x ){dr/dq x ) = e x /h u « 2 = ( 1/^2 ){.dr/dq 2 ) = e 2 /h 2 , «3 = ( 1/^3 )(.dr/dq 2 ) = e 3 //? 3 , where «*•«/ = 4/ (/c,/= 1,2,3); (1.2.7b) WWW.EBOOK777.COM §1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS Figure 1.2 (a) General orthogonal curvilinear coordinates; (b) cylindrical (polar) coordinates: x = rcostfi, y = rsin<^>, z — z, r = \OP'\; 2 ,i = h r ^ z = 1,r, 1; (c) spherical coordinates: x = (rsin 6) cos (f>, y = (rcosff) sin <j>, z = r cosd, r = \OP\; h i,2,3 = h rM = 1, Arsine. and since dr/dcji = (dx/dqffi + ( dy/dq x )j + ( dz / dq { )k, dr/ dq 2 = ■ ■ ■, the (normalizing) Lame coefficients h\ 22 are given by hi = \dr/dqi \ = [( dx/dq ,) 2 + [dy/dq { ) 2 + (&/<9t/i) 2 ] 1/2 , h 2 = dr/dq 3 = ■■■, (1.2.7c) • ,h 2 — ■■■. (1.2.7d) We notice that cos (u k ,x) = u k • i = [(1 /h k )(dr/dq k )\ •/= (1 /h k ){dx/dq k ), etc., or, generally, with x = x\, y = x 2 , z = x 3 , and i = i\, j = i 2 , k = / 3 , cos (u k ,x,) = u k -i, = (1 /h k )(dxi/dq k ). (1.2.7e) As a result of the above, and since dr/dq k = e k = /ia-h^, the arc length element ds, velocity v, and speed |v| of P are given, respectively, by (i) (>i) (hi) ds = \dr\ = 1^ 0 dr/dq k ) dq k = (/;f dqf + h 2 2 dq 2 2 + h 3 2 dq 3 2 )' /2 , (1.2.7f) v = dr/dt = Y ( dr/dq k )(dq k /dt) = Y v k e k = Y v k(h k u k ) = Y v (k) u k, (l-2.7g) |v| = v=(h l 2 v l 2 + h 2 2 v 2 2 +h 3 W) l/2 , (1.2.7h) where dq k /dt = v k : “contravariant” or generalized component of v along q k , (1.2.7i) V(A) = h k (dq k /dt) = h k v k : corresponding physical component (with units of length/ time ), WWW.EBOOK777.COM ( 1 . 2.79 CHAPTER 1: BACKGROUND Next, we define the generalized and physical components of the particle acceleration a as a k = a-e k = ( dv/dt ) • (■ dr/dq k ), a [k) =a-u k = a- (e k /h k ) = a k /h k . (1.2.7k) REMARKS (i) For an arbitrary vector h, in general orthogonal curvilinear coordinates, we have the following representations: h = Y b k e k = Y b k e k = Y b k (e k /h k ) = Y ( b k /h k )(e k /h k ) = Y b {k)U k where e k • e/ = gkl ~ = 0 , if kfl- = h k 2 if k = /; e k • ry = II k 1 kl Ik , e -e =g =g => 1 / b k , g kk 1 1 i 2 1 /g = h , Det(g k i) = hfh 2 2 hi 2 ', e k = e k /h k 2 -, b k = b ■ e k = b ■( h k u k ) = h k (b • «fc) = hb {k y, b k =b- e k = b- II ^5 that is. b (k) = b k h k : = b k /h k \ physical l components of b, b k = = b k /h k 2 : 3 II II (ii) Strictly speaking, q k should have been written as q k ; and, consequently, v k as v k \ (iii) In rectangular Cartesian coordinates/axes (this book), clearly, h k = 1 =>■ b( k ) = b k = b k . (iv) For the extension of the above to general curvilinear coordinates, see books on tensor calculus; for example, Papastavridis (1999, chap. 2, especially §2.10). From the first of (1.2.7k) we obtain successively (what are, in essence, the famous Lagrangean kinematico-inertial transformations, to be generalized and detailed in chaps. 2 and 3): a k = a- e k = (dv/dt) • ( dr/dq k ) = d/dt [v • (dr/dq k )\ — v • d / dt(dv / dq k ) | and, using the basic kinematical identities: (a) dr/dq k = dv/dv k [from (1.2.7g)] (b) d/dt(dr/dq k ) = Y d/dq,(dr/dq k )(dq,/dt ) + d/dt(dr/dq k ) = d/dq k (Y (' dr/dqi)(dq,/dt) + dr/dtj = dv/dq k ; i.e., d/dt(dv/dv k ) — dv/dq k = 0 = d/dt\v • (dv/dv k )] - v • (dv/dq k ) = d/dt[d/dv k (v 2 /2)\ - d/dq k (v 2 /2) (since v • v = v 2 ); WWW.EBOOK777.COM (1.2.71) §1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS and, invoking the second of (1.2.7k), we get, finally, the Lagrange an form: a (k) = a k /h k = (1/A*) [d/dt{dT/dv k ) - dT/dq k ], (1.2.7m) where T = v 2 /2 = {\/2)[h l 1 {dq x /dtf + h 2 2 {dq 2 /dt) 2 + h 3 2 (dq 3 /dt) 2 ] l/1 = (1/2)(A 1 2 v 1 2 + /i 2 2 v 2 2 + A 3 2 v 3 2 ) 1/2 : kinetic energy of a particle of unit mass (i.e., m = 1). (1.2.7n) Application (i) Cylindrical {polar ) coordinates [fig. 1.2(b)]. Here, x = rcosf, y = r sin z = z, and, therefore, ds 2 = ds 2 + dsf 2 + ds 2 = dr 2 + r 2 dcj> 2 + dz 2 , (1.2.8a) from which we immediately read off the following Lame coefficients: h i —> h r = 1, h 2 —> h ( j > = r, /? 3 —> A- = 1. (1.2.8b) Hence, the “unit kinetic energy’’ equals 2 T = (ds/dt) 2 = v 2 = [{dr/dt) 2 + r {df/dt) + (dr/tA) 1 ] = v, 2 + r 2 v^ + v z 2 , (1.2.8c) and so, by (1.2.71), the (physical) components of the acceleration are a W —> «(,.) = d/dt{dT/dv r ) — dT/dr = d 2 r/dt 2 — r{df/dt) 2 , (1.2.8d) <3(2) ->■ % i) = (l/p)[rfM(9r/av^) - dT/d(j>\ = (l/r){A/<A[r 2 (A<(>/Ar)]} = r{d 2 cj)/dt 2 ) + 2{dr / dt){df / dt), (1.2.8e) fl (3) —> «( z ) = d/dt{dT/dv z ) — /dz = d 2 z/dt 2 . (1.2.8f) (ii) Spherical coordinates. Here, x= (r sin 6) cos </>, y= (r cos 9) sin <j>, z = rcos9 [fig. 1.2(c)]. Using similar steps, we can show that —> a( r ) = d/dt{dT/dv r ) — dT/dr = d 2 r/dt 2 — r{d6/dt) 2 — r{d<f>/dt) 2 sin 2 9\ (1-2.8g) «(2) -► 0(9) = (i/r)[5r/5v e - ar/96»] = (1 /r){d/dt[r 2 (d9/dt)\ — r 2 {dcj)/dt) 2 sindcos d}; (1.2.8h) «( 3 ) —> = (1 /r sin 9) [d/dt{dT/dvf) — dT/d<j>\ = {l/r sm9){d/dt[r 2 (d(/)/dt) sin 2 0]}; (1.2.8i) v x = dx/dt = {dr/dt) sin 9 cos <f> + r{d9/dt) cos0cos</> — r{d<j>/dt ) sin 9 sin f, (1.2.8J) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND v y = dy/dt = ( dr/dt) sinf?sin</> + r(dO/dt) cost? sin <(> + r(d<p/dt) sin t?cos<(>, (1.2.8k) v z = dz/dt = (dr/dt) cos 9 — r(dO/dt) sinf?; (1.2.81) a x = d 2 x/dt 2 = [d 2 r/dt 2 — r(dd/dt) 2 — r(d<f>/dt)~] sin(9cos<(> + [r(d 2 0/dt 2 ) + 2(dr/dt)(d9/dt)\ cos 9 cos cj) — [r(d 2 (j)/dt 2 ) + 2 (dr/dt)(d(j>/dt)\ sin 8 sin cj) — 2r(d<j>/dt)(d9/dt) cost? sin ((>; (1.2.8m) a y = d 2 y/dt~ = [ d 2 r/dt~ — r(d0/dt) 2 — r(d<j>/dt)~] sint?sin</> + [r(rf 2 t?/iir) + 2(dr/dt)(d6/dt)\ cos0sin0 + [r(6? 2 </>/<ft 2 ) + 2 (dr/dt)(dcj)/dt)\ sin t? cos</> + 2 r(dcj>/dt)(d9/dt) cosOcosrf), (1.2.8n) a z = d 2 z/dt 2 = [d 2 r/dt 2 — r(dO/dt ) 2 ] cos (9 — [r(d 2 0/dt 2 ) + 2(dr/dt)(d9/dt)\ sin#; (1.2.8o) and, inversely, dr/dt = [x(dx/dt) + y(dy/dt) + z(dz/dt)]/(x 2 + y 2 + r 2 ) 1 ^ 2 , (1.2.8p) d8/dt = { \x(dx/dt) + y(dy/dt)\z — (x 2 + y 2 ) (dz/dt)} / (x 2 +y 2 )*^ 2 (x 2 + y 2 + z 2 ), (1.2.8q) d(f)/dt= [x(dy/dt) — y(dx/dt)]/(x 2 + y 2 ); (1.2.8r) and d 2 r/dt 2 = • • •, d 2 0/dt 2 = • • ■, d 2 (f)/dt 2 = • • ■, in complete agreement with (1.2.5). REMARK From now on, parentheses around subscripts (employed to denote physical compo¬ nents) will, normally, be omitted; that is, unless absolutely necessary, we shall simply write a r , ag, for U( r) , a^, a^s, respectively, etc. 1.3 BODIES AND THEIR MASSES Body or System A body or system is an ordinary three-dimensional material object whose points fill a spatial region completely; or a continuous connected three-dimensional set of mate¬ rial points, or mass points, or particles, such that any part of it, no matter how small, possesses the same physical properties as the entire object. The interactions of bodies, under the action of forces/fields, produces the various physical phenomena. WWW.EBOOK777.COM §1.3 BODIES AND THEIR MASSES Bodies are usually classified as solids, fluids, and gases. • The rigid body is a special solid whose deformation (or strain), relative to its other motions, can be neglected; and whose geometric form/shape and spatial material distribution are invariable. • The particle is a special rigid body whose rotation, relative to its other motions, can be neglected; it is small relative to its distance from other bodies, and its motion as a whole is virtually unaffected by its internal motion. It is a special localized continuum of infinite material density (see below). The complete characterization of a particle requires specification of its spatial posi¬ tion and of the values of its associated parameters (e.g., mass, electric charge). The former varies with time but the latter, since they describe the internal constitution of our particle, do not; if they did, we would have a more complex system. Whether one and the same body or system will be modeled as deformable con¬ tinuum, or rigid, or particle, etc., depends on the problem at hand. Below, we show such a problem to model correspondence for the system Earth: Problem Orbit around the Sun Tides and/or lunar eclipses Precession of the equinoxes Earthquakes etc. Mathematical Model Particle Rigid sphere Rigid ellipsoid Elastic sphere Mass To each body, B, that instantaneously occupies continuously a spatial region of volume V, we assign, or order, a real, positive and time-independent number expres¬ sing the quantity of matter in B, its mass m: a primitive concept with dimensions independent of the (also primitives) length and time. Symbolically, we have B —> m(B) = m = dm where (continuity hypothesis) (dm/dV)dV = pdV > 0, v Jv (1.3.1) p = [lim(d?n/zl V)\ Av ^ 0 = dm/dV\ mass density, or specific mass, of B (a piecewise continuous function of t and /•) (1.3.2) and m = constant, for a given body ( conservation of mass). The above imply that the mass is additive : the mass of a body, or system, equals the sum of the masses of its parts; with some intuitively obvious notation: m(B) = m(B\ + Bf) = m{B{) + m(B 2 ) = m { + m 2 ■ (1.3.3) REMARKS (i) For so-called “variable mass problems” (clearly, a misleading term); for exam¬ ple, rockets, chemical reactions, see Fox (1967, pp. 321-326) and, particularly, Novoselov (1969). WWW.EBOOK777.COM CHAPTER 1: BACKGROUND (ii) To describe several bodies, including possible gaps, via (1.3.1) and (1.3.2), we may have to assume that in some regions p = 0. (iii) Mathematically, mass additivity can be expressed as follows: Consider an arbitrary subset of the body B , b. If we can associate with b a nonnegative real number m(b), with physical dimensions independent of those of time and length, and such that m{b\ U bf) = m(bi) + mlbf) [U = union of two sets] for all pairs b\ and b 2 of disjoint subsets of b; and m(b) —> 0, as the volume occupied by b goes to zero; then we call B a material body with mass function m. The additive set function m(b ) is the mass of b; or the mass content of the corresponding set of points occupied by b. The above properties of m {...) imply the existence of a scalar field p = mass density of B. defined over the configuration of B. such that (1.3.1) holds. Impenetrability Axiom (and One-to-One Event Occurrence) Not more than one particle may occupy any position in space, at any given time. More generally (continuum form), if, during its motion, the material system initially occupies the spatial region V a , and later the region V, then the relation between V 0 and V is mutually one-to-one, and piecewise continuously differentiable (for the associated field functions). Discontinuities (e.g., rupture, impact) and accompanying loss of uniqueness can occur only across certain (two-dimensional) boundary sur¬ faces. Remarks on Particles, Bodies, Mathematical Modeling, and so on (i) A finite, or extended , body B or system 5 can be treated exactly, or approxi¬ mately, as a particle in the following three cases: (a) If B undergoes pure translation; that is, all its points describe congruent paths with (vectorially) equal velocities and accelerations. In this case, any point of B can play the role of that particle. (b) If the description of the kinetic properties of B requires only the investigation of the motion of its center of mass (§1.4). (c) If B is such that its dimensions are so small (or its distances from other bodies, its environment, are so large) that its size can be neglected; and its motion can be represented satisfactorily by the motion of either its mass center or any other inter¬ nal point of it. Such bodies we call small. • In cases (b) (always) and (c) (usually) that particle is the mass center. • Cases (a, b) are exact, while (c) is only approximate. • In case (a), that particle describes the motion of B completely, in (b) only partially (the motion about the mass center is neglected), and in (c) with an error depending on the neglected dimensions of B. From such a continuum viewpoint, a particle is viewed not as the building block of matter, but as a rigid and rotationless body ! As Hamel (1909, p. 351) aptly summarizes: “What one understands, in practice, by particle mechanics WWW.EBOOK777.COM §1.4 FORCE; LAW OF NEWTON-EULER (Punktmechanik) is none other than the theorem of the center of mass (Schwerpunktsatz). ’ ’ (ii) Both models of a body—that is, the one based on the atomistic hypothesis (body as a finite, discrete, set of material points, or particles; namely, small hard balls with no rotational characteristics) and the other based on the continuity hypothesis (body as a family of measurable sets, with associated additive set functions represent¬ ing the mass of that set)—have advantages and disadvantages; and both are useful for various purposes. The sometimes (in some engineering circles) fierce debate for/ against one or the other viewpoint, we consider counterproductive and petty hair¬ splitting; and so we will use both models as needed. Such dualisms are no strangers to physics (e.g., particles/corpuscules vs. waves/fields in atomic phenomena) and constitute a creative, dialectical, stress in it. Thus, we will view the rigid body (§1.8 If.) either as a (finite or infinite) set of particles whose mutual distances are constrained to remain invariable (i.e., fixed in time); or, more conveniently, as a rigid continuum, and accept the Newton-Euler law of motion for its differential mass elements as for a particle (§1.4, §1.6). In the discrete model, the building block is the single “sizeless,” but possibly quite “massive,” particle of mass m k > 0 (k = 1,2,...); while, in the continuum model, it is the differential element with mass dm = pdV > 0. In sum, we shall adopt the logically unorthodox, but quite fertile and successful, dialectical compromise: particle language and continuum notation ; and eventually (chap. 3 ff.) we will end up with ordinary differential equations. [In general, it is extremely difficult, if not impossible, to go by a limiting process from a statement about particles to one about continua; whereas, conversely, con¬ tinuum statements formulated in terms of Stieltjes’ integrals, like our earlier S (■■■)'■ St--) dm - XX- -)k m k (discrete), or (...) dm (continuum), lead to the same statements for discrete systems without much difficulty, almost automatically. See, for example, Kilmister and Reeve (1966, pp. 129-131).] 1.4 FORCE; LAW OF NEWTON-EULER [l]n the concept of force lies the chief difficulty in the whole of mechanics. (Hamel, 1952; as quoted in Truesdell, 1984, p. 527) Jeder weip aus der Erfahrung, was Schwerkraft ist; jede gerichtete Physikalische Grope, die sich mit der Schwerkraft in Gleichgewicht befinden kann, ist eine Kraft! [Approximate translation: Everyone knows from experience what gravity is; every directed physical quantity that can be in equilibrium with gravity is a force! (emphasis added).] (How Hamel used to begin his mechanics lectures; quoted in Szabo, 1954, p. 26) The fundamental law of mechanics [i.e. mass x acceleration = force] is a blank form which acquires a concrete content only when the conception of force occurring in it is filled in by physics. (Weyl, 1922, pp. 66-67) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Local Form of Newton-Euler Law To each and every material particle P of elementary mass dm and inertial accelera¬ tion a, of a body B or system S, we associate a total elementary force vector df acting on it, such that dma = df , (1.4.1) where df itself is the resultant of other “partial” elementary forces of various origins (to be examined later); that is, df = Y. d fk (* = 1 , 2 ,...). (1-4.2) Equation (1.4.1) is not simply a definition of one vector (df) in terms of another (dma), but is an equality of two physically very different vectors: one, the effect or kinetic reaction (dma), depending only on the properties of the particle P itself, and another, the cause (df), depending on the interaction between P and the rest of the universe — that is, on the action of the external world on the moving system, and the mutual, or internal, actions of the body parts on each other. Paraphrasing Elamel (1927, p. 3) slightly, we may state: The forces are determined by their “causes”; that is, by variables that represent the geometrical, kinematical, and physical state of the matter surrounding P (local causes) and away from it (global causes). This depen¬ dence is single-valued and, in general, continuous and differentiable; and, in addi¬ tion, these forces are objective — that is, independent of the frame of reference (see also Hamel, 1949, pp. 509-512). In practice, this leads to constitutive equations for the forces (stresses) that, when combined with the field, or ponderomotive, equations (1.4.1) lead to relations of the form: a = a(t,r,v; physical constants); (1.4.3) where a may also depend on the rs and v’s of other system (and even external) particles, but not on accelerations or other higher (than the first) d/dt(.. ^-deriva¬ tives. Such an (/-dependence would introduce an additional constitutive, or con¬ straint, equation of the form: dm a = df (...,«,...). However, and this does not contradict (1.4.1), such equations can occur as part of the solution process', namely, through elimination of variables from the complete set of equations of the problem; that is, elimination of forces related to the accelerations of other parts of the body, so that the acceleration of point P depends on, among other things, the accelerations of points Q, R .On this delicate and sometimes confusing point, see Hamel (1949, p. 49). In view of such difficulties in defining the force, a number of authors (mostly continuum mechanicians) consider it as a primitive concept — along with space, time, and mass. Force Classification [This also includes moments; and, in analytical mechanics, both forces and moments are replaced by system, or generalized, forces (§3.4).] The most important such classifications are as follows: Newton-Euler (or momentum) mechanics: Internal', originating wholly from within the system; in pairs. They depend on the spatial limits of the system. WWW.EBOOK777.COM §1.4 FORCE; LAW OF NEWTON-EULER External, originating, even partially, from outside the system. Only such forces appear in the corresponding equations of equilibrium/motion. Lagrangean (or energetic) mechanics: Impressed, depending, even partially, on physical (material) coefficients (chap. 3). Constraint reactions', depending exclusively on the constraints; geometrical and/or kinematical forces (chap. 3). Continuum mechanics: Surface , or contact : continuously distributed over material surfaces (and/or lines and points). Volume , or body : continuously distributed over material volumes. Usually, a given force is a combination of the above, and more. For example: Gravity: external, impressed, body; Stresses in rigid bodies: internal, reactions, surface; Stresses in elastic bodies: internal, impressed, surface; Dry rolling friction: internal or external, reaction, surface; Dry sliding friction: internal or external, impressed, surface. Other, more specialized force classifications are the following: potential/nonpotential, conservative/ nonconservative, gyroscopic / nongyroseopic, circulatory/noncirculatory, autonomous/nonautonomous, etc. They will be introduced later, if and when needed. Occasionally, forces are classified with the help of the momentum principles as follows: Linear or translatory loads: forces; Angular or rotatory loads: moments of forces and moments of couples; but such terminology is not uniform. For example, the authoritative Truesdell and Toupin (1960, p. 531) states that, in the general case, the ( total) torque consists of two parts: the moment of the force(s) and the couple', also, virtually alone among mechanics works, it refuses to use the term internal forces, opting instead for the term mutual (loc. cit., pp. 533-535). On Centers of Gravity and Mass, and Centroid The center of gravity ( CG ) of a material system in a parallel gravitational field is a point defined uniquely by v cg = S r dG / S dG ’ (1-4-4) where dG = elementary gravity force = g dm = pgdV = 7 d V: g = acceleration of gravity; p = density of matter; 7 = specific weight; dm = element of mass; dV = element of volume; and CG is independent of the orientation of the system , and through it passes the resultant gravity force, or weight, of the system, and: S {■■■)'• material summation, for a fixed time, and valid for discrete and/or contin¬ uous distributions (Stieltjes’ integral). This helpful notation, originated informally by Lagrange, is used a lot in the main body of this work. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND The center of mass , or inertial center , (CM) of a material distribution is defined uniquely by i'cm = r G = S r dm / S dm ' (1.4.5) The centroid (or geometrical center, or geometrical center of gravity) (C) of a figure is defined uniquely by r c = S rdV /S dV ' (1-4.6) • If g = constant, the gravitational field is uniform. Then, g = gu = constant, it = vertical unit vector (positive downward). • If p = constant, the body (matter) is homogeneous. In a uniform field: r CG = r CM = r G> For a homogeneous body: •‘cm =r G = r c \ For a homogeneous body in a uniform field: r CG = r CM = r c- (1.4.7a) (1.4.7b) (1.4.7c) REMARK In nonuniform fields, eq. (1.4.7a) is no longer true: the parts of the body closer to the attracting earth experience stronger gravity forces than those farther from it; and, therefore, upon rotation of the body, the point of application of the resultant of such forces changes relative to the body, that is, the center of gravity is no longer definable as a unique body-fixed point, independent of the orientation of the body relative to the field. The center of mass and centroid, however, are still defined uniquely by (1.4.5) and (1.4.6), respectively. Such complications may arise in problems of astronautics/ spacecraft dynamics; there, we replace the constant g with a central-symmetric grav¬ itational field. 1.5 SPACE-TIME AND THE PRINCIPLE OF GALILEAN RELATIVITY Galilean Transformations (GT) These are frame of reference transformations that leave the Newton-Euler law (1.4.1) form invariant. The most general such transformations have the following form (fig. 1.3): r' = A • r + bt + c {Direct/matrix notation) (1.5.1a) x k' = Y A k'k x k + b k ’t + c k t {Component notation), (1.5.1b) WWW.EBOOK777.COM §1.5 SPACE-TIME AND THE PRINCIPLE OF GALILEAN RELATIVITY Figure 1.3 On the geometry of Galilean transformations. where A = (A^f) is a proper orthogonal tensor with constant components — that is, A~ l = A t ; Det A = +1; and h = [h^) and c = (c^) are constant vectors — that is, F and F' are in nonrotating uniform motion ( uniform translation ) relative to each other, with velocity h, and t' = at + (3, (1.5.1c) where t is measured in F and t' in F', and a, (3 are constant scalars; a depends on the units of time, while (3 depends on its origin in the two systems of time measurement. Hence, if these units are taken to be the same, and these origins are made to coincide, then a = 1 and (3 = 0; in which case (henceforth assumed in this book), t' = t\ (1.5.Id) that is, in classical (Newtonian) mechanics there is, essentially, only one time scale. From the transformation equations (1.5.1a-d) we immediately obtain the follow¬ ing: d 2 r' / dt 2 = A • (d 2 r/dt 2 ) or a' = A-a, (1.5.2a) or, explicitly, with some easily understood notation, d 2 x'/dt 2 = cos (x 1 ,x)(d 2 x/dt 2 ) + cos {x' ,y)(d 2 y / dt 2 ) + cos(x', z){d 2 z/dt 2 ), etc.; (1.5.2b) that is, the accelerations of a particle P as measured in F and F' differ only by an ordinary (time-independent) geometrical transformation due to the, possibly, differ¬ ent orientation of their axes; and, therefore, they are physically equal: that is, un¬ affected by the relative motion of F and F 1 . Hence, we may take, with no loss in physical generality, the corresponding axes of F and F' to be ever parallel, in which case A = 1 (unit tensor), in which case (1.5.1a) simplifies to r' = r + bt + c => a'= a. (1.5.2c) Since dm\ F = dm\ F , = dm, and assuming that from dm a = df(t,r,v) and (1.5.2c) it follows that dma = df (t,r' — bt — c,dr/dt — b) = df ' (t,r ,dr / dt = v') = df (1.5.3) that is, df is also invariant under GT, and, therefore, as far as the law of motion (1.4.1) is concerned, there is no one ( absolute ) frame in which it holds, but, in fact, once WWW.EBOOK777.COM CHAPTER 1: BACKGROUND one such “inertial" frame is established, there is a whole family of them dynamically equivalent to it. More precisely, there is a (continuous linear) group that depends on ten (10) parameters: three for A [out of its nine components (direction cosines), due to the six orthonormality constraints, only three are independent], three for h, three for c, and one for (3 [equations (1.5.1c, d), a = 1, with no loss in generality]. This Galilean, or Newtonian, principle of relativity can be summed up as follows: an inertial frame—that is, one in which dm(d 2 r/dt 2 ) = df holds — is determined only to within a Galilean transformation (1.5.1a—d). REMARKS (i) The linear transformation (1.5.1c) can also be obtained by requiring that if a = d 2 r/dt 2 = 0 , (1.5.4a) then also d 2 r/d(f) 2 = 0 , (1.5.4b) for arbitrary values of r and dr/dt. Indeed, using chain rule, we find: dr/dt' = (dr / dt) / (dt' / dt) => d 2 r/d(f) 2 = [(dt'/dt)(d 2 r/dt 2 ) - (dr/dt)(d 2 f/dt 2 )}/(dt'/dt)\ (1.5.4c) and so, due to (1.5.4a), the requirement (1.5.4b) translates to (dr/dt)(d 2 1'/dt 2 ) = 0 , for arbitrary dr/dt\ (1.5.4d) that is, d 2 t'/dt 2 = 0 =>■ /' = at + /3, a,f3\ integration constants; Q.E.D. (1.5.4e) (ii) The logical circularity involved in the classical mechanics definition of inertial frames (i.e., “if dm a = df holds, the frame is inertial" and “if the frame is inertial frame then dm a = df holds”) can be resolved only by relativistic physics. Here, we are content to postulate the existence of frames in which dm a = df holds exactly (or, equivalently, of frames in which forceless motions are also unaccelerated motions', i.e., the position vectors are linear functions of time, and vice versa); and to call such frames inertial. For detailed discussions of this important topic, see any good text on the physical foundations of relativity; e.g., Bergmann, 1942; Nevanlina, 1968. 1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS An Axiom is a proposition, the truth of which must be admitted as soon as the terms in which it is expressed are clearly understood ... physical axioms are axiomatic to those only who have sufficient knowledge of the action of physical causes to enable them to see their truth. (Thomson and Tait, 1912, part 1, section 243, p. 240) WWW.EBOOK777.COM §1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS Conservation of Mass (Euler, Early 1 760s) dm(B)/dt = dm/dt = d/dt(^J dmj = d/dt pdV d/dt(pdV) = 0. (1.6.1a) (Henceforth, we shall, usually, omit the subscripts V, dV, etc., in the various integrals.) In the absence of discontinuities, the above leads to the local (differential) form: d/dt(p dV) = 0 =>■ pdV = constant = p 0 dV 0 [Material , or Lagrangean , or referential , equation of continuity ] (1.6.1b) where p 0 (dV 0 ) = density (element of volume) in some initial or reference configura¬ tion. Principle of Linear Momentum [Euler, 1750 (publ. 1752)] d/dt(^S v d n ^j = $ df or dp/dt=f , where p(B, t) = p = ^ v dm pv dV : Linear momentum of B , (1.6.2a) (1.6.2b) a system vector that depends on the frame, but not on the (fixed) origin in it; equivalent to Newton’s “quantitas motus”; and $ df =/. From the above, and invoking mass conservation [§1.3:(1.3.1)ff.), (1.6.1a, b)] and the definition of mass center (§1.4), we obtain p = mv G => ma G =f, (1.6.2c) where »” G /v G /a G are, respectively, the position/velocity/acceleration vectors of the center of mass of B, G. Equation (1.6.2c) shows that the motion of the center of mass G, of a body (or any material system, rigid or not), B, is identical to that of a fictitious particle of mass m located at G and acted upon by the body resultant on B,f; that is, by the vector sum of all (—> external) forces transported parallel to themselves to G. Thus, the motion of G is taken care of by this simple principle —> theorem. But the remaining problem of the motion of B about G (and, generally, of the motion of other body points) is far more difficult, and, unlike the motion of G, does depend on the specific material constitution of B (e.g., rigid, elastic), as well as on its motion (i.e., 1-, 2-, 3-dimensional); and, therefore, that problem necessitates additional con¬ siderations, such as the following. Principle of Angular Momentum [Euler, 1775 (publ. 1776)] d/dt(S(r> <v dm))= S( rxd f) or dH 0 /dt = M 0 , WWW.EBOOK777.COM (1.6.3a) CHAPTER 1: BACKGROUND where Ho{B , t) = Ho = $ ( r x vdm ): absolute angular momentum (or moment of momentum, or kinetic moment), about the fixed point 0, (1.6.3b) and M 0 = s (r x df): total moment about 0 (fig. 1.4). (1.6.3c) Other angular momenta, and their interrelations, are detailed in “Additional Forms of the Angular Momentum,” below. External and Internal Loads In the Newton-Euler approach to system mechanics, whether discrete or continuous, we classify body and/or surface forces and moments as internal or mutual (i.e., those due exclusively to internal causes) and external [i.e., those whose cause(s) lie, even partially , outside of the body or system]. Stresses are caused by one or more of the following: (i) deformations (solids); (ii) flows (gases, liquids); (iii) geometrical/kine- matical constraints [e.g., incompressibility, inextensibility (= incompressibility in one or two dimensions)]. Analytical mechanics necessitates a different force/moment classification (chap. 3). Principle of Action-Reaction (i) Discrete version. Let us consider a system of N particles {P k y k = 1,..., N}. Each particle P k is acted upon by a total external (to that system) force / k ext and a total internal force / ( r. inl due to the other N — 1 particles: fk.un = fkh with Iflk- i.e., f kk is, as yet, undefined(l) (1.6.4a) WWW.EBOOK777.COM §1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS Now, by Newton’s third law of motion ( action-reaction ) we shall understand the constitutive (i.e., physical) postulate; (a) f kl = —f lk and f kk = 0 (i.e., the particle cannot act on itself!) (1.6.4b) and (b) (r k — r/) x f kl = 0 (i.e., the internal forces are central and opposite', or oppositely directed pair by pair and collinear). (1.6.4c) [The second of (1.6.4b) is not included in the original Newtonian formulation. We follow Hamel (1949, p. 51).] In the discrete/particle model, so popular among physicists and such an anathema among certain mechanicians, this postulate, plus the principle of linear momentum, lead to the theorem of angular momentum for the external loads only. However, the converse is not necessarily true; that is, the angular momentum equation for a finite body dH 0 /dt = T / 0 externa i does not necessarily lead to (1.6.4b, c); other combinations of the internal forces may lead to the same effect (e.g., a sum of terms may vanish in a number of different ways). The converse may hold if we assume the validity of the angular momentum equation for any part of the system, or for any size subsystem. (ii) Continuum version. For every pair of particles P\ and P 2 , with respective positions iq and r 2 , the mutual forces and moments satisfy the following constitutive postulate: df (ti,t 2 ) = ~df (r 2 , fi) and dM(r\,r 2 ) = —dM{r 2 ,r\). (1.6.4d) Without (1.6.4d), or something equivalent that supplies knowledge of the internal loads, the problem of mechanics would, in general, be indeterminate (i.e., the adopted model would produce more unknowns than the number of scalar equations furnished by its laws). Additional Forms of the Angular Momentum Although the results derived below hold for any body or system, they become useful only for rigid ones. We define the following two kinds of (inertial) angular momen¬ tum (fig. 1.4): H. absolute = H. = £ (r - r.) x dm v = £ r /m x dm v: [v = dr/dt] Absolute angular momentum of body B, about the arbitrarily moving point • [because it involves the absolute (inertial) velocity v = dr/dt), (1.6.5a) and .relative = h » = S ^ ~ ^ X dm ( V “ = S * l* X dm V /•'' Relative angular momentum of body B, about the arbitrarily moving point • [because it involves the relative (inertial) velocity v — v. = v/.]. (1.6.5b) REMARKS (i) Although these kinematico-inertial definitions hold for any frame of reference (with r, r., v, v. denoting the positions and velocities relative to points fixed or moving with respect to that frame—see §1.7), they will normally be understood to refer to a specific inertial frame, unless explicitly stated to the contrary. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND (ii) Some authors define absolute angular momentum as in our (1.6.5a), but only for fixed points (i.e., v. = 0); in which case, clearly, (1.6.5a) and (1.6.5b) coincide. Unfortunately, here too, there is no uniformity of terminology and or notation in the literature; but, as will be seen in kinetics, some angular momenta are more useful than others. The connection between the above two angular momenta is given by the following basic theorem. THEOREM The angular momenta H. and h„ defined by equations (1.6.5a, b), are related by H. — It, = m(r G — r.) x v. = mr G /, x v.. (1.6.5c) PROOF Subtracting (1.6.5b) from (1.6.5a) side by side, and then utilizing the properties of the center of mass of B , G, we obtain H. - h. = g x (v - v/.) dm = £ {r /m x v.) dm = ^rx {dm v.) — ^ (r. x v.) dm = ( mr G ) x v. — r. x (mv t ), Q.E.D. (1.6.5d) Equations (1.6.5c, d) show immediately that, in the following three cases, the differ¬ ence between absolute and relative angular momentum disappears: (i) r G/ . = 0, i.e., • = G: H G = h G = gr /G x (, dmv /G ), (1.6.5e) (ii) v. = 0, i.e., • = fixed origin, say O: H 0 = h 0 = ^r x (dmv), (1.6.5f) (iii) r G /. parallel to v.. (1.6.5g) The first and second cases, (1.6.5e, f), are, by far, the most important; (1.6.5g) may be hard to check before solving the (kinetic) problem. Next, let us relate H. and It. with H a (which appears in the basic Eulerian form of the angular momentum principle). We have, successively, H o‘S' x {dmv) (introducing positions/velocities relative to •) = S [ 0 . + r/,) x dm (v. + v/.)] = ••• = /». + m{r. x v G ) + m{r G/ . x v.), (1.6.5h) — H. + m{r. x v G ) [thanks to (1.6.5c)]. (1.6.5i) The above leads easily to the following corollaries: (i) If • = fixed => r. — 0. then H 0 = H. + r. x {mv G ) = h. + m{r. x v G ) [r. = r./ 0 , H. = h,]; (1.6.5j) a slight generalization over (1.6.5f). (ii) If • = G, then H 0 = H G + r G x {mv G ) = h G + m{r G x v G ) [r G = r G/Oi V G = dr G /dt ; H G = h G ]. WWW.EBOOK777.COM (1.6.5k) §1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS By comparing (1.6.5h,i) with (1.6.5k), it can be seen that H. = H G + r G/ . x (mv G ), h. = H G + r G/ . x (mv G/ .). (1.6.51) (Interpret these “transfer” equations geometrically. What happens if • is fixed ; say, an origin 01) Finally, by applying the transfer equations (1.6.5h, i) between O and the arbitrarily moving points 1 and 2 , and then comparing, we can obtain the rela¬ tion between the absolute, relative, and absolute-relative angular momenta of a body! H\ *■—► H 2 5 *—* hi, h\ <—> hi- Additional Forms of the Principle of Angular Momentum With the help of the preceding kinematico-inertial identities/results, and the purely geometrical theorem of transfer of moments (hopefully well known from elementary statics) M. = M g + r G /, x / [where the force resultant / goes through G] = M g + r G j, x ( ma G ) [by the principle of linear momentum], (1.6.6a) the Eulerian principle of angular momentum ^rx {dm a) = d/dt[ Q r x {dm v)^ = ^ r x df\ that is, dH 0 /dt=M 0 (1.6.6b) [=> M 0 externa |, by action—reaction (plus, in the continuum version, of Boltzmann's axiom => symmetry of the stress tensor)], assumes the following forms: Center of Mass Form By (1.6.5k): dH 0 /dt = d/dt[H G + r G x {mv G )] = dH G /dt + m{r G x a G ), (1.6.6c) and by (1.6.6a), for • —> O: M 0 = M (j + r G x {ma G )\ (1.6.6d) and comparing these expressions with (1.6.6b), we obtain the fundamental form M g = dH G /dt (= dh G /dt). (1.6.6e) Absolute Form Using the above, and (1.6.51), we obtain, successively, M. = M g + r G/ . x {ma G ) = dH G /dt + r G/ , x {ma G ) = d/dt [H. - r G/ . x {mv G )\ + r G/ . x {ma G ) = dH./dt - v G/ . x (mv c ) - r G/ , x {ma G ) + r G/ . x (ma G ); WWW.EBOOK777.COM CHAPTER 1: BACKGROUND that is, finally, M. = dH./dt - v G/ . x ( mv G ) (using v G/ . = v G -v.) = dH./dt + v. x (mv G ) = dH./dt + v. x (mv G /.). (1.6.6f) Relative Form Similarly, using the above, and (1.6.51), we obtain, successively, M. = M g + r G/ , x ( ma G ) = dH G /dt + r G/ . x ( ma G ) = d/dt (h. - r G/ . x (mv G /.)) + r G/ . x (ma G ) = dh./dt - v G/ . x (mv G/ .) - r G/ . x (ma G/ .) + r G/ . x (ma G ); (1.6.6g) (1.6.6h) which, since it holds for any fixed point, is a slight generalization of (1.6.6b). These forms show clearly the importance of fixed points and of the center of mass, above all other points, in rotational dynamics, especially rigid-body dynamics. All these forms of the principle of angular momentum, and many more flowing from them, can be quite confusing, they are almost impossible to remember, and may be error-prone in concrete applications. They are stated here only for comparison purposes with the existing literature. From them, the most useful in both theoretical and practical situations, are, by far, (1.6.6b,e), and, secondarily, (1.6.6a) with (1.6.6e). We summarize them here: O'. fixed origin ; (1.6.6i) G\ center of mass; (1.6.6j) M. = dH G /dt + r G j, x (. ma G ), • : arbitrarily moving spatial point , (1.6.6k) or, compactly, Kinetic vectors (“torsor”) at G: ( ma Gl dH G /dt) ~Kinetic torsor at • : {ma G ,dH G /dt + r G /. x (. ma G )); and we are reminded that their left sides, by action-reaction (plus Boltzmann’s axiom, i.e., symmetry of stress tensor), include only external moments and couples. By comparing the absolute and relative forms of the principle of angular momen¬ tum, eqs. (1.6.6f, g) [or by d/dt(.. .), eq. (1.6.5c)], we can show that dH./dt — dh./dt = r G i, x (ma.) + v G /. x (mv.) r G/ . x (ma.) + v G x (mv.) = r G/ . x (ma.) + v G x (mv. /G ). (1.6.61) WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE Finally, crossing the local law of motion dma = df with r/. = r — r„ and then integrating over the body, etc., we obtain the following additional form of the principle of angular momentum: M. = dH 0 /dt — r. x ( ma G ) (= M 0 — r. x /, with / applied at •). (1.6.6m) 1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE (OR RELATIVE MOTION, OR MOVING AXES); ANGULAR VELOCITY AND ACCELERATION The theory of moving axes, a subject indispensable to rigid-body dynamics and other key areas of mechanics (including the transition to relativity), is based on the follow¬ ing fundamental kinematical theorem. Theorem (of Moving Axes) Let us consider two frames of reference in arbitrary relative motion, each represented by an ortho-normal-dextral (OND) basis and associated coordinate axes, rigidly attached to the frame; say, for concreteness but no loss in generality, one fixed or inertial F: (0 F - /, J , K/X, Y , Z) = (0 F - «r, u Y ,u z /X, Y, Z) = (0 F - u x ^ z /X, Y, Z) = (0 F - u k '/x k >; k' = 1,2,3 /X, Y,Z), (1.7.1a) and one moving or noninertial M : (0 M ~ h j,k/x,y,z) = (0 M - u x ,u y ,u z /x,y,z) = (0 M - u x ^ z /x,y,z) = (0 M ~u k /x k -, A-=1,2,3 /x,y,z), (1.7.1b) and an arbitrary (say free) vector p [fig. 1.5(a)]. Then its rate of change in F and M, dp/di and dp/dt, respectively, are related by dp/dt = dp/dt + a>xp, (1.7.2a) M (bj Figure 1.5 (a) Geometry of moving frames; (b) geometrical proof of (1.7.3c). WWW.EBOOK777.COM CHAPTER 1: BACKGROUND where (recalling the moving axes theory, §1.1) P = PxUx + PY U Y + Pz u z = Y Pk' U k' = PxUx+Pylty + Pz“z = Y PkU ^ [assumed instantaneous representation of p in F and M\; (1.7.2b) dp/dt = ( dp x /dt)u x + (dp Y / dt)u Y + (clp z /dt)u z = E 0 dPk'/dt)ti k '■ Absolute rate of change of p (or time flux)-, i.e., relative to F; (1.7.2c) dpldt = ( dp x /dt)u x + (i dpy/dt)u y + ( dp : /dt)u z = Y^ (dp k /dt)u k : Relative rate of change of p; i.e., relative to M; (1.7.2d) 01 = ! jJyUy UJyUy L0 Z U Z ~ ^ — CO v li x Ulyliy i0 Z U- = ^ ~ OJyUy = [(du r /dt) • u z \u x + [( du : /dt ) • u x ]u y + [(du x /dt) • u v ]u z : Angular velocity ( vector) of M movingframe relative to F Rxe d frame ; i.e., of ( 0 M — u k ') relative to ( 0 F — u k ); (1.7.2e) co x p = Transport rate of change of p relative to F. (1.7.2f) NOTATIONAL CLARIFICATION Here, partial derivatives, d(...)/dt, are, normally, associated with moving frame(s); while, for simplicity, primed subscripts signify fixed axes/components. To express this theorem in components, which is the best way to understand it, the simplest way is to choose the axes O f —XYZ and O m —xyz so that, instantaneously, either they coincide or are parallel. Then, since in such a case, (i dp/dt) x = {dp/dt) • Uy = dpx/dt = (dp/dt) ■ u x = ( dp/dt) x , etc., cyclically, (1.7.3a) (dp/dt) x = (dp/dt) • u x = dp x /dt = (dp/dt) • u x = (dp/dt) x , etc., cyclically, (1.7.3b) the theorem assumes the component form: dpx/dt = dp x /dt + u> r p- — ui-p Y , dpy/dt = dp y ,/dt + u> : p x — ui x p z , dp z /dt = dp : /dt + oj x p y — oj y p x ; (1.7.3c) and gives inertial rates of change, but expressed in terms of noninertial (relative) and transport rates. The above show clearly that (dp/dt) k f dp k /dt ( k = x, y, z ); (1.7.3d) even though, instantaneously, Px=Px, etc., cyclically, (1.7.3e) unless co x p = 0 (=> co = 0, or p = 0, or co parallel to p). WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE A geometrical interpretation of (1.7.3c) is shown in fig. 1.5(b): the moving axes Om— xyz momentarily coincide with the axes O m —XYZ\ the latter are always trans¬ lating relative to Op—XYZ —that is, they are “rotationally equivalent” to them. PROOF OF EQUATION (1.7.2a) By d/dt(.. .(-differentiating (1.7.2b), we obtain dp/dt = ( dp x /dt)u x + p x {du x /dt) + • • • = dp/dt + 5Z Pk{du k /dt). (1.7.4a) To transform the key second term in the above, we begin by d/dt(.. .(-differentiating the six geometrical orthonormality (=>• rigidity) constraints of these basis vectors u k ■ U/ = d ki ( k,l = x,y,z ), thus translating them into the following six kinematical constraints: (i du k /dt ) •/// + u k • ( duj/dt) = 0; (1.7.4b) that is, from the nine components of {du k /dt} only 9 — 6=3 are independent. Let us find them. By (1.7.4b) for k, l = x, du x /dt is perpendicular to u x , that is, it must lie in the plane of u r , Therefore, we can write du x /dt = l\u v + l 2 u z \ (1.7.4c) and, cyclically, du v /dt = l 2 u, + U u x , du z /dt = l^u x + l(,u y ; (1.7.4d) where /] 6 are scalar functions of time. Substituting these representations back into (1.7.4b) for k = x, l = y, and taking into account the geometrical constraints, we obtain {du x /dt) • u y + u x • (du v /dt) = 0 => ly + / 4 = 0; (1.7.4e) and, cyclically, (dUy/dt) • u : + u r • (du : /dt) = 0 => / 3 + l 6 = 0, (1.7.4f) (du z /dt) • u x + u z • (du x /dt) = 0 =>- l 5 + l 2 =0. (1.7.4g) Hence, (1.7.4c,d) can be rewritten in terms of the following three independent (unconstrained) /’s, or in terms of the three equivalent parameters w x , cj y , lj z : l\ = —I 4 = lo z , l 2 = —If, = u x , (5 = —U = w„ (1.7.4h) as du x /dt = Lu z u y — u> y u z = a) x u x , dUy/dt = ui x u z — u> z u x = to x u v , du z /dt = ui y u x — u) x u v = co x WWW.EBOOK777.COM (1.7.4i) CHAPTER 1: BACKGROUND where (O = U) X U X + UJyUy + jJ U- = u x [(diiy/df) • w z ] + u r [(du : /dt ) • h v ] + u : [(du x /dt) • /#,.] [a form that shows the cyclicity of the subscripts x, y, z] = —u x [(du z /dt) •«„] — u y [{du x /dt ) •«.] — u : [(du v /dt) •«*]. (1.7.4j) Finally, substituting these results into (1.7.4a), we obtain (1.7.2a): dp/dt = dp/dt + $>( 0 , x M 0 = dp/dt + uj x p k u^\ = dp/dt + uxp. (1.7.4k) REMARKS (i) Frequently, and with some good reason, the notation Sp/bt is employed for our dp/dt. Here, however, we chose the latter because in analytical mechanics 5{...) is reserved for virtual changes, under which St = 0 (chap. 2ff.). Other popular notations for the relative rate of change are d*p/dt (British authors; but some German authors use dp/dt for our u x p), ( dp/dt)M or (dp/dt) re i or d*p/df, or with a tilde over d (Soviet/Russian authors) d. Also recall remarks made regarding eq. (1.1.20i) about the overdot notation. (ii) The vector equation (1.7.2a) can be expressed in component form (i.e., it can be projected) along any axes, fixed or moving, by eqs. (1.7.3c), if 0 M —xyz and O m —XYZ momentarily coincide; and, if they do not, by ( dp/dt) x = cos(a;, X){dp x /dt) + cos(x, Y)(dp y /dt) + cos(x , Z)(dp z /dt) dp x /dt) = cos(x, X)(dpi/dt + W2P3 ~ ^ 37 * 2 ) + • • •, (1.7.5a) where the new axes <9^—123 coincide momentarily with Om—XYZ , but, in general, have an angular velocity m' = (w 1 ; 022 , 023 ) relative to them. (iii) The above show that as long as no rates of change are involved, the compo¬ nents of a vector along the various axes (fixed or moving) are related by ordinary coordinate transformations, with possibly time-dependent coefficients — that is, like the first line of (1.7.5a), or (1.7.5b), below; all such axes are mechanically (though not mathematically) equivalent. But when rates of change between such moving axes (—> frames) are compared, then, in general, a component of a vector derivative ( dp/dt) x does not equal the derivative of that component dp x /dt [(1.7.3d, e)]; these quantities are related by a frame of reference transformation — that is, like the second line of (1.7.5a). Mathematically, this is equivalent to an explicitly time-dependent coordinate transformation: a; = x(X, Y,Z;t),... X = X(x,y, z; t),... (recall dis¬ cussion following eq. (1.1.20k)). In such cases, to obtain equations like (1.7.3c), we begin with O m —XYZ and 0 M —xyz in arbitrary relative orientations, then we d/dt(.. ^-differentiate the component transformations, like p x = cos (x,X)p x + cos(a:, Y)py + cos(x, Z)p z , etc., cyclically, (1.7.5b) {not like p x = p x ) and then we make O m —XYZ and 0 M —xvz coincide. (iv) In kinematics, all frames are theoretically equivalent; and thus during the 17th century both Galileo and the Catholic church were ... kinematically correct! This is WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE expressed by the following geometrical, or Euclidean, and kinematical principle of relativity, any system of rectangular Cartesian coordinates can be replaced by any other such system that moves in an arbitrary fashion relative to the first; or, alternatively, the form of geometrical relationships must he invariant under the proper orthogonal group of rotations —and this, in effect, constitutes a definition of Euclidean geometry—that is, any two such sets of coordinates x k > and x k are related by x k' = A k'k(t)x k + (1.7.6a) where A k'k( 0 A k'l (0 = t>kh and Det(^ fc (0) = +1, (1.7.6b) and A k ' k (t), A k >(t ) are continuous functions of time, with first and second time derivatives. Such transformations include all frames/motions produced from the moving frame M by a continuous rigid-body movement (translations and rotations, but not mirror reflections). (v) If the moving triad u xyz is non-OND, then its inertial angular velocity is, instead of (1.7.4j), o) = [u x \{du y /dt) • u : ] + u v [{du~/dt) • u x \ + u : [{du x /dt ) • u y ]} J[u x • (u y x «_)]. (1.7.6c) [See, for example, Truesdell and Toupin (1960, p. 437). In case such angular velocity vector definitions seem unmotivated, another more natural one, based on the linear¬ ization of the finite rotation equation, is detailed in §1.10.] Corollaries of the Moving Axes Theorem Applying (1.7.2a) for a>, we get duo /dt = dus/dt + u) x u) = du>/dt = cx. Angular acceleration of moving axes relative to fixed axes. (1.7.7a) This result shows the special position of to in moving axes theory. From eq. (1.7.2a) and its derivation, we easily obtain the following general opera¬ tor form: d{.. .)/dt = d(.. -)/dt + u; x (...), (...): any vector. (1.7.7b) Applying (1.7.7b) to (1.7.2a), and invoking (1.7.7a), we obtain the following expres¬ sion for the second absolute rate of p, d/dt{dp/dt) = d 2 p/dt 2 : d 2 p/dt 2 = d(.. .)/ dt(dp/dt + u> x p) = [<9(. ..)/dt + us x (.. ,)\(dp/dt) + ( du/dt) xp + ux ( dp/dt) = ... = d 2 p/dt 2 + [a xp + co x (n> x/?)] + 2co x (dp/dt), WWW.EBOOK777.COM (1.7.7c) CHAPTER 1: BACKGROUND where d 2 p/dt 2 = ( d 2 p x /dt 2 )u x + ( d 2 p y /dt 2 )u y + ( d 2 p z /dt 2 )u z . (1.7.7d) In general, if a —> b = da/dt —> c = db/dt = d^a/dt 2 then we shall have for their components: b x = b x = da x /dt + cu y a z — u> z a y , (1.7.7e) c x = c x = db x /dt + ui v b z — u z b v = d/dt(da x /dt + u> v a z — co z a y ) + u y {da z /dt + u> x a y — u> y a x ) — LO z (da v /dt + u z a x — to x a z ), etc., cyclically. (1.7.7f) For example, application of (1.7.7c, d) to the moving basis vectors u xy z yields d 2 u x /dt 2 = d 2 u x /dt 2 + [a x u x + oj x (oj x u x )\ +2 uj x ( du x /dt ) = 0 + [a x u x + co x (e> x u x )\ + 0 = atxu x + a)x((ox u x ), etc., cyclically. (1.7.7g) Since (1.7.2a) is a purely kinematical result, the roles of the frames F and M can be interchanged. Indeed, from it, we immediately obtain dp/dt = dp/dt + (— cj) x p , (1.7.7h) where —co is the angular velocity of F relative to M. In particular, if p remains constant (i.e., fixed) relative to F, (1.7.2a) and (1.7.7h) yield dp/dt = (- u) x p ; (1.7.7i) that is, an observer, stationed in M, sees the tip of p rotate relative to that frame with an angular velocity —co. Application of (1.7.7i) to the fixed basis u X j^z gives dux/dt = (-u>) x u x = -(oj x ,ojy,u} Z ) x (1, 0, 0) = • • • = (0)«x + (—wz)«r + (ojy)u z , du Y jdt = (-iv) X Uy = ■ ■ ■ = ( UJ Z )u X + (0 )u V + (-UJ X )u Z , du z /dt = (-uj) xu z = ■■■ = (-w r )«x+ (wx)«r + (0 )u z ; (1.7.7j) and, therefore, (dux/dt) -Uy = —uj z (= —w z , for coinciding axes) ( dux/dt ) • u z = +uj y (= +uJ y , for coinciding axes), etc., cyclically. (1.7.7k) Alternative Definition of Angular Velocity (i) Below, we show that co = (1/2) \uk x (diik/dt)] (where k = 1,2,3 —> x,y,z), (1.7.8a) which can be viewed as an alternative to (1.7.2e, 6c) definition of angular velocity. WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE Indeed, using the fundamental equations (1.7.4i), we obtain, successively, [»fc x ( du k /dt)\ = ^2 [u k x (ox u k )\ = ^2 [(u k ■ u k )(o - (. u k • (o)u k )\ = (“fc • «*)) - 51 = ro ( 3 ) - ffl = 2<u, Q.E.D. (1.7.8b) From the above, and using the results of §1.1, we can show that the (inertial) angular velocity tensor of the moving frame to [i.e., the antisymmetric tensor whose axial vector is the (inertial) angular velocity of that frame to] can be expressed as to = (l/2)]T \(du k /dt) (g) u k — u k 0 (du k /dt )]. (1.7.8c) (ii) Next, if the (orthonormal) basis vectors u k are functions of the curvilinear coordinates q = (</|, q 2 , qfi —that is, u k = u k (q) — then, applying (1.7.8a), we find, successively (with all Latin subscripts running from 1 to 3; i.e., x,y,z), = 0/2){«, x (,du k /dqi){dqi/dt )) ^ c k (dq,/dt ), (1.7.9a) where ci = ^2 (1/2) [life x (du k /dqi)\ (“Eulerian basis” for to); (1.7.9b) that is, the dqj/dt are the (contravariant) components of to in the (covariant) basis ty. By formally comparing (1.7.8a) and the earlier equations (1.7.4i), (1.7.2e, 4j), with (1.7.9a, b) [i.e., to —> c t and du k /dt —> du k /dqj\, it is easy to conclude that duk/dqi = ci x u k , (1.7.9c) c, = u 1 [(du 2 /dqi)-u 3 ] + u 2 [(du 3 /dq,) ■ i/j] + u 3 [(du x /dq,)-u 2 \. (1.7.9d) We leave it to the reader to extend the above to the “rheonomic” case: u k = u k (q , t). EXAMPLES 1. The absolute (i.e., inertial) components of the angular acceleration of a rigid body rotating with angular velocity co B are (with the hitherto used notations) duB,x/dt = du Bx /dt + uiy lo Bz — ui z u> B v , etc., cyclically. (1.7.10a) What happens if a> B = wl 2. The conditions for a straight line with direction cosines (relative to moving axes) 4, /,,, l : to have a fixed inertial direction are dl x /dt + uj y l z — 'jj 7 l y = 0, etc., cyclically. (1.7.10b) How many of these three conditions are independent? Hint: l x 2 + l 2 + l 2 = 1. 3. The moving axis theorem (1.7.2a), applied to the generic vector p expressed in plane polar coordinates'. p=p r u r +pfill'/,, (1.7.10c) yields dp/dt = [dp r /dt — pfidtj)/dt)\u r + [dp^/dt + p r (d(f>/dt)\u rj> . (1.7.lOd) WWW.EBOOK777.COM 119 CHAPTER 1: BACKGROUND Apply (1.7.10d) for p = position vector of a particle r, and velocity vector of a particle v. Hint: The angular velocity of the moving polar ortho-normal-dextral triad u r,<j>.:=z > relative to the inertial one u XY7 , is at = (d(j)/dt)u : = ( d<f>/dt)u z . (1.7.lOe) Particle Kinematics in Moving Frames Velocities Application of the fundamental formula (1.7.2a) to the motion of a particle P, of inertial position vector 9? = v 0 + r (fig- 1-6) (i.e., for p —> r), yields v = dR/dt = d{r 0 + r)/dt = dr Q /dt + dr/dt = dr 0 /dt + (dr/dt + oxr), (1.7.11a) (since, in general, r is known only along the moving axes) or, rearranging, v = (dr 0 /dt + (ox r) + dr/dt (1.7.11b) or *’abs = Ttrans “F r re l, (1-7.11 c) where v abs = v = d^R/dt = (dX/dt)u x + ( dY/dt)u Y + ( dZ/dt)u z : Absolute velocity of P, (1.7.lid) v re i = dr/dt= ( dx/dt)u x + ( dy/dt)u y + ( dz/dt)u : : Relative velocity of P, (1.7.1 le) vtrans = dr 0 /dt + a) x r = dr 0 /dt + \x{du x /dt) + y(du r /dt) + z(du z /dt)]: Transport velocity of P. (1.7.Ilf) Figure 1.6 (a) Relative kinematics of particle P in two dimensions; (b) geometry of centripetal acceleration. WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE Clearly, if P is rigidly attached to M moving frame (e.g., if it is one of the particles of the rigid body M), then v rel = 0 and v = v trans ; that is, generally, v trans is the velocity of a particle rigidly attached to M and instantaneously coinciding with P. Accelerations Application of (1.7.2a) to (1.7.1 la-f) yields Aibs Ad A Arans A Aon where (1.7.12a) Abs = « = d 2< H/dt 2 = {d 2 X/dt 2 )u x + ( d 1 Y/dt)u Y + (d^ Z / dt 2 )u z : Absolute acceleration of P, (1.7.12b) Aei = dv K \/dt = d 2 r/dt 2 = ( d 2 x/dt 2 )u x + ( d 2 y/dt 2 )u y + (d 2 z / dt 2 )u z \ Relative acceleration of P, (1.7.12c) fl trans = d 2 r 0 / dt 2 + X X r + CO X (co x r) = d 2 r 0 /dt 2 + [. x(d 2 u x /dt 2 ) + y(d 2 u v /dt 2 ) + z(d 2 u : /dt 2 )\'. Transport (or drag) acceleration of P [ = Inertial acceleration of a particle fixed relative to M, and momentarily coinciding with P; its first term, d 2 r 0 /dt 2 = dv 0 ldt = dvo/dt + ojxvo, is due to the inertial acceleration of the origin of M\ its second , a x r, to the inertial angular acceleration of M; and its last term, co x (co x r) = (co ■ r)co — ui 2 r = —u) 2 r pi where r p = vector of perpendicular distance from co — axis (through O) to P, (fig. 1.6(b)),is called centripetal acceleration of T 5 ], (1.7.12d) A or = 2 to x v re i = 2a) x ( dr/dt ) = 2 \fdx/dt)(du x /dt) + (dy / dt)(du y /dt) + {dz / dt){du z / dtf): Coriolis (or complementary) acceleration of P [ = Acceleration due to the coupling between the relative motion of the particle P, v rel , and the absolute rotation (transport motion) of the frame M, co; it vanishes if v rel = 0, or if co is parallel to v re j]. (1.7.12e) If co = 0 and a = 0 — that is, if M translates relative to F — these equations reduce to v = v re i + Vo = dr/dt + dro/dt = dr/dt + dro/dt, (1.7.12f ) a = Ae l +«o = d 2 r/dt 2 + d 2 r 0 /dt 2 = d 2 r/dt 2 + d 2 r 0 /di 1 , (1.7.12g) WWW.EBOOK777.COM 122 CHAPTER 1: BACKGROUND Component Forms To appreciate eqs. (1.7.11) and (1.7.12) better, and prepare the reader for the key concept of nonholonomic coordinates , and so on (§2.9 If.), we present them below in terms of their components. In the general case of nonaligned axes we can project them on an arbitrary, fixed, or moving axis; that is, each of their terms can be resolved along any set of axes. (i) The position relation 9? = r a + r, with r 0 = (X 0 , Y 0l Z 0 ), reads X = X Q + cos(X, x)x + cos(3f, y)y + cos)!", r)z, etc., cyclically. (1.7.13a) (ii) The velocity equations (1.7.11a ff.) assume the following forms, along the fixed axes: dX/dt = dX 0 /dt + cos(T", x)(dx/dt + uj y z — uj z y) + cos (X,y)(dy/dt + u : x — ui x z) + cos(3f, z)(dz/dt + u) x y — u v x) = dX 0 /dt + d / dt(X — X 0 ), etc., cyclically; (1.7.13b) and, along the moving axes: v • u x = v x = v 0x + dx/dt + u) y z — oj z y, etc., cyclically, (1.7.13c) where v Qx = v Q • u x = cos (x : X)(dX Q /dt) + cos(x, Y)(dY 0 /dt) + cos (x,Z)(dZ Q /dt): component of inertial velocity of moving origin O , along the moving axis Ox [in general, not equal to the d/dt{.. .)- derivative of a coordinate , like dX Q /dt or dx/dt , and hence a quasi velocity (§2.9ff.)], etc., cyclically. (1.7.13d) (iii ) The acceleration equations (1.7.12a IT.) read, along the fixed axes: d 2 X/dt = d 2 X 0 / dt 2 + cos(X, x) [d/dt{dx/dt + uj y z — u> z y) + uj y (dz/dt + yco x — xco y ) — uj z (dy/dt + xu> z — zuj x )\ + ••• = d 2 X 0 /df + cos(3f, x){(d 2 x/dt 2 ) + [z(du y /dt) - y{dui z /dt)] + uj y (u x y — LO y x) — oj z (lo z x — oj x z) + 2[co y (dz/dt) — u) z (dy/dt)]} + ■■■ = d 2 X 0 /dt 2 + d 2 /dt 2 (X — X 0 ), etc., cyclically; (1.7.13e) = (d 2 X/dt\ el + (d 2 X/dt\ ans + (d 2 X/dt 2 ) cm , (1.7.13f) where ( d 2 X/dt 2 ) tel = cos {X, x){d 2 x / dt 2 ) + cos (X,y)(d 2 y/dt 2 ) +cos (X,z)(d 2 z/dt 2 ), (d 2 X/dt 2 ) tans = d 2 X 0 /dt 2 + cos(X,x) {[z{du] y /dt) - y(dio z /dt)\ + tu y (u} x y — oj y x) — cj z {uj z x — oj x z )} + • • •, (d 2 X/dt 2 ) cor = cos(X, x){2[u} y {dz/dt) - ut z (dy/dt)\ } + • • •, etc., cyclically; (1.7.13g) WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE and, along the moving axes: a • u x = a x = a Qx + [d/dt(dx/dt + u> y z — u z y) + ai y {dz/dt + vw , — xuj y ) — u> z {dy/dt + x uj z — zuj x )\ FC.rel “b dv.trails “1” dr.cor■ (1.7.13h) where ci x j*0j d jc jdt , «x,trans = «o,.x-+ [z(du y /dt) -y(du z /dt)\ +U)y(u x y-UyX) -uj z (cj z x-uj x z), cor — 2 [uj y (dz/dt) — u> : (dy/dt)], and (1.7.13i) a 0 x = ao • u x = cos (x,X)(d 2 Xo/dt 2 ) + cos(x, Y)(d 2 Y 0 /dt 2 ) + cos (x,Z)(d 2 Z 0 /dt 2 ), (in general, a quasi acceleration ), etc., cyclically. (1.7.13j) EXAMPLES 1. It is not hard to show that the conditions for a particle, with coordinates x, y, z, relative to moving axes, to be stationary relative to absolute space are u + dx/dt + zujy — yuj z = 0, etc., cyclically, (1.7.14) where («, v, w) = inertial components of velocity of origin of moving frame. 2. Plane Rotation Case. Let us find the components of velocity and acceleration of a particle P in motion on a plane described by the two sets of momentarily coincident rectangular Cartesian axes, a fixed O XY and a second O xy rotating relative to the first so that always OZ = Oz, with angular velocity to = (0,0, u> z = u> z = u). Here, momentarily, X = x, Y=y. (1.7.15a,b) Application of the moving axes theorem (1.7.2a), or (1.7.3c), (1.7.7e), with co XJ = 0 and c o z = w, yields the velocity components: dX/dt = dx/dt — yu>, dY/dt = dy/dt + jcw; (1.7.15c, d) and application of that theorem, or (1.7.3c), (1.7.7f), to the above gives the accel¬ eration components: d 2 X/dt 2 = d / dt(dx / dt — y u) — (dy / dt + x uj)u = d 2 x/dt 2 — y(duj/dt) — xui 2 — 2 {dy/dt)co (= relative + transport + Coriolis), (1.7.15e) d 2 Y/dt 2 = d/dt(dv/dt + xco) + ( dx/dt — yu)ui = d 2 y/dt 1 + x(du)/clt) — yu 2 + 2 {dx/dt)cj ( = relative + transport + Coriolis)', (1.7.15f) and similarly for higher d/dt{.. ^-derivatives. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND [Alternatively, we may start from the geometrical O-XY/O-xy relationship for a generic angle of orientation 0 = 0(t): X = (cos 4>)x + (— sin f)y, Y = (sin0)x + (cos0)_v, (1.7.15g) d/dt{.. .)-differentiate it, and then set 0 = 0 (df/dt = u f 0), thus obtaining (1.7.15c, d); then d/dt(.. .)-differentiate once more, for general 0, and then set 0—0 (to f 0, duj/dt = a f 0), thus obtaining (1.7.15e, f). The details of this straight¬ forward calculation are left to the reader. In this way we do not have to remember any kinematical theorems—differential calculus does it for us!] 3. Velocity and Acceleration in Plane Polar Coordinates via the Moving Axes Theorem [continued from (1.7.10c—e)]. Here, with the usual notations, r = ru,. and m = (d(/>/dt)u z = (d(f>/dt)u z , (1.7.16a) and, therefore, by direct d/dt (...)-differentiation and then use of (1.7.4i) — that is, treating the corresponding OND basis/axes through P , P — u r ii Ci) /r, as the moving frame — we obtain (i) v = dr/dt = ( dr/dt)u r + r(du r /dt) = ( dr/dt)u r + r(m x */,.) = ( dr/dt)u+ r{d(/)/dt)(u z x u r ) = ( dr/dt)u r + r{d(/>/dt/u^ = v r u r + rv^u^\ (1.7.16b) (ii) a = dv/dt = ( dv r /dt)u r + v r (du,./dt) + [d/rv^/dt/ju^, + (rv^/du^/dt) = ( dv,./dt)u r + vfco x u r ) + [d(rv <j> )/dt)]u (j> + (rv 0 )(<n x uf) = ( dv r /dt)u r + v r [{d(j)/dt)uf\ + [dirvfj / 'dt))u^ + {rv^-df/ dt)u,] = [ dv,./dt — {df/df/rvffu,. + [vfdf/dt) + d{rv^)/dt\u^ = [d 2 r/dt 2 — r{d(/>/dt) 2 ]ii r + {(dr / dt)/d(f>/dt ) + d / dt[r(d(f>/dt)}} u^, = [ d 2 r/dt 2 — r(d(f>/dt) 2 ]n r + \2{dr / dt){d(j) / dt) + r(d 2 (j)/dt 2 )\u^ = a [r) u r + (1.7.16c) 4. Velocity and Acceleration in Spherical Coordinates via the Moving Axes Theorem. Proceeding as in the preceding example, and since here r = ru,. (not the r of the polar cylindrical case) and to = (dc/>/dt)u z + (dO/dt/u^, u z = — sin 9u g + cos #«,., we can show that the velocity and acceleration are given, respectively, by v = ( dr/dt)u r + [r/dd/dt)]u g + [r/df/dt) sin^Ji/^ = v,.u r + rv g u g + v^u^, (1.7.17a) a = [ d 2 r/dt 2 — r(d9/dt)~ — r/df/dt)" sin 2 9\u r + [2/dr/dt){d9/dt) + r{d 2 9/dt 2 ) — r/dfi/dt) 2 sin#cos#]M fl + [ 2{dr/dt){d(/)/dt ) sin# + r{d 2 cj)/dt 2 ) sin# + 2r{dcj) / dt){d9 / dt) cos#]h^, = a {r) u r + a m u g + a^u^. (1.7.17b) The above are, naturally, in agreement with (1.2.8a ff.) WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 5. Inertial Angular Velocity of the Natural, or Intrinsic, OND Triad 0 M —u t u„Ub = 0 M —tnb\ Fre.net Secret Equations (fig. 1.7). We have already seen (§1.2) that dt/ds = n/p => dt/dt = (dt/ds)(ds/dt) = [(ds/dt)/p)n = (v t /p)n , (1.7.18a) also b = t x n. (1.7.18b) Next, d/dt( (. . .(-differentiating b-t = 0, we obtain 0 = ( db/dt ) • t + b • (dt/dt) = (db/dt) •t + b • [(v t /p)n] = ( db/dt ) • t\ (1.7.18c) and, similarly, d/dt(.. .(-differentiating b ■ b = 1 we readily conclude that (db/dt) -h = 0. (1.7.18d) Equations (1.7.18c, d) show that db/dt must be perpendicular to both t and b. Hence, we can set db/ds = —(1 /r)« =>■ db/dt = (db/ds) (ds/dt) = —(v t /r)n, (1.7.18e) where r = radius of torsion (or second curvature) of the curve C, traced by the moving origin 0 M = O, at O: positive (negative) whenever the tip of db/dt turns around t positively (negatively); that is, like a right- (left-)hand screw; or, according as db/dt has the opposite (same) direction as n. [Some authors use r for our 7/r; others use p K and p T for our p and r, respectively.] Now, the angular velocity of O tnb, relative to some background fixed triad Op—UxUyUz, is found by application of the basic formulae (1.7.4j), with the identi¬ fication 0 M —u x u y u z = O—tnb , and eqs. (1.7.18a-e). Thus, we find Tangent, u, —> u> x = u z • (du y /dt) = —u r • (du z /dt) = —n • (db/dt) = v ( /t; (1.7.1 8f ) Normal: u> n —> „ = u x • (du z /dt) = —• (du x /dt) = —b • (dt/dt) = 0; (1.7.18g) Binormal: u h —> co z = u y • (du x /dt) = — u x • (du y /dt) = n • (dt/dt) = v t /p. (1.7.18h) Binomial (b, z) Figure 1.7 On the geometry and kinematics of the Frenet-Serret triad O-tnb. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND In sum, the triad O-tnb rotates with inertial angular velocity: ( 0 = (v,/t )t + (0 )n + ( v,/p)b = v,(t /t + b/p). (1.7.18i) In general, a> ^ d6/dt, where 9 is some vector expressing angular displacement/rota- tion; that is, 6 is a quasi vector (§1.10, chap. 2). Further, from (1.7.18a-e) we also conclude that dn/ds = d/ds(b x/) = ( db/ds ) x t + h x ( dt/ds ) = “(I M(n Xt) + (1 /p)(b x n) = -(1 M(-b) + (l/p)(-f) = (-l/p)/+ (l/r)4. (1.7.18j) Equations (1.7.18a, e, j) (where 0 ^ p ^ +oo and —oo ^ r ^ +oo, r ^ 0) are the famous Frenet-Serret formulae for a space (or skew, or twisted ) curve. It is shown in differential geometry that: the “natural/intrinsic” curve equations p = p{s) and r = r(s) determine the spatial position of that curve to within a rigid displacement (i.e., a translation and a rotation). The F-S equations can also be written in the following memorable “antisymmetric form”: dt/dt= (0)f+(v f /p)n+(0)A, (1.7.18k) dn/dt = (— v t ] p)t + (0 )n + ( v t /r)b , (1.7.181) db/dt = (0)f + (— v t /r)n + (0 )b. (1.7.18m) The above allow us to calculate the torsion, l/r. From (1.7.18j, k, 1), with (...)' = d(.. .)/ds, we get b/T = t/p+ ( pt ')' = t/p + p't' + pt" = t/p + p'(n/p) + pr"', (1.7.18n) and so, dotting this equation with b , we find 1 j T = p(b ■ r"') = p[(t x n) • r'"] = p 2 [(r' x r") ■ /"], (1.7.18o) or, since [recalling (1.2.4c)] 1/p 2 = r" • r" = \r r '\ 2 , (1.7.18p) finally, Torsion = l/r= [(/ x r") • r"']/\r"\ 2 . (1.7.18q) With the help of the above, we can easily show that (i) The Frenet-Serret equations can be put in the following kinematical form: dt/dt = a) x t, dn/dt = a> x n, db/dt = a> x h (tu: kinematical Darboux vector, (1.7.18i)). (1.7.19a) (ii) If t, n , b can be expressed, in terms of their direction cosines along a fixed OND triad, as t={t\,t 2 ,h), n= (n u n 2 ,n 3 ), b = (b u b 2 ,b 3 ), (1.7.19b) then dt\/ds = n { /p , dni/ds = b\/r — t\/p, db\/ds = — «i/t; and similarly for the other components. WWW.EBOOK777.COM (1.7.19c) §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE (iii) The (inertial) angular acceleration of the F renet Serret triad, a = dm/dt, is given by a = [(dv t /dt)/r — (v 2 /T 2 )(dr/ds))t + [( dv t /dt)/p — (v 2 / p 2 )(dp/ds)\b. (1.7.19d) (iv) The (inertial) jerk vector of a particle, j = da/dt (or hyperacceleration, or velocity of the acceleration) is expressed along the Frenet-Serret triad as j = [d 2 v,/dt 2 - ( v, 3 /p 2 )\t + {v, 2 [d/dt{\/p)} + 3 v,{dv,/dt)/p)]}n + (v 3 / pr)b = [d 2 v,/dt 2 - (v 3 /p 2 )\t + [d/dt(v 3 /p)/v,]n+(v 3 / P T)\h = [d 2 v,/dt 2 - {v, 3 /p 2 )\t + [(3v 2 /p){dv,/ds) - (v, 3 / p 2 ){dp/ds)]n + (v 3 / pr)]b, (1.7.19e) where d(.. .)/dt = [d(...) / ds](ds / dt) = . .)/ds]\ that is, contrary to the acceleration, a = (dv,/dt)t + (v 2 /p) n, the jerk vector has t, n, and b components, and involves both p and r. (v) The following kinematic formulae hold for the curvature and torsion: k = l/p = |v x «|/|v| 3 = [v 2 a 2 — (v-a) 2 ] X 2 /v 3 , 7/r = [v (a xj)\/{a xjf = (v, a,j)/n 2 v 6 . (1.7.19f) HISTORICAL The theory of accelerations of any order (along general curvilinear coordinates) is due to the Russian mathematician/mechanician Somov (1860s), who also gave recur¬ rence formulae, from the (n — l)th order to the («)th order; and to the French mathematician Bouquet (1879). The second order shown above is due to the French mechanician Resal (1862), although the earliest such investigations seem to be due to a certain Transon (1845) (see, e.g., Schonflies and Griibler, 1902: 1901 1908). The jerk vector is called “acceleration du second ordre” (Resal), or "Beschleunigung a (2 *” (Schonflies/Grubler), where the ordinary acceleration (of the first order) is denoted by a : 1J = a. Clearly, such derivations are enormously aided with the use of vectors. These results allowed Mobius (1846, 1848) to give a geometrical interpretation to Taylor’s expansion (with some standard notations): Ar = r(t) - r(0) = vt + a (,) (t 2 /2) + « (2) (t 3 /1.2.3) + • • • = chord of particle trajectory between the times 0 and t. Particle Kinetics in Moving Frames Substituting the inertial acceleration a of a particle P of mass m, in terms of its moving axes representation, into its Newton-Euler equation of motion ma= f (= toted noninertial , or real, or objective, force on P ), (1.7.20a) and, rearranging slightly, we obtain its fundamental equation of relative motion (fig. 1 . 8 ): ma re \ f T ./'trails + ./cor ' WWW.EBOOK777.COM (1.7.20b) CHAPTER 1: BACKGROUND in words: mass x relative acceleration (« re i) = total real (/) plus inertial (/ lrans + /cor) f orce i (1.7.20c) where a re i = dv re \/dt = d 2 rldt 2 : apparent or Relative acceleration ofP, (1.7.20d) / trans = — wa tra ns = — m[d 2 r 0 /dt 2 + os x r+ a> x (to x r)]: total iniertial force of Transport on P = —m(d 2 i’ 0 /dt 2 ) [due to the inertial acceleration of the origin of the moving —m(<x x r) [due to the inertial angular acceleration of frame M\ — m[co x (to x r)] = —m[(a> • r)a> — ufr] = ■ ■ ■ = mafrp [centrifugal force on P, due to the inertial angular velocity of frame M; always perpendicular to the instantaneous axis of to, in the plane of P and that axis, and directed away from it (fig. 1.6(b))], /cor = ~ ma coT = -2 m(co x v re i) = -2 m[u x (dr/dt)\ : inertial force of Coriolis (or composite centrifugal force ) on P [due to the interaction of the relative motion of P (v re i = dr/di) with the absolute rotation of the moving frame (to); normal to both v rel , to, and such that v re i, to, and/ cor = 2 m(v rt \ x to), in that order, form a right-hand system]. (1.7.20f) WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE REMARKS (i) In classical mechanics, only / is a frame independent, or objective (or absolute) force; / trans and / cor are relative (i.e., frame dependent). At most, / can depend explicitly on relative positions (displacements), relative velocities, and time; but not on relative accelerations (as an independent constitutive equation). [In relativity all forces are relative, and hence can be eliminated by proper frame choice. On the classical objectivity requirements for /, see, for example, Pars (1965, pp. 11-12), Rosenberg (1977, pp. 12-16).] In addition, in general, the relative forces are not additive ; for example, the total force acting on a particle P due to two or more attracting masses, each exerting separately on it the absolute forces f x and/ 2 , equals if 1 ~b fl) A ( f trans ~b f cor) > if 1 ~b ./'trails “b f cor) ~b ( f 2 “b f trans “b ./cor) ■ -^S for the Coriolis “force”/ C0r = —2m(to x v rel ), even for the same problem (i.e., same m and/ ) that term obviously does depend on the particular noninertial frame used. This, how¬ ever, does not mean that its effects on people, property, and so on, are any less physi¬ cally/technically real than those of the real force/. [In fact, the study of such similarities between these forces led to the general theory of relativity (mid-1910s).] For the comoving (noninertial) observer, both / trans and/ cor are very real! Some of the most spectacular Coriolis effects occur in the atmospheric sciences (meteorology, etc.); that is, in phenomena involving the coupling between the motion of large liquid and/or gas masses and the Earth’s rotation about its axis. A prime such example is Baer’s law of river displacements: The inertia force on the northbound flowing water, along a meridian, presses against the right (left) bank in the northern (southern) hemi¬ sphere. The effects of this pressure are a stronger erosion of the right embankment; and a slightly but measurably higher water level at the right shore of the river. [In view of these realities, statements like the following cannot be taken seriously: “From the foregoing it is clear that the Coriolis-acceleration term arises from the description adopted, namely, via moving observers, and hence, contrary to popular belief it bears no physical significance ” [Angeles, 1988, p. 74 (the italics are that author’s)].] Finally, since / cor is perpendicular to v rel , its “relative power” / cor • v rel vanishes (more on such “gyroscopic forces” in §3.9). (ii) In the case of a finite body, v rel (a re i) in (1.7.20b) refers to the relative velocity (acceleration) of its center of mass G; and r is the position of G relative to the origin of the moving frame. Power Theorem in Relative Motion This constitutes the vector/particle form of theorems treated in detail in §3.9. Let us consider a system S in motion relative to the noninertial axes O xyz. To find its power equation in relative variables, we start with the equation of relative motion of a generic particle P of S, of mass dm [recall (1.7.20b fif.)] dm tf,. e | df 4“ r//'trans “b df co r> (1.7.21a) where 0 ,-ei = dv re i/dt = d 2 r/dt 2 , (1.7.21b) df = df + dR ( impressed + constraint reaction —see §3.2) (1.7.21c) ‘//'trans = —dm « tran s — —dm [d 2 r 0 /dt 2 + at x r + co x (tax r)], (1.7.21 d) df COI = —dma cor = —2 dm(yo x v re j) = —2 dm[u) x (dr/dt)\ (1.7.21e) 129 WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Now, the system power equation corresponding to the particle equation (1.7.21a) is S dm fl rel • Cel = S ^ ' Vrel + S ^rel ' Cel + S rf /cor ' Cel- (1.7.21f) Let us transform each of its terms: (i) S dm «rel • Vrei = S dm Cel ‘ (^Cel / dt) = d/ <9r( S ( 1/2 ) dm Cel • Vrei) = dT rel /dt, (1.7.21 g) or, since Vrei • {dv K \/dt) = Vrei 1 (dv K \/dt + UJ X V re l) = V re l • (<9v re l/ dt) , finally, S dm Orel • Vrel = dT rc] /dt = dT le \/dt. (1.7.21 h) (ii) We define S df‘ Cel = S d f ■ (dr/dt) = d'W/dt ; (1.7.21i) where, in general, no W exists (i.e., W is a quasi variable — more on this in §2.9). If $ dR • v re i = 0 , then d'W/ dt = $ dF • v re i. (iii) Clearly, S d f cot • Vrel = S [ - 2 dm (® X V re l)] • V re l = 0. (1 -7-2 lj) (iv) S d ftel ■ Vrel = ~S dm [“0 + * X r + m X (fi> X v)] • V rel . (1.7.21k) (a) - £ dm a 0 • v rel = -a Q ■ yfij dm v re i) = -m v Gjre i • «o (vc,rei = e drc /dt) . (1.7.211) (b) - $ dm [v rel • (a x r)] = -a • dm ( r x Cel)) = • #o,rei- (1.7.21m) (c) We have, successively, v re i • [w x (w x r)] = (w x r) ■ (v re i x ca) = —(uj x r) • [uj x (dr/dt)\ — —d/dt[(uj x r) 2 /2] = — d/dt[\uj x r| 2 /2] = —d/dt [\uj x r| 2 /2] , (1.7.21n) (i.e., as if during d/dt the vector uj remains constant) and, therefore, — $ dmv Te \ • [uj x (uj x r)] = d/dt(^Q dm[\uj x /*| 2 /2]^ = d/dt(^ $ dm [|cu x r\/ /2\^j. WWW.EBOOK777.COM (I. 7 . 210 ) §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE In view of (1.7.21g-o), eq. (1.7.21f) takes the following definitive form: dT K \/dt = d'W/dt - (mv G ,rei ) -«o —H 0 , re i • <x + d/dt(^^ dm [|u; x r| 2 /2]^. (1.7.21p) Specializations If O-xyz spins about a fixed axis through O, Ol, then a Q = 0 and eq. (1.7.21p) reduces to dT re \/dt = d'W/dt — Ho,re\ • tx + d/dt(Iuj 2 /2), (1.7.21q) where I = ^dmv 2 = moment of inertia of S about Ol. (1.7.21 r) If, further, uj = constant , then (1.7.21q) simplifies to dT Kl /dt = d'W/dt + ( dl/dt)uj 2 /2 . (1.7.21s) Finally, if d'W/dt = —dVo(r)/t, where Vo = Vo(f) = potential of impressed forces, then (1.7.21s) yields the conservation theorem: d/dt[T rei + (V 0 - Iuj 2 /2)\ = 0 => r rel + (V 0 ~ Iu 2 / 2) = constant. (1.7.21t) The above is a special case of the Jacobi-Painleve integral (§3.9). As with the equa¬ tions of motion, the “Newton-Euler” power equation (1.7.2lp) may be physically clearer than its Lagrangean counterparts, but the latter have the sam q form in both inertial and noninertial frames, and hence are easier to remember and apply. For further details and insights, see Hamel (1912, pp. 44(M143). The Angular Velocity Tensor Moving Axes Components Let us consider two OND frames/axes with common origin 0 F = O m = O (no loss in generality here), in arbitrary relative motion (rotation): one fixed 0 —u x u y u z I—XYZ and another moving 0—u x u y u z /—xyz ; or, compactly (in view of the heavy indicial notation that follows), O—u^j—x^ and 0—u k /—Xk, respectively. Now, d/dt(.. ^-differentiating their transformation relations, u k = ^2 A kk 'U k ', A kk , = u k • u k i = cos(x k , x k >) = cos(x k ,, x k ) = A k - k , (1.7.22a) and then employing their inverses, we find (since du k fdt = 0): du k /dt = ( dA kk fdt)u k - = Y ( dA kk fdt ) ( Y A di u ' ) = H , (1.7.22b) where O/k = Y1 A k 't(dA kk '/dt) = Y/, Aik’(dA kk '/dt) = Y^ {dA kk f/dt)A lk ' = ■■■ = Y { cos(x,, x h >) d/dt[cos(x k , x k ,)\} = I// • ( du k /dt ) = (du k /dt) • ui [= (l)th component of du k /dt\: Tensor of angular velocity of moving axes relative to the fixed axes ; but resolved along the moving axes. (1.7.22c) WWW.EBOOK777.COM 132 CHAPTER 1: BACKGROUND [As already pointed out (§1.1), this commutativity of subscripts in A _ constitutes one of the big advantages of the accented indices over other notations, such as A kh A ' k[ .\ Below we show that this tensor is antisymmetric : f2i k = —£2 k i- Indeed, d/dt(...)- differentiating the orthonormality conditions (constraints!), u k • u/ = ( £ A kk iu k i ^ • ^£ +//'«/') = • • • = £ A kk 'A, k t = S k/ , (1.7.22d) and then invoking the definition (1.7.22c) we obtain 0 = £ {dA kk ,/dt)A/ k J + £ A kk '(dAi k i/dt) [= u/ • ( du k /dt) + u k • ( du//dt)\ = fii k + ftki => f l/ k = —£lki , Q.E.D.; (1.7.22e) that is, due to the six constraints (1.7.22d), only three of the nine components of 14/ are independent. Hence, we can replace this tensor by its axial vector (1.1.16a ff.) W* = -££ (l/2)e krs n rs = - £££(1/2 )e krs [A rp ,{dA sp ,/dt)], (1.7.22f) and, inversely, Mrs = - £ Zkrs U k = - £ e rsk iO k . ( 1.7.22g) In extenso, and recalling the properties of e krs (§1.1), eqs. (1.7.22f) yield = Wx = — (l/2)(ei23^23 + £ 132 ^ 32 ) = f^23 = f?32 = — A 2k fdA/, k '/dt) [= — « 2 • (du 2 /dt) = —u y • ( du z /dt)\ = £ + 3 k i ( dA lk '/dt ) [= 1/3 • ( du 2 /dt ) = u : • ( du v /dt)\ {with 1,2,3 —> x,y,z; l',2',3' -> X, Y,Z: = ~[A Xy (dA Xz /dt) + A Yy (dA Yz /dt) + A Zy (dA Zz /dt)\ — A Xz {dA Xy /dt) + A Yz (dA Yy /dt') -y A Zz {ydA Zy fdify\ UJ 2 =Ul y = —(l/2) (£2311^31 +£213f2i3) = —1731 = I2l3 = — £ A 3k i (dA\ k '/dt ) [= —m 3 • ( dii[/dt ) = — u : • ( du x /dt)\ = Y^ A lk t (dA 2k fdt) [= U\ • ( du 2 /dt ) = u x • ( du./dt)\ { = ~[Axz(dA Xx /dt) + A Yz (dY Yx /dt) + A Zz (dA Zx /dt)] = A Xx (dA Xz /dt) + A Yx (dA Yz /dt) + A Zx (dA Zz /dt)}- : W 3 = UJ Z = —(l/2)(£3i2l7i2 + £3211721) = — I7l2 = 1721 = — £ A xk i{dA 2k i/di) [= —i/j • ( du 2 /dt ) = — u x • ( du y /dt)\ = Y^ A 2k fdA lk i/dt) [= «2 • ( du\/dt ) = u r • ( du x /dt)\ { = —\Axx{dA Xy /dt) + A Yx (dA Yy /dt) + A Zx (dA Zy /dt)\ = A Xy (dA Xx /dt) + A Yy (dA Yx /dt) + A Zy ( dA Zx /dt )}; (1.7.23a) (1.7.23b) (1.7.23c) WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE which are in complete agreement with (1.7.4j), and justify the name angular velocity tensor for (1.7.22c). In terms of matrices, the above assume the following memorable form: (12*/) 7 0 —CU3 0J2 \ ui 3 0 —cuj y — CU2 CJ\ 0 J ( 0 E A \k'{dA 2 k'/dt) E A \k'{dA2 k fdt)\ (1.7.23d) E A ik'{dA\ k f dt) 0 E A 2k' {dAyfr-r/dt) \E A 3k’{dA lk f/dt) E A 2k'{dA 2 k'/dt) o ) REMARKS (i) The formulae (1.7.22f ff.) can be combined into the following useful form: to k = ( du,./dt) •*/,, (1.7.24) where k,r,s= cyclic {even) permutation of 1,2, 3 (= x,y, r). (ii) The final expressions (1.7.23d) would have resulted if we had employed the following common angular velocity tensor definitions: 1 = ( du k /dt) ■ u, = ~{du,/dt) ■ u k = ^ {dA kk , / dt)A Ik , = - ^ {dA Ik ,/dt)A kk ,, (1.7.25a) but in connection with the also common axial vector definition-. UJ k = EE (1/2 )e krs f2 rs (1.7.25b) Then, we would have U 1 = (l/2) (£1231723 +£1321732) = 1723 = —1732 = E ( dA 2 k'ldt)A 2 k , = -^2 (, dA 3k fdt)A 2k etc. (1.7.25c) Fixed Axes Components Let us express the above inertial angular velocity tensor in terms of their components along the fixed axes 0—u x u y u z /—XYZ = 0—u k f—x k '. Dotting the representations of the position vector of a typical particle P, r = Y^ x k u k = E (1 -7.26a) with «/ and uy, respectively, and taking into account the orthonormality constraints of their basis vectors: I'k ■ Ui = (E A kk'«k') • (E A ii' u r) = E A kk ,A ik' = 6 k h (1.7.26b) Ilk' -U/' = (E A k'k«k) ■ (E A n U l) = E A k'k A l'k = 4'/', (1.7.26c) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND we easily obtain the component transformation equation x k' = Y A k'k*k *>x k = Y A kk' x k'- (1.7.26d) If the two sets of axes do not have a common origin, but [recalling fig. 1.6(a)] 9? = /* 0 + r, (1.7.26e) where SR = E **'«*', (1.7.26f) To = T m0 ving origin/fixed origin ^ ' b k U k ^ , b k ' ^k ' =>• h' = Y A k ' k b k b k = Y A kk fb k ', (1.7.26g) r = ^2 x k u k , (1.7.26h) then (1.7.26d) are replaced by X k' = E A k'k x k + b k ' = E ^ k ' k (x k + b k ) x k = Y A kk'{x k ' - b k 0 = E ^ k k lX k' - b k . (1-7.26i) Now, let us consider P to be rigidly attached to the moving axes. Then d/dt(.. .)- differentiating the x k ', while recalling that in this case x k = constant =>• dx k /dt = 0, we obtain, successively, dxu/dt = E {dA k , k /di)x k = E (dA k . k /dt) ( Y A ki' x i ') = Y Q k'i' x f > [which is none other than the familiar v = <o x r, resolved along the fixed axes] (1.7.26]) where Q k 'i' = Yj ( dA k ' k /dt)A kt i = Y, ( dA k : k /dt)A rk = Y{ cos ( x r,Xk) d/dt[cos(x k ',x k )]}: Tensor of angular velocity of moving axes relative to the fixed axes ; but resolved along the fixed axes [Note order of accented indices, and compare with order of unaccented indices in expression (1.7.22c, 25a).](1.7.26k) The components flgy, just like the f2 k i, are antisymmetric. Indeed, d/dt(.. .)-differ¬ entiating (1.7.26c), we obtain 0 = Y, {dA k ' k /dt)A/f k + Y, A k'k{dAi' k /dt) = Qf# + => n (k! = -f4y , Q.E.D. (1.7.261) u* = “EE (l/ 2 )^w4v = (1.7.26m) WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE and, inversely, Q's' = ~ £ k'r's' u k' = — £ r's'k lU} k '! (1.7.26n) or, in extenso, w l' = w V = = “ (l/2)( e l'2 , 3'^2'3' + Ei'3'2'^3'2') = — ^2'3' = ^3'2' {with 1,2,3 —> x,y,z; l',2',3' -► X, Y,Z: = — [A Zx {dA Yx /dt) + A Zy {dA Y y / dt) + A Zz (dA Yz /dt)\ — A Yx (dA Zx /dt) A Yy {dA Zy j dt) -\- A Yz (dA Zz /dt)'^', (1.7.27a) w 2' = u y' = w r= — (l/2)( £ 2'3'l'^3'l' + e2'l'3'^l'3') = ~^3'l' = ^l'3' = ~Yz A\'k(dA Vk /dt) = Y / Ay k (dAy k /dt) { = — + A Xy (dA Z y/dt) + /l X; (r/y4 z „/(7r)] = A Zx {dA Xx /dt) + A Zy (dA Xy ldt) + A Zz {dA Xz /dt )}; (1.7.27b) w 3 ' = w z' — W Z = ^(l/ 2 )( e 3 'l' 2 '^l' 2 ' + e 3 ' 2 'l'^ 2 'l') = — ^l' 2 ' = ^ 2 'l' = ~Y A Tk{dA Vk /dt) = Y A Vk {dA Vk /dt) { = [A Yx (dA Xx / dt) + A Y y(dA Xy /dt) + A Yz (dA Xz /dt)] = A Xx (dA Yx /dt) + A Xy (dA Yy / dt) + A Xz (dA Yz /dt)}\ (1.7.27c) or, finally, in the following memorable matrix form: ^ 0 —LUy 0J 2 ' \ Mt'/O = UJy 0 — UJ\ \— UJ2' (jJ\> 0 J 0 E {dA\' k /dt)A 2 ' k E {dAy k /dt)Ay k E {dA Vk /dt)A Vk 0 E {dA 2 ’k/dt)Ay k \E {dAy k /dt)Ay k E {dAy k /dt)A 2 ' k 0 (1.7.27d) or -h'l' — Y, Y A k 'k A-i'i^ki ^ ^ 4 / — A kk >A n ' fly. (1.7.27e) A Special Case If the axes and x k > coincide momentarily —that is, if instantaneously A k ' k = 6 k ' k (Kronecker delta), then eqs. (1.7.23) and (1.7.27) yield to x = dA Zv /dt = —dA Yz /dt , ui y = dA Xz /dt = —dA^/dt, to z = dA Yx /dt = —dA Xy /dt ; (1.7.28a) coy = dA Zy /dt — — dA Yz jdt , cuy : dA x ~/dt — — dA Zx /dt : cj z — dA Yx /dt = — dA Xy /dt. (1.7.28b) WWW.EBOOK777.COM 135 CHAPTER 1: BACKGROUND Rates of Change of Direction Cosines Let us calculate dAp k /dt in term of f2 k i, fi k 'f. (i) Fixed axes representation-. Multiplying both sides of (1.7.22c) with A/// and summing over /, we obtain E fiikA-n = E (' dA k i k / dt) (E A k i / A r/ 'j = E {dA k > k /dt){8 k >i>) = dA rk /dt. (1.7.29a) (ii) Moving axes representation-. Multiplying both sides of (1.7.26k) with A ks and summing over /', we obtain E f2 k ' t ' A t ' s = E (' dA k t k / dt) (E A kr A k ^ = E ( dA k i k /dt)(6 ks ) = dA k ' s /dt\ (1.7.29b) dA, II YAk'Ak = E ^4'/' i (1.7.29c) — A^/j + A k '2^2k + A k > 2 f2 3k =>■ dA k >\/dt = A k ' 2 uj 2 — A k ' 2 u 2 -, i.e., dA k > x/dt = = d. k 'yU z ~ A k ' Z Uy, dA k i 2 /dt = A k f 3 uji - A k :\LO y i.e., dA k 'y/dt = = A t z z w Y - ■ A k i x u> z , dA k i 3 /dt = A k '\U> 2 - A k ' 2 u> x \ i.e., dA k z z /d? = = A k t x u) y - - A k 'yU x (*' = Z, y,Z); (1.7.29d) + dr .id ii ^2' k-^k'2 ' + A'k^k'l' =>• dAy k /dt = A Vk u 2 ' — A 2 ' k L0 2 i; i.e. •; dA xk /dt = d zk u Y - A Yk u z , dA 2 ' k l dt = Ai' k Uy — A 3 z jt w 1 z; i.e. dA n-/ = A xk (x>z _ dA yk /dt = A 2 'kU\' — A^u^z; i.e ., dA Zkl dt = ^U' W X - dxk^Y (k = -UZA). (1.7.29e) Additional Useful Results (i) By d/dt{.. ^-differentiating the fixed basis vectors: 0 = du k i/dt = E [( dA k . k /dt)u k + A k , k {du k /dt)\ = • • •, (1.7.30a) it can be shown that du k /dt = E Q k' k u k ' , (1.7.30b) where O k > k = = yy (9x ; z / = • • ■ = dA k i k /dt (mixed “tensor”) (1.7.30c) Similarly, we can define the following mixed angular velocity “tensor”-. -<V = E = ^2(dx,/dx k ')f2 k , = E4(EEV A W Q pw') = • • • = e ^i'k‘ ■ WWW.EBOOK777.COM §1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE (ii) By d/dt{. . ^-differentiating x' k < = x k > = A k ' k x k , and noticing that v 'k = E Akk'V k ', it can be shown that the inertial velocity of a particle permanently fixed in the moving frame (i.e., dx k /dt = v k = 0 =>- v k * = 0) equals: v k = Y, f4/X/ (along the moving axes), (1.7.30d) dx k '/dt = v' k i = Y ftk'i'Xi’ (along the fixed axes). (1.7.30e) (iii) Let us define the following matrices : ft = (, fi k i ): matrix of angular velocity tensor, along the moving axes, (1.7.30f) ft' = matrix of angular velocity tensor, along the fixed axes, (1.7.30g) A = (A k > k ): matrix of direction cosines between moving and fixed axes. (1.7.30h) It can be shown that the earlier relations among them (i.e., among their elements) can be put in the following matrix forms [recalling that A -1 = A T and (.. .) T = Transpose of (...)]: ft = A t • (dA/dt) = — (dA/dt) T • A dA/dt = A • ft , (1.7.30i) JV = (dA/dt)-A T = -A-(dA/dt) T <*=> dA/dt = fl' - A, (1.7.30j) W = A • • A t O n = A t • Cl' ■ A . (1.7.30k) [(a) Equation (1.7.30j) expresses the following important general theorem: for an arbitrary (differentiable) orthogonal matrix (or tensor) A = A(t), dA/dt = (matrix of second-order antisymmetric tensor) • A; (1.7.301) and similarly for equation (1.7.30i). (b) Recall remarks on p. 84, below (1.1.19f), e.g. Ap 2 = A 2 y fA 2 y = A 12 /.] Angular Velocity Vector in General Orthogonal Curvilinear Coordinates [This section may be omitted in a first reading. For background, see (1.2.7a ft.).] In such coordinates, say q = [q \, q 2 , qf] = (( 71 , 2 , 3 ), the inertial position vector of a particle r becomes r = X(q)u x + Y(q)u Y + Z(q)u z = ^ x k fq)u e , (1.7.31a) and so the corresponding moving OND basis along q { 2 ,3 (i-e., the earlier x k ) is „ k = (dr/dq k )/\dr/dq k \ = (\/ h k )(dr / dq k ) (k = x,y,z), (1.7.31b) = (k, l = x,y,z). (1.7.31c) WWW.EBOOK777.COM with CHAPTER 1: BACKGROUND Next, d/dt(.. ^-differentiating (1.7.31b), we obtain, successively, d/dt(dr/dq r ) = d/dt(h r u r ) = (dh,./dt)u r + h r (du r /dt) = ( dh r /dt)u r + h r (o) x u r ) [by (1.7.4i)], (1.7.3Id) and dotting this with dr/dq s = h s u s (= e s , where r f s), in order to isolate u k , we get [d/dt(dr/dq r )] • ( dr/dq s ) = ( dh r /dt)hfu r ■ u s ) + h r h s [(a> x u r ) • «J = 0 + h r h s [(co • («,. x «,)] = h r h s (m ■ u k ) = h r h s u) k [dehnition of io k &\ where k, r, s = even (cyclic) permutation of 1, 2, 3 = x, y, z], that is, finally, Uk = (1 /h,.h s ) [d/dt(dr/dq r ) • (dr/dq s )\ { = ( du r /dt ) • u s = d/dt[(l/h r )(dr/dq,.)] • [(1 /h s )(dr/dq s )]}. (1.7.3le) Additional forms for these components exist in the literature; for example, with the help of the differential-geometric identities: du r /dq s = (1 /h r )(dh s /dq r )u s (r f 5 ), (1.7.31f) du,./dq r = -{\/h s ){dh r /dq s )u s - (1 /h k ){dh r /dq k )u k (rfsfkfr), (1.7.31g) and applying the second line of (1.7.3 le), we can easily show that wi = (l/h 2 )(dh 3 /dq 2 )(dq 3 /dt) - {l/h 3 )(dh 2 /dq 3 )(dq 2 /dt), (1.7.31h) w 2 = {1 / h 3 )(dh\ / dq 3 )(dqi / dt) - (l/h x )(dh 3 /dq x )(dq 3 /dt), (1.7.31i) w 3 = (l/h x )(dh 2 /dq x )(dq 2 /dt) - {\/h 2 ){dh x /dq 2 ){dq x /dt). (1.7.31J) [See Richardson (1992), also Ames and Murnaghan (1929, pp. 26-34, 94-98), for an alternative derivation based on the direction cosines between the moving and fixed axes: A k 'k = A kk t = u k '-u k = ( dr/dx k t) ■ [(1 /h k )(dr/dq k )\ = (1 /h)[{dr/dx k ’) ■ (^2 ( dr/dx r )(dx r /dq k 2j j = (1 /h) (X! (“*' • u i')i. dx i'/ dc lS) (since u k > ■ u v = 6 kr ) = (1 /h k )(dx k fdq k ), (1.7.31k) and their d/dt^.. ^-derivatives.] 1.8 THE RIGID BODY: INTRODUCTION The following material relies heavily on the preceding theory of moving axes (§1.7). The reason for this is that every set of such axes can be thought of as a moving rigid body, and, conversely, every rigid body in motion carries along with it one or more sets of axes rigidly attached to it, or embedded in it. To describe the translatory and WWW.EBOOK777.COM §1.8 THE RIGID BODY: INTRODUCTION Figure 1.9 Axes used to describe rigid-body motion. O-XYZ/IJK: fixed axes/basis; +-xyz/ijk: moving (body-fixed) axes/basis; +-XYZ/IJK: moving, translating but nonrotating axes/basis. angular motion of a rigid body B, we consider (at least) two sets of rectangular Cartesian axes and associated bases: (i) a fixed: that is, inertial, O XYZ/1JK or compactly 0-x k fiu k r, and (ii) a moving: that is, noninertial, and body-fixed set +-xyz/ijk or compactly ♦ -x k /u k , at the arbitrary body point ♦ (fig. 1.9). In the language of constraints (chap. 2), a free rigid body in space is a mechanical system with six degrees of global freedom; that is, six independent possibilities of finite spatial mobility: (i) three for the location of its body point ♦, say its O—XYZ coordinates X.=fi{i), Y.=f 2 (t), Z*=/ 3 (0; (1.8.1a) and (ii) three for its orientation —that is, of ♦ xyz relative to either O XYZ or ♦- XYZ\ where the latter are a translating frame at ♦ ever parallel to O XYZ —that is, one that is nonrotating but translating and hence is, generally, noninertial. Such “rotational freedoms” can be described via the nine direction cosines of ♦-xyz relative to +-XYZ (of which, as explained in §1.7, only three are independent); or via their three attitude angles: for example, their Eulerian or Cardanian angles 4>=m, e=f 5 {t), *p=f 6 (ty, (l.s.ib) or via a directed line segment called rotation “ vector ” [or via four parameter form¬ alisms (plus one constraint among them); for example, Hamiltonian quaternions, Euler-Rodrigues parameters, or complex numbers; detailed in kinematics treatises, also our Elementary Mechanics, ch. 16 (under production)]. With the help of the six positional system parameters, or system coordinates, fi 6 (t), the location/motion of any other body point P can be determined: '' = r{P, t) = r{P ; ,/ 6 ) = r.(f ], / 2 ,/ 3 ) + r /4 (P; / 4 , f 5 ,f 6 ), WWW.EBOOK777.COM 139 (1.8.2a) CHAPTER 1: BACKGROUND or, in components, X = Z* + cos(A, x)x/* + cos(X,y)y/+ + cos(X,z)z/,*, etc., cyclically, (1.8.2b) where »■/♦ = 0 >> In¬ constant rectangular Cartesian coordinates of P relative to +—xyz. (1.8.2c) or, in compact (self-explanatory) indicial notation, Xk' = *♦,*' + A k'k x k- (1.8.2d) In addition to ♦ -xyz and +-XYZ, we occasionally use other intermediate axes (or accessory axes, in Routh’s terminology) that, like +-XYZ, are neither space- nor body-fixed, but have their own special translatory and/or rotatory motion. 1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS (SUMMARY OF BASIC THEOREMS) Sections §1.9-1.13 cover material that is due to Euler, Mozzi, Cauchy, Chasles, Poinsot, Rodrigues, Cayley et al. (late 18th to mid-19th century). For detailed dis¬ cussions, proofs, insights, and so on, see for example (alphabetically): Alt (1927), Altmann (1986), Beyer (1929, 1963), Bottema and Roth (1979), Coe (1938), Gamier (1951, 1956, 1960), Hunt (1978), McCarthy (1990), Schonflies and Griibler (1902), Timerding (1902, 1908). The position, or configuration, of a rigid body B is known when the positions of any three noncollinear of its points are known; hence, six independent parameters are needed to specify it [e.g., 3x3 = 9 rectangular Cartesian coordinates of these points, minus the three independent constraints of distance invariance (i.e., rigidity) among them; or six coordinates for two of its points defining an axis of rotation, minus one invariance constraint between them, plus the angle of rotation of a body-fixed plane with a space-fixed plane, both through that axis]. If the body is further constrained, this number is less than six. It follows that the most general change of position, or displacement, of B is determined by the displacements of any three noncollinear of its points; that is, given their initial and final positions and the initial (final) position of a fourth, fifth, and so on, we can find their final (initial) positions with no additional data. Special Rigid-Body Displacements (i) Plane, or planar, displacement'. One in which the paths of all body points are plane curves on planes parallel to each other and to a fixed plane / [fig. 1.10(a)]: the body fiber P'PP" remains perpendicular to/, and the distance P+ remains constant, so that we need to study only the motion of a typical body section, or rigid lamina, b imagined superimposed on f and sliding on it. THEOREM Every displacement of a rigid lamina in its plane is equivalent to a rotation about some plane point / [fig. 1.10(b)], WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS iJ (...) = midpoint of (...) Figure 1.10 (a) Plane displacement of a rigid body, (b) The plane displacement of a rigid lamina on its plane is equivalent to a rotation about /; if / — > oo, that displacement degenerates to a translation. (ii) Translational displacement: One in which all body points have vectorially equal velocities. Translations can be either rectilinear or curvilinear, and can be represented by a free vector (three components). (iii) Rotational displacement: One in which at least two points remain /zxer/. These points define the axis of rotation-, and either they are actual body points, or belong to its appropriate fictitious rigid extensions. Rotations are, by far, the more complex and interesting part of rigid-body displacements/motions. The rotation is specified by its axis (i.e., its line of action) and by its angle of rotation-, and since a line is specified by, say, its two points of intersection with two coordinate planes — that is, four coordinates — and an angle is specified by one coordinate, the complete characterization of rotation requires 4+1 = 5 positional parameters. THEOREM Every translation can be decomposed into rotations. COROLLARY All rigid displacements can be reduced to rotations. The above special displacements (plane, translations, rotations) are all examples of constrained motions; that is, they result from special geometrical [or finite, or holonomic (chap. 2)] restrictions on the globed mobility of the body, as contrasted with local restrictions of its mobility [by nonholonomic constraints (chap. 2)]. EULER’S THEOREM (1775-1776) Any displacement of a rigid body, one point of which is fixed but is otherwise free to move, can be achieved by a single rotation, of 180° or less, about some axis through that point; that is, any displacement of such a system is equivalent to a rotation. Or: any rigid displacement of a spherical surface into itself leaves two (diametrically opposed) points of that surface fixed; and hence, in such a displacement, an infinite number of points, lying on the axis of rotation defined by the preceding two points, WWW.EBOOK777.COM CHAPTER 1: BACKGROUND remain fixed. (Under certain conditions this theorem extends to deformable bodies: one body-fixed line remains invariant.) To understand this fundamental theorem, let us consider a body-fixed unit sphere S B with center the fixed point ♦, representing the body, and let us follow its motion as it slides over another unit sphere S s concentric to S B but space-fixed and repre¬ senting fixed space. (This is the spatial equivalent of the earlier plane motion pro¬ blem where a representative rigid lamina slides over another fixed lamina.) Now, since this is a three degree-of-freedom system, its position can be specified by the coordinates of two of its points on S B , P , and Q [fig. 1.11(a)]: 2x2 = 4 coordinates [of which, since the distance between P and Q (= length of arc of great circle joining P and Q) remains invariable, only three can be varied independently]. Hence, to study two positions of the body—that is, a displacement of it—it suffices to study two positions of an arbitrary pair of surface points of it: an initial PQ and a final P'Q' [fig. 1.11(b)]. Then we join P and P', and Q and Q' by great arcs and draw the two symmetry planes of the arcs PP' and QQ'; that is, the two great circle planes that halve these two arcs. Their intersection, ♦C (which, contrary to the plane C-*~C, P^P', Q-* Q' A A CPQ, CP'Q': Congruent Triangles [CP = CP', CQ = CQ\ PQ = P'Q'] Figure 1.11 (a, b) The motion of a rigid body about a fixed point ♦ can be found by studying the motion of a pair of its points on the unit sphere with center ♦: from PQ to P'Q'; (c) special case of (b) where the planes of symmetry of the arcs PP' and QQ' coincide. WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS motion case, always lies & finite distance away), defines the axis of rotation; and their angle, x, defines the angle of rotation (around ♦C) that brings the spherical triangle CPQ into coincidence with its congruent triangle CP'Q'; and hence arc (PP 1 ) into coincidence with arc ( QQ'); and +PQ into coincidence with +P'Q ', and similarly for any other point of S B . In the special case where these two symmetry planes coincide [fig. 1.11(c)], the rotation axis is the intersection of the planes defined by ♦PQ and ♦P'6'- FUNDAMENTAL THEOREM OF GEOMETRY OF RIGID-BODY MOTION Any rigid-body displacement can be reduced to a succession of translations and rotations. Specifically, any such displacement can be produced by the translation of an arbitrary “base point,” or “pole,” of the body, from its initial to its final position, followed by a rotation about an axis through the final position of the chosen pole—and this is the most general rigid-body displacement. The translatory part varies with the pole, but the rotatory part (i.e., the axis direction and angle of rotation) is independent of it (fig. 1.12). COROLLARY FOR PLANE MOTION Any rigid planar displacement can be produced by a single rotation about a certain axis perpendicular to the plane of the motion; in the translation case, that axis recedes to infinity [fig. 1.10(b)]. THEOREMS OF CHASLES (1830) AND POINSOT (1830s, 1850s) Any rigid-body displacement can be reduced, by a certain choice of pole, to a screw displacement; that is, to a rotation about an axis and a translation along that axis. In special cases, either of these two displacements may be missing. In a screw displacement: (a) The axis of rotation is called central axis, and (for given initial and final body positions) it is unique, except when the displacement is a pure translation; (b) The ratio of the translation (/) to the rotation angle (x), which Figure 1.12 The most general displacement of the rigid body ♦PQ can be effected by a translation of the pole ♦, from ♦PQ to ♦ , P"Q"; followed by a rotation about an axis through ♦', from ♦ , P"Q" to ♦'P'Q'. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND equals the advance (p) per revolution (2ir), is called pitch of the screw: p/2ir = l/x => p = 2n(I/x); and (c) The translation and rotation commute. EXTENSION TO DEFORMABLE BODIES (Chasles’ Theorem + Deformation = Cauchy’s Theorem) The total displacement of a generic point of a continuous medium, say a small deformable sphere (fig. 1.13), is the result of a translation , a rigid rotation [of the local principal axes (or directions) of strain], and stretches along these axes; that is, the sphere becomes a general ellipsoid. Hence, rigid-body kinematics is of interest to continuum mechanics too; the latter, however, will not be pursued any further here. Rigid-Body Kinematics Thus far, no restrictions have been placed on the size of the displacements; the above theorems hold whether the translations and rotations are finite or infinitesimal. The finite case is detailed quantitatively in the following sections. Next, let us examine the important case of sequence of rigid infinitesimal displace¬ ments in time, namely, rigid motion. In particular, let us return to the motion about a fixed point (Euler’s theorem) and consider the case where the initial and final posi¬ tions of the arcs PQ (at time t) and P'Q' (at time t' = t + At) are very close to each other. Now, as At —> 0 the earlier (great circle) planes that halve the arcs PP' and QQ' coincide with the normal planes to the directions of motion of P and Q, respec¬ tively, at time t; and their intersection yields the instantaneous axis of rotation. Then the velocity of the generic body point P equals v P = v= /At)\ At ^ 0 = a) x r P/4t = m x r, (1-9.1) since Vp = v = | v| equals the magnitude of the angular velocity of that rotation, |o»|, times the perpendicular distance of P from the rotation axis. [Euler (1750s), Poisson (1831); of course, in components.] Hence, the instantaneous rotation of the body B about the fixed point ♦ is described by the single vector a>, which combines all three character¬ istics of rotation: axis, magnitude, and sense. As the motion proceeds, and since only the point ♦ is fixed, the axis of rotation (carrier of a >) traces, or generates, two general and generally open conical surfaces with common center ♦: one fixed on the body, the INITIAL INTERMEDIATE FINAL Figure 1.1 3 General displacement of a small deformable sphere: Translation —> Rotation —> Strain (Sphere —> Ellipsoid). WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS Figure 1.14 Rolling of body cone (P Po ih 0 de) on space cone (H er P Po | hode ). polhode cone; and one fixed in space, the herpolhode cone (fig. 1.14). Hence, the following theorem: THEOREM Every finite motion of a rigid body, having one of its points ♦ fixed, can be described by the pure (or slippingless) rolling of the polhode cone on the herpolhode cone; and, at every moment, their common generator (through ♦) gives the direction of the instantaneous axis of rotation/angular velocity. If ♦ recedes to infinity, these two cones reduce to cylinders and their normal sections become, respectively, the body and space centrodes. Velocity Field (Mozzi, 1 763) Since, for the first-order geometrical changes involved here (“infinitesimal displace¬ ments”) superposition holds, we conclude that the velocity of a generic body point P in general motion, v P = v, is given by the following fundamental formula of rigid body kinematics: v = v* + ft) x (r — r*) = r* + co x Y 1 + = v* + V/* [r ; /> = velocity of P relative to ♦ (both measured in the same frame)] (1-9.2) where ♦ is any body point (pole) (fig. 1.15); or, in terms of components (fig. 1.9) as follows: Space-Fixed Axes dX/dt = dX+/dt + wy(Z — Z*) — u z (Y — Y+), etc., cyclically, (1.9.2a) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Figure 1.15 Geometrical interpretation of eq. (1.9.2). or, equivalently, v x = v+j + - u> z Y/+, etc.,cyclically. (1.9.2b) Body-Fixed Axes v x = v«x + '^y z i* - w-V/«, etc., cyclically; (1,9.2c) where v+ x = cos(x,X)v* x + cos(x, Y)v+ Y + cos(x,Z)v 4z , etc.,cyclically; (1.9.2dl) and, inversely, v* x = cos(A, x)v+ iX + cos(A,y)v* :> , + cos(A,c)v* etc., cyclically. (1.9.2d2) The six functions of time v« ;w ,w w (or v* ; x,y,Zi w x,y,z ) characterize the rigid- body motion completely. The line-bound vectors co and r* constitute the torsor of motion, or velocity torsor, at ♦, from which the rigid-body velocity field can be determined uniquely. [Just as, in elementary statics, the resultant force / (or R) and moment M * of a system of forces constitute the force system torsor at ♦ (see “Formal Analogies ...” section that follows.] In the case of motion about a fixed point ♦, that torsor reduces there to (co, 0 ). Now, from the displacement viewpoint, the velocity transfer equation (1.9.2) states that: (i) The state of motion of the body consists of an elementary translation (di\ = r* dt) of a base point (or pole) ♦, and an elementary rotation (r// = oidt) about that point. Therefore, applying the earlier theorem of Chasles, we deduce that: WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS (ii) Any infinitesimal rigid (nontranslatory) displacement can be reduced uniquely to an infinitesimal screw ; that is, an infinitesimal translation plus an infinitesimal rotation about a (central) axis parallel to the translation. (The location of that axis and the pitch of the screw are given in the “Formal Analogies ..section below.) As the motion proceeds, that axis traces two (ruled) surfaces with it as common gen¬ erator: one fixed in space (F s ) and another fixed in the body (F B )—which constitute the “no fixed point” generalization of the herpolhode and polhode, respectively. Flence, the following theorem: (iii) The general finite motion of a rigid body can be produced by the rolling and sliding of r B over F s . (In plane motion, sliding is absent.) Next, we prove that (iv) The angular velocity vector to is independent of the choice of the pole. Applying the fundamental formula (1.9.2) for the two arbitrary and distinct poles ♦ and we have v = v* + (D x (r — r*) = v* + co x = »v + co' x (r — iv) = v*/ +co' x r/+i, (1.9.2e) where initially, we assume that co and co 1 are different and go through ♦ and ♦ 1 , respectively. We shall show that to = co'. (1.9.2f) Indeed, since »•/♦ = »•/♦'+*•♦'/♦ and tv = v* + to x iv/*, (1.9.2g) equating the right sides of (1.9.2e) we obtain to x r/* = to x /*♦</♦ + co' x rt+i =>■ to x »*/«.' = a>' x r/+i, (1.9.2h) from which, since r/+> is arbitrary, (1.9.2f) follows. [Since to is a body quantity (a system vector), it carries no body point subscripts (like v ), just like a force resultant. The only “insignia” it may carry are those needed to specify a particular body and/or frame of reference. Perhaps this supposed “base point invariance” of it may have given rise to the false notion that “to [of a body- fixed basis relative to a space-fixed basis] is a free vector, not bound to any point or line in space” (Likins, 1973, p. 105, near page bottom); emphasis added. A correct interpretation of (1.9.2e,f), however, shows that to is a line-bound, or sliding, vector, not a free one (just like the force on a rigid body); hence, to in eq. (1.9.2), is under¬ stood to be going through point ♦.] A USEFUL RESULT Let i"i and r 2 be the position vectors of two arbitrary points of a rigid body. Then, its angular velocity to equals to = (vj x v 2 )/(v[ • v 2 ), where v =dr felt. (1.9.2i) Formal Analogies Between Forces/Moments and Linear/Angular Velocities Comparing (1.9.2), rewritten as v 2 = v\ + r\/ 2 x co (1, 2: two arbitrary body points) with the well-known moment transfer theorem of elementary statics (with some, hopefully, self-explanatory notation): M 2 = M x + r 1 /2 x f - we may say that the WWW.EBOOK777.COM CHAPTER 1: BACKGROUND velocity v 2 is the moment of the motion, or velocity torsor (to , V[) about point 2; that is, m is the kinematic counterpart of the force resultant (/ or R), and hence is a line- hound, or sliding vector; while v ... is the counterpart of the point—dependent moment of the torsor M. Hence, recalling the (presumably, well-known) theorems of elementary statics, we can safely state the following: • An elementary rotation d% = m dt about an axis can always be replaced with an elementary rotation of equal angle about another arbitrary but parallel axis, plus an elementary translation dr = v dt , where v = at x /• is perpendicular to (the plane of) both axes of rotation, and r is the vector from an arbitrary point of the original axis to an arbitrary point of the second axis; that is, an elementary rotation here is equivalent to an equal rotation plus an elementary perpendicular translation there. • Several elementary rotations about a number of arbitrary axes can be replaced by a resultant motion as follows: (a) We choose a reference point 0 , and transport all these elementary rotations parallel to themselves to 0, and then add them geometrically there. Then, (b) We combine the corresponding translational velocities, created by the parallel transport of the rotations in (a) (according to the preceding statement), to a single translational velocity at 0. For example, two equal and opposite elementary rotations about parallel axes can be replaced by a single elementary translation per¬ pendicular to (the plane of) both axes. These formal analogies between forces/ moments and linear/angular velocities (also, linear/angular momenta), which are quite useful from the viewpoint of economy of thought (elimination of unnecessary duplication of proofs), are summarized in table 1.2. Table 1.2 Formal Analogies Among Vectors/Forces/Rigid-Body Velocities Forces/Moments Rigid-Body Velocities Vector Systems (On Rigid Bodies) (Instantaneous Geometry) Single vector a Moment of a about point O Vector couple (a, . a 2 = -Oi) => Constant moment Vector resultant R Vector torsor ( R, M Q ) Invariants: R ■ R, R ■ M... Vector wrench (or screw) ( RMc ) r = \R+ (R x M a )/R 2 [A = (r-R)/R 2 } Pitch = p = M c /R = R ■ Mo/R' • p = 0: Vector resultant R • p = oo: Couple Single force f (along line of action) Moment of f about 0 Force couple (/), f 2 = — f)) => Constant moment; or couple Force resultant R Force torsor (R, M 0 ) Invariants: R • R, R • M... Simplest Representation of Torsor Force wrench (R M c ) Central Axis of Wrench/Screw r = \R+ (R x M 0 )/R 2 p = M c /R= R-Mo/R 2 Pure force (resultant) R Pure couple Angular velocity m (about axis of rotation) Linear velocity of body point Ov 0 Rotational pair (», ,<o 2 = -o»i) => Constant translational velocity Rotation resultant to Motion torsor (to, v 0 ) 0'(R,o> at O) v o’ = v o + ro/o' x Invariants: a> ■ to, co-v... Motion screw (ox v c ) r = pro + (o x v 0 )/io 2 \p,= {r-(o)/ui 2 } p = v c /a> = co ■ Vo/u 2 Pure rotation oj Pure translation* Spatial Variation (or Transfer) Theorem: 0 Mo' = M 0 + ro/o' x R M 0 ' = M 0 + r 0 /o' x R *See also Hunt (1974). WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS Acceleration Field By d/dt(. . ./differentiating (1.9.2), we readily obtain the acceleration field of a rigid body in general motion : fl = «♦ + a x r/+ + co x (to x r/+) = «* + a x r/* + [(o> • r/+)co — 0J 2 r/+] = «♦ + («/*) lange „, + (^normal (= «♦ + «/♦)! O' 9 ' 3 ) or in terms of components (figure 1.9): Space-Fixed Axes a x — a *,x + { a Y^i* ~ a zY/+) T x xz+'j — uj Xj^ J, etc., cyclically. (1.9.3a) Body-Fixed Axes a x = a. rX + {ot y z/+ - a z y/.) + \f>x(. u x x /+ + Wyy /♦ + io z z/+) — u?x/f\ , etc., cyclically; (1.9.3b) where a*, x = cos(x, X)a+ x + cos(x, Y)a+ Y + cos(x, Z)o* z = cos(x, X)(d 2 X+/dt 2 ) + cos(x, Y){d 2 Y+ /dt 1 ) + cos (x,Z)(d 2 Z+/dt 2 ), etc., cyclically; (1.9.3c) and, inversely. a+ x = cos (X,x)a+ X + cos (X,y)a+ }y + cos{X,z)a+ z , etc., cyclically; (1.9.3d) and a x = ( dw/dt ) y ■ = dux/dt, etc., cyclically, (1.9.3e) a x = ( dao/dt ) • 3 1 5/ "^3 II ‘ •»» /dt) = duj x /dt — co • (co x /) = dui x /dt , etc., cyclically. (1.9.3f) Plane Motion The distances of all body points from a fixed, say inertial, plane /' remain constant; and so the body B moves parallel to/' (fig. 1.10a). [For extensive discussions of this pedagogically and technically important topic, see, for example. Pars (1953, pp. 336— 356), Loitsianskii and Lur’e (1982, pp. 227-261).] A rigid body in plane (but other¬ wise free) motion is a system with three global, or finite, degrees of freedom. As such, we choose (fig. 1.16): (a) The two positional coordinates of an arbitrary body point (pole) ♦ (that is, of a point belonging to the cross section of B with a generic WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Figure 1.16 Plane motion of a rigid body 6. plane / ever parallel to/') relative to arbitrary but /-fixed rectangular Cartesian coordinates O—XY, (X*, 7*); and (b) The angle between an arbitrary /-fixed line, say the axis OX, and an arbitrary 5-fixed line, say ♦/*, where P is a generic body point. (i) The velocity field (i.e., the instantaneous spatial distribution of velocity) Here, on = u-k = lo z K = uj K = (d(j)/dt)K (i.e., m is perpendicular to v), (1.9.4a) and so the general velocity formula (1.9.2) becomes v P = dvpjo = dr/dt = v = v* + v P /+ = v* + v/* = v* + oj x vpj+ = v* + oj x r/*, (1.9.4b) or, in components [along space-fixed (inertial) axes] {dX/dt,dY/dt, 0) = {dX./dt,dY./dt, 0) + (0,0, u) x (X /4 , T /4 ,0), => dX/dt = dX+/dt — ujY/+, dY/dt = dY+/dt + loX/+. (1.9.4c) The above show that, in plane motion, there exists—in every configuration—a point, either belonging to the body or to its fictitious rigid extension, called instantaneous center of zero velocity, or velocity pole (IC , or I, for short), whose velocity, at least momentarily, vanishes; that is, locally, at least, the motion can be viewed as an elementary rotation about that point (local version of fig. 1.10b). Indeed, setting in (1.9.4b,c)’ v —» vj = 0, i.e., choosing P = I, (1.9.4d) WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS we obtain its inertial instantaneous coordinates relative to our originally chosen pole ♦: X tj . = X,-X.= -( dY./dt)/u, Yj/. = Y, — Y. = +(dX./dt)/to. (1.9.4e) From these equations we conclude that, as long as to 0, / is located at a finite distance from the body and is unique; if lj = 0, then / recedes to infinity, and the motion becomes a translation-, and if we choose / as our pole — that is, ♦ = I —then (1.9.4b, c) yield dX/dt = —uiY/i, dY/dt = (jjX/j, or v — uirji [v 2 = (dX/dt) 2 + (dY/dt) 2 ]. (1.9.4f) [In the case of translation, eq. (1.9.4f) can be written qualitatively/symbolically as finite velocity = (zero angular velocity ) x (infinite radius of rotation)]. As the body moves, / traces two curves: one fixed on the body (space centrode) and one fixed in the plane (space centrode)-, so that the general plane motion can be described as the slippingless rolling of the body centrode on the space centrode, with angular velocity ui. (ii) The acceleration field Flere, a = da>/dt = ( duj/dt)k = ak = a K, (1.9.4g) and «•/'/♦ = 0, and so the general acceleration formula (1.9.3) becomes a P = a = «* + «/♦ = + a x r + to x (o x r/f) = «♦ + a x »■/♦ — lo 2 v/+, (1.9.4h) or, in components [along space-fixed (inertial) axes], (d 2 X/dt 2 ,d 2 Y/dt 2 , 0) = ( d 2 X./dt 2 , d 2 Y./dt 2 , 0) + (o,o,«) x (*■,♦, y/*,o) - w 2 (*■/♦, r/*,o), => d 1 X/dt 2 = d 2 X * /dt 2 — aY/+ — u?X/+, d 2 Y/dt 2 = d 2 T* /dt 2 + aI / ,-w 2 f / ». (1.9.4i) Along body-fixed axis ♦— xy, eq. (1.9.4h) yields the components (with some easily understood notation): a x = («*) x - ay/. - c?x/., a y = (a.) y + ax f . - cry/.-, (1.9.4j) where (a.) x = a. -i = cos (x,X)(d 2 X./dt 2 ) +cos(x, Y)(d 2 Y./dt 2 ), etc.; and similarly for the velocity field (1.9.4b), if needed. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Here, too, there exists an instantaneous center of zero acceleration , or acceleration pole, /', whose coordinates are found by setting in (1.9.4i) d 2 X/dt 1 = 0, d 2 Y/dt 2 = 0 and then solving for X/+, Y/+(P —> /') : X v/ + = X v -X 4f = [cv 2 {d 2 X./dt 2 ) - a(d 2 Y./dt 2 )\/{a 2 + w 4 ), Y rh = Y v - 7* = [w 2 (d 2 7,/* 2 ) + a(d 2 X./dt 2 )\/(a 2 + w 4 ). (1.9.4k) These equations show that as long as a 2 + w 4 0 (i.e., not both ui and a vanish), the acceleration pole /' exists and is unique. If ui, a = 0 (i.e., if the body translates), then I' (as well as I) recedes to infinity. Finally, with the choice ♦ = /' eqs. (1.9.4h,i) specialize to a = a* + <*/♦ = a x r/p + oj x (to x »•///) = a x r/ji — u> 2 r/p, (1.9.41) or, in components d 2 X/dt 2 = -olY/p - u?X / V , d 2 Y/dt 2 = +aX /7 , - w 2 T /7 ,. (1.9.4m) For the geometrical properties of I', the reader is referred to texts on kinematics. Additional Useful Results (i) Crossing 0 = v* + co x (/, — r 4 ) with <u, expanding, and so on, it can be shown that the position of the instantaneous velocity center is given by '"//♦ = r, - r* = (t» x t’*)/tu 2 ; (1.9.4n) and similarly for the location of the acceleration pole (ii) The location of the instantaneous center of zero velocity I, and zero accelera¬ tion in body-fixed coordinates ♦— xy, are given, respectively, by (fig. 1.17) Figure 1.17 Body-fixed axes in plane motion. WWW.EBOOK777.COM §1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS Xj = (1/w) \(dX+/dt) sin0 — (dY^/dt) cos<^>] = — {y+) y /w, (1.9.4o) yj = (1/w) [(dX+/dt) cos</> + (dY^/dt) sin<^>] =(v+) x /u, (1.9.4p) */' = [ W 2 (« 4 ), - a(a.) y ]/( U * + a 2 ), = [«(«♦), + u; 2 («*),]/(</ + a 2 ), (1.9.4q) where (v*) x = v* •/ = cos (x,X)(dX+/dt) +cos(x, Y)(dY+/dt ), etc. Contact of Two Rigid Bodies; Slipping, Rolling, Pivoting Let us consider a system of rigid bodies forced to remain in mutual contact at points, or along curves or surfaces of their boundaries. For simplicity and concreteness, we restrict the discussion to two rigid bodies, B' (fixed) and B (moving), in contact at a space point C; that is, a certain point P of the bounding surface of B, S, is in contact with a point P' of the bounding surface of B' , S 1 '; that is, then, C = P = P' (fig. 1 . 18 ). Now: (i) If C is fixed on both bodies, we call such a “bilateral constraint” (i.e., one expressible by equalities) a hinge, and we say that the bodies are pivoting about it. (ii) If, on the other hand, C is not fixed on one (both) of the bodies, we say that it is wandering on it (them). In this case, we call the relative velocity of P and P', which are instantaneously at C, the slip velocity there: Vp/pi = v P — v P f = v s . (1.9.5a) If we view the motion of C relative to B', C/B' , as the resultant of C/B and B/B ', then, since the velocities of the latter are tangent to the surfaces S and S', respec¬ tively, at C we conclude that lies on their common tangent plane there, p. Analytically, = y s ,T + v Sl N = Vs.T, (1.9.5b) Figure 1.18 Two rigid bodies in contact at a space point C. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND where v s t = component of v s along p 1 v sN = component of v, normal to p (= 0 ; i.e., contact is preserved; the two bodies cannot penetrate each other); (1.9.5c) and if, at that instant, B and B' separate , then v s N lies on the side of B 1 . Next, if the angular velocity of B relative to B\ at C, is to with components along and normal to p: a> T , oj n , respectively; that is, (o = (o T + a) N , (1.9.5d) then we can say that the most general infinitesimal motion of B relative to B\ B/B'. is a superposition of the following special motions: a pure slipping : v, f 0 , o)f — 0 , co N = 0 ; (1.9.5e) a pure rolling : = 0 , a> T f 0 , % = 0 ; (1.9.5f) a pure pivoting : v s = 0 , (Of —— 0 , oj n f 0 . (1-9.5g) If v s = 0 and m f 0 , the motion B/B' is an instantaneous rotation called rolling and pivoting ; which results in two (scalar) equations of constraint. In this case, the point C has identical velocities relative to both B and B'; and hence its trajectories, or loci, on the bounding surfaces of B and B', 7 and 7 ' respectively, are continuously tangent, and are traced at the same pace ; that is, if, starting from C, we grade them in, say cen¬ timeters, then the points that will come into contact during the subsequent motion will have the same arc-coordinates numerically. Such a B/B' rolling is expressed by saying that P and P' , both at C at the moment under consideration, have equal velocities relative to a (third) arbitrary body, or frame or reference; and the velocities of B about B' are the same as if B had only a rotation m about an axis through the “instantaneous hinge” C. If the locus of a> on B is the ruled surface Z, and on B' the also ruled surface Z 1 , then the slippingless motion B/B' can be obtained by rolling Z on Z' [The earlier curve 7 ( 7 ') is the intersection of Z with S(Z' with S 1 ')]. If B and B' are in contact at two points, say C and C' , and if 17 = 17 ./ = 0 , then the motion B/B' is an instantaneous rotation about the line CC'\ that is, co is along it. And if B , B' contact each other at several points C, C', C", ..., then slipping cannot vanish at all of them unless they all lie on a straight line. If, in addition, a> N = 0 (or co T = 0 ), we have pure rolling (or pure pivoting). In sum, slippingless rolling along a curve can happen only if that curve is a straight line carrying co (like a long hinge). Some Analytical Remarks on Rolling (i) The contact among rigid bodies is expressed analytically by one or more equations of the form f{t;qi,q 2 ,---,q n ) = 0, (1.9.6a) where q = (q u ..., q„) are geometrical parameters that determine the position, or configuration, of the bodies of the system; hence, their alternative name: system coordinates. Equation (1.9.6a) is called a holonomic constraint. WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION (ii) If, in addition to contact, there is also slippingless rolling, and possibly pivot¬ ing, then equating the velocities of the two (or more pairs of) material points in contact, we obtain constraints of the form a\dq x + a 2 dq 2 + • • • + a n dq n + a n+ \dt = 0, (1.9.6b) or, (roughly) equivalently, ci\{dq\/dt) + a 2 (dq 2 /dt) + • • • + a„(dq„/dt) + a n+l = 0, (1.9.6c) where a /c = cik{t,q) (k = 1,...,«). If (1.9.6b,c) is not integrable [i.e., if it cannot be replaced, through mathematical manipulations, by a finite (1.9.6a)-like equation], it is called nonholonomic. In mechanical terms, holonomic constraints restrict the mobi¬ lity of a system in the large (i.e., globally)', whereas nonholonomic constraints restrict its mobility in the small (i.e., locally). The systematic study of both these types of constraints (chap. 2) and their fusion with the general principles and equations of motion (chap. 3 ff.) is the object of Lagrangean analytical mechanics. 1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION; FINITE ROTATION The peculiarities of the algebra of finite rotations are just the peculiarities of matrix multiplication. (Crandall et al., 1968, p. 58) Recommended for concurrent reading with this section are (alphabetically): Bahar (1987), Coe (1938, pp. 157 ff.), Hamel (1949, pp. 103-117), Shuster (1993), Timerding (1908). The Fundamental Equation of Finite Rotation Since, by the fundamental theorem of the preceding section, the rotatory part of a general displacement of a rigid body is independent of the base point (pole), let us examine first, with no loss in generality, the finite rotation of a rigid body B about the ( body- and space-) fixed point O', and later we will add to it the translatory displace¬ ment of O. Specifically, let us examine the finite rotation of B about an axis through O, with positive direction (unit) vector it, by an angle \ that is counted positive in accordance with the right-hand (screw) rule (fig. 1.19). As a result of such an angular displacement, a generic body point P moves from an initial position P, to a final position Pj] or, symbolically, (*i, Pi) -► (''/>/’/)> ( 1 . 10 . 1 a) where p is the projection, or component, of the actual position vector of P, r, on the plane through it normal to the axis of rotation; that is, to n. Our objective here is to express »y in terms of ly, n, and x- To this end, we decompose the displacement Ar = ly — rj = pj — p t = Ap, which lies on the plane of the triangle APfif, into two components: one along p h P,B = Ar l , and one perpendicular to it, BP, = Ary. Ar = Ar\ + Ar 2 . (1.10.1b) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Figure 1.19 Finite rigid rotation about a fixed point O (axis n, angle x)- Now, from fig. 1.19 and some simple geometry, we find, successively, (i) Av x = —(APj - AB) = -0,. — Pi cos x) = ~Pi{ 1 - cos*) = -2p,-sin 2 (x/2); or, since Av x is perpendicular to both n x /y and n, and n X (« x v t ) = (n • i‘i)n — (it • «)r ; = OA — r t = PjA = —p h finally, Av\ = n x (n x r ,)2 sin 2 (x/ 2 ). ( 1 . 10 . 1 c) (ii) The component Ar 2 is perpendicular to the plane OAP h and lies along n x r,; and since the length of the latter equals |n x Vj | = |n||f,-| sincr = |r,-| sincr = [p ; |, and |p, | sinx = |CiP | = \BP f ] = \Ar 2 \ (the triangle APjP f being isosceles!), finally Ay 2 = (/i x Vj) sin X- (1.10.Id) Substituting the expressions (1.10.1c, d) into (1.10.1b), we obtain the following fundamental equation of finite rotation : Av = Vf — Vj = (n x Vj) sin x + nx (// x r,)2 sin 2 (x/2). (1.10. le) All subsequent results on this topic are based on it. Alternative Forms of the Fundamental Equation (i) With the help of the so-called “ Gibbs vector of finite rotation” 7 = tan(x/2)« = (71,72,73) = ( 7 a -, 7 y , 7 z ) = Rodrigues parameters, (l.i0.2a) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION relative to some background axes, say O—XYZ [Rodrigues (1840) — Gibbs ( late 1800s) ‘vector’] and, since by simple trigonometry, sinx = 2 sin(x/ 2 ) cos(x/ 2 ) = 2 tan(x/ 2 )/[l + tan 2 (x/ 2 )] = 27/(l+7 2 ), where 7= |y| = |tan(x/2)|, (1.10.2b) sin 2 (x/ 2 ) = tan 2 (x/ 2 )/[l +tan 2 (x/ 2 )] = (1 - cosx )/2 = 7 2 /(l + 7 2 ), ( 1 . 10 . 2 c) we can easily rewrite (l.lO.le) as Ar= [2/(1 +7 2 )]b xr, 4 yx(yx /*,•)]; (1.10.2d) and from this, since y x (y x r,) = — 7 2 r,- + (y • 17 ) y, we obtain the additional form r f = [2/(1 +7 2 )][y X *•/+ (r »-i)y] + [(l - 7 2 )/(l + 7 2 )h; (1.10.2e) which, clearly, has a singularity at 7 = ±/. Further, in terms of the normal projection of r, to the rotation axis n, r in , defined by i'i,„ = r i - (y • F)y/ 7 2 = n - [(y ® y) • n]h 2 , (i.io. 2 f) we can rewrite ( 1 . 10 . 2 e) successively as r f = n + [2/(1 + 7 2 )](y X r in - 7 \„) = r i + [2/(1 + 7 2 )] [y X r,- - rfri + (y ® y) • rj = n + P/(l + 7 2 )][yxr,- + yx(yx r,-)] = r i + [ 2 y/(l + 7 2 )] x (r,- + y x r,-); ( 1 . 10 . 2 g) that is, express 17 in terms of r, and the single vector y. {It is not hard to show that the components, or projections, of a vector a along ("along = «/) and perpendicular to ("perpendicuiar/normai = a n) another vector b (of com¬ mon origin) are «/ = («• b)b/b 2 , «„ = a — = a — (a • b)b/b 2 = [b x (a x b)\/b 2 }. Inversion of Eqs. (1.10.2e,g) Since a rotation —y should bring 17 back to r h if in (1.10.2g) we swap the roles of r, and rf and replace y with —y, we obtain the initial position in terms of the final one and its rotation: n = r f - [2y/(l + 7 2 )] x (iy - y x iy); (1.10.3) and thus avoid complicated vector-algebraic inversions. Rodrigues’ Formula (1840) Adding r, to both sides of (1.10.2e), we obtain n + r f = [2/(1 + 7 2 )][r,. + y x r t + (y • r,-)y], (1.10.4a) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and crossing both sides of the above with y, and then using simple vector identities and (1.10.2g) [or, adding (1.10.2g) and (1.10.3) and setting the coefficient of 2y/(l + 7 2 ) equal to zero, since it cannot be nonzero and parallel to y], we arrive at the formula of Rodrigues'. r f - c, = yx (r, + rf) = 2y x r m = 2n x r m tan(x/2), (1.10.4b) where 2r m = )',■ + iv = 2 {position vector of midpoint of PjPf ); (1.10.4c) or, rearranging, r f + r f x y = + y x r,. (1.10.4d) Finally, dotting both sides of this equation with y (or n), we obtain y»y = yr h (1.10.4e) as expected. (ii) With the help of the finite rotation vector l = X», (1.10.5a) which is, obviously, related to the earlier Gibbs vector y by y = tan(x/2)(z/x), (1.10.5b) and since 1 + 7 2 = l/ cos 2 (x/2), 1 - 7 2 = cos x/ cos 2 (x/2), (1.10.5c) the preceding rotation equations yield r f = 2cos 2 (x/2) [tan(x/2)(y x r,-)(l/x) + tan 2 (x/2)(y • r,-)(x/x 2 )] + cos x*t, (1.10.5d) or finally, r f = r/ cos x + (z x r,)(sin x/x) + (z * »h)z[(l " cosx)/x 2 ], (1.10.5e) a form that is symmetrical and (integral) transcendental function of y • y = \ The above can also be rewritten as Yf - r ; . = (sin x)(« x r,) + (1 - cosx)[« x (// x r,-)] = (sinx)(« x r,) + (1 - cos x) [(«•»•;)« - {n 2 )rfi (1.10.5f) or, slightly rearranged (since « 2 = 1), rf = r t cos x + [n x r,-) sin x + («• **,■)«( 1 — cos x) = r, + sin x(n x r,) + (cos x - 1) ['*/ - (»*,■ • n)n] = r t + sinx(n x r,-)+(cosx — 1) ( component of c,- perpendicular to n)\. (1.10.5g) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION REMARK The preceding rotation equations give the final position vector iy in terms of the initial position vector /*, and the various rotation vectors y, /, n (and x)- It is shown later in this section that, despite appearances, y is not a vector in all respects, but simply a directed line segment ; that is, it has some but not all of the vector character¬ istics (§1.1). This is a crucial point in the theory of finite rotations. Additional Useful Results (i) In the preceding rotation formulae: (a) For x = Firm (n = 1,2,3,...) they yield ly = i',-, (1.10.6a) that is, the body point returns to its initial position, as it should; and (b) If r,- • n = 0, and \ = 7t/ 2, then iy = n x iy, (1.10.6b) that is, n, r h /y form an orthogonal and dextral triad at O. (ii) By swapping the roles of /y and r, and replacing x with — x in (1.10.5g) (i.e., inverting it), we get e, = tycosx — (n x iy) sin\-+ (»*iy)n(l — cosx)- (1.10.6c) (iii) For small x, eqs. (1.10.5d, e) linearize to the earlier “Euler-Mozzi” formula: Vf = r t + i x r, => Ar = iy — r,- = / x r,-. (1.10.6d) Finite Rotation of a Line By using the rotation formulae, one can show that the final position of a body-fixed straight fiber joining two arbitrary such points P\ and P 2 , or 1 and 2 (fig. 1.20), is given by {'2/i) f = r 2 j ~ rij = ■■■ = (sin x)« x (r 2/1 ) ; + (cosx)(»-2/i); + (1 - cosx)[»- (^/OJb, (1.10.7a) where Initial position = (r 2 /i),- —> Final position = (r 2 / t h, (1.10.7b) and 1 * 2/1 = r 2 — r u for both i and/. (1.10.7c) Finite Rotation of an Orthonormal Basis By employing the finite rotation equations, let us find the relations between the two ortho-normal-dextral (OND) bases of common origin, O—uy (space-fixed) and WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Figure 1.20 Finite rotation of straight segment 12 , from ( 12 ), to ( 12 ) f . 0—u k (body-fixed), if the latter results from the former by a rotation x about an axis «; that is, symbolically, u k > u k . (1.10.8a) Applying the earlier rotation equations to this transformation, with /*, = u k ' and vj = u k , we obtain the following equivalent expressions: (i) u k = u k ' + (sinx)(« x u k <) + (cosx- !)«*',«, (1.10.8b) where Up n = u k i — ( u k t • n)n = u k f — (n 0 n) -u k i = (1 — n®n) • u k i = P • u k i = Component of u k t normal to n [P = projection operator , 1 = unit tensor (§1.1)]. (1.10.8c) (ii) u k = (cos x)«k' + (sinx)(» x u k ') + (1 - cosx)(« • u k <)n = [(cos x)l + (sin x)( n X 7 ) + (1 - cos x)(« ® »)] • u k ' = ( rotation tensor ) • u k < [examined in detail below] = up + (x«) x up (to the first order in x) ~ «k' + (X«) x u k , = up + / x up [Euler—Mozzi formula for small rotations], (1.10.8d) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION (iii) u k = u k ' + [2/(1+ 7 2 )] [y x u k , - 7 2 u k > + (y <8 7) • u k i] = » k ' + P/(i + 7 2 )][y x «*:' + yx(yx «*/)] = «*./ + [2y/(l + 7 2 )] x (u k > + y x «*/). (1.10.8e) To express the initial basis vectors u k ' in terms of the final ones u k , we simply replace in any of the above, say (1.10.8e), y with —y. The result is Uk' = u k - Py/(1 + 7 2 )] x {u k f - y x «*,). (1.10.8f) From the above, we can easily deduce that yu k = yu k (1.10.8g) as expected; or setting 7 = ^ kUk = Ik’Uk', (1.10.8h) in component form Ik = Ik'- (1.10.8i) The Tensor of Finite Rotation Let us express the earlier rotation equations in direct/matrix and component forms. Along the rectangular Cartesian axes O—XYZ = 0—X k , common to all vectors and tensors involved here, and with the component notations (k = X , Y,Z): r i = (X k ), r f = ( Y k ) , 7 = ilk- Rodrigues parameters) => 7 2 = ^ lk 2 = ( 7z ) 2 + ( 7y ) 2 + ( 7z ) 2 , n = ( n k : direction cosines of unit vector defining the axis of rotation), (1.10.9a) our rotation equations become r f = Rr i , Y k = Y J R ki x i = Yy*l( 1+7 2 )]*/, (1.10.9b) where, recalling (1.10.2efif.) and the simple tensor algebra of §1.1, the (nonsym- metrical but proper orthogonal) tensor of finite rotation , R = R{n, x) = ( Rki ) = {r k i /(1 + 7 2 )), has the following equivalent representations. WWW.EBOOK777.COM (1.10.9c) CHAPTER 1: BACKGROUND (i) Direct/matrix form (with N\ antisymmetric tensor of vector n ): R =1 cos XHf - N sin x + n <g n( 1 — cosx) 0 ( 0 -«Z n Y \ = 0 1 0 cosx + «z 0 -n x sinx 0 J \ ~n Y n x o ) "x 2 n x n Y n x n z N + n Y n x n Y 2 Hy« Z (i \ n z n x n z n Y 2 «z ) / C X + "x 2 (1 ~ cx) ~n z sx + n x n Y (1 - cx) n Y s\ + n x n z (1 - cx) \ = nzsx + n x n Y ( 1 - cx) cx + «y 2 ( 1 - cx) -«x^X + «y«zO - cx) V ~ n y s X + »y«z(l - cx) + «y«z(l - cx) cx + « z 2 (l - cx) / = R(n x ,n Y ,n z ;x), under n x 2 + n Y 2 + n z 2 = 1, (1.10.10a) where, as usual, c(...) = cos(...), s(.. .) = sin(...). (ii) Indicial (Cartesian tensor) form [with IV = (Nki), n = (n k )]: R k i = R k i(n r , x) = (Ski) cos x + (N k i) sin x + n k n,{ 1 - cos x) = (Ski) cosx + (5^ e w«,-) sinx + n*H/(l ~ cosx). ( 1 . 10 . 10 b) Occasionally, the rotation formula is written as Ar = R'-r, where Ar = rf — r,-, r = r (1.10.10c) and R' = (R' kl ) =R — 1 = ( R k i — Ski ); rotator tensor, R'ki = Rki-S k i = ---= (51 e krin r ) sinx + (n k n t - $ w )(l - cosx)- (1-lO.lOd) We notice that the representation (l.lO.lOd) coincides with the decomposition of R' kt into its antisymmetric part: (ekri n r ) sin x = N k i sin x , and symmetric part: (n k n, - 6 k ,)( 1 - cosx); of which, the former is of the first order in x, while the latter is of the second order, a result that explains the antisymmetry of the angular velocity tensor [(1.7.22e)]. WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION (iii) In terms of the Rodrigues parameters (a form, most likely, due to G. Darboux): ('«) / 1+7 X 2 - ( 7a 2 + lz 2 ) !{pixlY + lz) V ^{ixlz - 7 y) -{iXlY - lz) 1 + 7y 2 - {.lz + 7a- 2 ) 2(7a7z + 7a) 2(7a7z + 7a) \ 2(7a7z - 7a) 1 + 7z 2 - (7 a 2 + 7a 2 ) / (l.lO.lOe) The properties of R can be summarized as follows: (i) lim /?(«, x) = R(n, 0) = 1 , for all n ; x—>o that is, R(n, x) is a continuous function of X- (ii) R{n, x) • n = 11 ! n = axis of rotation. (1.10.11a) (1.10.11b) (iii) R(n,X i) -R(n,X 2 ) =R{n,X 1 + X 2 ) ■ (1.10.11c) (iv) R(n,x) -tf T («,x) = 1 , (1.10.lid) Sc 1! So 1 s' II 5 a 1 (1.10.lie) Also, since the elements of R, Ry, depend continuously and differentiably on three independent parameters — for example, Euler’s angles (§1.12) — we can say that the Figure 1.21 Plane rotation about Oz, through an angle x- WWW.EBOOK777.COM 164 CHAPTER 1: BACKGROUND rotation group is a continuous one; or a Lie group ; see, for example, Argyris and Poterasu (1993). Plane Rotation This is a special rotation in which y= (lx = 0,7r = 0,7z = tan(x/2)) = tan(x/2)« => n = K (1.10.12a) Then, with X k = X, Y and Y k = X', Y' (fig. (1.10.2g), and so on, specialize to 1.21), the rotational equations. A' = [(1 - 7 2 )/(l + r)]X - [27/(1 + 7 2 )] Y = -- = (cos x)X + (- sin x) Y, (1.10.12b) Y' = [2 7 /(l + r)]X + [(1 - 7 2 )/(l + 7 2 )] Y = ~. = (sin x)A + (cos x) Y, (1.10.12c) z' = z. (1.10.12d) Additional Useful Results (i) Alternative expressions of the rotation tensor: (a) Indicial notation: Rkl = Ski + ( £ krl n r) sin x + ( n k ni - 5 a ){ 1 - cos x) = Ski + N/d sin x + y~]Af fa lV ri (l - cosx) (1.10.13a) (b) Direct/matrix form [N = (N/f) antisymmetric tensor of vector n = (nf)\. R = 1 +Asinx + 21V-iVsin 2 (x/2) (1.10.13b) = 1 + (sinx)iV+ [2sin 2 (x/2)]iV 2 (1.10.13c) = 1 + (sin x)N + (1 - cosx)N 2 (1.10.13d) = J +2iVsin(x/2)[f cos(x/2) +iVsin(x/2)] (1.10.13e) [Notice that 1 — cos x = 2 sin 2 (x/2) and = n^n/ — Sy, or, in direct notation, N-N = n <S> n — 1. See also Bahar (1970)]. (ii) By swapping the roles of r , and /y and setting \ —>• — Xi nl the preceding rotation formulae, one can show that n=R l r f , (1.10.14a) where R 1 =1 —Asinx + 21V"IVsin 2 (x/2) = R J = R(n, ~x) ; (1.10.14b) that is, the rotation tensor is indeed orthogonal. WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION (in) Let P = ;Vtan(x/2): antisymmetric tensor of the Gibbs vector 7. By applying the Cayley-Hamilton theorem to P [i.e., every tensor satisfies its own characteristic equation (§1.1)], A( A) = |r- A7| = 0 => A{r) = -r 3 - [tan 2 (x/2)]r = 0, (1.10.15a) (since 7> P = 0 and Det P = 0), one can show that /? = 7 + 2cos 2 (x/2)(r + r 2 ), r = (i - ry 1 -(i + r) . (l.io.isb) Next, expanding (1.10.15b) symbolically in powers of P, we obtain the representa¬ tion R = (7 + P + ■■■)•(! + P) =1 +2r, to first r-order; (1.10.15c) => R'=R-l=2r , to first r-order. (1.10.15d) [Equations (1.10.15c, d) shed some light into the meaning of 7 and P, and prepare us for the treatment of angular velocity later in this section.] Similar results can be obtained in terms of N. The Mathematical Problem of Finite Rotation Usually, this takes one of the following two forms: (i) given x an d «, find R: or (ii) given R. find x an d n - Now, from the preceding indicial forms, we easily obtain (with k = X,Y, Z): (i) TrR = y R kk = cosxf+ sin x (EE £krk n r + (1 — cosx) (EE n k n^j = cosx(3) + sinx(0) + (1 - cosx)(l) = 2cosx + 1. (1.10.16a) ^■skl^krl^r (-) EE £ ski^kl — cos X (E E e skAl) + sin x (EEE + (1 - cosx) (EE £ skl n k n / S j = cosx(O) + sinx(E (~ 2 S rs)n r ) + (1 - cosx)(« x n) s = — 2(sinx)«s [Thanks to the e-identities (1.1.6b ff.)]. (1.10.16b) In sum, 7] = TrR = Rki- = 1 + 2cosx = First invariant of R , (1.10.16c) -EE £ skiRki = 2R S = 2{Axial vector ofR) s = 2(sin x) n s => Rk = (sin x) n k , (1.10.16d) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND or, explicitly, R { = (— 1 / 2 ){e 122 R 22 , + e 132^32) = (^32 — ^23,)/^- = ( s i n X) w l> ( 1 . 10 . 16 e) R 2 = (—l/ 2 )(e 2 3 i^ 3 i + £213-^13) = (^13 ~ ^ 3 i )/2 = (sin x) ,z 2? ( 1 . 10 . 16 f) Rt, = (—l/ 2 )(e 312 i?i 2 + £321^21) = (^21 — *i 2 )/2 = (sin x) n 3- ( 1 . 10 . 16 g) Now, the first problem of rotation is, clearly, answered by the earlier rotation for¬ mulae (1.10.10 ff.); while the second is answered by solving the system of the four equations (1.10.16c, e-g) for the four unknowns x; « 123 . Indeed, (i) From (1.10.16c), we obtain cos x = (Ii ~ l)/2 = (Tr /? — l)/2 . (1.10.17a) (a) From (1.10.16e g), if sinx ^ 0, «i = (^32 - ^23)/2sinx, n 2 = {R u ~ R 3 i)/2sinx, « 3 = (R 21 - ^i2)/2sinx, (1.10.17b) or, vectorially, n = (1 /n') [(/? 32 — /? 23 )/ + {Ru — Rji)J + (R21 ~ ^12)^"] i where «' = 2 sin x = • • • = [(1 + Tr/?) • (3 — Tr/?)] 1 / 2 : normalizing factor , (1.10.17c) (b) If sinx = 0^ then X = 0 or ±7r (or some integral multiple thereof); (b.l) If x = 0, then, as (1.10.11a) shows, R = {Ru) = {Ski) = 1\ that is, n becomes undetermined: no rotation occurs ; while (b.2) If x = ±tt =>• cosx = —1, then, as (1.10.10 ff.) show, R = {Ru) = (2 n k n, - 8u) (a symmetric tensor) /2»i 2 — 1 2/?| n 2 2«|H 3 \ = 2n 2 n\ 2n 2 2 — 1 2 n 2 n 2 , \ 2n 3 »i 2n 3 « 2 2 n 2 — 1 / or, explicitly, R n =W-\ =* m =±[{l + R n )/2}' /2 , 7? 22 = 2 n 2 — 1 =>■ n 2 = ±[(1 + /? 22 )/ 2 ] ^ , /? 33 = 2« 3 2 — 1 => « 3 = ±[(1 + /? 33 )/ 2 ] 1/,_ , (1.10.17d) (1.10.17e) (1.10.17f) (1.10.17g) and the ultimate signs of n l 2 3 are chosen so that (1.10.17e-g) are consistent with the rest of (1.10.17d): «1*2 - *12/2 - *2l/2, n l n 3 — /? 13 /2 — Rji/2, « 2 « 3 —/? 23 /2 —/? 32 /2. WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION The angle x can also be obtained from the off-diagonal elements of R as follows: multiplying (1.10.17b) with /q, n 2 , n 3 , respectively, adding together, and invoking the normalization constraint tp 1 + n 2 1 + n 3 = 1, we find si n X = (1/-)[ w i(-^32 " ^ 23 ) + ” 2(^13 ~ -^ 31 ) + ^ 3(^21 — ^ 12 )]■ (1.10.17h) Rotation as an Eigenvalue Problem (This subsection relies heavily on the spectral theory of § 1.1.) In view of the rotation formula r f = R-n, (1.10.18a) the earlier fundamental Eulerian theorem (§1.9: The most general displacement of a rigid body about a fixed point can be effected by a rotation about an axis through that point => that axis is carried onto itself: R • n = n) translates to the following algebraic statement: The real proper orthogonal tensor of rotation R has always the eigenvalue +1; that is, at least one of the eigenvalues of the eigenvalue problem (r f =) R- n = Ar,, (1.10.18b) equals +1; or, every rotation has an invariant vector, which is Euler’s theorem. Let us examine these eigenvalues more systematically. The latter are the three roots of |*-A2|=0 (A:Ai, 2 . 3 ), (1.10.18c) and it is shown in linear algebra that: (a) They all have unit magnitude [Since rp-rp = ( R ■ 77) ■ ( R ■r, ) = (77 • R T ) • ( R ■ 77) = r, ■ri = rp, the eigenvalue equation (1.10.18b) becomes r f . r f = = A 2 r, • 17 =>■ A 2 = 1 (for 17 7^ 0 )]; (b) At least one of them is real [From the corresponding characteristic equation: /1(A) = \R-X1\ = (-1) 3 A 3 + ••• + (DetR)\° = 0 , we readily see that limzl(A) = + 00 , and lim A (A) A—>+00 Hence, d(A) crosses the A axis at least once', that is, zl(A) = 0 has at least one real roof, and, by (i), that root is either +1 or —1.] (c) Complex eigenvalues occur in pairs of complex conjugate numbers [since the coeffi¬ cients of d(A) = 0 are real]; (d) I 3 (R) = / 3 — : Det R = lA^/l = R = AiA 3 A 3 = +1. [Initially, that is before the rotation, rj = R ■ ri = 77 => R = 1 => Det 1 = + 1, and since thereafter R evolves continuously from 1, it must be a proper orthogonal tensor, that is, \R\ = Det R = +1 — ^1(0). This expresses the “obvious” kinematical fact that, as long as we remain inside our Euclidean three-dimensional space, a right-handed coordinate system cannot change to a left-handed one by a continuous rigid-body WWW.EBOOK777.COM CHAPTER 1: BACKGROUND motion of its axes', such “polarity” changes, called inversions or reflections , require continuous transformations in a higher dimensional space; for example, right- handed two-dimensional axes can be changed to left-handed two-dimensional axes by a continuous rotation inside the surrounding t/tree-dimensional space.] Combining these results, we conclude that either: (i) All three eigenvalues of R are real and equal to +1; which is the trivial case of the identity transformation-, or, and this is the case of main interest (Euler’s theorem), (ii) Only one of these eigenvalues is real and equals +1 [ => zf(l) = \R — 1\ = 0]; while the other two are the complex conjugate numbers: cosx± / sin \ = exp(±;'x)- As a result of the above: (a) The direction cosines of the axis of rotation n = (n x ,n Y ,n z ) can be obtained by setting in eq. (1.10.18b) A = 1, r, = n: (R-\l)-n = 0^R-nmn, (1.10.19a) and then solving for n x Y z under the constraint n x 2 + n y 2 + n z 2 = 1; and (b) The invariants of R can be summarized as follows: I\ (R) = TrR = + R 22 + R 33 = Ai + A 2 + A 3 = 1 + exp(-H'x) + exp(—i'x) = 1 + 2cosx ; (1.10.19b) I 2 (R) = [(TrR) 2 - Tr(R 2 )}/2 = ( DetR)(TrR ~ l ) = (+l)(TrR T ) = (+l)(Tr *)=/,(*) [ = A| A 2 + A[A 3 + A 2 A 3 = (l)[exp(i'x)] + (l)[exp(-/x)] + exp(/x) exp(-ix) = 2cosx + l]; (1.10.19c) h(R) = DetR = AiA 2 A 3 = +1; (1.10.19d) that is, R has only two independent invariants. Composition of Finite Rotations Here we show that finite rotations are noncommutative; specifically, that two or more successive finite rotations of a rigid body with a fixed point O (or, generally, about axes intersecting at the real or fictitious rigid extension of the body) can be reproduced by a single rotation about an axis through O ; but that resultant or equivalent single rotation does depend on the order of the component or constituent rotations. Quantitatively, let the rotation vector carry the generic body point position vector from iq to r 2 ; and, similarly, let y 2 carry r 2 to r 3 . We are seeking to express the vector of the resultant rotation yj 2 (i.e., of the one carrying iq to r 3 ) in terms of its “components” yj and y 2 . Schematically, 7i 72 _ / y 12 (1.10.20a) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION By Rodrigues’ formula (1.10.4b), applied to iq —> r 2 and r 2 —> r 3 , we obtain = 7 i x ( i -2 + i-i), 1-3 ~t 2 = y 2 x (r 3 + r 2 ), ( 1 . 10 . 20 b) respectively. Now, on these two basic equations we perform the following opera¬ tions: (i) We dot the first of the above with y { and the second with y 2 : 71 • (r 2 - n) = yi • [yj x (r 2 + Tj)]= 0 =>• Vi -r 2 = yi -r u (1.10.20c) 72 • Ob - r 2 ) = y 2 • [y 2 X (r 3 + f 2 )] = 0 =>• y 2 • t 3 = y 2 • t 2 . (1.10.20d) (ii) We cross the first of (1.10.20b) with y 2 and the second with yj and subtract side by side: y 2 x (r 2 - i-i) - y 3 X (r 3 - r 2 ) = (y 3 + y 2 ) xr 2 -y 2 x c,-y, xr 3 = y 2 x [ yi x (t 2 + n)] - y { x [y 2 x (r 3 + r 2 )] = {yita-fo + 'i)] - (yi • y 2 )( r 2 + r i)} - biiyi • fa + r 2 )] - fa • 7 2 )fa + r 2 )} [expanding, and then rearranging while taking into account ( 1 . 10 . 20 c, d)] = [(72 • '*2 + 72 • 'h fa - fa • 7i) r 2 - fa • y 2 fa] - [(yj • r 3 + y x • r 2 )y 2 - fa • y 2 )r 3 - fa • y 2 )r 2 ] = [(72 • 'h + 72 • )7i - (7i • 72>2 - (7i • 72)**l] - [( 7 i • *3 + 7 i • **i )72 - ( 7 i • 72)^3 - ( 7 i • 72)^2] = [(72-'*3 + 72-'*i)7i] " [(71 •''3+71 -''1)72] ^ (71 *72)('"l -'h) = [(72 • r 1)71 - (7i •'h)72] + [(72-''3)71 - (7i •''3)72] - ( 7 i * 72)(''l -**3) = (72 x 71 ) x r x + (72 x 71 ) x r 3 - (y, -y 2 )fa - r 3 ) = (72 x 7 i) x fa +r 3 ) + (71 • 7 2 )fa - fa, ( 1 . 10 . 20 e) or, equating the right side of the first line with the last line of ( 1 . 10 . 20 e) and rearrang¬ ing, (7i + 72 ) xr 2 = y 2 xr 1 +7 1 x»'3 + (72 x 7 i) x fa +r 3 ) + fa • y 2 )fa - fa. ( 1 . 10 . 20 f) (iii) We add (1.10.20b) side by side and rearrange to obtain r 3 - n = yj x (r 2 + fa + y 2 x fa + r 2 ) = y x x r 2 + y x x r, +y 2 xc 3 + y 2 xr 2 => (7i + 72 ) x r 2 = r 3 - r x -y { x r, - y 2 x r 3 . (1.10.20g) (iv) Finally, equating the two expressions for fa + y 2 ) x r 2 , right sides of (1.10.20f) and (1.10.20g), and rearranging, we obtain the Rodrigues-like formula [i.e., a la (1.10.4b)] *•3 “''I = 7t,2-fa + fa, (1.10.20h) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND where ri ,2 = 7i— >2 = bi + 72 + 72 x 7i]/(1 — 7i * 72 ) = Resultant single rotation “vector,” that brings f] to r 3 . (1.10.20i) This is the sought fundamental formula for the composition of finite rigid rotations. [For additional derivations of (1.10.20h, i) see, for example, Hamel (1949, pp. 107 117; via complex number representations and quaternions), Lur’e (1968, pp. 101 104; via spherical trigonometry); also, Ames and Murnaghan (1929, pp. 82-85). The above vectorial proof seems to be due to Coe (1938, p. 170); see also Fox (1967, p. 8); and, for a simpler proof, Chester (1979, pp. 246-248).] In terms of the corresponding rotation tensors, we would have (with some ad hoc notations). r t —t rf\ rf = R\ • r,, (1.10.21a) rp —>• r f : r f = R 2 - r f , = R 2 (R\ ■ rf) = R IX ■ r l , (1.10.21b) where R\ 2= R-2 * R\ ( 7 ^ R\ • i ?2 = /? 2,1 ) : resultant rotation tensor. ( 1 . 10 . 21 c) REMARKS ON 7! 2 (i) Equation (1.10.20i) readily shows that the y’s are not genuine vectors; as the presence of y 2 x 7 ] there makes clear [or the noncommutativity in ( 1 . 10 . 21 c)], in general, finite rotations are noncommutative. Indeed, had we applied y 2 first, and y x second, the resultant would have been [swap the order of 7 ! and 7 2 in ( 1 . 10 . 20 i)] (72 + 7i + 7 i x 72)/(1 — 72 * 7i) = 72,1 = 72^1 ^ 7i,2 = 7i->2- (1.10.22a) For rotations to commute, like genuine vectors, the term 7 2 x 7 , must vanish, either exactly or approximately. The former happens for rotations about the same axis; and the latter for infinitesimal (i.e., linear) rotations: there, y 2 x y x = second-order quantity « 0. (ii) If 7 i *72 = 1, the composition formula ( 1 . 1 0.20i), obviously, fails. Then, the corresponding “resultant angle” 2 is an integral multiple of 7 r. (iii) From (1.10.20i) it is not hard to show that 1/(1 + 71, 2 2 ) 172 = (1 — 72 • 7 i)/[(l + 7 i 2 ) 1/,_ (l + T 2 “) 1/_ ], ( 1 . 10 . 22 b) and combining this, again, with ( 1 . 10 . 20 i) we readily obtain 71 , 2 /(1 + 7i,2 2 )' A = [7i + 72 + 72 x 7i]/[(l + 7C) 172 (1 + 72 2 )* 7 "] (1.10.22c) [which is the formula for the vector part of a product of two (unit) quaternions; see Papastavridis ( Elementary Mechanics, under production)]. Finite rotations may not be commutative, but they are associative: the sequence of rotations, expressed in terms of their 7 vectors—for example, 7 t — 7 2 — 73 —can be achieved either by combining the resultant of y x —*• y 2 with y 3 , or by combining y x with the resultant of y 2 ^ y 3 . In view of this, the sequence —y x ——> y 2 is equiva- WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION lent to the rotation y 2 , and also to the sequence —y x —> 2 . Therefore, if in the fundamental “addition” formula (1.10.20i) we make the following replacements: 72-^71,2, 71,2^ 72. (1.10.23a) we obtain the “subtraction” formula: 72 = [-7i + 7i,2 + 7i,2 x (— 7i)]/[l - (— 7i) ' 71 , 2 ]) (1.10.23b) or, finally, 72 = [ 71,2 — 7i +7i x 71 , 2 ]/(! +7i - 71 , 2 ): (1.10.23c) which allows us to find the second rotation “vector” from a knowledge of the first and the compounded rotation “vectors.” Similarly, to find y, from y 2 and y 12 , we consider the rotation sequence y X2 —> —y 2 , which, clearly, is equivalent to the rota¬ tion y x . Hence, with the following replacements: 7i~ > 7i,2. 72^-72: 7i,2 —* 7d (1.10.23d) in (1.10.20i) we obtain the “subtraction” formula: 7i = (7i,2 - 72 + 7 i,2 x 72 )/(1 + 72' 71 , 2 )- (1.10.23e) With such simple (and obviously nonunique ) geometrical arguments, we can avoid solving (1.10.20i) for y x , y 2 . (These results prove useful in relating y to the angular velocity to.) Infinitesimal (Linearized) Rotations Commute First, let us apply the infinitesimal rotation y x to r t [recalling (1.10.6d)]: ~*r\' = r, + dr t = r, + Zi x f,. (1.10.24a) Next, let us apply z 2 to rfi: '■| >y = r 1 + dr 1 = r, + Z 2 X r j = (c + Zi X V,) + 1 2 x (r,- + *i x r,-) = r i + (Zi + Z 2 ) x n + x 2 x (z, x r,). (1.10.24b) Reversing the order of the process—that is, applying i 2 first to r h and then to the result — we obtain r f = n + dr x =r x + Zj x r, = ('*,■ + Z 2 x r,) + zi x (r,- + z 2 x = r i + (/2 + Zi) X /*,• + Z| X (z 2 X *■/); (1.10.24c) and, therefore, subtracting (1.10.24c) from (1.10.24b) side by side, we obtain r f' ~ r f" = Z 2 x (Zi x r,) - zi x (z 2 x r,) = second-order vector in Zi,Z 2 i (1.10.24d) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND that is, to the first order in / l5 ii- r/ = r f ", Q.E.D. (1.10.24e) Similarly, for an arbitrary number of infinitesimal rotations Zi j 3 C 2 > - - ■ j t0 the first order: r f = r i + (Zi + li d-) x n- (1.10.24f) Angular Velocity (i) Angular Velocity from Finite Rotation Expanding the rotation tensor (1.10.1 Oe) [with (1.10.9b)] in powers of 7 XY z^ and since (with customary calculus notations) V = tan(x/ 2 )« = (x/ 2 )« + 0 (x 3 ) = z /2 + 0 (x 3 ), ( 1 . 10 . 25 a) we find R = / 1 -27 z 27 r \ 2 7z l -2 lx + V 27 y 27 X 1 / [Linear rotation tensor = R a ] o{j 2 ) /I 0 / 0 -27 z 27F \ 0 1 0 + 27z 0 -2y x \0 0 1 ) l -27 Y 27z 0 / + 0 ( 7 2 ) [Identity tensor] [Linear rotator tensor = R„' (recall (1.10. lOd, 15d))] (1.10.25b) / 0 -«z «y \ 7 + «z 0 -«x X+0{x 2 ), (1.10.25c) \-n Y «x 0 ( 0 -Xz Xy n 7 + Xz 0 -Xx + o( X 2 y, (1.10.25d) \-Xr Xx 0 , and, with the notations r,- = (X, Y,Z) = r, r f = r i + Ar i = ( X + AX , Y + A Y, Z + AZ) = r + Ar, (1.10.25e) we obtain, to the first order in the rotation angle, r+ Ar = R 0 -r= ( l+R 0 ')-r => Ar = Rj - r, (1.10.25fl) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION or, in extenso, /AX\ ( 0 -Xz Xy \ ( x \ H=i Xz 0 -Xz 4 (1.10.25f2) \iZ) ' V~Xr Xx 0 / U/ This basic kinematical result states that any orthogonal tensor that differs infinitesi¬ mally from the identity tensor, that is, to within linear terms, differs from it by an anti¬ symmetric tensor. Finally, dividing (1.10.25fl, 2) by At, during which Ar occurs, assuming continuity and with the following notations: Ytm{AX/At)\ At _^ = dX/dt = v z , etc., Y\m(xx/At)\ At ^ = w x , etc., we obtain the earlier found (1.9.1) fundamental kinematical equation of Poisson: or, in direct notation, v = dr/dt = il ■ r — <>> x r , where £2 = \xm(R 0 ' / At)\ A angular velocity tensor , a> = lim(27 / At)\ Ar „: angular velocity vector (axial vector of fl — a genuine vector!). (1.10.25g) (1.10.25h) (1.10.25i) (1.10.25j) As shown below [(1.10.26f)], (a) the velocities of the points of a rigid body moving with one point fixed are, at any instant, the same as they would be if the body were rotating in the positive sense about a fixed axis through the fixed point, in the direction and sense of co and with an angular speed equal to |o|; and, (b) since both r and v are genuine vectors, so is co (a fact that is re-established below). From all existing definitions of the angular velocity, this seems to be the most natural; but, in return, requires knowledge of finite rotation. (ii) co is a Genuine Vector Using the Rodrigues equation (1.10.4b): rf — r t = y x (r, + ly), (1.10.26a) let us prove directly that the angular velocity m, defined as co = lim(2y/zh)|^ r ^ 0 , where y = tan(x/2)«, (1.10.26b) is a genuine vector, even though y is not. PROOF With this in mind, we introduce the following judicious renamings: rj = r, Vf = l'/ + Ar = v + Ar. (1.10.26c) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Then, eq. (1.10.26a) yields Ar = y x [(rx Ar) + r] = y x (2r + Ar ) = (2y) x (r + Ar/2). (1.10.26d) Dividing both sides of the above by At, and then letting At —> 0 (while assuming existence of a unique limit as Ar —> 0), we obtain v = Xim{Ar/At)\ At ^ = lim[(2 y/At) x r]\ At _^ + lim[2y x {Ar/2)\\ At ^ = G)xr+0 = tt)Xr (v, r : vectors =>■ m : vector); Q.E.D. (1.10.26e) The physical significance of a> is understood by examination of the following case: \ = constant, in the direction and sense of the constant unit vector n. Then, with \ —^xAt=> y = [tan(xxlt/2)]«, and so (1.10.26b) specializes to: to = lim(27 /At) Q = n lim 2[tan{xAt/2)\ \ Jar-* 0 ■ ■ ■= xn ; (1.10.26f) i.e., here, m has the direction and sense of n (= instantaneous rotation axis), and length equal to the angular speed. Hence, Poisson’s formula, (1.10.25h), allows us to draw the conclusions following (1.10.25j). To complete the proof, let us next show that the line segments <u indeed commute. Dividing the composition of rotations equation (1.10.20i) 73 = 7 1,2 = (7i + 72 + 72 x 7i) /(1 -7i -72) (1.10.26g) by At/2, we get 2 y- i /At= [2yJ At + 2y 2 /At + (At/2)(2y 2 /At) x (2yJAt)] / [l - (At/2) 2 (2y 2 /At) • (2 yi /At )\; and then letting At — > 0, while recalling the earlier o-definition (1.10.26b, f), we find o) 2 = ^ 1,2 = + tn 2 — o 2 = o 2 \, (1.10.26h) that is, simultaneous rn’s obey the parallelogram law for their addition and decom¬ position, Q.E.D. (iii) to <-> y Differential Equation Let us consider a rigid body B with the fixed point O. Its instantaneous angular velocity m is related to its Gibbs “vector” y, which carries a typical Zi-particle from >•/= r(t) to iy = r(t + At), (1.10.27a) by a differential equation. The latter is obtained as follows: in the composition of rotations equation (1.10.20i) and in order to create the difference Ay there, we choose the rotation sequence 7i = —7 -> 72 = 7 + ^7, (1.10.27b) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION which, clearly, is equivalent to the single rotation y 12 = Ay, and occurs in time At. With these identifications in (1.10.20i), the earlier angular velocity definition yields m = lim(2dy/d?)|^ 0 = 2{lim(dy/d?) |^ 0 } = 21im [(l/dt){[(-y) + (y + Ay) + (y x Ay)\/{\ - (-y) • (y + dy)]}]|^ 0 = 21im{[(dy/df) + y x (dy/dt)] / [l + y • y + y • dy] }|^ 0 , or, finally, to = [2/(1 + 7 2 ))[dy/dt + y x (dy/dt)]. (1.10.27c) This remarkable formula, due to A. Cayley ( Cambridge and Dublin /., vol. 1, 1846), shows that, in general, m and dy/dt are not parallel! REMARK Equation (1.10.27c) also results if we apply to the formula for the subtraction of rotations (1.10.23c), the sequence y, = y — Ay — > y 2 = Ay', (1.10.27d) which is equivalent to yj 2 = y. Thus, we obtain Ay' = [y - (y - Ay) + (y - Ay) x y]/[1 + (y - Ay) • y] = (Ay + y x Ay) /(I + 7 2 — y • Ay), (1.10.27e) then divide by A t and take the limit as A t —> 0 to obtain c« = 2hm(dy7dt)|^ 0 = 2{[lim(dy/d?) + y x lim(dy/dt)]/(l + 7 2 - y • Ay)} |^ 0 = [2/(1 + 1 2 )][dy/dt + y x {dy/dt)], (1.10.27f) as before. The reader may verify that the sequence yj =y —> y 2 = Ay', which is equivalent to y { 2 = y + Ay, also leads to the same formula. (iv) Inversion of the Preceding Formula ca = to(y, dy/dt) First Derivation. Dotting both sides of that equation, (1.10.27c), by y yields ym= [2/(1 + 7 2 )] [y • {dy/dt)]\ (1.10.28a) while crossing it with y gives y x m = [2/(1 + 7 2 )]{y x (dy/dt) + y x [y x (dy/dt)]} = [2/(1 + 7 2 )]{y x (dy/dt) + [y • (dy/dt)\y - ^(dy/dt)}. (1.10.28b) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Eliminating y-(dy/dt) between (1.10.28a,b) produces y x ct) = [2/(1 + 7 2 )]y x (dy/dt) - [2 7 2 /(l + 7 2 )](dy/dt) + (y • co)y [expressing the first right-side term of the above via (1.10.27c)] = { t °- P/(i + 7 2 )](^yM)} ^ p7 2 /(i + 7 2 )K^yM) + (r <o)y = to — 2(dy/dt) + (y • to)y, (1.10.28c) or, rearranging, finally gives 2 (dy/dt) = to + (y • a>)y + to x y; (1.10.28d) which, for a given to(t), is a vector first-order nonlinear ( second-degree) differential equation for y(t) (and can be further reduced to a “Ricatti-type equation”). Equations (1.10.27d), and (1.10.27c) clearly demonstrate the one-to-one relation between to and dy/dt: if one of them vanishes, so does the other. Second Derivation. Applying the earlier rotation sequence y x =y^y 2 = Ay', (1.10.28e) which is equivalent to y { 2 = y + dy, both occurring in time At, to the composition formula (1.10.20i) we obtain y + Ay = (y + Ay' + Ay' x y)/(l — Ay' •y ), (1.10.28f) from which, subtracting y, we get Ay = [Ay' + (y • Ay')y + Ay' xy]/(l- y • Ay'), (1.10.28g) and from this, dividing by At and taking the limit as At —*• 0, while recalling that [eq. (1.10.27f)] to = 2\mfiAy'/At)\ At ^Q, we re-obtain (1.10.28d). For still alternative derivations of the to <-> y equations, via the compatibility of the Eulerian kinematic relation v = clr/dt = m / r with the d/dt(.. .)-derivative of the finite rotation equation /y = rj (y; r,) [eqs. (1.10.2-4)], see, for example (alpha¬ betically): Coe (1938, chap. 5; best elementary/vectorial treatment), Ferrarese (1980, pp. 122-137), Hamel (1949, pp. 106-107; pp. 391-393). (v) Additional Useful Results (a) Starting with y = «tan(x/2) => dy/dt = ( dn/dt ) tan(x/2) + n[(dx/dt)/2\ sec 2 (x/2), etc., and then using the to <-> y equation, we can show that to = ( dx/dt)rt + (sin x){dn/dt) + (1 — cosx)« x (dn/dt). (1.10.29a) WWW.EBOOK777.COM §1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION (What happens if n = constant ?) (b) Again, starting with y = n tan(x/2) = tan(x/2)(//x) = [tan(x/2)/x]z => dy/dt = ■ • ■ ,etc., and then using the to <-> y equation, we can show that a = (sin X /x)(dx/dt) + [(1 - cosx)/x 2 ][x x (dx/dt)} + [(1/X) - (sin x/x 2 )\(dx/dt)x = dx/dt + [(1 - cos x)/x 2 ][x x (dx/dt)] + [(X - sin x)/x’]{x x [x x (dx/dt)]}. (1.10.29b) (c) By inverting (1.10.29b), we can show that dx/dt = u> - (x x <»)/2 + (l/x 2 )[l - (x/2) cot(x/2)][x x (x x to)] . (1.10.29c) More in our Elementary Mechanics (§13.8 — under production). General Rigid-Body Displacement (i.e., no point fixed) We have already seen (§1.9) that the most general rigid-body displacement can be effected by the translation of an arbitrary base point or pole of it, from its initial to its final position, followed by a rotation about an axis through the final position of that point (see figs 1.12 and 1.22). Here, we show that the translational part of the above total displacement does depend on the base point, but the rotational part—that is, the rotation tensor — does not. Referring to fig. 1.22, let ll' = r p/t» PP"=r f/i , IP = r/\, l'P =r /v , l"P" =l'P" =r n », R\ = rotation tensor bringing l'P' to l'P"\ i.e., r^n = R\ -r/y. (1.10.30a) Figure 1.22 Most general rigid-body displacement; the rotation tensor is independent of the base point (or pole). r /1 -»• r /v -► r ir = ^ •'■/id fj = r\ + f \ fj, = r v + r /v r f = r v , + r /v , = r v , + R, ■ r jv (r v = r r ). WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Then, successively, PP" =P1+11' + l'P" = -r/i + r l7l + R t ■ r n ,, (1.10.30b) or since IP = l'P' (i.e., r/i = r^/), rfn = /•,'/! + (/?i -l)-r ,i . (1.10.30c) Had we chosen another base point, say 2, then reasoning as above we would have found (with some easily understood notations) r fn = r 2'a + ( r 2 -1) • r/i ■ (1.10.30d) But also, applying (1.10.30c) for P = 2, we have (since r 2 ' = r 2 ") r 2 'a = r i'n + ( R i ~ 1 )' r 2 n ■ (1.10.30e) Therefore, substituting (1.10.30e) in (1.10.30d) and equating its right side to that of (1.10.30c), we obtain r yn +(/?!- l)-r 2 /i + ( r 2 - l)-r /2 = r Vn + {R\ -l)-r n , from which, rearranging, we get (R 1 - 1) • (m - r 2 /i) = {R l -l)-r l2 = (R 2 -l)- r n , (1.10.30f) and since this must hold for all body point pairs P and 2 (i.e., it must be an identity in them), we finally conclude that Ri=R 2 =-=R. (1.10.30g) In words: the rotation tensor is independent of the chosen base point; it is a position- independent tensor. This fundamental theorem simplifies rigid-body geometry enormously and brings out the intrinsic character of rotation. (In kinetics, however, as the reader probably knows, such a decoupling between translation and rotation is far more selective.) 1.11 THE RIGID BODY: ACTIVE AND PASSIVE INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR; SUCCESSIVE FINITE ROTATIONS A 3 x 3 proper orthogonal tensor may be interpreted in the following consistent ways: (i) As the matrix of the direction cosines orienting two orthonormal and dextral (OND) triads, or bases, and associated axes; say, a body-fixed, or moving, triad t: t = (u k ) = relative to a space-fixed triad T; WWW.EBOOK777.COM (1.11.la) §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR T = («*,) = (ii) Then, since ( « x\ Uy \ u z) (1.11.1b) I = (/ • i)i + (I -j)j + {I • k)k = A Xx i + A Xy j + A Xz k, etc., cyclically, i = (i • /)/ + (* • J)J + (i • A - )A - = + ^y.v J + ^zx A, etc., cyclically, the two triads are related by T = A• t <^> t = A~‘.T = A t -T, (1.11.1c) where //•/ Ij I' k ) ^ Axx A Xy A Xz \ A = Ji Jj Jk = ''l Yx A Yy A Yz U Kj K-kj \Azx A Zy A Zz) = (A k 'k), A k ' k = cos(x k ',x k ) = u k <-u k [= cos (x k ,x k >) = A kk '}. (l.ll.le) The rotation of an OND triad, equation (l.ll.le), T —> t, constitutes the second interpretation of a proper orthogonal tensor. (iii) The third such interpretation is that of a coordinate transformation from the T-axes: 0—x k ' = O—XYZ to the t-axes: 0—x k = O—xyz (of common origin, with no loss in generality). In this interpretation, known as passive or alias (meaning otherwise known as), the point P is fixed in T-space and the t-axes rotate. Then [hg. 1.23(a)], OP = r = ^2 x k ,u k' = A/ + YJ + ZK = ^2 x k u k = xi + yj + :k. (1.11.2a) (a) PASSIVE (ALIAS) INTERPRETATION (b) ACTIVE (ALIBI) INTERPRETATION Figure 1.23 (a) Passive and (b) Active interpretation of a proper orthogonal tensor (two dimensions). WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and so we easily find x k > = r • u k , = ■ ■ ■ = A k'kXk, x k =r-u k = - = Y^ A kk'X k ' (= A k'k x k '), (1-11 -2b) or explicitly, in matrix form, (X) ^ A Xx A Xy A Xz \ Y = A Yx A Yy A Yz y UJ \ A Zx A Zy A Zz ) w r' = A • r Old axes New axes, (1.11.2c) ^ A Xx A Yx A Zx\ ( X \ y = A Xy A Yy A Zy Y \ z ; \ A Xz A Yz A Zz ) U/ r = A T • r' New axes Old axes. (1.11.2d) For example, in two dimensions [fig. 1.23(a)], the above yield ( X\ / cos x — sin %\ / x \ f x \ ( cos x sin x \ / X \ sin x cos x r' = A • r, J, sin x cos x : A t • r' (1.11.2e) (iv) Under the fourth interpretation, known as active or alibi (meaning elsewhere), the axes remain fixed in space, say T = t, and the point P rotates about O, from an initial position r, = XI + YJ + ZK to a final one »y = X' 1 + Y'J + Z'K. Then, following §1.10, and with A —> R (rotation tensor), (X'\ ( x ) (x\ Y' = R Y = A Y \z') \z) U / '/ = R • F Final position Initial position, (1.11,2f) (X) Y = R t (X'\ Y' = A J (x>\ Y' \z) \Z') \ z '! F = K T <7 Initial position Final position. (1.11 -2g) WWW.EBOOK777.COM §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR Equations (1.11.2f, g) hold about any common axes; and, clearly, the components of R depend on the particular axes used. For example, in two dimensions [fig. 1.23(b)], the above yield (X'\ = /cosx — sinx\ m (X\ = ( cos X sin X \ (X'\ \Y'J Vsinx cos X )\y) U/ l-smx cos X )\Y') r f = Rt,-, I", = R T • ry; (1.11.2h) and for the new triad (actually a dyad) i, j in terms of the old triad /, J [along the same (old) axes], they readily yield /cosx\ _ /cosx -sinx\/l\ /-sin x\ = /cosx -sinx\/0\ ysinx/ V sin X cosx/V 0 / V C0S X / V sin X cosx/W i = R • /, j = R-J. (1.11.2i) The passive and active interpretations are based on the fact that: The rigid body rotation relative to space-fixed axes ( active interpretation), and the axes rotation relative to a fixed body (passive interpretation) are mutually reciprocal motions. Hence [fig. 1.24(a, b)]: The coordinates of a rotated body-fixed vector along the old axes ( final position, active interpretation), equal the coordinates of the unrotated rigid body along the inversely rotated axes ( new axes, passive interpretation). It follows that if the body is fixed relative to the new axes and r' = XI + YJ, r = xi + yj , then the rotation equations—for example, (1.10.2e)—yields (with h + ^new (body-fixed) axes = ** and Vj > Told axes = * ) r' = [2/(1 +7 2 )][y X r+ (yc/yj + [(1 - 7 2 )/(l + 7 2 )]r. (1.11.3) A correct understanding of the above four interpretations—in particular, the inter¬ change of A with A t = A(—x) [and R with R T = R(—x)] in single, and, especially, successive rotations (see below)—is crucial to spatial rigid-body kinematics. Lack of it, as Synge (1960, p. 16) accurately puts it “can be a source of such petty confusion.” (a) ACTIVE INTERPRETATION (b) PASSIVE INTERPRETATION Figure 1.24 The final coordinates under x [active interpretation (a)] equal the new coordinates under —x [passive interpretation (b)], and vice versa. (X ^-rotated vector, old axes = Xjnrotaled vector,-\-rotated axes/ etc.) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND (i) (11) Below, we summarize these four interpretations of an orthogonal tensor A or R: A = (A k > k ) = («£/ • 14 ): Direction cosine matrix; (*) J = A j UJ w or T = A • t: Triad rotation; (X) ( x \ (hi) Y = A y Uv v z / or rold axes = r ' = A • r new axes = A • r. Passive interpretation (Vector fixed; axes rotated); { x '\ ( x \ (iv) Y' = R Y \Z') U/ or ly = R • r,: Active interpretation A = R {Vector rotated; axes fixed, and common). REMARKS (i) In the passive interpretation, we denote the components of A as A k > k , whereas, in the active one, we denote them, in an arbitrary but common set of axes, as R k/ (or This is an extra advantage of the accented indicial notation, especially in cases where both interpretations are needed. (ii) The passive interpretation also holds for the components of any other vector; for example, angular velocity. Successive Rotations Let us consider a sequence of rotations compounded according to the following scheme: T ^ T, ^ T 2 ^ ^ T n _, - T n = t A| A 2 A 3 A n _! A n (1.11.4a) Then we shall have the following composition formulae, for the various interpreta¬ tions. (i) Triad Rotation T=(A,.A 2 .A n ) • t O t=(A n T .A n _! T . A, t ).T; (1.11.4b) WWW.EBOOK777.COM §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR or, in extenso, (*) J (A| • A 2 • • ■ • A n ) ./' UJ U/ Initial triad (natural order Final triad, of component matrices) f 1 ] = (A n T -A n _ I T .- ••A, 7 ) J U / Final triad Initial triad. (ii) Passive Interpretation Here, with some easily understood ad hoc notations, we will have T 0 id axes = r' = Aj • r, = Aj • (A, • r 2 ) = ■ ■ ■ = (Aj • A 2 .A n ) • r , Tnew axes — T (A| • A2.A n ) • V (A n • A n _[ .A| ) * T , or, in extenso, ( x ) Y (Ai • A 2 • • ■ -A n ) (*\ y \z) \-J Old axes New axes, ( x ) ( X \ y = (A n T • A n _i T • • ••A, T ) Y U / New axes Old axes. (iii) Active Interpretation Here, choosing common axes corresponding to T; that is, /*,■ = XI + YJ + ZK -> r f = X'l + Y'J + Z'K (= Xi + Yj + Zk), WWW.EBOOK777.COM (1.11.4c) (1.11.4d) (1.11.4e) (1.11.4f) (111 4g) (1.11,4h) (1.11.41) CHAPTER 1: BACKGROUND we obtain, successively, r t = A, T • r n = A T , • (A t 2 • r f2 ) = ■ • • = (A, T • A 2 T .A n T ) • r f = (Rl T 'R2 T .RnV/, => jy = (A n • A n _].A!) Vj = (R n • Rn—1.Rl) •*';! or, in extenso, /x'\ ( X \ Y' (Rn • Rn-1 ‘ •• -Ri) Y [z'J U / Final position Initial position, ( X ) (X>\ Y = (R! T .R 2 T . • • -Rn T ) Y' UJ \Z'j Initial position (1.11.4]) (1.11.4k) (1.11.41) Final position. (1.11,4m) Body-Fixed versus Space-Fixed Axes The moving triad t and associated axes (O-xyz) may be considered as a rigid body going through a sequence of rotations, either about these body-fixed axes themselves, or about the space-fixed axes O-XYZ with which it originally coincided. Either of these two types of sequences may be used (although the tensors/matrices of rotations about body-fixed axes have simpler structure than those about space-fixed axes), and their outcomes are related by the following remarkable theorem: The sequence of rotations about Ox, Oy, Oz has the same effect as the sequence of rotations of equal amounts about OX, OY, OZ, but carried out in the reverse order. Symbolically, f R ] 1^2 (^y-fjxed lixcs (R 2 R[ (jpacp-fixedaxes' This nontrivial result will be proved in §1.12. Thus, for a sequence about space-fixed axes, eq. (1.11.4h) (which expresses the passive interpretation for a body-fixed sequence) should be replaced by /*\ y W (Si = (S n -S n—1 New axes S n T ) Si ) 1 n Y W n Y U/ Old axes, (1.11.4n) WWW.EBOOK777.COM §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR where the S k are the space-fixed axes counterparts (of equal angle of rotation) of the R k ; and similarly for the other compounded rotation equations. REMARKS (i) In algebraic terms, we say that such successive rotations form the Special Orthogonal (Unit Determinant) — Three Dimensional group of Real Matrices [=SO(3, R)], and are representable by three independent parameters; for example, Eulerian angles (§1.12). [By group, we mean, briefly, that (a) an identity rotation exists (i.e., one that leaves the body unchanged); (b) the product of two successive rotations is also a rotation; (c) every rotation has an inverse; and (d) these rotations are associative. See books on algebra/group theory.] (ii) Some authors call rotation tensor/matrix the transpose of this book’s, while others, in addition, fail to mention the distinction between active and passive inter¬ pretations. Hence, a certain caution is needed when comparing various references. Our choice was based on the fact that when the rotation tensor of the active inter¬ pretation is expanded a la Taylor around the identity tensor, and so on (1.10.25a ff), it leads to an angular velocity compatible with the definition of the axial vector (cn) of an antisymmetric tensor (1.1.16a ff.) f2: fl • r = u> x r; otherwise we would have f2 -r = — (o x r. Tensorial Derivation of the Finite Rotation Tensor Let us consider the following two rectangular Cartesian sets of axes, 0-x k t (= O—XYZ , fixed) and 0—x k (= O—xyz, moving), related by the proper orthogonal transformation: x k' = A k'k x k ^ x k = Yl A kk' x k'y A k'k = A kk' = cos (x k ',X k ). (1.11.5a) The corresponding components of the rotation tensor, R k t V and R kh respectively, will be related by the well-known transformation rule for second-order tensors (1.1.19j ff.): R k'l' — A k'k A l'l R kl ^ R-kl — ^2 A k'k A l'lRk'l'i (1.11.5b) or, in matrix form. R' = AR- A t <£> R = A T R - A, (1.11.5c) where R'= (**'/'), R = { R u) > A. = (Aw)- (1.11.5d) Here, choosing axes 0-x k in which R k/ have the simplest form possible, and then applying (1.11.5b, c), we will obtain the rotation tensor components in the general axes 0—x k ', R k 'i r , that is, eq. (1.10.10a). To this end, we select Ox k so that x\ = x is along the positive sense of the rotation axis «, while x 2 = y, x 3 = z are on the plane through O perpendicular to n (fig. 1.25). For such special axes, the finite rotation is a WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Figure 1.25 Tensor transformation of rotation tensor, between the general fixed axes O-XYZ and the special moving axes O-xyz; Ox axis of rotation. plane rotation of (say, right-hand rule) angle x about Ox, and, hence, there the rotation tensor has the following simple planar form: 10 0 R = I 0 cos x — sin x 0 sinx cos% / Now, to apply (1.11.5c) we need A. The latter, since i = A xX^ T A xY J T A X %K — ft “ nxl T tiyJ T becomes A = (1.11.5e) (1.11.5f) «X A Xy A Xz \ n Y A Yy a yz ; (1.11 -5g) «z A Zy A Zz / ...) = c(.. sin(. ..)=$(...), (1.11.5c) specia- lizes to R' = "x A Xy ( l 0 0 \ / n x n Y n z n Y A Yy A Yz b cx 1 to X X A Yy A Zv »z A Zy A Zz ) V° sx cx / \ Axz A Yz A Zz WWW.EBOOK777.COM §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR or, carrying out the matrix multiplications, and recalling that R\'\' = R xx , R { n' = R X y, and so on , Rxx = n X 2 + (^Xy 2 + A Xz ) COS X, R xy = n x n Y + (A Xy A Yy + A Xz A Yz ) cosx — (A Xy A Yz — A Yy A Xz ) sinx, Rxz = n x n z + (A Xy A 7) , + A X: A 7= ) cosx + (A- Zy A Xz — A Xy A Zz ) sinx; Ryx = n Y n x + (A Yy Axy + A Yz A X z ) cosx + (AxyAzz ~ A Yy A Xz ) sinx, R yy = n Y ~ + (A Yy 2 + A y 2 ) cosx, Ryz = n Y n z + (AY y A 7y + A Y: A Zz ) cos x — (A Y yA 7z — A Zy A Yz ) sinx; Rzx = n z n x + (A Zy A Xy + A Z: A Xz ) cos x — ( A Zy A Xz — A Xy A Zz ) sin x, Rzy — n z n Y + (A 7y A Yy + A Z: A Yz ) cosx + (A Yy A Zz — A Zy A Yz ) sinx, Rzz = n z + (A Zy 2 + A Zz 2 ) cos x- (1-11.5h) However, the nine A k ' k are constrained by the six orthonormality conditions: I • J = n x n Y + A Xy A Yy + A Xz A Yz = 0, J • K — A Yy-^Zy Yz^-Zz = 0, K • I = n z n x + A Zy A Xy + A Zz A Xz = 0; I ■ 1 = n x 2 + A x 2 + A Xz ~ = 1, J • J = fl Y A y y + A Yz ~ = 1 , K-K = n z 2 + A z 2 + A z 2 = 1; (l.ll.Si) and also n = u r x u z , or, in components, n x = A Y y A Zz — A Zy A Yz . n Y = A Zy A Xz — A Xy A Zz , n z = A Xy A Yz — A Yy A Xz . (i.n.sj) As a result of the above, it is not hard to verify that the R k 'i<, (1.11.5h), reduce to Rxx = n x~ + (1 — n x ~) cos x, Rxy = n x n Y + {-n x n Y ) cosx + («z) sinx, Rxz = n x n z + (-n x n z ) cosx + Or) sinx; Ryx = n Y n x + (~n x n Y ) cos x + (n z ) sinx, Ryy = n Y~ + (1 ~ n Y~) cos x, Ryz = »r«z + (~n Y n z ) cos x + (-«■y) sin x; Rzx = n z n x + (- n z n x ) cos x + (~n Y ) sinx, ^zr = »z«r + (-«z«y) cosx + («z) sinx, ^zz = + (1 — n z 2 ) cosx; (1.11.5k) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and when put to matrix form is none other than eq. (1.10.10a). We notice that the components are independent of the orientation of the O-xyz axes , as expected. Angular Velocity via the Passive Interpretation Let us consider a generic body point P fixed in the moving frame t: O ijk/O xyz, and hence representable by *•' = XI + YJ + ZK (space-fixed frame T: O—IJK/ O—XYZ) , (1.11. 6 a) r = xi + yj + zk (body-fixed frame; i.e. x,y,z = constant); ( 1 . 11 . 6 b) or, in matrix form, r' J = (X,Y,Z), r T = (x,y,z). (1.11.6c) According to the passive interpretation (1.11.2c) (with A replaced by the rotation tensor / matrix R), r = R-r, (1.11.6d) and, therefore, the inertial velocity of P, resolved along the fixed axes O XYZ equals v' = dr'/dt = (dR/dt) - r + R • ( dr/dt ) = (dR/dt) • r + R • 0 = (dR/dt) • (R t • r') = n'-r ' = o*' X r' , (1.11.6e) where [recalling (1.7.30fff.), with A —* R] Yl' = (dR/dt) • R t = angular velocity tensor of body frame t relative to the fixed frame T, but resolved along the fixed axes O-XYZ, (1.11. 6 f) ft* ' = axial vector of $Y; angular velocity vector of t relative to T. along T. (1.11. 6 g) The components of the angular velocity along the moving axes can then be found easily from the vector transformation (passive interpretation): v = inertial velocity of P , but resolved along the moving axes {not to be confused with the velocity of P relative to t, which is zero: drjdt = 0) = R t -v' = R t - [(dR/dt) -r\ = f l-r=oxr, (1.11.6h) where Cl = R t • (dR/dt) = angular velocity tensor of body frame t relative to the fixed frame T, but resolved along the moving axes O-xyz { = [R T • (dR/dt)] • (R t • R) = R t • [(dR/dt) • R T ] • R = R t • ff ■ R; a second-order tensor transformation, as it should be} , (1.11.6i) ft* = axial vector of angular velocity of t relative to T, along t [= R T • ft*']. ( 1.11 - 6 j) WWW.EBOOK777.COM §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR REMARK If R = R( 9l ,< 72 , ft) = R(ftJ, where the q a are system rotational parameters (e.g., the three Eulerian angles, § 1 . 12 ), then fY and to 1 can be expressed, respectively as follows: Tensor: ft' = ^ ft' a (dq a /dt ), Vector: to' = ^ to' a {dq a /dt ), (1.11.6k) where ft' a = (dR/dq a ) ■ R t and fl' a • x = (o' a x x , (1.11.61) for an arbitrary vector jc; that is, fY can be expressed in terms of the local basis {ft' a ', a = 1 , 2 ,3}; and similarly for ft and ft). Additional Useful Results 1. Consider the following two successive (component) rotations: First, from the “fixed” frame 0 to the moving frame 1, Ri / 0 = R ls and, next, from 1 to the also moving frame 2, R 2 /i = R 2 - Then, by (1.11.4a If.), the resultant rotation from 0 to 2 will be R = R] • R 2 . Now, let: fY/o,o = (dRi/dt) • Ri t : angular velocity tensor of frame 1 relative to frame 0 , along O-axes ; fY/o,i = R| T • (dRi/dt): angular velocity tensor of frame 1 relative to frame 0 , along 1 -axes ; ftz/ 1 , i = (dR 2 /dt) • R 2 t : angular velocity tensor of frame 2 relative to frame 1 , along 1 -axes ; fl 2/u2 = R 2 t • (dR 2 /dt): angular velocity tensor of frame 2 relative to frame 1 , along 2-axes ; fY/o,o = (dR/dt) -R 1 : angular velocity tensor of frame 2 relative to frame 0 , along O-axes ; fi 2 /o, 2 = R T • (dR/dt): angular velocity tensor of frame 2 relative to frame 0 , along 2-axes ; (1.11.7a) (this or some similar intricate notation is a must in matrix territory!) and therefore fY/o,o = Ri-tti/o.i- R i T ^ n I /o,i=Ri T -n 1 / o,o-R 1) fti/ 1,0 = Rt * fY/i. i • R| T <=> ftz/ 1 , i = Ri • S^ 2 /i,o • R| T ) ftz/ 1,1 = R2 • ftz/1,2 • R2 T ^ ftz/ 1,2 = R2 T • ftz/\. 1 ■ R2 ) ftz/t>, 1 = Rl T • ^2/0,0 ■ R| = R 2 • f^2/0,2 = R 2 T : angular velocity of frame 2 relative to frame 0 , but expressed along 1-axes; etc.; i.e., the multiplications Ri^-.^'R ] 7 convert components from 1-frame axes to 0-frame axes; while R] T (...) -R) convert components from 0-frame axes to 1-frame axes; and analogously for R 2 •(...) • R 2 T , R 2 T •(■■■)• R 2 - Then, and since R, R t , R 2 are orthogonal tensors, (a) fl 2/ 0,0 = (dR/dt) • R t = [d/dt(R, • R 2 )] • (R, • R 2 ) T = •■■ = (dRj/dt) -Rj+Ri-KdRj/dt) -Rj^.R , 1 = /o,o + Ri • ^ 2 / 1.1 ■ Ri T = fii/ 0.0 + ^ 2 / 1,0 WWW.EBOOK777.COM CHAPTER 1: BACKGROUND (theorem of additivity of angular velocities, along O-cuces ); (1.11.7b) ^2/0,1 = R] T - ^2/0,0 • Rl = R| T • J^l/0,0 ' Rl + ^2/1,1 = ^1/0,1 + ^2/1,1 (theorem of additivity of angular velocities, along 1-axes ); (1.11.7c) ^ 2 / 0,2 = R T • (dR/dt) [= R 2 T • rj 2/0 , i ■ R 2 = R T • ^ 2 / 0,0 • R] = (Ri-R 2 ) T -[d/dt(Ri-R 2 )] = • • ■ = R 2 t • [R! T • (dRi/dt)] • R 2 + R 2 t • (dR 2 /dt) = R 2 t • f 2 ]/o, 1 ■ R2 + ^2/1,2 = ^1/0,2 + ^2/1,2 (theorem of additivity of angular velocities, along 2-axes). (1.11.7d) (b) Next, d(.. ,)/dt-differentiating the above, say (1.11.7b), it is not hard to show that: d^ 2 /o,o/dt = dfl]/o,o/dt + d/dt(Ri • fi 2 /p 1 • Ri T ) = dn^o.o/dt + Ri • (dn 2/lj 1 /dt) • R] T + Ri • ! • n 2 /i! — n 2 /i j • • Ri T (theorem of non-additivity of angular accelerations, along 0-axes ); (1.11.7e) and similarly for dfi 2 /o, i/dt, dfi 2 /o, 2 /dt. The last (third) term of (1.11.7e) shows that if the elements of the matrices , fl 1 / 0 , 0 - ^ 2 / 1.1 tire constant, then, in general, the elements o/n 2/0 ,o will also be constant if f! 1 /o, 1 and f 2 2 /i,i commute , a well-known result from vectorial (undergraduate) kinematics. The extension of the above to three or more successive rotations is obvious. [As Professor D. T. Greenwood has aptly remarked: “Equations (1.11.7b—e) illus¬ trate how the use of matrix notation can make the simple seem obscure.”] 2. Matrix forms of relative motion of a particle , in two frames with common origin. By d/dt(.. ^-differentiating the passive interpretation (1.11.2c), Fixed axes Moving axes, (1.11.8a) we can show that (i) ( X ) f x\ d/dt Y = A- d/dt y + fi- y \z) K z / \ z ) { = A • [relative velocity + transport velocity]}. (1.11.8b) WWW.EBOOK777.COM §1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR ( X ) ( X ) d 2 /dt 2 Y = A- d 2 /dt 2 y + (dfi/dt) • y UJ \ z ) \ Z J ( X ) y + 2CI - d/dt y \ z j \ z j { = A- [relative acceleration (d 2 r/dr 1 ) + transport acceleration (a x r + a> x (a> x r)) + Coriolis acceleration (2a> x {dr/dt))]\ ; ( 1 . 11 . 8 c) we point out that, in the matrix notation, the d/dt vs. d/dt difference (§1.7) disappears. (iii) If the position of the origin of the moving axes, relative to that of the fixed ones, is r 0 = ( X D1 Y 0 ,Z 0 ) J , so that [instead of (1.11.8a)] (9 ■ (I) “■(;)■ then we simply add d/dt (X 0 ,Y 0 ,Z 0 ) T to the right side of (1.11.8b) and d 2 /dt 2 (A OI Y 0 ,Z 0 ) J to the right side of (1.11.8c). 3. Tensor of Angular Acceleration, and so on. (i) By d(.. .)/dt-differentiating (1.7.30i, j): dA/dt = A - Cl = Cl' • A, we can show that d 2 A/dt 2 = A • E =>- E = A T • (d 2 A/dt 2 ), (1.11.9a) where E = A+Cl-Cl = A+Cl 2 , (1.11.9b) A = Ml /dt : (Matrix of components, along the moving axes, of the) tensor of angular acceleration of the moving axes relative to the fixed ones (1.11.9c) { = d/dt[A T • (dA/dt)] = (dA T /dt) • (dA/dt) + A T • (d 2 A/dt 2 ) = -Cl-Cl+E} . (1.11.9d) [In fact, both A and E appear in (1.11.8c). Also, some authors call E the angular acceleration tensor, but we think that that term should apply to dfl/dt; that is, definition (1.11.9c).] (ii) Both E and A are (second-order) tensors; that is, E'(=^.' + r2'-0') = A-E-A t E = A t -E'-A, (1.11.9e) A'(= dJY/dt) = A • *4.- A t A = A T -.4.'-A; (1.11.9f) where, as before, an accent (prime) denotes matrix of components along the fixed WWW.EBOOK777.COM axes. CHAPTER 1: BACKGROUND (iii) The fixed axes counterpart of (1.11.9a) is: d 2 A/dt 2 = E'-A => E' = (d 2 A/dt 2 )-A T , (1.11.9g) (iv) It can be verified, independently of (1.11.9a-d) and (1.11.9e-g), that Jl 2 = $7 • T2 = —J7 t • 17 = —17 • 17 t = • • • = — (dA/dt) T • (dA/dt) = -(dA T /dt) • (dA/dt), (1.11.9h) (17') 2 = 17' -17' = —(17 , ) t • 17' = -17' • (17') t = • • • = -(dA/dt) • (dA/dt) T = -(dA/dt) • (dA T /dt). (1.11.9i) (v) Since d!7/dt is antisymmetric, and 17 • 17 is symmetric (explain this), show that the axial vectors of (the nonsymmetric) E and A coincide, and are both equal to none other than the vector of angular acceleration a; thus justifying calling A the tensor of angular acceleration. Finally, if the moving axes are fixed relative to a body B, then 17/17' and AI At are respectively, the tensors of angular velocity and acceleration of that body relative to the space-fixed axes; and if the earlier particle is frozen (fixed) relative to B (i.e., dx/dt = 0, d 2 x/dt 2 = 0, etc.), then (1.11.8b, c) give, respectively, the matrix forms of the well-known formulae for the distribution of velocities and acceleration of the various points of B (from body-axes components to space-axes components). [For an indicial treatment of these tensors, and recursive formulae for their higher rates, see Truesdell and Toupin (1960, pp. 439^140).] 1.12 THE RIGID BODY: EULERIAN ANGLES We recommend for concurrent reading with this section: Junkins and Turner (1986, chap. 2), Morton (1984). As explained already (§1.7, §1.11), the nine elements of the proper orthogonal tensor A (or R), in all its four interpretations, depend on only three independent parameters. A particularly popular such parametrization is afforded by the three (generalized) Eulerian angles. These latter appear naturally as we describe the general orientation of an ortho-normal-dextral (OND) body-fixed triad, or local frame t = \u k } = (/, j, k) relative to an OND space-fixed frame T = {iik 1 } = (/,./, K), with which it originally coincides, via the following sequence of three, possibly hypothetical, simple planar rotations (i.e., in each of them, the two triads have one axis in common, or parallel, and so the corresponding “partial rotation tensor” depends on a single angle): (i) Rotation about the (;)th body axis through an angle \(i) = Xi = </; followed by a (ii) Rotation about the (y)th body axis (j f i) through an angle Xu) = Xi = S', followed by a (iii) Rotation about the (k)th body axis (k f j) through an angle X(k) = Xt = The angles xi = f (about the original u, = «,-/), xi = S (about the (/-rotated i/ ; —> «■/), and X 3 = (about the 0-rotated u k —> u k «) are known as the / —> j —> k Eulerian angles. WWW.EBOOK777.COM §1.12 THE RIGID BODY: EULERIAN ANGLES Of the twelve possible such angle triplets, six form a group for which i fij k = i (two-axes group): 1 —► 2 —> 1, 1 —> 3 —> 1, 2 — > 1 —> 2 , 2 — > 3 —i- 2 , 3 —> 1 —> 3, 3 2 ^ 3; and six form a group for which i j k i ( three-axes group): 1—>2—> 3, 1 —s- 3 —!- 2, 2 —1 —^ 3, 2 —> 3 —> 1, 3 ^ 1 ^ 2, 3 —> 2 —»• 1. [Similar results, but with more complicated rotation tensors, would hold for rota¬ tions about the space-fixed axes {«*/:/,/, K}. If the partial rotations were about arbitrary (body- or space-fixed) axes, then, due to the infinity of their possible directions, we would have an infinity of angle triplets. It is the restriction that these rotations are about the body-fixed axes {u k } that brings them down to twelve.] Eulerian Angles The sequence 3 —> 1 —> 3, shown and described in fig. 1.26 [with the customary abbreviations: cos(...) = c(...), sin(...) = ,y(...)] is considered to be the classical Eulerian angle description, originated and frequently used in astronomy and physics, [although “In his original work in 1760, Euler used a combination of right-handed and left-handed rotations; a convention unacceptable today” Likins (1973, p. 97)]. (1973, p. 97)]. Using the passive interpretation and fig. 1.26, we readily find that the correspond¬ ing coordinates of the compounded transformation resulting from the above sequence of partial rotations about the nonmutually orthogonal axes OZ, Ox' , Oz" [i.e., the (originally assumed coinciding) space-fixed O—XYZ and body-fixed O—xyz] are related by (X\ fct -scp 0\ fx'\ Y = scj) ccj) 0 \Zj 0 1/ ( CCp — scj) °\ /! 0 0 ^ (*') = scf > ccj) 0 0 c8 -s9 // y u 0 1/ \0 s9 c9 Wv ( C(j) —scj) °\ Z 1 0 0 \ / ccj) —sip °\ (*\ = scj) ccj) 0 0 cO —s9 scj) ap 0 y u 0 1/ \o sO c9 / \o 0 1 ) \ Z J R( KA) R(t' ,0)- R(A" III 73 73 O • R«|, = = R, (1.12.1a) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND cf cip — sf c9 sip —ccp sip — scp cOcip scps9 srp cip + ccp cO sip —scp sip + ccp c6 cip —ccp s9 sO sip s9 cip c9 R or A = (A k ' k ) (=1, if 0,0,0 =0). (1.12.1b) Classical Eulerian Sequence: ( IJ.K ): 3(0) —► 1 (9) —> 3(0): (/, /, Ac) 0 < 0 ( precession, or azimuth, angle) < 27r, 0 < 5 [nutation (i.e., nodding), or pole, angle] < 7T, 0 < 0 [proper, or intrinsic, rotation angle; or (eigen-) spin] < 2w. 3(0); Precession 1(0); Nutation 3(0); Spin In sum: T = R 4 , • [Re • ( R 4 , -t)] = (10, ■ R e ■ 10,) -t = R t Figure 1.26 Partial, or elementary, rotations of classical Eulerian sequence: 0 —* 9 —» 0 (originally: O —xyz = O— x 0 y 0 z 0 = O— XYZ). REMARKS (i) Equation (1.12.1b) readily shows that if the direction cosines A k , k are known, the three Eulerian angles can be calculated from 0 = tan~*(— A l , 3 /A 2 ' 3 ), 0 = cos _1 (A 3 / 3 ), 0 = tan _1 (A 3 / 1 /A 3 » 2 ). (1.12.1c) (ii) If the origin of the body-fixed axes ♦ is moving relative to the space-fixed frame O XYZ, then in the above we simply replace X with X — A* and so on, cyclically. Then, x,y,z [or x/+,y/+,z/+ (§1.8)] are the particle coordinates relative to ♦-xyz. In this case, eq. (1.12.1b) shows clearly that a free (i.e., unconstrained) rigid body has six (global) degrees of freedom-. WWW.EBOOK777.COM §1.12 THE RIGID BODY: EULERIAN ANGLES cp 2 3 = X*, : inertial coordinates of base point (pole) ♦, q 4 5 6 = (j),9,’ip: Eulerian angles of body-fixed ♦— xyz relative to ♦— XYZ; and the constant x,y,r is the “name” of a generic body particle [more on this in chap. 2]. Inverting (1.12.1b) — while noting that, since all three component matrices ^ are orthogonal, the inverse of each equals its transpose (or using the passive inter¬ pretation equations in §1.11) — we readily obtain where R t = (R(|> • Re • R,) T = R\|/ T • Re T • R(|> T = R-+ • R-e • R-<j> ( 1 . 12 . 2 ) (1.12.2a) By adopting the active interpretation, we can show that (along arbitrary but common axes) (a) r f = R(k",fi).R(i\9) • R(£° = K,<j>)- rj = (Rq, • R e •R*) (1.12.3a) (b) n = (R\|/ • Re • R*) T • r j ■ = R(|) T • R 0 T • R\|/ T • ttf = (R_ 4> • R- e • R-+) • r f = [R(K : -</>)■ R(i\ — 9)-R(k",-fi)] .r <; (1.12.3b) while, by ; adopting the rotation of a triad interpretation, we can show that (a) T = (R<|) • Re • R\|/) • t, (1.12.4a) (b) t = (R—\|/ • R-e • R-4>) • T; (1.12.4b) where T = fiI,J,K) T , t = (ij ,a-) t . Next, we prove the following remarkable theorem. THEOREM (on Compounded Rotations about Body-fixed versus Space-fixed Axes) R + • R e • R* = 1>) • R(i - ', e) • R(*° = k, f) = R{K,(/>)-R(I,0)-R(K,il>). (1.12.5a) In words: the resultant rotation tensor of the classical Eulerian sequence about the body-fixed axes: (f>(k = k° = K) —> 9{i') —> f(k"), equals the resultant rotation of the reverse-order sequence about the corresponding space-fixed axes: m - e(i) - <kk). (i) To this end, we first prove the following auxiliary theorem. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Shift of the Axis Theorem Let us consider two concurrent axes of rotation described by the unit vectors n and //', and related by a rotation through an angle p about a third (also concurrent) axis described by the unit vector m; that is, n = R(//i, fi) • n = n • R t (/ii, p). (1.12.5b) Then, the corresponding rotation tensors about n and but with a common angle X, are related by the tensor-like (or, generally, “similarity”) transformation: R(«',X) = R (m,fj,)-R(n,x)-R T (m,n). (1.12.5c) PROOF Applying the rotation formula (1.10.10a) for n —> //' and x, we obtain, successively, R(«',X) = R[R(m,p) -fi,x] = (cos x) 1 + (sin x) [R(»i, p)-n)x 1 + (1 - cos x) [R(m, p) • «] ® [R(m, p) • n] [using the fact that, for any vector, v: (R • v) x 1 = R • (v x 1) • R T — see proof below] = (cosx)l + (sinx)[R(»»,p)-(« x l)-R T (/«,p)] + (1 - cos x) [R(m, m) • ( n ® ») • R T (/m, p)] [recalling that R(m,p) *R t (/m, p) = 1] = R(m, fj) • [(cosx)l + (sin x)( n x 1) + (1 - cosx)(«® «)] -R T (»»,p) = R(m,fi) • [(cosx)l + (sinx)N + (1 - cosx)(n® »)] -R t (»i, p) [recalling again (1.10.10a)] = R(»i, n) ■ R(«, x) • R T ( n h m), Q-E.D. (1.12.5d) [PROOF that (R • v) x 1 = R • (v x 1) • R t According to the passive interpretation , v and its corresponding antisymmetric tensor V = v x 1 transform as follows: R • v = components of v along the old axes =v', R • V ■ R t = components of V along the old axes = V'. Therefore, (R • v) x 1 = v' x 1 = V ' = R ■ V ■ R t = R • (v x 1) • R T , Q.E.D.] This theorem allows one to relate the rotation tensors about the initial (») and final (i.e., rotated) («') positions of a body-fixed axis. (ii) Now, back to the proof of (1.12.5a). Applying the preceding shift of axis theorem (1.12.5b, c), we get (a) R(k",-ip) = R (/', 9) • R(*\ V>) • R T (*’\ <t>), WWW.EBOOK777.COM (1.12.5e) §1.12 THE RIGID BODY: EULERIAN ANGLES where k" = R(i', 9) -k' . (1.12.5f) (b) R(i', 9) = R(tf, 0) • R(/, 0) ■ R J (K, 0), (1-12.5g) => R T (i', 0) = R (K, 0) • R t (/, 9) • R t (A', 0), (1.12.5h) where i' = R(JST, 0) •/; (1.12.51) (c) R(^0)=R(tf,0).R(tf,0).R T (JM), (1.12.5j) where k' = R(K,fi)-K. (1.12.5k) Substituting (1.12.5g, h, j) into the right side of (1.12.5e), while recalling that all these R’s are orthogonal tensors, yields R(*", 0) = [R(tf, 0) • R(/, 0) • R V, 0)] • [R(^i 0) • R(^"> 0) • R t (^> 0)] • [R(^i 0) • R t (^> 0) • R t (-^g 0)] = R(K, 0) • R(/, 9) • R(/f, 0) • R t (/, 9) • R T (tf, 0). (1.12.51) In view of (1.12.5g) and (1.12.51), the left side of (1.12.5a) transforms successively to R(*",0).R(i',0).R(tf,0) = [R (K, 0) • R(/, 9) ■ R(K, 0) • R t (/, 9) ■ R J (K , 0)] .[R(tf,0).R(/,0).R T (jr,0)] -R(Jf,0) = R(A',0)-R(/,6»)-R(A',0), Q.E.D. (1.12.5m) Generally, consider a body-fixed frame O-xyz originally coinciding with the space- fixed frame O XYZ. Then the sequence of rotations about Ox (first, xi) —> Oy ( second, \i) Oz (third, X 3 ) bas the same final orientational effect as the sequence about OZ (first, X 3 ) ~* OY (second, X 2 ) ~> OX (third, xi). [See also Pars, 1965, pp. 103-105.] Angular Velocity via Eulerian Angle Rates Let us calculate the vector of angular velocity of the body frame O-xyz relative to the space frame O XYZ, in terms of the Eulerian angles 0,0,0 and their rates = df/dt, ijj s = d9/dt, = dtp/df, both along the body- and the space-fixed axes. We present several approaches. (i) Geometrical Derivation By inspection of fig. 1.26 we easily find that o) = co^K + ujf) i + iO,p k . WWW.EBOOK777.COM (1.12.6a) CHAPTER 1: BACKGROUND But, again by inspection, along the space basis, K = (0)1 + (0)/ + (l)A - , i' = (cos </.)/ + (sin <p)J + ( 0 )if, k" = (- sin#),/' + (cos 9)k' = (— sin 9) [(— sin </>)/ + (cos cp)J] + (cos 9) K = (sin # sin 0)/ + (— sin# cos (/>)/ + (cos 9)K\ (1.12.6b) and along the body basis, K = y "(sin 9) + A: "(cos 9) = (i sin ip + j cos ip) sin # + A: cos #, i' = i cos ip —j sin ip, k” = k. ( 1 . 12 . 6 c) Inserting (1.12.6b, c) in (1.12.6a) and rearranging, we obtain the representations o — u)xl d - coy J ~b = to x i -\- ujyj co z k, (1.12.7a) where, in matrix form / (o C(p s4>s9 \ / w 0 \ CJy = 0 S(j> —cej) s9 u e \ w z / V 0 c9 / W/ Space axes E s(pace )(</>, 9) [no ^-dependence], ( Ux ) ( s9sip cip °\ (ujA OJy = s9 cil> —sip 0 lx) 8 ^ c9 0 V Body axes E b (ody) (6, VO [no (/.-dependence], (1.12.7b) (1.12.7c) Inverting (1.12.7b, c) (noting that, since the axes of are non-orthogonal, the transformation matrices E s , E b are nonorthogonal also; that is, their inverses do not equal their transposes), we obtain, respectively, ( —s</> c9 ccf> c9 s9\ ( U ' Y \ Ug = (1 / sin 9) C(ps9 s(j>s9 0 UJy \ J V A — C(j) o ) \ w z) E ,-‘(0,0), ( A cip °\ ( ^ (1 / sin 9) s9 cip —s9 sip 0 Uy \ —c9sip —c9 cip s9 / E b -‘(0,VO; (1.12.7d) (1.12.7e) WWW.EBOOK777.COM §1.12 THE RIGID BODY: EULERIAN ANGLES from which we can also calculate the u>x,y,z o w x y z (orthogonal!) transformation matrices. REMARKS (a) The transformations (1.12.7b-e) readily reveal a serious drawback of the 3 —1 —*• 3 Eulerian angle description, for 9 = 0 (or ±tt); that is, when Oz coincides with OZ (or —OZ), in which case the nodal line ON disappears, sind = 0, and, so, assuming </>, ip f 0, eqs. (1.12.7b, c) yield, respectively, — { c< P)we, u> Y = (s<p)vg, lo z = => lo x 2 + oj y ~ = ^6> 2 ; (1-12.7f) u x = (~Clp)w e , U)y = (~S1p)u) 8l U z = + U)$ => w x 2 +UJ y 2 = UJg 2 ', (1.12.7g) which means that knowing ui x Y, 7 /x,y,z(t) [say, after solving the kinetic Eulerian equations (§1.17)], we can determine u e uniquely , but not and Actually, all twelve generalized Eulerian angle descriptions mentioned earlier, Xi Xi -► X 3 > exhibit such singularities for some value(s) of their second rotation angle in which case, the planes of the other two angles become indistinguishable] From the numerical viewpoint, this means that in the close neighborhood of these values of x 2 > il becomes difficult to integrate for the rates < ixk/dt (k = 1,2,3). This is the main reason that, in rotational (or “attitude”) rigid-body dynamics, (singularity free) four-parameter formalisms are sought, and the reason that the classical Eulerian sequence 3 —> 1 —> 3 has been of much use in astronomy (where x,y, z have origin at the center of the Earth, and point to three distant stars) and physics; whereas other Eulerian sequences, such as 1 —> 2 —>3 or 3—> 2 —> 1 [associated with the names of Cardan (1501-1576) (continental European literature), Tait (1869), Bryan (1911) (British literature); and examined below] are more preferable in engineering rigid- body dynamics; for example, airplanes, ships, railroads, satellites, and so on. [Similarly, the position (<p,9,ip) = (0,0,0) represents a singular “gimbal lock”: the motions ui^ and uty are indistinguishable since each is about the vertical axis Z; only ufy + a;,/, is known. The lu 8 motion is about the X-axis, and so it is impossible to represent rotations about the T-axis; it is “locked out”; that is ( 0 , 0 , 0 ) introduces artificially a constraint, uj y = 0, to v = 0 that mechanically is not there (then, u> x = cu 9 , W>Y — 0, W Z = ^0 + 0 U> x = UJg, W z = Ufy + Ulf).] (b) Equations (1.12.7b, c) also show that the components lo x Y,z/x,y,z are quasi or nonholonomic velocities', that is, although they are linear and homogeneous combina¬ tions of the Eulerian angle rates ui$ = df/dt, to e = dO/dt, u)$ = dip/dt , they do not equal the rates of other angles. Indeed, if, for example, uj x = dd x /dt, where 0 X = 9 X (4>, 9, ip), then we should have d9 x /dt = {d9 x /df)[d(p/dt) + ( 39 x /d9)(d9 / dt ) + (d9 x /dip)(dip/dt) = (0 ){d(p/dt) + (ccp)(d9 / dt) + (scps9) (dip/dt) [by (1.12.7b)], (1.12.7h) that is, d9 x /d(p = 0, 89 x /d9 = c9, d9 x /dip >) II (1.12.7i) But, from (1.12.7i), it follows that, in general. d/d9(d9 x /d(p) = 0f d/df(d9 x /d9) = — scp. ( 1 - 12 .7j) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Hence, no such Ox exists; and similarly for the other ut’s. (An introduction to quasi coordinates is given in §1.14; and a detailed treatment is given in chap. 2.) (ii) Passive Interpretation Derivation (a) Body-fixed axes representation. Since m is a vector, we can express it as the sum of its three Eulerian angular velocities: ot = 4- u)fl T (1.12.8a) where ct>0 = ( d<p/dt)K , oj e = (dO/dt)i', a> y, = (dip/dt)k". (1.12.8b) Then, using the passive interpretation, (1.11.4h, 7a IT.), we can express (1.12.8a, b) along the (new) body axes basis (i,j,k). Since the Eulerian basis ( K,i',k") is non- orthogonal, we carry out this transformation, not for the entire o>. but for each of its above components o e ,co^, and then, adding the results, we obtain ( ( 0 \ m _ D T n T _ body components - Iv v)/ ^0 0 = R_q, • R e • 0 W/ (UK) W/ body components *'7 body components sip °\ p cip 0 0 1/ V- 0 w< / cip sip —SlP cip V 0 0 /°v 0 W/ (ijk) /! ° 0\ (°) ( (sO sip) \ 0 cO sO 0 = (sO cip)uj, 1 o V ( C0 H ) R 4 ‘ 0 VO/ /ocA 0 [oj {-sip)uj e V (o)w fl / (1.12.8c) (1.12.8d) (1.12.8e) Adding (1.12.8c-e), we obtain the body axes components, equations (1.12.7c), as expected. WWW.EBOOK777.COM §1.12 THE RIGID BODY: EULERIAN ANGLES (b) Space-fixed axes representation. Proceeding similarly, we find (!) = C(j) S(j> 0 —s<j> c4> 0 R<t> ( C(j) — S(j> °\ /l 0 ° \ + S(j> C(f) ° 0 c9 -s8 \o 0 1/ s9 c8 ) R(i> Re /o 0 V ( + I S(j) V° / s(p sb UJg + —C(j)s6 V c9 (1.12.8f) which is none other than (1.12.7b). Let the reader verify that the space -axes representation (1.12.8f) can also be rewritten as o> = R<i)- (° + R(j> • Re • R\|/ • I 0 while the body-axes representation (1.12.8c-e) can be rewritten as (°) { Uf) \ (o = • R _ 0 • R_§ • 0 + R-v|/ • R-e • 0 \“) (1.12.8g) (1.12.8h) (iii) Tensor (Matrix) Derivation We have already seen [(1.7.27e) and (1.7.30i k)] that the space-axes components of the angular velocity tensor (vector) are related to its body-axes components fl(<n) by the tensor (vector) transformation n' = r • fj • r t o n = r t • ft ' • r (m' = R-<u <« = R T -c</), (1.12.9a) where R, or A, is the matrix of the direction cosines between these axes; and also that = (dR/dt) • R t = R • (dR/dt) T [due to d/dt(R • R T ) = dl/dt = 0] ft = R t • (dR/dt) = — (dR/dt) T • R. dR'/dt = ft'-R [= (R-ft-R T )-R] = R-ft (1.12.9b) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND (a) Space-fixed axes representation. As we have seen, in the case of the classical Eulerian sequence </> —> 9 —> ip: R = R^ • R 0 • R^, and therefore (1.12.9b) yields, successively, il' = (dR/dt) • R t = d/dt(R + • R e • R v |,) • (R* • R e • R*) 7 = [(dR<|,/dt) • Re ■ R^ + R^ • (dRe/dt) • R^ + R^ • R 0 • (dR^/dt)] • (R^ 7 ■ Re T • R<|> 7 ) = (dR^/dt) • [R e • (R* • R/) ■ R e T ] • R/ + R^ • [(dRe/dt) • (R* • R/) • R e 7 ] • R/ + Rjj, • {R 0 • [(dR^/dt) • R/[ • R e 7 } • R* 7 = (dR*/dt) -R/ + R* • [(dRg/dt) • R 0 X ] • R,,, 7 + R (t> -R 0 - [(dR + /dt) .R + 7 ] • (R* • R e ) x [recalling the definition of tensor transformation (1.12.9a), and (1.12.9b)], (1.12.9c) = + R(j) • n'o • R/ + R<t> • Ro • Q\i • (R<t> • Ro) 7 [£!',(, e,v|) : “partial” rotation tensors, along the space-fixed axes], (1.12.9d) from which, after some long but straightforward algebra, we obtain [recalling (1.12.1a If.)] Qyy = Qxx = 0 , Qvt — —Qi'v = Qxy — —Qyx = — n»z = -[d(f)/dt + (c9){dip / dt )], Qyy = -Qyy = Qxz = -Qzx = -Wy = (stf>)(d9 / dt) - (c(f) s9)(di///dt ), Qyy = Qyy = 0 , Qyy — —Qyy = Qyz - —Qzy — ~a>x = -[(ctf>)(d9/dt) + (scps9)(dift / dt )], Qyy =Qzz = 0, (1.12.9e) which coincide with (1.12.7b), as expected. (b) Body-fixed axes representation. Proceeding analogously, we obtain £2 = R 7 • (dR/dt) = (R* • R 0 • R + ) 7 • [d/dt(R <t> • R e • R*)] = ■ ■ ■ = R/ • Re 7 • [R <t) 7 • (dR^/dt)] • R e • R^ + R + 7 • [R 0 7 • (dRe/dt)] • R + + R/ • (dR^/dt) = R vi/ 7 • Rq 7 • £2^ • Ro • R v + R v 7 • £2o • Ry + £2^ [£2^,e,vp : “partial” rotation tensors, along the body-fixed axes]. (1.12.9f) We leave it to the reader to verify that the above coincides with (1.12.7c). Alternatively, one can use the transformation equations (1.12.9a) to calculate £2 /<o from £2 '/a>'. (See also Hamel, 1949, pp. 735-739.) Cardanian Angles This is the Eulerian rotation sequence 3 —> 2 > 1 (fig. 1.27). The angles = 7(3) —> X 2 = (3(2) —► x 3 = a(l) are commonly (but not uniformly) referred to as Cardanian WWW.EBOOK777.COM §1.12 THE RIGID BODY: EULERIAN ANGLES Figure 1.27 Cardanian angles: xt = 7(3) —>■ xi = /3(2) —> X 3 = a(1 )■ (i) Rotation (OZ, xi = 7): O-XYZ (space axes) = 0-x 0 y 0 z 0 (initial body axes) —* O-x'y'z'. (ii) Rotation ( Oy \2 = PY O-x'y'z' —> 0-x"y"z". (iii) Rotation (Ox", y 3 = Q ) : 0-x"y"z" —> O-xyz (final body axes). angles. In vehicle and aeronautical dynamics, where such an attitude representation is popular, they are called yaw ( 7 ), pitch (0), and roll (a). Following the passive interpretation, we readily obtain (X) f Y = R- - R y • < Rp • Rql - y UJ V z 7 l \ Z J ( H —57 °\ / C/9 0 5/3 \ 0 0 ^ (*\ C7 0 0 1 0 0 COL —SOL y Vo 0 1 / V —5/3 0 c/3/ \o SOL COL J W R y • Rp R ( ( 1 . 12 . 10 a) / c/ 3 c 7 5a 5/3 C7 — ca 57 ca 5/3 C7 + 5a 57 \ c/357 5a 5/3 57 + ca C7 ca 5/3 57 — 5a C7 F \ 5a c /3 cac /3 / W ( 1 . 12 . 10 b) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and, inversely, since R a ,p, Y are proper orthogonal, fx y \- (R a T -Rp T .R y T ). (R-a - R-P ' R-y) * ( 1 . 12 . 10 c) Angular Velocity Tensors Using the basic relations (1.12.9a, b), we can show, after some long and careful but straightforward algebra, that (with u 1 = dy/dt, uip = d(3/dt, uj a = da/dt) Space axes —s~f C 7 c/3\ (“A CJ syc/3 up 0 -s/3 ) \u a J (1.12. lOd) i(pace) {if ft) [no a-dependence], Body axes —sf3 0 A / CU 7 c/3 sa COL 0 cac(3 —SOL 0 / \U a E b (ody) (A a) [no 7 -dependence]. (1.12. lOe) Inverting (1.12.lOd, e) (noting that, since the axes of w Q ,/ 3, 7 are non-orthogonal , the transformation matrices E s ( 7 , 0), E b (/3, a) are nonorthogonal also; that is, their inverses do not equal their transposes), we obtain respectively, ( u l^ / s/3c"f s[3sy c(3\ /u A \ Up = (1/cos /3) —syc/3 C 7 c/3 0 LUy \ w «y \ n S') 0 ) \u Z ) E s -‘(/3,7), / 0 sa ca \ (1 /cos/3) 0 cacfd —sa c(3 Uy yc/3 s/3sa 5/3 ca ) \u z J E b 1 (cr, /?); (1.12. lOf) (1.12. lOg) from which it is clear that the Cardanian sequence 3(7) — > 2 (/ 3 ) —> 1(a) has a singularity at (3 = ±(7 t/2 ). There, (1.12.lOd, e) become, respectively (for /3 = 7 r/ 2 ), u x = {si)up, wy = {ci)ufj, w z = w 7 -w Q => u x 2 + u Y 2 = Up 2 , (1.12.10h) u x = —l o 1 + u a , Lo y = ( ca)ojp, u> z = {—sa)u>p => u> 2 + u> y 2 = teg 2 ; (1.12. lOi) that is, a unique determination of u> 1 and uj a from u> z , or ui x , is impossible. WWW.EBOOK777.COM §1.13 THE RIGID BODY: TRANSFORMATION MATRICES Finally, using (1.12. lOf, g), we can obtain the transformation. For a complete listing of the transformations between u> x y z = 3 (body-fixed axes) and the Eulerian rates d\ 1 , 2 , 3 /^ = v i ,23 (and corresponding singularities), for all body-/space-axis Eulerian rotation sequences, see the next section. 1.13 THE RIGID BODY: TRANSFORMATION MATRICES (DIRECTION COSINES) BETWEEN SPACE-FIXED AND BODY FIXED TRIADS; AND ANGULAR VELOCITY COMPONENTS ALONG BODY FIXED AXES, FOR ALL SEQUENCES OF EULERIAN ANGLES Summary of Theory, Notations T = («/,/) T = {uy, u 2 ', M 3 ') T = (/,/, K) T : Space-fixed {fixed) triad. t = (i/a-) T = {ii 1 ,u 2 ,u 2 ) T = {i,j,k) T : Body-fixed {moving) triad. All triads are assumed ortho-normal-dextral (OND), and such that, initially, T = t. Eulerian angles (see §1.12): XuXi>X 3 (the earlier </>, 0, or a,/3, 7 ). 1. Basic Triad Transformation Formula T = Rt t= R t T, where R = {R/ck) = («*' • Uk ) [or {Ak'k)\ '■ Tensor / Matrix of rotation = R(«»xi) -R(«/tX 2 ) -R(«fc,X3) = ['(xi) J(x2),Mx3)] [Rotation sequence xi —► X 2 X '3 about the body -fixed axes m, —► i/ ; —> « A ] = R(«i',X3) •»(«/■',X 2 ) •«(«/',Xi) = [*'(X3) J J , (X2), i'(xi)] [Rotation sequence X 3 X 2 — 1 * Xi about the .v/?ace-fixed axes uy —♦ up —> «,/] [ 1 , ./,*;= 1,2,3; i',j',k' = l',2',3']; and, by the basic theorem on compounded rotations (§1.12), the inverse rotation R 1 = R t = R(«*, -X 3 ) • R {uj, -X 2 ) • R(«/, -xi) = R(«,-, -xt) • R(«/, -X 2 ) • R(«/c', -X 3 ) returns the body-triad t to its original position, i.e. realigns it with the space-triad T. How to obtain space-axis rotations; i.e., [k'(xi), j'{x 2 ); ^(xs)]. from a knowl¬ edge of body-axis rotations with the same rotation sequence : Xi ~* X 2 ~* X 3 ! he., from [/(Xi), j(x 2)1 k{xf)\, and vice versa. An example should suffice; by the above theo¬ rem, we will have [2(»),3(e),i(»)] = [i'(xj),3'(e),2'(x,)] WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and, therefore, swapping in the latter X 3 with xi (and vice versa), we obtain [1'(Xi), 3 '(x 2 )j2'(x 3 )], which appears in the listing below. Similarly, we have [2'(Xi),3'(x2),1'(X3)] = [1(X3),3(X2),2(xi)] and swapping in there X 3 with Xi (and vice versa) we obtain [ 1 (xi)i 3 (x 2 )> 2 (x 3 )]- Abbreviations: s t (.. .) = sin(x,), c ; (. ..) = cos(x,)- 2. Angular Velocity Components Body-fixed (moving) axes components: Cl = R t • (dR/dt) = —(dR/dt) T • R, [due to d/dt(R-R T ) = dl/dt = 0] Space-fixed (fixed) axes components: O' = (dR/dt) ■ R t = -R • (dR/dt) T ; with mutual transformations: O' = R • O • R t O = R t • O' • R, u' = R • u uj = R t • J where uj = axial vector of O, uj' = axial vector of O' [i.e. O • ( vector ) = uj x (vector), etc.]; Rotation tensor derivative: dR/dt = 0'-R [= (R-0-R T )-R] =R O. Listing of Transformation Matrices; and Angular Velocity Components (Body-fixed vs. Eulerian rates; and corresponding singularities. Notation: dX\,2,l/dt = v l,2,3) 1(a) [1(Xi),2(x2),3(x 3 )] = [3'(X3),2'(x2), l'(Xi)] [Singularity at x 2 = ±(V 2 )]: / C 2 C 3 ~C 2 S 3 Si \ S1S2C3 + J3C1 -s 1 ^ 2 5 3 + c 1 c 3 -sic 2 ; \-C 1 S 2 C 3 + S Y S 3 C!J 2 5 3 + J!C 3 C,C 2 ) W 1 = (c 2 c 3 )vj + (s 3 )v 2 + (0)v 3 uji = (—c 2 5 3 )vi + (c 3 )v 2 + (0)r 3 w 3 = ( J 2 ) v l + ( 0) v 2 + (l ) v 3 V 1 — ( c 2) *[( c 3) w l + 1 £3)^2 + (0 )o2 3 ] v 2 = ( c 2) 1 [( c 2‘ s 3) w l + (c 2 c 3 )w 2 + (0)w 3 ] v 3 — ( c 2) '[( —J 2 C 3) W 1 + ( s 2 s ’i) UJ 2 + (^ 2 )^ 3 ] ■ WWW.EBOOK777.COM §1.13 THE RIGID BODY: TRANSFORMATION MATRICES l( b ) [ 1 '(Xi)> 2 '(x 2 ), 3 '(x 3 )] = [ 3 (X 3 ), 2 (x 2 ), 1 (Xi)] [Singularity at x 2 = ±(V 2 )]: / C 2 C 3 SjSjC;, - C X S 3 C x S 2 C 3 + S { S 3 \ CiS 3 SiS 2 s 3 + CjC 3 C!S 2 s 3 -SiC 3 ; \ $2 s x c 2 C\C 2 ) W 1 = (l) v l + (0) v 2 + ( —J 2) v 3 U 2 = (0)vi + (ci)v 2 + (YiC 2 )v 3 w 3 = (0) v l + (~‘ v l) v 2 + ( c lC 2 )v 3 V 1 = { c l) *[( c 2) a; l + (■ sr l' s 2) c * J 2 + ( C 1 J 2) W 3] v 2 = ( c 2) * [(0) w l + ( c l c 2) w 2 + (—3 , lC 2 )W3] v 3 = ( c 2) * [(0) w l + ( 5 l) w 2 + ( c l) w 3]- 2(a) [2(xi)i3 (x 2 )j 1(X3)] = [l'(X3),3'(X2)>2'(xt)] [Singularity at X 2 = ±(V 2 )]: / c,c 2 ~ c iS 2 c 3 + sty 3 c l s 2 s 3 + 5 1 c 3 \ S 2 C 2 C 3 -c 2 s 3 V -S\C 2 SlS 2 C 3 + Ci S 3 —S[S 2 S 3 + CiC 3 J W 1 — (■ s 2 ) v l + (0)v 2 + (l)v 3 V| — (c 2 ) 1 [(0)^! + (c 3 )w 2 + ( — S 3 )o; 3 ] w 2 = ( C 2 C 3) V 1 + ( J 3) v 2 + (0) v 3 v 2 = ( c 2) ' [(O)t^i + {c 2 S 3 )u 2 + (c 2 C 3 )w 3 ] W 3 — ( —C 2 5 3 )vi + (c 3 )v 2 + ( 0 )v 3 v 3 — ( c 2) *[( c 2) w l + ( — S 2 C 3 V 2 + ( J 2- S 3) w 3]- 2(b) [2 '(xi),3'(x 2 ), l'(X 3 )] = [1(X3),3(x2),2(xi)] [Singularity at X 2 = ±(tt/ 2 )]: ( c x c 2 -S2 S\C 2 \ C 1 S 2 C 3 +S,s 3 C 2 C 3 S 1 S 2 C 3 - C[S 3 \C { S 2 S 3 - S!C 3 C 2 S 3 s l s 2 s 3 + c l c 3 / U\ — ( 0 )V[ + (—Sl)v 2 + (CiC 2 )v 3 w 2 = (l) v l + (0) v 2 + ( —5 2) v 3 w 3 — (O)l’l + ( c l) v 2 + (• s 'l c 2 ) v 3 V 1 = (c 2 ) 1 [ (^ 1 ^*2) F*- 7 1 + ( C 2 )u 2 + (SiS 2 )w 3 ] v 2 — (c 2 ) * [(—Sl c 2) w l + (0) w 2 + ( c lC 2 )u; 3 ] v 3 = ( c 2) * [( c l) w l + (0)w 2 + (^ 1 )^ 3 ]. 3(a) [3(xi), 1(X2),2(x 3 )] = [2'(x 3 )> l'(x 2 ),3'(xi)] [Singularity at x 2 = ±(tt/ 2 )]: ( —S\S 2 S 3 + C[C 3 -■SlC 2 s x s 2 c 3 + qs 3 \ C1S2S3 + .V[C 3 c x c 2 C[S 2 C 3 -T S\S 3 V c 2 s 3 s 2 c 2 c 3 ) Wi — ( — C 2 S 3 )vj + (c 3 )v 2 + (0)v 3 V[ — (c 2 ) 1 [(—J 3 )c* 2 1 + (0)tu 2 + (c 3 )w 3 [ w 2 = (* s ’2) v 1 3“ (0)v 2 + (1)^3 v 2 = (c 2 ) [(c 2 C 3 )o2i + (0)o; 2 + (c 2 S 3 )t<2 3 ] w 3 — ( C 2 C 3) V 1 + ( J 3) v 2 + (0) v 3 v 3 — ( c 2) 1 [(‘ s 2‘ s 3) w l + ( c 2) w 2 + ( — • ? 2 C 3) W 3]- WWW.EBOOK777.COM CHAPTER 1: BACKGROUND 3(b) [3'(xi), l'(X 2 ), 2 '(x 3 )] = [2(x 3 ), 1(x 2 ),3(xi)] [Singularity at x 2 = ±(V2)]: / S\S 2 S 3 + C\C 3 C 3 S 2 S 3 - .S' 1 c 3 C 2 S 3 \ S\C 2 c \ c i —s 2 ; \SlA 2 C 3 - Wi = (0)vj + (ci)v 2 + (SiC 2 )v 3 w 2 = (0)vi + (—i'i)v 2 + (C]C 2 )v 3 oi 3 = (l)v| + (0)v 2 + (—s 2 )v 3 C 1 J 3 C 3 + s l s 3 C 2 C 3 ) v i = ( c 2 ) 1 [(^i^ 2 )p^i + (cis 2 )u; 2 + (c 2 )oi 3 ] v 2 = (c 2 ) 1 [(ci c 2 )u; 1 + (— SiC 2 )u 2 + (0)oi 3 ] v 3 = (c 2 ) 1 [(-^ 1 )tDi + (ci)oi 2 + (0)oi 3 ]. 4(a) [1(xt),3(x 2 ),2(x 3 )[ = [2'(x 3 ),3'(x 2 ), 1 '(xi)] [Singularity at x 2 = ±(tt/2)]: / c 2 c 3 ~s 2 C 2 S 3 \ ClS 2 C3 + SiS 3 c,c 2 ClS 2 S 3 S[C 3 \SlS 2 C 3 - C\S 3 S\C 2 + c l c 3 J wi = (c 2 c 3 )vi + (—s 3 )v 2 + ( 0 )v 3 UJ 2 = (—S 2 )V] + ( 0 )v 2 + (l)v 3 w 3 = ( C 2 J 3) V 1 + ( c 3) v 2 + (b) v 3 V 1 = ( c 2 ) *[( c 3 ) a; l + ( 0 )oi 2 + (s 3 )oi 3 ] v 2 ~ ( c 2 ) *[( — C 2 J 3) W 1 + (0)oi 2 + ( C 2 C 3) W 3] v 3 = (c 2 ) * [(-y 2 C 3 )cu»i + (c 2 )oi 2 + (j 2 S 3 )oi 3 ]. 4(b) [l'(xi),3'(x 2 ),2'(x 3 )[ = [2(X3),3(x 2 ), 1(xi)] [Singularity at x 2 = ±(V 2 )]: / c 2 c 3 —C 1 J 2 C 3 + ^1^3 s l s 2 c 3 + C 1 J 3 \ s2 ClC 2 -SlC 2 V -c 2 s 3 C]S 2 S 3 + J 1 C 3 ~ s l s 2 s 3 + c l c 3 J Wi — (1)V] + (0)v 2 + (s 2 )v 3 w 2 — (0) v l + ( 5 l) v 2 + ( c l c 2 ) v 3 01 3 = (o)vj + (ci)v 2 + (—SiC 2 )v 3 V 1 = ( c 2 ) *[( c 2 ) w l + ( —c l 5 2 ) w 2 + (SiS 2 )oi 3 ] v 2 = (c 2 ) *[(0)011 + (SlC 2 )oi 2 + (CiC 2 )oi 3 ] v 3 = (c 2 ) *[(0)oii + (Ci)oi 2 + (—Sl)oi 3 ]. 5(a) [2(xi)> 1 (x 2 ),3(x 3 )] = P'fe), 1 '(x 2 ),2'(xt)[ [Singularity at x 2 = ±(tt/ 2 )]: / S Y S 2 S 3 + CjC 3 S] S 2 C 3 - C\S 3 s,c 2 \ c 2 s 3 c 2 c 3 —s 2 ; \ CiS 2 S 3 — Sic 3 oil = (c 2 s 3 )vi + (c 3 )v 2 + (0)v 3 w 2 = (c 2 c 3) v l + ( — ■ s 3) v 2 + (0)V3 oi 3 = (—S 2 )vj + (0)v 2 + (l)v 3 ClA 2 C 3 + SiS 3 C\C 2 J V 1 = ( c 2 ) * [(^3)^1 + ( c 3) w 2 + (0)013] v 2 = ( c 2 ) *[( C 2 C 3) W 1 + ( — c 2 s l)u 2 + (0)oi 3 ] v 3 = (c 2 ) *[(s 2 S 3 )oii + (s 2 C 3 )oi 2 + (c 2 )oi 3 ]. WWW.EBOOK777.COM §1.13 THE RIGID BODY: TRANSFORMATION MATRICES 5(b) [2'(xi), l'(X2),3'(x 3 )] = Pfe), 1(X2),2(xi)] [Singularity at x 2 = ±(tt/2)]: -S\S 2 S3 + C1C3 ~c 2 s 3 C\S 2 S 3 + Y1C3 ■Sl 3 , 2 C 3 + CjS-3 C 2 C 3 -CjS 2 C 3 +S!J 3 -■SlC 2 s 2 ClC 2 UJi = (0)V] + (cj)v 2 + (—^lC 2 )V3 w 2 = (l) v l + ( 0 )v 2 + (y 2 ) V 3 W3 = ( 0 )vi + ( 3 'i)v 2 + (CiC 2 )v 3 V 1 = ( c 2 ) 1 [(•Sl-S , 2 )wi + (c 2 )w 2 + (—CiJ 2 )w 3 ] v 2 = (c 2 ) 1 [(ciC 2 )Wi + (0)w 2 + (^lC 2 )cD 3 ] v 3 = (c 2 ) 1 [( — ■^l) w l + (0)w 2 + (ci)w 3 ]. 6(a) [3 (xi),2(x 2 ), Ife)] = [l'(X3),2'(x 2 ), 3'(xi)] [Singularity at x 2 = ±(tt/2)]: /C]C 2 C'l .S’ 2 .S' 3 - .S'l C‘3 C 1 Y 2 C 3 +Y 1 Y 3 \ ^iC 2 ^ 1 3- 2 y 3 + c 1 c 3 JiJ 2 c 3 - ; V ^2 iO\ = (—S 2 )vi + (0)v 2 + (l)v 3 w 2 = (c 2 ^3) v l + ( c 3)t 2 + ( 0 ) V 3 w 3 = ( C 2 C 3) V 1 + ( — ‘ S 3 )t 2 + (0) V 3 C 2 S 1 C 2 C 3 / v i = ( c 2 ) * [(0)wi + (53)^2 + (c 3 )w 3 ] v 2 = ( c 2 ) *[(0)^1 + (c 2 c 3 )w 2 + (— c 2 ^ 3 )cd 3 ] v 3 = ( c 2 ) 1 [( c 2 ) w l + ( s 2 s i)^2 + (j 2 C3)W3]. 6(b) [3'(xi)>2'(x 2 ), l'to)] = [1(X3),2 (x 2 ),3(xi)] [Singularity at x 2 = ±(tt/2)]: / CjC 2 -Y, C 2 Y 2 \ CiS 2 J 3 +SiC 3 -5 1 J 2 3'3 + C 1 C3 -C 2 ^ 3 ; \-Cj52C3 + J 1 S 3 w 1 = (0)v, + (i])v 2 + (CiC 2 )v 3 w 2 = (0)vi + (ci)v 2 + (—YlC 2 )V 3 w 3 = (l) v l + (0)v 2 + (s 2 )v3 S 1 J 2 C 3 + C 1 J 3 C 2 C 3 / V 1 = ( c 2 ) * [(— C 1 ‘ y 2 ) < ^’l + (SiJ 2 )w 2 + { c 2 ) ut \ v 2 = ( c 2 ) * [(■ s l c 2 ) w l + ( c l c 2 ) w 2 + (0)^] v 3 = ( c 2 ) * [( c l) w l + (~Yi)w 2 + (O)^]. 7(a) [l(xi),2(x 2 ). 1 (x 3 )] = [l'(X3)>2'(x 2 ), l'(Xi)] [Singularities at x 2 = 0 ,±tt]: / c 2 s 2 s 3 s 2 c 3 \ sis 2 ~s l c 2 S 2 + qc 3 -Jic 2 c 3 - ; \-ClS 2 C 1 C 2 ^3+^ 1 C 3 C l C 2 C 3 ~S l S 3 / w i = (c 2 )vi + ( 0 )v 2 + ( 1 ) V 3 UJ 2 = (*^ 2 *^ 3 ) Tj + (c 3 )v 2 + ( 0 )v 3 w 3 = (• S 2 C 3) V 1 + ( —J 3) v 2 + (0)V3 v i = fe) * [(0)^! + (y 3 )w 2 + (c 3 )w 3 ] v 2 = (^ 2 ) *[( 0 )wi + (s 2 c 3 )u} 2 + (— 52 ^ 3 ) 023 ] v 3 = (^ 2 ) *[(^ 2)^1 + {~ C 2 S 3 )^2 + ( —C 2 C 3) W 3]- WWW.EBOOK777.COM CHAPTER 1: BACKGROUND 7(b) [l'(xi),2'(x 2 )> I'fe)] = [1(X3) ; 2 (x 2 ), Uxi)] [Singularities at x 2 = 0, ±tt]: / c 2 c x s 2 \ S 2 Sj —^iC 2 ^3 + CiC 3 — CiC 2 $3 — J 1 C 3 ; \-S 2 Ci S\C 2 C 3 + C|.S '3 C\ C 2 C-j — .?| S 3 / V 1 = ( 5 2 ) 1 [(‘ y 2 ) aJ l + {~ s \ c 2)^2 + ( c l c 2 ) w 3] v 2 = ( 5 2 ) ' [(0) w l + ( c l^ 2 ) w 2 + (—^1^ 2 )w 3 ] v 3 = ( J 2 ) ' [(0) w l + (^1)^2 + ( c l) w 3]- w i = (l) v i + (0)v 2 + (c 2 )v 3 L0 2 = (0) V} + (Cj)v 2 + (5i3’ 2 )v 3 W 3 = (0)vi + (—Aj)v 2 + (Ci3’ 2 )v 3 8(a) [1 (Xi) i 3(x 2 ), 1(X3)] = [l'(X3),3'(x 2 ), l'(Xi)] [Singularities at x 2 = 0, ±tt]: / c 2 -s 2 c 3 S 2 S 3 \ C\S 2 c y c 2 c 3 -S\S 3 -c 1 c 2 5 3 -^ 1 c 3 ; \.?|.?2 ^ 1 C 2 C 3 + C 1 J 3 -S', C' 2 ,S ’3 - c, c 3 ) V 1 = ( J 2 ) 1 [(0) w l + (~ c 3) w 2 + ( 5 3 ) w 3] v 2 = (a 2 ) '[(0)^1 + ( S 2 S 3 )u> 2 + (a 2 C3)o23] W 1 = ( c 2 ) v l + (0)v 2 + (l) v 3 oj 2 = (—■s 2 C 3 )vi + (*y 3 ) v 2 + (0)v 3 w 3 = (‘ s 2‘ s 3) v l + ( c 3) v 2 + (0) v 3 v 3 — ( s 2 ) [(• y 2 ) w l + ( C 2 C 3) W 2 + {~ c 2 s 2 ) ljJ 'i}- 8(b) [l'(xi),3'(x 2 ), l'(X 3 )] = [1(X3),3(x 2 ), Uxi)] [Singularities at x 2 = 0, ±tt]: / C 2 C\S 2 s x s 2 \ *y 2 ^3 CiC 2 C 3 -SiS 3 -5 1 c 2 c 3 -c 1 ^3 ; \S 2 S 3 C 1 C 2 ^3 + 5 1 C 3 -S x C 2 S 3 + C\C 3 ) w i = (l) v i + (0)v 2 + (c 2 )v 3 w 2 = ( 0 )V[ + (*Sl) v 2 + ( —Ci 3 ’ 2 )V 3 W 3 = (0) V[ + (Cj)v 2 + (5 1 3’ 2 )v 3 V 1 = ( J 2 ) '[feVl + ( c l c 2 ) w 2 + ( — ■ s l c 2 ) w 3] v 2 = ( 5 2 ) *[(0) w l + (Sl^)^ + (CiS 2 )w3] v 3 = ( 5 2 ) '[(0) w l + ( — c l) w 2 + (■S’l)^]- 9(a) [2(xi), 1 (x 2 ))2(x 3 )] = P'Cto), l'(x 2 ),2'(xi)] [Singularities at x 2 = 0, ±tt]: /—S[C 2 S 3 + C{C 3 S’iS't SlC 2 C3 + Ci^ \ ■^3 c 2 -s 2 c 3 ; \ -t'l C 2 S 3 - 3, C 3 C X S 2 CjC 2 C 3 -S\S 3 J W 1 — (■y 2 5 3) v l + ( c 3) v 2 + (0) v 3 V 1 — ( s 2 ) * [feVl + (0) w 2 + (“ c 3) w 3] W 2 — (c 2 )vi + (0)v 2 + ( 1 )V 3 V 2 — (s 2 ) [(^ 2 C 3 )cDj + (0)cu 2 + (^ 253 ) 023 ] w 3 = ( — a 2 C 3 )v] + (5 3 )v 2 + (0)v 3 V3 = (^ 2 ) 1 [(—c 2 s 3 )cj x + (s 2 )uj 2 + (c z s 3 )uj 3 ]. WWW.EBOOK777.COM §1.13 THE RIGID BODY: TRANSFORMATION MATRICES 9(b) P'(xi), l'(X2),2'(x 3 )] = [2(x 3 ), 1(x 2 ),2(xi)] [Singularities at x 2 = 0 ,±tt]: (-S\C 2 Sl + C 1 C 3 s 2 s 3 c l c 2 s 3 +s l c 3 \ S\S 2 c 2 —C\S 2 \-SlC 2 C 3 -Ci5 3 U>\ = (0)vj + (ci)v 2 + (•S’l‘S'2) v 3 = (l) v l + (0) v 2 + ( c 2 ) v 3 w 3 = (0)Vi + (Si)v2 + ( — CiJ 2 )v 3 ^ cic 2 c 3 - s , 3 y V 1 = fe) ' [( — ‘ v l c 2) w l + ( 5 2 )w 2 + ( c lC 2 )w 3 ] V 2 = (s 2 ) * [(CV 2 V 1 + (0)w 2 + (sis 2 )w 3 ] v 3 = fe) * [(^l) w l + (0)^2 + (“ c l) w 3 ]- 10(a) [2(xi), 3 (x 2 ), 2(x 3 )] = [2'(x 3 ), 3'(x 2 ), 2'(xi)] [Singularities at x 2 = 0, ±tt]: / C X C 2 C 3 ~S X S 3 -Ci5 2 c l c 2 5 3 + 5iC 3 \ 5 2 C 3 c 2 5 2 5 3 \ C 2 C 3 - S\S 2 —5lC 2 Y 3 + CiC 3 / = (s 2 C 3 )V! + ( — 5 3 )v 2 + (0)v 3 w 2 = ( c 2 ) v l + (0)v 2 + (1)V 3 w 3 = (■ s 2‘ s 3) v l + ( c 3 ) v 2 + (0)v 3 V 1 = (* 2 ) ' [( c 3 Vl + (0)w 2 + (y 3 )w 3 ] v 2 ~ ($ 2 ) 1 [(—■ s '2- s ' 3) Cl; 1 + (0) w 2 + (Y2 c 3 ) w 3] v 3 = (^ 2 ) *[( —C 2 C 3 ) W 1 + ( S 2)U 2 + ( —C 2 S 3 )w 3 ]. 10(b) [2 , (xi),3 , (x 2 ),2'(x 3 )] = [2(x 3 )j 3(x 2 ), 2(xi)[ [Singularities at x 2 = 0, ±tt]: / C X C 2 C 3 ~S X S 3 Cl ^2 \"ClC 2 5 3 -SiC 3 iO\ = (0)V] + (—5j)v 2 + (C[Y 2 ) V 3 w 2 = (l) v l + (0)v 2 + (c 2 )v 3 t0 3 = (0)V] + (ci)v 2 + (5i5 2 )v 3 —5 2 c 3 YiC 2 C 3 + C]Y 3 \ c 2 s x s 2 ; s 2 s 2 -s { C 2 S 3 + c, c 3 J v l = fe) *[( —c l c 2 ) w l + (5 2 )w 2 + (—5!C 2 )w 3 ] V 2 = (j 2 ) [(—iV[ 5 2 )(U[ + (0)cu 2 + (C[Y 2 )w 3 ] v 3 = { s l) 1 [( c l) w l + (0)(U 2 + (5l)w 3 ]. 11(a) [3(xi)i l(x 2 )i 3(x 3 )] = [3'(x 3 ) ; l / (x 2 ).3'(xi)] [Singularities at x 2 = 0, ±tt]: ( ~S\C 2 s 3 + c x c 3 -5!C 2 C 3 - c x s 3 s x s 2 \ CiC 2 Y 3 +YiC 3 C\C 2 C 3 5^3 -CyS 2 \ \ s 2 s 3 = (5 2 5 3 )V[ + (c 3 )v 2 + (0)v 3 w 2 = (52 c 3 ) v 1 + ( — •5 3 ) v 2 + (0)v 3 w 3 = (c 2 )vi + (0)v 2 + (l)v 3 S 2 c 3 c 2 ) v l = (5 2 ) 1 [(5 3 )W] + (c 3 )w 2 + (0)w 3 ] v 2 = (® 2 ) 1 [(5 2 C 3 )wi + (—S 2 S 3 )l0 2 + (0)w 3 ] v 3 = (5 2 ) *[( —c 2 53)wi + (—c 2 e 3 )w 2 + (s 2 )u; 3 ]. WWW.EBOOK777.COM CHAPTER 1: BACKGROUND 11(b) [3'Od), l'(x 2 ),3'(x 3 )] = [3(x 3 ), 1 (X2),3 (xt)] [Singularities at x 2 = 0,±tt]: /-S l c 2 s 3 + CtC 3 -C l C 2 S 3 -S l C 3 S 2 S 3 \ S\C 2 c 3 + c | s 3 c x c 2 c 3 -s x s 3 -S 2 c 3 ; V -S’l - 5-2 CiS 2 c 2 ) U}\ = ( 0 )l’[ + (c i)v 2 + (^ 1 ^ 2 ) v 3 w 2 = (0)vi + (—■S , l)v2 + (cn?2) v 3 Ul 3 = (l)vi + ( 0 )V 2 + (^ 2 ) V 3 v i = ( s i ) 1 [( ^2)^1 + ( Cj c 2 ) uj 2 + (^2)^3] v 2 = (j 2 ) '[(ci^)^! + (— SiS 2 )u2 + ( 0 )cD 3 ] v 3 = ( J 2 ) * [( 5 l) w l + ( c l) w 2 + (0)w 3 ]. 12(a) [3(xi)j2(x 2 )j 3(x 3 )] = [3 '(x 3 ).2'(X2 ) j 3'(xi)] [Singularities at x 2 = 0,±tt]: (C\C 2 C 3 -SyS 3 C\C 2 S 3 .4 C 3 Cji 2 \ S\C 2 C 3 + Ci5 3 -^ 1 C 2 ^3 + C 1 C 3 S\S 2 ; V -*2C 3 W 1 = (—^2 c 3 ) v 1 + (• s, 3 ) v 2 + (0) v 3 w 2 = (^ 2‘ S 3 ) v 1 + (c 3 ) v 2 + (0) v 3 w 3 = (c 2 )vi + ( 0 )v 2 + (l)v 3 •^3 C 2 J v i = fe) *[(— c 3 ) w i + (^3)^2 + ( 0 )w 3 [ v 2 = fe) 1 [(• y 2 J 3 ) w l + ( s 2 c l)u 2 + ( 0 )w 3 ] v 3 = fe) * [{c 2 C 3 )ui\ + ( — C 2 ^ 3 )W2 + (^2)^3]- 12(b) [3 '(xi),2'(x 2 ),3 '(x3 )] = [3(x 3 ), 2 (x 2 )> 3(x0] [Singularities at x 2 = 0,±tt]: / C\c 2 c 3 - 5^3 -^ic 2 c 3 - c,5 3 ^ 2 c 3 \ C 1 C 2 J 3 + s l c 3 —^iC 2^'3 + C 1 C 3 S 2 S 3 i V -CiS 2 SiS 2 C 2 J w 1 = ( 0 )Vi + (^l)V 2 + ( —Ci5 2 )V3 w 2 = (0)V[ + (Cj)V2 + (•S'l^'b 0J 3 = ( 1 ) Vj + ( 0 ) V 3 + (c 2 ) v 3 V 1 = (^ 2 ) *[(^<^ 2)^1 + (~ s l c l) u 2 + fe)^] v 2 = (^ 2 ) '[(^l^Vl + ( c 1 5 2) w 2 + ( 0 )w 3 ] v 3 = (^ 2 ) * [(~ c l) w l + (■S , l)^2 + (0) w 3 ]- 1.14 THE RIGID BODY: AN INTRODUCTION TO QUASI COORDINATES As an introduction to quasi coordinates, and quasi variables in general (a topic to be detailed in chap. 2 ), we show in this section that the angular velocity, although a vector, does not result by simple d/dt(.. ^-differentiation of an angular displace¬ ment; its components along space-/body-fixed axes, say 04 , do not equal the total time derivatives of angles or any other genuine (global) rotational coordinates/para¬ meters, say 0kt that is, 04 f dO^/dt. This is another complication of rotational mechanics, one that is intimately connected with the noncommutativity of finite rotations; and it necessitates the hitherto search for connections of the 04 ’s with genuine angular coordinates and their rates, like the Eulerian angles <p, 9 , ip. WWW.EBOOK777.COM §1.14 THE RIGID BODY: AN INTRODUCTION TO QUASI COORDINATES Let us consider, for concreteness, the body-axes components of the angular velo¬ city tensor. From ft = A T - (dA/dt) = -(dA T /dt) - A, (1.14.1a) (1.7.30fff.) we have u x = A Xz (dA Xy /dt) + A Yz (dA Yy /dt) + A Zz (dA Zy /dt), etc., cyclically, (1.14.1b) or, multiplying through by dt and setting ui x dt = d6 x (just a suggestive shorthand!), dO x = Ax-dAxy + A Yz dA Yy + A Zz dA Zy , etc., cyclically. (1.14.1c) We shall show that 8(d9 x ) ~ d(89 x ) ^ 0, etc., cyclically, (1.14.2a) where, for our purposes, 6 (...) can be viewed as just a differential of (...), along a different direction from d(. . .); that is, with d(. . .) = d j(. . .) and 6(. ..) = d 2 (. ..), 8{d9 x ) = d 2 (di 9 X ), d{89 x ) = d x ( d 2 9 x ); (1.14.2b) and S9 X = A Xz SAx y + A Yz 5A Yy + A Zz SA Zy , etc., cyclically. (1.14.2c) Now, S (...)-differentiating d6 x and d (...)-differentiating 89 x , and then subtracting side by side, while noticing that 8{dA k , k ) = d(8A k , k ) (k' = X,Y,Z; k= x,y,z), (1.14.3a) we get 8(d9 x ) — d(69 x ) = 8A Xz dAx y — dA Xz bA Xy + 8A Yz dA Yy — dA Yz 6A Yy + bA Zz dA Zy — dA Zz 8A Zy . (1.14.3b) Next, in order to express 8A k * k , dA k ' k in terms of 89 k and d6 k , we multiply the components of dA/dt = A • L2 (1.7.30i) with dt, thus obtaining dA Xz = A Xx d9 y — A Xy d9 x => 8A Xz = A Xx 89 y — A Xy 60 x , (1.14.4a) dA Yz = A Yx d9 y - A Yy d9 x => 8A Yz = A Yx 89 y - A Yy 89 x , (1.14.4b) dA Zz = A Zx d9 y — A Zy d9 x => 8A Zz = A Zx 89 y — A Zy 89 x , (1.14.4c) dA Xy = A Xz d9 x — A Xx d9 z => 8A Xy = A X: 89 x — A Xx 89 : , (1.14.4d) dA Yy = A Yz d9 x — A Yx d9 z => 8A Yy = A Yz 89 x — A Yx 89 z , (1.14.4e) dA Zy = A Zz d9 x - A Zx d9 z => 8A Zy = A Zz 89 x - A Zx 89 z . (1.14.4f) Substituting (1.14.4a f) into the right side of (1.14.3b), and invoking the orthogon¬ ality of A = (A k ' k ) [e.g., (1.7.6a, b; 1.7.22d)], we find, after some straightforward algebra, the noncommutativity equation 6(d9 x ) - d(89 x ) = d9 y 89 z - d9 z 89 y . (1.14.5a) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Working in complete analogy with the above, we obtain 6{dO y ) - d(89 y ) = dO z 8Q x - d9 x 80 z , (1.14.5b) 8{d9 z ) - d{89 z ) = dO x 89 y - d0 y 80 x . (1.14.5c) These remarkable transitivity equations [because they allow for a smooth transition from Lagrangean mechanics (chap. 2ff.) to Eulerian mechanics (§1.15ff.)] show clearly that the 9 X y z are not ordinary (or genuine, or holonomic, or global; or as Lagrange puts it “variables finies”) coordinates, like the Eulerian angles </>, 9, ip; that is why they are called pseudo- or quasi coordinates. Their general theory, along with a simpler derivation of the above, are detailed in chap. 2. Similarly, we can show that in terms of space-axes components, the transitivity equations are 6{d9 x ) - d(69 x ) = d9 z 89y - d9 y 89 z , (1.14.6a) 8{d9y) - d(89y) = d9 x 89 z - d9 z 89 x , (1.14.6b) 8{dO z ) — d{89 z ) = d9y89 x — d9 x 89 y . (1.14.6c) In compact vector form, (1.14.5a, b, c) and (1.14.6a, b, c) read, respectively, 8 re ,{d0) - d(88) =d0 x 80, (1.14.7a) S(d9) - d(S8) =89 xd8, (1.14.7b) where dO = d9 x i + d9 y j + dd z k = d6 x l + d9yj + d6 z K => 89 = 89 x i + 80 y j + 89 z k = S9 X I + 89 y J + 80 z K => 8 re t(d9) = 6(dO x )i + S(d9 y )j + 8(d9 z )k,d(69) = d(89 x )i + d{89 y )j + d(89 z )k; that is, again, 9 is a quasi vector. Here (recall 1.7.30j), fl' = (dA/dt) • A t and dA/dt = fl' • A. [More in Examples 2.13.9 and 2.13.11 (pp. 368 ff.).] HISTORICAL Equations (1.14.5a-c), along with the systematic use of direction cosines to rigid- body dynamics, are due to Lagrange. They appeared posthumously in the 2nd edi¬ tion of the 2nd volume of his Mecanique Analytique (1815/1816). See also (alphabe¬ tically): Funk (1962, pp. 334-335), Kirchhoff (1876, sixth lecture, §2), Mathieu (1878, pp. 138-139). 1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY Introduction, Basic Definitions To get motivated, let us begin by calculating the (inertial) kinetic energy T of a rigid body B rotating about a fixed point 0\ the extension to the case of general motion follows easily. If a> is the inertial angular velocity of B. then, since the inertial velocity of a genetic body particle P, of inertial position r, is <u x r = v, we have, successively, 2 T = ^ dm v ■ v = ^ dm(oj x r) ■ (at x r) = £ dm [(co • co) (r • r) — (to • r) (to • #•)] (by simple vector algebra) = S dm[uj 2 r 2 — (to-r) 2 ], (1.15.1a) WWW.EBOOK777.COM §1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY or, in terms of components along arbitrary (i.e., not necessarily body-fixed) rectan¬ gular Cartesian axes O—xyz = 0—x kl in which r = (x,y,z) = (x k ), to = (lu x , u> y , u> z ) = M (* = 1,2,3), ^=SW(S>4(S> 2 )- = S’*[(SE hi^k^t) (E^ 2 ) ~ (EE^' W/X ^) = E E dki Uk u>i (Indicial notation) (1.15.1b) = OJ • I • (0 (Direct notation) (1.15.1c) = ft) T • I • to (Matrix notation; a>: column vector), (1.15.Id) where l() = I = (-lojd ) = (hi ), hi = S dm ( rl Ski ~ x k x,)\ Components of tensor of inertia of B, I, at O, along 0-x k , r 2 = x k 2 = E XkXk ’ or, equivalently, hi = Jo hi — Jkh where Jo =J = ( Jo,kl ) = (Jkl) , hi = S XkX > dm = Components of Binet’s tensor of B, J, at O, along O-x k , Jo = J\\ + J 22 + J33 = $ r 2 dm = Tr J. In direct notation, the above read / = ^ [(/■ • r)l — r (g> r\dm , J = ^ (r <8 r) dm. (1.15.2h) That I is a (second-order) tensor follows from the fact that, under rotations of the axes, T is a scalar invariant while (o is a vector (what, in effect, constitutes a simple application of the tensorial ‘‘quotient rule”). This means that the components of I along 0—x k ,I k h and along 0-x k /, 4'/', where x k < = JfA k i k x k (proper orthogonal transformation), are related by h'l' = E E Ak'kAinh-i = ^2 ,E A k'khiA/i' = (A -I•A T ) k ' l > [recalling eqs. (1.1.19j ff.)] ** hi = EE Akk'An'h’i' (Since, indicially, A k i k = A kk i) J . WWW.EBOOK777.COM (1.15.2e) (1.15.2f) (1.15.2g) (1.15.2a) (1.15.2b) (1.15.2c) (1.15.2d) (E w ***j (E w ^J 215 (1.15.21) CHAPTER 1: BACKGROUND Properties of the Inertia Tensor Clearly (and like most mechanics tensors), I is symmetric: fii = /«•; that is, at most six, of its nine components, are independent. In extenso, (1.15.2a, b) read (lxx Ixy 4z\ I = Iyx Iyy Iyz \ Izx Izy Izz ) (S dm O' 2 + ') — Q dm xy —S dm xz \ = K 1 to 1 dm (z 2 + x 2 ) ~ S dm z y — Q dm z x ~ S dm z y s dm ( x2 + y 2 ) / (1.15.3) The diagonal elements of I, l xx , Iyy, I Z z, are called moments of inertia of B about O—xyz', and they are nonnegative', that is, Ixx, yy ,zz > 0 - Th e off-diagonal elements of /. I xy = I yx , l xz = l 7X , I yz = I ly , are called products of inertia of B about O-xyz, and they are sign-indefinite', that is, they may be > 0 , < 0 , or = 0 . In view of the above, T can be rewritten as 2T = l xx U) x " + Iyy U)y~ ~\~ I ZZ LOf 2 ly-y UJy UJy 2 I yz OJ y CJ Z 2 Iy z (jJy LU z . ( 1 . 15 . 4 ) Now, evidently, the choice of the axes O-xyz is nonunique. Not only can they be non¬ body-fixed (in which case, the I ki are, in general, time-dependent); but even if they are taken as body-fixed, (1.15.3, 4) are still fairly complicated. Hence, to simplify matters as much as possible, and since the kinetic energy is so central to analytical mechanics, we, in general, strive to choose principal axes at O'. O—xyz —> <9-123; usually, but not always, body-fixed. Since I is symmetric, such (mutually orthogonal) axes exist always; and along them / becomes / (h 0 0 \ 0 I 2 0 \0 0 ij Principal axes representation of inertia tensor, (1.15.5) where the principal moments of inertia, at O, I\ 2 3 are the eigenvalues of E ^ kl W l = that is, they are the roots of its characteristic equation: D( A) = — Det(4; - A Skf) = 0 ; ^1,2,3 = A,2,3- (1.15.6a) (1.15.6b) Using basic theorems of the spectral theory of tensors [(1.1.17a ff.)] we can show the following: (i) At each point of a rigid body B there exists at least one set of principal axes. (ii) Since, by (1.15.lb d), the inertia tensor is not only symmetric, but also positive definite [i.e., fffflkia k ai> 0 , for all vectors a = (af) f 0 ], all three roots of (1.15.6b) are not only real but also strictly positive. WWW.EBOOK777.COM §1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY Further • If Aj ^ A 2 7 ^ A 3 Aj (all three eigenvalues distinct), then 0-123 is unique. • If Aj ^ A 2 = A 3 (two distinct eigenvalues), there exists a single infinity of such sets of principal axes; 01 and every line perpendicular to it are principal axes; that is, the direction of either 02 or 03 , in the plane perpendicular to 01, can be chosen arbi¬ trarily (e.g., a homogeneous right circular cylinder, with O on its axis of symmetry). • If \\ = A 2 = A 3 (only one distinct eigenvalue), there exists a double infinity: any three mutually perpendicular axes can be chosen arbitrarily as 0-123 (e.g., O being the center of a homogeneous sphere). Along principal axes, T, (1.15.4), with co = (uq, u> 2 , w 3 ), reduces to 2 T — T I2^2 T -^ 3^3 . (1.15.6c) The Generalized Parallel Axis Theorem (“Huygens-Steiner”) This explains how I changes from point to point, among parallel sets of axes. THEOREM Let O-xyz and G-xyz be two sets of mutually parallel axes, and let the coordinates of the center of mass of B , G, relative to O , be OG = r c = (x G , y G , zg) = (Gr, G 2i G 3 ) = (- a , -b, -c ), (1.15.7a) that is, a : b,c = coordinates of O relative to G-xyz. Then, the components of the inertia tensor of B at O , l 0 .ku and at G, are related by Direct notation : I 0 = 1 G + m{r c 2 l — r G < 8 > r G ) , (1.15.7b) Indicial notation: I 0 kl = I G k , + m [ G,.c)j S kl - G k G, ; or, in extenso, lo = Ig + m{b 2 + c 2 ) —mba —me a —mab m (c 2 + a 2 ) —meb —mac —mbc m(a 2 + b 2 (1.15.7c) (1.15.7d) PROOF We have, successively, (i) io,xx = S dm i(y~ b f + i z - c ) 2 ] = S dm(yl + ^ ~ 2b {S dm y) ~ 2b {S dm z'j + ^ dm(b' + c 2 ) = I G , XX + 0 + 0 + m(b 2 + c 2 ); etc., cyclically, for I 0iyy , / 0 , zz - (1.15.7e) (ii) Io,yz = ~S dm ^ y - b ^ z ~ = — ^dm yz + dtu + b (S dm “) — S dm bc = Ic.yz + 0 + 0 — mbc; etc., cyclically, for / 0 jcz , I 0iXy - (1.15.7f) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND More generally, it can be shown that between any two points A , B (with some ad hoc but, hopefully, self-explanatory notation), Ib = Ia + m(r A / B ~l — r A / B <g> r A / B ) + 2m[(r A / B -r G/A )l -(1/2 )(r A/B ® r G/A +r G/A ® r A/B )] ; (1.15.7g) which, when A —> G and r G / A — > 0, reduces to (1.15.7b) (see below). (See, e.g., Lur’e, 1968, p. 143; also Crandall et al., 1968, pp. 180-182, Magnus, 1974, pp. 200-201.) It should be noted that the transfer formulae (1.15.7b, g) are based on definitions of moments of inertia about points , like (1.15.2h), not axes, and therefore hold for any set of axes through these points; that is, they are independent of the axes orientation at A, B. If, however, these axes are parallel, certain simplifications occur; indeed, (1.15.7g) then yields the component form, IbM = 1A,kl + m \{ x A/B,\ + X A/B,2~ + X A/B, 2 )^kl — x A/B,k x A/B,l\ + 2rn \{x A / B \ X G / A ^\ + X A / B 2 X G / A 2 + X A / B T, X G / A j)S k/ — (1/2 )(x A / Bk x G / A j + x G / Ak x A / B/ )], (1.15.7h) where r A / B = (x A / BA , x A / B ^, x A / B ^) = coordinates of A relative to B, along axes B—xyz = B—x k , and r G / A = (x G / A I , x G / A 2 , x G / A 3 ) = coordinates of G relative to A, along axes A—xyz = A—x k (parallel to B—x k )', or, in extenso, with X A/B, 1 = X A/B: X A/B,2 = Va/B, x A/B,3 = Z A/Bi X G/A, 1 = X G/A, e tC., 7/1. \ \ 7.-1 .XX + m{y A / B 2 + Z A/B 2 ) + 2 m(y A / B )’g/a + z A/B z G/A ), etc., cyclically, (1.15.71) hi.xy = lA,xy ~ ™(yA/B x G/A + X A/Byc/A) ~ ™{ x A/ B yA/B ), etc., cyclically. (1.15.7j) If A —> G, then r G / A —> 0, r A / B —> r G / B , and the above reduces to ^B,kl = Ic,kl + ,n [{ x G/BA~ + X G/B,2~ + X G/B,3~)^kl ~ x G/B,k x G/B,l\ > (1.15.7k) from which, in extenso, h,xx = Iq,xx + m(yG/B 2 + z g/b~) = Ig,xx + + (“ C )~L etc., cyclically, (1.15.71) h,xy = hi.xy - ”K x g/b)’g/b ) = hi,xy ~ "»[(-«)(-*)], etc., cyclically; i.e., (1.15.7b—f). (1.15.7m) Ellipsoid of Inertia Let us consider a rectangular Cartesian coordinate system/basis O—xyz/ijk, and an axis u through O defined by the unit vector u = (u x) u y , u z ). Then, as the transfor¬ mation equations against rotations (1.15.2i) show, the moment of inertia of a body B about u, /„„ = /, will be (with k' = /' = u;k,l = x,y,z; A k i k = u k = u l 2 3, A k 'i = ui = ni^, etc.) I — I XX U X T iyyUy T I -y II - 2 U y U y 21 y-li yll - ~\~ 2 I y-U yll y . WWW.EBOOK777.COM (1.15.8a) §1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY [For nontensorial derivations of (1.15.8a), see, for example, Lamb (1929, pp. 66-67), Spiegel (1967, pp. 263-264)]. We notice that if a> = u>u = {u>u x , u>u v , touf) = (uj x , uj y . lo z ), then the kinetic energy of B , moving about the fixed point (9, eq. (1.15.4), becomes 2 T = Ilo 2 . (1.15.8b) Now, by defining the radius vector r = u/I 1/2 =xi + yj + zk [i.e., M = r = (1//) 1/2 ], (1.15.8c) we can rewrite (1.15.8a) as Ixx-^ lyyIzz'z' “I - 2/^xy + 2.1 y Z yz + 2 I zx zx = 1. (1.15.8d) Since I is positive [except when all the mass lies on «; then one of the principal moments of inertia, roots of (1.15.6b), is zero and the other two are equal and positive], every radius through O meets the quadric surface represented by (1.15.8d), in O-xyz , in real points located a distance r = (l//) 1 ^ 2 from O , and there¬ fore (1.15.8d) is an ellipsoid; appropriately called ellipsoid of inertia or momenta! ellipsoid. [A term most likely introduced by Cauchy (1827), who also carried out similar investigations in the theory of stress in continuous media (“stress quadric”).] If the axes are rotated so as to coincide with the principal axes of the ellipsoid — that is, 0—xyz—> O— 123 — then (1.15.8d) simplifies to Vf + h r 2~ + h r i 2 — 1) or h/(l//i ) 1/2 ] 2 + h/( 1 // 2 ) 1/2 ] 2 + ['- 3 /(l// 3) 1/2 ] 2 = 1, (1.15.8e) where r 123 are the “principal” coordinates of r, and (l//i, 2 , 3)*^ 2 are the semidia¬ meters of the ellipsoid. [Some authors (mostly British) define the radius of the momental ellipsoid along u (i.e., our r) as r = me 4 // 1/2 ~ 7 1/2 , (1.15.8c.l) where m = mass of body, and e = any linear magnitude (taken in the fourth power for purely dimensional purposes), so that the ellipsoid equations (1.15.8d) and (1.15.8e) are replaced, respectively, by I xx x 2 + I vy y 2 + I zz z 2 + 2 I xy xy + 2 I yz yz + 2 1 :x zx = me 4 , (1.15.8d. 1 ) I x r 2 + I 2 r 2 2 + I 2 r 2 = me 4 . (1.15.8e.l) Also, for a discussion of the closely related concept of the ellipsoid of gyration (introduced by MacCullagh, 1844), see, for example, Easthope (1964, p. 134ff.), Lamb (1929, p. 68 ff.)] However, it should be remarked that not every ellipsoid can represent an inertia ellipsoid; in view of the “triangle inequalities” (see below), certain restrictions apply on the relative magnitudes of the semidiameters, and hence the possible forms of the momental ellipsoid. Now, and these constitute a geometrical sequel to the discussion of the roots of the characteristic equation (1.15.6b): WWW.EBOOK777.COM CHAPTER 1: BACKGROUND • If I\ = I 2 = / 3 , the momental ellipsoid is a sphere. All axes through O are principal, and all moments of inertia are mutually equal. Such a body is called kinetically symmetrical about O. • If, say, / 2 = / 3 , the ellipsoid is one of revolution about Ox —all perpendicular diameters to Ox are principal axes. Such a body is called kinetically symmetric about that axis', or simply uniaxial (Routh). The above show that, in general, the ellipsoid of inertia, at a point, is nonunique. The ellipsoid of inertia of a body at its mass center G, commonly referred to as its central ellipsoid (Poinsot), is of particular importance: As the parallel axis theorem shows, if the moment of inertia about an axis through G is known, I G , then the moment of inertia about any other axis parallel to it is obtained by adding to I G the nonnegative quantity md 2 , where d is the distance between the two axes. Finally, the momental ellipsoid interpretation, plus the above parallel axis theo¬ rem, allow us to conclude the following extremum (i.e., maximum / minimum) proper¬ ties of the principal axes: • The principal axes of inertia, at a point O, are those with the larger or smaller moment of inertia than those about any other line through that point, /. Quantitatively, if h = 4,ax > h > h = 4,in (1.15.8f) [ =>■ (l/7i )'/ 2 < (I// 2) 1 / 2 < (I// 3 ) 1 / 2 ], something that can always be achieved by appro¬ priate numbering of the principal axes, then I mdx > I > Imin- (1.15.8g) • The smallest centra! principal moment of inertia of a body, say I G 3 = I Gm in , is smaller than or equal to any other possible moment of inertia of the body (i.e., moment of inertia about any other space point and direction there); that is, I G m in > I uu . Additional Useful Results (i) It can be shown that + h <^ 3 + hi h < h + (1.15.8h) that is, no principal moment of inertia can exceed the sum of the other two. Equations (1.15.8h) are referred to as the triangle inequalities (since similar relations hold for the sides of a plane triangle). Actually, this theorem holds for the moments of inertia about any mutually orthogonal axes (McKinley, 1981). (ii) Let pi 2 3 be the semidiameters (semiaxes) of the ellipsoid of inertia; that is, Pi 2,3 = (/ipp) -1 ^ 2 . Then, the third and second of (1.15.8h) lead, respectively, to the following lower and upper hounds for p 3 , if pi 2 are given, (p 2 -2 +pr 2 r 1/2 < p-s < iP 2 ~ 2 - pr 2 r 1/2 ; (i.i5.8i) and, cyclically, for pi 2 ; that is, arbitrary inertia tensors, upon diagonalization, may yield (mathematically correct but) physically impossible principal moments of inertia! As a result of the above, if two axes, say pi and p 2 , are approximately equal, the corresponding inertia ellipsoid can be quite prolate (longer in the third direction, WWW.EBOOK777.COM §1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY cigar shaped), but not too oblate (shorter in the third direction; flattened at the poles, like the Earth). (iii) The quantity Tr/ = I xx + f yy + /., (first invariant of I) depends on the origin of the coordinates, but not on their orientation. (iv) - I xx /2 < I yz < I xx /2, -I yy /2 < I zx < I yy /2, -IJ 2 < I xy < IJ 2. (1.15.8 j) (v) Consider the following three sets of axes: (a) O-XYZ: arbitrary “background” (say, inertial) axes; (b) G—XYZ = G—xyz: translating but nonrotating axes, at center of mass G; and (c) G-123: principal axes at G. By combining the transformation formulae for hh between parallel axes of differing origins (like O-XYZ and G-xyz) and arbitrary oriented axis of common origin (like G-xyz and G-123), we can show that Ixx = m (Y G ~ + Z G ~) + A x 2 1 { + A X2 2 l2 + A X3 2 I 3 , (1.15.8k) I yy = m(Z G - + X G ~) + A Y \~I\ + A Y2 1 2 + A Y3 ~I 3 , (1.15.81) Izz = m {X G ~ + Y g ) + A z 2 I\ + A Z 2 2 I 2 + A Z3 2 I 3 ; (1.15.8m) Ixy = — mX G Y G + A xx A Yl {I 3 — If) + A X2 A Y2 (I 3 — I 2 ), (1.15.8n) Iyz — ~ m Y g Z g + A Y \A Z \(J 3 — I\) + A Y2 A Z2 (I 3 — I 2 ), (1.15.8o) Izx = ~ m Z G X G + A zl A X i(I 3 — /J + A Z2 A X2 (I 3 — / 2 ); (1.15.8p) where = cos(Odf, Gl) = cos(Gx, Gl), etc., and X Gl Y G , Z G are the coordinates of G relative to O-XYZ. The usefulness of (1.15.8k p) lies in the fact that they yield the moments/products of inertia about arbitrary axes, once the principal moments of inertia at the center of mass are known. (vi) (a) If a body has a plane of symmetry, then (a) its center of mass and (/ 3 ) two of its principal axes of inertia there lie on that plane; while the third principal axis is perpendicular to it. (b) If a body has an axis of symmetry, then (a) its center of a mass lies there, and (0) that axis is one of its principal axes of inertia; while the other two are perpendicular to it. (c) If two perpendicular axes, through a body point, are axes of symmetry, then they are principal axes there. (But principal axes are not necessarily axes of symmetry!) (d) If the products of inertia vanish, for three mutually perpendicular axes at a point, these axes are principal axes there. [For a general discussion of the relations between principal axes and symmetry (via the concept of covering operation), see, for example, Synge and Griffith (1959, p. 288 ff.).] (e) A principal axis at the center of mass of a body is a principal axis at all points of that axis. (f) If an axis is principal at any two of its points, then it passes through the center of mass of the body, and is a principal axis at all its points. (vii) Centrifugal forces: whence the products of inertia originate. Let us consider an arbitrary rigid body rotating about a fixed axis OZ with constant angular velocity co. Then, since the centripetal acceleration of a generic particle of it P, of mass dm, equals v 2 /r = ui 2 r, where r = distance of P from OZ, the associated centrifugal force WWW.EBOOK777.COM CHAPTER 1: BACKGROUND df c has magnitude df c = dm(uj 2 r), and hence components along a, say, body-fixed set of axes 0—xyz{OZ — Oz) will be dfcx = dfc(x/r) = dm xui 1 , clf cy = dfi.{y/r) = dm yu? , df c ^ = 0; (1.15.9a) where x, y, z are the coordinates of P. Therefore, the components of the moment of df c along these axes are dM cx = y df cz — z df C y = —dm yzco 2 , dM cy = z df cx — x df cz = +dm xzur, dM cz = x df C j - y df^ x = 0. (1.15.9b) From the above, it follows that these centrifugal forces, when summed over the entire body and reduced to the origin O, yield a resultant centrifugal force f c \ f c ,x = S =U}2 S dm x = ^ mx G . fc,y = S d fc,y =Ujl S dm y = w 2 " 7 -V Gj fc, z =S d fc, = 0, (1.15.9c) where x G , y G are the coordinates of the mass center of B. G'; and a resultant cen¬ trifugal moment My. M c x = ^ dM c x = — u) 2 ^ dm yz= +url rz , M c y = ^ dM c v = ur ^ dm zx = — ui 2 I xz , M ( . z = gdM ( . z = 0. (1.15.9d) Equations (1.15.9c, d) show clearly that if G lies on the Z = z axis, then/ f vanishes, but M c does not. For the moment to vanish, we must have I yz = 0 and I xz = 0; that is, Oz must be a principal axis. In sum: The centrifugal forces on a spinning body tend to change the orientation of its instantaneous axis of rotation , unless the latter goes through the center of mass of the body and is a principal axis there. Such kinetic considerations led to the formulation of the concept of principal axes of inertia, at a point of a rigid body [Euler, Segner (1750s)]; and to the alternative term devia¬ tion moments, for the products of inertia. We shall return to this important topic in §1.17. 1.16 THE RIGID BODY: LINEAR AND ANGULAR MOMENTUM (i) The inertial, or absolute, linear momentum of a rigid body B (or system 5), relative to an inertial frame F, represented by the axes I-XYZ (fig. 1.28), is defined as p=^dmv = mv G ( G: center of mass of B). (1.16.1a) Substituting in the above [recalling (1.7.11a ff.), and with r G / 0 = r G , <n —> f2: angular velocity vector of noninertial frame —» axes O-xyz relative to inertial ones I-XYZ] Va = Vo + Vc/o = Vo + Vc,rei + ^ x r c (v G , re i = dr c /dt) (1.16.1b) WWW.EBOOK777.COM §1.16 THE RIGID BODY: LINEAR AND ANGULAR MOMENTUM Figure 1.28 Rigid body (6), or system (5), in general motion relative to the noninertial frame O-xyz; S2: inertial angular velocity of O-xyz (two-dimensional case). readily yields P = Ptmm + Preb (1.16.1c) /’trans = m ( v o + x r G ): Linear momentum of transport , (1.16. Id) p rel = mv G , rei = m(dr G / dt) : Linear momentum of relative motion. (1.16.1 e) If B is rigidly attached to the moving frame M, represented by the axes O-xyz (fig. 1.28), then, clearly, p te] = 0. (ii) The inertial and absolute angular momentum of B , relative to the inertial origin /, /y /abs = //,. is defined as H ^S [rj x dm ( drj/dt)\ = ^ [9? x dm (d^R/dt)\ [substituting 9? = r 0 /i + r = r 0 + r =>■ dW/dt = v 0 + v re j + Q x r, v re i = dr/dt] = mr 0 x (v 0 + ii x r G ) + mr G x v 0 + ^ dm [r x (Q x r)) + ^ dm (r x v rel ), (1.16.2a) or, since ^ dm [r x (Q x r)] = dm [r 2 i2 — (r <g> r) ■ O] = ^ dm [r 2 1 — (r ® r)] • fl = Io • , (1.16.2b) and calling H 0 , rel = S' x (dm r re i): Noninertial and absolute angular momentum of B, about O , (1.16.2c) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND we finally obtain the following general kinematico-inertial result: H r = H 0 + r Q x p + mr G x v 0 , Hq = I Q • + H 0?le i. (1.16.2d) Special Cases (i) If the body B is rigidly attached to the moving frame M , then v r ei = o, /> rel = 0, H 0 Te i = 0, Q = o) = inertial angular velocity of B , (1.16.3a) and, therefore, p = m(yo + <*> x r c ) = mva, Hj = I 0 - m + r 0 x p + tnr c x v 0 ■ (1.16.3b) (ii) If, farther, O = G, then r G = 0, r 0 = r 0 . and, therefore, p = mv G , Hr = I 0 • a) + r 0 x p. (1.16.3c) (iii) If 1 = 0 (i.e., rigid-body motion with one point, O, fixed), then r 0 = 0, v 0 = 0, and, therefore, p = m (a> x r G ) = mv G , H I = H 0 =Io- a> r x dm v (1.16.3d 1,2) It should be pointed out that the above hold for any set of axes, including non-body- fixed ones, at the fixed point O', but along such axes the components of Io will, in general, not be constant. Equation (1.16.3d2) would then yield, in components [omitting the subscript O and with r = (.x A .)], H k — ^2, ^ klUJh \t = s dm i( r • r ) 7 - ( r ® r )] h=S dm \ (X! XrXr ) Ski - XkX ‘ (1.16.4a) (1.16.4b) If the axes at O are body-fixed , then the I kl are constant; and, further, if they are principal, then Hk — 4 w k■ (1.16.4c) Linear Momentum of a Rotating Body To dispel possible notions that the linear momentum is associated only with transla¬ tion, let us calculate the linear momentum of a rigid body rotating about a fixed point ♦ . We have, successively, with the usual notations, p = mv G = m{m x r G /*) = m(a> x r G ) = m(u} x ,u y ,u) z ) x (x G ,y G , z G ) [components along any ♦-axes] = m(u y z G - u z y G , u z x G -u x z G , u x v G -u y x G ). (1.16.5a) WWW.EBOOK777.COM §1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION In particular, if the body rotates about a fixed axis through ♦ , say ♦" [recalling (1.15.9aff.)], then oj = u z k = (d<j)/dt)k, and so (1.16.5a) reduces to p x = — {my G )u> z = — m y u z , p y = (mx G )uj z = m x u z , p : = 0. (1.16.5b) These expressions appear in the problem of rotation of a rigid body about a fixed axis, treated via body-fixed axes ♦-xyz [see, e.g., Butenin et al. (1985, pp. 266-278) and Papastavridis (EM, in preparation)]. It is not hard to show that, in this case, the (inertial and absolute) angular momentum of the body //* = gr x dmv= (H x ,H y ,H : ), (1.16.5c) reduces to H x = I xz u> z , II v = I yz uj z , H z = I zz uj : = I : lu z . (1.16.5d) 1.17 THE RIGID BODY: KINETIC ENERGY AND KINETICS OF TRANSLATION AND ROTATION (EULERIAN "GYRO EQUATIONS") We recommend, for this section, the concurrent reading of a good text on rigid-body dynamics; for example (alphabetically): Grammel (1950), Gray (1918), Hughes (1986), Leimanis (1965), Magnus (1971, 1974), Mavraganis (1987), Stackel (1905, pp. 556-563). (i) The inertial kinetic energy of a rigid body B in general motion, T, is defined as the sum of the (inertial) kinetic energies of its particles: 2T(B,t) =2T=^dmv-v, (1.17.1) or, since v = v*+o>xr/* (♦: arbitrary body-fixed point), (1.17.2) 2T = S dm ( V * +(OX r /«) ' (’’♦ + m X ''/♦) = 2 ( 7 ’transl’n + ^rot’n + ^’g), (1-17.3) where 2T transPn = mv, • v* = mV, 2 : (twice of) Translators (or sliding) kinetic energy of B, (1.17.3a) 2r rot - n = ^dm(a) x ) • (<u x ) =«•/*•<«: (recalling §1.15) (twice of) Rotatory kinetic energy of B, (1.17.3b) T’cpi’g = x r G/ .) • = mv G/4 • v* = ( mr G/ *)' • v* = (dm Gj +/dt) • v*, WWW.EBOOK777.COM CHAPTER 1: BACKGROUND or T’cpPg = '”»* G/ * • (v* X to) = m Gj + • (v* x to): (1.17.3c) Kinetic energy of coupling between r* and to (where »i G /* = mr G /+) [= 0; e.g. if G = ♦ , or if v* and to are parallel [♦ on instantaneous screw axis of motion (§1.9)]; in which case, T decouples into translatory and rotatory kinetic energy]. (1.17.3d) • These expressions hold for any axes, either body-fixed or moving in an arbitrary manner, or even inertial. But if they are non-body-fixed , the components of r G /* and 7 * will, in general, not be constant. • We also notice that, in there, the mass m appears as a scalar ( nr. T transl > n ), as a vector of a first-order moment (/m g /* ■ T’cpPg), and as a second-order tensor (I* ■ 7rot' n ). • From (1.17.3b), we obtain, successively, grad w T mVn = dT voGn /do = £ dm v i * • (dv/* /dm) = $ #y* x(tax »y*) dm = £ jy* x (dm v/*) = ^♦.relative = [recalling (1.6.5b)]; (1.17.4) that is, the angular momentum is normal to the surface T IoVn = constant, in the space of the a/s. • If v* = 0 — for example, gyro spinning about a fixed point — (1.17.3b) yield 2 T =X 2T mt i n — I xx lu x ~ T * * * T 2I X y cc x uy T • • • = 77 * • to = /i* * o ^ 0; (1.17.4a) that is, since T is positive definite, the angle between 77 * = /j* and to is never obtuse : 0° < angle ( 77 * , to) < 90°. (1.17.4b) • Similarly, we can express T in terms of relative velocities; that is, with V = V* + J? X /y* + V/* 5re iative: V/*,relative = dr/Jdt , (1.17.5) where Q is the inertial angular velocity of the moving axes. Another Useful T-Representation We have, successively, 2 T = ^ dm »’•»’= ^ v • (dm r) = (v* + to x /y*) • (dm v) = v* • dm v) + ^ dp • (a) x »y*) (since dm v = dp) = V* •/> + ©• (^»y* X <//>) = V* •/) +to - 77 *, absolute- WWW.EBOOK777.COM (U7.6) §1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION Kinetic Energy of a Thin Plate of Mass m in Plane Motion on its Own Plane (fig. 1.17) Using plate-fixed axes ♦— xy, we find [with cos(...) = c(...), sin(...) = s (...)] = x G i + y G j = xi + yj, {x,y: constant) (1.17.7a) v* = (dX+ / dt)I + ( dY+/dt)J = ( dX/dt)I + ( dY/dt)J = (dX / dt){c(j) i — scj) j) + {dY / dt){sf i + ccj) j) = [{dX/dt)c4> + {dY/dt)sf\i + [ — {dX/dt)scf> + {dY / dt)c(f]j = v x i+v y j, a) = {d(f>/dt)K = {d<f>/dt)k; (1.17.7b,c) and so, successively, 2T tmmVn = m V* • v* = m[{dX/dt) 2 + {dY / dt ) 2 ]; (1.17.7d) T cpVg = in r G/4 • (v* x co) = m(x,y, 0) • \{v x , v y , 0) x (0,0, dcj)/dt)\ = m{d(j}/dt){VyX — v x y) = m{d<f) I dt) {[{dY / dt)x — {dX/dt)y\c<f> — [( dX/dt)x + {dY/dt)y\ s<fi}; (1.17.7e) 2r rofn = w-I.-oj =I. iZZ u z 2 =I{d(j>/df) 2 ; (1.17.7f) that is, 2 T = 2T{dX/dt, dY/dt , d^/dt) = m[{dX/dtf + {dY/dt) 2 ] + I{d<j>/dt) 2 + 2m{d(j)/dt){[{dY/dt)x — {dX/dt)y\ccf> — [{dX/dt)x + {dY/dt)y\s<f>}. (1.17.7g) An Application It is shown in chap. 3 that for this three degrees of freedom (DOF) (unconstrained) system, defined by the positional coordinates cp = X, q 2 = Y, q 2 = c\>, the Lagrangean equations of motion d / dt\dT / d{dq k / dt)] — {dT/dq^) = Qk [= system {impressed) force corresponding to q k ]; or, explicitly, angular equation (with M = total external moment about ♦): I{d 2 <j)/dt 2 ) + 777 { [{d 2 Y/dt 2 )x - {d 2 X/dt 2 )y]ccj) - [{d 2 X/dt 2 )xF{d 2 Y/dt 2 )y]s(j)} = M; (1.17.7h) which is none other than the (not-so-common form of the) angular momentum equation: Ua z + {r G /♦ x ma.) z = I{d 2 (j)/dt 2 ) + m[x{a.) y - y{a.) x ] = M, (1.17.7i) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND where (fig. 1.17), *•(?/♦ =x G i + y G j = xi + yj= ••• = (. xc(j> —yscj))I + (xsc/) + y cc/))J = XI + YJ, (1.17.7J) «♦ = ( d 2 X ♦ /dt 2 )I + (d 2 Y. /dt 2 )J = ( d 2 X/dt 2 )I + {d 2 Y/dt 2 )J = (d 2 X / dt 2 ) ( ccj)i — s</>./) + (d 2 7/ dr 2 ) (s<j> i + c</) j) = [( d 2 X/dt 2 )c(j) + (d 2 Y/dt 2 )scj)\ i + [-{d 2 X/dr)s4> + (d 2 Y/dt 2 )c<t>\ j = («*)x 1 + («♦),'./ = (1.17.7k) x, j-equations (with f x = components of total external force about x, j-axes, respectively): m[d 2 X/dt 2 — (d 2 (j) / dt 2 ){x s(j> + y c(j>) — (d(j>/dt ) 2 (x c</> — y scj))\ = f x . (1.17.71) m[d 2 Y/dt 2 + (d 2 4>/dt 2 )(x c(f> — y scj)) — (dcj)/dt) 2 (xs(j) + y c<j>)\ = f y , (1.17.7m) which are none other than m{a.) x -m[(d 2 cl)/dt 2 )Y + {d<f)/dt) 2 X} = f x , (1.17.7n) m{a.) y + m[(d 2 <t)/dt 2 )X - (dcj)/dt) 2 Y) = f y . (1.17.7o) For additional related plane motion problems, see, for example. Wells (1967, pp. 150-152). “British Theorem” It can be shown that the (inertial) kinetic energy of a thin homogeneous bar AB of mass in equals T = {m/6)(v A -v A +v B - v B + v A • v B ) = ( m/6)(v A 2 + v B 2 + v A ■ v B ). (1.17.8) (This useful result appears almost exclusively in British texts on dynamics; hence, the name; see, for example, Chorlton, 1983, pp. 165-166.) Principle of Linear Momentum; Motion of Mass Center Since v g = •’♦ + <*> x r Gj* ( + : body-fixed point), (1.17.9a) the principle of linear momentum (§1.6) m[dv G jdi) = f (total external force, acting at G ) (1.17.9b) WWW.EBOOK777.COM §1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION along body-fixe daxes (i.e., ca = f2) yields, successively, m d/dt(v+ + co x r G /f) = m(dv+ /dt+ co x v*) + m[(dco/dt) x r G /* + ca x (dr c /+ / dt )] [with dco/dt = at, c/r G( /*/c/t = v G y* = ca x r G / 4 ] = m(dv+ /dt) + m(co x v*) + a x (mr G ^) + co x [<o x (mr G /*)] = /, (1.17.9c) or, in terms of the center of mass vector of the mass moment m G/ + = mr G /+ [as in (1.17.3c, d)], m(dv+/dt) + m(a> x v*) + a x (m c /+ + co x (ca x m G /+) = f. (1.17.9d) Along body-fixed axes ♦— xyz, and with m G /+ = (m xyz ), = (v xyz ) there, the x-component of (1.17.9d) is m[dv x /dt + u) y v z — ut z v y ] + [m z (dco y /dt) — m y (dw z /dt)\ + \u) y (m y w x - m x ut y ) - u z (tn x iu z - m,u x )\ = f x , etc., cyclically. (1.17.9e) Special Cases (i) If ♦ = G, then m G /+ = 0 and, clearly, (1.17.9d) reduces to m(dv G /dt+ co x v G ) =/■ (1.17.9f) (ii) Along non-body-fixed axes at G, rotating with inertial angular velocity Q, (1.17.9b) yields m(dv G /dt + fl x v G ) =/; (1.17.9g) or, in components, with v G = (v G -. x . y f), m(dv G /dt) x = m(dv Gx /dt + 0,, v Gz — Q z v Gy ) = f x . etc., cyclically, (1.17.9h) where (. dv G /dt) x = component of a G along an inertial axis that instantaneously coin¬ cides with the moving axis Gx, and so on. In general, the v G . xyz are quasi velocities. Principle of Angular Momentum; Motion (Rotation) about the Mass Center Along body-fixed axes ♦— xyz, the principle of angular momentum [§1.6, with • (arbitrary spatial point) —> ♦ (arbitrary body point), and //♦.relative = A*], dh+/dt + r G /+ x [m(dv+/dt )] = A/ 4 , (1.17.10a) becomes dli./dt + to x h+ + m G /+ x (dv+/di) + m G /+ x (co x v*) = M* ; (1.17.10b) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND or, in components, with = {h x ^ y f), m a/+ = ml 'a/* = K, w ), »’♦ = (v x . yz ), and so on, dh x /dt + u) r h- — oj-h v + m r [dv./dt + to x v y — c o y v x ] — m z [dv y /dt + uj z v x — uj x v z \ = M x , etc., cyclically. (1.17.10c) [The forms (1.17.9d,e) and (1.17.10b c) seem to be due to Heun (1906, 1914); see also Winkelmann and Grammel (1927) for a concise treatment via von Mises’ (not very popular) “motor calculus.”] Special Cases (i) If ♦ = G, then m G = 0, and (1.17.10b) reduces to dh G /dt+w xh c =M G . (1.17.10d) (ii) Along non-body-fixed axes at G, rotating with inertial angular velocity 12, (1.17.10a) yields dh G /dt + f2 x h c = M c ; (1.17.10e) or, in components, ( dh G /dt) x = dh Gx /dt + Q v h G - — O z h Gy = M Gx , etc., cyclically. (1.17.lOf) (iii) If the axes are body-fixed, then Q = co; and if they are also principal axes, then, since (omitting the subscript G throughout) li = I-to.hk = hwk, k = 1,2,3, eqs. (1.17.1 Of) assume the famous Eulerian form (1758, publ. 1765): I\{du>\/dt) — ( I 2 — — M\, I 2 {du 2 /dt) - (/ 3 -Ii)ui 3 ui =M 2 , (1.17.11a) Ifiduj^/dt) — (/) — I 2 )u)\ui 2 = My, or, alternatively, deji/dt — [(/ 2 - I 2 )/Ii\lo 2 oj 2 = Mjl u etc., cyclically. (1.17.11b) From the above we readily conclude that: • A force-free rigid body in space can rotate permanently [i.e., dm/dt = 0 => o> = (a*!, 0,0), or (0, u> 2 , 0), or (0, 0, cufi = constant] only about a central principal axis of inertia. Or, if a free rigid body under no external forces begins to rotate about one of its central principal axes, it will continue to rotate uniformly about that axis; and, if a rigid body with a fixed point, and zero torque about that point, begins to rotate about a principal axis through that point, it will continue to do so uniformly about that axis. • The principle of angular momentum takes the “elementary” form M = d/dt(Iuj) only for principal axes of inertia, or if the body rotates about a (body- and space-) fixed axis. That is why a central principal axis was called a permanent axis (Ampere, 1823). WWW.EBOOK777.COM §1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION REMARKS (!) Equations (1.17.10a ff.) also hold for any fixed point O. (ii) The principles of linear and angular momentum are summarized as follows: the vector system of mechanical loads, or inputs [/at G, M G (moments offorces and couples )] is equivalent to the kinematico-inertial vector system of the responses, or outputs, ( ma G at G, dh G /dt)\ and this equivalence, holding about any other space point •, can be expressed via the (hopefully familiar from elementary statics) purely geometrical transfer theorem: M. = M G + r G/ . x / atG = M g + r G/ . x ma G = dh G /dt + m G/ . x a G . (1.17.12) (iii) In general, the direct application of the vectorial forms of the principle of angular momentum, either about the mass center G, or a fixed point O, and then taking components of all quantities involved about common axes in which the inertia tensor components remain constant, is much preferable to trying to match a (any) particular problem to the various scalar components forms of the principle. (iv) The relative magnitudes of the principal moments of inertia of a rigid body at, say its mass center G , I G . 12,3 = Z, 2,3 (he., its mass distribution there) provide an important means of classifying such systems. Thus, we have the following classifica¬ tion (§1.15: subsection “Ellipsoid of Inertia”): • If / = I 2 = / = /, we have a spherical top , or a kineticallv symmetrical body. Then, H g = h G = I G • (a = (II) • to = Ioi . • If I\ = I 2 f h, the body (or “top”) is symmetric, if / > / 3 , it is elongated , and if f < / 3 , it is flattened. • If /| f I 2 f f f I\, the body is unsymmetric. For further details and insights on these fascinating equations, see Cayley (1863, pp. 230-231), Dugas (1955, pp. 276-278), Stackel (1905, pp. 581-589). Energy Rate, or Power, Theorem for a Rigid Body By d/dt(.. ./differentiating the kinetic energy definition 2 T = S dm v • v, and then utilizing in there the rigid-body kinetic equation v = v* + a> x r/+, we obtain, suc¬ cessively, dT/dt = ^ dm v • ( dv/dt ) = ^ dm v • a = ^ dm (v* + m x r/+) • a = ^ dm v* • a + ^ dm(co x r /*) • a = dm a^j + <u • dm r/+ x a'j = v» • («7 a G ) + a> ■ {S dm [d/dt(r x v) + x v]| = v* • ( dp/dt ) -I (O' ( dH./dt+ v* x p), (1.17.13a) where (recalling the definitions in §1.6), p=^j dm v = m v G \ Linear momentum of body, (1.17.13b) dm(r /.* x v): Absolute (and inertial) angular momentum of body, about the body-fixed point ♦. (1.17.13c) WWW.EBOOK777.COM CHAPTER 1: BACKGROUND Invoking the principles of linear and angular momentum (§1.6), we can rewrite (1.17.13a) as dT/dt = v.-f + a-M ♦. (1.17.13d) On the other hand, the power of all forces, d'W/dt = ^ df-v, transforms, succes¬ sively, to d'W/dt = $df-(v. + ft) x »•/♦) = v* • (jSV/') +<»• ( S r /♦ x d f) = v.f+co-M.; ( 1 . 17.1 3e) that is, dT/dt = d'W/dt, (1.17.13f) which is the well-known power theorem, proved here for a rigid system. Special Case If v* = 0 (i.e., rotation about a fixed point), (1.17.13d f) reduce to dT/dt = d'W/dt = co -M.. (1.17.13g) If, in addition, M » = 0 ( torque-free motion), then d'W/dt = 0 and T = constant (energy integral), and M* = dH+/dt = dh+/dt = 0 =>■ //* = /;* = constant (angu¬ lar momentum integral). These two integrals of the torque-free and fixed-point motion form the basis of an interesting geometrical interpretation of rigid-body motion, due to Poinsot (1850s). For details see, for example (alphabetically): MacMillan (1936, pp. 204-216), Webster (1912, pp. 252-270), Winkelmann and Grammel (1927, pp. 392-398). Additional Useful Results (i) By multiplying the Eulerian (rotational) equations with w Xi> , jZ , respectively, and then adding them, we obtain the following power equation: d / dt[(Auj x T BiOy~ + Cuj z ~)/ 2\ — M x ui x T M x ujy T i.e., d/dt (Rotational kinetic energy ) = Power of external moments. (1.17.14) (ii) Plane motion : Principle of angular momentum for a rigid body B , about its instantaneous center of rotation I. We have already seen (1.9.4dff.) that the inertial coordinates of the instantaneous center (of zero velocity) /, relative to the center of mass G, are ri/o = (*>/g, Y,/ g , 0) = {-dYa/dt/w, +dX G /dt/u, 0). (1.17.15a) Therefore, application of the principle of angular momentum about /: Mi = I G (duj/dt) + ( r G/I x ma G )z, (1.17.15b) WWW.EBOOK777.COM §1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION yields, successively (with I G = mk 2 ), Mj = I G (du)/dt) + (m/co)[(dY G / dt, —dX G / dt, 0) x (d 2 X G /dt 2 ,d 2 Y G /dt 2 , 0)] = I G (duj/dt) + (m/J)[(dY G /dt)(d 2 Y G /dt 2 ) — {—dX G /dt){d 2 X G /df L )\ = I G (du>/dt) + (m/u>)[(dX G /dt)(d 2 X G /dt 2 ) + (dY G /dt)(d 2 Y G /dt 2 )] = (m/co) [k 2 u(duj/dt) + ( dX G /dt)(d 2 X G /df 2 ) + (dY G /dt)(d 2 Y G /dt 2 )] = (m/cu)(d/dt{(l/2)[k 2 uj 2 + ( dX G /dt ) 2 + {dY G /dtf}}) [noting that (dX G /dt) 2 + ( dY G /dtf = v G 2 = rw 2 , r = |r G// |] = (1/2 iv){d/dt[m(k 2 + r 2 )u) 2 }}, or, hnally, with f = m(k 2 + r 2 ) = mK 2 : moment of inertia of B about I (by the parallel axis theorem), M, = (l/2w)[^/ht(/ / w 2 )] = Ifdco/dt) + (1/2 )u(dl,/dt) = Ifdio/dt) + mr{dr/dt)u>. (1.17.15c) Special Cases (a) If B is turning about a fixed axis, or if I is at a constant distance from G, then dr/dt = 0 and (1.17.15c) reduces to M, = Ifidu/dt). (1.17.15d) (b) If the axis of rotation is mobile, but the body starts from rest, then, since initially u> = 0 and dr/dt = 0, the initial value of its angular acceleration is given by (1.17.15d): dco/dt= Mfilj. (1.17.15e) (c) If the body undergoes small angular oscillations about a position of equili¬ brium, then the term df/dt = 2mr(dr/dt) is of the order of the rate dr/dt, and therefore (df/dfixi is of the order of the square of a small velocity and so, to the first order (linear angular oscillations), it can be neglected; thus reducing (1.17.15c) to (1.17.15d), with f given by its equilibrium value. In sum, eq. (1.17.15d) holds if the instantaneous axis of rotation is either fixed, or remains at a constant distance from the center of mass', or if the problem is one of initial motion, or of a small oscillation. In all other cases of moments about I, we must use (1.17.15c). For further details and applications, see, for example (alphabe¬ tically): Besant (1914, pp. 310-314), Loney (1909, pp. 287, 346-347), Pars (1953, pp. 403-404), Ramsey (1933, part I, pp. 241-242), Routh (1905(a), pp. 103-104, 171- 172). Somehow this topic is treated only in older British treatises! Rigid-Body Mechanics in Matrix Form [Here, following earlier remarks on notation (§1.1), we denote vectors by bold italics, and matrices/tensors by bold, roman, upper case (capital) letters; for example, a, A (vectors), WWW.EBOOK777.COM CHAPTER 1: BACKGROUND A, B (matrices, tensors). This material (notation) is presented here not because we think that it adds anything significant to our conceptual understanding of mechanics, but because it happens to be fashionable among some contemporary applied dynamicists.] By recalling the tensor results of §1.1, and the earlier definitions and notations, (1.15.2a ff.), I = [(r-r)l — f ®f] dm = — ^ (r-r) dm = (l/2)(TrI)l - J (1.17.16al) [ => TrI = 2Tr J], J = £j(r®r)dm, (1.17.16a2) [/* = axial vector of tensor r and d (...) /dt is inertial rate of change], we can verify the following matrix forms of the earlier (§1.15-1.17) basic equations of rigid-body mechanics [while assuming that, in a given equation, all moments of inertia and moments of forces are taken either about the body’s center of mass, or about a body-and-space-fixed point (if one exists), and along body-fixed axes; and suppres¬ sing all such point-dependence for notational simplicity, except in eqs. (1.17.16bl 3) for obvious reasons]: (i) lo = Ig - m re • To = I G + m [(r G ■ r G )\ - r G <g) r G \ [r G = r G /o, etc., parallel axis theorem in terms of 1: (1.15.7b)],(1.17.16b!) =► Tr I 0 = Tr I G + 2mr G • r G , (1.17.16b2) Jo = Jg + m r G ®r G = (Tr I c /2) 1 I G + m r G ® r G [Parallel axis theorem in terms of J]; (1.17.16b3) (ii) dl/dt = O -1 -I-1 - O t = O -1 — I - r2 [recalling results of 1.1.20a ff.; to = axial vector of tensor S~2]; (1.17.16c) (iii) dl/dt = — (dj/dt) [= -(« • J - J • O)] (1.17.16d) (iv) H = I • to = — J • to + (Tr J)to (v) H = I • fi T — 0*1+ (Tr I) • $7 = (17 • I) T — IT • I + (Tr I) • Cl (1.17.16el) = J- r2 + r2-J = J- r2-(J-^) T =J-f2-^ T -J; (1.17.16e2) [H = axial vector of H (angular momentum tensor )] (vi) (vii) M = d/dt (I • to) = (dl/dt) • to + I • ( dto/dt ) [then invoking (1.17.16c)] = I • ( dto/dt ) + fl • (I - to) = I • ( dto/dt) + to x (J • to) (1.17.16fl) = —[J • (dto/dt) + fi • (J • to)] + (Tr J)(dco/dt); (1.17.16f2) M = (E • I) T — E • I + (Tr I) • (dO/dt) (1.17.16gl) = E-J-(E-J) t (1.17.16g2) [M = axial vector of M ( moment , or torque , tensor ); recalling (1.11.9a ff.): E = dJT/dt + f2-fi = «4-l-f2-fi]. Additional forms of the above are, of course, possible. WWW.EBOOK777.COM §1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION A Comprehensive Example: The Rolling Disk Let us discuss the motion of a thin homogeneous disk D (or coin, or hoop) of mass m and radius r, on a fixed, horizontal, and rough plane P (fig. 1.29). Kinematics Relative to the intermediate axes/basis G—xyz/ijk (defined so that k is perpendicular to D, at its center of mass G\ i is continuously horizontal and parallel to the tangent to Z>, at its contact point C; and j goes through G, along the steepest diameter of D, and is such that ijk form an ortho-normal-dextral triad), whose inertial angular velocity Si is Si = Si x i + Q y j + Q : k = (u>g)i + (w^ sin 9) j + (w^ cos 0)k, (1.17.17a) [where = d<j>/dt, = dO/dt , = dip/dt] the inertial angular velocity of D, co, equals oj — uj x i -|- uty j -|- c o z k — (u>g)i -|- (c sin $)y T - (cv^ cos 9 to^k = S2 + k. (1.17.17b) In view of the above, the rolling constraint v c = 0 , becomes v c = Vg + «> x r c/G = v x i + v y j + v.k + (w x , w y , u z ) x (0, -r, 0) = (v x + rw z )i + (■ v y )j + (v z - u x r)k = 0, (1.17.17c) C/C ■ Figure 1.29 Rolling of thin disk/coin D on a fixed, rough, and horizontal plane P. O-XYZ: space-fixed (inertial) axes; O-xyz : intermediate axes (of angular velocity o). A B = A, C: principal moments of inertia at G. For our disk: A = mr 2 /4, C = mr 2 / 2. WWW.EBOOK777.COM 235 CHAPTER 1: BACKGROUND from which it follows that v x + ru z = 0 => v x = — ruj z = — r (u^cosO + (1.17.17d) v y = 0, (1.17.17e) v z — u x r =0 => v z = roj x = rui g . (1.17.17f) These equations connect the velocity of G with the angular velocity and the rates of the Eulerian angles. Kinetics To eliminate the rolling contact reaction R, we apply the principle of angular momentum about C; that is, we take moments of all forces and couples [including inertial ones at G; i.e., ~m(dv G /dt) and — dh G /dt] about G (recalling 1.6.6a IT.) to give M c = dh G /dt + r G / c x \m{dv G /dt)]. (1.17.18a) But, with W = weight of disk, and sin(...) = s (...), cos(...) = c(. ..), we have (i) M c = r G / c x W = (0, r, 0) x (0, — Ws6, — Wed) = (—rWc8)i; (1.17.18b) (ii) dv G jdt = dv G /dt + fl x v G [with the ad hoc notation dv X y iZ /dt = fl VjV ,z] = a x i + a y j + a- k + (Q x , Q y , Q z ) x ( v x , v y , v z ) = (a x + Q v v z - Q z v y )i + ( a v + Q z v x — C2 x v z )j + ( a z + Q x v y — O v v x )k = (a x + v z s8 - v y c0)i + ( a y + v x c8 - v z ui g )j + (a z + VyU e - v x u < j > sd)k\ (1.17.18c) (iii) dh G /dt = dh G /dt + f2 x li G [with the ad hoc notation du} Xt y tZ /dt = a Xt y tZ \ = + {Aoiy)j + (Ca z )k] + (O x , Q y , Q z ) x (Au x ,Buj y , Cw z ) = (Acx x T GC2yLu z — AQ z oj y )i T (Acn y T AQ z ui x — GC2 x co z )j T (Go z T AC2 x uj y — Af2 v uj x )k = ( Aa x + Cco z u^ s8 — Au> y c8)i + ( Aa v + Au> x c8 — Cco z u> e )j T (Col z T Auj y uj(j — A jj x ujy, s())k. (1.17.18d) and so (1.17.18a) yields the three component equations of angular motion: mr(a z + v v — v x s9) + ( Aa x + Cuj z sd — Au y to^ cd) = — Wr c8, (1.17.18e) Aa y + Auj x u^cd - Cu : u> 0 = 0, (1.17.18f) — mr{a x + v, sd — v v c8) + (C a z + Au> y u e — Au x ui^ sO) =0. (1.17.18g) WWW.EBOOK777.COM §1.18 THE RIGID BODY: CONTACT FORCES, FRICTION The nine equations (1.17.18e, f, g) + (1.17.17d, e, f) + (1.17.17b, in components) constitute a determinate system for the nine functions (of time): qf>, 9, ip; u ^ (quasi velocities); v xyz (quasi velocities). We may reduce it further to two steps: (i) Using (1.17.17d, e, f) in (1.17.18e, f, g) (i.e., eliminating v xyz ), we obtain mr(ra x + ru> z sO) + Aa x + Cu-u^sO — AojyUj^cO = —WrcQ , (1.17.19a) Aa y + Au> x u)^ cO — Cco z cu g = 0, (1.17.19b) — mr(—ra z + na x uj < j > s9) + Ca z + Aco y ui g — Aco x uj ( j > s9 = 0. (1.17.19c) (ii) Using (1.17.17b) in (1.17.19a, b, c) (i.e., eliminating u xy ), we get three equa¬ tions of rotational motion in terms of 0, the rates of cp, 9, and the total spin U z = LO^, + U),pc9\ (A + mr 2 )(d 2 9/dt 2 ) + (C + mr 2 )u z (dcp /dt)s9 — A(df/dt) 2 c9s9 = — Wrc9, (1.17.20a) A d/dt [{d<p/di)s9\ + A{d(p/dt){d9/dt)c9 - Cto z (d9/dt) = 0, (1.17.20b) (C + mr 2 ){du> z /dt) — mr 2 {dtp / dt)(d9 / dt)s9 = 0; (1.17.20c) or, since A = B = mr 2 /A = (1/2) (wr 2 /2) = C/2, 9: 5r(d 2 9/dt 2 ) + 6 ruj z (d<p/dt) sin0 — r{dcp/dt) 2 sin 9 cos 9 + 4gcos 0 = 0, (1.17.21a) <p: 2u> z (d9/dt) — 2(d<p/dt)(d6/dt) cos 9 — (d 2 cp/dt 2 ) sin 0 = 0, (1.17.21b) u z . 3(du} z /dt) — 2{dcp/dt){d9/dt) sin0 = 0. (1.17.21c) These three nonlinear coupled equations contain an enormous variety of disk motions. For simple particular solutions of them, see, for example, MacMillan (1936, pp. 276-281); also Fox (1967, pp. 263-267). Once <p(t), 9(t), ip(t) have been found, the rolling contact reaction R = {R x . yz ) can be easily obtained from the principle of linear momentum: mac = m(dvc/dt + fl x va) = W + R=>R = ---=R(t). (1.17.22) The details are left to the reader. 1.18 THE RIGID BODY: CONTACT FORCES, FRICTION Recommended for concurrent reading with this section are (alphabetically): Beghin (1967, pp. 139-145), Kilmister and Reeve (1966, pp. 81-84, 141-143, 164-177), Peres (1953; pp. 62-66); also, our Elementary Mechanics (§20.1,2, under production). Introduction and Constitutive Equations The forces between two rigid bodies, B and B\, at a mutual contact point C (actually, a small area around C that is practically independent of the macroscopic shape of the bodies and increases with pressure), say from B to B\ , reduce, in general, to a resultant force R and a couple C; frequently, C can be neglected. Decomposing R WWW.EBOOK777.COM CHAPTER 1: BACKGROUND and C along the common normal to the bounding surfaces of B, B\, say from B towards B\ , and along the common tangent plane, at C, we obtain R = Rfri + Rj = Normal reaction (opposing mutual penetration) + Tangential reaction (opposing relative slipping), (1.18.1) C = C N + Cj = Pivoting couple (opposing mutual pivoting) + Rolling couple (opposing relative rolling). (1.18.2) These components satisfy the following “laws” (better, constitutive equations) of dry friction; that is, for a solid rubbing against solid, without lubrication: (i) As soon as an existing contact ceases, R = 0. (ii) Whenever there is slipping — that is, relative motion of B and B { (v c ^ 0) — R n points toward Bp, and R T and v c are collinear and in opposite directions: Rj x V (■ = 0, Rj • v'( ■ < 0, (1.18.3) and Rt = J(Rni v c ); (1.18.4) or, approximately (for small relative velocities), \R t /R n \ = p: coefficient of friction between B and Bp, a nonnegative constant. (1.18.5) Frequently, we use the following notation: R t = F and R n = N. (1.18.6) Then, with jf 7 ! = the above read R =Rn + Rj; Rn = Nti, Rj = Ft = —(1.18.7) where n = common unit normal vector, from B towards B u (1.18.7a) t = unit tangent vector, in direction of slipping velocity. (1.18.7b) (iii) When v c = 0 (no slipping — relative rest), R N points toward If , while R T can have any arbitrary direction and value on the common tangent plane, as long as l-^r/^jvl = \F/N\ < /x; or, vectorially, \R x n\ < p\R ■ n|; (1.18.8) with the equality sign holding for impending tangential motion. Actually, the p in (1.18.8) is called coefficient of static friction, p s ; and the p in (1.18.5) is called coefficient of kinetic friction, p K \ and, generally, ds > Pk- Here, unless specified otherwise, p will mean p K . WWW.EBOOK777.COM (1.18.9) §1.18 THE RIGID BODY: CONTACT FORCES, FRICTION The friction coefficient p is, in general, not a constant but a function of: (a) the nature of the contacting surfaces', (b) the conditions of contact (e.g., dry vs. lubricated surfaces); (c) the normal forces (pressure) between the surfaces; and (d) the velocity of slipping. Further, in the dry friction case (solid/solid, no lubricant), p increases with pressure, and decreases with v c ; and this dependence is particularly pronounced for small values of v c , so that, if p = p(v c ), then p < p„, where p 0 = p(0). In most such applications, we assume that /r is, approximately, a positive constant ( rough surface). Then the relation p = tan </> defines the “angle of friction.” If p « 0 ( smooth surfaces), then R k, R n = N, Rt ~ 0. If, on the other end, p —> oo (perfect rough¬ ness), then v c = 0 throughout the motion, and R can have any direction, as long as R N = N points toward . (iv) The contact couple C is included in the cases of small p and/or slippingless motion as follows: (a) If at a given instant and immediately afterwards m N = 0 (i.e., no instantaneous pivoting), then !CV| < |Cjy ]lnax |, C N<max = f p R N = limiting pivoting moment, (1.18.10) f p = pivoting friction/resistance coefficient. (1.18.10a) (b) If at a given instant m N f 0, or if it stops being zero at that instant, then IC# | = |Cjv, max |; (1.18.11) and C N and m N have opposite senses. (c) If m T = co rom „g = <» R = 0, then \C T \ < |C r , max |, C T max = f.R N = limiting rolling moment, (1.18.12) f r = rolling friction/resistance coefficient. (1.18.12a) (d) If at a given instant a> T f 0, or if it stops being zero at that instant, then \Ct\ = |CV, max |; (1.18.13) and C T and m T have opposite senses. The coefficients f p and f r have dimensions of length (whereas p is dimensionless!), and their values are to be determined experimentally. Theoretically, f p can be related to p,ifC N is viewed as resulting from the slipping friction over a small area around the contact point C —something requiring use of the theory of elasticity (no such relationship can be established for f). It turns out that f p is, generally, five to ten times smaller than fp in general, pivoting is produced faster than rolling. In closing this very brief summary, we point out that the above “friction laws” sup¬ ply only indirect criteria for relative rest or motion (rolling and slipping); that is, if, for example, we assume rest and the resulting equations are consistent with it, it means that rest is possible, not that it will happen. And if we end up with an inconsistency, it means that the particular assumption(s) that led to it is (are) false. Thus, to show that two contacting bodies roll (slip) on each other, all we can do is show that the assumptions of their slipping (rolling) lead to a contradiction. [For detailed examples WWW.EBOOK777.COM CHAPTER 1: BACKGROUND illustrating these points, see, for example (alphabetically): Hamel (1949, pp. 543-549, 629-639), Kilmister and Reeve (1966, pp. 165-177); also Poschl (1927, pp. 484-497).] Work of Contact Forces Under a kinematically possible infinitesimal displacement of B\ relative to B (assumed fixed) that preserves their mutual contact at C, the total elementary (first-order) work of the contact actions (of B on Bf is: d'W = R-dr c + C-dO (1.8.14) where dr c = elementary translatory displacement of the B , -11 xcd point, at contact, relative to B (= v c dt , in an actual such displacement), (1.18.14a) dO = elementary rotatory displacement of B t relative to B (= to dt, in an actual such displacement), (1.18.14b) • Since dr c preserves the B/B\ contact, it lies on their common tangent plane at C. Then: (a) if R T « 0 (i.e., negligible slipping friction), or (/3) if dr c = 0 (i.e., no slipping) and dQ = 0 (i.e., no rotating), then: d'W = 0. (1.18.15) • If dr c violates contact, but remains compatible with the unilateral constraints, it makes an acute angle with the normal toward B\. In this case, if R T « 0 =>■ R ss R N , and therefore d'W>0; (1.18.16) while for elementary displacements incompatible with the constraints, d'W<0. (1.18.17) • In a real, or actual, displacement d'W becomes d'W = (R • v c + C • <n) dt. From the earlier constitutive laws, we see that, as long as v c , co N , co T do not vanish , the pairs (Rti v c)i (Cjv,COv), (C t ,(O t ), are collinear and oppositely directed. Hence, frictions do negative work; that is, in general, d'W < 0. (1.18.18) WWW.EBOOK777.COM §1.18 THE RIGID BODY: CONTACT FORCES, FRICTION If, as commonly assumed, C«0, then d'W= (R t • v c ) dt = (F • Vc) dt = 0; if F = 0 (frictionless, or smooth , contact) = 0; if v c = 0 (slippingless, or rough , contact). (1.18.19) It should be stressed that, in all these considerations, the relevant velocities are those of material particles , and not those of geometrical points of application of the loads. WWW.EBOOK777.COM 2 Kinematics of Constrained Systems (i.e., Lagrangean Kinematics) I cannot too strongly urge that a kinematical result is a result valid forever, no matter how time and fashion may change the "laws" of physics. (Truesdell, 1954, p. 2) It is my belief that students have difficulty with mechanics because of an inadequate knowledge of kinematics. (Fox, 1967, p. xi) 2.1 INTRODUCTION As complementary reading for this chapter, we recommend the following (alpha¬ betically): General: Hamel (1904(a), (b)), Heun (1906, 1914), Lur’e (1968), Neimark and Fufaev (1972), Novoselov (1979), Papastavridis (1999), Prange (1935). Special problems, extensions: Carvallo (1900, 1901), Lobas (1986), Lur’e (1968), Stuckler (1955), Synge (1960). Research journals (see the references at the end of this book): Acta Mechanica Sinica (Chinese), Applied Mathematics and Mechanics (Chinese), Archive of Applied Mechanics (former Ingenieur Archiv; German), Journal of Applied Mechanics (ASME; American), Applied Mechanics (Soviet —> Ukrainian), Journal of Guidance, Control, and Dynamics (AIAA; American), PMM (Soviets Russian), ZAMM (German), ZAMP (Swiss); also the various journals on kinematics, mechanisms, machine theory, design, robotics, etc. In this chapter we begin the study of analytical mechanics proper with a detailed treatment of Lagrangean kinematics, i.e., the theory of position and linear velocity constraints (or Pfaffian constraints) in mechanical systems with a finite number of degrees of freedom-, that is, a finite number of movable parts; as opposed to contin¬ uous systems that have a countably infinite set of such freedoms. All relevant funda¬ mental concepts, definitions, equations — such as velocity, acceleration, constraint, holonomicity versus nonholonomicity, constraint stationarity (or scleronomicity) versus nonstationarity (or rheonomicity) — are detailed in both particle and system variables, along with elaborate discussions of quasi coordinates and the associated transitivity equations and Hamel coefficients', as well as a direct and readable (and very rare) treatment of Frobenius' fundamental necessary and sufficient conditions for the holonomicity, or lack thereof, of a system of Pfaffian constraints. WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS The examples and problems, some at the ends of the paragraphs and some (the more comprehensive ones) at the end of the chapter, are an indispensable part of the material; several secondary theoretical points and results are presented there. This chapter, and the next one on Kinetics, constitute the fundamental essence and core of Lagrangean analytical mechanics. 2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS Positions, Configurations, Motions Let us consider a general finite mechanical system 5 consisting of N ( = positive integer), free, or unconstrained , material particles. The position r of a generic 5-particle, P. at the generic time instant, t, relative to an “origin” fixed in a, say inertial, frame of reference, F, is defined by the vector function r=f(P,t)=r(P,t). (2.2.1) The collection of all these particle vectors, at a current instant t, make up a current system position , or current configuration of 5, C(f), and its evolution in time consti¬ tutes a motion of 5”. The latter, clearly, depends on the frame of reference. Thus, the complete description of a motion of 5, if the latter is modeled as a collection of N particles, requires (at most) knowledge of 3 N functions of time; for example, the 3 N rectangular Cartesian components = coordinates of the N r’s: (x l ,y l ,z 1 ;...;x N ,y N ,z N ) = ( x,y,z ) = (£i,... , £ 3 jv) = (2.2.1a) These numbers can be viewed as the rectangular Cartesian coordinates of the 3/V-dimcnsional position vector of a single fictitious, or figurative, particle represent¬ ing 5, in a 3/V-dimensional Euclidean space, Et, n , henceforth called the system’s unconstrained configuration space', and, therefore, a motion of 5 can be visualized as the path traced by the tip of that system position vector in E 3N . Equation (2.2.1) can be replaced by r =f(r 0 , t', t 0 ) = r(r„, t\ t 0 ), (2.2.2) where (fig. 2.1): r 0 = “reference position” of P at the “reference time” t = t„, is used to distinguish, or label, the various 5-particles; and the totality of r„’s constitutes the reference configuration of 5 at t B , C(t 0 ). For a fixed r a and variable t (i.e., a motion Reference configuration: C(t a ) Current configuration: C(t) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS of P), eqs. (2.2.1, 2) give the path of a particle P that was initially at r„. (The same equations for fixed t and variable r 0 would give us the transformation of the spatial region initially occupied by the system, to its current position at time t.) The one-to-one correspondence between r (and /) and r 0 (and t 0 ), of the same particle P — that is, the physical fact that “initially distinct particles must remain distinct throughout the motion” — requires that (2.2.2) has an inverse : r 0 = /~'(r, t\ t 0 ) = g(r, t a ; t) : reference configuration at (variable) time t a . (2.2.2a) Switching the roles of (r, t) and ( r 0 ,t 0 ), we can view (2.2.2a) as expressing the “current” position r 0 in terms of the “reference” position and time (r. t) and “current” time t a . From now on, for simplicity, we shall drop, in the above, the explicit ( r 0 ,t 0 ) and/or F-dependence [also, replace the rigorous notation//...) with r(...), as done frequently in engineering mathematics, except whenever extra clarity is needed], and write (2.2.1) simply as r = r(i). (2.2.3) REMARKS (i) For (2.2.2) and (2.2.2a) to be mutually consistent, we must have (2.2.2) for t=t 0 ^ (2.2.2a): r = f(r 0 ,t\t 0 ) => r 0 =f(r 0 ,t 0 -,t 0 ) =f~\r,t-,t 0 )- (2.2.2a) for t a = t => (2.2.2): r 0 =f~\r,t;t 0 ) => r =f~'(r,t;t ) = f(r 0 ,t\t 0 ); (2.2.2b) hence, also r=f[f{r 0 ,ti\t 0 ),t-,t l ] =f(r 0 ,t;t 0 ), (2.2.2c) where t\ is another reference time. (ii) In continuum mechanics, (r 0 , t) and (r, t) are called, respectively, material (or Lagrangean) and spatial (or Eulerian ) variables; with the former preferred in solid mechanics (e.g., nonlinear elasticity), and the latter dominating fluid mechanics (e.g., hydrodynamics). (See, e.g., Truesdell and Toupin, 1960, and Truesdell and Noll, 1965.) (iii) For systems with a finite number of particles, the dependence on the latter is, frequently, expressed by the discrete subscript notation (i.e., r„ —> positive integer denoting the “name” of the particle): r P = r P (t) = {xp(t),y P (t),z P (t)} (P = 1,... ,N). (2.2.4) The simpler continuum mechanics notation, eqs. (2.2.1, 3), dispenses with all un¬ necessary particle indices, and allows one to concentrate on the system indices (as we begin to show later), which is the essence of the method of analytical mechanics. It also allows for a more general exposition; for example, a unified treatment of systems containing both rigid (discrete) and flexible (continuous) parts. Constraints If the N vectors r, and/or corresponding (inertial) velocities v = dr/dt , are functionally unrelated and uninfluenced from each other ( internally) or from their environment (externally), apart from continuity and consistency requirements, like (2.2.2b,c) — WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS something we will normally assume — that is, if, and prior to any kinetic considera¬ tions, the r’s and v’s are free to vary arbitrarily and independently from each other, then S is called (internally and/or externally) free or unconstrained ; if not, 5 is called (internally and/or externally) constrained. In the latter case, certain configurations and/or (velocities =>) motions are unattainable, or inadmissible; or, alternatively, if we know the positions and velocities of some of the particles of the system, we can deduce those of the rest, without recourse to kinetics. [Outside of areas like astron¬ omy/celestial mechanics, ballistics, etc., almost all other Earthly systems of rele¬ vance, and a lot of non-Earthly ones, are constrained — hence, the importance of analytical mechanics, especially to engineers.] Such restrictions, or constraints, on the positions and/or velocities of S' are expressed analytically by one or more (< 3N) scalar functional relations of the form f(t,r u ...,r N ;v u ...,v N ) = 0, or, compactly, f(t,r,v) = 0. (2.2.5) These equalities are assumed to be; (i) continuous and as many times differentiable in their arguments as needed (usually, continuity of the zeroth, first-, and second-order partial derivatives will suffice), in some region of the (x,y,z-dx/dt,dy/dt,dz/dt;t); (ii) mutually consistent (i.e., kinematically possible, or admissible); (iii) independent [i.e., not connected by additional functional relations like F(f ,/ 2 ,...) = 0]; and (iv) valid for any forces acting on S, any motions of it, and any temporal boundary! initial conditions on these motions (see also semiholonomic systems below). Following ordinary differential equation terminology, we call (2.2.5) a first-order (nonlinear) constraint, or nonlinear velocity constraint. With few exceptions [as in chaps. 5 and 6, where generally nonlinear constraints of the form /(r, r, a, t) =0 (a: accelerations) are discussed], the velocity constraint (2.2.5) is the most general constraint examined here. [Other, perhaps more suggestive terms, for constraints are conditions (Victorian English: equations of condition', German: bedingungen), and connections or couplings (French: liaisons', German: bindungen', Greek: avvbea[ioi', Russian: svyaz’).] Special Cases of Equation (2.2.5) (i) Constraints like </>(?,r) = 0, or [recalling (2.2.1a)], </>(?,£) = 0, (2.2.6) are called finite, or geometrical, or positional, or configurational, or holonomic. [From the Greek: holos = complete, whole, integral; that is, finite, nondifferential; and nomas = law, rule, (here) condition, constraint. After Hertz (early 1890s); also C. Neumann (mid-1880s).] (ii) Again, with the exception of chapters 5, 6, and 7, all velocity constraints treated here have the practically important linear velocity, or Pfajfian, form /= S(B.v) + B = 0, (2.2.7) where B = B{t,r), B = B(t,r) are known functions of the r’s and t, and Lagrange’s symbol S (• • •) signifies summation over all the material particles of S, at a given instant, like a Stieltjes’ integral (so it can handle uniformly both continuous and discrete situations). Those uncomfortable with it may replace it with the more famil¬ iar Leibnizian f (...). WWW.EBOOK777.COM 246 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Multiplying (2.2.7) by dt , which does not interact with S (■ ■ •), we obtain the kinematically possible, or kinematically admissible, form of the Pfaffian constraint, fdt=^(B-dr) + Bdt = 0. (2.2.7a) Degrees of Freedom A system of N particles subject to h (independent) positional constraints: Mt,r)= 0 (H=l,...,h), (2.2.8) and m (independent) Pfaffian constraints: f D = S(B d -v) + B d = 0 (D=l,...,m), (2.2.9) that is, a total of h + m constraints, is said to have a total of 3 N — (h + m)(> 0) degrees of freedom (DOF). This is a fundamental concept whose significance to both kinematics and kinetics (of constrained systems) will emerge gradually in what follows. [Quick preview: DOF = Number of independent components of system vector of virtual displacement (§2.3-7) = Number of kinetic (i.e., reactionless) equations of motion of system (chap. 3).] Holonomicity versus Nonholonomicity A positional constraint like (2.2.6), since it holds identically during all system motions, can always be brought to the velocity form (2.2.7) by d(.. .)/r/t-differentiation: d(f>/dt = (df/dr) • v + df/dt = 0; (2.2.10) that is, B —> dtp/dr = grad 0 (normal to the E-iN -surface </> = 0) and B —> df/dt. However, the converse is not always true: the velocity constraint (2.2.7) may or may not be (able to be) brought to the positional form (2.2.6); that is, by integration and with no additional knowledge of the motion of the system; namely, without recourse to kinetics. If (2.2.7) can be brought to the form (2.2.6), then it is called completely integrable , or holonomic (H); if it cannot, it is called nonintegrable, or nonholonomic (NH); or, sometimes, anholonomic. This holonomic/nonholonomic distinction of velocity constraints is fundamental to analytical mechanics; it is by far the most important of all other constraint classifications. [The term anholonomic, more consistent than the term nonholonomic seems to be due to Schouten (1954).] Schematically, we have Holonomic Integrable or Holonomic _ Nonintegrable or Nonholonomic _ WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS Hence, a H velocity constraint, like (2.2.10), is actually a positional constraint disguised in kinematical form. Before embarking into a detailed study of H/NH con¬ straints, we will mention some additional, secondary but useful, constraint classifica¬ tions. Scleronomicity versus Rheonomicity Velocity constraints of the form /(r, v) = 0 => df/dt = 0 (2.2.7b) are called stationary, otherwise (i.e., if df /dt f 0), they are called nonstationary. If all the constraints of a system are stationary, the system is called scleronomic; if not, the system is called rheonomic. [From the Greek: scleros = hard, rigid, invariable; rheo = to flow; and the earlier nomos = law, rule, decree, (here) condition; that is, scleronomic = invariable constraint, rheonomic = variable/fluid constraint. After Boltzmann (1897-1904).] For positional constraints and Pfaffian constraints, stationarity means, respectively, <f>(r) = 0 and $B(r)-v = 0. (2.2.11) Catastaticity versus Acatastaticity Pfaffian constraints of the form $B(t,r)-v + B(t,r) = 0, (2.2.11a) are called acatastatic ; while those of the form $B(t,r)-v = 0 [i.e, B(t,r) = 0] (2.2.11b) are called catastatic. It is this classification [due to Pars (1965, pp. 16, 24) and, obviously, having meaning only for Pfaffian constraints], and not the earlier one of scleronomicity versus rheonomicity, that is important in the kinetics of systems under such constraints. REMARKS (i) The reason for calling the second of (2.2.11) scleronomic, instead of $B(r)-v + B(r) = 0, (2.2.11c) that is, for requiring that scleronomic constraints linear in the velocities be also homogeneous in them (i.e, have B = 0 =>■ catastaticity), is so that it matches the kinematic form generated by d/dt{. . ^-differentiating the scleronomic positional constraint (first of 2.2.11): <j>{r) = 0 => df/dt = $ (df/dr) • v = 0. (2.2. lid) Geometrical interpretation of this requirement: Otherwise, the corresponding con¬ straint surface, in “velocity space,” would be a plane with distance from the origin proportional to B. That term, representing the (negative of the) velocity of the WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS constraint plane normal to itself, is clearly a rheonomic effect. (Remark due to Prof. D. T. Greenwood, private communication.) (ii) Clearly, every scleronomic Pfaffian constraint is catastatic (B = 0); but cata- static Pfaffian constraints may be scleronomic [B = B{r), second of (2.2.11)] or rheonomic [B = B(t,r), (2.2.11b)]. Bilateral versus Unilateral Constraints Equality constraints of the form (2.2.5) are called bilateral, or two-sided , or equality, or reversible, or unchecked (after Langhaar, 1962, p. 16); while constraints of the form f{t,r,v)> 0 or f(t,r,v) < 0 (2.2.1 le) are called unilateral, or one-sided, or inequality, or irreversible. Physically, bilateral constraints occur when the bodies in contact cannot separate from each other: for example, a rigid sphere moving between two parallel fixed planes, in continuous contact with both. In the unilateral case, the bodies in contact can separate: for example, a sphere in contact with only one plane, or a system of two particles connected by an inextensible string—their distance cannot exceed the string’s length. Following Gantmacher (1970, p. 12), we can state that the general motion of a unilaterally constrained motion may be divided into segments, such that: (i) in certain segments the constraint is “taut” [(2.2.lie) with the =sign; e.g., particle on a light, inextensible, and taut string], and motion occurs as if the constraint were bilateral', and (ii) in other segments, the constraint is not taut, it is “loose,” and motion occurs as if the constraint were absent. Concisely, a unilateral constraint is either replaced by a bilateral one, or is eliminated altogether. Hence, in what follows, we shall limit ourselves to bilateral constraints. REMARKS (i) A small number of authors call all constraints of the form (2.2.6) holonomic, as well as those reducible to that form; and call all others nonholonomic. According to such a definition, bilateral constraints like (2.2.lie) would be nonholonomic! The reader should be aware of such historically unorthodox practices. (ii) The equations (j){r, t) = 0 and dcj)/dt = S (<9</>/<9r) • v + dcjr/dt = 0 restrict a system’s positions and velocities; equation dcj>/dt = 0 is the compatibility of veloci¬ ties with (j> = 0. Similarly, the equation d 2 (f)/dt 2 = [d/dt(dcj)/dr) ■ v + ( df/dr ) •«] + d / dt(dcj) / dt) = 0 is the compatibility of accelerations with (j> — 0, d<j>/dt = 0; and likewise for higher such derivatives. (iii) In the case of unilateral constraints, if at a certain time t: f > 0, then, as explained earlier, that constraint plays no role in the system’s motion. But if / = 0, then, as a Taylor expansion around t shows, motion that satisfies either of these two relations may occur; in the former case df /dt = 0, and in the latter df /dt > 0. Thus, the simultaneous conditions / = 0 and df /dt < 0 allow us to detect a possible incompatibility between velocities and / > 0. Usually, such conditions occur in impact problems (chap. 4; also Kilmister and Reeve, 1966, pp. 67-68). (iv) Geometrical!physical remarks: In a system S consisting of several rigid bodies, and its environment (i.e., other bodies/foreign obstacles, massless coupling elements: WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS e.g., springs, cables) the following conditions apply: (a) Every condition expressing the direct contact of two rigid bodies of S , or the contact of one of its bodies with a foreign obstacle (environment) that is either fixed or has known motion (i.e., its position coordinates are known functions of time only), results in a holonomic equation of the form (2.2.6); and the corresponding contact forces are the reactions of that constraint. (b) If, further, at those contact points, friction is high enough to guarantee us (in advance of kinetic considerations) slippinglessness, then the positions and velocities there satisfy (2.2.7)-like Pfaffian equations (usually, but not always, nonholonomic). These conditions express the vanishing of a component of (relative) slipping velocity in a certain direction; and, therefore, there are as many as the number of indepen¬ dent such nonslipping directions. (c) If, in addition, friction there is very high , so that not only slipping but also pivoting vanishes, then we have additional (usually nonholonomic) (2.2.7)-like equations; that is, linear velocity constraints arise quite naturally and frequently in daily life. [Nonslipping and nonpivoting are maintained by constraint forces (and couples), just like contact. All these constraint forces are examples of passive reactions; for more general, active , constraint reactions, see, for example, §3.17.] (v) Holonomic and/or nonholonomic constraints due exclusively to the mutual interaction of the system bodies are called internal (or mutual)', while those arising, even partially, from the interaction of the system with its environment are called external. The associated constraint reactions are called, respectively, internal (or mutual) and external. (vi) Finally, we repeat that such holonomic and/or nonholonomic constraints express restrictions among positions and velocities independently of the equations of motion and associated (temporal) initial/boundary conditions, and before the complete solution of the problem is carried out. Solving the problem means finding r = /•(/): known function of time; then v = clr/dt = v(t): known function of time; and these r’s and v’s automatically satisfy the constraints. Under such a viewpoint, integrals of the system, like those of linear/angular momentum and energy, assuming they exist, do not qualify as constraint equations. The (bilateral) constraints, discussed above, are summarized as follows: General first-order constraints /(') = 0 : f{t,r) = 0: fir, v) = 0: f{t,r,v) = 0: Holonomic (integrable) and scleronomic (stationary) Holonomic (integrable) and rheonomic (nonstationary) Nonholonomic (if nonintegrable) and scleronomic (stationary) Nonholonomic (if nonintegrable) and rheonomic (nonstationary) Pfaffian velocity constraints S B(t,r)-v+ B(t,r) = 0: SB(r)-v + B(r) = 0: $B(t,r)-v = 0: S B W-v = 0: Rheonomic and acatastatic Rheonomic and acatastatic Rheonomic and catastatic Scleronomic and catastatic (There is no such thing as scleronomic and acatastatic Pfaffian constraint.) 249 WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Figure 2.2 Plane pursuit problem: a dog (D) moving continuously toward its master ( M ). Example 2.2.1 Plane Pursuit Problem — Catastatic but Rheonomic ( or Nonstation¬ ary) Pfaffian constraint. The master ( M) of a dog ( D ) walks along a given plane curve: R = R(t) = {X = X(t), Y = T(/)}. Let us find the differential equation of the path of D: r = r(t) = {x = x(t),y = v(t)}, if D moves, with instantaneous velo¬ city v, to meet M, so that at every instant its velocity is directed toward M (fig. 2.2). We must have: v = parallel to R — r = v[(/? — r)/\R — r|] = ve, or, in components, dx/dt = v[(X — x)/\R — r\], dv/dt = v[( Y — y)/\R — r|]; (a) or, eliminating v between them, \Y(t) — y\(dx/dt ) — [df(r) — x](dy/dt) = 0. (b) It is not hard to show that this pursuit problem in space leads to the following constraints (with some obvious notation): [ Y{t ) — y\(dx/dt) — [X{t) — x)(dy/dt) = 0, (c) \Z{t) — z](dx/dt) — \X{t) — x\{d:/dt) = 0, (d) [Z(0 — z\(dy/dt) — [T(t) — y\(dz/dt) = 0. (e) See also Hamel (1949, pp. 770-773). Example 2.2.2 Acatastatic Constraints. Let us consider the rolling of a sphere 5 of radius r and center G on the rough inner surface of a vertical circular cylinder A of radius R(> r). Let us introduce the following convenient intermediate axes/ basis G 123/ G-ijk (fig. 2.3): Let </> be the azimuth , or precession- like, angle of the plane G-13, and z = vertical coordinate of G (positive upward from some fixed WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS TOP VIEW: Figure 2.3 Rolling of a sphere on a vertical circular cylinder. G7: vertically upward; G3: horizontally intersects the (vertical) cylinder axis; G2: horizontal, so that G-123 is orthogonal-normalized-dextral (OND). plane, perpendicular to the cylinder axis). Then, v G = inertial velocity of G = (v, = dz/dt , v 2 = (R — r)(d<j>/dt) = (R — r)w^,, v 3 = 0); (a) or, alternatively, if OG = zK + (R — r)(—k), then (with d(j)/dt = ujf) v G = d(OG)/dt = ( dz/dt)K + (R — r)(—dk/dt) = ( dz/dt)i + (R — r){u^j). (b) If co = inertial angular velocity of sphere = {to i, cd 2 , cd 3 ), then the inertial velocity of the contact point C, v c , is v c = V G + to x r c/G = (v 1 ,v 2 ,v 3 ) + (w 1 ,0) 2 ,w 3 ) x (0,0, -r) = (v 1 -w 2 r, v 2 +uj { r, v 3 ). (c) Therefore: (i) If the cylinder is stationary (i.e., fixed), the rolling constraint is v c = 0, or, in components, v l —u) 2 r=0 => w 2 = (dz/dt)/r, v 2 + w\r = 0 => u>\ = [1 — (i?/r)]uty, v 3 = 0. (d) (ii) If the cylinder is made to rotate about its axis with an (inertial) angular velocity Q = (2 (?) = given function of time, the rolling constraint is v c = SI x r c/0 = (QK) x (-JM) = (Qi) x (-Rk) = ( QR)j = (0,QR,0), or, in components [invoking (c)], Vj — uj 2 r = 0 => u 2 = (dz/dt)/r= v z /r, v 2 + (jj\r = Q(t)R =$■ u>\ = ojfj, + (i?/r)w r , v 3 = 0, WWW.EBOOK777.COM (<=> CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS where oj r = Q — = relative angular velocity of cylinder about meridian plane G-13. The first of the constraints (e) is nonstationary and acatastatic, even if Q = constant. [As explained in §2.5 If., the virtual form of that constraint is 6p 2 + 68 p- = 0, where dp 2 = v 2 dt and dd\ = co\ dt; and this coincides with the virtual form of the catastatic second of the constraints (d). In general, p 2 and 9 X are “quasi coordinates” — see §2.9 ff] First and second of the constraints (d) in terms of the Eulerian angles of the sphere <P, 0, T, relative to the “semiinertial” (translating but nonrotating) axes G XYZ We have, successively (recalling §1.12,13), V[ = dz/dt = v z , co 2 = cos(2, X)cVy T cos(2, T)cuy T cos(2, Z)tu z = (— sin 4>)ojx T (cos T (0 )(u z = (— sin</>)[cos A>(d0/dt) + sin <P sin &(d'F/dt)\ + (cos 0) [sin <P{d©/dt) — cos <P sin ©(d'F/dt)] = ■ ■ • = sin(«P - (j)){d©/dt) - cos($ - (j>) sin ©(d'F/dt), (f) that is, the familiar co Y component but with (j> replaced by d> — </>; v 2 = {R~ r)(dcj)/dt), toi = cos(l, X)cux T cos( 1, Y)lo y T cos( 1, Z)oJz = (0)cuy + (0)wy + (l)w z = dd>/dt + cos 0( d'F/dt). (g) Therefore, the first and second constraints (d) transform to Vi — u) 2 r = dz/dt — r[sin(fp — (/){d©/dt) — cos(<P — (j>) sin 0(d < F/dt)\ = 0, (h) v 2 + uqr = (R — r){d(f)/dt) + r[d$/dt + cos ©(d'F/dt)] =0; (i) and similarly for the first two of (e). Example 2.2.3 Acatastatic Constraints. Let us consider the rolling of a sphere S of radius r and center G on a rough surface of revolution with a vertical axis. Let us introduce the convenient frame/axes/basis G 123/G ijk shown in fig. 2.4. Lurther, let </> be the azimuth, or precession- like, angle of the meridian plane (and of plane G-23); and 8 be the nutation -like angle between the positive surface axis and the common (outward) normal. Then, with dr//dt = co^, dd/dt = co e , we will have (i 0 = inertial angular velocity of G —123 = (f2 1 ,f2 2 ,f2 3 ) = (-a^sinfl, u^cosfl), (a) v G = inertial velocity of G = (v 3 , v 2 ,v 3 ) = (pco s , Rco^ = psindco^, 0), (b) where p = radius of curvature of meridian curve of parallel surface at G. WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS Figure 2.4 Rolling of a sphere on a vertical surface of revolution. C3: along common normal, outward; Cl: parallel to tangent to meridian curve, at contact point C; C2: parallel to tangent to circular section through C (or, so that G-123 is OND). If m = inertial angular velocity of sphere = (uii,ui 2 , w 3 ), then the inertial velocity of the contact point C, v c , equals v c — v g + 03 x r C/G = (vi,v 2 ,v 3 )+ (w 1 ,W 2 ,w 3 ) X (0,0, -r) = (vj -u 2 r, v 2 +uj l r, v 3 ). (c) Therefore: (i) If the surface is stationary, the rolling constraint is v c = 0 , or, in components, Vi — uJ 2 r = 0, v 2 + u\r= 0, v 3 = 0. (d) (ii) If the surface is compelled to rotate about its axis with (inertial) angular velocity Q = Q(t) = given function of time, the rolling constraint is v c = i2 x r c/0 = (0,Q(R - rsind),0), (e) or, in components, V; — tu 2 r = 0 => u> 2 = V\/r = pug/r, (f) v 2 + W|T = Q(R — rsin0) =>■ uji = (R/r)u> r — Qsin9, v 3 = 0, (g) where ui r = Q — luj, = relative angular velocity of surface about meridian plane G—13. The first constraint (g) is nonstationary and acatastatic, even if Q = constant. [As explained in §2.5 ff., the virtual form of that constraint is 8p 2 + 60\r = 0, where dp 2 = v 2 dt and d6 x = df, and it coincides with the virtual form of the catastatic second constraint (d). In general, p 2 and 6\ are “quasi coordinates”— see §2.9 ff.] WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS SPECIALIZATIONS (i) If the surface of revolution is another sphere with radius p a = p — r = constant , since then Vi = (p„ + r)uj s , v 2 = [(p a + r) sin Q\w$, the constraints (f) and the second of (g) reduce, respectively, to w 2 — [(Po + r )/ r \ u S — [(Po/ r ) + 1]<U0, (h) w i = [(Po/ r ) + 1] sin day — 12 sin d. (i) (ii) If the surface of revolution is another sphere with radius p 0 = p — r, that is free (i.e., unconstrained) to rotate about its fixed center with inertial angular velocity (o' = (u\, u/ 2 ,u/ 3 ), then, reasoning as earlier, we obtain the catastatic constraint equations Vi - uj 2 r = pota'i, v 2 +u>i r=—p 0 u\, v 3 = 0. (j) However, if the u>\, uj' 2 , uj' 2 are prescribed functions of time, then the first and second of (j) become nonstationary (and acatastatic). For additional such rolling examples, including the corresponding Newton-Euler (kinetic) equations, and so on, see the older British textbooks: for example, Atkin (1959, pp. 253-259), Besant (1914, pp. 353-359), Lamb (1929, pp. 162-170), Milne (1948, chaps. 15, 17). Example 2.2.4 Problem of Ishlinsky (or Ishlinskii). Let us consider the rolling of a circular rough cylinder of radius R on top of two other identical circular and rough cylinders, each of radius r, rolling on a rough, fixed, and horizontal plane (fig. 2.5). Let O-xyz and O-x'y'z' be inertial axes, such that O-xy and O-x'y' are both on that plane, while their axes Ox and Ox' are parallel to the lower cylinder generators Figure 2.5 Rolling of a cylinder on top of two other rolling cylinders. Transformation equations: x = x' cosx - y'sin %, Y = x ' sin X + y' cosx- WWW.EBOOK777.COM §2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS and make, with each other, a constant angle x- To describe the (global) system motion, let us choose the following six position coordinates: (i) (x,y) = inertial co¬ ordinates of mass center of upper cylinder G (as for its third, vertical, coordinate we have z = 2r + R): (ii) 9 = angle between +Ox and upper cylinder generator; (iii) ip, ip l ,ip 2 = spin angles of the upper and two lower cylinders, respectively. Finally, let r, and r 2 be the position vectors of the contact points of the lower cylinders with the upper one, relative to G, and v,, v 2 be the corresponding (inertial) velocities. The rolling constraints are v G + co x r i = Vj and v G + m x r 2 = v 2 . (a) Let us express them in terms of components along O-xyz. We have v G = (dx/dt,dy/dt, 0) = (v*, v^O) co: inertial angular velocity of upper cylinder = ((dep/dt) cos 9, ( dcp/dt) sin 9,d9/dt) = (u;^ cos 9, sin 6, uj g ), »'i = (-Lv-rcpi) cot 9,-{y-r<p{),-R), r 2 = [r<P 2 - 0cosx - x sinx)] cot (9 - x)i' + [r<p 2 ~ (y cos x~x sin x)\j' - Rk' = ({ r <t> 2 + x sin x - y cos x) cos 9/ sin(6> - x), (rcp 2 + xsinx — jcos x) sin 9/ sin(0 — x), — R), = (0,2rwi,0) [where to 1>2 = dcp\ 2 /dt \, v 2 = (-2 nv 2 sin x, 2 ru 2 cos x, 0). (b) Substituting the above into (a), we obtain the following four constraint components: Ft - R^<t, sin 9 - uoircpi - y) = 0, v y + Rw^ cos 9 + u) e (rcp l — y) cot 9 — 2no l = 0; v x sin(0 — x) — Rco^ sin 9 sin(0 — x) — u e (rcp 2 + x sin x — y cos x) sin 9 + 2ruj 2 sin x sin(0 — x) = 0, v v sin(0 — x) + R^<p cos 9 sin(0 — x) + w s(t(/> 2 + x sin x — v cos x) cos 9 — 2rw 2 cosxsin(0 —x) =0. (c) For further details, see, for example, Mei (1985, pp. 33-35), Neimark and Fufaev (1972, pp. 99-101). It can be shown (§2.11, 12) that these constraints are non- holonomic. Therefore, the system has n = 6 globed DOF , and n — m = 6 — 4 = 2 local DOF (concepts explained in §2.3 ff). Example 2.2.5 When is Rolling Holonomic? So as to dispell the possible notion that all problems of (slippingless) rolling among rigid bodies lead to nonholo- nomic constraints, let us summarize below the cases of rolling that lead to holo¬ nomic constraints. It has been shown by Beghin (1967, pp. 436-438) that these are the following two kinds: (i) The paths of the contact point(s) of the rolling bodies are known cdtead of time ; that is, before any dynamical consideration of the system involved and as function of WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS its original position, on these bodies. Consider two such bodies whose bounding surfaces, 5) and S 2 , are described by the curvilinear surface (Gaussian) coordinates (u 1; v t ) and (u 2 , v 2 ), respectively, in contact at a point C. Their relative positions, say of Si relative to S 2 , are determined by the values of these coordinates at C and the angle </> formed by the tangents to the lines u\ = constant and u 2 = constant (or v 1? v 2 = constant) there. Knowledge of the paths of C on both Si and S 2 translates to knowledge of the four holonomic functional relations: «i = «i(vi), u 2 = u 2 {v 2 ), (j) = (j){u u u 2 ), 5i(m,) = s 2 (u 2 ) ± c; (a) where .s, and s 2 are the arc lengths (or curvilinear abscissas) of the contact point paths S[ and S 2 , and c is an integration constant. It follows that, out of the five surface positional parameters, U\ , vj, u 2 , v 2 , </>, only one is independent; the other four can be expressed in terms of that one by finite (holonomic) relations. (ii) The bounding surfaces S[ and S 2 are applicable on each other; they touch at homologous points and their homologous curves (trajectories of the contact point C on them) join together there. This is expressed by the condition of contact, and by Ml = U 2 , V! =v 2 , (j) = 0, (b) at C; that is, again, a total of four holonomic equations. This condition is guaranteed to hold continuously if it holds initially and, afterwards, the pivoting vanishes. Such conditions are met in the following examples: (a) Rolling of two plane curves (or normal cross sections of cylindrical surfaces Si and S 2 ) on each other, and expressed by .sy = ,s 2 ± c. (b) Rolling of a body on a fixed surface, which it touches on only two points. For example, the rolling of a sphere on a system made up of a fixed circular cylinder and a fixed plane perpendicular to it [fig. 2.6(a)], (If the cylinder rotates about its axis in a known fashion, the trajectories of the contact points on both plane and cylinder are known, but they are unknown on the sphere and, hence, such rolling is non- holonomic.) (c) Rolling of two equal bodies of revolution whose axes are constrained to meet and, initially, are in contact along homologous parallels, or meridians [fig. 2.6(b)], The pivoting of such applicable surfaces vanishes. (a) (b) Figure 2.6 Examples of holonomic rolling: (a) rolling of a sphere on a fixed circular cylinder and a fixed plane perpendicular to it; (b) rolling of a cone on another equal fixed cone. WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY 2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY Let us examine the differences between holonomic and nonholonomic constraints, in some mathematical detail, for the simplest possible case: a single particle, with (inertial) rectangular Cartesian coordinates x, y, z, moving in space under the Pfaffian equation adx + bdy + cdz = 0, (2.3.1) where a, b, c = continuously differentiable functions of x, y, z. [The Pfaffian expression a dx + bdy + cdz is a special differential form of the first degree. The total or Pfaffian differential equation (2.3.1) is a specialization of the Monge form: 0 = f (x, y, z; dx, dy, dz) = stationary and homogeneous in the velocity components [dx/dt, dy/dt, dz/dt), and hence (since t is absent) only path restricting. The Monge form is, in turn, a specialization of the general first-order partial differ¬ ential equation: F[f,x,y,z\ dx/dt, dy/dt, dz/dt) = 0.] Now, the constraint (2.3.1) may be nonholonomic or it may be holonomic in differential (or velocity) form; specifically, if (2.3.1) can become, through multiplica¬ tion with an appropriate integrating factor, p = p[x,y,z), an exact, or perfect, or total differential d<j> = df[x,y,z ) of a scalar function <f> = (f>[x,y,z): p(a dx + b dy + c dz) = df, (2.3.1a) from which, by integration, we may obtain the (rigid and stationary) surface: <f>(x,y,z) = constant, or z = z[x,y), (2.3.1b) then (2.3.1) is holonomic; if not, it is nonholonomic. [Since, as is well known, the two-variable Pfaffian a[x,y) dx + b[x,y) dy has always an integrating factor (in fact, an infinity of them), eq. (2.3.1) is the simplest possibly nonholonomic constraint. More on this below.] In particular, if p = 1 (i.e., dcf) = adx + bdy + cdz), the integrable Pfaffian dcj) is exact. Then, a = dcj)/dx, b = dcj)/dy, c=d(f/dz, (2.3.2) and so the necessary and sufficient conditions for (2.3.1) to be exact are that the first partial derivatives of a, b, c, exist and satisfy (by equating the second mixed (^-derivatives): da/dy = db/dx, da/dz = dc/dx, db/dz = dc/dy. (2.3.3) Equations (2.3.3) are sufficient for (2.3.1) to be completely integrable = holonomic; but they are not necessary: every exact Pfaffian equation is integrable, but every integrable Pfaffian equation need not be exact', in general, a p f 1 may exist, even though not all of (2.3.3) hold. In mechanics, we are interested in the holonomicity (= complete or unconditional) integrability, or absence thereof, of the constraints. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Let us now make a brief detour to the general case', the system of m Pfalhan constraints in the n{> m) variables x = (xq,..., x n ), d'6o = a Dk{x)dxk = 0 [D = 1,..., m(< «)], (2.3.4) where rank(a Dk ) = m (i.e., these equations are linearly independent in a certain x-region), is called completely (or unconditionally ) integrable, or complete, or holonomic, if either (i) it is immediately integrable, or exact', that is, if the m d'Oi/s are the exact, or total, or perfect, differentials of m functions 4>d = <t> d M : y a Dki x ) dx k = d<j> D (x)\ (2.3.4a) or (ii) each d'dp, although not immediately integrable, nevertheless admits a (nonzero) integrating factor d> D (x)', that is, if the 2m (not all zero) functions {<P D (x), <f>D{x)\ D = 1,... , m{< n)} and (2.3.4) satisfy d>\ d'9\ = <b\ (an dx i + • • • + a\ n dx n ) = df \ (x), . (2.3.4b) tn d 0 m — (ain i dx\ • • • -f- a mn dx n ) — d(b m (.r). or, compactly, <P D d'0 D = d> D fffao k dx] < ) = dcf> D (x), where the {dfo} are (linearly) independent. Summing (2.3.4b), over D, we also obtain its following consequence: y, d>o d'9 D = y <P D ^ y a Dk dx^J = y df k =d<t> — y (90/ dx k ) dx k => y <b D a Dk = d(p/dx k . Clearly, in both cases, (2.3.4a, b), the constraints (2.3.4) are equivalent to the holonomic equations ffx) = C x ,...,f m {x) = C m , (2.3.4c) where the m constants {C D ; D = 1 ,m} are fixed throughout the motion of the system. (Elaboration of this leads to the concept of semiholonomic constraints, treated later in this section.) If the constraints (2.3.4) are nonintegrable, neither immediately nor with integrating factors, they are called nonholonomic; and the mechanical system whose motion obeys, in addition to the kinetic equations, such nonholonomic constraints, either internally (constitution of its bodies) or externally (interaction with its environment, obstacles, etc.), is called a nonholonomic system. An alternative definition of complete integrability of the system (2.3.4), equivalent to (2.3.4b), is the existence of m independent, that is, distinct, linear, combinations of the m d'9o that are exact differentials of the m independent functions/^*): /Tn d 9 1 + • • • + [nm d 9 m — dfy ,..., Pmi d 9\ • • * 4- Pmm d 9 m — df m , (2.3.4d) where p, DD t = Pdd'( x ), compactly, y p DD > d'9 D ' = df D o d'9 D = y M dd > dfjy (D, D 1 = 1,..., m), (2.3.4e) [where ( M DD i ) is the inverse matrix of (pdd 1 ), an d both (m x m) matrices are nonsingular] and, hence, yield the m independent integrals (hypersurfaces): WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NON HOLONOMICITY f\ = Ci,... ,f m = c m ; that is, the system of eqs. (2.3.4) is completely integrable if there exists an m-parameter (n — m) -dimensional manifold that solves them. [Frobenius (1877) has shown that if m = n, or n — 1, then the system (2.3.4) is always completely integrable — more on this later.] Finally, calling the determinant of the coefficients p DD i the multiplicator of (2.3.4) [i.e., \p,DD'\ = 0)], and generalizing from the single constraint case (2.3.1), we can state that every multiplicator has always the form p F(f ,..., f D ), where F(.. .) is an arbitrary differentiable function of the /’s; that is, there exists an infinity of multiplicators. From the above, it immediately follows that in the case of a single Pfaffian equation in the n variables x = (xi,.... x n ) (i.e., for m = 1), complete integrability, in a certain x-domain, means that there exists, locally at least, a one-parameter family of (n — 1) -dimensional manifolds f(x) = <p(x) — constant = 0, which solves that equation. [We remark that the solutions of d'6 = a k (x)dxk = 0 are always one-dimen¬ sional manifolds, or curves: x k = x k (u), where u = curve parameter. And, generally, if the x are functions of the m(< n) new variables (), then Xk = x k (u\,..., u m ) is called an m-dimensional solution manifold of d'6 = 0, if, upon substitution into it, identical satisfaction results.] Problem 2.3.1 Verify that the sufficient (but non-necessary !) conditions for the complete integrability of the system of m Pfaffian equations [essentially the dis¬ crete version of (2.2.9) for a system of N particles], fn dt = Yl ( a Dk dx k + b Dk dy k + c Dk dz k ) + e D dt = 0, (a) where D = 1,... ,m(< 3 N), k = 1 and ( a,b,c,e) = continuously differenti¬ able functions of (x,y,z, t), are that da Dk /dx l db Dk /dy l dc Dk /dz, da D i/dx k , db D i/dy k , dc m /dz k , da Dk /dy , db Dk /dz, dc Dk /dt = = db DI /dx k , = dc Dl /dy k , de D /dz k ; da Dk ldzt = dc m /dx k , da Dk /dt = de D /dx k ; (b) db Dk /dt = de D /dy k ; (c) (d) for all k, l = 1,... ,N, for a fixed D. [In fact, the (obvious) choice: a Dk = df D /dx k , b D k = df D /dy k , c Dk = d(j) D /dz k , e D = d(j) D /dt; <f> D = x,y,z) satisfies (b-d).] Then, (a) simply states that df D = 0; and the latter integrates immediately to the holonomic constraints: <p D = (f> D (t-,x,y,z) = (constant) D . Introduction to Necessary and Sufficient Conditions for Holonomicity Let us, for the time being, postpone the discussion of the general case and return to the single Pfaffian equation in three variables, eq. (2.3.1), and find the necessary and sufficient conditions for its holonomicity. Assuming that this is indeed the case, then from (2.3.1) and the second of (2.3.1b) we readily see that dz = (dz/dx) dx + ( dz/dy) dy = (— a/c ) dx + (—b/c ) dy (2.3.5) must hold for all dx, dy, dz. Therefore, equating the coefficients of dx and dy of both WWW.EBOOK777.COM 260 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS sides, we obtain [assuming c f 0, and that z(x,y) is substituted for z in a, b, c] dz/dx = — (a/c) and dz/dy =—(b/c), (2.3.5a) and since d/dy(dz/dx) = d/dx(dz/dy), we obtain d/dy(a/c) = d/dx(b/c), or, explicitly, c[da/dy + (da/dz)(dz/dy)\ — a[dc/dy + {dc/dz){dz/dy)\ = c[db/dx + (db / dz)(dz / dx)\ — b[dc/dx+ (dc/dz) (dz/dx)], and inserting in it the dz/dx- and <9z/<9y-values from (2.3.5a), and simplifying, we finally find I = a(db/dz — dc/dy) + b(dc/dx — da/dz) + c(da/dy - db/dx) = 0. (2.3.6) Equation (2.3.6), being a direct consequence of the earlier mixed partial derivative equality, is the necessary and sufficient condition for (2.3.1) to be holonomic. If I = 0 identically (i.e., for arbitrary x,y,z ), then (2.3.1) is holonomic; if / ^ 0 identically, then (2.3.1) is nonholonomic. REMARKS (i) The form I is symmetric in (x,y,z) and ( a,b,c ); that is, it remains unchanged under simultaneous cyclic changes of (x,y,z) and (a,b,c ). (ii) Alternative derivation of equation (2.3.6): The mixed partial derivatives rule applied to (2.3.1a) readily yields d(pb) /dx = d(fia) / dy, d(/j,c) / dx = dffia) / dz, dffic) /dy = d(ffi) /dz. Multiplying the above equalities with c, b, a, respectively, and adding them together, we obtain (2.3.6); so, clearly, the latter is necessary and sufficient for the existence of an integrating factor (for further details, see, e.g., Forsyth, 1885 and 1954, pp. 247 ff.). (iii) A special case: If a = a(x,y), b = b(x,y), and c = 0, then, clearly, 7 = 0; which proves the earlier claim that the two-variable Pfaffian equation a(x,y) dx + b(x,y) dy = 0 is always holonomic, that is, for nonholonomicity, we need at least three variables. (iv) A special form: If (2.3.1) has the equivalent form dz = (—a/c) dx + (—b/c) dy = A(x,y,z) dx + B(x,y,z) dy = A[x, y, z(x, y)] dx + B[x, y, z(x, y)\ dy = A*(x,y) dx + B*(x,y) dy, (2.3.7) (or, similarly, dx = • • •, dy = • • •; depending on analytical convenience and/or avoid¬ ance of singularities), then the mixed partial derivative rule dA*(x,y)/dy = dB*(x,y)/dx, (2.3.7a) due to the chain rule (one should be extra careful here): dA*/dy = dA/dy+ ( dA/dz)(dzjdy) = dA/dy+ (dA/dz)B, (2.3.7b) dB*/dx = dB/dx+ (dB/dz)(dz/dx) = dB/dx + (dB/dz)A, (2.3.7c) WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NON HOLONOMICITY finally yields dA/dy+(dA/dz)B=dB/dx+(dB/dz)A; (2.3.7d) whose identical satisfaction in .a, y, z, is the necessary and sufficient condition for the complete integrability, or holonomicity, of (2.3.7). It is not hard to verify that (i) replacing, in (2.3.7d), A with — a/c and B with —b/c, we recover (2.3.6); and, conversely, (ii) since (2.3.7) can be written in the (2.3.l)-like form: A dx + Bdy + (—1) dz = 0, replacing, in (2.3.6), a, b , c, with A, B, —1, respec¬ tively, we recover (2.3.7d). If, in (2.3.7), dA/dz = 0 and dB/dz = 0, then (2.3.7d) reduces to dA/dy = dB/dx. Finally, the sole analytical requirement here is the con¬ tinuity of all partial derivatives appearing in these conditions (but not those of the nonappearing ones, such as dA/dx and dB/dv). Example 2.3.1 Let us test, for complete integrability, the following constraints: (i) dz = (z) dx + (z 2 + a 2 ) dy; (ii) dz = z( dx + .v dy). (i) Here, A = z and B = z 2 + a 2 , and therefore (2.3.7d) yields (l)(z 2 + a 2 ) = (2 z)z => z 2 = a 2 , that is, no identical satisfaction; or, our constraint is not completely integrable — it is nonholonomic. Then, the original equation becomes dz = z dx + 2 z 2 dy; and so (a) if a = 0, then z = 0 is a constraint integral; but (b) if a ^ 0, then there is no integral. For complete integrability, we should have an infinity of integrals depend¬ ing on an arbitrary integration constant. (ii) Here, the test (2.3.7d) gives xz = z + xz =>■ z = 0; that is, no identical satis¬ faction, and therefore no holonomicity. As the original equation shows, this is the sole integral. Problem 2.3.2 Show that the constraint of the plane pursuit problem (ex. 2.2.1): [F(t) — y](dx/dt) — [A(0 — x](dy/dt) = 0, (a) or, equivalently, [Y(t) — y\dx — \X{t) — x] dy + (0) dt = 0, (b) is holonomic if and only if [X(t) — x\ / [Y (t) — y\ = (dX / dt) / (dY / dt) [= (dx/dt)/(dy/dt)\. (c) Problem 2.3.3 Show that under a general one-to-one (nonsingular) coordinate transformation (x,y,z) (u, v, w)\ x = x(u. v, w), y = •••, z = ■■■, I = [d(u,v,w)/d(x,y,z)\-l'; WWW.EBOOK777.COM (a) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS where (with subscripts denoting partial derivatives) dO = a dx + bdy + c dz = pdu + qdv + r dw, (b) I = I(x,y, z) = a(b, - c y ) + b(c x - a z ) + c(a y - b x ), (c) I' = I'{u, v , w) =p(q w - r v ) + q(r u -p w ) + r{p v - q u ), (d) and d(u,v,w)/d(x,y,z) = Jacobian of the transformation (f 0); that is, / and I' vanish simultaneously; or, the holonomicity of dd = 0, or absence thereof, is co¬ ordinate invariant, and hence an intrinsic property of the constraint (a proof of this fundamental fact, for a general 1 1 fa 111 an system, will be given later). [Incidentally, the transformation law (a) also shows that scalars like / are not necessarily invariants (/ f /', in general); in fact, in the more precise language of tensor calculus, they are called relative scalars of weight +1, or scalar densities', see, e.g., Papastavridis (1999, pp. 46-49).] Geometrical Interpretation of the Pfaffian Equation (2.3.1) The latter, rewritten with the help of the vectors dr = (dx, dy, dz) and h = (a, b, c) as h- dr — 0, (2.3.8) means that, at each specified point Q(x,y,z), dr must lie on a local plane perpendi¬ cular to the “constraint coefficient vector” h there; or, that the particle P can move only along those curves, emanating from Q, whose tangent is perpendicular to h. Such curves are called kinematically admissible, or kinematically possible. If (2.3.1,8) is holonomic, then all motions lie on the integral surface (2.3.1b); that is, (2.3.6) is the necessary and sufficient condition for the existence of an orthogonal surface through Q , for the field h = ( a,b,c ) [actually, a family of surfaces <f> = <f>(x,y,z) = constant, everywhere normal to h — see below]. We also notice that, with the help of h, the condition (2.3.6) takes the memorable (invariant) form: / = h- curl h = 0; or, symbolically, a b c d/dx d/dy d/dz a b c 0; (2.3.8a) that is, at every field point. It is parallel to the plane of its rotation, or perpendicular to that rotation and tangent to the surface <f> = constant there [W. Thomson (Lord Kelvin) called such fields doubly lamellar ]; while (2.3.7d), with h —> H = (A, B, — l), becomes H ■ curl H = dA/dy + B(dA/dz) - dB/dx - A(dB/dz ) = (l/c 2 )/z • curl h = 0. (2.3.8b) Vectorial Derivation of Equation (2.3.8a) We recall from vector analysis that a (continuously differentiable) vector is called irrotational, or singly lamellar, if (a) its line integral around every closed circuit WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY vanishes, or, equivalently, (b) if its curl (rotation) vanishes, or (c) if it equals the gradient of a scalar. Now: (i) If h = ( a,b,c ) is irrotational, then there is a ej> = cj)(x,y,z ) such that h = grad <j>, and, therefore, h ■ dr = grad 0 • dr = do = exact differential. (ii) If h f irrotational, still an integrating factor (IF) p, = p(x,y,z ) may exist so that // h = grad <f>. Then, as before, // h • dr = grad o • dr = def> = exact differential. (iii) Conversely, if p = IF, then p h = grad <j> = irrotational; and “curling” both sides of this latter, we obtain: 0 = curl(grad 0) = p curl h I- grad p x h, and dotting this with h. 0 = p(h • curl It), from which, since pfi 0, we finally get (2.3.8a). In this case, since h and grad <j> are parallel: It = (1 / p) grad 0 = v(grad 0), and, therefore, curl h = curl (a grad (f>) = grad v x grad 0, so that h ■ curl h = v grad <b • ( grad a x grad (f> ) = 0; (2.3.8c) that is, the doubly lamellar field h is perpendicular to its rotation curl It. [This condi¬ tion is necessary for the existence of an IF. For its sufficiency, see, for example, Brand (1947, pp. 200,’230-231), Sneddon (1957, pp. 21-23); also’Coe (1938, pp. 477-478), for an integral vector calculus treatment.] These derivations are based on a general vector field theorem according to which an arbitrary vector field can be written as the sum of a simple and a complex (or doubly) lamellar field: h = grad f + v grad (f). Finally, if the Pfaffian constraint is, nonholonomic, then (2.3.1,7) yield one¬ dimensional “nonholonomic manifolds”; that is, space curves orthogonal to the field h (or H), and constituting a one-parameter family on an arbitrary surface. Accessibility The restrictions on the motion of the particle P in the two cases 7=0 (holonomic) and If 0 (nonholonomic) are of entirely different nature. If I = 0, then P is obliged to move on the surface </> = <j)(x,y,z) = 0. If, on the other hand, 7^0, then the constraint (2.3.1) does not restrict the (x,y,z), but does restrict the direction (velocity) of the curves through a given point (x,y,z). The cumulative effect of these local restrictions in the direction of motion (velocity) is that the transition between two arbitrary points is not arbitrary, P can move (or be guided through) from an arbitrary initial (analytically possible) position, to any other arbitrary fined (analytically possible) position, while at every point of its path satisfying (2.3.1, 8); that is, the particle can move from “anywhere” to “anywhere,” not via any route we want, but along restricted paths. As Langhaar puts it, the particle is “constrained to follow routes that coincide with a certain dense network of paths ” (1962, pp. 5-6); like kinematically possible tracks guiding the system. In sum: (i) Holonomic constraints do reduce the dimension of the space of acces¬ sible configurations, but do not restrict motion and paths in there; in Hertz’s words: “all conceivable continuous motions [between two arbitrary accessible positions] are also possible motions.” (ii) Nonholonomic constraints do not affect the dimension of the space of acces¬ sible configurations, but do restrict the motions locally (and, cumulatively, also globally) in there; not all conceivable continuous motions (between two arbitrary accessible positions) are possible motions (Hertz, 1894, p. 78 ff.). These geometrical interpretations and associated concepts are extended to general systems in §2.7. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Degrees of Freedom The above affect the earlier DOF definition: they force us to distinguish between DOF in the large (measure of global accessibility, or global mobility) and DOF in the small (measure of local/infinitesimal mobility). We define the former, DOF(L ), as the number of independent global positional (or holonomic) “parameters,” or Lagrangean coordinates = n (= 3 in our examples, so far); and the latter, DOF(S) = /, as n minus the number of additional (possibly nonholonomic) inde¬ pendent Pfaffian constraints: f = n — m(> 0). In the absence of the latter, DOF(L) = DOF(S): f = n. This fine distinction between DOFs rarely appears in the literature, where, as a rule, DOF means DOF in the small. {For enlightening exceptions , see, for example, Sommerfeld (1964, pp. 48-51); also Roberson and Schwertassek (1988, p. 96), who call these DOFs, respectively, positional(L) and motional(S); and the pioneering Korteweg (1899, p. 134), who states that “Die anzahl der Freiheitsgrade sei bei ihr eine andere (kleinere) fur unendlich kleine wie fur endliche Verruckungen” [Translation: The number of degrees of freedom is different (smaller) for infinitesimal displacements than for finite displacements.]} As explained later in this chapter (§2.5 flf.), DOF(S) = f equals the number of independent virtual displacements of the system', and this, in turn (chap. 3), equals the smallest, or minimal, number of kinetic (i.e., reactionless) equations of motion of it. In view of this, from now on by DOF we shall understand DOF in the small ; that is, DOF = DOF(S) = n — m = /, unless explicitly specified otherwise. The concept of DOF in the large is more important in pure kinematics (mechanisms). Finally, in the general constraint case, all these results hold intact, but for the figurative system "particle” in a higher dimensional space — more on this later. Semiholonomic Constraints We stated earlier that if 7 = 0, the Pfaffian constraint (2.3.1) is holonomic; that is, it can be brought to the form df/dt = 0 =>■ <j) = constant = c. (2.3.9) Such situations necessitate an additional, albeit minor, classification of holonomic constraints into proper holonomic, or simply holonomic, and improper holonomic, or semiholonomic ones. In both cases, the constraints are finite (i.e., holonomic), but, in the proper case, the constraint constants have a priori fixed values, independent of the system’s position/motion; whereas, in the semiholonomic case, those constants depend on the arbitrarily specified values of the system coordinates at some “initial” instant; that is, semiholonomic constraints are completely integrable velocity (Pfaffian) constraints =>■ (generally) initial condition-depending holonomic con¬ straints. In the proper holonomic case, the initial values of the coordinates must be determined in conjunction with the given constraints and their constants; that is, they must be compatible with the latter. However, semiholonomic constraints, being essentially holonomic, can be used to reduce the number of independent global/ Lagrangean coordinates; and, thus, differ profoundly from the nonholonomic ones. Clearly, the proper/semiholonomic distinction applies to rheonomic holonomic constraints, like (f>{x, y , z, t) = c. For further details, see (alphabetically): Delassus (1913(b), pp. 23-25: earliest extensive discussion of semiholonomicity), Moreau (1971, pp. 228-232), and Peres (1953, pp. 60-62, 218-219). WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY Critical Comments on Nonholonomic Constraints The concept of nonholonomicity (in mechanics) has been around since the 1880s, and has been thoroughly studied and expounded by some of the greatest mathema¬ ticians, physicists, and mechanicians, for example (approximately chronologically): Voss, Hertz, Hadamard, Appell, Chaplygin, Voronets, Maggi, Boltzmann, Hamel, Heun, Delassus, Caratheodory, Schouten, Struik, Goursat, Cartan, Synge, Vranceanu, Vagner, Dobronravov, Lur’e, Neimark, Fufaev, et al. Direct definitions of nonholonomicity and analytical tests have been available, on a large and readable scale, at least since the 1920s. And yet, on this topic, there exists widespread mis¬ understanding and confusion; especially in the engineering literature. For example, some authors state that constraints that can be represented by equations like </>(r, r) = 0, or <j)(x,y,z,t ) = 0, are called holonomic, and that all others are called nonholonomic; for example, Goldstein (1980, p. 12 ff), Kane (1968, p. 14), Kane and Levinson (1985, p. 43), Likins (1973, pp. 184, 295), Matzner and Shepley (1991, pp. 23-24). Under such an indirect, vague, negative definition, inequality constraints like (j> > 0, or (perhaps?!) holonomic ones, but in velocity form, like d(j)/dt = (dtj>/dr) • v + d(j>/dt = 0, (2.2.10) would be called nonholonomic! Or, we read blatantly contradictory and erroneous statements like “With nonholonomic systems the generalized coordinates are not independent of each other, and it is not possible to reduce them further by means of equations of constraint of the form f(q j,..., q n , t) = 0. Hence it is no longer true that the qf s are independent ” (Goldstein (1980, p. 45), emphasis added). Others call nonholonomic all velocity constraints that cannot be written in the above form <j> = 0, which is correct; but they fail to supply the reader with analytical (or geometrical, or even numerical) tools on how to test this; for example, Roberson and Schwertassek (1988, p. 96), Shabana (1989, pp. 123, 128). The more careful of this last group talk clearly about integrability, exactness, and so on, but restrict them¬ selves to only one velocity constraint; for example, Haug (1992, pp. 87-89). Still others mix nonholonomic coordinates (quasi coordinates, etc.) with nonholonomic constraints, and exactness of Pfaffian forms with (complete) integrability of a system of Pfaffian equations, without ever supplying clear and general definitions, let alone analytical tests. And this results in defective definitions of the concept of DOF; for example, Angeles (1988, pp. 80, 103). Even the (otherwise monumental) treatise of Pars (1965, pp. 16-19, 22-24, 35-37, 64-72, 196) is limited to an introduction to the subject, albeit a careful and precise one. Finally, there is the recent crop of texts on “modern” dynamics, where the problem of nonholonomicity is “solved” by ignoring it altogether; for example, Rasband (1983). Only Neimark and Fufaev (1967/1972) discuss the nonholonomicity issue clearly, competently, and in sufficient generality and completeness to be useful. We hope that our treatment complements and extends their beautiful work. Extensions/Generalizations of the Integrability Conditions (May be omitted in a first reading) (i) Single Pfaffian Equation in the n Variables x= (jq,..., x n ): d'9 = a^dx% = 0, a^ = a^ix). WWW.EBOOK777.COM (2.3.10) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS It can be shown that the necessary and sufficient condition for the complete integr- ability = holonomicity of (2.3.10) is the identical satisfaction of the following “symmetric” equations: hip = a k (da,/dx p - da p /dx ,) + a,{da p /dx k - da k /dx p ) + a p (da k /dxi — dai/dx k ) = 0, (2.3.10a) for all combinations of the indices k, /, p = 1[For example, one may start with the integrability condition of (2.3.1), (2.3.6) (i.e., n = 3) and then use the method of induction; or perform similar steps as in the three-dimensional case; see, for example, Forsyth (1885 and 1954, pp. 259-260).] Further, it can be shown (e.g., again, by induction) that out of a total of n[n — \){n — 2)/6 equations (2.3.10a), equal to the number of triangles that can be formed with n given points as corners, only rij = ( n — 1)(« — 2)/2 are independent. For n = 3, that number is indeed 1: eqs. (2.3.6) or (2.3.8a). Also, if a k / 0, it suffices to apply (2.3.10) only for / and p different from k. Finally, with appropriate extension of the curl of a vector to ^-dimensional spaces, (2.3.10) can be cast into a (2.3.8a)-like form (see, e.g., Papastavridis, 1999, chaps. 3, 6). Problem 2.3.4 (i) Specialize (2.3.10a) to the acatastatic constraint {n = 4): a(t,x,y,z) dx + b(t, x,y,z) dy + c(t,x,y,z) dy + e(t,x,y,z) dt = 0. (a) (ii) Show that (a) is holonomic if, and only if, the symbolic matrix ( a b c e \ d/dx d/dy d/dz d/dt , (b) a b c e J has rank 2 (actually, less than 3); that is, all possible four of its 3 x 3 symbolic subdeterminants, each to be developed along its first row, vanish. (iii) Further, show that if all such 2x2 subdeterminants of (b) vanish, then (a) is exact. (iv) Specialize the preceding result to the catastatic case e = 0; verify that, then, we obtain (2.3.6). Problem 2.3.5 For the Pfaffian equation (2.3.10), define the (« + 1) x n matrix / a. P = Oil \ a\ n (a) \ a n i ... a im J where a k / = da k /dxi — dai/dx k (= —a & ); k, l = 1Clearly, a n = • • • = a nn = 0. Now, it is shown in differential equations/dilferential geometry that for the holonomicity of (2.3.10), it is necessary and sufficient that the rank of P equal 1 or 2. Show that (i) rank P = 1 (i.e., all its 2 x 2 subdeterminants vanish) leads to the exactness conditions a ki — 0; (b) WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY (ii) rank P = 2 (i.e., all its 3x3 subdeterminants vanish) leads to the earlier complete integrability conditions (2.3.10a) ctj. Q/p -T- ci i ci pf- -T- a p a k j — 0. (c) Problem 2.3.6 Show that for n = 3, equations (b,c) of the preceding problem become, respectively, a k i = 0 (k, 1= 1,2,3), (a) and nT#23 + a 2 a 2 \ + a 3 a i2 — 0 (2.3.6)]. (b) Problem 2.3.7 Consider the Pfaffian equation (2.3.10). Subject its variables x to the invertible coordinate transformation (with nonvanishing Jacobian) x —> xin extenso: Xfc = x k (x k >) o x k > = x k ,(x k ) (k, k' = 1,..., n). (a) Show that the requirement that, under that transformation, the Pfaffian form d'O remain (form) invariant, that is, d'O —> (d'O)' = a k ' dx]j = d'O (= 0), a k > = a k ’(x'), (b) leads to the following (covariant vector) transformations for the form coefficients: a k < = ( dx k /dx k ')a k & a k = Y (dx k ,/dx k )a k . . (c) Problem 2.3.8 Continuing from the previous problem, define the antisymmetric quantities a M = da k /dx l - dajdx,, (= -a lk ), (a) a k <v = da k ,/dx r - da v /dx k , (= -a n ,), (k',l' = 1 ,... ,n). (b) Show that under the earlier invariance requirement d'O —> (d'O)' = d'O, the above quantities transform as (second-order covariant tensors): a k 'i' = EE (dx k /dx k ')(dx,/dx r )a k , & a k , = EE (dx k : /dx k ) (dxf /dxi)a k 'i>. (c) Problem 2.3.9 Continuing from the preceding problems, assume that the x (and, therefore, also the x') depend on two parameters iq and uy. x k = x k (ui,u 2 ) and x' k = x' k (u u u 2 ). (a) Introducing the simpler notation d'O = dO and (d'O)' = d0', show that d 2 (d\0) — d\(d 2 0) = d 2 (d\0') — d\(d 2 0'), (b) WWW.EBOOK777.COM 267 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS where d\9 = d\Xk = ’y^ ak[(dxk/dui)dui], did = Y (ikd 2 x k = y^akKdxk/ duijdui], are equivalent to EE (i dx k /du x ){dx,ldu 2 )a kl = EE (dx kl /du 1 )(dx r /du 2 )a kr . (c) (ii) If the Pfaffian Constraint (2.3.10) has the Equivalent, (2.3.5, 7)-like, Special Form: dz = y^ b k (x,z) dx k {k = 1,... ,ri), (2.3.10b) then, proceeding as in the three-dimensional case, or specializing (2.3.10a), we can show that the necessary and sufficient integrability conditions are the n(n — l)/2 independent identities [replacing n with n + 1 in the earlier «/, following (2.3.10a)]: dbk/dxj + ( db k /dz)b, = db,/dx k + (db,/dz)b k (k,l = 1,...,«). (2.3.10c) Here, too, only the existence and continuity of the partial derivatives involved is needed. (iii) General Case of m(< n ) Independent Pfaffian Equations in n Variables [In the slightly special total differential equation form, with x = {xd,Xj)\. dx D = E b D f(x) dxj or dx D /dxj = b D[ (x) (general form), (2.3.11) where ( here and throughout this book) D = 1(for Dependent) and / = m+ 1(for Independent), b DI = given (continuously differentiable) functions of the m x D = (v 1; ... ,x m ), and the (n — m) x k = (x m+ i,... ,x n ). (2.3.11a) The system (2.3.11) is called holonomic or completely integrable [i.e., functions x D (xi) can be found whose total differentials are given by (2.3.11)], if, for any set of initial values x I o , x D o , for which the b DI are analytic, there exists one, and only one, set of D functions x D (x r ) satisfying (2.3.11) and taking on the initial values x D o at xj 0 . It is shown in the theory of partial (total) differential equations—see references below—that: For the system (2.3.11) to be holonomic, it is necessary and sufficient that the follow¬ ing conditions hold: db DI /dx r + y b D ' I f{db DI /dx D f) = db Dr /dx, + y b r yj(db Dr /dx DI ) [D,D' = 1,... ,m\ /, I' = m + 1,...,«], (2.3.11b) WWW.EBOOK777.COM §2.3 QUANTITATIVE INTRODUCTION TO NON HOLONOMICITY identically in the x D , xfs [i.e., not just for some particular motion(s)] and for all combinations of the above values of their indices; if they hold for some, but not all, m values of D , then the system (2.3.11) is called “partially integrable.” Now, and this is very important, as the second (sum) term, on each side of (2.3.1 lb), shows, the integrability of the Z>th constraint equation of (2.3.11) depends, through the coupling with b D 'p and b D n, on all the other constraint equations of that system; that is, each (2.3.11b) tests the integrability of the corresponding constraint equation (i.e., same D) against the entire system — in general, holonomicity/non- holonomicity are system not individual constraint properties. Geometrically, integrability means that the system (2.3.11) yields a field of (.n — m)-dimensional surfaces in the n-dimensional space of the x’s; that is, mechani¬ cally, the system has (n — m) global positional/Lagrangean coordinates, namely, DOF(L) = DOF(S) = n- m. Further: • With the notation bm = b D i(x D ,xf) = boj[xj)(xj),Xi\ = Pdi( x i) = Pdu (2.3.11c) and since, by careful application of chain rule to the above, d/3 DI /dx r = db DI /dx,i + ^ {db DI /dx D ,){dx D fdx r ) = db DI /dx r + ^2 (db D1 /dx D ')b D , v , [if x D = x D (xj), then dx D = (dx D /dx r ) dx, = ff b D! (x) dxj] the holonomicity conditions (2.3.1 lb) can also be expressed in the following perhaps more intelligible/ memorable (“exactness”) form: dfoi/dx/' = dpDi'/dx, (/' = m + 1,... ,n); (2.3.lid) • It is not hard to verify that the system (2.3.11b, d) stands for a total of m(n — 1)(« — 2)/2 identities, out of which, however, only m(n — m)(n — m — l)/2 = mf{ f — l)/2 are independent [/ = n — m; as in the general case of the first of (2.12.5)]. • In the special case where b m = b DI (xj) [Chaplygin systems (§3.8)], (2.3.11b) reduce to the conditions: dboj/dxji = db DI fdxj [compare with (2.3.1 Id)], (2.3.lie) which, being uncoupled, test each constraint equation (2.3.11) independently of the others. Last, we point out that all these holonomicity conditions are special cases of the general theorem of Frobenius, which is discussed in §2.8-2.11. • Equations (2.3.11b, d) also appear as necessary and sufficient conditions for a Riemannian (“curved”) space to be Euclidean (“flat”) =4> vanishing of Riemann- Christoffel “curvature tensor”; and in the related compatibility conditions in non¬ linear theory of strain — see, for example, Sokolnikoff (1951, pp. 96-100), Truesdell and Toupin (1960, pp. 271-274). • Historical : The fundamental partial differential equations (2.3.11b) are due to the German mathematician H. W. F. Deahna [J. fur die reine und angewandte Mathematik ( Crelle’s Journal) 20 , 340-349, 1840] and, also, the French mathemati¬ cian J. C. Bouquet [Bull. Sci. Math, et Astron., 3(1), 265 ff., 1872], For extensive and readable discussions, proofs, and so on, see, for example, De la Valee Poussin (1912, WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS pp. 312-336), Levi-Civita (1926, pp. 13-33), and the earlier Forsyth (1885/1954). Regrettably, most contemporary treatments of Pfaffian system integrability are written in the language of Cartan’s “exterior forms,” and so are virtually inaccessible to the average nonmathematician. 2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS So far, we have examined constraints in terms of particle vectors, and so on. Here, we begin to move into the main task of this chapter: to describe constrained systems in terms of general system variables. Let us assume that our originally free, or uncon¬ strained, mechanical system S, consisting of N particles with inertial position vectors [recalling (2.2.4)] rp = rp{t) = {x P (t), yp(t),zp(t)} (P=l,...,N), (2-4.1) is now subject to h(< 3N) independent positionaljgeometrical/holonomic (internal and/or external) constraints <t>H{t,r P ) = <j> H {t,r) = <t> H {t\x,y,z) = 0 [H = 1,..., h{< 32V)], (2.4.2) or, in extenso, 0i(*;-*i!j ; i> z i> • • • j ■A'a'iLa'i^aO ~ 0, y ti z i) • • •) x Ni Jjvi z n) — (2.4.2a) where independent means that the (/>!,...,</>/, are not related by a(ny) functional equation of the form $(</>!,..., cj) h ) = 0 {In that case we would have, e.g., (j) h = F(t;(j>i, 1 ), so that one of the constraints (2.4.2, 2a), i.e., here </)>, = 0, would either be a consequence of the rest of them [if F(t; 0,..., 0) =0, while </>/, = 0], or it would contradict them [if F(t; 0,..., 0) ^ 0, while (p h = 0]}. At this point, to simplify our discussion and improve our understanding, we rename the particle coordinates (x,y,z) as follows [recalling ( 2 . 2 . 1 a)]: x l — £li y 1 — Z l=^3i-"I X N — £lN-2i Jv — &AT-1! Z N — C 3 V) (2.4.3) or, compactly, x p = &P-2, yp = C 3 T -1 ; z P = iip (P = 1, • ■ • j -^0; (2.4.3a) in which case, the constraints (2.4.2a) read simply = 0 [H=\,...,h(<3N)-, * = 1,..., 3N\. (2.4.3b) Therefore, using the h constraints (2.4.2a, 3b), we can express h out of the 3N coordinates £= ( x,y,z ), say the first h of them (“dependent”) in terms of the remaining n = 3N — h (“independent”), and time: = S rf (t;^ + i,...,^ 3 A r) =S d {t\ii) [d=l,...,h ; i = h + 1,..., 3N]; (2.4.4) WWW.EBOOK777.COM §2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS and so it is now clear that our system has n {global) DOF , h down from the previous 37V of the unconstrained situation. Further, since for h = 37V (i.e., n = 0) the solu¬ tions of (2.4.2a) would, in general, be incompatible with the equations of motion and/or initial conditions, while for h = 0 (i.e., n = 3N) we are back to the original unconstrained system; therefore, we should always assume tacitly that 0 < h < 37V or 0 < n < 37V. (2.4.5) Now, to express this / 2 -parameter freedom of our system, we can use either the last n of the £’s [i.e., the earlier = (£ /l+1 ,... , £ 3A /)], or, more generally, any other set of n independent (or unconstrained, or minimal), and generally curvilinear , coordinates, or holonomic positional parameters q = Vh = <h (7),- ■ -,?» = q n {t)\ = {qk = qk{i)\ k = or, simply. q = (qi,---,q n ), related to the via invertible transformations of the type 6 = 60;?) ?/t = ?*0;6)- (2.4.6) (2.4.6a) [The reader has, no doubt, already noticed that sometimes we use for the totality of the independent £’s; i.e., (£ A+1 , • • •, 6w)> and sometimes for a generic one of them; and similarly for other variables. We hope the meaning will be clear from the context.] In view of (2.4.6a), eq. (2.4.4) can be rewritten as 6 i = s d 0;6) = s 4*;60;?)] = s At;q), ( 2 . 4 . 6 b) that is, in toto, £* = q), * = 1,..., 37V; and so (2.2.4), (2.4.1) can be replaced by x P = x P {t, q), y P = y P (t, q), z P = z P (t, q), or rp = r P (t,q), (2.4.6c) or, finally, by the definitive continuum notation, r = r(t,q). (2-4.7) Let us pause and re-examine our findings. (i) The n = 37V — h independent positional parameters q = q(t) are, at every instant t, common to all particles of the system (even though not every particle, necessarily, depends on all of them); that is, the q s are system coordinates; but once known as functions of time they allow us, through (2.4.7), to calculate the motion of the individual particles of our system S. The c/s are also called holonomic (or true, or genuine, or global), independent (or unconstrained, or minimal) coordi¬ nates, although they might be constrained later (!); for short, Lagrangean coordi¬ nates', and the problem of analytical mechanics (AM) is to calculate them as functions of time. Most authors call them “generalized coordinates” (and, similarly, “general¬ ized velocities, accelerations, forces, momenta, etc.”). This pretensorial/Victorian terminology, introduced (most likely) by Thomson and Tait [1912, pp. 157-60, 286 IT.; also 1867 (1st ed.)], though inoffensive, we think is misguiding, because it WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS directs attention away from the true role of the cfs: the key word here is not general¬ ized but system (coordinates)! The fact that they are, or can be, general—that is, curvilinear (nonrectangular Cartesian, nonrectilinear)—which is the meaning intended by Thomson and Tait, is, of course, very welcome but secondary to AM, whose task is, among others, to express all its concepts, principles, and theorems in terms of system variables. Nevertheless, to avoid breaking with such an entrenched tradition, we shall be using both terms, generalized and system coordinates, and the earlier compact expression, Lagrangean coordinates. (ii) The ability to represent by (2.4.7) the most general position (and, through it, motion) of every system particle (i.e., in terms of a finite number of parameters), before any other kinetic consideration, is absolutely critical (“nonnegotiable”) to AM; without it, no further progress toward the derivation of (the smallest possible number of) equations of motion could be made. (iii) Further, as pointed out by Hamel, as long as the representation (2.4.7) holds, the original assumption of discrete mass-points/particles is not really necessary. We could, just as well, have modeled our system as a rigid continuum ; for example, a rigid body moving about a fixed point, whether assumed discrete or continuum, needs three cf s to describe its most general (angular) motion, such as its three Eulerian angles (§1.12). In sum, as long as (2.4.7) is valid, AM does not care about the molecular structure/ constitution of its systems. [However, as n —> oo (continuum mechanics), the descrip¬ tion of motion changes so that the corresponding differential equations of motion experience a “qualitative” change from ordinary to partial.] (iv) Even though, so far, r has been assumed inertial, nevertheless, the cf s do not have to be inertial; they may define the system’s configuration(s) relative to a non- inertial body, or frame, of known or unknown motion, and that (on top of the possible curvilinearity of the cf s) is an additional advantage of the Lagrangean method. (As shown later, the r’s may also be noninertial.) For example, in the double pendulum of fig. 2.7, cf> x , cj) 2 , 0 X are inertial angles, whereas 0 2 is not. If the constraints are stationary (—> scleronomic system ), then we can choose the cf s so that (2.4.7) assumes the stationary form [recalling (2.2.2 ff.)]: r P = r P {cf) or r = r(r 0 , q) = r(q); (2.4.7a) and, therefore, scleronomicity/rheonomicity (= absence/presence of dr/dt) are ^-dependent properties, unlike holonomicity/nonholonomicity. Figure 2.7 Inertial and noninertial descriptions of a double pendulum: OA, AB. Coordinates: <^ 1<2 : inertial; 8 X = fa: inertial; 0 2 = <j> 2 ~ <j>V- noninertial; O, A, C: collinear. WWW.EBOOK777.COM §2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS Analytical Requirements on Equations (2.4.6a-c, 7) The n cf s are arbitrary, that is, nonunique, except that when the representations (2.4.6c, 7) are inserted back into the constraints (2.4.2, 2a) they must satisfy them identically in the q' s, which, analytically, means that <Mo£*)=0 => (/> H [t;Ut;q)\= 0 => ^2(d(/) H /dQ(d^dq k ) = d(f) H /dq k = 0, where H — 1,..., * = 1,..., 37V; k = 1, ...,n(= 3N - //); (2.4.8) and where, due to the constraint independence and to (2.4.5), the Jacobians of the transformations cj> H £* and £* q k must satisfy rank(df H jdt/f) = /;, rank(d^/dq k ) = n (2.4.8a) [and since \d^/dq k \ 7 0 =>■ rank(d£,-/dq k ) = n\, in the region of definition of the £ and t. In addition, the functions in the transformations (2.4.6a, b) must be of class C 2 (i.e., have continuous partial derivatives of zeroth, first, and second order, at least, to accommodate accelerations) in the region of definition of the q's, and t. Last, conditions (2.4.8a) imply that the representations (2.4.6a, b) have a (non- unique) inverse: <lk = qtcifO = cj k {t,x,y,z) =q k = q k (t,r). (2.4.8b) Additional “regularity” requirements are presented in §2.7. Example 2.4.1 Let us express the above analytical requirements in particle vari¬ ables. Indeed, substituting into (2.2.8) and (2.2.10): v = dr/dt = ^ (dr/dq k )(dq k /dt) + dr/dt (k = 1,..., n), (a) we obtain, successively, 0 = df H /dt = s (d4> H /dr) • ( dr/dq k )(dq k /dt ) + dr/dtj + dcj> H /dt = i^ft/dr) ■ (dr/dq k fj ( clq k /dt) + (S • (dr/dt) + df H /dt ) = ( d$ H /dq k )(dq k /dt ) + d$ H /dt, (b) from which, since the holonomic system velocities dq k /dt are independent, d<P H /dq k = 0, (c) d<P H /dt = 0. (d) Constraint Addition and Constraint Relaxation The n cf s (just like the It s) are independent ; that is, we cannot couple them by nontrivial functions d>(q) = 0, independent of the problem’s initial conditions, and such that upon substitution of the cf s from (2.4.8b) into them they vanish identically WWW.EBOOK777.COM 274 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS in the <fs and t (i.e., <P[t, q(t, £)] = $(?,£) = 0 is impossible). Thus, as in differential calculus, when all the cf s except (any) one of them remain fixed, we are still left with a “nonempty” continuous numerical range for the nonfixed cf s; and these latter correspond to a “nonempty” continuous kinematically admissible range of system configurations (a similar conception of independence will apply to the various q- differentials, dq, 6q, .... to be introduced later). However, upon subsequent imposi¬ tion of additional holonomic constraints to the system, the n cf s will no longer be independent, or minimal. To elaborate: in the "beginning,” the system of particles is free, or unconstrained (“brand new”); then, its cf s are the 3 N £’s. Next, it is subjected to a mix of constraints; say, h holonomic ones like (2.4.2,2a), and m Pfaffian (possibly nonholonomic) ones like (2.2.7,9). Now, the introduction of the n = 3N — h cfs, as explained above, allows us to absorb, or build in, or embed, the h holonomic constraints into our description; the representations (2.4.6c, 7) guaran¬ tee automatically the satisfaction of the holonomic constraints, and thus achieve the primary goal of Lagrangean kinematics, which is the expression of the system’s configurations, at every constrained stage, by the smallest, or minimal, number of positional coordinates needed [which, in turn (chap. 3) results in the smallest number of equations of motion. The corresponding embedding of the PfafRan constraints, which is the next important objective of Lagrangean kinematics (to be presented later, §2.11 flf.), follows a conceptually identical methodology, but requires new “non¬ holonomic, or quasi, coordinates”]. Specifically, if at a later stage, h\< n) additional, or residual, or non-built-in, independent holonomic constraints, say of the form * H <t,q) = 0 (2.4.9) are imposed on our already constrained system, then, repeating the earlier proce¬ dure, we express the n q' s in terms of n' = n — h' new positional parameters q' = {qk'\k' = l,..., n')\ qic = qkifqic'), rank(dq/dq') = n', (2.4.10) so that, now, (2.4.7) may be replaced by r = r (t, q) = r[t, q(t , q')} = r(t, q'); (2.4.11) the representation (2.4.7) still holds, no matter how many holonomic and nonholo¬ nomic constraints are imposed on the system; but then our cfs will not be indepen¬ dent: they have become the earlier £’s. This process of adding holonomic constraints to an already constrained system, one or more at a time, can be continued until the number of (global) DOF reduces to zero: 3N — (h + h' + h" + •••)—> 0. Also, no matter what the actual sequence (history) of constraint imposition is, it helps to imagine that they are applied succes¬ sively, one or more at a time, in any desired order, until we reach the current, or last, state of “constrainedness” of the system. It helps to think of a given constrained system as being somewhere “in the middle of the constraint scale”: when we first encounter it, it already has some constraints built into it; say, it was not born yester¬ day. Then, as part of a problem’s requirements, it is being added new constraints that reduce its DOF(L), eventually to zero; and, similarly, proceeding in the opposite direction, we may subtract some of its built-in constraints, thus relaxing the system and increasing its DOF(L), eventually to 3 N. [Usually, such a (mental) relaxation of WWW.EBOOK777.COM §2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS one or more built-in constraints is needed to calculate the reaction forces caused by them (—> principle of “relaxation,” §3.7).] In sum: Any given system may be viewed as having evolved from a former “relaxed” (younger) one by imposition of constraints; and it is capable of becoming a more “rigid” (older) one by imposition of additional constraints. For example, let us consider a “newborn” free rigid body. The meaning of rigidity is that our system is internally constrained; and the meaning of free(dom) is that, when presented to us and unless additionally constrained later, the system is exter¬ nally unconstrained; that is, at this point, its built-in constraints are all internal: hence, n = 6. If, from there on, we require it to have, say, one of its points fixed (or move in a prescribed way), then, essentially, we add to it three external (holo- nomic) constraints; that is, n' = n — 3 = 6 — 3 = 3. If, further, we require it to have one more point fixed, then we add two more such constraints; that is, n" = n' — 2 = 3 — 2= 1. And if, finally, we require that one more of its points (noncollinear with its previous two) be fixed, then we add one more such constraint; that is, n" = n" — 1 = 1 — 1 = 0. But if, on the other hand, we, mentally or actually, separate the original single free rigid body into two free rigid bodies, then we sub¬ tract from it six internal built-in constraints (in Flamel’s terminology, we “liberate” the system from those constraints) so that this new relaxed system has n + 6 = 6 + 6= 12 (global) DOF. Equilibrium, or Adapted, Coordinates Frequently, we choose, in E 3N , the following “equilibrium,” or “adapted (to the constraints)” curvilinear coordinates: Xi = 4>i(t;x,y,z) (=0),..., Xh = 4>h(t’,x,y,z) (=0); Xh+\ = <t>h+\(t\x,y,z) (^0),..., X'3 N = hN(f,x,y,z) (^0); or, compactly, Xd = 4>d(t\x,y,z) (=0) (d= 1,...,/*); X, = 4>i(t-,x,y,z) #0) (i = h + 1,..., 31V), (2.4.12) and X3JV+1 = 4>3N+i = t (/ 0); where <f> d = (0 1; ..., </>/,) are the given constraints, and (j) t = ... ,4>3n) are n new and arbitrary functions, but such that when (2.4.12) are solved for the 3 N + 1 (t;x,y,z), in terms of (/;%[,... ,X 3 n)> and the results are substituted back into the constraints <f> d = 0, they satisfy them identically in these variables. In terms of the latter, which are indeed a special case of q s, the constraints take the simple equilibrium forms: Xd = (Xi = 0,...,Xh = 0), (2.4.12a) and so (2.4.7), with q —> \h reduces to *■ = r {U X/i+i > • ■ • i X 3 n) =r(t,Xi)- (2.4.12b) 275 WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Clearly, the earlier choice (2.4.4) corresponds to the following special x-case (assum¬ ing nonvanishing Jacobian of the transformation): Xd = Xd(t,0 =Zd-Bd(t,&) =0 (d= 1 Xi = xM=ii^ 0 (i = h+\,...,3N). (2.4.12c) In practice, the transition from £ to q, Xi is frequently suggested “naturally” by the geometry of the particular problem. However, the general method described above [but in differential forms; i.e., as d\d = dcj) d (= 0 ) and d\i = dcj), 0 )] will allow us, later (§2.11 If.), to build in Pfafhan (possibly nonholonomic) constraints. Finally, such equilibrium q, x,’s extend to the case of the earlier described «'(> 0) additional constraints. There we may choose the new equilibrium coordinates: x'd 1 = d>d' (= 0 ) (d'= x'v = ®i' #0) {i' = h'+l,...,n), (2.4.12d) so that r = r(t,q)^>r(t,x'i')- (2.4.12e) Excess Coordinates Sometimes, in a system possessing n minimal Lagrangean coordinates, q = (< 71 ,..., q n ), we introduce, say for mathematical convenience, e additional excess, or surplus, Lagrangean coordinates q E = (q„ + i,..., q„ +e )- Since the n + e positional coordinates q and q E are nonminimal—that is, mutually dependent— they satisfy e constraints of the type F E (t;qi,...,q n ;q n +u---,<ln+e) = F E (t;q,q E ) =0 (E = 1 ,... ,e); (2.4.13) and then we may have r = r(t,q,q E ). (2.4.13a) If we do not need the q E s, we can easily get rid of them: solving the e equations (2.4.13) for them, we obtain q E = q E (t;q), and substituting these expressions back into (2.4.13a) we recover (2.4.7). For this to be analytically possible we must have (see any book on advanced calculus) \dF E /dq E ,\^0 (E= E' = n+1,...,n + e). (2.4.13b) Example 2.4.2 Let us consider the planar three-bar mechanism shown in fig. 2.8. The O-xy coordinates of a generic point on bars OA\ and AiAt, can be expressed in terms of the angles <j>i and d> 3 , respectively; similarly, for a generic point P on A\A 2 , such that A\P = l, we have x = /[ cos^! + /cos^> 2 ) y = h sin^! + /sin^ 2 - (a) So, (j>i, 4> 2 , <^ 3 express the configurations of this system; but they are not minimal (i.e., independent). Indeed, taking the x, y components of the obvious vector equation OAi + A t A 2 + A 2 A 3 + AjO = 0, WWW.EBOOK777.COM §2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS we obtain the two constraints: F\ = l\ cos (j)\ T / 2 cos cf)2 T I3 cos (/>2 — L — 0, F 2 = l\ sin (j>\ + l 2 sin — h sin ft, = 0. (b) Therefore, here, n = 1 and e = 2; knowledge of any one of these three angles deter¬ mines the mechanism’s configuration. However, it is preferable to work with the representation (a), under (b), because if we tried to use the latter to express x and y in terms of either f j, or f 2 , ° r ft, °nly (wherever the corresponding Jacobian does not vanish), we would end up with fewer but very complicated looking equations of motion. It is preferable to have more but simpler equations (of motion and of constraint); that is, requiring minimality of coordinates, and thus embedding all holonomic constraints into the equations of motion, may be highly impractical. [See books on multibody dynamics; or Alishenas (1992). On the other hand, minimal formulations have numerical advantages (com¬ putational robustness).] Another “excess representation” of this mechanism would be to use the four O-xy coordinates of A x and A 2 , (x^yf) and (x 2 ,y 2 ), respectively. Clearly, these latter are subject to the three constraints (so that, again, n = 1 but e = 3): ( x i)~ + (ki ) 2 = (A) - ; ( x 2 — x \Y + (k2 - y\Y = ( 4 ) 2 ; (L — x 2 ) 2 + (0 — y 2 ) 2 = (h) 2 - (c) Example 2.4.3 Let us consider the planar double pendulum shown in fig. 2.9. The four bob coordinates x t , y, and x 2 , y 2 are constrained by the two equations ( x i ) 2 + (yiY = (A) 2 ) (-T2 — ^i ) 2 + (k 2 ~ ji) - = (^2)“; (a) that is, here N = 2 =>■ 2 N = 4, and so the number of holonomic constraints = F[ =2 =£- n = 2 N — FI = 2. A convenient minimal representation of the pendulum’s WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS configurations is x 1 =/ 1 cos 0 1 , >'] = /| sin <p \; x 2 = l\ cos^! + / 2 cos<j> 2 , y 2 = A sin + U sin ^ 2 - (b) 2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS; IN SYSTEM VARIABLES Velocity and Acceleration We begin with the fundamental representation of the inertial position of a typical system particle P in Lagrangean variables (2.4.7): r = r(t,q). (2.5.1) [Again, the inertialness of ris not essential, and is stated here just for concreteness. The methodology developed below applies to inertial and noninertial position vectors alike; and this, along with the possible curvilinearity (nonrectangular Cartesianness) and possible noninertialness of the coordinates, are the two key advantages of Lagrangean kinematics (and, later, kinetics) over that of Newton-Euler. This will become evident in the Lagrangean treatment of relative motion (§3.16).] From this, it readily follows that the (inertial) velocity and acceleration of P, in these variables, are, respectively, v = dr/dt = ^2 ( dr/dq k ){dq k /dt ) + dr/dt = ^ v k e k + e 0 , (2.5.2) a - dv/dt = E {dr/dq k )(d 2 q k /dt 2 ) + EE (d 2 r/dq k dq,){dq k /dt)(dq,/dt) + 2^ {9 2 r/dqkdt){dq k /dt) + d 2 r/dt 2 = E w ^+EE V*V/ e kJ + 2^2 v k e k,o + <?o,o, (2.5.3) WWW.EBOOK777.COM §2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS where dq k /dt = v k , v= (vi,... III j* ii (2.5.2a) d 2 q k /dt 2 = dv k /dt = w k , W= (Wj,.. . ,w„) = (w k ;k = 1,...,«) [but, in general, a f ^ w k e k + M’ 0 e 0 ; see (2.5.4—6) below)], (2.5.3a) associated with these cf s; and the fundamental (holonomic) basis vectors e k , e 0 , also associated with the cf s, are defined by e k = e k (t,q) = dr/dq k , e 0 = e 0 (t,q) = dr/dt (or, sometimes, e„ +1 , or e t ); (2.5.4) and the commas signify partial derivatives with respect to the cf s, t: e kj = de k /dq, = de,/dq k = e lik [i.e., d/dq,(dr/dq k ) = d/dq k (dr/dq,)], (2.5.4a) e k fi = de k /dt = de 0 /dq k = e ok [i.e., d/dt(dr/dq k ) = d/dq k {dr/dt)}; (2.5.4b) we reserve the notation a k for the representation a = Jf a k e k + a o e o- Also, note that with the help of the formal (nonrelativistic) notations: t = q 0 = q n+ i =>■ dt/dt = dq 0 /dt = dq n+l /dt = v 0 = v„ + \ = 1 7 (2.5.5a) and d 2 t/dt 2 = dv 0 /dt = dv n+ i/dt = m ’ 0 = w n+ \ = 0, (2.5.5b) we can rewrite (2.5.2,3), respectively, in the “stationary” forms: v = E v a e m a = w a e a + ^ ^ v a v p e a ^, (2.5.6) where, here and throughout the rest of the hook , Greek subscripts range from 1 to n+ 1 (=“ 0 ”). The v k = dq k /dt are the holonomic (and contravariant, in the sense of tensor algebra) components, in the (/-coordinates, of the system velocity or, simply, Lagrangean velocities or, briefly, but not quite accurately, “generalized velocities.” The key point here is that the velocity and acceleration of each particle, v and a, respectively, are expressed in terms of system velocities v = dq/dt and their rates w = dv/dt = d 2 q/dt 2 , which are common to all particles, via the (generally, neither unit nor orthogonal) “mixed” = particle and system, basis vectors e k , e 0 . The latter, since they effect the transition from particle to system quantities, are fundamental to Lagrangean mechanics. HISTORICAL These vectors, most likely introduced by Somoff (1878, p. 155 If.), were brought to prominence by Heun (in the early 1900s; e.g., Heun, 1906, p. 67ff, 78 IT.), and were called by him Begleitvektoren « accompanying, or attendant, vectors. Perhaps a better term would be “H(olonomic) mixed basis vectors” (see also Clifford, 1887, p. 81). From the above, we readily deduce the following basic kinematical identities'. (i) dr/dq k = dv/dv k = da/dw k = ■ ■ ■ = e k , (2.5.7) WWW.EBOOK777.COM 279 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS that is, [with (...)' = d(.. .)/dt\, dr/dq k = dr/dq k = df/dq k = • • • = e k (“cancellation of the (over)dots”); (2.5.7a) and (ii) d/dt(dr/dq k ) = d/dt ( dv/dv k ) = de k /dt = d/dq k ( dv/dt ) = dv/dq k , or, with the help of the Euler- Lagrange operator in holonomic coordinates: E k (...) = d/dt(.. . / dq k ) -d... /dq k = d/dt(.. . / dv k ) - d... /dq k , (2.5.9) finally, E k (v) = d/dt(dv/dv k ) — dv/dq k = 0. (2.5.10) In fact, for any well-behaved function/ = f(t,q), we have / = df/dt = ^2 ( df/dq k )(dq k /dt ) + df/dt, f = d 2 f / dt 2 = • • •, => df/dq k = df/dq k = df/dq k = ■ ■ ■; (2.5.8) and E k (f ) = d/dt ( df/dq k ) - df/dq k = d/dt ( df/dv k ) - df/dq k = 0 . (2.5.11) The integrability conditions (2.5.7, 10) are crucial to Lagrangean kinetics (chap. 3); and, just like (2.5.2, 3), have nothing to do with constraints; that is, they hold the same, even if holonomic and/or nonholonomic constraints are later imposed on the system, as long as the q's are holonomic (genuine) coordinates (i.e., q / nonholonomic or quasi coordinates; see §2.6, §2.9). Admissible and Virtual Displacements Proceeding as with the velocities, (2.5.2), we define the (first-order and inertial) kinematically admissible, or possible, and virtual displacements of a generic system particle P, respectively, by dr = ^2 ( dr/dq k ) dq k + ( dr/dt ) dt = ^ e k dq k + e 0 dt, (2.5.12a) 6r = ^2 ( dr/dq k ) Sq k = ^ e k 6q k ; (2.5.12b) whether the ^-increments, or differentials, dq, 6q, and dt are independent or not (say, by imposition of additional holonomic and nonholonomic constraints, later). As the above show: (i) if dq k = ( dq k /dt ) dt = v k dt, then dr = v dt; (ii) if all the dq' s and dt (Sq’s) vanish, then dr = 0 (Sr = 0); and (iii) d(dr)/d(dq k ) = d(Sr)/d{Sq k ) = e k . These identities (in unorthodox but highly instructive notation) are useful in prepar¬ ing the reader to understand, later, the nonholonomic coordinates. WWW.EBOOK777.COM §2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS REMARKS ON THE VIRTUAL DISPLACEMENT Let us, now, pause to examine carefully this fundamental concept. First, we notice that Sr is the linear (or first-order) and homogeneous, in the Scf s, part of the “total virtual displacement” Ar, which is defined by the following Taylor-like r-expansion in the first-order increments Sq , from a generic system configuration corresponding to the values q, but for a fixed time t: Ar = r(t,q + Sq) — r(t,q) = Sr + (1/2) S 2 r + ■ ■ ■. (2.5.13) In other words, Sr is a special first position differential, mathematically equivalent to dr with t = constant —> dt = 0 (i.e., completely equivalent to it for stationary constraints); hence, the special notation S (...): dr —> Sr, dq —> Sq, and dt —+ St = 0 (isochrony, always ). (2.5.13a) One could have denoted it as d*r, or (dr)*, or z, and so on; but since we do not see anything wrong with S(. ..), and to keep with the best traditions of analytical mechanics [originated by Lagrange himself and observed by all mechanics masters, such as Kirchhoff, Routh, Schell, Thomson and Tait, Gibbs, Appell, Volterra, Poincare, Maggi, Webster, Heun, Hamel, Prange, Whittaker, Chetaev, Lur’e, Synge, Gantmacher, Pars et al.], we shall stick with it. Readers who feel uncomfor¬ table with 6(...) may devise their own suggestive notation; dr and dq won’t do! The above definitions also show the following: (i) Sr is mathematically equivalent to the difference between two possible/admis¬ sible displacements, say d\r and d 2 r, taken along different directions but at the same time (and same dt); that is, skipping summation signs and subscripts, for simplicity, d 2 r — d x r = \(dr/dq) d 2 q + ( dr/dt) dt] — [( dr/dq ) cfq + (dr/dt) dt] = ( dr/dq)(d 2 q — cfq) ~ (dr/dq) Sq = Sr. (2.5.14) (ii) For any well-behaved function/ = /(/): Sf = (df/dt) St = 0;butif / = f(t,q ), then Sf = (df /dq) Sq f 0 [even though, after the problem is solved, q = <?(/)!]. (iii) The virtual displacements of mechanics do not always coincide with those of mathematics (i.e., calculus of variations). For example, even though, in general, dr Sr, for catastatic systems [i.e., dr = e k (t, q) dq k , Sr = e k (t, q) Sq k ] the equality dq k = Sq k ^ dr = Sr is kinematically possible [and in (q, //space dr and Sr are “orthogonal” to the /-axis, even though dt 0,St = 0]; whereas, in variational calculus we are explicitly warned that dq (parallel to the /-axis) / Sq (perpendicular to it). These differences, rarely mentioned in mechanics and/or mathematics books, are very consequential, especially in integral variational principles for nonholonomic systems (chap. 7). As definitions (2.5.12, 13), and so on, show, the (particle and/or system) virtual displacement is a simple, direct, and, as explained in chapter 3 and elsewhere, indis¬ pensable concept — without it Lagrangean mechanics would be impossible! Yet, since its inception (in the early 18th century), this concept has been surrounded with mysticism and confusion; and even today it is frequently misunderstood and/ or ignorantly maligned. For instance, it has been called “too vague and cumbersome to be of practical use” by D. A. Levinson, in discussion in Borri et al. (1992); “ill- defined, nebulous, and hence objectionable” by T. R. Kane, in rebuttal to Desloge (1986); or, at best, has been given the impression that it has to be defined, or “chosen properly” (Kane and Levinson (1983)), in an ad hoc or a posteriori fashion to fit the WWW.EBOOK777.COM 282 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS facts, that is, to produce the correct equations of motion. For an extensive rebuttal of these false and misleading statements, from the viewpoint of kinetics, see chap. 3, appendix II. Others object to the arbitrariness of the Sq' s. But it is precisely in this arbitrariness that their strength and effectiveness lies: they do the job (e.g., yielding of the equations of motion) and then, modestly, retreat to the background leaving behind the mixed basis vectors e k . It is these latter [and their nonholonomic counter¬ parts (§2.9)] that appear in the final equations of motion (chap. 3), just as in the derivation of differential (“field”) equations in other areas of mathematical physics. For example, in continuum mechanics, for better visualization, we may employ a small spatial element (e.g., a “control volume”), of “infinitesimal” dimensions dx, dy, dz, to derive the local field equations of balance of mass, momentum, energy, and so on; but the ultimate differential equations never contain lone differentials — only combina¬ tions of finite limits of ratios among them; that is, combinations of derivatives. Moreover, differentials, actual/admissible/virtual, in addition to being easier to visua¬ lize than derivatives, are invariant under coordinate transformations; whereas deriva¬ tives are not. [Such invariance ideas led the Italian mathematicians G. Ricci and T. Levi-Civita to the development of tensor calculus (late 19th to early 20th century); see, for example, Papastavridis (1999).] For example, taking for simplicity, a one (global) DOF system, under the transformation q —> q = qft, q), we find, successively, Sr = e Sq = ( dr/dq ) Sq = (dr/dq)[(dq/dq') Scf] = [(dr/dq)(dq/dq')\ Sq' = e' Sq' , that is, e' = dr/dq' = (dq/dq')e e = dr/dq = {dq'/dq)e'. (2.5.15) But there is an additional, deeper, reason for the representation (2.5.12b): the position vectors r(t,q) and (possible) additional constraints, say ip H i(t,r) = 0 —> ip H ft,q) = 0, cannot be attached in these finite forms to the general kinetic principles of analytical mechanics, which are differential, and lead to the equations of motion (§3.2 ff). Only virtual forms ofr and = 0 — special first differentials of them, linear and homogeneous in the Sq's [like (2.5.12b)] — can be attached, or adjoined, to the Lagrangean variational equation of motion via the well-known method of Lagrangean multipliers (§3.5 ff.); and similarly for nonlinear (non- Pfaffian) velocity constraints, an area that shows clearly that nonvirtual schemes (as well as those based on the calculus of variations) break down (chap. 5)1 Hence, the older admonition that the virtual displacements must be “small” or “infinitesimal.” For example, to incorporate the nonlinear holonomic constraint f(x,y) = A ' 2 + y 2 = constant to the kinetic principles, we must attach to them its first virtual differential, S<j> = 2(x Sx + y Sy) = 0; which is the linear and homoge¬ neous part of the total constraint change, between the system configurations (x,y) and (a + Sx, y + Sy): Acj) = <f>(x + Sx, y + Sy) - </>(x,y) = [Sf + (1/2) <S 2 (/>] forsman ^ Sf = 0. But in the case of the linear holonomic constraint cj> = x + y = constant, that total constraint change equals Ac/) = Sf = Sx + Sy = 0, no matter what the size of Sx, Sy; and both equations, f = 0 and Sf = 0 , have the same coefficients (—> slopes). In sum: As long as we take the first virtual differentials of the constraints, the size of the Sq' s is inconsequential, whether they are one millimeter or ten million miles! It WWW.EBOOK777.COM §2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS is the holonomic (or “gradient,” or “natural”) basis vectors { e k ; k = 1that matter. As Coe puts it: “We often speak of displacements, both virtual and real, as being arbitrarily small or infinitesimal. This means that we are concerned only with the limiting directions of these displacements and the limiting values of the ratios of their lengths as they approach zero. Thus any two systems of virtual displacements are for our purposes identical if they have the same limiting directions and length ratios as they approach zero” (1938, p. 377). Coe’s seems to be the earliest correct and vectorial exposition of these concepts in English; most likely, following the exposi¬ tion of Burali-Forti and Boggio (1921, pp. 136 ff.). See also Lamb (1928, p. 113). The earlier mentioned indispensability of the virtual displacements for kinetics will become clearer in chapter 3. Nevertheless, here is a preview: it is the virtual work of the forces maintaining the holonomic and/or nonholonomic constraints (i.e., the constraint reactions ) that vanishes, and not just any work, admissible or actual; in fact, the latter would supply only one equation. This vanishing-of-the-virtual-work- of-constraint-reactions (principle of d’Alembert-Lagrange) is a physical postulate that generates not just one equation of motion (like the actual work/power equation does), but as many as the number of (local) DOFs-, and, additionally, it allows us to eliminate/calculate these constraint forces. A more specialized virtual displacement —> virtual work-based postulate is used to characterize the more general “servo/ control” constraints (§3.17). Example 2.5.1 Differences Between Kinematically Admissible I Possible and Virtual Displacements. (i) Let us assume that we seek to determine the motion of a particle P capable of sliding along an ever straight line / rotating on the plane O—xy about O. The config¬ urations of / and of P relative to that plane are determined, respectively, by <j) and r, </> (fig. 2.10). Since r = r(r, </>): position of P in O—xy, we will have, in the most general case, dr = (dr/dr) dr + (dr/dcfi) dfi. kinematically admissible displacement of P , in O-xy , (a) Sr = (dr/dr) Sr + (dr/d(f>) Sf. virtual displacement of P, in O—xy. (b) Figure 2.10 On the difference between possible/admissible and virtual displacements (ex. 2.5.1: a, b). WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (a) If the rotation of l is influenced by the motion of P relative to it, then <j> is another unknown Lagrangean coordinate, like r, waiting to be found from the equations of motion of the system P and 1 (i.e., n = 2). Then dr and Sr are given by (a, b), respectively, and are mathematically equivalent, (b) If, however, the motion of / is known ahead of time (i.e., if it is constrained to rotate in a specified way, unin¬ fluenced by the, yet unknown, motion of P), then <j> = f(t): given function of time =>■ d(j) = df(t) = [df(t)/dt\ dt = u(t) dt ^ 0, but S(j> = Sf(t) = uj(t) St = 0. As a result, (a, b) yield dr = (dr/dr) dr + (dr/d</>) d<f> = (dr/dr) dr + (dr/d(f>) u(t) dt = dr(t, r; dt, dr), (c) Sr = (dr/dr) Sr = Sr(t, r: Sr). (d) (ii) Let us consider the motion of a particle P along the inclined side of a wedge W that moves with a given horizontal motion: x =f (t) (fig. 2.11). Here, we have MMi = M- i M 2 = (dr/dx) dx = (dr/dx)[df (t)/dt\ dt = (dr/dt) dt ~ dt-, (e) MM 3 = M { M 2 = Sr= (dr/dq) Sq - Sq\ (f) MM 2 ~ dr = (dr/dq) dq + (dr/dt) dt; but Sx = 0. (g) (iii) Let us consider the motion of a particle P on the fixed and rigid surface <j>(x,y,z) = 0 or z = z(x,y). Then, r = r(x,y,z) = r[x,y,z(x,y)\ = r(x,y), and the classes of dr and Sr are equivalent. But, on the moving and possibly deforming surface (j>(t;x,y,z) =0 or z = z(x,y; t), r = ■ ■ ■ = r(t;x,y), and so dr ^ Sr: Sr still lies on the instantaneous tangential plane of the surface at P, whereas dr does not. Figure 2.11 On the difference between possible/admissible and virtual displacements (ex. 2.5.1: b). WWW.EBOOK777.COM §2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS Example 2.5.2 Lagrangean Coordinates and Virtual Displacements. Let us deter¬ mine the Lagrangean description r = r(r 0 -,t,q) and corresponding virtual displace¬ ments Sr = ■ ■ ■ for the following systems: (i) Two particles, Pi and P 2 , are connected by a massless rod of length l, in plane motion (fig. 2.12). For an arbitrary rod point P(X, Y), including P x and P 2 , we have X = X l +xcos(j) = X(x-,X 1 ,<f>), Y — Y) + x sin </> = F(x; F b </>), (a) or, vectorially, r = f*i + xi = r(x; X \, Yi , </>), 0 < x < l. (b) Therefore, r 0 = xi', while the (inertial) positions of P x and P 2 are given, respectively, by r l =r(0-,X u Y l ,(f>) = X l I+Y l J, (c) r 2 = r(I ; X h Y), (f>) = (X x + /cos (j))I + (Y { + / sin (j>)J = (X { I + Y X J) + /(cos0/ + sin c/rJ) = r t + //. (d) Hence, this is a (holonomic) three DOF system: q x = X x , q 2 = Y), q 3 = </>. From (a) we obtain, for the virtual displacements, SX = 6X\ + x(— sin0) SY = SY\ + x(cos </>) 6<f>] (e) or, vectorially, Sr = Sri + x hi = Sri + x[(6<f>k) x i] = Sr x + (x 6(f)) j. (f) (ii) A rigid body in plane motion (fig. 2.13). For this three DOF system we have X = A* + xcos cj) — y sin c/) = X (x, y; A*, </>), Y = Y* + xsin (f) + y cos cf) = Y (x, y; Y+,<f>), WWW.EBOOK777.COM (g> CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (and Z = Z* = 0, say), or, vectorially, v = XI+YJ = r(x, y;X*, 3^,0); (h) that is, r c = xi + yj and q { = X+, q 2 = Y+, q 2 = cj). Therefore, the virtual displace¬ ments are 8X = 8X. - xsin<t>6<t>-ycosif> 8<f> = 6X. - 64>(Y - 7*), 8Y = (57* + x cos <j) 8<\> — v sin cf> 8<j> = 8Y+ + 8cj>(X — X»), (i) (and 8Z = <5Z* = 0), or, vectorially, 8r = 5r+ + 8<t> x (r — r 4 ), 6<j) = 84>k. (j) The extension to a rigid body in general spatial motion (with the help of the Eulerian angles, §1.12, and recalling discussion in § 1 . 8 ) is straightforward. 2.6 SYSTEM FORMS OF LINEAR VELOCITY (PFAFFIAN) CONSTRAINTS Stationarity/Scleronomicity/Catastaticity for Positional/Geometrical (=> Holonomic) Constraints in System Variables We begin by extending the discussion of §2.2 to general system variables, inertial or not. Positional constraints of the form cj){q) = 0 (=> dcj)/dt = 0) are called stationary, otherwise — that is, if q) = 0 (=> dcj)/dt ^ 0 ) — they are called nonstationary, and if all constraints of a system are (or can be reduced to) such stationary (nonstation¬ ary) forms, the system is called scleronomic ( rheonomic). Clearly, such a classification is nonobjective —that is, it depends on the particular mode and/or frame of reference used for the description of position/configuration: for example, substituting r(t, q) into the stationary constraint cp(r) = 0 turns it to a nonstationary constraint, cj>[r(t, q')\ = tj>(t,q) = 0 (and this is a reason that certain authors prefer to base this classification only for constraints expressed in system variables); or, a constraint that WWW.EBOOK777.COM §2.6 SYSTEM FORMS OF LINEAR VELOCITY (PFAFFIAN) CONSTRAINTS is staLionary when expressed in terms of inertial coordinates ( q ) may very well become nonstationary when expressed in terms of noninertial coordinates ( q')\ under the frame of reference (i.e., explicitly time-dependent!) transformation q —» q'{t,q) <=> q —► q(t,q'), the stationary constraint <j>(q) = 0 transforms to the nonstationary one <j>(t,q') =0. Hence, a scleronomic constraint <f>(q) =0 remains scleronomic under all coordinate (not frame of reference) transformations q —> q'(q) q' —► q(q')\ that is, its scleronomicity under such transformations is an objective property. Stationarity/Scleronomicity/Catastaticity for Pfaffian Constraints in System Variables Next, let us assume that the h holonomic constraints have been embedded into our system by the n = 3N — h Lagrangean coordinates q. To embed the additional, m(< n) mutually independent and possibly nonholonomic Pfaffian constraints (2.2.9) into our Lagrangean kinematics and kinetics: first, we express them in system variables. Indeed, substituting v from (2.5.2) into (2.2.9), we obtain the Pfaffian constraints in system (holonomic) variables: Id = § (Bd -v) + B d = ■ ■ ■ = ^ c Dk v k + c D = 0 (D = 1 ,... ,m), ( 2 . 6 . 1 ) where cat = c Dk (t, q) = $ B d • ( dr/dq k ) = £ B D ■ e k , (2.6.1a) c d — c D,n+ 1 — c d,o = c ,d(L q) = B D • (dr/dt) + B D = B D ■ e 0 + B D \ (2.6.1b) and rank(c Dk ) = in. Similarly, substituting dr from (2.5.12a) into the differential form of (2.2.9), f D dt = 0, we obtain the kinematically admissible, or possible, form of these constraints in (holonomic) system variables: d'do =f D dt= cat dq k + codt = 0; (2.6.2) with d'Oo- not necessarily an exact differential', that is, do may not exist, it may be a “quasi variable” (§2.9) and, in view of what has already been said about virtualness, namely, dt —» 8t= 0 , the virtual form of these constraints in particle variables is 6'9 D = ^B D -Sr = 0, (2.6.3) and, accordingly, invoking (2.5.12b), in system variables, S'0 D = ^ c Dk Sq k = 0 . (2.6.4) The above show that, as in the particle variable case, the virtual displacements are mathematically equivalent to the difference between two systems of possible displacements, d\q and d 2 q, occurring at the same position and for the same time, but in different directions: apply ( 2 . 6 . 2 ) at (t, q), for d\q ^ d 2 q, and subtract side by side and a (2.6.4)-like equation results. And, as in (2.5.12a,b), once the constraints have been brought to these Pfaffian forms, the size of the Sq's does not matter; it is the constraint coefficients c Dk that do. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Now, if in (2.6.1-2), (i) dc Dk /dt = 0 => cat = c Dk {q) (2.6.5a) and (ii) c D = c Dn+ \ = c D q = 0, (2.6.5b) =>• constraint: f D = Y c Dk{q)vk = 0, for all D = 1, ... ,777 and k = 1, ... , 77 , these constraints are called stationary, other¬ wise they are nonstationary, and a system with even one nonstationary constraint is called rheonomic; otherwise it is scleronomic. The inclusion of (2.6.5b) in the stationarity definition is made so that the velocity form of stationary position con¬ straints coincides with that of the stationary velocity constraints: <t>D{q) = 0 => df D /dt = Y ( d<j) D /dq k )v k = Y <t>Dk{q)vk = 0. (2.6.5c) If only c D = c D n+ 1 = c D q = 0, for allD, but dc Dk /dtf 0 =>■ c Dk = c Dk (t, q) even for one value of D and k, the Pfaffian constraints are called catastatic [« calm, orderly (Greek)]; otherwise they are called acatastatic. We notice that stationary constraints are catastatic, but catastatic constraints may not be stationary; we may still have dc D k/dt f 0 for some D and k. As mentioned earlier (2.2.1 la ff.), it is the castastaticl acatastatic classification, having meaning only for Pfaffian constraints, that is the important one for analytical kinetics, not the stationary/nonstationary one. Finally, as (2.6.1b) shows, the acatastatic coefficients c D result from the nonsta¬ tionary part of v (i.e., dr/dt ), and the acatastatic part of (2.2.9) (i.e., B D ). From this conies the search for frames of reference/Lagrangean coordinates where the Pfaffian constraint coefficients take their simplest possible form; a problem that, in turn, leads us to the investigation of the following. Transformation Properties of c Dk and c D , under a General Frame-of-Reference Transformation The latter is mathematically equivalent to an explicitly time-dependent coordinate transformation: q-^> q = q(t,q ) and t—* t' = t. Then (2.6.1-lb) become f D =Y c Dk (X (dq k /dq k ,)v k f + dq k /dt ) + c D = ... = Y c Dk' v k' + c ' D (=0) (£,&' = 1,...,7Z; D= 1,..., 777 ); (2.6.6) where c Dk ' = Y^ ( dq k /dq k ’)c Dk (covariant vector-like transformation in k), (2.6.6a) c' D = Y, ( dcjk/dt)c Dk + c D (covariant vector-like transformation in t = 77 + 1, with q' n+ \ = t' = t => dt'/dt = 1). (2.6.6b) The above readily show that: (i) if dq k /dt = 0 [i.e., q = q(q')] (= coordinate trans¬ formation', in the same frame of reference), then c' D = c D ; and (ii) we can choose a frame of reference in which c' D = 0; that is, catastaticity/acatastaticity (and statio- narity/nonstationarity) are frame-dependent properties. WWW.EBOOK777.COM §2.6 SYSTEM FORMS OF LINEAR VELOCITY (PFAFFIAN) CONSTRAINTS Holonomicity versus Nonholonomicity The m(< n) constraints (2.6.1) are independent if the m x n constraint matrix (c Dk ) has maximal rank (i.e., m) at each point in the region of definition of the cf s and t. Now, if these constraints are completely integrable = holonomic [i.e., either they are exact: c Dk = dh D /dq k and c D = dh D /dt, where h D = h D (t,q) (=0); or they possess integrating factors, as explained in §2.2], then there exists a set of n “equili¬ brium,” or “adapted (to the constraints)” system coordinates x = (Xti---jXn) i n which these constraints take the simple uncoupled form: Xi = h{t, q) = 0,..., x,n = h m {t, q) = 0; (2.6.7a) Xm +1 = q) + 0,... ,Xn = K(f q) ± 0; (2.6.7b) where, as in §2.4, the n — m functions h m+] (t, q),..., h n (t, q) are arbitrary, except that when (2.6.7a, b) are solved for the n q’s in terms of the (n — m) xi = (Xm+i, ■ ■ •, Xn) and these expressions are inserted back into the m holonomic constraints hi(t,q) = 0,... ,h m (t,q) = 0, they satisfy them identically in the xf s and t. The X/’s are the new positional system coordinates of this 3 N — (h + m) = (3 N — h) +m = n — m = n' (both global and local) DOF : q ->■ q' = (xm+i? ■ • ■ ,Xn) = {q\, ■ ■ ■, q'„’)- (2.6.7c) This process of adaptation to the constraints via new equilibrium coordinates can be repeated if additional holonomic constraints are imposed on the system; and with some nontrivial modifications it carries over to the case of additional nonholonomic constraints (§2.11: essentially, by expressing this adaptation ... idea in the small; i.e., locally, via “equilibrium quasi coordinates”). The importance of this method to AM lies in its ability to uncouple constraints, and thus to simplify significantly the equations of motion (chap. 3). If, on the other hand, the constraints (2.6.1) are noncompletely integrable = non¬ holonomic , then the number of independent Lagrangean coordinates (= number of global DOF) remains n, but the system has n — m = f DOF (in the small, or local case); that is, under the additional m nonholonomic constraints [(2.6.1), (2.6.2)], the n q s remain independent (unlike the holonomic case!), but the n v/dq/Sq 's do not —or, if the differential increments 6q are arbitrary (if, for example, we let q k become q k + Sq k while all the other cf s remain constant ), then they will no longer be virtual ; that is, they will not be compatible with the virtual form of the constraints (2.6.4); and similarly for the v’s, dcf s. [Of course, if m = 0, then the n cf s are independent and their arbitrary increments 6q are virtual; that is, both q’s and bcf s satisfy the existing (initial) h holonomic constraints. For example, in the case of a sphere rolling on, say, a fixed plane: (a) if the plane is smooth (i.e., m = 0), both the arbitrary <?’s and the arbitrary (q + dqf s, are kinematically possible; while (b) if the plane is suffi¬ ciently rough so that the sphere rolls on it (i.e., m f 0, and the additional (rolling) constraints are nonholonomic), only the q’s are still arbitrary (independent), the (q + dqf s are not—or, if they are, the sphere does not roll. For details, see exs. 2.13.4, 2.13.5, 2.13.6.] To find the number of independent Sq’s under the additional m (holonomic or nonholonomic) constraints (2.6.1, 2, 4) we must now turn to the examination of the following. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Introduction to Virtual Displacements under Pfaffian Constraints (Introduction to Quasi Variables) In this case, the particle virtual displacement is still represented by (2.5.12b): Sr = J2 ( d, ’/ dc lk) Sq k = Y2 e k Sq k , (2.6.8) but, due to the virtual constraints (2.6.4), out of the n Sq's only n — m are indepen¬ dent; that is, if, now, all n Sq's vary arbitrarily, the resulting Sr, via (2.6.8), will not be virtual—denoting a differential increment of a system coordinate by Sq does not necessarily make it virtual; it must also be constraint compatible. For example, solving (2.6.4) for the first m Sq's, Sq D = ( Sq { ,..., Sq m ) = Dependent Sq's, (2.6.9a) in terms of the last n — m of them. Sqj = (Sq m+ 1 ,..., 6q„) = Independent Sq's, (2.6.9b) we obtain Sq D = b oi 6c li ( D = 1. ■ • •, m; I = m+ 1,... ,n), (2.6.9) where b DI = b DI {q, t) = known functions of (generally, all) the q s and t. Substituting (2.6.9) into (2.6.8), we obtain, successively, Sr = Y^ e k Sq k = e D 6q D + Y^ e i s <h = ^2 e D (5Z bm bq <) + J2 e < hqf ' hnally either Sr = Sq k , under ^ c Dk Sq k = 0 (6q k , nonarbitrary), (2.6.10a) or 6r = ^2f}jSqj (Sqj, arbitrary), (2.6.10b) where P, = ej + ^2 S>Di e D = dr/dqj + ^ b DI ( dr/dq D ) [see also (2.11.13a ff.)]; (2.6.10c) that is, the most general particle virtual displacement under (2.6.4) can be expressed as a linear and homogeneous combination of the “ narrower" basis {Pj\ I = m + 1,...,«}, whose vectors are, in general [and unlike the e k s —recalling (2.5.4a If.)], nongradient, or nonholonomic : [Ijfdr/dq, ^ dfij/dqr f d[l r /dq, (/,/' = m + 1,...,«); (2.6.11) as can be verified directly by using (2.6.10c) in (2.6.11). The number of independent Sq's, here n — m = f, equals the earlier defined number of local DOFs; and, inversely, we can redefine the number of DOFs in the small, henceforth called simply DOF, as the smallest number of independent parameters qi = Vj/dqj/Sq / needed to determine v/dr/Sr , for all system particles and any admissible, and so on, local motion; that is, the number of DOFs (in the small) = minimum number of independent “local positional”, or motional , parameters. Just as the number of DOFs in the large, F = n (here), is the minimum number of WWW.EBOOK777.COM §2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS independent positional parameters needed to determine the configurations of all system particles in any admissible, and so on, global motion. REMARKS (i) The / bqi can, in turn, be expressed as linear and homogeneous combinations of another set of /-independent motional parameters, say r\p. 6qj = Hir(t,q) Vr (I, I' = m + 1,...,«); in which case (2.6.10b) becomes Sr = Y l } ! (£ H »' J lr) = E (E H "'fir)vr = Y bl ' 7 h' = Y (2.6.10d) (ii) As already mentioned, the importance of these considerations lies in kinetics (chap. 3), where it is shown that the number of independent kinetic equations of motion (= equations not containing forces of constraint) equals the number of inde¬ pendent 6q’s. Problem 2.6.1 Show that due to the m Pfaffian constraints (2.6.1) (expressed in terms of the notation dq k /dt = v k ): Y dDfcbt + c D = 0 CD = 1,, m; k = 1,..., n), (a) or, equivalently, in the (2.6.9)-like form, in the velocities, v D = Y b Di v i + b D (I = m + l,... ,n), (b) the additional holonomic constraint q) = 0 satisfies the following (n — m) + 1 conditions: dcb/dq, + ^ t> DI {d(j)/dq D ) = 0 and dd/dt + ^ b D (d(j)/dq D ) = 0; (c) which, in terms of the notation eft, q D , q,) = <j>[t, q D (t, qf, cji] = <j> 0 {t,qi) = 0, read simply d(/) o /dqi = 0 and df 0 /dt= 0 , (d) respectively (compare with example 2.4.1.). Before embarking into the detailed study of nonholonomic constraints and asso¬ ciated “coordinates” (to embed them), and the most general v/r/r/<5r-representations in terms of n — m arbitrary motional system parameters, of which the previous V/ = q I /dq I /6q I are a special case, let us pause to geometrize our analytical findings; and in the process dispel the incorrect impressions, held by many, that analytical mechanics is, somehow, only numbers (analysis), no pictures — an impression initiated, ironically, by Lagrange himself! 2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS Configuration Spaces As explained in §2.2, before the imposition of any constraints, the configurations of a mechanical system S' are described by the motion of its representative, or figurative, particle P(S) = Pin a (clearly, nonunique) 3A-dimensional Euclidean, or noncurved/ WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS flat, space, E 3N , called unconstrained, or free, configuration space. [Briefly, Euclidean, or noncurved, or flat, means that, in it, the Pythagorean theorem (“distance squared = sum of squares of coordinate differences”) holds globally, that is, between any two space points, no matter how far apart they may be; see, for example, Lur’e (1968, p. 807 ff.), Papastavridis (1999, §2.12.3).] The position vector of P, in terms of its rectangular Cartesian coordinates/com¬ ponents relative to some orthonormal basis of fixed origin O, in there, is [recall (2.4.3 ff.)] «=Ki=£i(0>--->6w = 6*(0]- (2-7.1) However, as detailed in §2.4, upon imposition on 5 of h holonomic constraints and subsequent introduction of n = 3N — h Lagrangean coordinates q=[q i = q\(t),... ,q n = q„(t)}, or simply q = (q l ,... ,q n ), the above assumes the parametric representation £ = §(<,?) = [6 =C\(t;q),...fi 3N = ^ N {t;q)}, (2.7.1a) which, in geometrical terms, means that, as a result of these constraints, P can no longer roam throughout E iN , but is forced to remain on its time-dependent n-dimen- siona! surface defined by (2.7.1a), called reduced, or constrained configuration space of the system; actually the portion of that surface corresponding to the mathematically and physically allowable range of its curvilinear coordinates q. In differential-geo- metric/tensorial terms, that space, described by the surface coordinates q, when equipped with a physically motivated metric, becomes, at every instant t, a generally non-Euclidean (or curved, or nonflat) metric manifold, M n {t) = M n , usually a Riemannian one, embedded in E 3N ; and this explains the importance of Riemannian geometry to theoretical dynamics. [Riemannian manifold means one in which the square of the infinitesimal distance (“line element”) is quadratic, homo¬ geneous, and (usually) positive-definite in the coordinate differentials dq k . In dynamics, the manifold metric is built from the system’s kinetic energy (§3.9). See, for example, Lur’e (1968, pp. 810 ff.), Papastavridis (1999, §2.12, §5.6 ff.)] Schematically, we have Configuration spaces < Unconstrained: Esn (Euclidean manifold) Constrained: M n (Non-Euclidean manifold) [N = number of particles, h = number of holonomic constraints, n = 3N - h] Now, as S moves in any continuous, or finite, way in the ordinary physical (three- dimensional and Euclidean) space, or some portion of it, P moves along a contin¬ uous M„-curve, q = q(t). The relevant analytical requirements on such c/’s (§2.4) are summarized as follows: (i) The correspondence between the q n-tupies and some region of M„ must be one-to- one and continuous (additional holonomic constraints would exclude some parts of that region from the possible configurations). (ii) If As = displacement, in M n , corresponding to the ^-increment Aq, we must have \\m{As / Aq k ) f 0, as Aq k —> 0 (k = \,..., n); or dq k /ds (= “direction cosines” of unit tangent vector to system path in M„) = finite. The q's are then called regular. (See also Langhaar, 1962, p. 16.) WWW.EBOOK777.COM §2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS Event Spaces Instead of the “dynamical” spaces Ein and M n , we may use their (formal and nonrelativistic) “union” with time t = q 0 = q n+ \, symbolically, E 3N+l = £ 3A , x T(ime) and M n+1 = M„ x T(ime). (2.7.2) These latter are called (unconstrained and constrained, respectively): manifolds of configuration and time (or of extended configuration), or “ geometrical ” space-time manifolds, or film spaces', or, simply, event spaces. M„(M„ +1 ) is suitable for the study of scleronomic (rheonomic) systems. (One more such “generalized space,” the phase space of Lagrangean coordinates and momenta, is examined in chap. 8.) Constrained Configuration Spaces and their Tangent Planes The h stationary and holonomic constraints define, in £ 3Ar , a stationary (nonmoving) and rigid (nondeforming) n-dimensional surface M n ; while h nonstationary holonomic constraints define, in E m , a nonstationary (moving) and nonrigid (deforming) n- dimensional surface M„(t). However, these same nonstationary constraints also define, in £ 3A r +1 , a stationary and rigid (n + \)-dimensional surface M n+l ; hence, the relativity of these terms! The equations t = constant define 00 1 privileged surfaces M„(t) in M„ +1 . Thus, the motion of the system can be viewed either as (i) a stationary curve in the geometrical space M n+ p, or (ii) as the motion of the representative system point in the deformable, or “breathing,” dynamical space Further, through each M„-point q(t) there passes a (n — l)-ple infinity of kinematically possible system paths, on each of which the “rate of traverse” dq/dt is arbitrary; and through each M n+ \ -point (q, t ) there passes an 77 -ple infinity of such paths, but these latter, since there is no motion in M n+l , are not traversed. The kinetic paths of a system in M„ and M„ + i are called its trajectories/orbits and world lines, respectively. Additional M„/M n+l differences are given below, in connection with nonholonomic constraints. Next, and as differential geometry teaches, (i) the set of all (n + l)-ples (dq a ) make up the tangent point space (hyperplane) to M n+l at k- T n+ 1 (£); while (ii) the vectors {E a = E a (P); a = 1,... , 77 + 1}, defined by dP = dl; = dq = J 2 E„ dq a : vector of elementary system displacement determined by P(q) and P(q + dq) (each E a being tangent to the coordinate line q a through P) constitute a “natural” basis for the tangent vector space associated with, or corresponding to, T „+1 (£); and similarly for M„. For simplicity, we shall denote both these point and vector spaces by T n+l (P), UP). REMARKS (i) Without a metric, these tangent spaces are affine. After they become equipped with one, they become Euclidean; properly Euclidean if the metric is positive definite, and pseudo-Euclidean if the metric is indefinite. As mentioned earlier, in mechanics the metric is based on the kinetic energy, and, therefore, it is either positive definite or positive semidefinite. (ii) It is shown in differential geometry that the condition that dE a =J 2 (" ')apE/3 be an exact differential [i.e., d/dqry(dE a /dqp) = d/dqp(dE a /dq 1 )\ leads to the requirement that be a Riemannian manifold. For details, see, for example, Papastavridis (1999, p. 135). WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Pfaffian Constraints Let us begin with a system subjected to h holonomic constraints (2.4.2), and, there¬ fore, described by the n = 3N — h holonomic coordinates q. Then, a motion of the system in the physical space E 3 corresponds to a certain curve in M n (trajectory or orbit)/M n+1 (world line ) traced by the figurative system particle P; and, conversely, admissible M„/M n+ \ curves represent some system motion. Now, let us impose on it the additional m Pfaffian constraints: Kinematically admissible form : d'0 D = Cuk dq k + Codt = 0, (2.7.3) Virtual form : S'do = Cok Sq k = 0. (2-7.4) As a result of the above, we have the following geometrical picture: (i) At each admissible M„ + i-point P = ( q,t ), the m constraints (2.7.3) define (or order, or map, or form), the \{n + 1) — m] = [(« — m) + 1]-dimensional “element” T(n+\)-m{P) = T(n-m)+\(P) = T I+ i(P): tangent space (plane) of kinematically admis¬ sible displacements (motions), of the earlier tangent plane T„+1 (P), on which the kinematically admissible displacements of the system, dq, and dt lie. Therefore, at every P, only world lines with velocities v a = dq a /dt on that plane are possible—the system can only move along directions compatible with (2.7.3). (ii) At each such point P, the m constraints (2.7.4) define the (n — m) -dimensional plane T n _ m (P ): tangent space of virtual displacements (motions), or virtual plane, on which the virtual displacements of the system, Sq, lie. Clearly, T n _ m (P ) is the inter¬ section of T(„- m ) + ] (P) with the hyperplane dt—+6t = 0 there; symbolically, T„- m (P) = T [n _ m)+1 \ St=0 =V n . m (P) ( V for virtual). {And a manifold M n /M„ +l whose tangential bundle (i.e., totality of its tangential spaces) is restricted by the m nonholonomic equations (2.7.3) [assuming that (2.7.3), (2.7.4) are nonholonomic] is called nonholonomic manifold M nn _ m /M n+ln _ m . Some authors call the so-restricted bundle, T ln _ m - j+i or T n _ m , nonholonomic space embedded in M„, or M n+i . See also Maiffer (1983-1984), Papastavridis (1999, chap. 6), Prange (1935, pp. 557-560), Schouten (1954, p. 196).} (iii) The given constraint coefficients (c Dk , c D ) define, at P, an (m + 1 (-dimensional kinematically admissible constraint plane ( element ) C m+ \(P) perpendicular to T m + 1 (P) (with orthogonality dehned in terms of the kinetic energy-based metric); while the (c D f) dehne an ;n-dimensional virtual constraint plane (i.e., of the virtual form of the constraints) C,„(P) perpendicular to V n _ m (P). Sometimes, C m (P) is referred to as the orthogonal complement of V n _ m (P) relative to T„(P). The c D / ( can be viewed as the covariant (in the sense of tensor calculus) and holonomic components of the m virtual constraint vectors c D = ( c D k ), which, by (2.7.4), are orthogonal to the virtual displacements bqk'. c D ■ 6q = CDk^qk — 0- Hence since the c D are independent, they constitute a basis (span) for the earlier space C m (P). These two local planes are frequently called the nulI[V n _ m (P)\ and range[C„(P )] spaces of the m x n constraint matrix (c D „). These geometrical results are shown in fig. 2.14 (see also fig. 3.1). Let us consolidate our findings: (i) Under n initial holonomic constraints, a system can go from any admissible initial M„/M n+l - point, P„ to any other final such point, P f , along any chosen (-lying path joining P, and P f . WWW.EBOOK777.COM §2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS Figure 2.14 Virtual displacement ( V n _ m ) and constraint (C m ) hyperplanes in configuration space (see also fig. 3.1). (ii) If the additional m Pfaffian constraints (2.7.3, 4) are holonomic, disguised in kinematical form, the local tangent planes become the earlier local tangent planes to reduced, or “smaller,” configuration/event manifolds M n _ m /M^ n+l \_ m , inside M n /M n+ \. These reduced but finite surfaces contain all possible system motions through a given P, —the system can go from any admissible initial M n _ m /M( n+l y m - point, P h to any other final such point, Pj, along any chosen (M n j M, I+ \)-lying path joining P f and P<\ that is, DOF(local) = DOF (global) = n — m. (iii) On the other hand, if the additional m Pfaffian constraints are nonholonomic, we cannot construct these M n _ m /Mi n+ n_ m . The global configuration/event mani¬ folds of the system are still M n /M n+U but these constraints have created, in there, a certain path-dependence : any (M„/M„ +1 )-point Pr (in the admissible portions of M„/M n+ 1 ) is, again, accessible from any other (M„/M, 1+1 )-point Pj but only along a certain kinematical family, or “ network ,” of tracks that is “narrower” than that of case (i); that is, the transition Pj —> Pf is no longer arbitrary because of direction-of- motion constraints, at every point of those paths. Or, under such constraints, all conhgurations/events are still possible, but not all velocities (and, hence, not all paths)', only certain M„/M n+ \ -curves correspond to physically realizable motions — the system is restricted locally, not globally; that is, n = DOF{global) DOF(local) = n — m. We continue this geometrical interpretation of constrained systems in §2.11. Kinetic Preview, Quasi Coordinates The importance of these considerations, and especially of the concept of virtualness, to contained system mechanics arises from the fact that most of the constraint forces dealt by AM (the so-called “passive,” or contact, ones; i.e., those satisfying the d’Alembert-Lagrange principle, chap. 3) are perpendicular to the virtual displa¬ cement plane V n _ m , and so lie on the virtual constraint plane C m . And this, as detailed in chapter 3, allows us to bring the system equations of motion into their simplest form; that is (i) to their smallest possible, or minimal, number ( n ), and (ii) to a complete decoupling of them into (n — m) purely kinetic equations—that is, WWW.EBOOK777.COM free ©books www.ebook777.c0im 296 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS equations not containing constraint forces — by projecting them onto the local virtual hyperplane, and (m) kinetostatic equations — equations containing constraint forces — by projecting them onto the local constraint hyperplane, which is perpendi¬ cular to the virtual hyperplane there. This is the raison d’etre of virtualness, and the essence of Lagrangean analytical mechanics. In all cases, under given initial/bound¬ ary conditions and forces, the system will follow a unique path (a trajectory, or orbit) determined, or singled out among the problem’s kinematically admissible paths, by solving the full set of its kinetic and kinematic equations. Schematically, our strategic plan is as the following: Minimal number of eqs. of motion (n) Kinetic (n — m) Uncoupling of eqs. of motion: Now, if the m Pfaffian constraints are holonomic, their uncoupling (and that of the corresponding equations of motion) is easily achieved by “adaptation to the constraints,” as explained in §2.4 and §2.6; but, if they are nonholonomic this “adaptation” can be achieved only locally, via “equilibrium” nonholonomic co¬ ordinates, or quasi coordinates. We begin the study of these fundamental kinematical concepts by first examining one of their important features: the possible commutativity/noncommutativity of the virtual and possible operations, S(.. .) and d(...), respectively, when applied to this new breed of “coordinates”; that is, we investigate the relation between d[8(quasi coordinate )] and 8[d (quasi coordinate)}. 2.8 NONCOMMUTATIVITY VERSUS NONHOLONOMICITY; INTRODUCTION TO THE THEOREM OF FROBENIUS Let us recall the admissible (d) and virtual (6) forms of the Pfaffian constraints (2.7.3, 4) (henceforth keeping possible non-exactness accents only when really necessary!): d9 D = ^c Dk dq k + c D dt = 0 and S9 D = ^2c Dk 5q k = 0, (2.8.1) where D = 1,..., m; I = m + 1, ...k (and all other small Latin indices) = 1 Now, 6(.. .)-varying the first of (2.8.1), and d(. ..)-varying the second, and then subtracting them side by side, we find, after some straightforward differentiations and dummy index changes, d(S9 D ) - 8{d9 D ) = ^ ~2c Dk [d(8q k ) - S(dq k )\ +£(£ {dc Dk /dq l - dc Dl /dq k ) dq, +{dc Dk /dt - dc D /dq k ) dt^j 6q k + C{)[d(8t) — 8(dt)\, (2.8.1a) WWW.EBOOK777.COM §2.8 NONCOMMUTATIVITY VERSUS NONHOLONOMICITY or, since the last term is zero [St = 0 =>■ d(St) = 0, and, during 8(.. .) time is kept constant =>■ S(dt) = 0], and with the earlier notations q 0 = q n+] = t => 8q 0 = Sq„ +l = St = 0,c D = c D0 = c Dn+i , and Greek subscripts running from 1 ton + 1 (or from 0 ton): d{S0 D ) - S(dO D ) = 22c D k[d(Sq k ) - 5(dq k )\ ( 2 . 8 . 2 ) + EE ( dc Dk /dq a - dc D Jdq k ) dq a Sq k . A final simplification occurs with the useful notations d(S.. .) — 6(d.. .) = D {...), and C D fja = dc D p/dq a - dc Da /dq, 3 = -C D a0 , (2.8.2a) F d = EE C D ka dq a Sq k : Frobenius’ bilinear, or antisymmetric, covariant of the Pfaffian forms (2.8.1). (2.8.2b) Thus, (2.8.2) transforms to DSo = '22 c ° k + F D . (2.8.2c) Problem 2.8.1 Starting with eqs. (2.5.12a,b): dr =22 e k dq k + e 0 dt, 6r = 22 e k Sq k , and repeating the above process, show that blr = E Dc lk e k- (a) (b) From the above basic kinematical identities, we draw the following conclusions: (i) ifc fl to = o, identically in the q’s and t, and for all values of D, k, a then, since Dq k = d{Sq k ) — S(dq k ) = 0, (2.8.3a) the q k being genuine = holonomic coordinates, it follows that D6 d = d{Sd D ) - S{dO D ) = 0; (2.8.3b) that is, the do are also holonomic coordinates, the ddo/SOo are exact differentials. In this case (2.8.1) may be replaced by m holonomic constraints; which, in turn, may be embedded into the system via n’ = n — m new equilibrium coordinates, as explained in §2.4. (ii) If F d ^ 0, then D0 d ^ 0; or, more generally, we cannot assume that both d(dq k ) = S(dq k ) and d(69 D ) = S(d6 D ) hold; it is either the one or the other. (As detailed in chap. 7, this realization helps one understand the fundamental differences that exist between variational mathematics and variational mechanics. See also pr. 2.12.5.) If we assume (2.8.3a) for all holonomic coordinates, constrained or not, then D6o ^ 0; that is, the do are nonholonomic coordinates; and, as Frobenius’ theorem shows (see below), the constraints (2.8.1) are nonholonomic. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (iii) If, however, F D = 0, since the dq/bq are not independent, it does not neces¬ sarily follow that C D ka = 0. To make further progress — that is, to establish necessary and sufficient holonomicity/nonholonomicity conditions in terms of the constraint co¬ efficients, c Dk and c D . we need nontrivial help from differential equations/dififerential geometry; and this leads us directly to the following fundamental theorem of Frobenius (1877). First, let us formulate it in simple and general mathematical terms, and then we will tailor it to our kinematical context. Theorem of Frobenius The necessary and sufficient condition for the complete (or unrestricted) integrability = holonomicity of the Pfaffian system: X D = Y J ^ DK dx K = Q [D= l,...,m(<F); K,L= 1,...,F], (2.8.4) where X nK = X DK (x ]..... x F ) = X nk (x) = given and well-behaved functions of their arguments, and rank(X DK ) = m(<F); that is, for it to have m independent integrals/ D (x) = C D = constants, is the vanishing of the corresponding m bilinear forms: ( dX DK /dx L - dX DL /dx K )u K v L , (2.8.5) identically (in the x’s) and simultaneously (for all D" s), for any/all solutions u = (u ] ...., u F ), and v = (v 1; ..., v F ) of the m constraints ^dkVk = 0; that is, for any/all rj K —> u K ,v K satisfying Y Xdk u k = 0 and ^ Xdk v k = ^ Xdl v l = 0- (2.8.6) [Also, recall comments following eqs. (2.3.lie).] [If the system (2.8.4) is completely integrable, then, since its finite form depends on the integration constants C D (i.e., ultimately, on the initial values of the x’s), then it is semiholonomic (§2.3).] Adapted to our kinematical problem — that is, with the identifications F —► n + 1, u K —> 6q k , v L —> dq a , x — > t, q, and recalling that q n+l = t satisfies the additional holonomic constraint Sq n+ 1 = 6t = 0 — Frobenius theorem states that: If Fd =d{Sf} D ) -5(d8 D ) = d[y^cpk8q k ) - of ~Fq j = E(E ( dc Dk /dq a - dc Da /dq k ) dq a ) 6q k = EE C D ka dq a Sq k = 0, (2.8.7) for arbitrary dq a = clq k , dq ll+l = dq 0 = dt and 6q k , solutions of the constraints: E c Da dq a = °Dk dq k + c D dt = 0 and Y c Dk % = 0, (2.8.1) then these constraints are holonomic. WWW.EBOOK777.COM §2.8 NONCOMMUTATIVITY VERSUS NONHOLONOMICITY The above show that since our dq' s and Sq's are not independent, the vanishing of the F D ' s does not necessarily lead to C D ka = 0, (2.8.8) as holonomicity conditions. For this to be the case, eqs. (2.8.8) are, clearly, sufficient but not necessary, they would be if the dq's and Scf s were independent; namely, unconstrained. This observation leads to the following implementation of Frobenius’ theorem: we express each of the («) nonindependent dcf s and Scf s as a linear and homoge¬ neous combination of a new set of n — m independent parameters (and dt, for the dq' s), insert these representations in F D = 0, and then, in each of the so resulting m bilinear covariants (in these new parameters), set its n — m coefficients equal to zero. We shall see in §2.12 that, in the general case, this approach leads to a direct and usable form of Frobenius’ theorem, due to Hamel. But before proceeding in that direction, we need to examine in sufficient detail the necessary tools: nonholonomic coordinates, or quasi coordinates (§2.9), and the associated transitivity relations (§2.10). REFERENCES ON PFAFFIAN SYSTEMS AND FROBENIUS' THEOREM (for proofs, and so on, in decreasing order of readability for nonmathematicians): Klein (1926(a), pp. 207-214): introductory, quite insightful. De la Vallee Poussin (1912, vol. 2, chap. 7): most readable classical exposition. Guldberg (1927, pp. 573-576) and Pascal (1927, pp. 579-588): outstanding handbook summaries. Forsyth (1890/1959, especially chaps. 2 and 11): detailed classical treatment. Lovelock and Rund (1975/1989, chap. 5): excellent balance between classical and modern approaches. Cartan (1922, chaps. 4-10): the foundation of modern treatments. Weber [1900(a), (b)]: older encyclopedic treatise (a) and article (b, pp. 317-319). Heil and Kitzka (1984, pp. 264-295): relatively readable modern summary. Chetaev (1987/1989, pp. 319-326): happens to be in English (not particularly enlighten¬ ing). Frobenius (1877, pp. 267-287; also, in his Collected Works, pp. 249-334): the original exposition; not for beginners. Hartman (1964, chap. 6): quite advanced; for ordinary differential equations specialists. Outside of Lovelock et al., we are unaware of any contemporary readable exposition of these topics in English; i.e., without Cartanian exterior forms, and so on. Example 2.8.1 Necessary and Sufficient Condition(s) for the Holonomicity of the Single Pfqffian Constraint (2.3.1) via Frobenius’ Theorem: dO = a(x,y,z)dx + b(x,y,z)dy + c(x,y,z ) dz = adx + bdy + cdz = 0, (a) or, since it is catastatic, 59 = a 5x + b Sy + c 5z = 0. (b) 299 WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS By ^-varying (b), and ^-varying (a), and then subtracting side by side, we find, after some straightforward differentiations: d{59) — 8{d9) = a[d{Sx ) — h(dx)] + b[d{8y) — 8{dy)} + c[d{8z) — 8{dz)\ + [{da 8x — 8a dx) + {db 8y — 8b dy) + {dc 8z — 8c dz )] = {da/dy — db/dx)(dy 8x — 8ydx) + {da/dz — dc/dx){dz8x — Szdx) + {db/dz — dc / dy){dz 8y — 8zdy) [substituting into this, dz = {—a/c) dx + {—b/c) dy and 8z = (— a/c) 8x + (— b/c) 8y, solutions of the constraints (a, b), respec¬ tively; since here n — m = 3 — 1 = 2 = number of independent differ¬ entials (for each form of the constraint); we could, just as well, substitute dx = • • • dy + • • • dz and 8x = ■■ ■ 8y H- 8z, or dy = ■ ■ ■ and 8y = ■ ■ ■ ] = ■ ■ ■ = ( da/dy — db/dx)(dy 8x - 8ydx) + {da/dz — dc/dx){—b/c){dy8x — 8ydx) + {db/dz — dc/dy){—a/c){dx8y — 8xdy) = \{da/dy - db/dx) + {b/c){dc/dx - da/dz) + {a/c){db/dz — dc / dy)]{dy 8x — 8ydx). Setting d{89) — 8{d0) = 0, and since now the bilinear terms dySx and Sydx are independent, we recover the earlier holonomicity condition (2.3.6). Vectorial Considerations Equations (a)/(b), in terms of the vector notation h={a,b,c ), dr = {dx,dy,dz), and 8r = {8x, 8y, 8z), (d) state that h ■ dr = 0 and h ■ 8r = 0; (e) that is. It is perpendicular to the plane defined by the two (generally independent) directions dr and 8r, through r = (. x,y,z ). On the other hand, the second of (c) states that d{59) — S{d9) = curl h ■ {dr x Sr) = 0, (f) that is, curl h is perpendicular to the normal to that plane; and, hence, excluding the trivial case dr x 8r = 0, curl h lies on that plane. Accordingly, b and curl h are perpendicular to each other: h curl It = 0. i.e., (2.3.8a). (g) Example 2.8.2 The Two Independent and Catastatic Pfaffian Constraints: d9 = a{x,y,z)dx + b{x,y,z) dy + c{x,y,z)dz = adx + b dy + cdz = 0, (a) dO = A{x,y, z) dx + B{x , y, z) dy + C{x,y,z) dz = Adx + Bdy + Cdz = 0, (b) WWW.EBOOK777.COM §2.9 QUASI COORDINATES, AND THEIR CALCULUS when taken together (i.e., n = 3, m = 2) will always make up a holonomic system; even if each one of them separately (i.e., n = 3, m = 1) may be nonholonomic! Solving (a) and (b) for any two of the dx, dy, dz in terms of the third, say dx and dy in terms of dz , we obtain dx = e(x, y, z) dz and and, similarly, since (a) and (b) are catastatic, dy =f(x,y,z)dz; (c) 6x = e(x,y,z ) & Therefore, we find, successively, d{50) - 5{d6) = and Sy=f{x,y,z)8z. (d) = ••• = (■■ ■ )(dy6x — 6ydx ) + (• • -)(dz8x — Sz dx) + (• • -)(dz8y — 8zdy ) = [using (c) and (d)] = ■■■ = (•• -)(dz8z — 8zdz) = (• • -)0 = 0; (e) and, similarly, d{5&) - 6(dO) = •■■ = (•••)(*&- fe<fe) = (•■■)() = 0; Q.E.D. (f) Proceeding in a similar fashion, we can show that: a system of n — 1 (or n) indepen¬ dent Pfaffian equations, in n variables [like (2.8.1) with m = n — 1 or n] is always holonomic. This theorem illustrates the interesting kinematical fact that additional constraints may turn an originally (individually) nonholonomic constraint into a holonomic one (as part of a system of constraints); see also §2.12. 2.9 QUASI COORDINATES, AND THEIR CALCULUS Let us, again, consider a holonomic system S described by the hitherto minimal, or independent, n Lagrangean coordinates q = (q x ,... ,q n ), and hence having kinema¬ tically admissible I possible system displacements {dq, dt) = (, dq i,..., dq n ; dt). Now, at a generic admissible point of S”s configuration or event space {q, t ), we can describe these local displacements via a new set of general differential positional and time parameters ( d9,dt) = ( d0 l: ..., d0 n ; d6 n+ \ = dd 0 ), defined by the n + 1 linear, homo¬ geneous, and invertible transformations: d0 k = ^ a k i dq/ + a k dt , dd n+x = dd 0 = dq n+l = dq Q = dt , (2-9.1) rcmk(a k {) = n => Det[a k [) f 0, (k, l = 1,... ,n), (2.9.1a) where the coefficients a k / and a k = a k n+x = a k0 are given functions of the q's and t (and as well-behaved as needed; say, continuous and once piecewise continuously differentiable, in some region of interest of their variables). Inverting (2.9.1), we obtain dq t = A[ k d0 k + Aj dt , dq„+i = d6 n+l = dd 0 = dt, (2.9.2) rank(Aj k ) = n => Det(Aj k ) f 0, (k, l = 1,..., n), (2.9.2a) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS where the “inverted coefficients” A /k and A\ = A/„ +1 = A m become known functions of the q’s and t, and are also well-behaved. Clearly, since the transformations (2.9.1) and (2.9.2) are mutually inverse, their coefficients must satisfy certain consistency, or compatibility, conditions; so that, given the as, one can determine the A’s and vice versa. Indeed, substituting dq t from (2.9.2) into (2.9.1), and d6 k from (2.9.1) into (2.9.2), and with 8 k/ = Kronecker delta (= 1 or 0, according as k = l, or k ^ /), we obtain the inverseness relations: ^ ^ ^ ^ A r i G-kr &kl i ^ ^ &kr A /• — ^ ^ A r Clk r Clki (2.9.3a) ^ ^ Alr&rk = ^ ^ ClykA-lr &kh ^ ^ Alr&r = y ^ Clr-A-iy = Aj. (2.9.3b) Further, with the help of the unifying notations a k = a kn+] and A t = A /n+l , the definitions a n+xk = 6 n+]k (= 0) and A ll+l l = 8 n+1 j (= 0), and recalling that Greek subscripts have been agreed to run from 1 to n + 1, the transformation coefficient matrices in (2.9.1) and (2.9.2) take the (« + 1) x (n+ 1) “Spatio-Temporal” forms: a = /«ii • &\n ( a k , a k a n i ' ^nn a n,r 7+1 V o 1 \ o ■ ■■ 0 1 ) »s 0 1 (2.9.4a) ( A \ A = V 0 M n 0 l l,n+l 1 n,n +1 A k i A k 0 1 As Ay 0 1 = {A h \ (2.9.4b) 1 / Then (2.9.1, 2) assume the simpler (homogeneous) forms: = ^ dqp = ^2 Afa d6 1 , (2.9.5) while the consistency relations (2.9.3a) read simply a A 1 or ^ ' aps A$y — ^ ~ A^apg (2.9.6a) and from this we obtain the “spatio-temporally partitioned” matrix multiplications: a S 3y 0 1 As Ay 0 l a sAs a s A x ay 0 1 1 0 that is, a§A§ — 1 and a^Ay -I- a-j- — 0; (2.9.7a) WWW.EBOOK777.COM §2.9 QUASI COORDINATES, AND THEIR CALCULUS and, similarly, the consistency relations (2.9.3b) read (aA) T = A T a T = 1 or from which that is, As^s — 1 ^ ' Ay; Ufip ^ @S(3 ^7 S ^ A-S^S , A x as + a x and A x a s + a x — 0. Let us recapitulate the notations used here: (2.9.6b) (2.9.7b) (i) Matrices are shown in roman and bold; vectors in italic and bold; (ii) (.. .) T = transpose of square matrix (...); (iii) a s ,A s = (nx n) spatial , or catastatic, submatrices of a and A, respectively; and a x ,A T = (n x 1) temporal, or acatastatic, submatrices of a and A, respectively; (iv) 1 = square unit, or identity, matrix (of appropriate dimensions); (v) 0 = zero matrix (column or row vector of appropriate dimension); and (vi) Here, commas in subscripts — for example, a^+i ,A/ = A /n+1 — are used only to sepa¬ rate the spatial from the temporary of these subscripts, for better visualization; that is, no partial differentiations are implied, unless explicitly specified to that effect. Thus, for example, for (3 —> k and 7 —> / eqs. (2.9.6a) yield ^ ^ ttkr^rt T i 6/tf > 'y ' ctkfAfi 6 £/, i.e., first of (2.9.3a), for {3 —* n + 1 and 7 —»/ they yield 'y \ tin+\,r A r t T a t 24-1 5 1 An+iy 4+i,o i.e., 0 + 0 0, while for (3 —* k and 7 —> n + 1 they yield ^ j/ ^r,n+ 1 T dk,n+\ 4 ,«+1 b, i.e., second of (2.9.3a), and similarly with (2.9.6b). Specializations, Remarks (i) If (a/a) is an orthogonal matrix — that is, if a ki = A i k and Det(a /f/ ) = ±1, (2.9.8a) then the spatial parts of (2.9.3a, 3b) are replaced, respectively, by Y a kr a lr = Y a h a kr = 4 / and Y a ri a rk = Y c ‘ rk a >' 1 = Skl ’ ( 2 - 9 - 8b ) and, similarly, for the full (« + 1) x (« + 1) a and A matrices. (ii) As shown in chap. 3, and foreshadowed below, it is the spatial/catastatic submatrices as and A s that enter the equations of motion; not the temporary/acata¬ static submatrices a x and A x . The latter, however, enter the rate of energy, or power, equations (§3.9). In what follows, we shall have the opportunity to use all these, WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS mutually equivalent and complementary notations; primarily the indidal and secon¬ darily the matrix ones. All have relative advantages/drawbacks, depending on the task at hand. Velocities and Virtual Displacements Just as we defined new general kinematically admissible/possible system displace¬ ments via (2.9.1, 2), etc., we next define the following: (i) The corresponding general system velocities (dO —> oj dt); u k = Y a ki{dqi/dt) +a k = Y a kidi + a k = Y ak ' v ‘ + ° k ’ u) n+ \ = u) 0 = dq n+i /dt = dt/dt = 1 (isochrony ), (2.9.9) or, compactly, U P~Y a h( d q y /dt) = Y a h v v (2.9.9a) and, inversely, d q,/<lt = qi = vi = Y A ikU k + A h dq n+l /dt = bj n+l =dt/dt=\, (2.9.10) or, compactly, dq~ 1 ldt = q 1 = v 1 = Y^A 1 Qtjjp\ (2.9.10a) and (ii) The corresponding general system virtual displacements ( d6 —> 60, d0 n+ 1 50, 1+]l =6t = 0): 69 k = Y a k' 6 dt, S0 n+X = Sq n+ i = St = 0; (2.9.11) and, inversely, Sqi = Y A ’ k S9k ’ Sq "+ l ~ Sq ° ~ S9 «+ l ~ S9 ° = St=0. (2.9.12) If the dO and dt describe an actual motion, then d0 k = to k dt. But it would be incor¬ rect to set 60 k = ui k 6t, because of the ever present (better, ever assumed) virtual time constraint 5t = 0; whereas, in general, 56 k f 0! Next, let us examine the integrability of these Pfalfian forms (not constraints!) (2.9.1, 11), of our hitherto n DOF system. Bilinear Covariants, Integrability, Quasi Coordinates Indeed, proceeding as in §2.8, and assuming that d(6q k ) = 6(dq k ), constraints or not, we find that the Frobenius bilinear covariants of (2.9.1, 11), d(60 k ) — 6(dO k ), equal d{60 k ) - 5{dO k ) = EE {da kl /dq s - da ks /dq,) dq s 6q, + Y ( da k i/dt - da k /dq,) dt6q, = Y dipkiSqi■ WWW.EBOOK777.COM (2.9.13) §2.9 QUASI COORDINATES, AND THEIR CALCULUS Now, with the help of these expressions, and since the dq s and 8q's are (as yet) unconstrained (i.e., m = 0), like the q s, we can enunciate the following “obvious” theorems, in increasing order of specificity: • The necessary and sufficient conditions for th e particular Pfaffian/onn (not constraint!) d0 k = ^ a k / dqi + a k dt, or in virtual form 89 k = ^ a kl 8q h (2.9.14) to be an exact differential — that is, for the hitherto shorthand symbols d9 k and 89 k to be the genuine (first and total) differentials of a bona fide function 9 k = 9 k (q,t) (—» holonomic coordinate) —is that its bilinear covariant (2.9.13), vanish. • The necessary and sufficient condition for a Pfaffian form (2.9.14) to be the exact differential of 9 k [since its n 8q' s in (2.9.13) are arbitrary] is that its associated n Pfaffian forms di>kl = ^2 ( 9a «/ dc ls ~ da ks /dqi) dq s + ( da kl /dt - da k / dq,) dt, (2.9.15) all vanish; that is, dipu = 0 for all / (= 1,...,«). • The necessary and sufficient condition for a Pfaffian form (2.9.14) to be the exact differential of 9 k [since the n dq’s and dt in (2.9.15) are arbitrary] is that the following n(n+ l)/2 integrability (or exactness) conditions hold: da k //dq s — da ks /dqi = 0 and da k i/dt — da k /dqi = 0, (2.9.16) identically in the q's and t, and for all values of /, s (= 1,...,«). [For additional insights and details, see, for example, Hagihara (1970, pp. 42—46), Whittaker (1937, p. 296 ff.).] Hence, if (2.9.16) hold for all k= 1,..., n, the n 9's are just another minimal set of Lagrangean coordinates, like the q's: 6 k = 9 k (q u ..., q,,-, t); and u k = d6 k /dt are the corresponding holonomic Lagrangean (generalized) velocities. But if, and this is the case of interest to AM, da k //dq s — da ks /dqi ^ 0 or da k i/dt — da k /dcji ^ 0, (2.9.17) even for one I, s, then ut k is not a total time derivative, and d0 k is not a genuine differential of a holonomic coordinate 9 k , only the dO k /56 k /u) k are defined through (2.9.1, 9, 11). Such undefined quantities, 9 k , are called pseudo- or quasi coordinates [a term, most likely, due to Whittaker (1904)], or nonholonomic coordinates’, and the ut k , depicted by some authors by symbolic (.. ^'-derivatives, like uj k = d'6 k /dt = 6 k = 9 k , etc., (2.9.18) instead of d9 k /dt, are called quasi velocities. From now on we shall assume, with no loss in generality, that all (2.9.17) hold, and therefore all 9 k are quasi coordinates. {We notice that, the isochrony choice d9 n+l = dq n+x = dt, resulting in [recalling (2.9.4a, b)] — d n -\-\,k 0; ^ 7 t+l,H+l ^n+l,n+\ 1) (2.9.19a) and ^ n-\-l,k — ^n+\,k ^5 (2.9.19b) guarantees that 9 n+ \ remains holonomic.} WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS REMARKS (i) Let us consider, for simplicity, the catastatic version of (2.9.9), u k = '^2a kl (t 1 q)v, = u k (t,q,q= v). (2.9.20) If the q’s are known/specified functions of time t, then integrating (2.9.20) between an initial instant t 0 and a current one t we obtain the line integral Qk{t) - e k {t 0 ) = Uk[r,q(T), v(r)] dr = a ki[r,q{r)]vi(T)\ dr, (2.9.20a) similar to the work integral of general mechanics and thermodynamics. Since this is the integral of an inexact differential, as calculus/vector field theory teach, 0 k [t) depends on both t (current configuration) and the particular path of integration! history followed from t 0 to t; it is point- and path-dependent. If it was a genuine global coordinate, it would be point-dependent , but path-independent. 6 k (t ; tf) is a functional of the particular curves/motion {g(r), t a < t < t}\ (ii) As will be explained in §2.11, the satisfaction of (2.9.16) guarantees that 9 k , as defined by (2.9.14), is a holonomic coordinate; and that property will hold even if, at a later stage, the dq k /6q k /v k become holonomically and/or nonholonomically con¬ strained. One the other hand, if 9 k is originally [i.e., as defined by (2.9.14)] nonholo- nomic, then upon imposition on the latter’s right side of a sufficient number of additional holonomic and/or nonholonomic constraints, later, it will become holo¬ nomic; but that would be a different Pfaffian form. In sum: once a holonomic coordinate, always a holonomic coordinate; but once a nonholonomic coordinate, not always a nonholonomic coordinate. (iii) The local transformations a v = Ap v Ep Ep = ^2 a v pa v , where [recalling discussion in (§2.7)] each Eg is tangent to the coordinate line dqp at (q, t) and all together they constitute a holonomic basis for the local tangent space T„+1 > and the coefficients satisfy the earlier (2.9.3a, 3b), define a new but, generally nonholonomic basis there: that is, ff a I; dO r .\ nonexact differential =>■ daJdOp f dap/dO n [where the nonholonomic gradients, d/dOp, are defined in (2.9.27 ff.)]. And, in view of 55 ^ E 0 = 51 v 0 E 0 = 55 ^(55 a ^ a v) = 55 (55 a v3 v a) a v = 55 ^ = 55 w /3 a /3> the up are simply the nonholonomic components of the system velocity vector, while the vp are its holonomic components. [The system basis {a, ( } plays a key role in the geometrical interpretation of Pfaffian constraints (§2.11.19a ff.)] (iv) The precise term for the 6 k s is “nonholonomic (local) system coordinates,” and for the u k s “nonholonomic system velocity parameters,” or “( contravariant ) nonholonomic components of the system velocity ” (Schouten, 1954/1989, pp. 194— 197). We shall call them collectively quasi variables’, and their symbolic calculus, if proper precautions are taken, is quite useful. As Synge puts it: “In the theory of quasi-coordinates in dynamics, however, it pays to live dangerously and to use the notation d6 k [in our notation]. Otherwise we shall be depriving ourselves of a very neat formal expression of the equations of motion” (1936, p. 29). On the symbolic calculus of quasi variables, see also Johnsen (1939). WWW.EBOOK777.COM §2.9 QUASI COORDINATES, AND THEIR CALCULUS Example 2.9.1 The most common example of quasi velocities in mechanics is the components of the (inertial) angular velocity of a rigid body moving, with no loss in generality here, about a fixed point O, resolved along either space-fixed (inertial) axes O XYZ, w Y ,a)y,w z ; or body-fixed (moving) axes O xyz.uj x ,uj y ,Lj z . If cp —> 9 —> ip are the three Eulerian angles 3 —> 1 —> 3, then for body-axes, and with the convenient notations s(. ..) = sin(...) and c(.. .) = cos(...), and d<j>/dt = cd6/dt = u>g, dip/dt = we have (§ 1 . 12 ) uj x = ( sipsh+ (cip)uig + (0 )uty, (a) iOy = (cips8)uj <l> + {-sip)u e + (0)w* (b) LO z — (c9)u)0 + (0)ujg + (c) that is, with k = x — > 1, y —>2, z —>3; and / = </>—» 1, 6 — >2, ip — >3, the nonvan¬ ishing elements of ( a ki ) are a n = sips9, a l2 = cip-, a 2 \ = cipsO, a 22 = —sip-, a 2 \ = cO, u 33 = 1. (d) Clearly, not all (2.9.16) hold identically here. For example, da i2 /dq 2 f da\ 2 /dq 2 : d(cip)/dip f 9(0)/ 88: —sipfO; (e) except in the special ( nonidentical!) case: ip = 0, 2-7T. If we set cu x = d9 x /dt, then = UJ X \8(t),iP(t) ; cu^(t), oj s {t)\ dr + 9 X ( initial): path dependent ; (f) that is, 9 X is an (angular) quasi coordinate, and co x an (angular) quasi velocity; and similarly for 9 y ,9 z -,lu v ,uj z -, that is, they are quasi variables (if the (p, 9, ip are uncon¬ strained). However, if we impose additional constraints, for example, (p = constant , 9 = constant (fixed-axis rotation), then (a-c) reduce to u x = 0 , uj v = 0 , u z = dip/dt =>■ u x ,L 0 y ,u z -. holonomic velocities-, (g) 9 z (t) — 9 Z ( t 0 : initial) [dip(r)/dr\ dr = ip(t) — ip{t 0 \ initial ): path independent .(h) Problem 2.9.1 Let the reader verify that the corresponding space-fixed components 9 X ,9 Y ,9 Z and lo x ,u> y ,lu z (such that u x =d0 x /dt, etc.) are also, respectively, quasi coordinates and quasi velocities; and that under additional constraints they too may become holonomic variables. Particle Kinematics in Quasi Variables Due to the 9 <-> q transformation relations (2.9.1, 2, 9, 10, 11, 12), the (inertial) velocity, acceleration, kinematically admissible/possible displacement, and virtual displacement, of a typical system particle, obtained in §2.5 in holonomic variables, assume the following quasi-variable representations, respectively: WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (i) Velocity: (ii) Acceleration: a = ■■■ = Y^ s k (du> k /dt) + terms not containing (dw/dt)' s; = Y^ e k d> k + terms not containing cb’s, (2.9.22) (iii) Kinematically possible/admissible displacement: dr = E C/t (E Afci dO/ A.fc dt'j €q dt — • • • — ^ ^ dQfc £q dt 5 (2.9.23) (iv) Virtual displacement: Sr = E e k (E A kl 69 ‘) = ■ ■ ■ = E Ek S0k\ (2.9.24) where the fundamental, generally nongradient, n + 1 particle and system vectors s k and s„ +1 = £q, corresponding to the 0’s, nonholonomic counterparts of the gradient vectors e k and e n+l = e 0 , which correspond to the q’s [recalling (2.5.4-4b)], and defined naturally by (2.9.21-24), obey the following basic (covariant vector-like) transformation equations: = E ( dv l/ djJ k)ei = Y A lk e h (2.9.25a) e k = Y, {9u)i/9v k )^i = E a,kEl [comparing with (2.9.11, 12)]; (2.9.25b) (2.9.26a) e 0 = Y^ a k^k + «o = — E AkCk e ° [recalling (2.9.3a, 3b)]. (2.9.26b) Clearly, if the e vectors are linearly independent (and \a k/ \, \A k i\ 0), so are the e vectors; even if the q’s and/or dq/dt= v’s get constrained later. And, as with the (^-representation (2.5.12b), so with (2.9.24): the size of the 60’s is unimportant; it is the e’s that matter, because they are the ones entering the equations of motion (chap. 3)! Quasi Chain Rule, Symbolic Notations The above, especially (2.9.24), suggest the adoption of the following very useful symbolic quasi-chain rule for quasi variables: dr/dO k = Y (9r/dq,)(dvi/duj k ) = Y ( dr / dt li) [d{dqi)/d{dO k )} = Y ( dr / d di) [d(6q,)/d(60 k )\, WWW.EBOOK777.COM §2.9 QUASI COORDINATES, AND THEIR CALCULUS or, simply, dr/d9 k = Y A^dv/dq ,): i.e, (2.9.25a); (2.9.27) and, inversely, dr/dq k = Y (dr/d9,)(duj,/dv k ) = Y a ik(dr/d9,): i.e., (2.9.25b). (2.9.28) Similarly, for a general well-behaved function / =f(q, t), and recalling (2.9.12), we obtain, successively, (i) for its virtual variation 5f : Sf = Y fa = E ( 5 //%t) (E ^k/duti) 60 ,) = Y (Qf/Wi) (2.9.29) that is, df/dO, = Y C Of/ 8 *' ) (fok/&>t) = Y Mdf/dq k ) , (2.9.30a) and, inversely, df/dq k = Y (9f/de,) (a j,/dv k ) = Y Mdf/dd,) ; (2.9.30b) and (ii) for its total differential df [recalling (2.9.2)]: df = Y ( d f/ d( la) dc l3 = Y ( d f/ dc lk ) dq k + ( df/dt) dt = Y ( d f/ d dk) (E Akt + Ak dt ) + ( d f / dt ) dt = E(E A kiW/dq k )) do, + (£ A k {df/dq k ) + 3//0f) dt = Y W/Wi) do, + (d//d0 o ) dt, (2.9.31) where we have introduced the additional symbolic notation [recalling that 0 0 = 0„+l — U n+ 1 = 1 ] ; d.../dd n+l = Y( d ---/ d( la)(dvp/duj n+l ) = Y ( d ■ ■ ■ /dq k )(dv k /duj n+ i) + {d... /dt)(dv n+l /du) n+l ) = Y A k(d.../dq k ) + d.../df, (2.9.32) instead of the formal extension of (2.9.30a) for 9 : —* 9 n+l . This latter we shall denote by d.../d(t)\ d.../d(t) = Y( d ■ ■ ■ / d< lk){dv k /duj n+l ) = Y A k{d.../dq k ); (2.9.32a) so that (2.9.32) assumes the final symbolic form <9... /d6„ + i = d... /89 0 = d... /d(t) + d... /dt. WWW.EBOOK777.COM (2.9.32b) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Inversely, we have d.../dt= ^ ~2(du a /dv n+x )(d.../d6 a ) = d... /d9„ +l + Y a k (d... /d6 k ), (2.9.32c) and, comparing this with (2.9.30a, b), we readily conclude that 8... /d{i) = Y A k (d... /dq k ) =-22 a k (d... /d0 k ). (2.9.32d) Such (by no means uniform) symbolic notations are useful in energy rate/power theorems in nonholonomic variables (§3.9). Some Fundamental Kinematical Identities From the above (2.9.21 ff.), we readily obtain the following fundamental kinematical identities, nonholonomic counterparts of (2.5.7-10), and like them, holding indepen¬ dently of any subsequent holonomic and/or nonholonomic constraints. (i) dr/d9 k = dr/d9 k = dr/dO k = dr/duj k = ■ ■ ■ = e k , or dr/dO k = dv/doj k = da/duj k = ■■■ =e k ; (2.9.33) (ii) dq k /d9, = dq k /d9, = dq k /d0, = dq k /du, = ■■■ = A kh or dq k /ddi = dv k /du>i = dw k /du>i = ■••= A k p (where dv k /dt = w k ) (2.9.34) (iii) dOk/dqi = duj k /dv, = duj k /dw, = ■■■ = a kh (2.9.35) with formal extensions for 9 n+ \ = q n+i = t. The du k /dt = d 1 9 k /dt 2 are called (not quite correctly) quasi accelerations', while the 9/lo/lj/ ... are referred to, collectively, as (system) quasi variables. (iv) We have, successively, d(dr/d9 k )/dt = d(dv/dto k )/dt = ds k /dt = ^ik e /) / dt = Y^ [( dA lk /dt)ei + A Ik (dei/dt )] = Y ( dAi k /dt)ei + Y A lk (dv/dq,) [recalling (2.5.7,10)]. (2.9.36) But by partial (^/-differentiation of v(q, v, t) = v[q, v(q, u >, t),t\ = v*(q, u, t ), we hnd dv*/dq, = dv/dq, + Y (dv/dv,.)(dv r /dq,) = dv/dq, + Y (9v r /dq,)e n WWW.EBOOK777.COM §2.9 QUASI COORDINATES, AND THEIR CALCULUS and so E A ik{dv/dcj!) = Y A ik{dv*/dq,) ~YY A ik( dv r/dqi)e r = dv*/d6 k — YY a ik(dv,./dqi)e r . Therefore, returning to (2.9.36), we see that it yields d&k/dt — dv*/dQfc — ^ ^ ^dA.jfcjdt — E A r k{dvi/dq r )\ e, ± 0, (2.9.36a) that is, unlike the H coordinate ease (2.5.10), E k *(v*) = (i dv*/dd k y - dv*/dd k = d/dt(dv*/du k ) - dv*/dO k ^ 0. (2.9.37) This nonintegrability relation is a first proof that, in general, the s k basis vectors are nongradient , or nonholonomic. More comprehensible and useful forms of E k *(v*) are presented in the next section. [Some authors call the s k vectors “partial velocities.” However, in view of (2.9.33), they could just as well have been called partied positions , or partial accelerations , or even partial jerks (recall that da/dt =j= jerk vector , and therefore dj/du k = s k ), etc. Perhaps a better term would be nonholonomic mixed basis vectors (i.e., nonholonomic counterpart of Heun’s Begleitvektoren ).] A Useful Nonholonomic-Variable Notation Frequently, for extra clarity, we will be using the following “(.. .)*-notation”: With its help: (i) Equations (2.9.21), (2.9.22), (2.9.33) become, respectively, v(t,4,v) = Y e k(t,q)v k + e 0 (t,q ) = Y E k{Uq)Uk + So(t,q) = v*(t,q,uj); (2.9.39) a(t, q , v, w) = Y^ e k(U q) w k + no other q = tv-terms = Y^ £ /c(h q)u k + no other w-terms = a*(t, q , u, u); (2.9.40) dr/dO k = dv*/duj k = da*/du> k = ■■■ = s k ; (2.9.41) (ii) The quasi-chain rule (2.9.30a) and its inverse (2.9.30b) generalize , respectively, to dj*/de, = Y (dT/dq k )(dv k /dui) = Y Mdf*/dq k ), (2.9.42a) and dj*/dq k = Y (df*/ae,)(di 0 ,/dvk) = Y «ft(0/W); WWW.EBOOK777.COM (2.9.42b) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS also, we easily obtain the related chain rules [recall derivation of (2.9.37), and (2.9.42a)] df*/dq k = df/dq k + Y {df/dvi){dv,/dq k ), (2.9.43a) df*/dOi = Y A ki(df*/dq k ) = Y A ki df/dq k + Y ( df/dv,){dv r /dq k ) (2.9.43b) (iii) The following genuine (i.e., ordinary calculus) chain rule , and its inverse, hold: dfo/cko, = Y (df/dv k )(dv k /a^,) = Y A ki(9f/dv k ), (2.9.44a) df/dv k = Y (dr/^di^/dvk) = Y a, k (df*/dio,). (2.9.44b) We notice the difference between (2.9.42a, b) and (2.9.44a, b); the former are non- vectoricil transformations, just symbolic definitions; while (for those familiar with tensors) the latter are genuine covariant vector transformations. (iv) Finally, invoking (2.9.11, 12, 42a, b), it is not hard to see that Y (dff/m k ) 80 k = Y (5/7 dq k ) 6q k . (2.9.45) Some Closing Comments on Quasi Coordinates The theory of nonholonomic coordinates and constraints is, by now, a well estab¬ lished and well understood part of differential geometry/tensor calculus and mechanics, with many fertile applications in those areas. Its long and successful history has been created by several famous mathematicians, such as (chronologi¬ cally): Gibbs, Volterra, Poincare, Fleun, Hamel, Synge, Schouten, Struik, Vranceanu, Vagner, Kron, Kondo, Dobronravov et al. And yet, we encounter con¬ temporary statements of appalling ignorance and confusion, like the following from an advanced “Tract in Natural Philosophy” devoted to rigid kinematics: “It appears that the reason why many a book on classical dynamics follows Kirchhoff’s approach is a lack of understanding of the kinematics of rigid bodies. Thus, one finds extensive discussions on ill-defined — or, sometimes, totally undefined — esoteric quantities such as quasi-coordinates and virtual displacements ,” (Angeles, 1988, p. 2, the italics are that author’s). 2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS So far, our system remains a holonomic (H) one, with n = 3 N — h DOF. Now, to be able to either (i) embed to it additional Pfaffian (possibly nonholonomic) constraints in their “simplest possible form” or, even if no such additional constraints are imposed, (ii) express the equations of the problem in quasi variables, or (iii) do both, we need to represent the right sides of the Frobenius bilinear covariants of the Pfaffian forms of its quasi variables, (...) dqSq [recall (2.9.13)], in terms of the latter’s differentials, (...) d6 69. [By simplest possible form we mean uncoupled from each other; and, as WWW.EBOOK777.COM §2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS detailed in chap. 3, this leads to the simplest possible form of the equations of motion.] To this end, we insert expressions (2.9.2 and 12) into the right side of (2.9.13), and group the terms appropriately. The result is the following generalized transitivity, or transpositional, equations (Hamel’s Ubergangs-, or Transitivitdts- gleichungen ): d{S0 k ) — S(dO k ) = y a k i[d(Sq/) — 6(dq /)] + EE l k a(i d0 p S0 a = E a kll d ( S( ll) ~ S (dqt)\ + E E E d6 0 S6 r [ since b °n+ 1 = St = 0] = E a k‘ i d + E E E de s 66 r + Y 7 k r dt SO,., ( 2 . 10 . 1 ) (again, we recall that all Latin (Greek) indices run from 1 to n (1 to n + 1)) where the so-defined 7 ’s, known as Hamel ( three-index ) coefficients, are explicitly given (and sometimes also defined) by E = EE {da kp /dq e - da ke /dq p )Ap r A es = EE 0 da kb /dq c - da kc /dq b )A hr A cs + E ( da kb/dt - da k ,n+\ldqb)A hr A n+Xs + E ( da k, n +i/dq c - da kc /dt)A n+X/ A cs + (da Kn+l /dt - da kin+l /dt)A n+hr A n+l s , (2.10.1a) or, due to A n+Xr = 6 n+Xr = 0 which leads to the vanishing of the last three groups/ sums of terms, finally, l k rs = E E ( da kb/dq c - da kc /dq h )A br A cs -, ( 2 . 10 . 2 ) and E,n +1 = -7^+1/ - 7*1 = E E ( da kp/ dc le - da k Jdqp)Ap r A En+l , (2.10.3) or, with a kn+ \ = a k , A kn+] = A k , and since A„ +l: „ +1 = 6 n+ M+1 = 1, finally, e-EE (da kb /dq c da kc /dq b )A br A c -\- ^ i()a kb /<)t da k /dq b )A br . (2.10.4) [The 7 ’s are a significant generalization of coefficients introduced by Ricci (mid- 1890s), Volterra (1898), Boltzmann (1902) et al.; and, hence, they are also referred as “Ricci/Boltzmann/Hamel (rotation) coefficients.” See, for example, Papastavridis (1999, chaps. 3, 6 ).] It is not hard to show [with the help of (2.9.3a, b)] that (2.10.1) inverts to d{Sq k ) - S(dq k ) = Y A kl [[d{S0,) - S/d0,)\ -EE 7 ^ d0 s 60 r — 7 ^ dt <5d r |. (2.10.5) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS For an actual motion, dividing both sides of (2.10.1) and (2.10.5) with dt [which does not interact with 8 (...)], we obtain, respectively, the (system) velocity transitivity equation and its inverse: (S6 k ) - 8uj k = 55 «jy[(<5<7/) - 8v,\ + 55 7 » Mr + 55 7*r (2-10.6) (%)■ -«v t = 55 ^,{[w- m-55 E -55 7 U^}- ( 2 - 10 - 7 ) Properties of the Hamel Coefficients (i) Clearly, these coefficients depend, through the transformation coefficients ag £ and Ag e , on the particular v <-> u> choice; that is, they do not depend on any particular system motion. (ii) The 7 * contain the contributions of (a) the acatastatic terms a k and A k , and of (b) the explicit time-dependence of the homogeneous coefficients of the v <+> u> trans¬ formation. Hence, for scleronomic such transformations (i.e., a k = 0 => A k = 0, and da k i/dt = 0 => dA kl /dt = 0) they vanish; but for catastcitic ones, in general, they do not. In fact then, as (2.10.4) shows, they reduce to 7 * = 55 ( da k b/di)A br (for catastatic Pfaffian transformations). (2.10.4a) (iii) The matrix y k = ( 7 fc „) is, obviously, antisymmetric', that is, 7 fe „ = — 7 k sr => y k rr : diagonal elements = 0 (k,r,s = 1 ,...also n+ 1 ). ( 2 . 10 . 8 ) To stress this antisymmetry in r and s, we chose to raise /c; that is, we wrote j k rs instead of y rks , or 7 krs , or y rsk , and so on. [Nothing tensorial is implied here, although this happens to be the tensorially correct index positioning; see, for example, Papastavridis (1999, chaps. 3, 6 ).] Hence, each matrix y k can have at most n(n — l )/2 nonzero (nondiagonal) elements. (iv) From the above, we readily conclude that Yl- f-1 A \ Yl~\~ 1 A H~\~ 1 I - 1 A H~\~ 1 A T e/t H =" 7 kl 0? T k,n -fl 7 n+ljc 0? T n+l,n+l 0 [k, l = 1,... e,P = 1— ,n-n + 1], (2.10.9) and from this (recalling that a n+i k = 8 n+ik = 0 ), that {86 „ + 1 )' — 8u> n+l = d/dt(8q n+l ) — 8(dq„ +l /dt ) = d/dt(8t) — 8(dt/dt ) = 55 E 7" +1 » w, 89,. + 55 7” +1 r , n+1 89,. = 0 + 0 = 0 , ( 2 . 10 . 10 ) which, essentially, states that d{89 n+l ) - 8(d9„ +l ) = d(8t) - 8(dt) = rf(0) - 6{dt) = 0-0 = 0, (2.10.10a) as it should, and also shows that ( 2 . 10 . 1 ) and ( 2 . 10 . 2 ) also hold for k = n + 1 . (v) In concrete problems, the analytical calculation of the nonvanishing 7 ’s is best done, as Hamel et al. have pointed out, not by applying (2.10.1a^4), which WWW.EBOOK777.COM §2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS are admittedly laborious and error prone, but by reading them off as coefficients of the bilinear covariant (2.10.1,6), in terms of the general subindices: °, * = 1 ■•..,«;«+ 1 : d(89 t ) — 8{d6f) = -h (7*0*) dQ+ 80 o -\ -. (2.10.11) Also, this task is independent of any particular assumptions about d(6q) — 8(dq); and, hence, assuming that for cdl holonomic coordinates d(8q k ) = 8{dq k ), or equiva¬ lently ( 8q k )' = 8(q k ) = 8v k (Hamel viewpoint — see also pr. 2.12.5), even if they (or their differentials) become constrained later , we may safely and conveniently calculate all the nonvanishing 7’s from the simplified, and henceforth definitive, transitivity equation: d(69 k ) - 8{d9 k ) = EE j k rs de s 8e r +^^ k r dtse r . (2.10.12) Finally, dividing the above with dt, and so on, we obtain its velocity form: W-4 = EE 60 r + 7*r 60,-, (2.10.13) a representation useful in Hamilton’s time integral “principle” in quasi variables (chap. 7). Unfortunately, the transitivity equations, and their relations with the 7’s, are nowhere to be found in the English language literature (with the exception of Neimark and Fufaev, 1967 and 1972, p. 126. IT.); although the definition of the 7’s via (2.10.1a, 2) appears in a number of places. This unnatural situation produces an incomplete understanding of these basic quantities. REMARK (A PREVIEW) As will become clear in chapter 3, the expression for the system kinetic energy (and the Appellian “acceleration energy”) are simpler in terms of quasi variables, such as the w’s and dui/df s, than in terms of holonomic variables like the v’s and dv/dt’s. And this leads to formally simpler equations of motion in the former variables than in the latter; for example, the well-known Eulerian rotational rigid-body equations (§1.17) are simpler in terms of such quasi variables than, say, in terms of Eulerian angles and their (...)'-derivatives. But there is a catch: to obtain such simpler-look- ing Lagrange-type equations of motion — that is, equations based on the kinetic energy and its various gradients — we must calculate the corresponding 7’s; some¬ thing that, even with utilization of (2.10.11-13) and other practice-based short cuts, requires some labor and skill. On the positive side, however, the 7’s supply an important “amount” of understanding into the kinematical structure of the parti¬ cular problem; and Appellian-type equations in quasi variables may not contain the 7’s, but they have other calculational difficulties. In sum, there is no painless way to obtain simple-looking equations of motion in quasi variables. Problem 2.10.1 Verify that the transitivity equations, say (2.10.12), can be rewritten as d(60 k ) - 8{dO k ) = EE 7 \ s (d9 s S9 r — 89 s d9 r ) + 7 k r dt89 n (a) where X] X] ’ means that the summation extends over r and s only once ; say, for s<r. [We point out the following interesting geometrical interpretation of (a): each of its WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS double summation terms is proportional to a 2 x 2 determinant, which, in turn, equals the area of the infinitesimal parallelogram with sides two vectors on the local “9 S 9, .-plane,” at its origin ( q , t) in configuration/event space, of respective rectangular Cartesian components (dQ s , dd r ) and (66 s , 69 r ) there; with the factor of proportionality being y k rs . That parallelogram is the projection of the generalized parallelogram with sides d6 = (d0 { ,..., d6 n ) and 69 = (69 1 ,..., 69„), at (q, t ), on the “6 s 6 r -plane” (see, e.g., Boltzmann, 1904, pp. 104-107; Webster, 1912, pp. 84-87, 381-383; also Papastavridis, 1999, §3.14).] Other Expressions for the 7's By ^-differentiating (2.9.3a) and then rearranging so as to go from the (da/dq )'s to the (dA/dq)’s, we obtain E ( da kb/dq c )A br = ~Y a kh(dA hr /dq c ), (2.10.14a) E ( da kc/dq h )A cs = ~Y a kc(dA cs /dq b )-, (2.10.14b) then, substituting the above into (2.10.2), and renaming some dummy indices, we obtain the equivalent 7-expression: 7 k rs = YY a kb[Acr(dA hs /dq c ) - A cs (dA br /dq c )]. (2.10.15) For j —> h + 1, the above yields an alternative to the (2.10.3), (2.10.4) expression for k _ k 7 r,«+1 = 7r- Problem 2.10.2 Show that yet another 7-expression is A = EE (A hr A cs A cr A hs ) (dei kh /dq c ). (a) and similarly for 7* v , + i = 7* (see also Stiickler, 1955; Lobas, 1986, pp. 34-36). Some Transformation Properties of the 7's (i) With the help of the following useful notation: a be = da k b/dq c - da kc /dq b = -a k cb , (2.10.16a) a k b,n+ 1 = a \ = da kh /dt - da k /dq b (2.10.16b) [recalling (2.9.16); also similar notation in (2.8.2a)], the 7-definitions (2.10.2) (2.10.4) are rewritten, respectively, as 7 \s = E E ak bcA br A c „ r y l \. = Y, E ak bcAb r A c + E ak bA br ■ (2.10.17a, b) With the help of the inverseness conditions (2.9.3a, 3b) and a number of dummy index changes, it is not too hard to show that (2.10.17a,b) invert, respectively, to ^ he ^ \ ^ 7 rs®rb®sc ; ^ h ^ ^ ^ ^ 7 rs®rb®s 7 ^ ^ 7 r@rb‘ WWW.EBOOK777.COM (2.10.18a,b) §2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS The above transformation equations show that if the a\ c and a\ vanish [recall con¬ ditions (2.9.16)], so do the ~/ k rs and 7 *; and vice versa; that is, the vanishing of 7 * ... constitutes the necessary and sufficient condition for d0 k /69 k to be an exact differential, and hence, for 9 k to be a holonomic coordinate. If the dq/8q/v are unconstrained, as is the case so far (i.e., m = 0 ), this new set of exactness conditions in terms of the 7 ’s does not offer any advantages over (2.9.16); the a k kc and a k k are easier to calculate than y k rs and 7 *. As shown in the next section, the real value of the 7 ’s, in questions of holonomicity, appears whenever the dq/6q/v are constrained (m f 0 ). REMARK For those familiar with tensors, the transformation equations (2.10.17a, b) show that the 7 * and a k transform as covariant tensors in their two subscripts; that is, both are components of the same geometrical entity, the a' s, its holonomic components in the local “coordinates” dq/8q, and the 7 ’s, its nonholonomic components in the local “coordinates” d9/89, at (q, t). In precise tensor notation, using, for example, accented ( unaccented) indices for nonholonomic ( holonomic ) components, summation convention over pairs of diagonal indices of the same kind (i.e., both holonomic, or both cs k 7 « - nonholonomic), and the k' 7 rV notational changes: A hr —> A f —> Af, a k rs —> 7 *„ (= holonomic components), (= nonholonomic components), the transformation equations and FF be with k ' ->■ a be (2.10.17a) read k' = AfA k' ■'7 rsi (2.10.17c) and similarly for (2.10.17b) (2.10.18b). Such elaborate notation is a must in advanced differential-geometric investigations of nonholonomic systems. Fortunately, it will not be needed here. (ii) The invariant definition of the 7 ’s via the transitivity equations (2.10.1) and ( 2 . 10 . 12 ) readily shows that, contrary to what one might conclude by casually inspecting their derivative definition via (2.10.2-4), these nontensorial coefficients, known in tensor calculus as geometrical objects of nonholonomicity (or anholonomicity), are independent of the original holonomic coordinates q, and thus express geometric properties of the local/differential basis dO/89/ui. In particular, it follows that if the 7 ’s do (not) vanish, when based on some ( q,t ) frame of reference, they will (not) vanish in any other frame ( q',t ), obtainable from the original frame by an admissible transformation. (iii) However, under a local transformation d9 k d9 k ', that is, at the same (q, t)- point, the 7 ’s, do change, in the earlier mentioned nontensorial fashion. [(a) For further details on tensorial nonholonomic dynamics see, for example, Dobronravov (1948, 1970, 1976), Kil’chevskii (1972, 1977), MaiBer (1981, 1982, 1983-1984, 1991(b), 1997), Papastavridis (1999), Schouten (1954), Synge (1936), Vranceanu (1936); and references cited there, (b) For transitivity equation-based proofs of these statements, see, for (ii): ex. 2 . 12 . 2 , and for (iii): ex. 2 . 10 . 1 ; and for a derivative definition-based proof, see, for example, Golab (1974, pp. 140-141).] Noncommutativity of Mixed Partial Quasi Derivatives Below we show that the second mixed partial symbolic quasi derivatives of an arbitrary well-behaved function / =f(q,t,...), in general, do not commute: d/d9 k (6f/d9,) f d/d9,(df/d9 k ). (2.10.19) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Invoking the basic quasi-derivative definition (2.9.30a, b), we obtain, successively, d 2 f/de k do, = d/de k {df/dei) = J^a rk {d/dq r {^2 A sl {df/dq s ))} = EE {8 2 f/dq r dq s ) + A rk {dA sl /dq r )(df/dq s )] = EE A rk A sl {d 2 f/dq r dq s ) + a bsA rk {dA sl /dq r )) (df/d6 b ), and, analogously (with k —> / and / —> k in the above). 5 2 //90 ,&) k = d/d0i{df/d0 k ) =■■■ = EE A r iA sk (d 2 f/dq r dq s ) + a b s A,i{dA sk /dq r )) (8f/89 h ), and therefore subtracting these two side by side, and recalling the 7 -dehnition (2.10.15), we obtain the following alternative transitivity /noncommutativity relation: d 2 f/d9 k 89 l -d 2 f/d9 l d9 k = d/do k {df/dOi) - d/d9,(df/do k ) = E{ EE“4T, (dA sl /dq r ) - ^ r/ (5^/%)] | ( df/d6 h ) = E^E9//^ a ); ( 2 . 10 . 20 ) which expresses noncommutativity in terms of (9... /50)-derivatives, rather than (d.../6.. ^-differentials, as ( 2 . 10 . 1 ) and ( 2 . 10 . 12 ) do. REMARK In the theory of continuous (or Lie) groups, it is customary to write X k f for our df/89 k , (2.9.30a); that is, d.../dd k = X k - •• = E {d.../dq,){dv,/duj k ) = A lk {8... /dq,). (2.10.21) The differential operators X k are called the generators of that group. In this notation, equation ( 2 . 10 . 20 ) is rewritten as [*it, X,]f = £ 7 b k ,(X h f), (2.10.22) where [X k ,X,] = X k X, — X,X k = Y^ 1 h k i(X b ): commutator of group. For further details, see texts on Lie groups, and so on; also Hamel (1904(a), (b)), Hagihara (1970), McCauley (1997). Problem 2.10.3 Extend (2.10.20) to the case where one or both of 9 k , 9, are the (0„+i)th “coordinate”, that is, 9 — > t. Problem 2.10.4 The choice / —> q r in (2.10.20), and then use of (2.9.34), yields the symbolic identity d 2 q r /d9 k 89, - d 1 q,./d9 l 89 k = ^ 7 b kl {dq r /89 b ) = ^ A rb f b kl . (a) WWW.EBOOK777.COM §2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS Solving (a) for the 7’s, derive the following alternative symbolic expression/definition for 7: 7 \i = Y a br(d 1 <lr/dQk dOi ~ d 2 q,./de l d6 k ) . (b) HINT Multiply (a) with a sr and sum over r, and so on. Nonintegrability Conditions for a Nonholonomic Basis Since (2.10.20) holds for an arbitrary /, let us apply it for f —> r = r(t,q). In this case, df/d6 h — > dv/d6 h = z h , and thus we obtain the basic nonintegrability conditions for the nonholonomic basis {s k ; k = 1 d&,/dd k - dz k /d0 1 = Y £h, (2.10.23) or, compactly, [e^, £/] = J h ki E h — commutator of basis {e^}. (2.10.23a) In differential geometry, such bases are called nonholonomic, or noncoordinate, or nongradient; that is, they are not parts of a global coordinate system; like the {e k = dr/dq k } for which, clearly [recalling (2.5.4a)], de,/dq k - de k /dq, = [e k , e,] = 0. (2.10.23b) In sum: the vanishing of the 7’s is the necessary and sufficient condition for the corresponding basis to be holonomic; or gradient, or coordinate. We leave it to the reader to show that (2.10.23) also hold for k, l = n + 1; that is, e^t. A Fundamental Kinematical Identity Here, with the help of (2.10.23), we will complete the derivation of the basic identity (2.9.37). Indeed, since z k = dv/du k = dv*/du> k = z k (t, q), and [recalling (2.9.21)] v = v*(t, q,uS) = e k Lo k + £„ +1 = E k UJ k + £ (h we obtain, successively, (i) d / dt(dv / du> k ) = d/dt(dv*/dui k ) = dz k /dt = Y ( d£ k/dqi)vi + ds k /dt [recalling the inverse quasi chain rule (2.9.30b)] = Y (5Z a ri(9 £ k/90r)) v/ + de k /dt [recalling (2.9.9)] = Y ( ds k /d6 r )(u) r - a r ) + ds k /dt = E (wk - E ( ds k /d9 r )a r + ds k /dt. (2.10.24a) (ii) dv/dQ k = dv*/d0 k = Y^ (' dz r /d6 k )uj r + dz 0 /dd k . WWW.EBOOK777.COM (2.10.24b) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Therefore, subtracting the above side by side, and recalling (2.9.32a, d), we obtain de k /dt - dv/dd k = Y (■ de k /dO r - ds r /d9 k )uj r + (ds k /dt - ds 0 /dd k ) ~53 {ds k /d9,.)a r = Y ( d£ kl de r ~ ds r /d9 k )u r + (de k /d9 0 - 53 A s(9e k /dq s fj - (ds 0 /d9 k - Y ( ds k /d9 r )a = E ( ds k/ dS p ~ 9sp/d9 k )u 0 -zMz a rs (ds k /d9 r )^ - Y a r (ds k /d9 r ) {for the first sum we use (2.10.23), with / —> k, k —► (3, b —> r [recalling (2.10.9)]; and by the second of (2.9.3a) the last two sums add up to zero} = E (E (2.10.24c) and so, finally, E k *(v) = E k *(v*)\ Hamel vector of nonholonomic deviation of a particle = d/dt(dv/du> k ) - dv/d9 k = d/dt(dv* / dui k ) — dv*/d9 k = ds k /dt — dv/d9 k = EE 7V £ '- W /3 — EE YikUfir +53 7 n+i,* w «+i 8 r [swapping k and /] = - E E 7 « ujiEr ~ E tY,- = - E (E 7« w /+ Yk )+ = ~Yh T k *r\ (2.10.25) where h' k = 53 Yu + + Yk = E 7 C3 +9 • Two-index Hamel symbols. (2.10.25a) This fundamental kinematical identity, in its various equivalent forms, like the tran¬ sitivity equations ( 2 . 10 . 1 , etc.), shows clearly the difference between holonomic and nonholonomic coordinates (not constraints): for the former, E k (v) = 0; while for the latter, E k *(v) = E k *(v*) f 0. It is indispensable in the derivation of equations of motion in quasi variables (§3.3). Problem 2.10.5 Transitivity Relations for System Velocities. (i) Show that for the general nonstationary transformation (with <37 = v/) W>k = 53 a kl V l + a k v l = E A ' kUJ k + A h ( a ) the following transitivity identities hold: Ei{u k ) = d/dt(duj k /dv,) - duj k /dqi = Y (E E/ 3 +?)+■/ = E (E T k rs U s + 7 A () a H EE 53 hk r a rl- ( b ) WWW.EBOOK777.COM §2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS (ii) Then show that, in the stationary case, (b) specializes to E,(uj k ) = d/dt(du k /dv,) - duj k /dq, = EEAow; (c) that is, the first line of (b) with ft —> s. (iii) Show that, as a result of the above, the transitivity equations (2.10.13), become (69 k y - Sco k = EE 7 « W, 60, + E 7 k r Mr - E l,k >■ 66 >- = EE E l {aJ k )A lr 66 r = EE (dvi/du> r )Ei(uj k )S9 r , (d) where the (9vi/duj r )Ei(uj k ) can be viewed as the nonlinear generalization of the h k r (§5.2). Problem 2.10.6 By direct (//^-differentiations of Sr = E k-M k and dr = ^ e k d0 k , respectively (assume stationary systems, for algebraic simplicity but no loss in generality), and then use of de k = d (E Aik e ?j = E e t + Ai k def), (a) and dei = E (de,/dq r ) dq r = EE (■ de t /dq r )A rs d9 s , dA/ k = E ( dAik/dq ,.) dq r = EE {8A lk /dq r )A rs d9 s , (b) and similarly for hr/, = <5( ^4 /a^/) = ..., and then recalling the 7-definitions, obtain the following basic particle/vectorial transitivity equation : <*(&■) - W = E { [<W - <W] + E E 7« d °r Ms } £ A3 ( C ) or, dividing by dt, its equivalent velocity form (Sr)' -Sv = E {[(%)' - Su) k\ + EE 7 k rsU r S9sj s k . (d) Replacing in the above r with ft = 11, extends it to the nonstationary / rheonomic case. [Note that (c) and (d) are independent of any (/(<5</) — <5(d<7) assumptions. Therefore, since d(Sr) - S(dr) = E [^(^/) - S(dqi)]ei = EE [d(&y,) - <5(^/)]a«£ k , (e) if we assume d(Sq/) — 6(dq t ) = 0 (Hamel viewpoint), then d(<5r) — 6(dr) = 0, and this leads us back to the transitivity equations (2.10.12) and (2.10.13).] Example 2.10.1 Local Transformation Properties of the Hamel Coefficients. Let us find how the 7’s transform under the admissible (and, for simplicity, but with no loss in generality) stationary quasi-variable transformation 9 —> 8': d9 k ' = E a k'k d9 k <s> d9 k = E A kk ' dd k >, WWW.EBOOK777.COM (a) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS where a k ' k = o-k'k( c j), A kk' = A k k'{q), and all Latin indices run from 1 to n. We find, successively, d(60 k/ ) - 8(dO k i) = d(^2 <*k'kMk) ~ $(^2 a k'k d °k) ^ ^ [ dci^'k 80k 4 - cik'k d(80fc) ficL^'k dO^ Q-k'k 8(^d0k)\ = E{(E {da k , k /dq p ) dq^j 89 k + a k , k d{86 k ) ~ (E ( da k'k/dq P ) H,) de k - a k ’ k 8{dO k ) | [recalling that dq p = ^ A pr d9 r , etc.] = E a k , k [d(S0 k ) - 8(d9 k )} + EE E [( da k'k/dq p )A pr d6 r 86 k - (i da k . k /dq p )A pr dO k 89,] = E a k'k (E E l\c d9 c 89^j + EEE \{da k ' k /dq p )A pr - (, da k , r /dq p )A pk ] d9,.89 k = E E E a k'kl k hc A cc' dO c ) (E A bb , 89 h ^j + E E E [( da k'k/dqp)A pr -(da k , r /dq p )A pk \ (E A rr , </0 r ')(E =EEEEE ( [ a k'k A rr' A U'l k lr ) ^r' + EEEE C da k ' k /d9 r - da k , r /d9 k )A rr ,A kl , d9 r , 89,, ; (b) and since, by definition, </(<%,) - «$(<%,) = EE 1 k ' l , r ,d6 r ,89 l ,, (c) we conclude that l k l'r' = E E E a k'k A U'Arr'l k lr + E E ( da k'k/dOr ~ dd k , r /89 k )A k ,, A,,.,. (d) In tensor calculus language, the transformation equation (d) shows that the j k , r do not constitute a tensor, if 7 k lr = 0 (i.e., if the 9 k are holonomic coordinates), it does not necessarily follow that 7 k ,, r , = 0; and that is why these quantities are called, instead, components of a geometrical object. However, if the second group of terms (double sum) in (d), which looks (symbolically) like a Hamel coefficient between the d9 k and (19 k ,, vanishes, the 7^,. transform tensorially. In such a case, we call the d9 k and d9 k , relatively holonomic ; that happens, for example, if the coefficients a k , k are constant. For futher details, and the relation of the 7’s to the Christoffell symbols (§3.10) and the Ricci rotation coefficients, both of which are also geometrical objects, see, for example (alphabetically): Papastavridis (1999), Schouten (1954), Synge (1936), Vranceanu (1936); also, for an alternative derivation of (d), see Golab (1974, pp. 141-142), Lynn (1963, pp. 201-203). a kl' WWW.EBOOK777.COM §2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES We have developed all the necessary analytical tools of Lagrangean kinematics. In the following sections, we will show how to apply them to the handling of additional Pfaffian (possibly nonholonomic) constraints. For quick comparison, when working with other references, we present below the following, admittedly incomplete, but hopefully helpful, list of common 7 -notations in the literature: (i) Our notation (also in Papastavridis, 1999): 7 a b c = y h ac (sometimes, for extra clarity, a subscript dot is added between a and c, directly below b). (ii) Authors whose notation coincides with ours: Dobronravov (1948, 1970, 1976), Golomb and Marx (1961), Gutowski (1971), KiFchevskii (1972, 1977), Koiller (1992): la \. (iii) Authors whose notation differs from ours: Butenin (1971), Fischer and Stephan (1972), Neimark and Fufaev (1967/1972), Whittaker (1937; but his a kt is our <%): 7 abc ; Corben and Stehle (1960): j acb ; Nordheim (1927): y cba ', Rose (1938): j hac ', Pasler (1968): — 7 bac ; Djukic (1976), Funk (1962), Lur’e (1961/1968), Mei (1985), Prange (1935): 'y c h a ; Kilmister (1964, 1967): 7 “/; MaiBer (1981): A c b a ; Desloge (1982): a abc ; Stiickler (1955): /3 abc ; Heun (1906): (3 ach ; Winkelmann and Grammel (1927): (3 cab ; Morgenstern and Szabo (1961): f3 bac ; Hamel (1904(a), (b)): /3 acb ; Hamel (1949): (3 b a ' c ; Schaefer (1951): 0 c b a ; Vranceanu (1936): w b c \ Wang (1979): K A B C ; Schouten (1954): 2Q c h a ; Levi-Civita and Amaldi (1927): rj b \ ca . 2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES, AND THEIR GEOMETRICAL INTERPRETATION Let us, now, assume that our hitherto holonomic n (= 3N — h )-DOF system is sub¬ jected to the additional m independent Pfaffian constraints [recalling (2.7.3 and 2.7.4)]: Kinematically admissible!possible form: X! °Dk dq k + c D dt = 0 , ( 2 . 11 . 1 a) Virtual form: c ° k Sqk = °’ ( 2 . 11 . 1 b) Velocity form (with dq k /dt = v k ): c Dk v k + CD = 0 ; ( 2 . 11 . 1 c) where D = 1,..., m (< n ), k = 1and the constraint independence is expressed by the algebraic requirement rank(c Dk ) = m. Since additional holonomic constraints (in any form) can always be embedded, or built in, with a new set of fewer cf s, we can, with no loss of generality, assume that all constraints ( 2 . 11 . 1 ) are nonholo¬ nomic. Now, and in what constitutes a direct and natural extension of the method of holonomic equilibrium coordinates (§2.4) to the embedding Pfaffian constraints, we introduce the following equilibrium quasi variables (Hamel’s choice): WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Kinematically admissible!possible form: dO D = a Dk dc lk + aD dt (= 0) , (2.11.2a) dd, = Y^ a Ik dq k + a, dt {f 0), (2.11.2b) d() n+] = d0 o = dq n+x = dq 0 = dt (^ 0); (2.11.2c) Virtual form: S6 d = ^2 a Dk Sq k (=0), (2.11.2d) M I = '52a lk 6q k (^0), (2.11.2e) © x III + III + (2.11.2f) Velocity form: OtD = ^ a Dk v k + a D (— 0); (2.11.2g) aJi = a lk v k +aj (^0), (2.11.2h) O II III O > III + Ill o 3 III + K 3 (2.11.2i) where (here and throughout the rest of the book): D = 1 ,...,m (< n) = Dependent , / = m+ 1 = Independent [additional dependent ( independent ) indices will be denoted by D'. D", .. . ( l',I ", ...)]; and the coefficients a k i, a k are chosen as follows: (i) a Dk = c Dk and a D = c D [i.e., 0 D = Xd, recall (2.6.2—4; 2.8.1)], (2.11.3) (ii) The a Ik and a r for the dq/6q/v in are arbitrary , except that when eqs. (2.11.2) are solved (inverted) terms of the independent d0/86/u>, respectively; that is, Kinematically admissible!possible form: dq k = ^ A k t d6[ + A/ dt (yf 0), (2.11.4a) © 2b o ^3 III + ^3 III o ^3 III + (2.11.4b) Virtual form: 8q k = Y2 A ki Mt 0) , (2.11.4c) © II III o <3> III + <3? III o '-O III + <-0 (2.11.4d) Velocity form: v k = ^2 AkjUJ I Al > (2.11.4e) o' dk II ill o 3 + 3 = III O > III + (2.11.4f) and then these results are substituted back into (2.11.1a-c) and (2.11.3), they satisfy them identically. Other choices of 0’s and a’s are, of course, possible (see special forms/choices, below), but Hamel’s choice (2.11.2) is the simplest and most natural, because then our Pfaffian constraints assume the simple and uncoupled form: WWW.EBOOK777.COM §2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES Kinematically admissible!possible form: Virtual form: Velocity form: dO j) — 0, (2.11.5a) b II (2.11.5b) lo d — 0; (2.11.5c) and, as a result (already described in §2.7 and detailed in ch. 3), the equations of motion decouple into n — m kinetic equations (no constraint forces) and m kineto- static equations (constraint forces). Constrained Particle Kinematics In view of the constraints (2.11.5), the particle kinematical quantities (2.9.23-26) reduce to the following: Kinematically admissible!possible displacement: dr = ^ £[ d9j + e„ +1 dt = ^ £/ d9j + £ 0 dt; (2.11.6a) Virtual displacement: 6r = £ £ i 66 1 ; (2.11.6b) Velocity: v = ^ £ I U] + £„ + i = ^ £/ U>1 + £ 0 ; (2.11.6c) Acceleration: a = ^ £/ 0/ + terms not containing w. (2.11.6d) Special Forms/Choices of Quasi Variables 1. Once we have chosen the equilibrium quasi variables dd/69/ui, we can move to any other such set dO' / SO' /a/, defined via linear (invertible) transformations of the following type: dO k ' = £ cik'k dOk + &k' dt — ^ ^ cifcti dOj cifc' dt 0), (2.11.73.) d0 {n+l y = dO n+l = dq n+l =dt 0); (2.11.7b) and, inversely \{a k ' k ), {A kk i)\ nonsingular matrices], d9 k = ^ A kk f d6 k ' + A k dt —> d0 D = 0 and d9 k f 0; and similarly for 69 k ', u> k >. WWW.EBOOK777.COM (2.11.7c) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 2. If the Pfaffian nonholonomic constraints are given in the quasi-variable forms: X a D'k db k + a D ' dt = 0, or X a D'k S9 k = 0, or X c ‘ D ' k Uk + a °' = °> (2.11.8a) then, proceeding a la Hamel again, we may introduce new quasi variables by dO ^ ~ a pi/- d0 k a d' dt = 0, dO j' = ^ ) ci,> k dO k -T- a,> dt / 0; (2.11.8b) or fiQp' = X a D'k = 0; 80 1 ' = X a I'k 7^ Oj (2.11.8c) or UJD 1 = X a D'k u} k + = 0, W/' = X a l'k^>k + a V 7^ 0> (2.11.8d) where, again, the coefficients ap k , a // are arbitrary; but when (2.11.8b-d) are solved for the dO/80/ui in terms of the dO'/80'/to', and the results are substituted back into (2.11.8a), they satisfy them identically (see also their specialization in item 4, below). 3. Frequently, the Pfaffian constraints (2.11.1) are given, or can be easily brought to, the special form [recalling (2.6.9-11), and, using the notation dq k /dt = v k ]\ dq D = X b oi dc li + b D d t, or 8q D = X b Di or v D = X b Di^i + b D, (2.11.9) where the coefficients b DI , b D are known functions of q and t\ that is, the first m (or dependent) dqa/8q[)/v[) are expressed in terms of the last n — m ( independent ) dq,/8q,/v /. [In terms of the elements of the original mxn constraint matrix ( c Dk ) = (%), we, clearly, have ( b DI ) = — (ct DD 'fi l (a DI ), and so on. See also pr. 2.11.2.] Now, the transformations (2.11.9) can be viewed as the following special choice of d0/80/cu: cl0 D = dq D — X b Di dc li ~ b D dt = 0, dO, = dq, / 0, d0„ +l = dq n+l = dt / 0; (2.11.10a) 80 D = 8q n - X b DI 8qj = 0, 80, = 8q, / 0, 80 n+x = 8q n+l =8t= 0; (2.11.10b) u D = Vd ~ X b Di y i - b D = 0, ujj = v/ / 0, ui n+] = v„ +1 = dt/dt =1/0. (2.11.10c) The above invert easily to dq D = d6 d T ^ ^ b di d6j T dt = ^ ^ ^/)/ dOj T bp dt : WWW.EBOOK777.COM §2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES dq I = dOj, dq n+l = dd n+l = dt; (2.11.11a) Sq D = 69 d + "22 6>di 6qi = ^ b DI 6q ,, 6qi=60j, 6q„ + i = 69 n+ i = 6t = 0; (2.11.11b) v d = + 22 b DI uJi + b D = 22 b DI u}j + b D , v/ = w / , v„ +1 = v 0 = w„ +1 = w 0 = dt/dt= 1. (2.11.11c) Comparing (2.11.10, 11) with (2.11.2, 4) we readily conclude that, in this case, the (mutually inverse) transformation matrices a and A [recalling (2.9.4a ff.)] have the following special forms: a = (i 0 1 -bn+1 N 0 A = 1 \ b 0 1 b n+ l^ 0 v 0 0 1 J v 0 0 1 J that is, ( 2 . 11 . 12 ) (2.11.12a) (2.11.12b) where b = ( b DI ), b n+1 = {b D ^ n+x = b D ); and, of course, satisfy the consistency rela¬ tions (2.9.3a, b). For a slight generalization of the choice (2.11.10c), see pr. 2.11.2. Particle Kinematics In this case, the particle kinematical quantities [recalling (2.5.2 ff.) and (2.11.6a If.), and that e n+ \ = e 0 = dr/dt] specialize to dr = 22 e k dq k + e n+l dt = 22 e n dq D + 22 e, dq, + e n+l dt = 22 e ° (5Z ^ DI ^ c li + b D dt'j + "22 e i d< h + e n+x dt = *22 Pi d< h +Pn+i dt = ^2 Pidqi + P 0 dt, (2.11.13a) *■ = ■■■ = £/»/«?/, (2.11.13b) v = £ P/Vi + P„+i = v(t,q,vj) = v 0 , (2.11.13c) a = 22 Pi^’i + terms not containing V/ = a(t , q , vj, v k ) = a 0 , WWW.EBOOK777.COM (2.11.13d) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS where (s/ —► Pi): Pi = e i + ^ b DI e D , P n+l = p 0 = e„ + i + Y^ b D e D = e 0 + ^ b D e D . (2.11.13e) REMARK It should be pointed out that under the quasi-variable choice (2.11.9), and, according to an unorthodox yet internally consistent interpretation [advanced, mainly, by Ukrainian/Soviet/Russian authors, like Suslov, Voronets, Rumiantsev; and at odds with the earlier statement (§2.9) that the q s are always holonomic coordinates], the < 7 /, and hence also the q D , are no longer genuine = holonomic coordinates, but have instead become quasi-, or nonholonomic coordinates; even though one could not tell that very well from their notation. To avoid errors in this slippery terrain, some authors have introduced the particular notation (q) (Johnsen, 1939); we shall use it occasionally, for extra clarity. Thus, specializing (2.9.27), while recalling the first of (2.11.12b), we can write dr/d{q I ) = Y (dr/dq k )(dv k /dv,) = Y ( dr / d( bD)(dvD/dv 1 ) + Y {dr/dq v ){dv v /dvj) = dr/dq , + Y h Di(dr/dq D ) = Y A diC D + Y A i'/ e i' = Y^ b DI e D + Y^ bri e ]' = Y1 bDi e D + e i = Pi': (2.11.14a) and analogously for P n+l = P Q . Similarly, with the helpful notation [(2.11.13c)]: v = v(t, q, v) = • • • = v a (t, q, Vj) = v 0 , chain rule, and recalling (2.11.9), we obtain dv 0 /dv, = dv/dv, + Y {dv/dv D ){dv D /dv ,) = e, + Y e obDi = Pi', (2.11.14b) that is, the fundamental identities (2.9.33) specialize to dr/diq,) = dv a /dv I = dajdvj = ■■■ = Pj = P^t.q) (2.11.14c) [not to be confused with the analogous holonomic identities (2.5.7, 7a)]. Equation (2.11.14a) gives rise to the special symbolic quasi chain rule (see also chap. 5): d.../d[q,) = Y( d ---/ dl lk){dv k /dv I ) = Y( d "' / dc JD)(dv D /dv,) + Y( d --- / dqi')(dv r /dvi) = d.-./dqj + Y b D i(d.../dq D ); (2.11.15a) which, when applied to v D , yields dv D /d(q /) = Y (dv D /dq D '){dv D '/dv,) + Y (dv D /dq r )(dv r /dv,) = dv D /dq, + Y b D 'i(dv D /dq D -). (2.11.15b) WWW.EBOOK777.COM §2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES Generally, applying chain rule to f=f(t,q,v) =f[t,q,v D (t,q,v I ),v I ] = f 0 (t,q,v,) =f 0 , (2.11.15c) we obtain the useful formulae dfo/dv, = df/dv, + Y (df/dv D )(dv D /dv,) = df/dv, + Y b D i{df/dv D )\ (2.11.15d) and dfo/dq i = df/dq, + Y ( df/dv D ){dv D /dq, ); (2.11.15e) while (2.11.15a,b) are seen as specializations of df 0 /d{q,) = dfo/dq, + Y ( df 0 /dq D )(dv D /dv,) = dfo/dq, + E bmidfo/dqf] [notation, not chain rule!]. (2.11.15f) Problem 2.11.1 With the help of the above symbolic identities [recall (2.11.12 ff.)] show that: (i) dq k /dO, = dv k /duj, -> dq k /d{q,): dqD/dfjD') = A D d' = boD'i dq D /d{q,) = A D , = b D ,, dq,/d{q D ) = A, D = 0, dq,/d{q,') = A, r = 6„f. (a) (ii) dd k /dq, = dw k /dv, -> d{q k )/dq,: d(q D )/dq D ' = a DD ’ = <W, d{q D )/dq, = a DI = -b DI , d(q,)/dq D = a, D = 0, d[q,)/dq r = a IV = 8„:. (b) [Notice that dq D /d{q,) = b D , f d{q D )/dqi = -b D ,.] (iii) d... /d0 11+ 1 —> d... /d{q n+i ) [recall (2.9.32 ff.), and since A kn+l = A k \ = Y A ^ d ■ ■ ■ ! dq ^+ 9 ■ ■ ■ / dt = Y A °( d ■ ■ ■ +E A ^ d ■ • • / dqi )+ 9 ■ ■ • / dt = Y b D {d.../dq D ) + 0 + d.../dt = d.../d{t) + d.../df, (c) which for r yields the earlier (2.11.13e). (iv) dr/dfio ) = Y ( dr/dq k )(dv k /dv D ) = Y e k A kD = ■■■ = dr/dq Dl i.e., p D = e D . (d) (v) dp,/{q ,0 f dp,,/d{q,y, (e) which is a specialization of (2.10.23), and shows clearly that the basis {p,} is non¬ gradient. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 4. Occasionally, the constraints appear in the (2.11.9)-like form, but in the quasi variables dO/89/uj [special case of (2.11.8a)]: dOf) = B di dOj + B d dt ; or 89 D = B DI 89 /; or ui D = B di loi + B Dl (2.11.16) where the coefficients B DI , B D are known functions of q and t. To uncouple them, proceeding as before, we introduce the following new equi¬ librium quasi variables dO' / 80 ' /uj' (to avoid accented indices, we accent the quasi variables themselves): d9 'd = d9 0 — 'y ' BDi d9j — B dt = 0, dO 1 j = d9j ^ 0, dO’ nJr \ = d0 n+ j = dt ^ 0; (2.11.17a) 89' D = 89 d — B di 80j = 0, 89'j = 80j 0, 80' n+ \ = 89 n+i = 8t = 0; (2.11.17b) u 'd = u D — ^ B di loj — B d = 0, lo'j = ujj ^ 0, u/ n+1 = w„ +1 = dt/dt = 1 ^ 0; (2.11.17c) which invert easily to d9 j) — d9 i) T y ^ B i) j d9 / T Bd dt — y ( Bdi dO / T Bd dt , d0[ = dO' j, d@n+l — ^0 — dt , (2.11.18a) 89 d = 89' D + ^2 Bdi SO'i = ^ B D i 89'j, "hs II qS" o II III + III + (2.11.18b) U>D — u'd + ^ B D j Lo'i + B d — ^ B D 1 Oj'j + B d , UJi = u'j, w „+l — u'n+l — dt/dt — 1. (2.11.18c) Clearly, (2.11.16) (2.11.18) bear the same formal relation to (2.11.8a) that (2.11.9) (2.11.11) bear to (2.11.2) (2.11.4). In sum, the possibilities are endless and, in practice, they are dictated by the specific features and needs of the problem at hand. The essential point in all these descriptions is that, ultimately, they express the n dq/8q/v in terms of n — m independent parameters d9 I /89 I /ijj I ', and if the nonholonomic constraints are in coupled form, either among the dq/8q/v or among another set of n quasi variables dO/89/uj then, following Hamel, we introduce new equilibrium quasi variables dO'/80 '/lu' such that d0' D /80 ' D /uj' d = 0 and dOj/89'jju'j ^ 0. And, as already stated, this uncoupling of the Pfaffian constraints is the main advantage of the method. Problem 2.11.2 Consider the homogeneous Pfaffian constraints, U> D = ^ <-<Dk v k — ^ a DD' v D' + ^ a DI’ v I’ (= 0), u>i = V/ = ^ 8 ID 'V D ' + ^ 8/j'Vj' (= ^ fifth1 7 ^ o) ; WWW.EBOOK777.COM (a) (b) §2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES where D, D' = 1,..., m,I, I 1 = m + l,... ,n; that is, (with some easily understood notations) a =>- a s = ( a k ,) = ( a DD') ( a DI')\ ( ( a DD') ( a DI ') (a ID i) (aw) ) \ (0/ d ') (<)//') (i) Verify that its inverse (assuming that a is nonsingular) equals , i^DD') (Ajj/')\ ( (flDZ)') ~( a DD') ( a D'I ‘) A => A s = ( A kt ) = (c) (d) V (A/d') (A//') / y (0 /d') (<5//0 / (ii) Extend the above to the nonhomogeneous case; that is, wd=X) a Dk v k+ a D (=0), uji = V/ (^ 0). (iii) Verify that the earlier particular choice (2.11.9 If.) is a specialization of the above. Geometrical Interpretation of Constraints (May be omitted in a first reading.) We begin by partitioning the mutually inverse n x n matrices of the virtual transformation between bq <-> 66, a s = (a k i) and A s = (A kl ), into their dependent and independent parts: a s = (2.11.19a) A s = (A d | = (A kD | A kI ). (2.11.19b) Clearly, / 3 dAd 3 dAi\ /I 0\ a s A s = = . (2.11.19c) V a iAo a iA! ) \0 1 / Next, we partition these submatrices in terms of their dependent and independent (column) vectors as follows [with (.. .) T = transpose of (...), and using strict matrix notation for vectors and their dot products, instead of the customary vector notation used before and after this subsection]: /V\ a D = a I W) (a m+ ?\ V ) a D T — («i, • ■ ■, «,„), a D T — (a D \,... a Dn ), (2.11.20a) a i T = (« m +u---'«»)' «/= («/u •••«/«). (2.11.20b) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Ad — (Ai, ■ • • ,A m ), ( A ' T ) U,v A| ( A m +\ :••••> A/;) , — {A\d, ■ ■ ■, A„ D ), (2.11.20c) A, t = {A u ,...,A nI ), (2.11.20d) Also, since (a D • a D T ) 1 • (a D • a D T ) = 1 and A d t • a D T = 1, it follows that Ad T — ( a D -a D T ) ' -a D- (2.11.20e) Now, in linear algebra terms, the virtual form of the constraint equations X] a DkSq k = 0 [rank(a Dk ) =m{<n)\, (2.11.21a) [we note, in passing, that rank(a Dk ) mxn = rank(a Dk \a D )^ mx ^ n+l ^] or, in the above- introduced matrix notation, a D • Sq = 0 (one matrix eq.), a D T • Sq = 0 [m vector (dot product) eqs.], (2.11.21b) state that every virtual displacement ( column ) vector Sq 1 = (Sq l ,..., Sq„), at the point (q, t), lies on the local (n — m)-dimensional tangent/null/virtual plane of the (virtual form of the) constraint matrix a D = (%.), T„_ m (P) = V n _ m (P) = V„_ m (§2.7, suppressing the point dependence); or, equivalently, that Sq is always orthogonal to the local m-dimensional range space/constraint plane ofa D T ,C m (P) = C,„ (which is orthogonally complementary to V n _ m ). Next, in view of our quasi-variable choice, that is, Sq = Aj • S6j, where S0j T = (S6 m+U ..., S9 n ), the (n — m) vectors (A m+1 ,..., A n ) = {A f } constitute a basis for while the m constraint vectors (« 1; ... ,a m ) = {a D } constitute a basis for C,„. Or, all ( Sq k ) satisfying (2.11.21a, b), at (q 7 t), form a local vector space V n _ m , which is orthogonal to the local vector space C m built (spanned) by the m constraint vectors a D T = (fl^i, • ■ ■, flz>„)- More precisely, expressing (2.9.3a, b) in the above matrix/vector notation, we have (i) yi a ik^ki’ — a t T • Ap — S n ’ (I, /' — m + 1,... ,«), (2.11.22a) or a, • A, = 1, or A I T -a ] T = l; (2.11.22b) that is, the columns of aj T and A t , or the rows of aj and Aj T , namely, the vectors {«/} and { Aj }, are mutually dual , or reciprocal , bases of V n _ m ; and (ii) a Dk^kD' — a D T • A/)' — S DD ' (D 7 D' — 1 ,.. (2.11.22c) or a D -A D = l, or A D T -a D T = l 5 (2.11.22d) WWW.EBOOK777.COM §2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES that is, the columns of a D T and A D , or the rows of a D and A D T , namely, the vectors {a D } and {A D }, are mutually dual bases of C m . Clearly, if the { a D } are orthonormal , so are the {A D }, and the two bases coincide; and similarly for the bases {«/}, {Aj}. Likewise, from (2.9.3a, b) we obtain (iii) ^ a Dk A kI = a D T -A, = S D1 = 0 (D = 1,..., m; I = m + 1, or a D • Aj = 0, or A t T • a D T = 0; that is, the vectors {a D } and {Aj} are mutually orthogonal. (iv) J2 a/kA kD = a, T -A D = 6 W = 0 (/ = «?+ 1,..., n; D = 1,..., m ), (2.11.22g) or a I -A D = 0, or A D T -a I T = 0. (2.11.22h) that is, the vectors {«/} and { A D } are mutually orthogonal. Equations (2.11.22f) and (2.11.22h) state, in linear algebra terms, that the “virtual displacement matrix” A t is the orthogonal complement of the “constraint matrix” a D . [Hence, the projections of an arbitrary system vector M = (M x ,..., M n ) on the local mutually orthogonal (complementary) subspaces V n _ m and C m , are, respec¬ tively, Null/ Virtual space projection P ...(•• •): Y A kI M k = (A[ T ■M) i = A, t -M = M i = ^V(Null)(-^0 = P V (Virtual) (M ); (2.11.23a) • ,«), (2.11.22e) (2.11.22f) Range/constraint space projection P ...(•• •): Y A kD M k = (A D T • M) d = A d 7 • M = M d = f 3 /?(Range)(^) = ^C(Constraint) (^)-] (2.11.23b) The above hold, locally at least, for any velocity constraints, be they holonomic or nonholonomic. However: (a) If the constraints are nonholonomic, the corresponding null and range spaces are only local; at each admissible point of the system’s con¬ strained configuration (or event) space; but (b) If they are holonomic, then these spaces become global; that is, the hitherto /(-dimensional configuration space is replaced by a new “smaller” such space described by n — m Lagrangean coordinates, as detailed in §2.4 and §2.7. Tensorial Hors d'Oeuvre These projection ideas, originated by G. A. Maggi (1890s) and elaborated, via tensors, by J. L. Synge, G. Vranceanu, V. V. Vagner, G. Prange, G. Ferrarese, P. MaiBer, N. N. Poliahov et al. (1920s 1980s), are very useful in interpreting the general problem of AM [i.e., of decoupling its equations of constrained motion into those containing the forces resulting from these constraints and those not containing these forces], in terms of simple geometrical pictures of the motion of a single “particle” in a generalized system space. They have become quite popular among multibody dynamicists, in recent decades; but, predominantly as exercises in linear WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS algebra/matrix manipulations, that is, without the geometrical understanding and insight resulting from the full use of general tensors. To show the advantages of the powerful tensorial indicial notation, over the noncommutative straightjacket of matrices, we summarize below some of the above results. With the help of the summation convention [over pairs of indices, one up and one down, from 1 to n; and where, here, capital indices (accented and/ or unaccented), signify nonholonomic components], we have the following: (a) Equations (2.11.21a, b), and their inverses: 89° = a D k 8q k = 0, 8q k = A k , 6(t. (2.11,24a) (b) Equations (2.11.22a, b): a I k A k I ' = <5/<) (2.11.24b) (c) Equations (2.11.22c, d): 44 = 4 , (2.11.24c) (d) Equations (2.11.23a): P v {M) = A k I M k = M I , (2.11.24d) (e) Equations (2.11.23b): P c (M) = A k D M k = M D . (2.11.24e) The summation convention explains why, in order to project the (covariant) M k , above, we dot them with At, and A k D , instead of a\, a D k , respectively. [Briefly, the a 1 (A,) build a nonholonomic contravariant (covariant) basis in V n _ m , while the a D (A D ) build a nonholonomic contravariant (covariant) basis in C m .] Last, a higher level of tensorial formalism may be achieved, if, as described briefly in (2.10.17c), we use accented (unaccented) indices to denote nonholonomic (holo- nomic) components; for example, successively: a k , —> ct) —> A k k , A k , —> A k , —> A k k >; so that a-A = 1 reads A k k A k ,i = 8 k and similarly for the other equations. For further details on tensorial nonholonomic dynamics, see, for example, Papastavridis (1999) and references cited therein. 2.12 CONSTRAINED TRANSITIVITY EQUATIONS, AND HAMEL'S FORM OF FROBENIUS' THEOREM Constrained Transitivity Equations Let us begin by examining the transitivity relations (2.10.1) under the Pfaffian con¬ straints (2.11, 2a flf.), dd D = 0, 86 d = 0, and their implications for the latter’s holo- nomicity/nonholonomicity. Indeed, assuming d(6q k ) = 8(dq k ) for all k=l whether the dq/8q are constrained or not (what is known as the Hamel viewpoint , WWW.EBOOK777.COM §2.12 CONSTRAINED TRANSITIVITY EQUATIONS see pr. 2.12.5), the general transitivity equations (2.10.1) reduce to d(66 D ) - 6(d6 D ) = EE 7 V df) r 89, + I n i dtS0 h (2.12.1a) d(89,) - 8(d9 k ) = EE 'l I ri "dO I »8e r + ^2'y I I 'dt66it. (2.12.1b) From the above we conclude that, even though uj D (t) = 0 (or a constant), or d9 D (t) = 0, or 69 D (t) = 0, from which it follows that (66 D )' = 0 or d(89 D ) = 0, yet, in general, d(89 D ) — 8(d9 D ) f 0 => —8(d9 D ) f 0! Specifically, as (2.12.1a) shows, -8(d9 D ) = EE ^ D n' d9[' 89j + ^ 7 ^ dt89; f 0 (in general); (2.12.1c) that is, we cannot assume that both d(Sq k ) = 8(dq k ) and d(89o) = 8(d9o)(= 0)! This is a delicate point that has important consequences in time-integral variational prin¬ ciples for nonholonomic systems (see Hamel, 1949, pp. 476-477; and this book, chapter 7; also pr. 2.12.5). The Frobenius Theorem Revisited (and Made Easier to Implement) We have already stated (§2.8) that the necessary and sufficient condition for the holonomicity of the system of m Pfaffian constraints d9 D = E a Dk dq k + a D dt = 0, 69 D = E a Dk 8q k = 0 (D = 1,... ,m), (2.12.2) that is, for the existence of m linear combinations of the d9 D = 0, or 89 D = 0, that equal m independent exact differential equations df\ = 0 ,..., df m = 0 => f\ = constant ,..., f m = constant , is the identical vanishing of their Frobenius bilinear covariants [recall (2.9.13)] d(89 D ) - 8(d9 D ) = EE (da Dk /dq l - da Dl /dq k ) dq, 8q k + ^2(da Dk /dt - da D /dq k ) dt 8q k , (2.12.3) for all dq k , dt, 8q k solutions of (2.12.2). From this fundamental theorem we draw the following conclusions: (i) If the dq k , dt, 8q k are unconstrained, that is, if m = 0, then the identical satisfaction of the conditions a°ki = da Dk /dq, - da D ,/dq k = 0, a D k = da Dk /dt - da D /dq k = 0, (2.12.3a) for all k, / = is both necessary and sufficient for the holonomicity of 9 D (§2.9). (ii) But, if the dq k , dt, and 8q k are constrained by (2.12.2), then the vanishing of d(89 D ) — 8(d9 D ) does not necessarily lead to (2.12.3a). To obtain necessary conditions for the holonomicity of the system (2.12.2), we must express the dq k , dt, 8q k , on the right side of (2.12.3), as linear and homogeneous combinations of n — m independent parameters (Maggi’s idea); that is, we must take the constraints (2.12.2) themselves into account. Indeed, substituting into (2.12.3) the WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS general solutions of (2.12.2) [recalling (2.11.4)]: dq k = E A kI ddj + A k dt, 6q k = E A kI 60 j, (2.12.4) we obtain (2.12.1a). From this, it follows that (and this is the crux of this argument), since the 2 (n — m) differentials dO k , 80j are independent/unconstrained, the conditions 7 V = 0 and 'y°i, n +i = l°i = 0 , (2.12.5) for all D = 1,... ,m; I, I 1 = m + 1,... ,n {i.e. maximum total number of distinct/ independent such j D ’s is [m(n — m){n — m—\)/2]+m(n — m) = m(n — m)(n — m+l)/2 = mf(f+ l)/2 ,/ = n — m}, are both sufficient and necessary for the holonomicity of the Pfaffian system (2.12.2). • Since [recalling the 7 -definition (2.10.2 ff.)] (2.12.5) can be rewritten as A- = EE (da nh /dq c - da Dc jdq h )A hl A cV = EE aD bcA hI A cI , = 0, (2.12.5a) ^, = EE ( da Db /dq c - da Dc /dq h )A h[ A c + E ( da Dh /dt - da D /dq h )A hI = EE a ^hc^blA c + E aD b^bl = 0) (2.12.5b) we readily recognize that the (identical) vanishing of (all) the 7 fl .’s does not neces¬ sarily lead to the vanishing of (all) the a D kc , a n k , while the vanishing of all the latter leads to the vanishing of all the 7 z> . ’s; that is, (2.12.3a) lead to (2.12.5, 5a, b) but not the other way around. Hence, (2.12.3a) are sufficient for holonomicity but not necessary, whereas (2.12.5, 5a, b) are both necessary and sufficient. • Since, as (2.12.5a, b) make clear, each y D ( 7 "°) depends, in general, on all the coefficients a Dk ,A kI (a Dk ,A kI ;a D ,A k ), the holonomicity/nonholonomicity of a(ny) particular constraint, of the given system ( 2 . 12 . 2 ), depends on all the others; that is, on the entire system of constraints. In other words: eqs. (2.12.5) check the holo¬ nomicity, or absence thereof, of each equation d0 D , S0 D = 0 against the entire sys¬ tem; that is, there is no such thing as testing an individual Pfaffian constraint, of a given system of such constraints, for holonomicity; doing that would be testing the new system consisting of that Pfaffian equation alone (i.e., m = 1) for holonomicity. In short, holonomicity/nonholonomicity is a system property. As Neimark and Fufaev put it “the existence of a single nonintegrable constraint (in a system of constraints) does not necessarily mean a system is nonholonomic, since this constraint may prove to be integrable by virtue of the remaining constraint equations” (1972, p. 6 , italics added). However [and recalling (2.10.16a-18b)], we can see that the identical vanishing of all coefficients y k rs and 7 * n+l = 7 * (for all r,s = 1 ,...,«) in d{86 k ) - 6{dd k ) = ■■■ + ( 7 *.) d6 69+ ( 7 *.) dtSO, (2.12.5c) independently of the constraints d0 D ,60 D = 0 (or, as if no constraints had been applied; and which is equivalent to a k rs = 0 , a k r = 0 , identically), is the necessary and sufficient condition for that particular 9 k to be a genuine/Lagrangean coordinate ; that is, a kr = d6 k /dq r , a k = d0 k /dt. Let us recapitulate/summarize our findings: (i) Pfaffian forms (not equations), like d6 k = a k /(q) dqi {k = 1 ,..., n'\ l=\,...,n\ n and n unrelated) ( 2 . 12 . 6 ) WWW.EBOOK777.COM §2.12 CONSTRAINED TRANSITIVITY EQUATIONS (for algebraic simplicity, but no loss in generality, we consider the stationary case), are either exact differentials, or inexact differentials. If their dq s are unconstrained , then the necessary and sufficient conditions for d6 k to be exact, and hence for 9 k to be a holonomic coordinate , are a k rs = 0 (r,s= (2.12.7) In this case, each of the n' forms d9 k is tested for exactness independently of the others; k, in (2.12.7), is a free index, uncoupled to both r and s. If n = n', then, as already stated, conditions (2.12.7) can be replaced by l\s = 0 (r,j= (2.12.8) but since calculating the 7 ’s requires inverting ( 2 . 12 . 6 ) for the n dq's in terms of the n dO' s, eqs. (2.12.8) offer no advantage over eqs. (2.12.7). If eqs. (2.12.7) hold, then dd k remains exact no matter how many additional con¬ straints may be imposed on its dq’s later. For, then, we have d(69 k ) - 8(d9 k ) = EE (■ da kr /dq s - da k Jdq r ) dq s 8q r = 0; (2.12.9) that is, if 9 k is a holonomic coordinate, it remains holonomic if additional constraints be imposed among its dq, 8q's, later. This is the meaning of Hamel’s rule: d(8q k ) = 8(dq k ), for all cf s, constrained or not. If eqs. (2.12.7) do not hold, d9 k is inexact; but it can be made exact by additional constraints among its dq, 8q' s; that is, if 9 k is a quasi coordinate, it may become a holonomic coordinate by imposition of additional appropriate dq, 8q constraints. For example, let us consider the Pfaffian form (not constraint) d9 = a(x, y, z ) dx + b(x, y, z) dy + c(x, y, z) dz. Under the additional constraints y = constant =>■ dy = 0 and z = constant =>■ dz = 0, it becomes dd = a{x, y, z) dx = f(x)dx = exact differential, even if, origin¬ ally, x was a quasi coordinate. (ii) Pfaffian systems of constraints d0 D = ^2 a Dk dq k = 0 , 89 D = ^ a Dk 8q k = 0, (2.12.10a) are either holonomic or they are nonholonomic. The necessary and sufficient condi¬ tions for holonomicity are d(89 D ) - 8(d9 D ) = EE y n , v d9 r 80f = 0, (2.12.10b) or, since the d9i, 89j are independent, r y D ir = 0 {D = 1,... ,m\ I,I 1 = m + 1,... ,n); (2.12.10c) that is, the “dependent” q’s relative to their “independent” indices (subscripts) should vanish; or, the components of the dependent (constrained) Hamel coefficients along the independent (unconstrained) directions vanish. {This, more easily implementable, form of Frobenius’ theorem seems to be due to Hamel (1904(a), 1935); also Cartan (1922, p. 105), Synge [1936, p. 19, eq. (4.16)], and Vranceanu [1929, p. 17, eq. (9'); 1936, p. 13].} WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS In closing this section, we repeat that Frobenius’ theorem is about the integra- bility of systems of Pfaffian equations, like (2.12.2), not about the exactness of individual Pfaffian forms, like (2.12.6). Example 2.12.1 Special Case of the Hamel Coefficients, via Frobenius’ Theorem. Let us calculate the Hamel coefficients corresponding to the special constraint form dq D = £ bm d<h ' Sq ° = Y bm 6qi ’ (a) where b DI = h DI {q), and formulate the necessary/sufficient conditions for their holonomicity. We begin by viewing (a) as the following special Hamel choice [sta¬ tionary version of (2.11.10a—12b)]: dq D - Y b Dl dq I = °> d9, = - dq, f 0, + III g- + III g. ik © (b) 8q D — Y, b DI bc lt = 89, = 8q, f 0, fi@n +1 — fitfn +1 — fit 0, (c) Md + £ b Di d9j = Y^ b D , d9,, ^3 II ^3 III + ^3 III + ^3 (d) 89d + £ b DI 8q, = Y^ 8di fiqi, 8q, = 89,, © II III + III + (e) also, since here dq, = d9,, we can rewrite the system (a) as where dqk — £ BkI dqi = £ ^kt dbfy (B a ) ^ u\,m+l • • • b\rfi b m ,m +1 * • fifWl 1 ■ 0 0 •• 1 / (f) Since 9, = q,, we shall have 'y I a g = 0; while (2.12.9), with k^> D and (f), becomes, successively, d(89 D ) - 8{d9 D ) = YY a D rs dq s 8q r = YY a ^{Y dqi) (£ B it' 6c h') ==££££ (a D rs b s , b rV ) dq, 8q r = - = ££ 7 D ridq, 6q,< = £ £ 7 D iv dqr 8q, = Y, £ 7 °//' d8 r 89, , (g) where (expanding the sums in r and s, with D, D ', D" = 1, ..., m\ /, I' = m + 1, .. ., n) l D i'i = £ £ t^D'D" t>D"i t>D'r + £ cPd'i b d'i' + £ ti D rn' b,y, + a D i'i\ (h) or, since cP D i D « = a DD ' D « — a DD " D ', where commas denote partial differentiations WWW.EBOOK777.COM §2.12 CONSTRAINED TRANSITIVITY EQUATIONS relative to the indicated cf s and [by (2.11.12—12b)] a DD ' —* S DD ', a DI —> —b DI , ci id —> 0, a n t —> S rr : (0)b D "ib D 'i' + [0 — {—db DI /dq D ')\ b D 'i' + El [( -,% £/7 %d') ~ 0]^ 7 + [{-db D i</dqi) - (-db DI /dq r )\, or finally, 7 D /7 = \dbpi/dq,' + ( db DI /dfopfep'/'l - [db Dr fdq r + ^ (db Dr >/dq D >)b r y r ] = —w D jh = = Voronets (or Woronetz) coefficients ; (i) clearly, a specialization of 7 Z /'/. Thus, (g) becomes d(se D ) - = EE 7 D I'I dqi Sqr = EEA ' dqj bqj', (j) and, since the dqj and <5^/ are independent, by Frobenius’ theorem, the necessary and sufficient conditions for the holonomicity of the system (a) are H’V = 0, (k) which are none other than the earlier Deahna-Bouquet conditions (2.3.11b ff.). REMARKS (i) With the help of the symbolic notation (2.11.15a), we can rewrite (k) in the more memorable form, 7 D i'i = w D W = db DI /d(q r ) - db Dr /d(q I ). (1) (ii) In the special “Chaplygin (or Tchapligine) case” (§3.8), where b DI = b D Mm+ 1 ,- • -,qn) = b DI (q D ), the above reduce to l D ri = db DI /dq r - db Dr /dq, = t D lv . (m) Problem 2.12.1 Continuing from the previous example, show that eqs. (i) for the Voronets coefficients also result by direct application of the definition (2.10.2) 7 V = EE (da Dk /dq r - da Dr /dq k )A kI A rI , (a) to the constraints (ex. 2.12.1 :a) in the equilibrium forms (ex. 2.12.1 :b e). HINT Flere [recalling again (2.11.12—12b)]: a DD ' = b DD ', a DI = ~b DI , a ID = 0, a ir = 6 n >; Add 1 = bf) r y, A di = bph A[ D = 0, Ajy = 6 n >. Problem 2.12.2 Continuing from the above, show that in the general nonstationary case dq D = ^2 b oi dc h + b D dt , dq, = ^ Sir dq r = dq u (a) bq D = E bDI Sqi = E b]V 6qi ' = bqi ~ b ° l = bDl( d’ ^)> h ° = b ° q )> ( b ) WWW.EBOOK777.COM 339 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS the 7 D ii' remain unchanged, but we have, the additional nonstationary Voronets coefficients: 1 D I,n +1 = 7 D I = ~W D I,n+\ = = db D /dq, + Y (db D /dq D ')b D '! = db D /d(q,) - db DI /d(q n+ i ) db DI /dt + Y ( db DI /dq D ')b D - (c) [recalling the symbolic (2.9.32ffi), (2.11.15): A k —> b D , and ]C (d ■ ■ ■ /dq D )b D = a.../5(t)]. REMARK In concrete problems, use of the above definitions to calculate the iv-coefficients is not recommended. Instead, the safest way to do this is to read them off directly as coefficients of the following bilinear difference/covariant: d(S0 D ) - 6(dO D ) = • • • + 7 D IV dO r <50/ + • • • + 7 D j dttoj + • • • == • • • — w jj! defat + * ■ * — w / dt + • • •. (d) Problem 2.12.3 Continuing from the preceding problem, verify that: (i) in the catastatic Voronets case, the w D n i remain unchanged, while w D I = db DI /dt, and (ii) in the stationary Voronets case, the w D r[ ' remain unchanged, while it’ 23 / = 0. Problem 2.12.4 Continuing from the above problems, verify that (i) O II C"~ (/ = m + 1,..., n; (3,e= 1-- n; n + 1); (a) (ii) 7 V = 0 (D,D' = 1,... , 777 ; e = 1- n + 1); (b) (recall that 8 / = qj is a holonomic coordinate). Problem 2.12.5 Continuing from the above example and problems, consider again the nonstationary constraints in the special form (2.11.10a ff.): d c lD — E ^di dqi + bo dt, btlD = E b Dl v ° = E b D/ vi + b D , (a) where b DI = b DI {t,q), b D = b D (t,q), and, as usual, v k = dq k /dt. Show by direct (//^-differentiations of the above, and assuming that d(6qj) — 6(dqj) = 0, that d(Sq D ) - b{dq D ) = EE w L //< dqi* bqi + / ' w°i dt bqj, (b) or, dividing both sides by dt, {& c Id) — ^(,9d) = d) ~ 8 v d = E (E 11 ' Vr = E vJ?/ ( c ) WWW.EBOOK777.COM §2.12 CONSTRAINED TRANSITIVITY EQUATIONS that is, in general, and contrary to the hitherto adopted Hamel viewpoint (§2.12), d(8q D ) ^ 6 ( dq D ), as if the q D are no longer holonomic coordinates! REMARKS The alternative (and, as shown below, internally consistent) viewpoint exhibited by (c) [originally advanced by Suslov (1901-1902), (1946, pp. 596-600), and continued by Levi-Civita (and Amaldi), Neimark and Fufaev, Rumiantsev, and others], is based on the following assumptions: (i) If the n differentials/velocities dq/hq/v are unconstrained, then we assume that the Hamel viewpoint holds for all of them; that is, d(6q k ) = S ( dq k ) (k = 1 ,...,»). (ii) But, if these differentials/velocities are subject to m (a)-like constraints, then we assume that the Hamel viewpoint holds only for the independent of them, say the last n — m, but not for the dependent of them, that is for the remaining (first) nr. Suslov viewpoint: d(8q,) — 8(dq,) = 0, but d(8q D ) — S(dq D ) ^ 0. (d) Let us examine this quantitatively, from the earlier generalized transitivity equations (2.10.1, 5): d(89 k -) — 8(d9, k ) = Y, a k i[d(8qi) — 8(dqj)\ + EE 7 \ s dd s 89,. + E 7 \dt86 n (e) d{8q k ) - 8{dq k ) = ^zl«{[J(^) -8{d9i)\ -EE 7 l rs d9 s 89,. -E 7 Uhd,.}. (f) (a) Hamel viewpoint'. d(8q k ) = 8{dq k ), always. Then, since d9 D , 89 D = 0, (e) yields d{89 D ) - 8{d9 D ) = EE 1 D IV dOp 89, + 7 D I dt 89/ [by (pr. 2.12.4: b)[ (g) = -EE w D ji' dq,< 8q, —^ w D I dt 8q, [by (ex. 2.12.1: g, j)[; (h) d(89,) - 8(d9,) = EE 7 I VV ! d9,» 89r + Y y v dt89 r = 0 [by (pr. 2.12.4: a)]. (i) (b) Suslov viewpoint [for the Voronets-type constraints (a)]. Since here, A dd i = 8 DD >, A id = 0, Ajp = 8,p, eq. (f) yields, successively, (1) 0 = d(8q,) - 8(dq,) = 4{[« - S(d9 r )] ~ E E d’ri" d9j» 89, v - Y 7 V dt69 v ) = [d{89j) - 8(d9j)\ - EE 7 I i'i" dOp' 89p -E 7 1 p dt 89ji = d{89[) - 8(d9j) [by (pr. 2.12.4: a)]; (j) (2) d{8q D ) - 8{dq D ) = E 'W{i<*(«M - - EL 7 D ' IV d9p 89, - Y l D 'i dt89 ,} = \d{89 D ) - 8{d9 D )\ - EE 7 D ,pd9p89, -E 7 D ,dt89j = [d(89 D ) - 8(d9 D )] + EE w D jp d9p 89, + Y w °i dt Mi [by (ex. 2.12.1: i), (pr. 2.12.2: c)[ -EE w D ,p dcjp 8q, + E w°,dt8q, [by (b)[, WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS and comparing the last two expressions of d(8q D ) — 8(dq D ), we immediately con¬ clude that d(S0 D ) - 8{d9 D ) = 0 . (k) Hence: In the Suslov viewpoint we must assume that d(89 k ) = 8 (d9 k ) (k = 1,..., n). Both viewpoints are internally consistent; but, if applied improperly, they may give rise to contradictions/paradoxes. Hamel’s viewpoint, however, has the advantage of being in agreement with variational calculus (more on this in §7.8). Problem 2.12.6 Consider the special stationary dO dq transformation: dd D =y^ a DD ' dq D i (= 0 ) and d9j =dq k 0 ; the 9j are holonomic coordinates), (a) where a DD t = a DD fq x ,..., q m ) = a DD fq D ). Show that, in this case, the Hamel co¬ efficients are (i) 7 ( da Dd '/dq d « - da Dd «/dq dl )A d , D ,A d « D «, (b) where D, D', D", d' , d" = 1,... ,m and dq D t A D s D d9 D ; and (ii) 7 k rs = 0 , for any one of k,r,s greater than m. (c) Example 2.12.2 Transformation of the Hamel Coefficients under Frame of Reference Transformations. Let us again consider, for algebraic simplicity but no loss in generality, the stationary Pfaffian constraint system: d9 n = a Dk dq k = 0, 89 D = Y a Dk % = 0. (a) Further, let us assume that (a) is nonholonomic; that is, ^ D U ’ f 0. Now we ask the question: Is it possible, by a frame of reference transformation q —> q'(t, q), to make the constraints (a) holonomic? In other words, is it possible to find new Lagrangean coordinates q k > = q k '(t,q k ), in which the corresponding (dq k > d9 k ) Hamel co¬ efficients "fiq 1 ) 0 ]]' = 7 ,D n' vanish? Below we show that the answer to this is no; that is, if a system of constraints is nonholonomic in one frame of reference, it remains nonholonomic in all other frames of reference obtainable from the original via admissible frame of reference transformations. Indeed, we find, successively [with 7 {q) D w = 7 D w], 0 f d(89 D ) - 8(d9 D ) = EE 1 n r,d9,b9 r = EE a D rs dq s 8q r [dq s /dq s ') dq s i + (dq s /dt) dt'j ' ( dcj r / dq r f 8q r '^j = EEEE [(dq s /dq s ’)(dq r /dq,.')a D rs \dq s > 8q r - + EEE [( dq s /dt)(dq,./dq l j)a D rs \ dt 8q r , -EE a D r ' s i dq s ’ 8q r t + E a D r ’ dt8q r > [where a D r > s > = da Dr fdq s i — da Ds '/dq r ', etc.] WWW.EBOOK777.COM §2.12 CONSTRAINED TRANSITIVITY EQUATIONS = E E aD ''' s ' (E J ^ s ' 1 + A s’ dtj A,.: i' 80 ['^ ^ a D r i A r iji 80j^j dt = EEEE {a n r i s ’A s ’iA r ’ji) dOj 80i‘ +E(EE + E aD r'A,'i^j dt 80 j* = EE r )' D Vi dO, 80 r + ^ A(b) from which, comparing with the first line of this equation, we readily conclude that 7 (?) Z /'/ = l{q') D i'i and ffiq') D i> = 0 ; (c) that is, the Hamel coefficients remain invariant under frame of reference transforma¬ tions', or, these coefficients depend on the nonholonomic “coordinates” 0 k but they are independent of the particular holonomic coordinates frame used for their derivation. Incidentally, this derivation also demonstrates that the 7 -definition (2.10.1) is both practically and theoretically superior to the more common (2.10.2-4). REMARKS (i) We are reminded that the transformation properties of the 7 ’s under local transformations: dO k <=> dO k ', at (q,t), have already been given in ex. 2.10.1. (ii) The reader can easily verify that if, instead of (a), we had chosen a general nonstationary dO <^> dq transformation, we would have found 7 {q') D r = l{q) D i', instead of the second of (c). Also, then, c10,. = a rs dq s + a r dt = E a «(E (dq s /dq s ') dq s f + ( dqjdt ) dt^j + a r dt = ^ a n .t dq,.' + a' r dt, from which we can readily deduce the transformation relations among the coeffi¬ cients a(q), a(q') [recall ( 2 . 6.6 ff.)]. Problem 2.12.7 (see Forsyth, 1890, p. 54.) Verify that a system of n independent Pfaffian constraints in the n (or even n + 1) variables; that is, d0 k = E a ki dq, = 0 (k, l = 1 ,... ,n), (a) is always holonomic. Problem 2.12.8 Alternative Formulation of Frobenius’ Theorem. It has been shown, by Frobenius and others (see, e.g., Pascal, 1927, p. 584), that the Pfaffian system: d0 D = E a Dk (q) dq k = ° {D=l,...,m;k=\,...,n), (a) WWW.EBOOK777.COM 343 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS is holonomic if, and only if, each of its m (n + m) x (n + m) antisymmetric “Frobenius matrices”: 0 D D a 12 a \3 ' ' a D \n an ■ d m \ D a 21 0 a D a • • a °2n a n ■ d m 2 r, D a n\ a D n 2 ■ 0 a ln ■ &mn a n a \2 d\n 0 ■ ■ 0 d m 1 dm2 ■■■ ' &mn 0 • • 0 where a D kl = da Dk /dq l - da Dl /dq k = a DkJ - a D lk = -a D lk (e.g., a D u = ~cP 2 1 , a D n = —a\ | =>■ a D \\ = 0, etc.), has rank 2m. Apply this theorem for various simple cases: for example, m = 0 (i.e., dq k uncon¬ strained), m = 1 (one constraint), and m = 2 (two constraints). Example 2.12.3 Geometrical Interpretations of the Frobenius Conditions (May be omitted in a first reading.) In terms of the earlier (2.11.20a ff.) m constraint vectors a D = (a D i, ■ ■ ■ ,a D „) and n — m virtual vectors A k = (A n ,... ,A ln ) (in ordinary vector, nonmatrix notation), Frobenius’ conditions first of (2.12.5) assume the following forms: (i) First interpretation: From the a D we build the antisymmetric tensor: 0 °ki) = (- a\) = ( da Dk /dq, - da D ,/dq k ). (a) These can be viewed as the holonomic (covariant) components of the “ rotation or curlling) of a D "\ a D kt = —(curl a D ) kl . Also, we recall that A k/ = dv k /du>i. As a result of the above, (2.12.5): {da Dk /dq, - da m /dq k )A kI A u , -EE aD kiA k iAn' — 0, (b) assumes the (covariant) tensor transformation form, in k, I: 7V = EE (dv k /du,)(dvi/duj r )a D k i = A r ■ curla D ■ A, = ^ ^[A v )\curla D ) lk (Aj) k = 0; (c) that is, the (covariant) components of the curl of the dependent!constraint vectors a D along the independent nonholonomic directions Aj should vanish. (ii) Second interpretation: The Frobenius conditions (first of 2.12.5), rewritten with the help of the alternative expression (2.10.15) and the quasi chain rule WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS (2.9.30a) as ^ = ££ {a Dh [A c i(dA hr /dq c ) - A cI ,{dA bI /dq c )}} — £ a Db(dA h i>/ddj - dA hI /d6ji) = 0, (d) state that the constraint vectors a D should be perpendicular to the (n — m) [n — — l)/2 vectors: A w = {^2 A ci( dA bi'/dq c ) ~ ^rA cI ,(dA hI /dq c )^ = {dA^jdQ, - dA hI /dd r ) = -A VI , that is, 1°iv — a D ' A iv = 0 - ( e ) Similarly for the nonstationary/rheonomic Frobenius conditions (second of 2.12.5): d A 7 / = 0. For further details, including the precise positioning of indices, as practiced in general tensor analysis (and not observed in the above discussion!), see, for example, Papastavridis (1999, §6.9). 2.13 GENERAL EXAMPLES AND PROBLEMS Example 2.13.1 Introduction to the Simplest Nonholonomic Problem: Knife, Sled, Scissors, and so on. Let us consider the motion of a knife S, whose rigid blade remains perpendicular to the fixed plane O-xy, and in contact with it at the point C(x,y), and whose mass center G lies a distance b 0) from C along the blade (fig. 2.15). The instantaneous angular orientation of 5 is given by its blade’s angle with the +Ox axis <j>. Let us choose as Lagrangean coordinates: q\ = x, q 2 = y, q 2 = <j>. If v = ( dx/dt , dy/dt, dz/dt = 0) = (y Y , v y , 0) = (inertial) velocity of C, and u = (cos </>, sin (j>, 0): unit vector along the blade, then the velocity constraint is v x u = 0 =>■ (sin 4>)v x + (— cos <j})v y = 0, or dy/dx = tan (/). (a) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Since this is a stationary (and, of course, catastatic) constraint, we will also have, for its kinematically admissible/possible and virtual forms, respectively, (sin f) dx + (— cos <j>) dy = 0 and (sin</>) 8x + (— cosf) 6y = 0. (b) Other physical problems leading to such a constraint are: (i) A racing boat with thin, deep, and wide keel, sailing on a still sea. Since the water resistance to the boat’s longitudinal motion is much larger than the resistance to its transverse motion, the direction of the boat’s instantaneous velocity must be always parallel to its keel’s instantaneous heading; (ii) a lamina moving on its plane, with a short and very stiff razor blade (or some similar rigid and very thin object: e.g., a small knife) embedded on its underside. Again, the lamina can move only along the instantaneous direction of its guiding blade; (iii) a sled; (iv) a pair of scissors cutting through a piece of paper; (v) a pizza cutter, etc. Application of the holonomicity criterion (2.3.6) or (2.3.8a) to (a), (b), with h = (sin <j>, — cos cj), 0) and dr = (dx. dy, df) [as if x,y,<j> were rectangular Cartesian right-handed coordinates] yields I = h- curl h = It - [(<9/cbc, d/dy, d/df) x (sin </>, — cos <j), 0)] = (sin</>, — cos </>, 0) • (— sin </>, cos (/>,0) = — 1 f 0; (c) that is, the constraint (a), (b) is nonholonomic. This means that, although the general (global) configuration of 5 is specified completely by the three independent coordi¬ nates x,y,cj>, not all three of them can be given, simultaneously, small arbitrary variations; that is, although there is no functional restriction of the type f(x,y,4 ») = 0, there is one of the type g(dx,dy,dcf)-,x,y,(j)) = 0, namely the Pfaffian constraint (a), (b). Put geometrically: the blade has three global freedoms (x,y,(f>), but only two local freedoms (any two of dx.dy.df). Since n = 3 and m = 1, this is the simplest nonholonomic problem; and, accordingly, it has been studied extensively (by Bahar, Caratheodory, Chaplygin, et al.). The independence of x,y, </> can be demonstrated as follows: we keep any two of them constant, and then show that varying the third results in a nontrivial (or nonempty) range of kinematically admissible positions: (i) keep x and y fixed and vary <j) continuously; the constraint (a), (b) is not violated [fig. 2.16(a)]; (ii) keep y and (f> fixed. Varying x we can achieve other admissible configurations with different x’s but the same y and <j>\ but to go from one of them to another we have to vary all three coordinates [fig. 2.16(b)]; (iii) similarly when x and <f> are fixed and y varies [fig. 2.16(c)], The precise kinetic path followed in each case, among the kinematically possible/ admissible ones, depends on the system’s equations of (constrained) motion and on its initial conditions. WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS (a) (c) (2) Figure 2.1 6 Global motions of a knife showing the independence of its three positional coordinates. We can always, through a suitable finite motion, bring the knife to a position and orientation as close as we want to any specified original position and orientation; that is, the relation among x, y, <j> is nonunique. An Ad Hoc Proof of the Impossibility of Obtaining a Relation f(x,y,f) = 0 Let us assume that such a constraint exists. Then c/-varying it, and with subscripts for partial derivatives, yields df = f x dx +f y dy+f 0 df = 0, (d) or, taking into account the constraint in the form: dy = (tan f ) dx, df = {f x +f y tan <j>) dx + (/ 0 ) df = 0, (e) where now dx and df are independent. Equation (e) leads immediately to U = 0 => / = / {x, v) and f x +f y tan f= 0. (f) By {d/df) -differentiating the second of (f), while observing the first of (f), we obtain f y {\/cos 2 f) = 0, (g) from which, since in general 1/cos 2 f f 0, it follows that f y = 0 => / =/(x). But then the second of (f) leads to f x = 0 =>- / = constant {independent of x,y,f), and as such it cannot enforce the constraint f{x, y, f) = 0. Hence, no such/ exists (with or without integrating factors). WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS However, if the knife was constrained to move along a prescribed path, on the O-xy plane, the system would be holonomic! In that case, we would have in advance the path’s equations, say in the parametric form: x = x{s) and y = y(s) (s = arc length), (h) from which <j> could be uniquely determined for every s [i.e., 0 = 0(s)], via dy/dx = dy/ds j dx/ds = y'(s)/x'(s) = tan 0(F). (i) This is somewhat analogous to the basic variables of Lagrangean mechanics q k , dq k /dt = v k , which, before the problem is solved, are considered as independent , and then, after the problem is completely solved, become dependent through time. Example 2.13.2 The Knife Problem: Hamel Coefficients. Continuing from the preceding example: in view of the constraint (a), (b) there, and following Hamel’s methodology (“equilibrium quasi velocities,” §2.11), let us introduce the following three quasi velocities: w, = (- sin 4>)v x + (cos f)v y + (O)v^ (= 0), u 2 = (cos 0) v Y + (sin (j))v y + (O)v 0 = v (^ 0), w 3 = (0)v x + (0)v > , + (l)v^ = 0), (a) where v = velocity component of the knife’s contact point C; and hence v x = vcos 0, v v = vsin0, and the constraint is simply uq = 0. Clearly, since 9(cos cfj/dff 9(0 )/dx and 9(sin cjf/dff 9(0) /dy, (b) uj 2 = v is a quasi velocity, that is, v f total time derivative of a genuine position coordinate, or of any function of x,y, 0. Inverting (a), we obtain v x = (—sin0)w! + (cos (j))u) 2 + (0)w 3 , v y = (cos + (sin0)w 2 + (0)w 3 , v <l> = (0) w i + (0) w 2 + (l) w 3- ( c ) If a and A are the matrices of the transformations (a) and (c), respectively, then we easily verify that a = A, and Deta = Det A = — sin 2 0 — cos 2 0 = — 1 (i.e., nonsin¬ gular transformations). Further, we notice that (a), (c) hold with u ; 12 ,3 and v x , v y , v$ replaced, respectively, with dd l 23 = iv l23 dt and ( dx,dy,df ) = {v x , v y , v^) dt: and, since they are stationary, also for 69 x 2 3 and bx, by, <50. Next, by direct ^/^-differentiations of 69 u d9 x , and then subtraction, we find, successively, d[69f) ~ 6{d9 y ) = d[{— sin0) 6x + (cos 0) by + (0) 6f\ ~ <5[(— sin 0) dx + (cos 0) dy + (0) df\ = (— sin <p)(dbx — bdx) + (cos </>)(dby — bdy) — cos <f>d(j)bx — sin (f>d<j>by + cos <f> dx b<f> + sin 0 dy be/) = 0 + 0 — cos<£(1) d9 2 [{— sin0) 69 x + (cos0) b9 2 + (0) <5d 3 ] — • • • WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS [i.e., expressing dx, 6x, dy, 6y, df, Sc/) from (c), with replaced with dO l2 3 , S9 X23 ] and so we, finally, obtain the differential transitivity equation: d(69i) - S(dO x ) = dd 2 S9 3 - d9 3 Sd 2 , (d) [i.e., d(S9 1 ) — 8{dO x ) f 0, even though S9 X = 0 and d6 x = 0]; and, also, dividing this by dt, which does not couple with 6 (...), we obtain its (equivalent) velocity transi¬ tivity equation: (59 1 ) --6u) 1 = (0)69 l + (-l)u> 3 S9 2 + (l)u) 2 69 3 (fO). (e) Similarly, after some straightforward differentiations, we find (59 2 ) '-Su) 2 = (1)u) 3 S9 1 + (0)S9 2 + (-1)uj 1 S9 3 (= 0), (f) (69 3 )-- 6u) 3 = (0)69 1 + (0)69 2 + (0)69 3 (= 0). (g) From (e, f, g) we readily read off the nonvanishing Hamel’s coefficients: 7V(Z)=1;/,J' = 2,3): 7 23 = ~l\i = -1; (h) (I = 2- k,l = 1,3): 7 2 i3 = -7n = 1 ■ « REMARKS (i) Since not all y D n ’ —> (k, l = 2,3) vanish, we conclude, by Frobenius’ the¬ orem (§2.12), that our constraint, in any one of the following three forms: Velocity: = (-sin0)v x + (cos0)vj, + (O)v0 (=0), (j) Kinematically admissible: d6 x = (— sin c/>) dx + (cos 0) dy + (0) dc/> (= 0), (k) Virtual: S6 X = (—sin </>) Sx + (cos</>) Sy + (0) 8</> (= 0), (1) is nonholonomic. (ii) The fact that upon imposition of the constraints S6 x = 0, uq = 0, the transitivity equation (f) yields (69 2 )' — 6lo 2 = 0 does not mean that dd 2 = (cos</>) dx + (sin^) dy + ( 0 ) dc/) = vdt (^ 0 ), (m) or S9 2 = (cos c/>) Sx + (sin </>) Sy + (0) Scj) 0), (n) are exact; it does not mean that 0 2 is a genuine (Lagrangean) coordinate. For exactness, we should have 7 2 W = 0 (k, I = 1,2,3) => ( S9 2 )' — Su> 2 = 0, independently of the constraints ui l /d9 l /S9 l = 0. [We recall (§2.12) that Frobenius’ theorem tests the holonomicity, or absence thereof, of a system of Pfaffian equations of constraint ; whereas the exactness, or inexactness, of a particular Pfaffian form , like d0 2 and d9 3 {f 0) is a property of that form; that is, it is ascertained by examination of that form alone, independently of other constraint equations. In sum: constraint holonomicity is a system (coupled) property; while coordinate holonomicity is an individual (uncoupled) property .] WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (iii) Since cc 3 = dtp/dt = v r3 is a genuine velocity, 7 3 w = 0 (k,l= 1,2,3); as expected. Hamel Viewpoint versus Suslov Viewpoint So far, we have assumed Hamel’s viewpoint; that is, d(Sx) = 8(dx), d(8y ) = 8(dy), d(8cj)) = 8(d(f>)\ (o) and d{80i) f 8{dO x ), in spite of the constraint 80\ = 0 and dQ\ = 0 [and that even if di^86\) = 0, still -6(d0i) f 0!]. Let us now examine the Suslov viewpoint: with the analytically convenient choice, q D = y and q, = x,(j), we can rewrite the constraint as d9 x = dy — (tan </>) dx = 0 and 89 x = 8y — (tan </>) 8x = 0 [instead of (a)], or dy = (tan </>) dx + (0) dcf) and 8y = (tan f) 8x + (0) <5</>; (p) and, therefore, the corresponding transitivity equations become [instead of (d)-(g)] Dependent: d(8y ) — 8{dv) = d^xtan^) — <5(fi(xtan</>) = • • • = \d{8x) — 8{dx)\ tan tf>+ (1/cos 2 (f>)(d(t>8x — dx8(j)) = (1/cos 1 (j))(d4> 8x — dx8(j>) f 0, (q) Independent: d(8x) — 8(dx) = 0, d(8(j>) — 8(d(f) = 0; (r) from which we readily read olf the sole nonvanishing Voronets symbol: = -vrV = 1/ cos 2 ^>. (s) Under Hamel’s viewpoint, using the same variables, front <5y = (tan^>) <5x (i.e., 89 x =0) it follows that d(8y) = d(8x) tan <(>+(1/cos 2 <(>) df8x [i.e., d(80 x ) =0]; but from dy = (tan f) dx (i.e., dd x = 0) it does not follow that 8(dy) = 8{dx) tan </> — (1/cos 2 ^) 8cf>dx [i.e., 8(d9 x ) f 0]. Problem 2.13.1 Consider a knife (or sled, or scissors, etc.) moving on a uniformly rotating turntable T (fig. 2.17). In T-fixed (moving) coordinates O-xyf, its constraint is (sin^)^ + (— cos cj))v v = 0 [v x = dx/dt , v y = dy/dt]. (a) Show that in inertial (fixed) coordinates 0-XY<P, where X = (cos 6)x + (— sin 9)y + (0)0, Y = (sin 9)x + (cos 9)y + (0)0, <f> = (0)x+(0)y + (!)</> + 9, (b) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS and 6 = wt, to: constant angular velocity of O-xy relative to O-XY [i.e., say, X = X(x,y,t), etc.], the constraint takes the (acatastatic) form (with v v = dX/dt, Vy = dY/dt), (sin <P)v x + (— cos <P)v Y + w[(cos <P)X + (sin <P)Y] =0. (c) Example 2.IB.3 Rolling Disk—Vertical Case. Let us consider a circular thin disk D , of center G and radius r, rolling while remaining vertical on a fixed, rough, and horizontal plane P (fig. 2 . 18 ). (The general nonvertical case is presented later in ex. 2 . 13 . 7 .) This system has four Lagrangean coordinates (or global DOF ): the (x,y,z = r) coordinates of G, and the Eulerian angles cj) (precession) and ijj (spin). The constraints z = r (contact) and 6 = 7r/2 are, clearly, holonomic (H). The velocity constraint is v c = 0 (where C is the contact point); or, since along the fixed axes O-XYZ [with the notation dx/dt = v x , dy/dt=v y \ dcj)/dt = uu, d'f/dt = i tty]: v G = (v x ,v y , 0), CJ = (-Uty sin <j>, u^cos(/>, w 0 ), and r c/G = (0,0, -r), => v c = v G + O) x r c / G = ■ ■ ■ = (v x - rui^ cos cj>, v y - rw^ sin <j>, 0) = 0, (a) or, in components, in the following equivalent forms: Velocity: v x = r uy, cos </> and v y = ruj^ sin 0, (b) Kinematically admissible: dx = (r cos <fr) dip and dy = (r sin <p) dip, (c) Virtual: 6x = (rcosqi) Sip and 6y = (rsin^) Sip. (d) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS X - % cosp^ r\pcosp. y = V 0 xinj-ry simp; u «r (cosp, simp), n = (sirtp. cosp) Figure 2.18 Rolling of vertical disk on a fixed plane. As shown below, these constraints are nonholonomic (NH). Hence, the disk is a scleronomic NH system with / = n — m = 4 — 2 = 2 DOF in the small. It is not hard to see that imposition, on (b-d), of the additional H constraint dcp = 0 =>■ <j) = constant, say tp = 0 , would reduce them to the well-known H case of plane rolling : dx = r d'tp =>■ x = r ip + constant, and dy = 0 =>■ y = constant. {Also, the problem would become H if the disk was forced to roll along a prescribed O-XY path. For, then, the rolling condition would be [with s: arc-length along (c)] ds = rdip =>■ s = rip + constant, and (c) would yield the parametric equations x = x(s) and y = y(.s); that is, for each s there would correspond a unique x,y,ip, and cp [from (b-d)], and that would make the disk a 1 ( global) DOFH system.} Ad Hoc Proof of the Nonholonomicity of the Constraints (b-d) Let us assume that we could find a finite relation f(x,y,<p,ip ) = 0 , compatible with (b-d). Then (with subscripts denoting partial derivatives), we would have df = f x dx +f y dy +4 dtp +f^dip= 0. (e) Substituting dx and dy from (c) into (e) — that is, embedding the constraints into it — yields {rf x cos tp + rf y sin <p +4) dip + (/ 0 ) dtp = 0, (f) which, since now dip and dtp are independent, gives 4 = 0 f =f(x,y,ip) and rf x cos <p + rf y sin 0+4 = 0. (g) Next, (9/<9</>)-difTerentiating the second of (g) once, while taking into account the first of (g), yields — rf x sin tp + rf y cos tp = 0, (h) and repeating this procedure on (h), while again observing the first of (g), produces —rf x cos tp — rf. sin tp = 0. (i) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS [Further (d/<90)-differentiations would not produce anything new.] The system (h), (i) has the unique solution, fx = 0 and f y = °, (j) due to which the second of (g) reduces to = 0. It is clear that the above result in / = constant, and such a functional relation, obviously, cannot produce the con¬ straints (b d) — no f{x,y,(j>,tp) exists. Geometrically, this nonholonomicity has the following consequences: Starting from a certain initial configuration, we can roll the disk along two different paths to two final configurations with the same contact point—namely, same final (x,y), but rotated relative to each other, that is, with different final (0,0). If the constraints were H, then 0 and 0 would be functions of (x,y) and the two final positions of the disk would coincide completely. Proof that the Constraints (b-d) are NH via Frobenius’ Theorem Let us rewrite the two constraints (c, d) in the equilibrium forms: Kinematically admissible : dd l = dx — (rcos0) dfi = 0, dd 2 = dy — (rsin0) dif = 0, (k) Virtual: 89\ = 8x — (rcos0) Sift = 0, 89 2 = 8y — (rsin0) Sip = 0. (1) It follows that the corresponding bilinear covariants (2.8.2 ff.) are d{89 ]) — 8{d9f) = • • • = (rsin0)(d0<50 — dip8cf>), (m) d(89 2 ) — 8{d9 2 ) = ■ ■ ■ = (—r cos <f>)(d(p8ij.’ — dfidfi), (n) and, clearly, these vanish for arbitrary values of the independent differentials dc\>,8c\),dfi,8fi, if sin0 = O and cos0 = 0. But then the constraints (c) reduce to dx = 0 => a: = constant and dy = 0 => y — constant, which is, in general, impossible. Hence, the constraints are NH [one can arrive at the same conclusion with the help of the 7 ’s (§2.12), but that is more laborious]. Problem 2.IB.2 Continuing from the previous problem (vertically rolling disk), show that its velocity constraints can be expressed in the equivalent form: v G • u = v x cos 0 + v y sin 0 = ru>^, (a) v G -n = — y Y sin0+ v_ F cos0 = 0, (b) where u and n are unit vectors on the disk plane (parallel to O-XY) and perpendi¬ cular to it, respectively (fig. 2.18). [Notice that (b) coincides, formally, with the knife problem constraint.] Example 2.13.4 Rolling Sphere — Introduction. Let us consider a sphere of center G and radius r, rolling without slipping on a fixed, rough and, say, horizontal plane P (fig. 2.19). The complete specification of a generic sphere configuration requires five independent (minimal) Lagrangean coordinates. As such, we could take the (inertial) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS coordinates of G ( X , Y), and the three Eulerian angles (</>, 9, ip) of body-fixed axes G-xyz relative to translating (nonrotating) axes G-XYZ. The contact constraint is expressed by the holonomic (H) equation, Z = vertical coordinate of G = r. The rolling constraint is found by equating the (inertial) velocity of the contact point of the sphere C with that of its (instantaneously) adjacent plane point, which here is zero; us = inertial angular velocity of sphere. Using components along O-XYZ axes throughout, we find v c = v g + m x r c/G = dPc/dt + (*> x (— rK) = ( v Xi v Yi 0) + ( w Xi w z) x (0,0, —r) = • • • = (v x — rujy, v Y + ruj x , 0) = 0. Hence, the rolling conditions are Vj - ruy = 0, v Y + rujx = 0; (a) or, expressing the space-fixed components u> x Y in terms of their Eulerian angle rates (§ 1 - 12 ), Vx _ r(sin(j)u)g — sin 9 cost/) uifi) = 0, v Y + r(cos()iw e + sindsin(/uu^,) = 0; (b) or, further, in kinematically admissible form, dX — r(sm(j)d9 — sin 9 cos (f>dip) = 0, dY + r(cos cj)d9 + sin 9 sin (f>dip) = 0; (c) or, finally, since these constraints are catastatic, in virtual form, 8X — r(sin $ 69 — sin 9 cos <j> 6ip) = 0, <5T+ r(coscj)89+ sin 9 sin (f> 8i/j) = 0. (d) [Absence of pivoting would have meant the following additional constraint: ( <u ) normal to sphere atC ^ij> T COS 0 UJ t p 0, or d4> + cos 9 dip = 0 =>■ d(f>/dip =—cos 9 = h(9)\. WWW.EBOOK777.COM (e) §2.13 GENERAL EXAMPLES AND PROBLEMS As shown later, the constraints (a-d) are nonholonomic (NH). [We already notice that (c), for example, do not involve d<p, and yet the constraints feature sm<j> and cos (/>.] Mathematically, this means that it is impossible to obtain them by differen¬ tiating two finite constraint equations of the form F(X, Y, <p, 9, ip) =0 and E(X, Y, <p, 9, ip) = 0; that is, the coordinates X,Y,(p,9,ip are independent. But their differentials dX, dY, dtp , d6, dip, in view of (a-d), are not independent; that is, in general, only three of them can be varied simultaneously and arbitrarily. We say that the sphere has five DOF in the large, hut only three DOF in the small: f = n — m = 5 — 2=3. (Had we added pivoting, we would have / = 2.) Kinematically, the above mean that the sphere may roll from an initial config¬ uration, along two different routes, to two final configurations, which have both the same contact point and center location (i.e., same X, Y), but different angular orientations relative to each other (i.e., different cp,9,ip). If the constraints (a-d) were holonomic—for example, if the plane was smooth —it would be possible to vary all X , Y, (p, 9 , ip independently and arbitrarily without violating the (then) con¬ straints; namely, the sphere’s rigidity and the constancy of distance between G and C. Further, the sphere can roll from any initial configuration, with the sphere point Cj in contact with the plane point P h to any other final configuration, with the sphere point Cf in contact with the plane point Pf. To see this property, known as acces¬ sibility (§2.3), we draw on the plane a curve ( 7 ) joining C, and Pf, and another curve on the sphere (6), of equal length to ( 7 ), joining C, and Cf. Now, a pivoting of the sphere can make the two arcs ( 7 ) and (6) tangent, at C, = P h Then, we bring Cj to Pf by rolling ( 6 ) on ( 7 ). A final pivoting of the sphere brings it to its final configuration (see also Rutherford, 1960, pp. 161-162). A Special Case Assume, next, that the sphere rolls without pivoting, and also moves so that 9 = constant = 9 0 . Let us find the path of G. With 9 = constant =>• dQ = 0, the roll¬ ing constraints (c) reduce to dX + r(sin0 o ) cos <p dip = 0, dY + r(sin9 0 ) sirup dip = 0; (f) and the no-pivoting constraint (e) to dcp/dip = — cos 9 a = constant. (g) This leaves only n — m = 5 — 4 = 1 DOF in the small. Taking <p as the independent coordinate and eliminating dip between (f), with the help of (g), yields dX = r(tan 9 0 ) cos <p dcp, dY = r(tan 9 0 ) sin (p d(p, (h) which integrates readily to the curve (with X 0 and Y„ as integration constants): X — X 0 = r(tan0 o ) sin/ Y — Y 0 = — r(tan 9 0 ) cos cp; (i) that is, G describes, on the plane Z = r, a circle of radius r tan0 o . [These considerations also show how imposition of a sufficient number of WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS additional holonomic and/or nonholonomic constraints turns an originally non- holonomic system into a holonomic one.] Example 2.IB.5 Rolling Sphere on a Spinning Table — Introduction. Let us extend the previous example to the case where the plane P is not fixed, but rotates about a fixed axis OZ perpendicular to it with, say, constant (inertial) angular velocity 12. In this case, the rolling condition expresses the fact that the contact points of the sphere and the plane, C, have equal inertial velocities: ( ^c) sphere = Oc)pl an e : V G + « x r c/G = Qx V c/0 (= Q X p)' (a) or, in terms of their components along inertial (background) axes O-XYZ / O-IJK: (vx,vy,0) + (u}x,to Y ,toz) x (0,0, —r) = (0,0,0) x ( X , Y, 0), (b) from which we easily obtain the two rolling conditions: v y — rujy = — Of, Vy T rujx = QX. (c) Next, expressing wj, u> Y in terms of their Eulerian angles (between translating/ nonrotating axes G-XYZ and sphere-fixed axes G-xyz ) and their time rates, as in the preceding example, we transform (c) to v x — r(sin (puj e — sin 9 cos </> ojfi) + QY = 0, v Y + r(cos (picg + sin 9 sin <p ujfi) — QX = 0. (d) The O-proportional terms in (d) are the acatastatic parts of these constraints, and arise out of our use of inertial coordinates to describe the kinematics in a noninertial frame; had we used plane-fixed (noninertial) coordinates, the constraints would have been catastatic in them. It is not hard to see that the kinematically admissible (possible and virtual forms of these constraints are, respectively (note differences between them resulting from constraint 6t = 0), iIX — r(sin <f>d9— sin 9 cos <j) dip) + (Q Y) dt = 0, dY + r(cos 4>d9 + sin 9 sin <j) dip) — (QX) dt = 0; (e) 6X — /-(sin tp69 — sin 9 cos </> Sip) = 0, 6Y + r(cos cp 89 + sin 9 sirup 6ip) = 0. (f) Example 2.13.6 Rolling Sphere on Spinning Table — the Transitivity Equations. Continuing from the preceding example, let us show that its rolling constraints (c-f); as well as those of its previous, stationary table case) are nonholonomic; that is, the system has n = 5 DOF in the large , and / = /; — /// = 5 — 2 = 3 DOF in the small. In view of the structure of these constraints, we choose the following equilibrium quasi velocities (with the usual notations: dX/dt = v x , ■ ■ ■, dcp/dt = ...): WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS Dependent: Ui = v x — rujy + QY = v x — r(sin<j)ujg — cos</>sin 0uty) + QY = v x — rcj 4 + QY (=0), (a) oj 2 = Vy+ ru> x — C2X = v Y + r(cos (j>uig + sin (f> sin 9 uX) — QX = v Y + rui 3 — QX (= 0), (b) Independent: w 3 = u> x = (cos <j>)u) e + (sin cj) sin 9)u ^ (^ 0), (c) UJ 4 = u> Y = (sin (j))u s + (—cost/) sin (^ 0 ), (d) w 5 = w z = ( 1 )^V + (cos 9)u>^ ( 7 ^ 0 ), (e) u) 6 = dt/dt= 1 ( isochrony ). (f) Recalling results from §1.12, we readily see that these partially decoupled equations invert to '’1 = Vx = uj\ H- y CJ4 — £2 Y (without enforcement of constraints u> x 2 = = 0), (gl) = Vy = UJ2 — ? ^3 + QX (without enforcement of constraints uq 2 = = 0), (g2) v 3 = ^ = (— cot# sin 0)u; 3 + i (cot 9 cos (j))uj 4 + u 5 , (g 3 ) v 4 = UJg = (cos </>)w 3 + (sin 4 >)ui 4 , (g 4 ) V 5 (sin (j)/ sin0)u; 3 + (- -cos (j>/ sin 9)lo 4 , (g 5 ) v 6 = dt/dt = U) 6 =l. (g6) The virtual forms of (a-g 6 ) are as follows [note absence of acatastatic terms in (hi, 2 )]: Dependent: 69 { = 6X— r69 Y = 6X+ (—r sine/)) 69+(r cos ft sin 9) 6ip = 6X — r69 4 (=0), (hi) 69 2 = 6Y + r69 x = 6Y + (rcosc/)) 69 + (rsin <?!> sin0) 6ip = 6Y + r69 3 (=0), (h2) Independent: 69 2 = 69x = (cos cj)) 69 + (sin cj)sin 9) 6 i/j (=/= 0), (h3) 69 4 = 69 Y = (sin cjs) 69+ (—cos cj) sin 9) 6tp (^ 0), (h4) 69 5 = 69 z = (1) 6<j) + (cos 9) 6 r/> (^ 0), (h5) 69 6 = 6 q 6 = 6 t = 0 (isochrony)-, (h 6 ) 6 q\ = 6 X = 69\ + r69 4 , (il) 6 q 2 = 6 Y = 69 2 -r69 3 , (i2) 6 q 3 = 6 (j> = (— cot 9 sin (js) 69 3 + (cot 9 cos 0 ) 69 4 + 69 5 , (i3) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 8q 4 = 80 = (cos (j>) 80 3 + (sin c/>) 89 4l (i4) 8q 5 = 8ip = (sin^/ sin#) 89 3 + (— cos 0/sin#) 89 4 , (i5) 8q 6 = 8t = 89 6 = 0. (i6) Now we are ready to calculate Hamel’s coefficients from the transitivity equations (§2.10): (86 k ) ' - 6u) k = ^ Y ^ r P “P S0 r = YY 6d >■ + Y ^ ' S9 " 0) where k,r,s = /? = 1, - - -, 6; J k r = 1 k r ,„+i = 7^6- By direct differentiations, use of the above, and the indicated shortcuts [and noting that, even if Q = Q(t) = given function of time, still 8Q = 0], we obtain, successively, (89 1 )' — Scoi = (8X — r89 Y ) — 8(v x — raj Y + QY) = [(<Wf)' - 8v x ] - r[(89 Y y - Su Y \ -Q8Y = 0 — r[(89 4 y — 8 uj 4 ] - Q8Y [invoking the rotational transitivity equations (§1.14 and ex. 2.13.9), and (i2)] = —r(c Oz 80x — ^x 89 z ) — Q(892 — y 89 3 ) = —r(uj 5 80 3 — u> 3 89 5 ) — Q(89 2 — r89 3 ), or, finally, (89 1 )’ — 8uj\ = (—r)uj 5 86 3 + (r)u> 3 89 5 + (-Q) 89 2 + (r(2) 89 3 \ (kl) (89 Y)' — 8lu 2 = (8Y + r89 x )' — <5(vy + ruj x — QX) = [(h7)- - 8vy} + r[(89 x y - 8u x ] + Q8X = 0 + r[(89 3 y - 8 cj 3 ] + Q 8X [invoking again the rotational transitivity equations and (il)] = v(iOy89/ — (jJ/ 80 y ) 4 - C2(80\ 4 - y89 4 ) = y(co 4 89 3 — cu 3 89 4 ) + Q(89\ 4 ~ y 89 4 ) , or, finally, (<5#2)" — 8 co 2 = (— r)ui 3 89 4 + (y)lu 4 89 5 + (Q) 89\ + (r£2) 664 ] (k2) and, again, the rotational transitivity equations (with X — >3, Y -► 4, Z - -> 5) give ( 86 3 )' — 8 cj 3 = w 4 89 5 — qj 5 89 4i (k3) (89 4 )' — 8 ui 4 = uj 5 89 3 — u > 3 89 5 , (k4) (89 Y) — 8 uj 3 = lo 3 89 4 — cu 4 89 3 . (k5) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS Comparing (j) with (kl-5) we readily find that the nonvanishing 7 ’s are 1 1 1 1 _ 1 r, 1 1 _ 1 r, / n \ 7 35 = -7 53 = -C 7 26 = “7 62 = 7 2 = ““, 7 36 = ~7 63 = 7 3 = ' ^ l 11 ) 7 2 45 = - 7 2 54 = -r, 7 2 16 = - 7 2 61 = 7 2 1 = Si, 7 2 46 = - 7 2 64 = 7 2 4 = r Q; (12) 7 3 45 = - 7 3 54 = 7 4 53 = - 7 4 35 = 7 ? 34 = - 7 ? 43 = ~ 1 [= -1 {permutation symbol )]. (13) Here, Dependent) = 1,2 and I, I'independent)) = 3,4,5. Therefore, l D w ; 7 1 35 = —^ 0,7 2 45 = —^0; 7°/ : 7 ' 3 = 7 2 4 = ^ ^ 0; (m) and so, according to Frobenius’ theorem (§2.12), the system of Pfaffian constraints u>i = 0 and uj 2 = 0 is nonholonomic, in both the catastatic (rolling on fixed plane) and acatastatic (rolling on rotating plane) cases; that is, for any given Q = Q(t). Example 2.13.7 Rolling Disk on Fixed Plane. Let us consider a thin circular disk (or coin, or ring, or hoop), of radius r and center G, rolling on a fixed horizontal and rough plane (fig. 2.20). A generic configuration of the disk is determined by the following six Lagrangean coordinates: X, Y,Z: inertial coordinates of G; Eulerian angles of body-fixed axes G-xyz relative to the cotranslating but nonrotating axes G-XYZ (similar to the rolling sphere case). /' = costj) I + siiuj) J. j' = cos8 m a , + sin8 K. k' = -sin8 u N + cos8 K; u N = -snuj) I + cost/) J, u n = i' Figure 2.20 Geometry and kinematics of circular disk rolling on fixed rough plane. Axes: C-nNZ = C-x’NZ: semifixed; C-nn'z = G-x'y'z 1 : semimobile; G-xyz: body axes (not shown, but easily pictured); C-XYZ-. space axes. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS [In view of the complicated geometry, we avoid all ad hoc, possibly shorter, treat¬ ments, in favor of a fairly general and uniform approach. An alternative description is shown later.] The vertical coordinate of G, clearly, satisfies the holonomic constraint Z=rsind, (a) and this brings the number of independent Lagrangean coordinates down to five'. X , Y, <j>, 9, ?/>; that is, n = 5. The rolling constraint becomes, successively, 0 = v c = v G + co x r c / G (to: inertial angular velocity of disk) = V G - !'(,) X ) = v G — r(u> x ’i' + ujy'j 1 + u z 'k') x j' (semimobile o-decomposition) = v G — ru> x ’{i' x j') — ruj z i(k' x j') = v G -rUx'{k')-ru) z '(-i), (b) from which we obtain the constraint components along the two “natural" (semi¬ fixed) directions n(i') and N(u n ): (i) 0 = v c -i' = v G •/' + ru> z i or v G] „ + rw,< = 0 ; (cl) (ii) 0 = v c • u N = v G ■ u N - ru> x '{k' • u N ) + ru z >(i' ■ u N ) = v G ■ u N - ru x '{k' • u N ), or v GN — rw x i cos( 7 r /2 + 9) = v GtN + ruj x t sind = 0. (c2) The third semifixed direction component gives the earlier constraint (a): 0 = v c -K = v G -K-ru x '(k' •K) + ru z '{i'-K) = v G • K - rw x >(k' ■ K), or, since w x i = ui e , 0 = t g ,z — rtog cos 9 =>■ dZ — r cos 9 d9 =0 => Z — r sin 9 = constant —> 0. Equations (cl, 2) contain nonholonomic velocities. Let us express them in terms of holo¬ nomic velocities exclusively. It is not hard to see that, with v G = (X, Y,Z ) = (vx, Vy. vz), (i) u> x i = Log, u y ' = (sind)a; 0 , uy = (cos 0 )w^ + uy; (dl) (ii) v Gi „ = v G -i' = v G -u„ = (y x l + v Y J + v z K) • (cos(j)I + sin </>/) = (cos^)vx + (sin</>)v y ; (d2) (iii) Vg,n = v g -u n = (v x I + v Y J + v z K) • (— sind>7 + cos 4>J) = (— sind>)vx + (cos^)vy. (d3) With the help of (d 13), the constraints (cl, 2) take, respectively, the holonomic velocities form: (i) v c -i = v c> „ = (cos^)vx + (sin 0 )v y + r(oty + cos 0 c^) = 0 , (el) (ii) v c ~u N = v CjA r = (— sin 0 )vx + (coscjfivy + rsinOug = 0 . (e 2 ) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS In view of (el, 2), we introduce the following equilibrium quasi velocities: Dependent: w i = v c,n — v G,n + ruJ z' = (cos</>)vy + (sin<^)vy + (rCOS 0 )cU 0 + (r)u>^ (= 0 ), (fl) u> 2 = v c,n = v g,n + rsinduv = (— sin</>)vy + (cos</>)vy + (rsm9)ajg (= 0 ); (f 2 ) Independent (semimobile components of o>): = W„ = W x ' = Ug (^0), (f3) UJ 4 = w„' = Uy> = (sin 6)u}^ 0), (f4) W 5 = U) z = LJ Z ' = (cos 9)u>q 1 + Ufy (^ 0). (f5) These catastatic, and partially uncoupled, equations invert easily to V] = v x = (cos^)wi + (— sin0)(u 2 + (r sin 0 sin </>)w 3 + (— rcos<fi)u) 5 , (gl) v 2 = v Y = (sin<^)u;i + (cos^>)w 2 + (—rsin0cos^>)w 3 + (—rsin0)w5, (g2) v 3 = (l/sin0)u; 4 , (g3) v 4 =uj e = u} 3 , (g4) V5 = kty = u 5 - (cot 6)u} 4 . (g5) Below, we show that the constraints u>i = 0 and w 2 = 0 are nonholonomic; that is, n = 5 global DOF, m = 2 —> / = n — m = 3 local DOF. Indeed, by direct d- and ^-operations on (fl g5), and their virtual forms (which can be obtained from the above velocity forms in, by now, obvious ways), and combination of simple shortcuts with some straightforward algebra, we find, succes¬ sively, (MiY - < 5^1 = [(Sp Gi „y - <5v G] „] + r[( 66 5 Y - &u 5 ] [where dp Gn = v Gj „<*] = ■ • • = {[(cos(/>) + (sin ^>) 6 Y\ — 6[(cos0)vy + (sin0)v y ]} T r( uj 4 69 3 — (u 3 69 4 ) = • • • = {w 0 [(— sin^) 6 X + (cos 0) 6 Y] — 6 cj)[(— sin0)vy + (cos^)vy]} T /' ( UJ 4 (50 3 — CU 3 b0 4 ) = (^ Sp G jf - v Gi y <5<)>) + r(uj 4 69 3 - w 3 <50 4 ) [where dp GN = v G A , <fr] = [(cu 4 / sin 0) 6p G2 ^ — Vg,n(^® 4 / sin 0)] -t- r(c o 4 69 3 — co 3 <50 4 ) = (cu 4 / sin 9)(6p GN + rsin9 69 3 ) — (69 4 / sin0)(v GA r + /-sin0w 3 ) = (1 / sin 9) (uj 4 6 p CN - v CjN 69 4 ) [where dp c , N = v CjA r dt\ = (1 / sin 9) ( 0 J 4 69 2 — tu 2 69 4 ) = 0 (after enforcing the constraints 60 2 ,uj 2 = 0), (hi) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (69 2 ) — 6 u >2 = [{ 8 p Gi n) — ^ v g,n\ + [(rsin#60 3 ) — <S(r sin0(u 3 )] [since w 3 sin 9 = uj s sin 9 is integrable, the second bracket term vanishes] = [(— sin (j>) 8 X + (cos (j>) 8 Y\ — 6 [(— sin^)v z + (cos</>)vy] = — lj^cos cj) 6 X + sin (j> 8 Y) + 6 tj>[( cos(f>)v x + (sin^)vy] = 8 p G „ + 8 <j> v Gt „ = — (u> 4 / sin0)(^0! — rS9 5 ) + (<50 4 / sin#)(u 7 — ruj 5 ) = — ( 1 / sin 0 )(u ; 4 89 x — 07 ^a ) + ( r / sinff)(w 4 89 5 — u 5 89 4 ) = (r/ sin 6 )(u >4 89 5 — u > 5 89 4 ) ^ 0 (even after enforcing the constraints £ 0 1 ,u 7 =O); (h 2 ) (80 3 y — Slot, = 0 (independently of constraints) =>- 6> 3 = holonomic coordinate, (h3) ( 89 4 )' — 8 u >4 = (sin 989)' — 6(sin9ut^) = (cos 9) 8 cj) — 89) = (cot 9) (w 3 89 4 - 0 J 4 89 2 ), (h4) ( 89 $)’ — 8 cj 5 = ( 8 ip + cos 98(f))' — 8 (w^ + cos = (sin 9){u>^ 89 — ujg 8 c/)) = 0 J 4 89 2 — cn 3 894 ', (h5) and since Z = r sin 9, with uj 6 = v z — r cos 9 ujg => 89 6 = 8 Z — r cos 9 89, we get (89 6 y- 8 uj 6 = ( 8 Z - rcos989)'~ 8 (v z — rcos9aj e ) = { 8 Zy- r(cos9)'89 — r cos 6(89)' — 8 v z + r(— sin0) 89tog + r cos 9 8 ujg = 0, (independently of the other constraints) as expected. From the above, we immediately read off the nonvanishing 7 ’s: 7*24 = 7^ 42 = 1/sin 9- 7 2 4 i = - 7 2 i 4 = 1/ sin 6 », 7 2 54 = ~ 7 2 45 = r/ sin 6 >; 7 4 43 = - 7 4 34 = cot0; 7 5 34 = -7 5 43 = 1- (i) Flere, Dependent) = 1,2 and /, I'(independent) = 3,4,5. Therefore, 7V : 7 2 54 = ''/sin 6 >/ 0 ; (j) and so, according to Frobenius’ theorem (§2.12), the system of Pfaffian constraints 07 = 0 and u > 2 = 0 is nonholonomic. We also notice that to calculate all nonvanishing 7 ’s, we must refrain from enforcing the constraints u 7 , 89 j = 0 and uj 2 , 89 2 = 0, in the earlier bilinear covariants. Rolling Constraints via Components along Space Axes With reference to tig. 2.20, we have, successively, r c / G = — (rcos9)u N — ( rsin9)K = (— rcos9)(— sin (/> / + cos </>/) + (— rsin9)K = (r cos 9 sin <j>)I + (—r cos 9 cos tj>)J + (—r sin 9)K, (kl) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS o) = oj x I + u Y J + w z lf [recalling formulae in §1.12] = [(cos (j))ug + (sin cj) sin 9)to^\l + [(sin (j))u e + (— cos 4> sin d)co^]J +[w, i> + (cos 9)u)^,\K, (k 2 ) and, of course, V'c; = V X I + V Y J + V Z K. (k3) Substituting these fixed-axes representations into the constraint (b): 0 = v c = v G + a) x r c / G , and setting its components along I,J,K, equal to zero, we obtain the scalar conditions: v x + r( cos (j> cos 9u) z — sin 9 to Y ) = v x + r[(cos cj) cos d)co >0 — (sin 0 sin 9) lu b + (cos 4>) u>^\ = 0, v y + r(sindo;x + sin</>cosdu; z ) = v Y + r[(sin c/)COS 9)u) ( j > + (cos</)sind)u ; 9 + (sin^)u;^] = 0, v z — r cos 9 (cos (j)Lu x + sin (j)uj Y ) = v z — rcos9ui s = 0 => Z = rsia9 (i.e., holonomic). (k4, 5,6) We leave it to the reader to verify that (k4, 5) are equivalent to the earlier (el f2); and, also, that they can be brought to the (perhaps simpler) form, [{X/r) + sin 0 cos d]' + (cos = 0, [(Y/r) + cos ^ cos 0]' + (sin^)ufy = 0. (k7) Constraints and Transitivity Equations in Terms of the (Inertial) Coordinates of the Contact Point of the Disk (X c , Y c ) [This is a popular choice among mechanics authors (e.g., Hamel, 1949, pp. 470 ff., 478-479; Rosenberg, 1977, pp. 265 ff.) but our choice — that is, in terms of the coordinates of the disk center, G — shows more clearly the connection with the Eulerian angles.] Taking the fixed-axes components of the obvious relation r G = r c + v G / c , and then d/dt(. . ^-differentiating them, we obtain (consulting again fig. 2.20, and with v c,x — dX c /dt , v C Y = dY c /dt) (i) X = X c — (rcos9) sin</> => v x = v CjX — (rcost/)cos9 )u>j, + (rsm(f>sm9)u}g, (11) (ii) Y= Y c + (r cos 9) cos (/) => v Y = v c Y - (rsin^cos#)^ — (rcos^sind)o; e ; (12) and substituting these v x , v Y expressions into (k4, 5), respectively, we eventually obtain the simpler forms v cx + (r cos(/>)uty = 0 and v CY + (r sin0)w^ = 0. (13) (The above can, also, be obtained by ad hoc knife problem-type considerations.) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS In view of (13), we introduce the following new equilibrium quasi velocities: Dependent: = T CjX + (rcos^Wy, (=0), (ml) W 2 = Vc,Y + ( r sin 4>)u)^ (= 0); (m2) Independent: w 3 = 0) => q 3 .. = 0, (m3) w 4 = w e (^0) => 7 4 .. = 0, (m4) w 5 = (cos0)v CiZ +(sin0)v Cj y [= by (13)]; (m5) or, instead, the equivalent but simpler, knife-type, quasi velocities: Dependent: Qi = {- sin4>)v c ,x +{cos(j))v CiY [= 0, by (13)], (nl) Q 2 = r + (cos <f>) Vc,x + (sin d 1 ) Vc, r = t + W 5 (= 0 ); (n 2 ) Independent: 0 3 = ( 7 ^ 0 ), (n3) Q A = ui A = uj e (y^O), (n4) ^5 =^ 5 = (cosd)v c ,x +(sind)Tc,r [=-r + Q 2 =-r u$). (n5) Inverting the above yields vi = v c ,x = (— sin 4>)Q\ + (cos <j))Q 5 , (ol) v 2 = v CY = (cos^)^ + (sin</))i 2 5 , (o 2 ) v 3 = W 0 = f2 3 , (o3) V 4 = UJg = Q 4 , (o4) v 5 = = ( l / r )(^2 — ^ 5 )- (° 5 ) By direct d/d-dififerentiations of (nl-5), use of (ol-5), and the obvious notation d0 k = Q k dt; k= 1,...,5, we obtain the corresponding transitivity equations as follows: {80 { )' — 8 Q 1 = [(— sind) 8 X C + (cosd>) 8 Y C \ - <5[(— sind>)v cz + ( cos ( t ) ) v c,Y} = ■ ■ ■ = cos 4>{ v c,x 8<j> — 8X C ) + sin 4>{vcj 8<f> — w^SYc) = cos </>[(— sind>f2[ + cos <j)Q 5 ) 80 3 — f2 3 (— sin c/)80\ + cos cj)80 5 )] + smd>[(cosd>f 2 [ + sind>f2 5 ) 80 3 — ^(cosd’iS©! + sind>h@ 5 )] = Q 5 80 3 -Q 3 80 5 , (pi) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS (<50 2 ) — <5f2 2 — [r fi'ip T (cos <j>) 8X G -\- (sin (j>) (iF^j — 8\rui^ T (cos (jf^c.x T (sin </>) Vc,r] = • • • = sin f(v G x 8f — 8X G ) -t- cos <j>(ca^, 8Y g — v G y 8(f) = sin <)>[(— sin (/>£?! + cos </> 0 5 ) <50 3 — f2 3 (— sin</>5@i + cos <j><50 5 )] + cosc/)[Qt,(cos(/)6&\ + sin f8Qf) — (cos^Oi + sin^0 5 ) 60 3 ] — f2 3 60\ — C2\ <50 3 (= 0, upon imposition of the constraints <50,, Q x = 0), (p2) (<50 3 )‘ — 6Q 3 = (8(f)' — 8ui^ = 0 (0 3 = holonomic), (p3) (<50 4 )' — 8Q 4 = (66)' — 8 uig = 0 (0 4 = holonomic) , (p4) (<50 5 )’ — <5f2 5 = [(cos</>) <LF C + (sin0) <5F C ]' - 5[(cos^)v C x + (sin<j>)v C y] = (-r&tpy - «(-r^) + (Se 2 y-SQ 2 = O+Q 3 80 X -Q x 80 3 — Qj 80 x — Q\ <50 3 (= 0, upon imposition of the constraints ^0 I ,f2 1 =O); (p5) that is, just like the knife problem (ex. 2.13.2), all the 7 ’s are either ±1 or 0. Finally, here, Dependent ) = 1,2 and /, I'independent) = 3,4,5. Therefore, l D n' ■ 735 = -753 = 1 ^0; (j) and so, by Frobenius’ theorem (§2.12), the constraint system Q x = 0 and f2 2 = 0 is nonholonomic. Problem 2.13.3 Rolling Disk in Accelerating Plane. Continuing from the preceding example, show that if the plane translates (i.e., no rotation), relative to inertial space, with a given velocity ( v x (t ), Vy(t), v z (t)), and the new inertial axes O-XYZ are chosen so that OZ is always perpendicular to the translating plane, and X, Y, Z are the new inertial coordinates of the center of the disk G, then the rolling constraints take the rheonomic form (cos<£)[Fy - v x (t)\ + (sin <j>)[V Y - v Y (t)\ + (rcos0)w 0 + (r)u^ = 0, (a) (— sin 4>)[V x - v x (t)\ + (cos<^)[F y - v Y (t)\ + (/• sin 6 ») cd 0 = 0 , (b) where V x = dX/dt, V Y = dY/dt; = d<j>/dt, and so on; that is, they are the same as in the fixed plane case, but with v G replaced with v G — v c (where v c is the inertial velocity of contact point of disk with plane). Example 2.13.8 Pair of Rolling Wheels on an Axle. Let us discuss the kinematics of a pair of two thin identical wheels, each of radius r, connected by a light axle and able to turn freely about its ends (tig. 2 . 21 ), rolling on a fixed, horizontal, and rough plane. For its description, we choose the following (six —>) five Lagrangean coordinates: (X, Y,Z = r): inertial coordinates of midpoint of axle, G; f: angle between the O-XY projection of the axle (say, from G" toward G') and +OX; ifi', ill"', spin angles of the two wheels. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Figure 2.21 Rolling of two wheels on an axle, on fixed plane. Here, the constraints are v c > = 0 and v c „ = 0, where C' and C" are the contact points of the two wheels. However, due to the constancy of G"G' (and C"C' = 2b) and the continuous perpendicularity of the wheels to the axle, these conditions translate to three independent component equations, not four; say, the vanishing of v c » and v c » along and perpendicularly to the axle (the “natural” directions of the problem). Let us express this analytically: since V C' — V G' + °V x r C'/G' — ( V G + 0>A x r G'/G ) + °V x r C'/G' \(D W ' and m A : inertial angular velocities of first wheel and axle, respectively] = (vx, vy, 0) + (0,0, ty,) x ( b cos 4>, b sin </>, 0) + (ay cos <£, ay sin <j>, w 0 ) x (0,0, -r) = (v x — b a^ sin (j) — r ay sin 0, v Y + b cos (j> + r ui y cos <j>, 0); (a) and similarly, for the second wheel [whose inertial angular velocity is co w » = (ty,» cos </>, uy< sin u^)], v C " = (vy + buifiSmcj) — ray/ sin </>, v Y — bto^ cos(/) + ray/ cos (j), 0); (b) the constraints are (with u ... for unit vector): 0 = v c > = v c / • u„ = v c , • (cos 4 >, sin <j>, 0) = v c> = v c » • u n = v c „ • (cos <j>, sin qb, 0), or v C ',n = v C",n = V_yCOS(/) + v y sin<() = 0; (cl) and v C ',t = v c' • u t = v c' • (— sin 0, cos 4 >, 0) = — v x sin< j>+ v Y cos(/) + bco ( / ) + ray = 0, (c2) v C ",t = v C " ■ u, = v c » • (— sin qb, cos cf), 0) = — v x sin 4> + v Y cos <f> — bui^ + ry» = 0. (C3) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS [The above can also be obtained by simple geometrical considerations based on fig. 2 . 21 .] By inspection, we see that (c2, 3) yield the integrable combination 2buj^ + r(uj^ - = 0 =>■ 2b 4> = c - r(ip' - ip"): [c = integration constant, depending on the initial values of ft"]. (d) Hence, we may take X, Y, ip', ip", as the minimal Lagrangean coordinates of our system, subject to the two knife-like nonholonomic (to be shown below) constraints v C',n = v C",n = v x cos </> + v y sin </> = 0; (el) Vc',t = -Vysin^-I- vy cos (j) T (r/2)(uy + uy<) = 0; (e2) that is, n = 4, m = 2 =>■ f = n — m = 4 — 2 = 2 DOF in the small , and 4 DOF in the large. In view of (el, 2), we introduce the following equilibrium quasi velocities: uq = (cos(j>)vx + (sin^)vy (=0); (fl) w 2 = (-sin</>)v A - + (cos^)vy (^ 0); (f2) (^0); (f3) uq = 2(—Vjy sin cf> + Vy cos </>) + r{ut^i + uy,») (= 0); (f4) w 5 = 2 bto^ + r (— uty») (= 0; tu 5 = holonomic velocity); (f5) which invert easily to v x = (cos (f))iOx + (— sin <p)ui 2 , (f6) Vy = (sin0)uq + (cos</>)w 2 , (f7) w 4 , = (0)uq + ( 0 )(jJ 2 + ( 1 )^ 3 , (f8) = (l/2r) (—2cu 2 — 2r u> 2 + uq + uq), (f9) = (1 /2r)(—2 w 2 -f- 2ruq -T uq. — uq). (flO) Comparing the above with the quasi velocities of the knife problem (ex. 2.13.2), to be denoted in this example by lo k , we readily see that we have the following corre¬ spondences: K K K / y, 1 \ CDi —S- CD 2-> w 2 —> 1, Uq —> U> 3. (ill) Hence, and recalling the transitivity equations of that example, we find {86 — <Suq = {89 k 2 )' — 8uj k 2 = lo k 2 86 k x - u> K x 89 K 3 = uq 86 2 — uq 89 2 (^ 0), (gl) (86 2 )' — 8uj 2 = {89 k [)' — 8uj K \ = lo k 2 89 k 2 — u/q 89 k 2 = uq 89 2 — uq 89\ {=/= 0), (= 0, after enforcement of the constraints 89 \, uq = 0), (g2) {80^)' — 8oj 2 = 0 ( 0 3 = (j) = holonomic coordinate), (g3) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS ( 504 ) - 5w 4 — 2[(502) — boj 2 \ + r\(8i/j + 6 i/j ) — 5(afy' T ^ 5 //')] = 2(uq50 3 — w 3 50[) + 0 (= 0, after enforcing 50! ,04 = 0), (g4) (50 5 ) ' - 5w 5 = 2b[{8cj)y - 5w 0 ] + r[(5V>' - 5w")' - 5(uy - uy,)] [=0 =>- 05 = 2 bcf> + r(ip' — ip") = holonomic coordinate], (g5) The above immediately show that the nonvanishing 7 ’s equal ±1, as in the knife problem; and since here D = 1,4; /, I' = 2,3 and IV '■ 7*23 = — 7*32 = 1 7^ Oj (h) the system of Pfaffian constraints uj x = 0 and w 4 = 0 is nonholonomic. For additional wheeled vehicle applications, see also Lobas (1986), Lur’e (1968, pp. 27-31), Mei (1985, pp. 35-36, 168-175, 437^139), Stuckler (1955—excellent treatment). Example 2.13.9 Transitivity Equations for a Rigid Body in General (Uncon- trained) Motion. As explained in §1.8 IT., to describe the general spatial motion of a rigid body B we employ, among others, the following two sets of rectangular Cartesian axes [and associated orthogonal—normalized-dextral (OND) bases]: (i) a body-fixed set ♦ xyzj* ijk (noninertial), where ♦ is a generic body point (pole); and (ii) a space-fixed one O-XYZ/O IJK (inertial), where O is a generic fixed origin. Frequently (recalling §1.17, “A Comprehensive Example: The Rolling Coin”), we also use other “intermediate” axes/bases that are neither space- nor body-fixed: +-x'y'z'/+-i'j'k'\ for example, axes + XYZ translating, or comoving, with B but nonrotating (i.e., ever parallel to O-XYZ). Let us examine the transitivity equations associated with the translation of B with pole ♦, and its rotation about ♦ (earliest systematic treatment in Kirchhofif, 1883, pp. 56-59). (i) Rotation. As shown in §1.12, the transformation relations among the spatial and body components of the inertial angular velocity of B , a>, and the Eulerian angles (and their rates) between ♦ -xyz and O-XYZ (or ♦ XYZ) are [with s... = sin..., c... = cos...] as follows: Body axes components (assuming sin0 0): lo x = (s6 sip)^ + (cip)u e , uj y = (sOcip)^ + (~sip)uj e , u z = (c9)^ + oty; (al) => u<t, = {sip/s9)u x + (ctp/s0)u)y, ivg = (cip)u x + (~sip)u y , uty = (— cot 9s(j>)io x + (— cot 0 np)u> y + u> z . (a 2 ) Space axes components (assuming sin 0^0): U X = {cf)uj e + (sc/)s9)u)^ Wy = WH + (-c 0 j 0 )w^ UJ Z = w# + (c 0 )u^; (bl) = r > 1 jj 0 — ( — cot 0 S(p)ui z T (cot 0 cc/djcoy T tog — (ccp)cu % T (sfijcoy , uty = ( s(p/s9)u x + {—c(j)/s9)io Y . (b 2 ) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS Since these transformations are stationary, they also hold with the oj xy ,, z -x,y,z replaced by the d9 xyrXY .7 or 69 xyz . X j^z and the replaced by or 6(j >,..., respectively. Transitivity Equations (a) Body axes: Differentiating/varying the first of (al), while invoking the “d<5 = 6d ” rule for cf>, 9, ip, we obtain, successively, (69 x )' — 6u> x = [(s9sip) 6(f> + (cip) 66\ — 6[(s9si/))+ (cip)u>g\ = c9 sip(u>g 6<j> — 69) + s9 cip(uj^ 6cf> — 6tp) + sip(uig 6ip — 69 ), and substituting u>^,... /6(j ),... in terms of u x ,... /69 x ,..., from (a 2 ), we eventually hnd (69 x )' — 6 uj x = ui z 69 r — ui y 69 z , (cl) and similarly, for the other two, (69 v )' — 6u> y = oj x 69 _ — uj, 69 x , (c2) (69 z )' — 6 lo z = ui y 69 x — uj x 69y. (c3) Hence, the nonvanishing 7 ’s are l X yz = -7 \y = 1, I y zx = “7 V xz = 1, l\y = “7 %x = lj ( d l) or, compactly [with k,r,s —> x,y,z : 1 , 2 ,3], 7 \s = £ krs =(k-r)(r- s)(s - k)/2 = ±1, according as k,r,s/x,y,z are an even or odd permutation of 1,2,3; and = 0 in all other cases: Levi-Civitapermutation symbol (1.1.6 ff.). (d2) (b) Space axes: Applying similar steps to (bl, 2), we eventually obtain (69 x ) — 6 ujx — coy 69 x — uiz 69y , (el) and similarly, for the other two, (69 Y )' — 6 lo y = ui z 69 x — ui x 69 z , (e 2 ) (69 z ) — 6 coz = 69 y — cuy 69 x ■ (c3) Hence, the nonvanishing 7 ’s are X X , Y Y , Z Z , / ri N 7 yz — ~7 zy — ~ 1) 1 zx — 7 xz — 1) 7 xy — ~7 yx — ~ b (11) or, compactly (with k , r, s —> X, Y, Z: 1,2,3), 7 K rs = ~ £ krs = £ rks ■ ( f 2 ) The above show clearly that the orthogonal components of m, aj xyz and u x ,y,z are nonholonomic; while the nonorthogonal components are holonomic (and, WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS again, this has nothing to do with constraints, but is a mathematical consequence of the noncommutativity of rigid rotations). REMARK These transitivity relations and 7 -values, (cl-f2), are independent of the particular oJ x _ y ^x,Y,z ^ relationships (al-b 2 ); they express, in component form, in¬ variant noncommutativity properties between the differentials of the vectors of infinitesimal rotation and angular velocity. A direct vectorial proof of these proper¬ ties is presented in the next example. (c) Intermediate axes: Such sets are the following axes of ex. 2.13.7: (i) G-nn'z = G-x'y'z' , with OND basis G i'j'k' = G u n ii n k; and (ii) G-nNZ , with OND basis G-u n u N K (. u„ = i' = unit vector along + nodal line); and they are called by some authors semimobile ( SM ) and semifixed ( SF ), respectively. Below, we collect some kinematical data pertinent to them. Since their inertial angular velocities are (consult fig. 2.20) vi^M = vi K T luq i — u)q i T (sin O j T cos 0 k ) = (u e )i' + sin 9)j' + (w 0 cos0)*' = UJ,pK + UJg U„ = V) SF + U)g U n = Vi" + Vig H„, (gl) 0 ) SF = Vi" = K, (g2) we will have the following relations for the rates of change of their bases: du n /dt = a' x u„ = u)<pu N = uj^cosdu,/ — sin6 k')-, (g3) du n t/dt = a' x «„/ = (—u^cosd)w„ + ojgk'-, (g4) dk'/dt = a' x k' = (a^sin 6 )u„ — cogu n r, (g5) dii„/dl = a" x */„ = u} 0 ti N ; (g6) du N /dt = a" x u N = — (g7) dK/dt = ca" xK = 0 . (g8) Finally, the body angular velocity along the SM axes, thanks to the second line of (gl), equals VI = Vi' + u^k' = ( ujg)i' + (cu^sind)/ + (u>^ + o^cos 0)k' = w x < 1 + Uyj' + ay k (g9) and since this is a scleronomic system, (g9) holds with uyy z ', replaced with d0 x ' y < z < and 66 x 'y z i ; and replaced with d</>, hd, t/i/' and 6</),69,6iIj, respectively. From the above, by straightforward differentiations, we obtain the rotational transitivity equations in terms of semimobile components'. ( 69 x ')' — 6u> x i = 0 (9 x i = 9 = holonomic coordinate => 7* = 0), (69 y t)' — 6u) v i = cot 9(u) x ' 69 v ' — ay 69 x i), ( 69 : i )' — 6u > z 1 = (ay 69 x i — uj x i 69 r '); WWW.EBOOK777.COM (hi) (h 2 ) (h3) §2.13 GENERAL EXAMPLES AND PROBLEMS and hence the nonvanishing 7 ’s are (assuming cot 0 = finite) = cot 9, 7 x'y' = -7 'y'x' = 1 - (h4) Note that (hl^t) are none other than (h3-5) and (i) of ex. 2.13.7. (ii) Translation of pole (or basepoint) ♦. Let us assume that ♦ has inertial position: OP = p = Px I + Py J + Pz^i (h5) and, therefore, inertial velocity: v = dp/dt = ( dpx/dt)I + ( dp Y /dt)J + ( dpz/dt)K = v x I + v Y J + v zK [along space- axes: v x = dp x /dt = v • 7, etc.; Px,Y,ziyx,Y,z)'- holonomic coordinates (velocities) of ♦.] = v x i + v y j + v z k = ( dp x /dt)i + ( dp y /dt)j + ( dp z /dt)k [along body- axes: v x = dp x /dt = v • i, etc.; Px, y ,z( v x,j,z): nonholonomic coordinates (velocities) of ♦.] (h 6 ) Clearly, the above velocity components are related by the following vector transfor¬ mations: V x = COS(x, X) V X + COS(x, Y) Vy + cos(x, Z)v z , etc. v x = cos(X, x)y,. + cos(3f,y)v > , + cos(X,z)v,, etc., (h7) and, since this is a scleronomic system, their differentials are related by dp x = v x dt = (v • /) dt = dp • i = cos(x, X) dp x + cos(x, Y) dp Y + cos(x, Z) dp z , etc, Sp x = bp • i = cos(x, X) Sp x + cos(x, Y) bp Y + cos(x, Z) 6 p z , etc. (h8) Next, since di = dQ x i, Si = 56 x i etc., (il) where dO = a> dt = d9 x i + dO y j + dO z k = dfK + dO u n + dfik = elementary ( inertial) kinematically admissible/possible rotation vector , hence SO = 59 x i + 59 v j + 59- k = 5(j)K + 59 u„ + 5ipk = ( inertial) virtual rotation vector, we find by direct calculation [with (...)' = inertial rate of change, for vectors] {SPxY = ( Sp-i)' = {Spy-i + 6p- (/)■ = {Spy -i + 5p-{(0 x 1 ), 5{dp x /dt) = 5v x = 5{v • i) = 5v ■ i + v • Si = 5v • 1 + v • {50 x i), (i2) and subtracting the above side by side, while noting that {Sp)' — Sv = 5{dp/dt) — Sv = Sv — Sv = 0, we obtain the x-component of the pole velocity transi¬ tivity equation : (5p x )' — 5v x = 5p • {a x i) — v • {50 x i) = (5p x 0 ) • i — (v x 50) ■ i = (5p x a) — v x 50) • i; WWW.EBOOK777.COM ( 13 ) CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS and similarly for its y and z components. Hence, our pole transitivity equation can be written in the following vector form [with d (...) and S re i (...) denoting differentials of vectors, and so on, relative to the moving, here body- fixed, axes]: d(8p)/dt — 5 re i(dp/dt) = 5p x to v x 80, (i4a) or Sreiv = d(8p)/dt + co x Sp+v x 80. (i4b) In component form, along ♦-xyz, (i4a) reads (5p x y - 8v x = (u z 5p y ~u y 8Pz) - (Vy 86 z - v z 86y), (i5) ( 5p y y - 8v v = (u} x 8p z -L0 : 8p x ) - (V- 80 x — Tv 56 z ), (i 6 ) (8p z y - II ~u x 8p y ) - (v„ 56 y - v y 8o x y, (i7) and, therefore. the nonvanishing 7 ’s are [with accented (unaccented) indices for the components 8p, )] 7 X y'z = - -7 V = 1 and 7 V = - -*/ , = 1 1 z y A 5 (18) 7 * z'x = - '7 V = 1 and I y zx' = - - 7 ^ = 1 , (i9) 7 Z 'x'y = - -7 Z yx' = 1 and t 1 xy' = - Z 1 -7 y'z = 1- (ilO) Semifixed axes +-u„u n K. Here, we have v« = iPnY = *’ • = ■S . II cos^ ’ + (p Y )'sm4, (Jl) v N = E (Pn) )' = V-U N = ~{px)' sin 4 +(p Y y cos 4 , (J2) v z = E iPz) = v-K = (PzY => 7 Z 0 (i.e., v z = holonomic velocity). (j3) We leave it to the reader to show that (recalling the earlier semimobile axes kine¬ matics) (6p„Y - 6v„ = — [aty 8p x - {p x )'S(f>\ sin 4 + 8p Y - (prY 84} cos 4 = ^ 6p N - v N 84 = (1 / sin 4) (u y Sp N - v N 86 ,,), (j4) (Spn) ' - Sv n = -[^0 8p x - (px)' 84) cos 4+ [(p Y )' 84 - Sp Y ) sin 4 = -Up8p n + (p n y 54= (1/sin 4)[{p„y 86 y - Wy 8p„\; (j5) and hence that the nonvanishing 7’s are (with some, easily understood, ad hoc notation; and assuming that sin 8 ^ 0) 7 n Ny = -7 n yN = V sin 6, j N yn = -7 N ny = 1/ sin 6. (j6) [Recalling ex. 2.13.7 (rolling disk problem), eqs. (d2, 3), (hi, 2), (i), etc.] For related discussions of the rigid-body transitivity equations, see also Bremer (1988(b)) and Moiseyev and Rumyantsev (1968, pp. 7-8). WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS Example 2.13.10 Cardanian Suspension of a Gyroscope. Let us consider a gyroscope suspended a le Cardan (fig. 2.22). The rotation sequence cp = (^[precession ) —> q 2 = 9(nutation ) —> cp = f(spin) (i.e., 3 —> 2 —> 1, in the Eulerian angle sense of §1.12) brings the original axes G-XYZ, through the intermediate position G-x'y'z' (outer gimbal), to the also intermediate position G-xyz (inner gimbal). Now: (i) The inertial angular velocity of the outer gimbal co 0 , along outer gimbal- fixed axes , is U)o,x' = o, tx>o,y' = 0, <x> 0 ,z' = (a) (ii) the inertial angular velocity of the inner gimbal coj, along inner gimbal-fixed axes, is Vi,x = ~^0 sin 6 , U) hy = uj Si w /iZ = cos 6 ; (b) and (iii) the inertial angular velocity of the gyroscope co, along inner gimbal-fixed axes, is lo x = — (jOj, sin 6 , uiy = ug, co z = u )0 cos 9. (c) Let us find the transitivity equations corresponding to these quasi velocities. Equations (c) can be rewritten as pfi = p-h = ( — sin 9 ) 10 ^ + (0)wg + (l)u )0 [f 0), (d) uj 2 = oj y = (O)lo 0 + (\)uj s + (O)u >0 [f 0), (e) P>3 = P4 = ( cos 9)^0 + (0)Wfl + (0)aty (^0); (f) Figure 2.22 Kinematics of Cardanian suspension of a gyroscope. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS and their inverses are readily found to be vi = ufy = (0)04 + (0)^ + (l/cos0)w z , (g) v 2 = io g = ( 0 )w Y + (1 )u> y + ( 0 )c j z , (h) v 3 = = (1 )uj x + (0)w v + (sin 9/ cos 8 )u> z . (i) From these stationary relations, and assuming d(8cp < ) = S(dq k ) (k = x,y, r), we obtain, successively, (i) d(S8 x ) - S(d8 x ) = d[(- sin 9) 8<j> + Sip] - <5[(- sin 8 ) dcp + dip] = • • • = (cos 9)(dcp89 — ddScp ) = • • • = d8 z 68 y — dd y 68 : ; (j) (ii) d(89 y ) — 8(d9 y ) = 0 (=>■ 9 y = holonomic coordinate); (k) (iii) d(S9 z ) — S(dd z ) = h[(cos 9) Sep] — <5[(cos 9) dcp] = ■ ■ ■ = (sin 9) {dcp 89 - d9 Sep) = ■ ■ ■ = (tan 9) ( d8,88 y - d9 y 68 ,); (1) and so the nonvanishing 7 ’s are (assuming 8 ^ ± ir/2) l x yz = -1% = 7 23 = -7 ‘32 = +1, (m) 7 % == 7 3 23 = -7 3 32 = tan 6 ». (n) Example 2.IB. 11 An Elementary ad hoc Vectorial Derivation of the Rotational Rigid-Body Transitivity Equations. Let us consider, with no loss of generality, a free rigid body B rotating with (inertial) angular velocity m about a fixed point O. Then, as is well known (§1.9 ff.), and since this is an internally scleronomic system, the (inertial) velocity/kinematically admissible displacements/virtual displacements of a typical 5-particle of (inertial) position vector r, are, respectively, v = uj x r => dr = dO x r, 5r = SO x r, (a) where dO = ut dt, and d(...)/5{...) are kinematically admissible/virtual ( inertial ) var¬ iation operators. Now, d(.. .)-varying the last of (a), S(.. .)-varying the second, and then subtracting the results side by side, while invoking (a) and the rule d(8r) — 8(dr) = 0, we obtain, successively, 0 = d[8r) — 5(dr) = [d[8Q) x r + SO x dr] — ]5(d0) x r + dO x Sr] = [d(S0) x r+ SO x (dO x /■)] — [S(dO) x r + dO x (SO x /•)] = [d(50) — S(dO)] x r + [SO x (dO x r) — dO x (SO x r)] [and applying to the second bracket (last two triple cross-products) the cyclic vector identity, holding for any three vectors a,b,c: ax(bxc) + bx(cxa) + c x (a x b) = 0, with the identifications: a —» SO, b —> dO , c —> r] = ]d(50) — S(d0)] x r + (SO x dO) x r, from which, since r is arbitrary, we finally get the fundamental and general inertial rotational transitivity equation: d(50) — S(d0) = dO x SO. (b) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS Dividing the above with dt, which does no interact with these differentials (and noting that, by Newtonian relativity, dt = dt), we also obtain the equivalent tran¬ sitivity equation in terms of the angular velocities: d(50)/dt — S(o = (oxSO. (c) Next, let us find the counterparts of (b, c) in terms of relative differentials/variations, i.e. relative to moving (here body-fixed) axes, to be denoted by d(... )/S re i( .. .). Applying the well-known kinematical operator identities [(§1.7 ff.)] d(...) = d(...)+ddx (...), 5(...) = 6 rel (...) + 60 x (...), (d) (which immediately yield dr = 0 and S re i r = 0, as expected) to 56 and d0, respec¬ tively, and then substracting side by side, we find, successively, d(56) — 5(dd ) = 8(58 ) — 5 re i(dd) + ( d0 x 56 — 56 x dd ) = 8(56) — 5 re i{dd) + 2(dd x 56), (e) or, invoking (b) for its left side and rearranging slightly, we get, finally, 8(56) — 5 re i(dd) = 56 x d6; (f) and dividing by dt, we also obtain its velocity equivalent, d(58)/dt — 5 re ico = 58 x a). (g) The kinematical identities (f, g) are the noninertial counterparts of (b, c). The difference between (b, c) and (f, g) often goes unnoticed in the literature. To understand it better, let us write them down in component form, along space- fixed axes ♦ - A YZ and body-fixed axes ♦-xvz. Only the first equations are shown (i.e., X, x); the rest follow cyclically: Space-fixed (inertial) axes: d(56 x ) — 6(dO x ) = dOy 50/ — d0/ 50y . or ( 50 x ) — 5lo x = ccy 56/ — io/ 50y : (hi) Body-fixed (noninertial) axes: d(56 x ) — 6(dd x ) = d0 : 59 v — dd v 66 z , or (58 x )' — 5ui x = w z 56 — ai y 69 z ; (h2) which, naturally, coincide with equations (cl-f2) of ex. 2.13.9, and §1.14. [When dealing with derivatives/differentials of components, we may safely use the same notation (...)' /d(.. .)/6 (...) for both space and body such changes; here, the intended meaning is conveyed unambiguously]. Additional Special Results (i) Applying the second of (d) for to, and then equating the resulting fou-expres- sion with that obtained from (c), we get d(56)/dt — a> x 56 = <5 re /tu +56 x a), or, simplifying, d(58)/dt = 5 re im', or, equivalently (multiplying with dt), d(50) = 5 re i(dd). (i) WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS (ii) Starting from (c), and then invoking the first of (d), we obtain, successively, 8a> = d{8Q)/dt — a) x 56 = [d(56)/dt+co x 56} — oj x 56 = d(86)/dt [= 5 re i(o + 89 x <u, by (g)]; that is, 5(o = d(58)/dt, or, equivalently, 5(dd) = d(56)\ (j) which is “symmetrical" to (i). (iii ) Applying the first of (d) for <u yields dm=d(o, or, equivalently, d(dd) = d(dO); (k) but the second of (d) shows that 5w ^ 5rei(o, or, equivalently, 5{d6) ^ 5 re i(dd). (1) Problem 2.13.4 Rigid-body Transitivity Equations. Using the results of the preceding example and its notations, show that, for a rigid body rotating about a fixed point, d(8r) — 5 re i(dr) = (56 x dO) xr / 0, (a) or, equivalently (dividing by dt = dt), d(5r)/dt — 8 re iv = (86 x m) x /• ^ 0; (b) even though d(8r) — 8(dr) = 0; that is, the rule d(6.. .) = 6(d.. .) is not frame- invariant] Example 2.13.12 A Special Rigid-Body Transitivity Equation — Holonomic Coordi¬ nates. Continuing from the above examples, we show below that, for a rigid body rotating about a fixed point, the following transitivity/nonintegrability identity holds: E k ((o ) = d/dt(d(o/dv k ) — dco/dq k = m x (d(o/dv k ). (a) For such a system (with k = 1,2,3; and q k = angular Lagrangean coordinates', e.g., Eulerian angles <j>, 9, ip) we will have (o = m(q k , dq k /dt = v k ) = c o(q , v) = linear and (for our system, also) homogeneous function of the v^-’s = 'y^ (da>/dv k )v k (by Euler’s homogeneous function theorem) = E CkVk, (b) [definition of the c k s; also, recalling (1.7.9a, b)] from which it follows that dd = oj dt = c k dq k . WWW.EBOOK777.COM §2.1 3 GENERAL EXAMPLES AND PROBLEMS and since this is a scleronomic system se-'EckSq* ( c ) and so the basis (quasi) vectors c k = dco/dv k (independent of the v k s) can also be defined symbolically by Ck = 90/dq k = d{dO)/d(dq k ) = d{50)/d(6q k ). (d) Now, let us substitute the above representations into the earlier (inertial) transitivity equation d(86)/dt — 8ct) = oj x 86. (e) We find, successively, (i) Left side [we assume that (8q)‘ = 5(dq/dt) = <5v]: d{86)/dt — 8(o = d/dt(^^{d(o/dv k )8q^j — ^ [{dm/dq k )8q k + (dm /dv k )8v k ] = • - = [d/dt{d(o/dv k ) - dco/dq k \8q k = ^ E k {m) 8q k . (f) (ii) Right side: ai x 86 = tt) x ( y^(dco/dv k )8q k ) = ^ [to x (dco/dv k )\8q k , (g) and therefore (since the 8q k are independent—but even if they were constrained that would only affect the equations of motion) equating the right sides of (f) and (g), the identity (a) follows. In terms of the earlier c k vectors, (a) reads dc k /dt = to x c k + dco/dq k = ^ (c/ x + dc,/dq k )v,. (h) Finally, applying the first of (d) of ex. 2.13.11 to dto/dv k , and inserting the result into (a, h) produces the following interesting result: E k ,rei{(o) = d/dt{dm/dv k ) — doj/dq k = 0 or dc k /dt = dco/dq k . (i) Problem 2.13.5 Using the well-known kinematical result du k /dt = a) x u k , (a) where {u k = u k {q)} is, say, a body-fixed basis rotating with inertial angular velocity <u (like the earlier i. j, A'), with the co-representation (b) of the preceding example: (D = co{q k , v*) = co(ry, v) = ^ c k v k , (b) show that du k /dqi = Ci x u k [note subscript order], (c) i.e., (1.7.9c). Clearly, such a result holds for any vector h = h{q) rotating with angu¬ lar velocity co: db/dqi = c/ x h = {da>/dv k ) x h. Also: (i) d / dt{db / dq{) = d/dqfdb/dt ), and (ii) db/dvi = 0. WWW.EBOOK777.COM CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Problem 2.IB.6 By direct substitution of the representations (16 = mdt = Y^ Ck dq k and 89 = Y^ c k 5q k (a) into the earlier inertial rotational transitivity equation [ex. 2.13.11: eq. (b)]. d{59) - 5{d9) =d0 x 59, (b) and some simple differentiations, show that dc k /dq, - dc,/dq k = c, x c k . (c) This nonintegrability relation shows clearly that the basis {c^} is nonholonomic (nongradient); whereas if c k = 89/8q k , then (1(86) = 5{d9) =>- 9 = genuine angular coordinate. Simplify (c) if the {c/J are an orthogonal-unit-dextral basis (see also Brunk, 1981). Example 2.13.13 A Special Rigid-Body Transitivity Equation — Nonholonomic Coordinates. Continuing from ex. 2.13.11, let us substitute the (fully non¬ holonomic) representations (o = y^(da>/duk)wk = 'Y, E k u} k = co(q,u)), d9 = ^^E k d9 k , 59 = s k 59 k , (a) where, as usual, 9 k = quasi coordinates, u> k = d9 k /dt = quasi velocities, and E k = dca/dcok = 8{d9) / 8{dO k ) = 8{59)/8{59 k ) = 89/89 k : nonholonomic basis, (b) into the fundamental inertial rotational transitivity equation d{59)/dt — So = co x 59. (c) We find, successively, (i) Left side: d(59)/dt— 5co\( 8 (o/ 8 uj k )'59 k + { 8 (o / 8(o k ){59 k )'] - Y [{ 8 ( 0 / 8 q k ) 5q k + ( 8 (o/ 8 uj k ) 8 oj k ] [and setting 8 q k = E A k i 89/ = ( 8 v k / 8 iO /) 89/ (definition of the A k /)\ = Y [( do) / duJ k :)'-Y A ik{ d< °/ dc li) Sd k +Y {dco/8uji)[{89,y-8 uj,) [recalling the 8 ... /89 k definition (2.9.30a); and setting (as in pr. 2.10.5) {89/)' — 5lo/ = Y; h > k89 k (definition of the h' k )] = Y [{ d( °/ d ^k )' ~ 8(o/89 k ] 89 k + YJ2 {8(o/dui)h l k 89 k = Y + E h l k ( 8 (o/dui)^ 89 k . (d) WWW.EBOOK777.COM §2.13 GENERAL EXAMPLES AND PROBLEMS (ii) Right side: a, x 59 = a) x f y ^(d(o/duk)b6k) = Yj [to x (do/dw*)] (e) and, therefore, equating the right sides of (d) and (e), we obtain the identity d / dt(da) / dcj k ) — dm/dd k + ^ h l k (da)/dui/) = ®x (dco/dui k ), (f) or, in terms of the quasi vectors s k = e k (q), ds k /dt — dco/dO k = cj x ( dco/duj k ) — Yj h l k E/. (g) Finally, since d£ k /dt = dE k /dt + a> x s k , (g) takes the body-axes form: ds k /dt— dm/d9 k = d/dt(d(o/d6 k ) — dco/d9 k =E k * rel (m) = — Y^ h l k £i, (h) which is a special case of the transitivity equation (2.10.25). [Here too, we point out the differences between the notation-. do)(q, u>)/dd, = Y [dto(q, uf)/dq k ] (dv k /daj,), (i) and the vector transformation (by chain rule): da)(q,io)/du>i = E [da)(q,v)/dv k \(dv k /duj,) or £, = Y A ki e k] (j) See also Papastavridis, 1992. Problem 2.13.7 By direct substitution of the representations d9 = mdt = Y e k d0 k and 59 = ^^£ k 69 k (a) into the earlier inertial rotational transitivity equation [ex. 2.13.11, eq. (b)] d(60) - 5{d9) = dO x 59 (b) and some simple differentiations, show that ds k / 861 - d£,/dd k + Y V b ki = «/ x £ k , (c) where these special Hamel coefficients rj h kl are defined by d(69 k ) — 6(d9 h ) = E E 1 At d9,69 k . Example 2.13.14 Angular Acceleration. Let us consider intermediate axes ♦ - u k rotating with inertial angular velocity Q = E ®k- u k- If the inertial angular velocity of a rigid body, resolved along these axes, is m = E u k u k then its inertial angular acceleration equals a = dai/dt = dco/dt+ Q x a) = dm/dt — a>„ x cu, (a) where dco/dt = E {dui k /dt)u k , and co 0 = co — Q = angular velocity of body relative to the intermediate axes. WWW.EBOOK777.COM 380 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS Applying this result to the earlier case of semimobile axes ♦ -i'j'k' = +-u n ii n k (ex. 2.13.9) where co = (i uj s )u„ + (w 0 sin 9)u„i + cos 6)k = Q + k = Q + co 0 , (b) [with the customary notations: = d(j>/dt , lo 8 = cW/dt , = di//dt\ that is, m 0 = ui^ k, we find, after some straightforward calculations, oc = a„u„ + (x n 'U n i + (y.^k. (c) where cx n = dujflIdt -\- sin d, cx n ' = (dui, ; , jdt) sin 0 cos 6 — ujq ctk = ( dtOfj, / dt) cos 9 + du>^/dt — co e sin 9. (d) Let the reader repeat the above for the semifixed axes +-u„u N K, where co = ( ui^K + u)gii n ) + oj^k = (t o^K + uigii u ) + — sin 9 i/.y + cos 9K) — {j jJ e) u n + (—ufy sin 0 )tiyr + + ufy cos 0 ) A" = ui^K + co 0 = fl + (D 0 . (e) [For matrix forms of rigid-body accelerations, see (1.11.9a ff.); also Lur’e (1968, pp. 68-72).] WWW.EBOOK777.COM 3 Kinetics of Constrained Systems (i.e., Lagrangean Kinetics) Where we may appear to have rashly and needlessly interfered with methods and systems of proof in the present day generally accepted, we take the position of Restorers, and not of Innovators. (Thomson and Tait, 1867-1912, Preface, p. vi) [A] work of which the unity of method is one of the most striking characteristics_That which most distinguishes the plan of this treatise from the usual type is the direct application of the general principle to each particular case. (Gibbs, 1879, 3rd footnote, emphasis added; the work/treatise Gibbs refers to is Lagrange’s Mecanique Analytique, and the “general principle” is Lagrange’s Principle (§3.2)) [T]he author . . . again and again . . . experienced the extraordinary elation of mind which accompanies a preoccupation with the basic principles and methods of analytical mechanics. (Lanczos, 1970, p. vii) 3.1 INTRODUCTION This is the key chapter of the entire book; and since it is based on chapter 2, it should be read after the latter. We begin with a detailed coverage of the two fundamental principles, or pillars, of Lagrangean analytical mechanics: (i) The Principle of Lagrange (and its velocity form known as The Central Equation ); and (ii) The Principle of Relaxation of the Constraints. From these two, with the help of virtual displacements , and so on (§2.5 ff.), we, subsequently, obtain all possible kinetic energy-based (Lagrangean ) and acceleration energy—based ( Appellian ) equations of motion of holonomic and/or Pfaffian (possibly nonholonomic) systems; in holonomic and/or nonholonomic variables, with/without constraint reactions; such as the equations of Routh-Voss, Maggi, Flamel, and Appell, to name the most important. Next, applying standard mathematical transformations to these equations, we obtain the theorem of work-energy in its various forms; that is, in holonomic and/ or nonholonomic variables, with/without constraint reactions, and so on. This con¬ cludes the first, general, part of the chapter (§3.1-12). The second and third parts apply the previous Lagrangean and Appellian methods/principles/equations, WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS respectively, to the rigid body (§3.13—15) and to noninertial frames of reference (or moving axes ) (§3.16). The chapter ends with (i) a concise discussion of the servo-, or control, constraints of Beghin-Appell (§3.17); and (ii) two Appendices on the historical evolution of (some of) the above principles/equations of motion, and their relations to virtual displacements and the confusion-laden principle of d’Alembert-Lagrange. As with the previous chapters, a large number of completely solved examples and problems with their answers and/or helpful hints, many of them kinetic continua¬ tions of corresponding kinematical examples and problems of chapter 2, have been appropriately placed throughout this chapter. For complementary reading, we recommend (alphabetically): Butenin (1971), Dobronravov (1970, 1976), Gantmacher (1966/1970), Hamel (1912/1922(b), 1949), Kilchevskii (1977), Lur’e (1961/1968/2002), Mei (1985, 1987(a), 1991), Mei and Liu (1987), Neimark and Fufaev (1967/1972), Nordheim (1927), Pars (1965), Poliahov et al. (1985), Prange (1935), Synge (1960). As with chapter 2, we are unaware of any other single exposition, in English, comparable to this one in the range of topics covered. Only Hamel (1949), Mei et al. (1991) and Neimark and Fufaev (1967/1972) cover major portions of the material treated here. 3.2 THE PRINCIPLE OF LAGRANGE (LP) We begin with a finite mechanical system S consisting of particles {P}; each of mass dm, inertial acceleration a = dv/dt = d 2 v/dt 2 , and each obeying the Newton-Euler equation of motion (§1.4): dma = df, (3.2.1) where df = total force acting on P. As explained in chapter 2, the continuum notation for particle quantities, employed here, simplifies matters, since it allows us to reserve all indices (to be introduced below) for system quantities. The Force Classification Now, and here we start parting company with the Newton-Euler mechanics, we decompose df into two parts: (i) a total physical, or impressed, force dF, and (ii) a total constraint force, or constraint reaction, dR : df = dF + dR. (3.2.2) Let us elaborate on these fundamental concepts: (i) By constraint reactions, on our particle P, we shall understand (external and/or internal) forces, due solely to the (external and/or internal) geometrical and/or kine¬ matical constitution of the system 5; that is, forces caused exclusively by the pre¬ scribed (external and/or internal) constraints of S, and whose raison d’etre is the preservation of these constraints. As a result, such forces are (a) passive (i.e., they appear only when absolutely needed; see below), and (b) expressible only through these constraints (since, by their definition, they contain neither physical constants nor material functions/coefficients). Therefore, these reactions become fully known only after the motion of 5 (under possible additional, nonconstraint forces and initial conditions) has been found. Examples of constraint reactions are: inextensible WWW.EBOOK777.COM §3.2 THE PRINCIPLE OF LAGRANGE (LP) cable tensions, internal forces in a rigid body, normal forces among contacting (rolling/sliding/pivoting/nonpivoting) rigid bodies, and rolling (or static) friction. (Generally, constraints and their reactions are classified, on the basis of the precise physical manner by which they are maintained, as passive, or as active. Except §3.17, where the latter are elaborated, this chapter deals only with passive constraints / reactions.) (ii) By physical or impressed forces, on our particle P, we shall understand all other (external andjor internal, nonconstraint) forces acting on it, which means that [since the total force on P is determined through variables describing the geometrical! kinematical and physical state of the rest of the matter surrounding that particle (recalling §1.4)] the impressed forces depend, at least partially , on physical, or mate¬ rial, constants, unrelated to the constraints, and which can be determined only experimentally. Examples of such constants are: mass, gravitational constant, elastic moduli, viscous and/or dry friction coefficients, readings of the scale of a barometer or manometer; and examples of physical/impressed forces are gravity (weight), elas¬ tic (spring) forces, viscous damping forces, steam pressure, slipping (or sliding, or kinetic) friction [see remark (iii) below]. In other words, the impressed forces are forces expressed by material, or consti¬ tutive, equations, that contain those physical constants, and are assumed to be valid for any motion of the system; physical means physically ( functionally) given — it does not mean that the values of these forces are necessarily known ahead of time! In sum: Impressed forces are given by constitutive equations, white reactions are not; but, in general, both these forces require, for their complete determination, knowledge of the subsequent motion of the system (which, in turn, requires solution of an initial- value problem; namely, that of its equations of motion plus initial conditions). Impressed forces are also, variously, referred to as {directly’) applied, active, acting, assigned, given, known (where the last two terms have the meaning described above — see also remarks (iii) and (iv) below). In addition, the great physicist Planck (1928, pp. 101-103) calls our impressed forces “ treibende ” (driving, or propelling), while the highly instructive Langner (1997-1998, p. 49) proposes the rare but conceptually useful terms “ urgente ” (urging) for the impressed forces, and “ cogente ” (cogent, con¬ vincing) for the constraint forces. We follow Hamel (1949, pp. 65, 82, 517, 551), who calls impressed forces “ physikalisch gegebene ” (physically given) or “ eingeprdgte ”; also Sommerfeld (1964, pp. 53-54), who calls them “forces of physical origin.” REMARKS (i) From the viewpoint of continuum mechanics, practically all forces are physical (i.e., impressed); for example, an inextensible cable tension can be viewed as the limit of the tension of an elastic cable, or rubber band, whose modulus is getting higher and higher (—> oo); and a rigid body can be viewed as a very stiff, practically strain¬ less, deformable body. But there is also the exactly opposite viewpoint: kinetic and statistical theories of matter explain macroscopic phenomena, such as friction, visc¬ osity, rust, by the motion of large numbers of smooth molecules, atoms, and so on. Their 19th century forerunners (Kelvin, Helmholtz, et al.) even tried to reduce the internal potential energy of bodies to the kinetic energy of a number of spinning “molecular gyrostats” strategically located inside them — see, for example, Gray (1918, chap. 8). And there is, of course, general relativity, which, continuing tradi¬ tions of forceless mechanics, initiated by Hertz et al., set out to geometrize gravity completely; that is, replace tactile mechanics by a visual mechanics, albeit in a four-dimensional “space.” For the modest purposes of macroscopic earthly WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS mechanics, the impressed/constraint force division is both logically consistent and practically useful (economical), and so we uphold it throughout this book. (ii) The decomposition (3.2.2), what Hamel (1949, p. 218) calls “d’Alembertsche Ansatz” (~ initial proposition), is the hallmark of analytical mechanics. Expressing system accelerations as partial/total derivatives of kinetic energies with respect to system coordinates, velocities, and time (§3.3) is a welcome but secondary character¬ istic of Lagrangean analytical mechanics; the primary one is the decomposition (3.2.2) and its consequences with regard to the equations of motion. By contrast, the Newton-Euler mechanics decomposes df into (a) a total external force df e (= force originating, even partially, from outside of our system S ), and (b) a total internal, or mutual, force df t (= force due exclusively to the rest of S, on its generic particle P ): df = df e + df i . (3.2.3) The connection between (3.2.2) and (3.2.3) is easily seen by decomposing dF(dR) into an external part dF e (dR e ) and an internal part dF^dR,). and then rearranging a la (3.2.3); that is, successively, df = dF + dR = (dF e + dF,) + ( dR e + dR t ) = ( dF e T dR e ) -f- ( dp,- T d R : - 'j ~ d f e 4- d f i , where df e = dF e + dR e and d '/, = dF t + d /?,. The decompositions (3.2.2) and (3.2.3), although physically different, may, for some special systems, coincide. For example, in a free (i.e., externally unconstrained) rigid body all external forces are impressed (i.e., external reactions = 0), and all internal forces are reactions (i.e., interned impressed forces = 0). The coincidence of external forces with impressed forces and of internal forces with reactions in this popular and well-known system is, probably, responsible for the frequent confusion and error accompanying d’Alembert’s principle (detailed below), even in contemporary dynamics expositions. (iii) Rolling friction should be counted as a constraint reaction because it is expressed by a geometrical/kinematical condition, not by a constitutive equation; while slipping friction should be counted as an impressed force because, according to the well-known Coulomb-Morin friction “law,” it depends both on the contact condition (through the normal force, which is in both cases a constraint reaction) and on the physical properties of the contacting surfaces (through the kinetic friction coefficient). (That slipping friction is governed by a physical inequality does not affect our force classification.) The above apply to the (possible) rolling/slipping and pivot¬ ing/non-pivoting couples. The difference between rolling and slipping friction, from the viewpoint of analy¬ tical mechanics (principle of virtual work, etc.), has been a source of considerable confusion and error, even among the better authors on the subject. (iv) The force decomposition (3.2.2) is completely analogous to that occurring in continuum mechanics. For instance, in an incompressible (i.e., internally con¬ strained) elastic solid, the total stress (force) consists of a “hydrostatic pressure” or “reaction stress” term (constraint reaction), plus an “elastic stress” term (impressed force) expressed by a constitutive equation/function of the elastic moduli WWW.EBOOK777.COM (3.2.4) (3.2.4a) §3.2 THE PRINCIPLE OF LAGRANGE (LP) (material constants) and the strains (motion —> deformation), and it is assumed to be valid for any motion of that system. In general, the values of the stresses, both “incompressible/pressure” and “elastic” parts, for specific initial and boundary conditions, are found after solving that particular “initial- and boundary-value problem”; namely, the equations of motion of the solid plus its initial and boundary conditions. HISTORICAL The fundamental decomposition (3.2.2) seems to have been first given by Delaunay (1856); see, for example (alphabetically): Rumyantsev (1990, p. 268), Stackel (1905, p. 450, footnote 11a); also Hamel (1912, pp. 81-82, 301-302, 457^158, 469^170), Heun [1902 (a, d)], Pars (1953, pp. 447^148), Webster (1912, pp. 41^12, 63-65). Example 3.2.1 Let us Find the Most Important Internal I External and Impressed! Constraint Forces in a Diesel-Powered Electric Locomotive, Rolling on Rails. These are as follows: (i) Gravity and air resistance (drag ) are both external (their cause lies outside the system locomotive), and impressed (both depend partially on the physical constants: g = acceleration of gravity and p = air density, respectively). (ii) Pressure of burnt diesel fuel is internal (it originates within the engine’s cylin¬ ders) and impressed (depends on the gas temperature, density, etc.). (iii) Forces on connecting rods and other moving parts of the engine: (a) If these bodies are considered rigid, the forces are internal reactions; (b) If they are considered flexible, say elastic, these forces are internal but impressed (and to calculate them we must know their elastic moduli). (iv) Forces between axles and their wheel bearings are internal (for obvious rea¬ sons) and impressed (due to the relative motion among them — no constraints). (v) Friction forces between wheels and real are external (caused, partially, by an external body, the rail) and reactions (due to the slippingless rolling of wheels), and this holds for both their tangential (friction) and normal components; however, in the case of slipping (skidding), the friction changes to an external impressed force (it depends, partially, on the wheel-rail friction coefficient). Example 3.2.2 Let us Identify and Classify the Key Forces on a Person Walking up a Rough Hilly Road. The external forces needed to overcome the (also exter¬ nal) forces of gravity and air resistance are those generated by the road friction. The latter are reactions, since there is no relative motion (i.e., constraint) between the walker’s shoes and the road surface. Arguments of the Forces In classical (Newtonian) mechanics, the force df on a particle P, of a system S, can depend, at most, on its position, velocity, and time; and on those of other particles of S, or even outside of S; and also, on material functions/coefficients. But, as an independent constitutive equation (i.e., not by some artificial control law), df cannot depend on the acceleration a of P (and/or its higher time derivatives). This, however, does not preclude the occurrence of such a dependence by elimination; in the course of solving the equations of motion, and so on, of a problem, it is possible to relate WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS functionally a force with an acceleration; but that is a mathematical coupling, not an independent physical one. [Pars (1965, pp. 11-12; also 24-25) has shown that if clf depended on «, then the initial state of P, that is, its initial position and velocity, would not determine its future uniquely; see also Rosenberg (1977, pp. 10-17); and Hamel (1949, p. 49). But in other areas of classical physics, for instance electrodynamics (e.g., radiation damp¬ ing), such a non-Newtonian explicit «-dependence does not create inconsistencies.] Lagrange's Principle Dotting each of (3.2.1) and (3.2.2) with the corresponding particle’s inertial virtual displacement Sr (§2.5 IT.) and then summing the resulting equations over all system particles, for a fixed generic time, we obtain g dm a ■ Sr = g dF • Sr + g dR • Sr , (3.2.5) or, rearranging, g (dm a - dF) • Sr + $ (-dR) -Sr= 0; (3.2.6) where [recall (§2.2.7 IT.)] the material sum S (■ ■ ■) is to be understood as a Stieltjes’ integral extending over all the continuously and/or discretely distributed system particles and their geometric/kinematic/inertial/kinetic variables. Equations (3.2.5, 6) do not contain anything physically new; that is, they result from (3.2.1, 2) by purely mathematical transformations. To make further progress towards the derivation of reactionless equations of motion, one of the key objectives of analytical mechanics, we now postulate that (for bilateral, or equality, or rever¬ sible, constraints) -S'W R = g (-dR) • Sr = -gdR- Sr = 0; (3.2.7) in words: at each instant, the (first-order) total virtual work of the system of (external and internal) “lost” (or forlorn, or accessory) forces {—c//f}, — S'W R , vanishes. Then, equations (3.2.5, 6) immediately reduce to the new and nontrivial principle of d’Alembert in Lagrange’s form , or, simply and more accurately, principle of Lagrange ( LP ) for such constraints: ^ dm a • Sr = ^ dF • Sr or ^ (dm a — dF) • Sr = 0; (3.2.8) what Lagrange calls “la formule generate de la Dynamique pour le mouvement d’un systeme quelconque de corps.” This fundamental differential variational equation states that during the motion of a constrained system whose reactions, at each instant, satisfy the physical postulate (3.2.7), the total (first-order) virtual work of (the negative of) its “inertial forces” —{—dm a) = {dm a}, 61 = $ dm a-Sr, (3.2.9) equals the similar virtual work of its ( external and internal) impressed forces {<7E}, S'W= gdF-Sr- (3.2.10) WWW.EBOOK777.COM §3.2 THE PRINCIPLE OF LAC RANGE (LP) that is, S'W R = 0 => SI=S’W. (3.2.11) The entire Lagrangean kinetics is based on LP, equations (3.2.7—11). Let us, therefore, examine them closely. • Another, equivalent, formulation of the above is the following: during the motion, the totality of the lost forces {—dR = df — dm a} are, at each instant, in equilibrium-, not in the elementary sense of zero force and moment, but in that of the virtual work equation (3.2.7) (see also chap. 3, appendix 2). • Here, we must stress that the above equations, and associated virtual work conception of equilibrium, are the contemporary formulation and interpretation of d’Alembert’s principle; and they are due, primarily, to Heun and Hamel (early 20th century). As such, they bear practically zero resemblance to the original workless exposition of d’Alembert (1743). The latter postulated what, again in contemporary terms, amounts to equilibrium of the {— dR} in the elementary (i.e., Newton-Euler) sense of zero resultant force and moment: S (-dR) = 0 => g (dm a - dF) = 0. $rx(-dR) = 0 => $rx(dma-dF) = 0. (3.2.12) It is not hard to see that for a rigid body (what d’Alembert dealt with) (3.2.7) specializes to (3.2.12). Indeed, substituting into (3.2.7) the most general rigid virtual displacement, 5r = dr. + 56 x (r — r.) [where ♦ = generic body point, and SO = (first-order/elementary) virtual rigid body rotation (recalling §1.10 ff.)] and simple vector algebra, we obtain, successively, -S'W R = $ (-dR) • [hr* + SO x (r - r .)] = [£(-<«) Sr. + [^(/--r,) x (-dR) ■SO = (-R) ■ Sr. + M. (-R) -S0 = 0, (3.2.13) from which, since Sr. and SO are arbitrary, (3.2.12) follows [and if $ (—dR) = 0, then M.(—R) = M origin (—R)]. If, further, the rigid body is free, that is, uncon¬ strained, then, as explained earlier, all its external (internal) forces are impressed (reactions) (i.e., {df e } = {dF} and {dff = {dR}), and the above lead to the Eulerian principles of linear and angular momentum (recall §1.8.18): S‘<f. = S dm a and s (r-r.) x df e =s (r — r.) x dm a. (3.2.14) It follows that, in studying the statics of free rigid bodies via virtual work, we only need include their external = impressed forces; and that is why here the methods of Newton-Euler and d’Alembert-Lagrange coincide and supply conditions that are both necessary and sufficient for equilibrium (see also Hamel, 1949, pp. 80-83). This preoccupation of d’Alembert, and many others since him, with the special case of (systems of) rigid bodies and elementary vector equilibrium (3.2.12), has diverted attention from the far more general scalar virtual work equilibrium (3.2.7), which constitutes the essence of LP. • In LP it is the sum S' W R = S dR ■ Sr that vanishes, and not necessarily each of its terms dR • Sr separately; although this latter may happen in special cases. WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS For example, as explained above, in a free rigid body (3.2.7) reduces to S'W R —> (S' ^internal forces = although individually dfj • Sr may not vanish. • While the dm a are present wherever a mass is accelerated, the dF may act only at a few system particles. • In general, S'W R and S'W are not the exact (or perfect, or total) virtual differ¬ entials of some system “work/force functions” W R and W, respectively; that is, in general, they are quasi variables, and that is the purpose of the accented delta S' (recall §2.9 ff.). The same holds for SI, but here, for convenience, we will make an exception and leave it unaccented. • For unilateral (or inequality, or irreversible ) constraints, LP is enlarged from (3.2.7-11) to gdR-Sr= S(dma-dF)-Sr> 0 => $ din a-Sr > $ dF • Sr, (3.2.15) or S'W R > 0 => SI > S'W. (3.2.15a) For example, in the case of a block resting under its own weight on a fixed horizontal table, the sole impressed force on the block, gravity, cannot perform positive virtual work; while the normal table reaction cannot perform negative virtual work: S'W R = -S'W> 0. Lagrange's Principle as a Constitutive Postulate It must be stressed that LP, eqs. (3.2.7-11), is what is known in continuum mechanics as a constitutive postulate for the nonphysical part of the df s, namely, the constraint reactions {dR}: like Hooke’s law in elasticity, or the Navier-Stokes law in fluid mechanics; hence, applying LP to a free (i.e., unconstrained) particle is like, say, applying the theory of elasticity to a rigid body! As such, LP is not a law of nature, like the Newton-Euler equation (3.2.1) (and its Cauchy form, in continuum mechanics), but subservient to them; if (3.2.1) can be likened to a constitution article, LP is a secondary law (say, a state law). Just as in continuum mechanics, where not all parts of the stress need be elastic, here in analytical mechanics too, not all con¬ straint reactions need satisfy (3.2.7) (see §3.17). Those reactions that do, which is most of this book, we shall call ideal (or perfect, or passive, or frictionless). In view of these facts, the frequently occurring expression “workless, or nonwork¬ ing, constraints” must be replaced by the more precise one, virtually workless con¬ straints. Indeed, under the most general kinematically admissible/possible particle displacement (§2.5) dr = ^ e k dq k + e 0 dt , where e k = dr/dq k , e 0 = dr/dt (= e n+x ), (3.2.16) the corresponding (first-order, or elementary) work of the constraint reactions is d'W R = gdR-dr=--- = (d'W R ) x +(d'W R ) 2 , WWW.EBOOK777.COM (3.2.16a) §3.2 THE PRINCIPLE OF LAGRANGE (LP) where {d'W R ) x = Y Rkdq k , Rk = S dR ' e k > (3.2.16b) {d'W R ) 2 = R 0 dt , R 0 = $dR e o (= i?„ +1 ); (3.2.16c) while, under an equally general virtual displacement Sr = e k ^k, the correspond¬ ing work is S'W R = g dR ■ Sr = • • • = Y R k % = 0; (3.2.16d) and therefore, since (d' W R ) ] and S'W R are mathematically equivalent ( dq ~ Sq), {d'W R \= 0 => d'W R = (d'W R ) 2 = dR • (dr/dt) dt f 0. (3.2.16e) [In view of (3.2.16 If.), it is, probably, better to think of first-order virtual work as projection of the forces in certain directions; and to forget all those traditional (and confusion-prone) definitions of it like “work of forces for a constraint compatible infinitesimal movement of the system.”] In sum: in general, the constraint reactions are working; even when virtually nonworking. Actually, that is why the whole concept of virtualness was invented in analytical mechanics. For example, let us consider a particle P constrained to remain on a rigid surface S, which undergoes a given motion. Then, the virtual work of the normal reaction exerted by S on P is zero, while the corresponding d'W R is not; but, if S is stationary, then both S'W R and d'W R vanish. From the viewpoint of continuum mechanics, the need for LP, or something equivalent, for the constraint reactions is relatively obvious. Below, we present a simple such mathematical argument from the viewpoint of discrete mechanics. In an A-particle system with equations of motion [discrete coun¬ terparts of (3.2.1)], m P ap = Fp + R P (P=l,...,N), (3.2.17a) and assuming that the impressed F P 's are completely known functions of t, r, v (something that may not always be the case: e.g., sliding friction), we have 3 N + 3N = 6N unknown scalar functions: (i) the 3N position vector components/ coordinates {xp(t),y p (t),zp(t): rectangular Cartesian components of />} —> {d~x P /dt~ = a Px , d yp/dt = a Pv , d^zp/dt = a Pz \ rectangular Cartesian com¬ ponents of a P = d 2 r P /dt 2 }, plus (ii) the 3N reaction force components {R Px ,Rp v: Rp-}. Against these unknowns, we have available: (i) the 3 N scalar equations of motion (3.2.17a), and (ii) a total of h + m scalar equations of constraint (recall §2.2 If.): h geometric: f H (t,r P ) = 0 {H = 1, P = 1,..., N), (3.2.17b) m velocity (possibly nonholonomic): fi)(t,rp, v P ) =0 (D = 1,... ,m\ P = 1,..., N); (3.2.17c) that is, a total of 3TV + h + m (differential) equations. Therefore, to make our problem determinate, we need 6 N — (3 N + h + m) = (3TV — h) —m = n — m = / (= # DOF in the small) additional scalar equations. And here is where LP comes in: as shown later in this chapter, the single energetic but variational equation WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 6' W R = 0 =>■ 61 = 6' W produces precisely these f needed independent scalar equa¬ tions (unlike the single actual, nonvariational, work/energy theorem, which always produces only one such equation!); and the latter, along with initial/boundary con¬ ditions make the above constrained dynamical problem determinate, or closed. This simple argument, number of equations = number of unknowns [probably originated by Lur’e (1968, pp. 245-248) and Gantmacher (1970, pp. 16-23)], shows clearly the impossibility of building a general constrained system mechanics without additional physical postulates , like LP, or something equivalent (it would be like trying to build a theory of elasticity without Hooke’s law, or something similar relating stress to strain!), and thus lays to rest frequent but nevertheless erroneous claims that “analytical mechanics is nothing but a mathematically sophis¬ ticated rearrangement of Newton’s laws.” In sum, analytical mechanics is both mathematically and physically different from the momentum mechanics of Newton-Euler. Schematically: Lagrangean analytical mechanics = Newton-Euler laws + d’Alembert’s physical postulate. As Lanczos puts it: “Those scientists who claim that analytical mechanics is nothing but a mathematically different formulation of the laws of Newton must assume that [LP] is deducible from the Newtonian laws of motion. The author is unable to see how this can be done. Certainly the third law of motion, “action equals reaction,” is not wide enough to replace [LP]” (1970, p. 77). The above also show clearly that trying to prove LP is meaningless; although, in the past, several scientists have tried to do that (like trying to prove Hooke’s law in elasticity!). These considerations also indicate that if we choose to decompose the total force df according to some other physical characteristic, then we must equip that mechanics with appropriate constitutive postulates for (some of) the forces involved, so as to make the corresponding dynamical problem determinate. Thus, in the Newton-Euler mechanics, where, as we have already seen, df is decomposed into external and internal parts, the system equations of motion — that is, the principles of linear and angular momentum — thanks to the additional constitutive postulate of action-reaction, contain only the external forces (and couples); with¬ out that postulate, the equations of motion would involve all the forces, and the corresponding problem would be, in general, indeterminate. And in the case of matter-electromagnetic field interactions (e.g., electroelasticity, magneto-fluid- mechanics), we must, similarly, either know all forces involved, or supplement the equations of motion (of Newton-Euler and Maxwell) with special electromechanical constitutive equations, so that we end up again with a determinate system of equations. More on Lagrange's Principle as a Constitutive Postulate Here is what the noted mechanics historian E. Jouguet says about the physical nature of Lagrange’s Principle (freely translated): In sum, therefore, Huygens and Jacob Bernoulli implicitly admit that the forces devel¬ oped by the constraints in the case of motion, are, like the forces developed by the con¬ straints in the case of equilibrium, forces that do no work in the virtual displacements WWW.EBOOK777.COM §3.2 THE PRINCIPLE OF LAGRANGE (LP) compatible with the constraints. There is here a new physical postulate. It could be quite possible that the property of not doing work be true for the constraint forces during equilibrium and not for the constraint forces during motion; the reaction of a fixed surface on a point could be normal if the point was in equilibrium, and inclined if the point was moving; the reaction of a surface on a point could be normal if the surface was fixed and oblique if it was moving or deformable. This new postulate expresses, to use the lan¬ guage of Mr. P. Duhem [a French master (1861-1916), particularly famous for his contributions to continuum thermodynamics/energetics (in the tradition of Gibbs), and the history/axiomatics of theoretical mechanics], that the constraints, that have already been supposed [statically] frictionless, are also without viscosity. (1908, pp. 195-196), and The dynamics of systems with constraints rests therefore on the property of forces generated, during the motion, by the constraints, of not doing work in the virtual displacements compatible with the given constraints. This is an experimental property, and at the same time an experimental property distinct from those that we have found for the forces developed by the constraints in the case of equilibrium, because it intro¬ duces the condition that the constraints are without viscosity. (1908, p. 202, emphasis added). When Are the Methods of Newton-Euler (NE) and d'Alembert-Lagrange (AL) Equivalent? Since there is only one mechanics, this is a natural question, but not an easy one. To begin with, since NE divides forces into external and internal (“apples”), while AL divides them into impressed and reactions (“oranges”), we should not be surprised if, for general mechanical systems and forces, no such equivalence exists, or should be expected, at all stages of the formulation and solution of a problem. Equivalence at the highest level of the fundamental principles may exist only for special systems and problems: that is, those for which (i) the internal forces (NE) coincide with those of constraint (AL), and (ii) the external forces (NE) coincide with the impressed ones (AL). The only such system that we are aware of, satisfying both (i) and (ii), is the earlier-examined free rigid body, and there we saw that LP leads to the NE principles of linear and angular momentum. For other systems where the internal forces may be (wholly or partly) impressed, for example, an elastic body, the NE principles do not follow from LP; the latter, as an independent axiom , says nothing about impressed forces. However, for a given system and forces, both meth¬ ods of NE and AL do the job pledged by all classical descriptions of motion, which is, given (i) the external (NE) and impressed (AL) forces, along with (ii) the system’s state at an “initial” instant (i.e., initial configuration and velocities = initial condi¬ tions), and (iii) appropriate constitutive postulates for its internal forces (NE) and constraint reactions (AL), respectively (and possibly other additional geometrical/ kinematical/physical facts intrinsic to that problem), then both NE and AL are theoretically equally capable in predicting the subsequent motion of the system and its remaining unknown forces (although both approaches may not be equivalent laborwise, or from the important Machian viewpoint of conceptual economy). On these fundamental issues, see also the masterful treatment of Hamel (1909; 1927, pp. 8-10, 14-18, 23-27, 38-39; 1949, chap. 4 and pp. 513-524). WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS The above can be summarized in the following: (0 Force decomposition: Newton-Euler (NE) D’Alembert-Lagrange (AL) Reactions External impressed External reactions Internal impressed Internal reactions Impressed external Impressed internal External reactions Internal reactions (ii) Whence the need for d’Alembert’s principle: Unknown forces NE: Internal {dff} — > Discrete: action-reaction s dfi = 0, S r x dfi = 0 [—> Continuum: Boltzmann’s axiom-, i.c., symmetry of stress tensor] AL: Reactions {///?} —► Lagrange’s principle ^ dR • Sr = 0 (iii) Consequences: NE: Linear momentum-. Angular momentum: AL: Lagrange’s principle: S d -f‘+S d f °= S dm a - s«.=s dm a =>■ f e = ma G (G = mass center) S [r x (dfi + df e )\ = S (r x dm a) => S>'* d fc = d / dt [S ( ‘- X dm v)j [S'W R = g dR ■ hr = 0] + [dm a = dF + dR] => ^ dF • br = ^ dm a ■ Sr (iv) Unknown force retrieval: NE: Principles of rigidification and cut AL: Principle of constraint relaxation (Befreiungsprinzip, see below and §3.7) (v) Coincidence of NE with cl’AL: free rigid body Free: External forces = Impressed forces; i.e., {df e } = {dF} ({dR e } = 0) Rigid: Internal forces = Constraint reactions; i.e., {dfi} = {dR} ({dF,} = 0) [Briefly (a) Rigidification principle: If a system is in equilibrium under impressed and constraint forces, it will remain in equilibrium if additional constraints are imposed on it so as to render it partly or wholly rigid; that is, deformable bodies in equilibrium can be treated just like rigid ones — both satisfy the same (necessary) conditions; (b) Cut principle'. We can replace the action of two contiguous parts WWW.EBOOK777.COM §3.2 THE PRINCIPLE OF LAGRANGE (LP) of the body by corresponding force systems (=>■ free body diagrams). Both prin¬ ciples are due to Euler. For details, see books on statics; also Papastavridis (EM, in prep.).] Example 3.2.3 Plane Mathematical Pendulum: Comparison Between Principles of Moment (Original d'Alembert) and Virtual Work (Lagrange). Let us consider the motion of a mathematical pendulum, of length / and mass m, about a fixed point O on a vertical plane. (i) According to the original formulation of the principle (first by Jakob Bernoulli and then by d’Alembert), the string reaction S on the oscillating particle P must be in equilibrium; that is, its moment about O must vanish: M q Erx5 = fl =>■ S must be parallel to the string OP (r = OP) . (a) As a result, the second part of the principle—that is, impressed forces minus inertia forces must be in equilibrium , yields (with W = weight of P) rxW = rx(ma) => — (JF)(/sin</>) = {ni[l(d 2 (f>/dt 2 )]}(l) => d 2 (j)/dt 2 + (g/l) sin (f> = 0. (b) (ii) According to Lagrange’s formulation of the principle, the virtual work of S must vanish: S'W R = S • <)r = 0 => S must be perpendicular to the virtual displacement of P. (c) and since the latter is along the instantaneous tangent to P’s circular path about O, we conclude that S must be parallel to OP, as before. Hence, the second part of the principle—that is, virtual work of impressed forces minus that of inertia forces must vanish, yields W-6r=(ma)-6r => -(Wsincf)(lScf) = {m[l(d 2 (f/dt 2 )]}(lScf) =>■ d 2 (j>/dt 2 + (g/l) sin <^> = 0; (d) that is, the moment condition (b) and the virtual work condition (d) differ only by an inessential factor 6<p, and thus they produce the same reactionless equation of motion. In view of the extreme similarity, almost identity, of these two approaches in this and other rigid-body problems, we can see how, over the 19th and 20th centuries, various scientists came to confuse the zero moment method of James (Jakob) Bernoulli-d’Alembert {i.e., S r x (dF — dm a) =0, in our notation} with the zero virtual work method of Lagrange {i.e., S br ■ (dF — dm a) = 0}, and to view the former as equivalent to the latter. (Also, the fact that the string tension S is not zero—that is, that the constraint reaction is in equilibrium, not in the elementary sense of zero moment and force, but in the sense of zero virtual work, demon¬ strates clearly one of the drawbacks of the original d’Alembertian formulation of the principle.) Example 3.2.4 Motion of an Unconstrained System Relative to its Mass Center G, via Lagrange’s Principle (Adapted from Williamson and Tarleton, 1900, WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS pp. 242-293). Substituting r = r G + v/ G =>■ a = a G + a/ G into LP, (3.2.8), and regrouping, we obtain 0 = Sr G • ( dm a G - dF) J + br G • dm a /G + «g • (S dmSr /G \ + § ( dma /G - dF) • Sr /G , (a) from which, since S dm r/ G = 0=> S dm 8r j G = 0 and S dm a / G = 0, and the br G , 8r/ G are unrelated, we obtain (i) Sr G [S ( dm a ° ~ dF "> = 0 S dm a G = S dF ’ that is, m a G = F (Principle of linear momentum ), (b) if 6r G is unconstrained, and (ii) ^ (dm a/ G — dF) • br / G = 0, under the constraint dm br/ G = 0. (c) Combining, or adjoining, the second of (c) into the first of (c) with the vectorial Lagrangean multiplier k = k(t) (see §3.5), we readily get dma/ G = dF + kdm, (d) and, summing this over the system, we obtain S dma /a = s dF + dmj 0 = F + km => k = —F/m , so that, finally, (d) becomes dma/ G = dF ~ dm(F/m) (= dF — dma G ,as expected). (e) (f) Example 3.2.5 Sufficiency of the Statical Principle of Virtual Work (PVW) for the Equilibrium of Ideally Constrained Systems Deduced from LP. In analytical statics (i.e., LP with a = 0), the PVW states that in a bilaterally constrained and originally motionless system (in an inertial frame), the vanishing of b'W is a neces¬ sary and sufficient condition for it to remain in equilibrium in that frame. In con¬ crete applications, what we really employ is the sufficiency of the principle; that is, if b'W = 0, then the originally motionless system remains in equilibrium. Here, we will start with LP as the basic axiom, set S' W = 0, and then derive sufficient conditions to maintain equilibrium; that is, go from kinetics to statics. Most authors proceed inversely—that is, go from statics to kinetics—and that makes the detection of the importance of the various constraints more difficult. (i) Necessary conditions'. If the system is in (inertial) equilibrium, then a = 0, and therefore 6’W= $dF-6r = 0 (^S'W R =^dR-6r=0), (a) for tj < t < tf, where tftf) = initial (final) time and 1/ — t t = r. WWW.EBOOK777.COM §3.2 THE PRINCIPLE OF LAGRANGE (LP) (ii) Sufficiency conditions : If S'W = 0, for U<t< tf, then LP gives SI = dm a ■ Sr = 0, for < t < tf. (b) Let us investigate the consequences of (a, b) for equilibrium. Substituting into (b) the particle displacement dr — e 0 dt = (v — e 0 ) dt =[v — (dr/dt)]dt, which is mathemati¬ cally equivalent to its virtual displacement, and cancelling dt(ffi 0), we obtain dm a • v = ^ dm a • e 0 , (c) and since 2T = S dmv-v =>• dT/dt = S dm a ■ v, we are readily led to the following rheonomic-type power equation : dT/dt = ^ dm a • e {) = ^ dF • e 0 + ^ dR ■ e 0 . (d) Integrating the above between t t and t{< tf), and setting T, = T(t/), T =T(t) yields AT=T - which also follows from S'W dt sions: T = ^ dm a- e 0 ) dt ; (e) = 0. Equation (e) leads to the following conclu- (a) If e 0 = 0, then AT = 0, and since v,- = v{t/) = 0 =>■ T, = 0, it follows that T = 0 for some time 1 — /,■ (< t f - ti) ; and from this, since T = positive definite in the v • v = v 2 , we conclude that then all the v’s vanish for / — /,■ (< tf — t { ); that is, the system remains in equilibrium in that time interval. Conversely, if AT = 0 for any t > t h then (e) leads, for arbitrary systems, to e 0 = 0. In this case, (c) gives v = 0; that is, equilibrium [while (a) yields S dF ■ v = S dF ■ e 0 = 0], The consequences of this in the presence of additional Pfaffian constraints are discussed below. (b) If e 0 f 0, then, in general, AT f 0; that is, the system moves away from its original equilibrium configuration, even though 6'W = 0, for n<t< tf, and v,- = 0. Weaker special assumptions for equilibrium result for the following conditions: (c) If e 0 f 0, but tf [S dm a ■ e 0 ) dt = 0; or (d) If e 0 f 0 but a ■ e 0 =0, for r ; < t < tf. Comparison with Gantmacher Gantmacher’s formulation of the PVW is as follows: “For some position (compa¬ tible with constraints) of a system to be an equilibrium position, it is necessary and sutficient that in this position the sum of the works of effective forces [our impressed forces] on any virtual displacements of the system be zero” and “If the constraints are nonstationary, then the term ‘compatible with constraints’ signifies that they are satisfied for any t if in them we put [our notation] r = r,- and v = 0” and “It is then assumed that [our] equation (a) holds for any value of t if in the expression for dF we put all r = r t and all v = 0” (1970, p. 25). Let us relate this formulation with ours. By (2.5.2) v = J2 e k v k + e 0 . Hence, if v = 0: (i) If the v k ’s are unconstrained, and since e k f 0, then v k = 0 =>■ q k = constant and e 0 = 0—that is, the constraints are stationary —then the system will remain in equi¬ librium. WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS (ii) If, on the other hand, the v*/s are constrained , then, invoking the convenient repre¬ sentations (2.11.9, 13c, e), we have v —> v 0 = ^2 Pi v i + /?o = 0 =>• v,- = 0 => c[, = constant , and /? 0 = 0; P 0 = e 0 + ^2 b D e D = 0 => e Q = 0, and b D = 0; v D = b DI v, + b D => v D = 0 => q D = constant (=> q k = constant). In the light of the above, the PVW can be reformulated as follows: An originally motionless system remains in equilibrium if and only if (i) S'W = 0 and (ii) its holonomic constraints are stationary (e Q = dr/dt = 0) and its Pfaffian constraints are catastatic (a D = 0 or b D = 0). (The latter, however, may be nonstationary; and this explains Gantmacher’s statement: “Note that in this case the virtual displace¬ ments ... may also be different for different t.”) REMARKS (i) That “compatibility with constraints (during equilibrium)” leads to the above conclusions about them can be seen more clearly as follows. Let our system be subject to h holonomic constraints and m Pfaffian (holonomic and/or nonholonomic constraints): <t>H{t,r) = 0, f D = $B D (t,r)-v + B D (t,r) = 0 {H = 1,..., h\ D = 1,..., m). (f) By d/dt(.. ^-differentiating the above, to make them explicit in both velocities and accelerations, we readily obtain [recalling dot-product-of-tensor-definition [(see 1.1.12d ff.)], in the first sum in (g2) below]: (a) d<j) H /dt = $ (d(/> H /dr) • v + dcp H /dt = 0, (gl) (b) d 2 cj) H /dt 2 = \(d 2 (j) H /dr dr): (v ® v) + ( dcj) H /dr) •a + 2(9 2 cpu/dt dr) • v] + d 2 ^ H /dt 2 = 0, (g2) (c) df D /dt = SiK dB D/dr) • v + (dB D /dt)] ■v + B D -a } + S (^B D /dr) ■ v + dB n /dt = 0. (g3) Now, since compatibility requires that, for t, < t < tf, eqs. (f-g3) should hold with v = 0 and a = 0 in them (just like the equations of motion), we readily obtain from the above the following conditions on these constraints: (pH = 0, (hi) d<t> H /dt = 0 => d(P H /dt = 0, (h2) d 2 ct> H /dt 2 = 0 =► d 2 cf) H /dt 2 = 0; (h3) Jd = 0 =>■ B d = 0, (h4) df D /dt = 0 => dB D /dt = 0; (h5) that is, for t, < t < tf , the holonomic constraints must be stationary, and the Pfaffian ones must be catastatic, as found earlier. WWW.EBOOK777.COM §3.2 THE PRINCIPLE OF LAGRANGE (LP) (ii) If we assume Earth to be inertial , then an Earth-bound system is scleronomic. But if we assume it to have a given motion, then our system is rheonomic. In both cases, the contact (nongravitational) forces from the Earth to that system are exter¬ nal reactions. If, finally, the Earth interacts with our system, then the two taken together constitute a scleronomic system whose internal forces are impressed (see also Nordheim, 1927, pp. 47-49). Example 3.2.6 Nonideal Constraints. Let us consider a particle P of mass m, moving under an impressed force F and subject to the velocity constraint fit, r, v ) = 0. (a) If the reaction created by (a) is R , then the equation of motion of P is m a = F + R. (b) To relate the constraint equation to the reaction, so as to incorporate (a) into (b), we d/dt{.. .)-differentiate the former: / = 0 => df/dt = df/dt + ( df/dr ) • v + {df/dv) • a = 0, (c) and, therefore, ma- {df/dv) = — m[df/dt + (df/dr) • v]; (d) but, also, from (b), m a • (df/dv) = F • ( df/dv) + R • ( dfjdv). (e) Equating the right sides of (d, e), thus eliminating the acceleration, and rearranging, we obtain R ■ (df/dv) = - [m{df / dt) + m{df/dr) • v + F • {df,/dv)\. (f) Now, the most general solution of (f), for R , is R = -{df/dv) [m{df/dt) + m{df/dr) ■ v + F ■ {df/dv)}/{df/dvf + T, (g) where T = arbitrary vector orthogonal to df/dv. The above shows that, generally, the constraint reaction consists of two parts: (i) one parallel to df/dv. N = -{df/dv) [/ n{df/dt) + m{df/dr) v + F- {df/dv)]/{df/dvf = A {df/dv) (h) [where A = Lagrange an multiplier — see Lagrange’s equations of the first kind, (§3.5)]; and (ii) one normal to it, T. If T = 0, the constraint (a) is called ideal ; and in that case, clearly, the equation of motion of the particle (b), under (a), becomes ma = F— [F • {df/dv) + m{df/dr)-v + m{df/dt)] [{df/ dv) / {df/ dv) 1 ]. (i) To make the problem determinate, we, usually, introduce a constitutive equation between N and T. For example, in the common case of dry (solid/solid) sliding friction, we postulate the following relation between their magnitudes: T = pN = p\\{df/dv)\, p = coefficient of kinetic friction. (j) WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS Then, and with (g, h), eq. (b) becomes ma = F + X (df/dv) — p\X(df/ dv)\u, (k) where u = v/\v\. For further details and applications of (k) see Poliahov et al. (1985, pp. 152-170). Problem 3.2.1 Continuing from the preceding example, show that if the con¬ straint (a) has the holonomic form <l>(t,r)= 0, (a) then (h) and (i) reduce, respectively, to N = —(df/dr) [m(d<j>/dt) + m(df/dr) • v + F • (d<j>/dr)] /(df/dr) 2 = X(df/dr) (b) and ma = F— [F • (df/dr) + m(d<j>/dr) • v + m(d<fi/dt)] [(9</>/<9r)/(<9</>/dr)~]. (c) (See also Lagrange’s equations of the first kind, in §3.5.) Introduction to the Principle of Relaxation of the Constraints (PRC) Before we embark into a detailed quantitative discussion of Lagrange’s Principle (LP) and its derivative equations of motion, let us discuss briefly the second pillar of analytical mechanics, the principle of relaxation of the constraints ( Befreiungsprinzip ; Hamel, 1917). LP allows us to get rid of the constraint forces and, eventually, obtain reactionless equations of motion; and, historically, this has been considered (and is) one of the advantages of the method, especially in physics. However, in many engineering problems we do need to calculate these reactions, and thus the question arises: How do we achieve this with such a reaction-eliminating Lagrangean form¬ alism? Here is where PRC comes in: to retrieve a(ny) particular, external and/or internal, “lost” reaction we, hypothetically, free, or relax, the system of its particular, external and/or internal, geometrical and/or motional, constraint(s) causing that reaction; that is, we, mentally, allow the formerly rigid, or unyielding, constraint(s) to deform, or become flexible, relaxed, so that the former reaction becomes an impressed force that depends on the deformation of the violated constraint via some constitutive equation. Then we calculate its virtual work, add it to S'W, and apply LP: (SI = S'W) relaxed tem ; and so on and so forth, for as many reactions as needed (one, or more, or all, at a time). Last, since in our model the constraints are rigid, we enforce them in the final stage of the differential equations of motion. The mathematical expression of PRC is the very well-known and widely applied method of “ undetermined ,” or Lagrangean, multipliers (§3.5). REMARKS (i) Another, mixed, method is, first, to use LP to calculate the reactionless equa¬ tions (and from them the motion), and to then use the method of Newton-Euler (NE) to calculate the external and/or internal reactions. This may be practically expedient, WWW.EBOOK777.COM §3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO-INERTIAL IDENTITIES but it is not logically/conceptually satisfactory; it makes Lagrangean mechanics look incomplete. (ii) The counterpart of PRC in the NE method is the following: if, for example, we want to calculate an internal force — that is, one that, due to the action-reaction postulate, drops out of the force/moment side in the NE principles of linear/angular momentum — then, applying Euler’s cut principle, we choose an appropriate new free-body diagram so that the former internal force(s)/moment(s) becomes external, and then apply to these new subsystems, the NE principles. 3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO INERTIAL IDENTITIES Elere we transform (3.2.9), 61 = S dm a • 6r, from particle variables to system vari¬ ables; both holonomic and nonholonomic. (Actually, 61 is the negative of the virtual work of the “inertial forces” {—dm a}. We hope that this slight deviation from traditional terminology will not cause any problems.) Understandably, this relies critically on the kinematical results of chapter 2 and, therefore knowledge of that material is absolutely necessary. To obtain the most general system equations of motion from LP, we must use the most general expressions for a and 6r. We recall (§2.5 ff.) that these are (with k, l = 1,...,«) 6r = ^ e k 6q k = holonomic variable representation = ^ £/ 69/ = nonholonomic variable representation (= 6r *) ^ ^ £i66[, under the constraints 69 D = 0: D + 1,..., m\ I = m+ 1 (3.3.1) where the fundamental mixed basis vectors { e k } and {s,} are related by e k = dr/dq k = ^ a, k £, o £, = dr/89, = ^ A kl e k . (3.3.1a) 1. Holonomic System Variables Substituting the first of (3.3.1) into 61 we obtain, successively, SI = S dm a ' = S dm a - e k 6q,^j = ■ ■ ■ = ^ E k 6q k , (3.3.2) where E k = ^ dm a • e k : holonomic (Ar)th component of system inertial “force” = ^ dm a - ( dr/dq k ) = ^ dm a - ( dv/dq k ) = ^ dm a - ( da/dq k ) = ^ dm a • ( dr/8q k ) = ^ dm a • ( dv/dv k ) = ^ dm a ■ ( da/dw k ) WWW.EBOOK777.COM (3.3.3) CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS Now, E k transforms, successively, as follows: E k = dm a • e k = ^ dm(dv/dt) • (dv/dv k ) = d/dt^£jdm v • (dv/dv k ) — ^ [dm v • (d/dt)(dv/dv k )\ [recalling identity (2.5.10): E k (v) = d/dt(dv/dv k ) - dv/dq k = 0] = d/dt^^Jdm v • (dv/dv k ) — ^dm v ■ (dv/dq k ), (3.3.4) or, finally, with the help of the (inertial) kinetic energy T = S ( l / 2 )(dmv-v) = T(t,q,q) = T(t,q,v ) [since v = v(t,q,v)\, (3.3.5) we obtain E k = d/dt(dT/dv k ) - dT/dq k = d/dt{dT/dq k ) - dT/dq k = E k (T), (3.3.6) where E k (...) = d/dt{d... /dv k ) — d.. ■ /dq k = d/dt(d... /dq k ) — d... /dq k . (holonomic Euler-Lagrange operator) /f . (3.3.6a) Equation (3.3.6) is a kinematico-inertial identity, that is, it holds always, indepen¬ dently of any possible additional constraints, as long as the q’s are holonomic coor¬ dinates. Its cardinal importance to Lagrangean mechanics lies in the fact that it expresses system accelerations in terms of the partial and total derivatives of a scalar energetic function of the system coordinates and velocities, T(t,q,v), AS IF the q's and q’s = v’s (and t) were independent variables. That is why we have reserved the special notation E k {T) = E k when that operator is applied to the kinetic energy, even though E k (.. .) can be applied to any function of the q’s, v’s, and t. Also, (3.3.1-6a) clearly show the indispensability of virtual displacements (i.e., the e k vectors) to Lagrangean mechanics/equations of motion [i.e., the particular T-based expression for the system inertia/acceleration given by (3.3.6)], whether the constraint reactions are ideal or not. In sum: no e k s, no Lagrangean equations, that is, for an arbitrary particle/system vector z k f e k . Qclma-Zkf {d/dt){dT/dv k ) — dT/dq k . (3.3.6b) This should put to rest once and for all false claims that “one can build Lagrangean mechanics without virtual displacements.” The <5(...) is not the issue; the e k (—> projections) are! Let us collect the key kinematico-inertial identities involved here: (a) g dm v • e k = g dm v ■ ( dv/dv k ) = dT/dv k = dT /dq k = p k {t, q , v) = p k : Holonomic (k)th component of system momentum-, (3.3.7a) (b) g dm v • ( de k /dt ) = g dm v • ( dv/dq k ) = dT/dq k = r k (t, q , v) = r k : Holonomic (k)th component of “associated, or momental, inertial force”; (3.3.7b) (c) E k = dp k /dt - r k . (3.3.7c) WWW.EBOOK777.COM §3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO-INERTIAL IDENTITIES [p k = dT / dq k is the only kind of momentum that there is in analytical (Lagrangean and Hamiltonian) mechanics; and, as shown later, it comprises both the linear and angular momentum of the Newton-Euler mechanics.] 2. Nonholonomic System Variables Substituting the second of (3.3.1) into 61, we obtain, successively with a = a* = particle acceleration in nonholonomic variables (and similarly for other quantities): 51 = ^ dm a- 6v = Qdm a* ■ s k 69 ^ = • • • = "Y, 4 59 k , (3.3.8) where I k = ^ dm a* • s k = nonholonomic (k) th component of system inertial force = ^ dm a* • ( dv*/duj k ) = ^ dm a * • (da* /du> k ), recalling (2.9.35,43) = 4 (t,q,u,u) [since a* = a*(t,q,u,ij) and e k = s k (t,q)). (3.3.9) From the invariance of 61: E k 6q k = J2 4 69 k , and [recalling (2.9.11, 12)] 6q k = A kl 69 1 69/ = ai k 5q k , we readily obtain the basic (covariant vector¬ like) transformation equations: 4 = E ^/ k E/ E k = Y, a ft //. (3.3.10) The above expresses the nonholonomic inertial components in holonomic variables. To express them in terms of nonholonomic variables, we transform (3.3.9), succes¬ sively, as follows: I k = S d m a * ‘ E k = S dm(dv*/dt) • (dv*/du) k ) = d/dt(^^J dm v* • (dv*/du> k )^j - ^ [dm v* • d / dt(dv* / dco k )\ [adding and subtracting ^ dm v* • ( dv*/d9 k ), and regrouping] = d/dt(^^J dm v* • (dv*/du k )^j - ^ dm v* • (dv*/d9 k ) — ^ dm v* • [(d/dt)(dv*/duj k ) — dv*/d9 k \; (3.3.11a) or, invoking the nonintegrability identity (2.10.24, 25) [Greek subscripts run from 1 to n + 1 (time)], E k *(v*) = d/dt(dv*/duj k ) — dv*/d9 k = dr. k /dt - dv*/d9 k = -EE i\i w/£,- - E V* [since ui„ +l = uj 0 = dt/dt = 1] = “EE VtaHA = ~EE yk a u a (dv*/duj r ), (3.3.11b) introducing the (inertial) kinetic energy in cpiasi variables T = ^ l/2(dm v* • v*) = T(t,q,ul) = T* [since v* = v*(t, q, w)] (3.3.11c) and recalling the symbolic cpiasi chain rule (2.9.32a, 44a) dT*/d9 k = Y (dT*/d qi )(dv,/duj k ) = Y A lk (dT*/dq,), (3.3.1 Id) WWW.EBOOK777.COM 401 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS and the (nonholonomic Euler—Lagrange operator) k E k *{...) = d/dt{d.../du k )-d.../dO k , (3.3.1 le) we finally obtain the nonholonomic (system) variable counterpart of E k \ 4 = d / dt{dT* / duj k ) - dT*/d6 k + EE 7 r kl (dT*/dLj r )u, + Y Y k (dT*/du> r ) = d/dt(dT*/du k .) - dT*/dO k + YY. YUdTVdu, K = E k *{T*) — r k = E k * — r k [note difference from (3.3.6)], (3.3.12) where [recalling (2.10.25a)] -r k = dm v * - E k*( v *) = E E 7 r ka(dT*/duj r )u a = Y h r k (dT*/du r ) = —(System nonholonomic deviation , or correction, term) fc . (3.3.12a) We summarize the key kinematico-inertial identities below: (a) g dm v* ■ s k = g dm v* ■ (dv*/du> k ) = dT*/du k = P k (t, q, u) = P k : Nonholonomic (A’)th component of system momentum , (3.3.13a) (b) y k = E k *(v*) = d& k /dt — dv*/dO k = ■■■ = EEto = —(Particle nonholonomic deviation , or correction, term)^, (3.3.13b) Q dm v* • (de k /dt) = ^ dm v* • (dv*/d9 k ) + ^ dm v* • y k = dT*/d6 k + EE 7 r ak(dT*/du r )u a = dT*/dO k - EE 7 r ka (dT*/dLo r )io a = dT*/d0 k + r k [note difference from (3.3.7b)], (3.3.13c) -f t = EE 7 r ka {dT*/du r )u a = -r M - A,o, (3.3.13d) where - T Kn = E E 7 ^{dT'/dw^wu (3.3.13e) - r k0 = -r M+1 = ^ 7 r k (dT*/du> r y. “nonholonomic rheonomic force”. (3.3.13f) With the help of the above, 4 , (3.3.12), can be rewritten in the momentum form: 4 = dP k /dt - dT*/de k + YY, 7 r kaP r w«. (3.3.14) [Originally due to Hamel [1904(a),(b)], but for stationary/scleronomic transforma¬ tions; that is, with a replaced by, say, / = 1,...,«.] (c) SI =S dm a ' 6r = ^2 E k % = E 7/ - ^ (3.3.15) WWW.EBOOK777.COM §3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO-INERTIAL IDENTITIES where E k = gdm a-e k = d/dt(dT/dv k ) - dT/dq k = E k (T) = ^ a, k I h (3.3.15a) 4 = S dm a *' £ k = d/dt(dT*/du k ) - dT*/dd k + ^ ^ Y ka (dT* / du r )u a = E k *(T*) - r k = E k * -r k = Y J A i k Ei\ (3.3.15b) that is, it is E k = E k (T) and I k that transform like covariant vectors; the E k * = E k *(T*) do not (or, the terms E k * and r k , considered separately, do not transform as covariant vectors; but taken together , as E k * — T k = I k , they do!). 3. Acceleration, or Appellian, Forms The above expressions for the inertia vector E k (or I k ) are based on the kinetic energy T (or T*), because for their derivation we used the velocity identities e k = dv/dv k (or s k = dv*/du k ). Let us now find expressions for these vectors using the acceleration identities e k = da/dq k = da/dw k (or e k = da*/du k ). The results will turn out to be based on a scalar function that depends on the accelerations in a similar way that T (or T*) depend on the velocities. [The choice e k = dr/dq k does not seem to lead to any useful expression for E k \ while the choice z k = dr*/dd k = A lk e I = A /k (dv/dvi) will be examined later.] (i) Holonomic variables We have, successively, E k = ^ din a ■ e k = Q dm a • ( da/dq k ) = dS/dq k = dS/dw k , (3.3.16a) where S = ^ (1/2 )(dm a ■ a) = ^ (1/2 )(dm a 2 ) = S(t, q, q , q) = S(t, q , v, w): “Gibbs-Appell function,” or simply Appellian, in holonomic variables [or “acceleration energy” (Saint-Germain, 1901)]. (3.3.16b) (ii) Nonholonomic variables Similarly, we obtain 4 = ^ dm a* • s k = dm a* • (da*/du> k ) = dS*/dui k , (3.3.17a) where S* =^l (1/2 )(dm a * • a*) = ^ (1/2 )[dm(a*) 2 } = S*(t , q,uj,u): Appellian, in nonholonomic variables. (3.3.17b) To relate the above, we apply chain rule to 5 = S*. We obtain, successively, dS/dq k = dS/dw k = ^ ( dS*/du,)(du l /dq k ) = a lk (dS*/du,), (3.3.17c) WWW.EBOOK777.COM 404 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS and, inversely, dS*/du k = Y ( dS/dq,)(dq,/dw k ) = Y A, k {dS/dq,) = Y A ik(dS/dw,)- (3.3.17d) which are none other than the transformation equations (3.3.10). In sum, we have the following theoretically equivalent expressions for E k and I k : (i) E k = £ dm a ■ e k (= Y a ik I l) = d/dt(dT/dv k ) — dT/dq k [Lagrange (1780)] = Y aik[E,*(T*)-r,] = Y a ik(dS*/dd},) = dS/dq k = dS/dw k [Appell (1899)]; (3.3.18a) (ii) 4 = S dm a * ' Ek ( = E A ‘ kE >) = d/dt(dT*/du k ) - dT*/d9 k + EE 7 r ka {dT*/du, r )u )a = E k *(T*) - r k [Volterra (1898), Hamel (1903/1904)] = Y A ik[d/dt{dT/dv t ) - dT/dq ,\ [Maggi (1896, 1901, 1903)] = Y A ik{dS/dq,) = dS*/du k [Gibbs (1879)]. (3.3.18b) REMARKS (i) We can define (n + l)th, or (0)th, “temporal” holonomic and nonholonomic components of the system inertia vector by (with dq n+ \/dt = dq 0 /dt = dt/dt = v„+i = v 0 = 1 ) E„ + i = E 0 = ^ dm a • e„ +1 = ^ dm a • e 0 = ^ dm a • ( dr/dt ) = • • • = d/dt(dT/dv 0 ) - dT/dq 0 = d/dt(dT/di) - dT/dt, (3.3.19a) I n+l = I 0 = ^ dm a* • £„ +1 = Q dm a* • s 0 = ^ dm «* • (dr*/d9 0 ) = ■ ■ ■. (3.3.19b) However, such nonvirtual components will not be needed in the equations of motion; they could play a role in the formulation of “partial work/energy rate” equations (§3-9). (ii) Here, as throughout this book [e.g. (2.9.38ff.), ch. 5], superstars (.. .)* denote functions of t, q, u>, u ,...: f(t, d, ?>?>•• •) =f[t, Cl, g(t, q, w), q{t, q,u,Lb),...} =f*(t, q, A Special Case Let us find E k and I k for the following special quasi-velocity choice (recalling 2.11.9 ff.) Vo = Y h r)r( t ;C])v ! + b n (t,q), v, = Y s rr v r = v i, (3.3.20a) WWW.EBOOK777.COM §3.4 VIRTUAL WORKS OF FORCES: IMPRESSED (SW) AND CONSTRAINT REACTIONS (5W R ) and its inverse ^D = ^b DI ( t ,q) Vl + b D (t,q), 0Jf = v,. (3.3.20b) Here, clearly [recalling (2.11.12b), and with 6. = Kronecker delta ]; Add' = A DI = b D j, A ID = 0, A n i = 6 n i , (3.3.20c) and so the Maggi form 4 = S A lk E) specializes to I k = A Dk E D + A lk Ej =^4>' = ^2 b DD 'E D + )Ej = E d ,, (3.3.20d) 4 ' = ^ b DI 'E D + ^ bjf'E, = E,' + ^2 b DI 'E D . (3.3.20e) In sum, for the special choice (3.3.20a, b) I k takes the following form, in terms of holonomic Lagrangean ( T ) and Appellian (S) variables (with D=l,...,m; I = m + 1 ,..., n as usual): I D — E D = ( dT/dv D y - dT/dq D = dS/dv D = dS/dw D - (3.3.20f) i i = e i + Y1 h Dt E D = [{dT/dv,)' - dT/dq,] + ^ b DI [{dT/dv D )' - dT/dq D ] [Chaplygin (1895, publ. 1897), Hadamard (1895)] (3.3.20g) = dS/dq, + ^2 b DI {dS/dq D ) = dS/dwj + ^ b DI {dS/dw D ). (3.3.20h) The specialization of I k , for (3.3.20a, b), to nonholonomic variables [due to Chaplygin (1895/1897), in addition to his equations (3.3.20g); and Voronets (1901)] and other related results, are given in §3.8. We have expressed the {total, first order) virtual work of the {negative of the) inertial “forces,” 61, in system variables. The kinematico-inertial identities obtained are central to analytical mechanics, and that is why they were deliberately presented before any discussion of system forces and constraints; because, indeed, they are independent of the latter. These identities also show clearly the importance of the kinetic energy (primarily) and the Appellian (secondarily) to our subject, and so these quantities are examined in detail later (§3.9.11, 13-16). Now, let us proceed to express the virtual works of the real forces, namely, 6'W and 6'W r , in system variables. This will be considerably easier than the task just completed. 3.4 VIRTUAL WORKS OF FORCES: IMPRESSED (<5'I/V) AND CONSTRAINT REACTIONS (<5'l/V R ) 1. Holonomic Variables Substituting 6r=J2 e kbqk into the earlier expressions for 6'W and 6'W R (3.2.7, 10), we readily obtain 5'W= SdF.8v= SdF.^ekbqk) = ■■■ = ^ Q k Sq k , (3.4.1a) s ' Wr - S dR • Sr =S dR ' (H Ck Sqk ) = " ■ = X! Rk 6qk ' WWW.EBOOK777.COM (3.4.1b) CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS where Q k = s dF -e k : Holonomic (Ar)th component of system impressed force, (3.4.1c) R k = s dR • e k : Holonomic (A:)th component of system constraint reaction. (3.4.Id) 2. Nonholonomic Variables Substituting Sr = J2 s k S9 k (= <$*'*) into S'W and S'W R , we, similarly, obtain 6’ W = S dF ■ Sr* = S dF - (E S °k) = • • ■ = E 0 a- 60 k , (3.4.2a) S' W R = $ dR ■ Sr* = $ dR ■ (J2 s k 60 k ) = ■ ■ ■ = E A k 69 k , (3.4.2b) where © k = s dF • e. k : Nonholonomic (/c)th component of system impressed force, (3.4.2c) A k = s dR ■ s k : Nonholonomic (Ar)th component of system constraint force. (3.4.2d) Here too, these are ever valid definitions/results, no matter how many constraints may be imposed on the system later. 3. Transformation Relations From the invariance of the virtual differentials S'W and S'W R , we obtain the follow¬ ing transformation formulae for the various system forces; that is, from s'w = Y Qk % - E 2a- (E A * S6 >) we conclude @1 = E A klQk and, inversely, Qk = E a ‘k £ = £ ®'(E a lk foil)) 'y ~ Qk &qk, (3.4.3a) E Qk(9v k /duJi) (3.4.3b) E ( 9 w // 9 v a -) 6 >/ ; (3.4.3c) and, similarly, from S'W R = ■ ■ ■, we conclude d/ — E AkiRk and, inversely, R k = E a ik^i E R k( dv k/dui) (3.4.3d) Y( d ui/dv k )A, . (3.4.3e) WWW.EBOOK777.COM §3.4 VIRTUAL WORKS OF FORCES: IMPRESSED (SW) AND CONSTRAINT REACTIONS (SW R ) [These formulae can also be obtained from the e k s k transformation equations (2.9.25a, b) as follows: &, = SdF-s, = $dF. (J2 A kl s k ) = E A kl (SdF-e k ) = E A kl Q k , Qk = S dF • e k = S dF ' (E a ' kEl ) = E a ‘ k (S dF • £ /) = E a lk & A Rheonomic, or “temporal,” (n+ 1 )th nonvirtual force components can also be defined by Qn+i — Qo = f/T • C f; . ] = s dF • e 0 = ^ dF • ( dr/dt ) (holonomic impressed), (3.4.4a) ^71+1 = ^0 e n+\ = S dR • e 0 = S dR • {dr/dt) (holonomic reaction); (3.4.4b) @ n+ 1 = ©o = S dF • £ «+i = S dF ■ £ o = S dF • (9r*/dd n+ 1 ) (nonholonomic impressed), (3.4.4c) A n+ \ =A 0 = ^dR- £„+i = ^dR-s 0 = S dR - {dr*/dB n+x ) (nonholonomic reaction); (3.4.4d) and, recalling (2.9.26a, b), we can easily deduce the following transformation equa¬ tions among these components: Q/i+i — Qo — ^ AF • e n +i ^ dF • ^ (i k ,n+\£ k + = E a >^+ 1 £a ) + S dF ' £ »+! = E a k,n+\ 0 k + ®n+ 1, (3.4.4e) -^n +1 — *0 = S dF * ^ n +l S dF ' . ‘"Tv.'i ■ I + £72+1^ ^ dR * T ^dR'Efj^ i ^ ' ^k,n} i -4 / c T /I l i, (3.4.4f) and, conversely, 0 o = S dF ' s ° = S dF ' (E + e ») = E (5 dF ■ e k) +S dF ' e o = E + 2o, (3.4.4g) = iS ^ • £ o = ^ dR ' (E ^ AeA + e °) = E Ak {S dR • ca ) + S dR • e ° = E ^+j«o- ( 3 - 4 - 4h ) Problem 3.4.1 With the help of the second of each of (2.9.3a, b), prove the addi¬ tional forms of the above transformation equations: &,,+ 1 = -E a k,n+\®k +Qn+ 1 , or, simply, 00 = -E Uk@k + G°’ ( a ) A „+1 = - E a k,n+ \ A k + ^,,+ 1 , or, simply, A 0 = - E a k A k + Rq- (b) WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS REMARK A little analytical reflection will show that all these transformations can be con¬ densed in the formulae [with Greek subscripts running from 1 to n+ 1, recall (2.9.6a, b)]: ®ot — ^2 ^ Qfi ~ U ap®ai (3.4.5a) A-a. ^ ^ -^-fia^-fi R-fi ^ ^ &a.fiA-or (3.4.5b) A Special Case For the earlier particular case (3.3.20a IF.: A dd i = 8 DD ', A DI = b DI , A n i = 8jp), the above transformation equations specialize to 5 II © @D' — 55 A kD 'Q k — 55 ^dd'Qd + 55 (0)2/ = Qd'i i- e -> @ d = Qd, (3.4.6a) &r = A ki'Qk = 55 ^ di'Qd +55 = Qf + 55 ^dvQd-, i.e., &! = Qi + 55 b diQd = Qi,o = Qio\ (3.4.6b) and, similarly, @o = ■ ■ ■ = 55 boQo + Qo = Qo,o■ (3.4.6c) Example 3.4.1 Virtually Workless Forces. The following are examples of forces that do zero virtual work: (i) Forces among the particles of a rigid body; generally, the forces among rigidly connected particles and/or bodies. (ii) Forces on particles that are either at rest (e.g., a fixed pivot, or hinge, about which a system body may turn, or a joint between two system bodies), or are constrained to move in prescribed ways; that is, their (inertial) motion is known in advance as a function of time. (iii) Forces from completely smooth curves and/or surfaces that are either at (inertial) rest or have prescribed (inertial) motions. (iv) Forces from perfectly rough curves and/or surfaces, either at rest or having pre¬ scribed motions. See also Pars (1965, pp. 24-25), Whittaker (1937, pp. 31-32). Example 3.4.2 If z is a virtual displacement, then S dR-z = 0. Let us show the converse'. If for a kinematically admissible/possible vector z we have S dR • z = 0, then z is a virtual displacement; that is, z = ^2eiS9, (I = m+ 1. (a) The proof is by contradiction: Let z = Sr + y Sr), where the relaxed part y may be, at most, y = s D 6'9 d + s 0 6't (. D=l,...,m ; 6'6 D , 8't: components of j). (b) Substituting (b) into LP we get, successively, 0= gdR-z = $dR 6r+ $dR y= ■■■ = 0 + 55 A D 8'0 D + A 0 6't, WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS from which, since the m + 1 6'6 D and 6't are independent, we obtain A D , A 0 = 0. But, clearly, due to the constraints 66 D = 0 and 6t = 0, this is impossible. Thus, if we assume that y ^ 0, we are led to a contradiction. Hence, y = 0, and z is a virtual displacement expressible by (a). 3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS Let us now proceed to the final synthesis; that is, the formulation of equations of motion in general system variables. We begin with the “constraint reaction part” of LP, eq. (3.2.7), and so on: S'W R = SdR-8r = Y, R k 6q k = Y J A k S6 k = 0. (3.5.1) If the n 6q s are unconstrained (or independent , or free), so are the n 66' s. Then, (3.5.1) leads to Rk = 0, A k = 0. (3.5.2) If, however, the n 5q s are constrained by the m (< n) Pfaffian, holonomic and/or nonholonomic, constraints 66 D = E ° Dk 6q k = 0 (3.5.3) then, introducing m Lagrangean (hitherto) undetermined multipliers —A R = —A o(t) (the minus sign is only for algebraic convenience—see multiplier rule, below), and invoking (3.5.3), we can replace (3.5.1) with 6'Wr + Y, (-^) S0 D = E A k M k + E (-**) 60 d = e R k %+ E E (—X D )a Dk 6q k — 0, (3.5.4) where, now, the n 6q's and 66's (can be treated as if they) are free. Therefore, (3.5.4) leads immediately to the following: (i) in holonomic variables, E (E - E An = 0 => R k = E X D a Dk [= R k {q , t)]; (3.5.5) (ii) in nonholonomic variables (with I = m + 1,..., n), E A k b ° k - E An S0 D = E ( a d ~ An) 60 D + Y {A ’~ °) 60 ' = °- ( 3 - 5 - 6 ) and from this to the nonholonomic counterpart of (3.5.5), A D = X D (1 • 66 d = 0) and Aj = 0 (0 • SOj = 0); (3.5.7) that is, the m Lagrangean multipliers associated with the m “equilibrium” constraints lo d = 0 or 66 d = 0 are, in effect, the first m nonholonomic (covariant) components of the system reaction vector in configuration space. We also notice that when¬ ever 66 k = 0, A k ^ 0, and vice versa (k=l,...,n; and even «+l), that is, WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS A d 89 d = (Yl Z ))(0) = 0 and A t 86 } = (0 ){88 I ) = 0, so that 8'W R = 80 d + ^2Ai66 i = 0 + 0 = 0, (3.5.8) in accordance with LP. The advantage of the nonholonomic (3.5.7, 8) over the holonomic (3.5.5) is that, in the former, constraints and reactions decouple naturally; whereas in the latter they are coupled; that is, in general, Sq k ^0, R k ^ 0 ^ R k 6q k ^ 0, but R k 8q k = 0. (3.5.9) Finally, substituting the first of (3.5.7) into (3.5.5), we recover the earlier transfor¬ mation equations (3.4.3e): R k = Y A D a Dk = Y A i a ik > as expected. REMARK In the case of unilateral constraints 89 D > 0 (if, originally, they have the form 80 D < 0, we replace 88 D with — 80 D ), from the “unilateral LP” 8'W R > 0 and (3.5.4) we conclude that Y ^d 80 d > 0; and since the 80 D are positive or zero, the X D must also be positive or zero. In sum: If the unilateral constraints are chosen so that 86 * > 0 is possible/admis¬ sible, then the corresponding reaction A* is positive or zero (see also §3.7). The Lagrangean Multiplier Rule, or Adjoining of Constraints This fundamental mathematical theorem [one of the most useful mathematical results of the 18th century, initiated by Euler, but brought to prominence by Lagrange — see Hoppe (1926(a), p. 62)] states that: The single (differential) variational equation 8'M = ^2 M k 8q k = 0, (3.5.10a) where M k = M k (q , q, q,..., t) and the n 8q’s are restricted by the m (<«) indepen¬ dent Pfaffian constraints 80 d = Y^ a D k 8q k = 0 [rank (a Dk ) = m\, (3.5.10b) is completely equivalent to the new variational equation i’m + Y. (~X D ) 80 d = 8'M + EE ( — A D a Dk) bdk ~ 0) or E (^k ~ Y, X D a Dk ) 8q k = 0, (3.5.10c) where the n 8q’s are (better, can be viewed as) unconstrained; that is, (3.5.10a, b) are equivalent to the n equations M k — 22 ^ ° a Dk WWW.EBOOK777.COM (3.5. lOd) §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS which, along with the m constraints (3.5.10b), in velocity form ^E a D k<lk + a D = 0, (3.5.lOe) make up a system of n + m equations for the n + m unknown functions q(t) and A (t). INFORMAL PROOF Let us define the m X D ’s by the m nonsingular equations M D , = J2 X D a DD' (D,D'= (3.5.lOf) that is, eqs. (3.5.lOd) with k —> D'. For such A’s eq. (3.5.10c) reduces to E ( M/ _ H x d<*di) Hi = °> (3.5.lOg) where the (n — m) Sqfs are now free. From the above, we immediately conclude that Mi = E A d ci di'i (3-5. lOh) that is, eqs. (3.5. lOd) with k —> I. [References on the multiplier rule: Gantmacher (1970, pp. 20-23), Hamel (1949, pp. 85-91), Osgood (1937, pp. 316-318), Rosenberg (1977, pp. 132, 212-214). For a linear algebra based proof, see, for example, Woodhouse (1987, pp. 114-115).] Example 3.5.1 Lagrange’s Equations of the First Kind. The multiplier rule applied to 6'W r = gdR-Sr = 0, (a) where the Sr are restricted by (i) the h holonomic constraints (// = 1..... /;) ‘MbO = 0 => Sfif = g (dfn/dr) ■ Sr = 0, (bl) and (ii) the m Pfaffian (possibly nonholonomic) constraints [D=l,...,m (< n = 3N— h) and B D = B D (t , r)] Sb d -v + B d = 0 =» SB D -Sr = 0, (b2) leads, with the help of the h + m Lagrangean multipliers \i H = /i//(t) and \ D — \d( 0, to S{ dR -H dH(d 0 H /dr) - E A d Bd) -Sr= 0, (cl) from which, since the Sr can now be treated as free, we obtain the constitutive equation for the total constraint reaction on the typical particle P due to all system constraints: dR = E dHi^H/dr) + E ^pBp- (°2) Then, the Newton-Euler/d’Alembert particle equation dm a = dF + dR becomes the famous Lagrange’s equation of the first kind : dma = dF + fi H (d(j) H /dr) X d B d . (c3) WWW.EBOOK777.COM 411 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS In the more common discrete notation, the constraints (bl,2) become (with P = 1,..., N = number of system particles) S(p H = Y (d^H/drp) • 6r P = 0 and Y B DP • 6r P = 0, (c4) respectively, while the equation of constrainted motion (c3) assumes the form m P a P = Fp + Rp = F P + ^ dttid&it/dr P ) + ^ X D B DP . (c5) To understand the relation between the particle reactions dR, R P and their system counterparts R k , A k , we insert (c2) into their corresponding definitions (3.4.Id, 2d). We find, successively, (i) Rk ~S dR - e k = S dr-rid&Ff/dr) + Yj X D B D ^j ■ e k (dc/) H /dr) • ( dr/dq k )j + *■>■«) ~ Yj dHidrf’H/dqk) + Ys ^D^Dki (dl) and, comparing with the second of (3.5.5), R k = J2 ^D a Dk , we readily conclude that q)/dq k = S i d Ml, r)/dr\ ■ [dr{t, q)/dq k \ = 0, (d2) and Bpk = S B ° 4 e k = a Dk ; (d3) recall ex. 2.4.1 and (2.6. Iff.). REMARK The above also show that, as long as the quasi variables are chosen so that 0 = S B °' Sr = S9d = Y a ° k Sqk ^ S B '>' ek = aDk '- ( el ) the multipliers X D in (c2, 3) coincide with those in the second of (3.5.5). Indeed, e k — dotting (c3) and then S -summing, we obtain the “Routh-Voss” equations [see (3.5.15) below]: S dm a - ek = S dF ' ek+ J2 m(S (d(/) H /dr) • e,)j + 5>(S B D -e^j, (e2) or Ek = Qk + 0 + Yj X D a Dk ■ ( e 3) (ii) A k = S dR ' Ek = S (5Z dH^H/dr) + Y = Y ^niS ( df H /dr) • (dr/dd k )^j + Y Xd {S B °' E k ) = "Yj dH(d(j) H /d(j) k ) + Y^ Wok, WWW.EBOOK777.COM <fi) §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS and, comparing with (3.5.7), A D — A D and Aj = 0, we readily conclude that d&n/dOk = £ ( d(j) H /dr) -s k = g (d(p H /dr) ■ (E A lk e^j = • • • = E A ik(d ( t > H /dqi) = 0, (f2) and (with k = 1 D, D' = 1,. ... in; I = m+ 1 B'nk — S ^ D ‘ Ek ~ S • (E A ik e i) = E ^ lk (S ' e i ) = E A ik a Di = that is, B'dd' = $ ' £ d' = Sdd'i B'di = $ B d • £j = b D[ = 0. (G) Some of the above can also be obtained from the virtual forms of the constraints. Thus, we find, successively, (a) 0 = ty H = S Wtf/dr) ■ hr = g {d(p H /dr) • (E e k bO^j = • • • = E (d<p H /dd k ) bQ k = E (WH/dd D ) b0 D + E (dcpn/dd,) SBj = 0 + E (d<P H /de,) 66, =► d0 H /dO, = 0. (gl) (b) 0 = S0 D = S b d Sr = S B d • (E «* «**) = • • • = E ^ = E B'od 1 69 d’ + E! B'di 69, = 0 + E B'di 69, => B di = 0. (g2) HISTORICAL The terms Lagrange’s equations of the first kind (and second kind—see below) seem to have originated in Jacobi’s famous Lectures on Dynamics (winter 1842/1843, publ. 1866), and have been widely used in the German and Russian literature. They are not too well known among English and French authors (see, e.g., Voss, 1901, p. 81, footnote #220). Example 3.5.2 Lagrange’s Principle and Multipliers: Particle on a Surface (Kraft, 1885, vol. 2, pp. 194-195). Let us consider a particle P of mass m moving on a smooth surface (p(x,y, z, t) = 0, where x,y,z are inertial rectangular Cartesian coordinates of P , under a total impressed force with rectangular Cartesian compo¬ nents (V, Y,Z). According to LP, the motion is given by (with the customary notations dx/dt = v x , d 2 x/dt 2 = dv x /dt = a x ,...) (m a x — X) Sx + (m a v — Y) by + (m a z — Z) bz = 0, (a) under the (virtual form of the surface) constraint bcf> = 0: (<9</>/ dx) bx + (dcp/dy) by + (dcp/dz) bz = 0. (b) By Lagrange’s multipliers, (a) and (b) combine to the unconstrained variational equation, [m a x — X — X(d(p/dx)\ bx + [m a y ~Y — \{d(p/dy)\ by + [m a z — Z — A (d(p/dz)\ bz = 0, WWW.EBOOK777.COM (c) CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS and this leads directly to the three Lagrangean (Routh-Voss) equations of the first kind: ma x = X + \(d(f)/dx), ma y = Y + X(df/dy), ma z = Z + X(d<j)/dz). (d) Eliminating the multiplier A among (d) we obtain the two reactionless equations (with subscripts denoting partial derivatives): (ma x — X)/<f> x = {ma y — Y)/f y = ( ma : — Z)!f z (= A). (e) Next: (i) either we solve the system consisting of any two of (e), plus the constraint <f) = 0, for the three unknown functions x(t), y(t ), z(t), and then calculate A — > A(?) from (d), or (e), if needed; (ii) or we solve the system consisting of (d) and (j> = 0 for the four unknown functions x(t), y(t), z(t), and X(t). Equations (e) can also be obtained as follows: in view of (b), only two out of the three virtual displacements are independent, here n = 3 and m = 1. Taking Sx as the depen¬ dent virtual displacement, and solving (b) for it in terms of the other two (assuming that (j) x f 0), we obtain Sx = ~{(j)y/(j) x )5y - (<j) z /(j) x )5z, (f) and substituting this into (a), and regrouping terms, we get the new unconstrained variational equation of motion [ma y - Y - ( ma x - X)(f y /f x )\ 6y + [m a z — Z — (m a x — X) (cp z / ff)] Sz = 0. (g) The above, since Sy and Sz are now free, leads immediately to the two reactionless = kinetic equations, ma y = Y + (nia x — X)(f v /f x ), ma z = Z + ( ma x — X)((j) z /f x ), (h) which are none other than the earlier eqs. (e). REMARKS (i) Equations (h) can be, fairly, called “Maggi —*■ Hadamard equations of the first kind”; and the extension of this idea to holonomic system variables and correspond¬ ing Pfaffian constraints yields “ELadamard’s equations (of the second kind)” (§3.8). (ii) Equations (b-d = “ adjoining of constraints”) and equations (f h = “em¬ bedding of constraints”) embody the two available ways of handling constrained stationary problems in differential calculus; although, there, the former is discussed much more frequently than the latter! Specialization Let the reader verify that if the surface constraint has the special form z = f(x,y), then: (i) eqs. (d, e) reduce, respectively, to WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS Routh-Voss equations: ma x = X + X(df/dx), ma y = Y + \(df/dy), ma z = Z — X, (i) Kinetic Maggi —> Hadamard equations (of the first kind): (ma x - X)/f x = ( ma y - Y)/f y = ( ma z - Z)/(-1) (= A), (j) => (772 a x — X) + (m a : — Z)f x = 0, (m a v — Y) + (m a z — Z)f y = 0. (k) (ii) Substituting into (k): a z = z = ■ ■ ■ = xf x + yf y + (A) 2 /™ + (fflyy + 2 xyf xy ; that is, using the constraint and its (...)'-derivatives to eliminate 2 and its derivatives from them, we obtain the two kinetic equations in x,y and their derivatives alone: *(1 +fx 2 ) + yfxfy + itffxfxx + 2-Xyfxfxv + (jffxfyy = i X + fx Z)/m, (1) +fy 2 ) + xfxfy + ( yffyfyy + 2.xyf.J\, + {x) 2 f y f xx = (7 +f y Z)/m, (m) [which are the “Chaplygin-Voronets”-type equations of the problem (see §3.8)]. (iii) Solving the last of (j) for A, and then using into it the earlier expression z = • • •, we obtain the following (kinetostatic) expression: A = Z — = Z — m xfx + yfy + ( x) 2 f XX + (yffvy + Xyf (n) which, once the motion has been found: x = x(t), y = y(t), yields the constraint reaction A = A(t). (iv) Finally, substituting (n) into the first and second of (i), we recover (k, 1), respectively. Example 3.5.3 Let us apply the results of the preceding example to a particle P of mass m moving under gravity on a smooth vertical plane that spins about a vertical of its straight lines, Oz (positive upward), with constant angular velocity <u (Kraft, 1885, vol. 2, pp. 194-195). Choosing inertial axes O-xyz so that Ox coin¬ cides with the original intersection of the spinning plane and the horizontal plane O-xy through the origin, we have, for the impressed forces, X = 0, 7 = 0, Z = +mg\ (a) and, for the constraint, y/x = sin(uV)/cos(uV) =>■ cj>{t, x,y, z) = ycos(ut) — vsin(wt) = 0. (b) Therefore, equations (e) of the preceding example yield (. mix. — 0)/[— sin(wt)] = (my — 0)/ cos (cot) = (mz — mg)/ 0 (= —A), or, rearranging (to avoid the singularity caused by f z = 0), (mx—0) cos(otf) = (my — 0)[— sin(utf)] => xcos(wt) +y sin(wt) = 0, (c) (mx — 0)(0) = (mz — mg) [— sin(u;t)] => z = g, (d) (my — 0)(0) = (mz — mg) cos(wt) =>■ z = g. (e) WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS Let us, now, solve (c-e). Using plane polar coordinates (r, 4>)\ x = rcos(wt) =>■ x = ■ ■ • => x = • • • and y = r sin(wt) => y = •••=> y = •• •, we can rewrite (c) in the simpler form r — ui 2 r = 0. (f) The solution of (d) = (e), with initial conditions 2 ( 0 ) = z 0 and z(0) = v 0 , is z = (l/2)gt 2 + v 0 t + z 0 , (g) while that of (f), with initial conditions r(0) = r 0 and r(0) = v r o , is 2 ui r = (u r 0 + v, ;0 )e w ' + (ur 0 - v ri0 )e~ wt . (h) Equations (g, h) locate P on the spinning plane at time t, and, with the help of (b), specify its inertial position at the same time. (See also Walton, 1876, pp. 398-411.) Example 3.5.4 Lagrange’s Equations of the First Kind; Particle on Two Surfaces. Let us calculate the reactions on a particle P moving in space under the two con¬ straints (where x,y,z are the inertial rectangular Cartesian coordinates of P) (f>i = x 2 + y 2 + z 2 — / 2 = 0 and <j) 2 = z — y tan 9 = 0; (a) that is, n = 3 — 2=1: for example, the bob of spherical pendulum of (constant) length /, forced to remain on the plane (j> 2 = 0, that makes an angle 9 with the plane z = 0. Using commas followed by subscripts to denote partial (coordinate) derivatives, we find, from (a), 64>i = <t>\ iX + </>i r Sy + 4>\~ 6z = 0, S(f> 2 = <\> 2 x & x + 02,y + 02,z &z = 0. (b) Solving (b) for the two excess virtual displacements in terms of the third, say 6y and 8z in terms of 8x, we obtain where Sy = —(2x/J) Sx and Sz = —(2xtan 9/J) 6x, 01, y 01,z 02,y 02,z 2(_v + ztan 9) (f0, assumed). (c) (d) Substituting Sy and Sz from (c) into the principle of d’Alembert-Lagrange for the particle reaction — that is, R x Sx + R y Sy + R~ Sz = 0, (e) results in [R x - (2 x/J)R y - (2x tan 9/ J)R Z ] Sx = R' x Sx = 0, (f) from which, since Sx is independent, we obtain R' x = 0; that is, RJ (R y + tan 9R-) = x/(v + z tan 9). (g) Further, the ideal reaction postulate for R: R= (dfi /dr) + A 2 (df 2 /dr), (h) WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS with (a) and in components, yields R X = + A 2 </> 2 , JC = • • • = 2\ { X , (i) R y = A| (f> ly + = • • • = 2A l v — A 2 tan0, (j) R~ = A[(/>i z + A 2 </ , 2 ,z = • • • = 2A(Z + A 2 ; (k) which, of course, are consistent with (g). Finally, since / ^ 0, we can use any two of (i-k) to express A ] 2 uniquely, in terms of R xyz l for instance, solve (j, k) for A ] 2 in terms of R r z . (See also Routh, 1891, p. 35.) Example 3.5.5 Lagrange’s Equations of the First Kind; and Elimination of Reactions. Let us consider a system of N particles, moving under the h + m = M (possibly nonholonomic but ideal) constraints Mt,r,v) = 0 (D=l,...,M<3N), (a) and, therefore (recalling ex. 3.2.6), having Lagrangean equations of motion of the first kind (we revert to continuum notation for convenience), dm a = dF + E A D {df D /dv). (b) Now, to obtain reactionless = kinetic equations of motion, we will combine (b) with the acceleration form of (a). To this end, we (...)'-differentiate (a) once, thus obtaining dfo/dt = dfo/dt + £ [(df D /dr) • v + (df D /dv) ■ a] = 0, (c) and from this, rearranging, we get S ( d/n/dv ) • a = ( df D /dr) • v - df D /dt. (d) Now, to be able to use (d) in (b), so as to eliminate a, we dot the latter with df D /dv and sum over the particles (with D,D' = 1,..., M ): S ( d fo/9v) a = S [( dF/dm) • (q f D /dv)\ + ^ V (S ( d fo'/ dv ) • ( df D /dv)/dm S j, (e) and, comparing the right sides of the above with (d), we readily conclude that EMS [dfn'/d(dmv)\ • (df D /dv)} = S { (iF ' l d fD/d(dmv)}} - £ (df D /dr) • v - dfo/dt. (f) Since rank[df D /d(dmv)\ = rank(df D /dv) = M, and therefore Det lS [df d'/ d{dmv)\ • ( df D /dv ) | f 0, (g) the M linear nonhomogeneous equations (f) can supply uniquely (locally, at least) the A/)’s as functions of the r’s, v’s, and t. Finally, substituting the so-calculated \ D ’s back into (b), we obtain N second-order equations for the v = r(t). (See also exs. 3.10.2, 5.3.5, and 5.3.6; and Voss, 1885.) WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS Problem 3.5.1 Continuing from the preceding example, find the form that (f) takes if the constraints (a) have the holonomic form <t>H{t,r) = 0 . (a) HINT Calculate f D = d(f) D /dt = 0, and then show that df D /dv = dcj) D /dr. Let us now turn to the second, and more important, “reactionless part” of LP, (eqs. 3.2.8, 11), and express it in system variables. 1. Holonomic Variables In this case LP, SI = S'W, assumes the form E Ek Sqk = E & k 6qk ’ or, explicitly, J2{[d/d t (d T /dq k ) - dT/dq k ] - Q k }Sq k = 22 { [d/dt{dT / dv k ) - dT/dq k \ - Q k } Sq k = 0. (3.5.11) This differential variational equation is fundamental to Lagrangean analytical mechanics; all conceivable/possible Lagrangean equations of motion are based on it and flow from it. (a) Now, if the n Sq’s are independent (i.e., m = 0 =>f = n DOF), (3.5.11) leads immediately to Lagrange’s equations of the second kind'. E k = Q k , or explicitly (re¬ calling the kinematico-inertial results of §3.3 in holonomic variables), E k = d/dt(dT/8q k ) - dT/dq k = d/dt(dT/dv k ) — dT/dq k = Q k [Lagrange (1780)], (3.5.12) = dS/dq k = dS/dv k = dS/dw k = Q k [Appell (1899)]. (3.5.13) Further, substituting 6q k = 22 A k i S0 t (k,l = 1,..., n) into (3.5.11) readily yields Z A klEk — E AkiQ k (i.e., 7/ = 0/, hut in holonomic variables) [Maggi (1896,1901,1903)]. (3.5.14) However, in this unconstrained case, neither Appell’s equations, (3.5.13), nor Maggi’s equations, (3.5.14), offer any particular advantages over those of Lagrange, (3.5.12); their real usefulness/advantages over eqs. (3.5.12) lie in the constrained case (see below). Equations (3.5.12) are rightfully considered among the most important ones of the entire mathematical physics and engineering; we shall call them simply Lagrange’s equations. (b) If the n Sq’s are constrained by (3.5.3): ffa Dk 6q k = 0 (D=\,...,m\ k = 1,...,«), that is, / = n — m = number of DOF, then application of the multiplier WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS rule, between these constraints and (3.5.11), leads immediately to the Routh-Voss equations E k — Qk + E ^ D a Dk (— Qk + Rk )> (3.5.15) or, explicitly, as in (3.5.12, 13), E k = d/dt(dT/dq k ) - dT/dq k = Q k + E X D a Dk [Routh (1877), Voss (1885)], (3.5.16) = dS/dq k = Q k + A o a Dk (Appellian form of the Routh-Voss eqs.). (3.5.17) The corresponding Maggi form is presented below. [Equations (3.5.15) are not to be confused with the other, more famous, equations of Routh of steady motion, etc. (§8.3 ff.)] CAUTION Some authors (e.g., Elaug, 1992, pp. 169-170) state, falsely, that if the n cf s are independent, the n bq's are arbitrary, and then (3.5.12, 13) follow from (3.5.11). But as we have seen (§2.3, §2.8, and §2.12), if the additional constraints (3.5.3, 10b) are nonholonomic the cf s remain independent, whereas, obviously, the bq' s are no longer arbitrary, that is, (3.5.12, 13) do not always hold for independent q s. 2. Holonomic —> Nonholonomic Variables In this case LP, 61 = b'W, assumes the form £ 4 M/t = £>* M/t- (3-5.18) (a) If the n <5(9’s are unconstrained (i.e., if m = 0=^f=n — m = n = # DOF), then (3.5.18) leads to I k = 0 k , or, due to the kinematico-inertial identities (3.3.10 If.) for I k , to the following three general forms: 4 = ^ A ik Ei = A ik Qi , or, in extenso, ^ [, d/dt(dT/dq ,) - dT/dq,]A lk = ^ A lk Q, {Maggi form: holonomic variables), (3.5.19a) = E Ai k {dS/dqi) = E ^ikQi {Appellian form of Maggi form: holonomic variables), (3.5.19b) = dS*/ddi k = 0 k [Gibbs (1879): nonholonomic variables, but no constraints!], (3.5.19c) = d/dt{dT*/du k ) - dT*/d6 k + EE 7 r UdT*ldw r )u> a = 0, = E k *(T*) -r k = 0 , [Volterra (1898), Hamel (1903-1904)]. (3.5.19d) Equations (3.5.19a, b) have no advantages over Lagrange’s equations (3.5.12); but equations (3.5.19c, d) may be truly useful for unconstrained systems in quasi WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS variables, for example, a rigid body moving about a fixed point [—> Eulerian rota¬ tional equations (§1.17)]. (b) If the 89' s are constrained by 89 D = 0, but 69, ^ 0 (i.e., if f = n — m = # DOF), then the multiplier rule applied to (3.5.18) yields the following two groups of equations: Kinetostatic (i.e., reaction containing) equations: Id = + A d [=&d + \d {D = 1,..., m)\, (3.5.20a) Kinetic (i.e., reactionless) equations: I, = 0, (/ = m + (3.5.20b) [and in view of the constraint 1 • 89 n+l = 1 • 6t = 0, we also have 7„ +1 = 0 n+ \ + A n+1 , but that nonvirtual relation is more of an energy rate-like equation (as in §3.9)]. Alternative Derivation of Equations (3.5.20a, b) First, with the help of the Kronecker delta (hopefully, not to be confused with the virtual variation symbol 8 ...), we rewrite the constraints 89 D = 0 as 0 = 89 n = ^ 6 dd . 89 d , = ^ 8[)[)' 89 d , + ^ 8 DI 89, = ^ S Dk 86 k . (3.5.20c) Then, using the method of Lagrangean multipliers, we combine them with (3.5.18): (1) we multiply each constraint 89 D (y= 0) with —\ D {^ 0) and sum over D ; (2) we multiply each “nonconstraint” 89,{^= 0) with — X,(= 0) and sum over /; and, (3), we add the so-resulting two zeros to (3.5.18), thus obtaining E (A - & k - E 6 ™) 6 °k = 0- (3.5.20d) Since the 89 k can now be viewed as unconstrained, (3.5.20d) decouples to the two sets of equations: k = D'\ I d > — & D ' = ^ X D 8 DD i = A D i (Kinetic equations ), (3.5.20e) k = I: I, — ©i = ^2 X D 8 d / = X, = 0 (Kinetostatic equations). (3.5.20f) Here, too, as with the unconstrained case (3.5.19a-d), we have the following three general forms for (3.5.20a) and (3.5.20b): • Kinetic equations (with /,/' = m + 1,..., n; k = 1,..., n; 7 / = 7 /«+i): I I = 2L = '22 AklQki WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS or, in extenso, E [d/dt(dT/dq k ) - dT/dq k ]A kI = Y A kiQk [Maggi (1896, 1901, 1903): holonomic variables], (3.5.21a) = E A ki{dS/dq k ) = Y A kiQk (Appellian form of Maggi form: holonomic variables), (3.5.21b) = dS*/dcb, = 0j [Appell (1899-1925): special cases of nonholonomic variables], (3.5.21c) = d/dt{dT 7<9w 7 ) - dT*/dO, + EE Yn'{dT*/dui k )uj r + Y 7 k i(dT*/du k ) = 0, [Hamel (1903-1904): “Lagrange-Euler equations”]; (3.5.21d) • Kinetostatic equations (with D = 1,..., m; I = m + 1,...,«; A* = 1 Yd = 7 Vi) : Id = E A kDEk = E A kDQk + A D ! or, in extenso, Y [d/dt(dT/dq k ) - dT/dq k ]A kD = Y A koQk + a d, (3.5.22a) = E A ko(dS/dq k ) = Y^ A kDQk + a Di (3.5.22b) = dS* / du>£) = 0 d + Ad [Cotton (1907): special variables] (3.5.22c) = d/dt(dT*/dcj D ) - dT*/d0 D +EE YDiidT^d^io, + Y 7 k D (.dT*/du k ) = 0 d + A d [Stiickler (1955); special case by Schouten (late 1920s, 1954)]. (3.5.22d) REMARKS (i) In the absence of constraints, the above n equations in the ca’s, plus the n transformation equations q k = J2 a u IjJ i + A k- constitute a system of 2 n first-order equations in the 2 n unknown functions ui k = u> k (t) and q k = q k (t). [Or, after using the lo <-> q equations in them, thus expressing the ui k s in terms of the q k s, they constitute a set of n second-order equations for the n unknowns q k {t)-] In the presence of m constraints oj d = 0, the n — m kinetic equations plus the n transformation equations q k = J^ A ki U} i + A k constitute a system of 2n — m first-order equations in the 2 n — m functions u I = tu I (t) and q k = q k {t). Or, equivalently, substituting tjjj = a Ik q k + fl/ (fi 0) into the n — in kinetic equations, we obtain a system of n — m second-order equations for the q k = q k (t); and then, pairing them with the m constraints a nfik + «n = 0. we hnally obtain a system of (n — in ) + in = n second-order reactionless equations for the q k = q k (t). Further, it can be shown that there exists a nonsingular linear transformation q k = A k /LO/ + A k , or WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS q a = Y^ (recalling that dq n+] /dt = w„ +1 = dt/dt = 1) that brings the non¬ negative kinetic energy 2T = M a pq a qp to the following sum of squares form: 2 T -2 = u « 2 = E + w„+i 2 ; (3.5.23a) in which case, since P k = dT*/du> k = uj k and P„+ 1 = dT*/doj n+l = w„ +1 = 1, dT*/dd k = {dT*/dqi)A Ik = 0, the nonholonomic system inertia assumes the Eulerian form (recall inertia side of Eulerian rigid-body rotational equations, §1.17): I k dui k /dt T 'y ( y ( 'y ka ut r cu a , (3.5.23b) and that is why Hamel called his equations “Lagrange-Euler equations.” (However, by choosing the u> k s so as to nullify the dT*/dd k s, we probably end up complicat¬ ing the Y ka ’s.) (ii) The advantage of nonholonomic variables in the Hamel “equilibrium form” co D = 0 is that then both constraints and equations of motion decouple naturally into n — m purely kinetic (i.e., reactionless) equations (69j f 0; Aj = 0 =>- 7/ = 0f) and m reaction-containing, or kinetostatic, equations (S0 D = 0; A D f 0 => Id — 0d + A D ). In holonomic variables, by contrast, both (Pfaffian) constraints and (Routh-Voss and Appell) equations of motion are coupled. Solving the n — m kinetic equations (plus constraints, etc.) constitutes the lion’s share of the difficulty of the problem. Once this has been achieved, then the reactions A D follow immediately from the (now) algebraic equations: A D = A D (t) = I D (t) — 0 D (t). (iii) When using Hamel’s equations under the constraints uj d = 0, we must enforce the latter after all partial differentiations have been carried out, not before; otherwise, we would not, in general, calculate correctly the key nonholonomic terms (k,r = 1 a = 1+ 1; /' = m + 1,...,«) -r k = YY1 7 r ka(dT*/du r )u a = YY1 7 r ki'(dT*/du r )uj r + Y Y k (dT*/duj r )-, (3.5.24a) and, unfortunately, this drawback holds for both kinetic and kinetostatic equations. Let us see why. Expanding T* a la Taylor around lu d = 0, we obtain T* = T* 0 + ^2 (9T*/8uj d ) 0 uj d + quadratic terms in iv D , (3.5.24b) where T* a = T*(q, u> D = 0 ,u} u t) = T* 0 (^w/H); (3.5.24c) and, generally, (.. .) 0 = (... ,uj d = 0,...) (a useful notation, to be utilized frequently, for extra clarity); and, therefore, {dT*/du} D ) 0 f dT\/dco D = 0, (3.5.24d) (dT*/doj I ) 0 = dT* 0 /diu 1 d/dt[(dT*/du,) 0 ] = d/dt(dT* 0 /du I ), (3.5.24e) {dT*/d6 k ) 0 = Y A rk (dT*/dq,) 0 = Y A rk {dT* 0 /dq r ). (3.5.24f) WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS In view of these results, — T k , (3.5.24a), transforms to l D ki'{dT*/du} D ) 0 u r + EE 7 7 ki'(dT* 0 /duj I »)uj I ' + E 7 D k(dT*/du D ) 0 + 7 MdT^/diUj)-, (3.5.24g) an expression that shows clearly that the presence of the first ( double ) and third (, single ) sums generally necessitates the use of T*, instead of T* a . However, with the help of the above expression we can obtain conditions that tell us when we can use the constrained kinetic energy T* a in Hamel’s equations right from the start. Let us do this, for simplicity, for the common case of the kinetic such equations of a scleronomic system. Then, -r^-r 7 = EE 7 D II ,(dT*/dco D ) 0 u I ,+ EE T* a /duj I »)u> I r, (3.5.24h) and (3.5.24d, e) make it clear that the sought conditions will result from the (iden¬ tical) vanishing of the first sum in (3.5.24h); that is, EE 1 D II ,(dT*/du D ) o u> r = 0. (3.5.241) But (as made clear in §3.9), 1T * = E E M *kM)u k u, = E E ( d2r */ du} k dui)uj k uj, => (dT*/dLe D ) 0 = J2(d 2T */ d “Ddu; I )u I , (3.5.24j) and so (3.5.24i) reduces to EEE 7 D IV {d 1 T*/du} D dujr'jujjnoj,, = 0; (3.5.24k) and from this we easily conclude that the necessary and sufficient conditions for the use of T*„ in Hamel’s equations are E 7 D IV {d 2 T*/dLo D dur) = 0. (3.5.241) For example, in the case of a single Pfafhan constraint, uq = 0 (i.e., m = 1), (3.5.241) yields y l w (d 2 T*/duj { dwj») = 0 (/,/',/" = 2,... ,n), (3.5.24m) which means that either all “nonholonomic inertial coefficients” d 2 T*/du l dujj" = M* |/» vanish [i.e., T* consists of an wpfree part and an ^-proportional part; or 7)// = 0, which means that constraint is holonomic (by Frobenius’ theorem, §2.12)]. The consequences of (3.5.241) are detailed in Hamel (1904(a), pp. 22-29); see also Hadamard (1895). In sum, in using the Hamel equations, even if we are not interested in constraint reactions, we must begin with the unconstrained kinetic energy T* = T*(q,uj D ,Lc I ,t), carry out all required differentiations, and then enforce the constraints wo = 0, at the end; and, a constraint uid = 0 can be enforced ahead of time in T*-terms that are quadratic in that uj d \ namely, in (.. .)to D 2 terms. WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS This inconvenience is a small price to pay for such powerful and conceptually insightful equations. Similarly, a detailed analysis of (3.5.24g) shows that it is possible to have r k = 0 (i.e., Hamel equations —> Lagrange’s equations) even though not all 7 ’s are zero. An analogous situation occurs in the Maggi equations, even in the kinetic case—that is, 1 i=Yl Ak,Ek = [ d /dt(dT/dq k ) - dT/dq k \A kI = A kI Q k , (3.5.24n) since k= 1we have to calculate T = TfijCpqnfij)', the “reduced,” or con¬ strained, kinetic energy To = T{t,q,q D = b nl q, + b n ,q,) = T 0 (t,q,q >), (3.5.24o) obviously will not do. This seems to be a drawback of all T-based (i.e., Lagrangean) equations. No such problems appear for the kinetic Appellian equations: there, with the convenient notation S* = S*(t,q,u} D ,u> I ,u> D ,u)i) = original, or unconstrained, or relaxed, Appellian -*■ S*(t,q,u D = 0 ,u,,w D = 0, w/) = = S* 0 = constrained Appellian, (3.5.25a) and the help of the Taylor expansion (with some obvious calculus notations) S* = S* a + ^2 [(9S*/duj D ) 0 uj D + (dS*/du D ) 0 tjj D \ + quadratic terms in uj d ,lj d , (3.5.25b) we get the general results [similar to (3.5.24c, e)] (dS*/du I ) 0 = dS* 0 /du I and (dS*/dcu D ) 0 ± dS* 0 /dcu D = 0. (3.5.25c) Therefore, if we are not interested in finding constraint reactions, we can enforce the constraints lo d = 0 and lu d = 0 into S* right from the beginning; that is, start work¬ ing with S* 0 , and thus save a considerable amount of labor. This property, due to the first of (3.5.25c), marks a key difference between the equations of Appell and Hamel, and their corresponding special cases. Special Case If all constraints on the q's are holonomic and have the equilibrium form 0d c 1d = constant = q Do , then {dS*/d^) 0 -+ Ej{T)\ 0 - Ej(T 0 ), where T a = T(t,q D = constant, q h q D = 0, qj) = T 0 (t, q h q,), and, similarly for the impressed forces, Qj = Q\(t,q, q) —> — Qioi aQ d so the kinetic equations become E, = E,(T n ) = dS 0 /dq, = Q Io . (iv) Comparison between Lagrange’s equations of the first and second kind, and their respective constraints. Those of the first kind, eq. (ex. 3.5.1: c3), constitute a set of 3 N + (h + m) = [(3iV — h) + m\ + 2h = (n + m) + 2 h WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS scalar equations, for the 3 N + (.h + m) unknown functions: {x P ,y P ,z P ; P=l,...,N}, {p ff ; H = 1,..., h} and {\ D ; D = 1,... ,m}. Once the positions (and hence accelerations) and multipliers become known functions of time, (ex. 3.5.1: c2) supply the reactions. Those of the second kind, actually the Routh-Voss equations (3.5.15, 16), con¬ stitute a set of n + m = (3 N — h) + m equations for the n + m unknowns: {q k ; k=l,...,n} and {X D ; D = l,... ,m}. Once the p’s and A’s have been found as functions of time, then r P = r P (t, q) —* r P (t) [—> a P = a P {t)], and, again, (ex. 3.5.1: c2) supply the reactions. From the latter and the (now) known A’s, we can calculate the p’s. In sum, in the second-kind case we have 2 h fewer equations, which is the result of having absorbed the h holonomic constraints into that description with the n = 31V — h q s [see remark (v) below]. Also, even in the presence of additional holonomic and/or nonholonomic constraints, we still work with the unconstrained kinetic energy T. However, and this is a general comment, the ultimate judgement regarding the relative merits of various types of equations of motion must be shaped by several, frequently intangible/nonquantifiable considerations (in the sense of the famous Machian principle of Denkokonomie), in addition to the mere tallying of their num¬ ber of equations, and so on (“bean counting”). (v) Purpose for appearance of the multipliers. That the multipliers p. H , of the h holonomic constraints <j> H (t,r) = 0, are not present in Lagrange’s equations of the second kind (and in the Routh-Voss equations) is no accident: the m A^’s (and this is a general remark) express the reactions of whatever constraints have not been taken care of by our chosen p’s; that is, they are due to the additional holonomic and/ or nonholonomic constraints not yet built in (or embedded, or absorbed) into our particular q’s description. Then, the multipliers appear as coefficients in the virtual work of the reactions of these additional constraints. (vi) Apparent indeterminacy of Lagrange’s equations. Let us consider a system with equations of motion E k = d/dt(dT/dq k ) - dT/dq k = Q k . (3.5.26a) Since, as explained earlier, all possible constraints are already built in into the chosen p-description, the corresponding system constraint reactions R k have been elimi¬ nated from the right side of (3.5.26a); the Q k are wholly impressed. However, occa¬ sionally, the latter depend on constraint reactions: for example, the sliding Coulomb-Morin friction F on a particle sliding on a rough surface — according to our definition, an impressed force — is given by -FN(v/\v\), (3.5.26b) where N = normal force from surface to particle (clearly, a contact constraint reac¬ tion), p = sliding friction coefficient, and v = particle velocity relative to the surface. In such a case, if we embed cdl holonomic constraints into our p’s, and hence into our T WWW.EBOOK777.COM 426 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS and Qf s, the resulting Lagrangean equations (3.5.26a) will, in general, constitute an indeterminate system; that is, the total number of equations, including constitutive ones like (3.5.26b), will be smaller than the number of unknowns involved. Such an indeterminacy [what Kilmister and Reeve (1966, p. 215) call “failure” of Lagrange’s equations] can be easily removed by relaxing the system’s constraints, and thus generating the hitherto missing equations (see also “principle of relaxation” in §3.7). Similar “failures” would appear if one used minimal quasi velocities to embed all nonholonomic constraints (see also Rosenberg, 1977, pp. 152-157). (vii) We have presented the four basic types of equations of motion: Routh-Voss, Maggi, Hamel, and Appell. They can be classified as follows: Kinetic energy- based equations of motion Holonomic variables: Routh-Voss (coupled) Maggi (uncoupled: kinetic, kinetostatic) Nonholonomic variables: Hamel (uncoupled: kinetic, kinetostatic) Acceleration- based equations of motion Holonomic variables: Appell (coupled) Nonholonomic variables: Appell (uncoupled: kinetic, kinetostatic) Additional special cases and/or combinations of the above—for example, equations of Ferrers, Hadamard, Chaplygin, Voronets, et al.—are presented in §3.8. • From all the equations of constrained motion given earlier, only those by Hamel (and their special cases—see §3.8), through their 7 -proportional terms (recall Hamel’s formulation of Frobenius’ theorem, §2.12), can distinguish between genu¬ inely nonholonomic Pfaffian constraints and holonomic ones disguised in Pfaffian/ velocity form. All other types, that is the equations of Routh-Voss, Maggi, Appell (and their special cases—see §3.8), hold unchanged in form whether their Pfaffian constraints are holonomic or nonholonomic; that is, those equations cannot detect nonholonomicity, only Hamel's equations can do that. • On the other hand, only Appell’s equations preserve their form in both holo¬ nomic and nonholonomic variables', and, in the kinetic ones, the nonholonomic con¬ straints can be enforced in the Appellian function right from the start. (viii) The terms kinetic and kinetostatic, in the particular sense used here (brought to mainstream dynamics by Heun and his students, in the early 20th century), and observed by some of the best contemporary textbooks on engineering dynamics, for example, Butenin et al. (1985, vol. 2, chap. 16, pp. 330-339), Loitsianskii and Lur’e (1983, vol. 2, chap. 28, pp. 345-384), Ziegler (1965, vol. 2, pp. 146-152), are not well known among English language authors, and so one should be careful in comparing various references. (ix) Finally, we would like to state that we are not partial to any particular set of equations of motion; all have advantages and disadvantages; all are worth learning! All such conceivable equations (whose combinations and special cases are practi¬ cally endless; see also §3.8) flow out of the differential variational principles of analytical mechanics; that is, the principles of Lagrange and of relaxation of the constraints, in their various forms (see also §3.6 and §3.7). These principles, being invariant, constitute the sole physical and mathematical glue that holds all these (coordinate and constraint-dependent) equations of motion together—and they keep reminding us that, in spite of appearances, there is only one ( classical ) mechanics! WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS Geometrical Interpretation of the Uncoupling of the Equations of Motion into Kinetic and Kinetostatic The Routh-Voss equations, E k = d/dt(dT/dq k ) - dT/dq k = Q k + ^ A D a D k, (3.5.27a) represent an equation among (covariant) components of vectors at a point (q) in configuration space, or a point (t,q) in event space. Now, we recall from §2.11, eq. (2.11.19a IT.), that the n — in vectors A/ T = (A u ,..., A nI ) span, at that point, the null, or virtual, hyperplane (or affine space) Nj = V, of the constraint matrix A d = (a Dk ); while the m vectors A D T = (A lD ,... ,A nD ) span its orthogonal comple¬ ment, the range, or constraint, hyperplane (or affine space) C m . Therefore, multi¬ plying (3.5.27a) with A kl (A kD ) and then summing over k, from 1 to n, means projecting that equation onto the local virtual ( constraint) space; and since the con¬ straint reactions R k = Aare perpendicular to the virtual space, they disap¬ pear from the kinetic Maggi equations. Indeed, we have, successively, (i) ^ A kI E k = ^ A kI Q k + ^ ^ A D a DkA k i = ^ A k iQk + ^ A D 6 di = ^ A kl Q k + 0, that is, ^ A k ,E k = ]T A kf Q k or I, = (3.5.27b) (ii) A kD' E k = A kD'Qk + A D a Dk^kD' = ^ A k D'Qk + ^ A D Sdd 1 = ^ A kD 'Qk + X D t, that is, A k oEk = A kD Q k + A D or I D = 0 D + X D . (3.5.27c) Tensorial Treatment (Kinetic complement of comments made at the end of §2.11; may be omitted in a first reading.) In the language of tensors (whose general indicial notation begins to show its true simplicity and power here), the //, 0/, A k = 0 ( I D , 0 D ,A D = X D ) are covar¬ iant components of the corresponding system vectors along the contravariant basis A 1 (A d ), which is dual to the earlier basis Aj(A d ). Dotting the vectorial Routh-Voss equations [fig. 3.1(a)] E = Q + R, (3.5.28a) where (with summation convention) R = R k E k = (A fl a° k )E k (i.e., R is per¬ pendicular to the virtual local plane) with Aj = A'^Ek —that is, projecting it onto the virtual local plane — yields E-A, = Q-A l +R-A Ii (3.5.28b) WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS (a) (b) Figure 3.1 (a) Geometrical interpretation of uncoupling of equations of motion ("Method of projections" of Maggi); and (b) its application to the planar mathematical pendulum. or, since R- A, = X D (A D ■ Aj) = X D 6 D j = 0, finally E ■ Aj = Q- Aj, i.e., A k t E k = A k ,Q k (kinetic Maggi), or I I = 0 I - (3.5.28c) while dotting them with A D = A k D E k —that is, projecting it onto the constraint local plane—yields E-A d = Q-A d + R-A d , (3.5.28d) or, since R - A D = X D '(A D ■ A D ) = X D : 6 D D = X D , finally A k D E k = A k D Q k + X n (kinetostatic Maggi), or I D = 0 D + X D . (3.5.28e) For the planar mathematical pendulum of length /, mass m, and string tension 5 [fig. 3.1(b)], Aj = dr /d(j> = along tangent, A D = dr/dr = along normal , and so (3.5.28b, d) become E-Aj = Q-Aj\ ml(d 2 (f>/dt 2 ) = —mg sincj) (kinetic Maggi eq.), (3.5.28f) E ■ A D = Q-Aj) + R- A D : ml(d(f>/dt) = —mg cos cj)+ S (kinetostatic Maggi eq.). (3.5.28g) These geometrical considerations demonstrate the importance of the method of pro¬ jections of Maggi, over and above that of the Maggi equations. His method can be applied to any kind of multiplier-containing (mixed) equations. Example 3.5.6 Lagrange’s Equations (Williamson and Tarleton, 1900, pp. 437- 438). Let us consider a scleronomic system described by the Lagrangean equations d/dt(dT/dv k ) — dT/dq k = Q k (k = !,...,«). (a) WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS Now, the change of the system momentum p k = dT/dv k during an elementary time interval dt is ( dp k /dt)dt , and this, according to (a), equals Q k dt + [dT/dq k ) dt. Since the system is scleronomic, dT/dq k = quadratic homogeneous function of the v k ’s (see also §3.9), and therefore if the system is at rest, it vanishes. Hence, the result: The elementary change of a typical component of the system momentum consists of two parts: one due to the corresponding impressed force, and one due to the [possible ) previous motion. Problem 3.5.2 Lagrange’s Equations: 1 DOF. Let us consider the most general holonomic and rheonomic 1 DOF system; that is, n = 1 and m = 0, with inertial (double) kinetic energy 2 T = A(t, q)q 2 + 2B(t, q)q + C[t, q), [A, C > 0, always) and hence Lagrangean (negative) inertial force E q {T) = [dT/dq)' — dT/dq = (1/2) [2Aq + [dA/dq)q 2 +2[dA/dt)q + 2[dB/dt) -dC/dq], (i) Show that the new Lagrangean coordinate x, defined by x = [A[t,q)) xl2 dq x{t, q) q=q(t,x), (a) (b) (c) reduces 27" to 2T = x 2 + 2b[x, t)x + c(x, t ), (d) where b[t,x) = (A l/2 [B/A + dq/dt) ) (e) k ) evaluated at q=q(t,x) c(t,x) = (A(dq/dt) 2 + 2B(dq/dt) + cl , (f) ) evaluated at q=q(t,x) and generates the following (negative) Lagrangean inertial force: E X [T) = ( dT/dx)' — dT/dx = d 2 x/dt 2 + db/dt — (l/2)(<9c/c>x); (g) that is, no [dx/dt)-proportional (i.e., damping/friction) terms. Such coordinate trans¬ formations may prove useful in nonlinear oscillation problems. (ii) Show that in the scleronomic case, i.e., when B, C = 0 and hence 2T = A(q)q 2 , the inertia forces (b) and (g) reduce, respectively, to A[d 2 q/dt 2 ) + [\/2)[dA/dq)[dq/dt) 2 and d 2 x/dt 2 . (h) Problem 3.5.3 Lagrange’s Equations: 1 DOF. Let us consider a 1 DOF system with kinetic and potential energies 2 T = A[q)[dq / dt) 2 and V = V(q), (a) WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS respectively, capable of oscillating about its equilibrium position q = 0. Show that the period of its small amplitude (i.e., linearized, or harmonic) vibration equals [with (••■)' = d(...)/dq] 2n[A(0)/V"(0)} 1/2 . (b) HINT Here, A( 0) > 0, V(0) = 0, V'(0) = 0, V"(0) > 0; and, as shown in §3.9 If., Q = -dV/dq = - dV/dq = -V’. Expand T and V a la Taylor about q = 0, and keep only up to quadratic terms in q and q, etc. Problem 3.5.4 Lagrange’s Equation: 1 DOF. Continuing from the preceding problem, show that if q = q a is an equilibrium position, instead of q = 0, then (b) is replaced by 2n[A(q 0 )/V"(q 0 )] 1 ' 2 . (a) Problem 3.5.5 Lagrange’s Equations: Pendulum of Varying Length. Show that the planar oscillations of a mathematical pendulum of varying, or variable, length / = /(f) = given function of time, on a vertical plane, are governed by the (variable coefficient) equation +glsin(f) = 0 => d 2 (f>/dt 2 + 2(j/ l)(d(j> / dt) + (g/l) sin^ = 0, (a) where <j> = angle of pendulum string with vertical. For the treatment of special cases, see for example, Lamb (1943, pp. 198-199). Example 3.5.7 Lagrange’s Equations: Planar Double Pendulum; Work of Impressed Forces. Let us consider a double mathematical pendulum in vertical plane motion, under gravity [fig. 3.2(a)]. Below we calculate the components of the system impressed force by several methods. (a) (b) (C) Figure 3.2 (a-c) Double planar mathematical pendulum, under gravity; calculation of impressed system forces. WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS (!) From the Q k -definitions (§3.4). Here, with q x = 0i, q 2 = 02 and some obvious notations, we have I*! = (/j cos0i, /[ sin0 l7 0), r 2 = cos0j + / 2 cos <j> 2 , l\ sin0! + l 2 sin0 2 ,0), (a) F\ = (™ig,0,0), F 2 = (m 2 g. 0,0), (b) and, therefore, we obtain Qi = S dF 1 ( dr / d di) = F i • (9fi/dqi) + F 2 • (dr 2 /dq x ) = ■ ■ ■ = —m x gl i sin0! - m 2 gl\ sin = —(m, + m 2 )gl\ shn^, (c) Q 2 = S dF ' ( dr / d( h) = P\ ■ {dr\/dq 2 ) + F 2 • (, dr 2 /dq 2 ) = • • • = —m 2 g l 2 sin 0 2 . (d) (ii) Directly from virtual work. Let us find Q 2 , that is, S'W for 8<j>\ =0 and <50 2 f 0: (8'W) 2 = Q 2 8(j> 2 . Referring to fig. 3.2(b), we have (8'W) 2 = (m 2 g)S(l 2 cos 0 2 ) = —m 2 gl 2 sin 0 2 8(j> 2 => Q 2 = — m 2 gl 2 sin0 2 . (e) Similarly, to find Q\ —that is, 8'W for <50] f 0 and <50 2 = 0: (b'W) { = Q\ <50j, referring to fig. 3.2(c), we find (8'W) { = ( m x g ) 6(li cos 00 + (, m 2 g ) 8(l x cos 00 = (m x + m 2 )g 8(1 x cos 00 = ~(m\ + m 2 )gl\ sin0! <50] => Q x = ~(m x + m 2 )gli sin0]. (f) (iii) From potential energy (see also §3.9). Here, the total potential energy of gravity (^impressed forces), V = F(0i,0 2 ), is V = —(m\g)(l\ cos 00 - (m 2 g)(l\ cos0, +/ 2 cos0 2 ) = -(m x +m 2 )gl[ cos0! -m 2 gl 2 cos0 2 , (g) and since S'W = —8V, we obtain £?i = -dV/dfi = -(mi +m 2 )gli sin0!, g 2 = -dV/dcf) 2 = -m 2 g/ 2 sin0 2 . (M) REMARK Had we chosen as system positional coordinates (fig. 3.3) <7i = #1 = 01 and q 2 = 6» 2 = 0 2 - 0! = 0 2 - 0,, (j) then (g) would assume the form F = F(0i, 0 2 ) = -(m, + m 2 )gli cos9i - m 2 gI 2 cos (9 X +9 2 ), (k) and the corresponding Lagrangean forces would be Q x = -dV/d9 x = -(mi + m 2 )gl x sin 0, - m 2 gl 2 sin (0, + 0 2 ), Q 2 = —8V/d9 2 = — m 2 g l 2 sin(9i +9 2 ). WWW.EBOOK777.COM 431 ( 1 ) (m) CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS I Figure 3.3 Double planar mathematical pendulum under gravity; alternative coordinates. Example 3.5.8 Lagrange’s Equations: Planar Double Pendulum; Derivation of Equations of Motion. Continuing from the preceding example (and its figures), let us first calculate the kinetic energy of the pendulum. We find, successively, jci = /| cos 0! => jcj = —/ 1 sin 0!, (a) y\ = l\ sin =>j>i =hh cos0 1; (b) X 2 = 1 1 cos (j)\ + 12 COS (j) 2 =>■ x 2 = sin^! — l 2 f 2 Sin(j) 2: ( c ) y 2 = !\ sin (f> x + l 2 sin^ 2 => y 2 = h<Pi cos^] + l 2 f 2 cosfy, (d) vi 2 = (.t 1 ) 2 + (j,) 2 = ••• = /! W, (e) v 2 ~ = (x 2 ) 2 + (yi)~ = ■■■ = i\~{4> i) 2 + 2 / 1 / 2 cos (</>2 — f 2 + 4“(<fe) 2 ; (f) 2 T = m x vf + m 2 v 2 = ■ ■ ■ = {m x + m 2 )l\-(<j) i) 2 + 2m 2 l x l 2 cos{<f> 2 — <pi)fi<j> 2 + m 2 / 2 2 (</> 2 ) 2 ; (g) and by the preceding example, Qi = -( m i+m 2 )glism(j> l , Q 2 =-m 2 gl 2 sin f 2 . (h) From the above, we obtain dT/dfi = {ni\ + m 2 )l x ~(j)i + m 2 lf 2 cos(cf> 2 — <t>\)<j> 2 , (dT/dfaY = {m x +m 2 )l x 2 ij)\ + m 2 l\l 2 cos (</> 2 — f\)cf> 2 — m 2 I\l 2 sin(</> 2 — fi)(4> 2 — 4 >i) 4> 2 , dT/dfi = m 2 lf 2 sin (</> 2 — (f> x )f x f 2 . WWW.EBOOK777.COM (i) (j) §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS Therefore, Lagrange’s equaLion for q x = fa\ (dT/dfa)' — dT/dfi = Q x , becomes after some simple algebra, (in\ T mi)l\~(d(f>\/dt ) —|— 77^2/1 /^z cos ((/>2 — $1 ) (d(f 2 !dt ) — m 2 l\l 2 sin (</>2 — 4>\)(dfa/dt) 2 + (in x + m 2 )gl\ sin^q = 0 . (k) Similarly, we find Lagrange’s equation for q 2 = fa'. m 2 l 2 2 {d 2 (j) 2 /dt 2 ) + m 2 l x l 2 cos(fa — (j)\){d 2 ({)\/ dt 2 ) + m 2 l\l 2 sin (</> 2 — (j)\){d(j)\ / dt) 2 + m 2 gl 2 sin fa = 0 . ( 1 ) The above constitute a set of two coupled nonlinear second-order equations for fa(t) and <j> 2 (t). Constraints (i) Assume, next, that we impose on our system the constraint /1 =yi = h sin^i =0 [=> fa(t) = 0 =>84>\ = 0]; (m) that is, we restrict the upper half OP\ to remain vertical, so that the double pendulum reduces to a simple pendulum P\ P 2 oscillating about the fixed point P x . Since df/dfa = l\ cos fa = !\ and df/d(j) 2 = 0 [=> <5/j = {l\ cosfa)5fi + {0)S(f>2 = (l\)8fa + ( 0 )^ 2 ], the equations of motion in this case are (k) and ( 1 ), but with the terms A] /1 cos</>i = Ai l\ and and Ai -0 =0 (where X\ = multiplier corresponding to the con¬ straint df\ = 0 ) added , respectively, to their right sides; that is, in general, it is not enough to simply set in these two equations f 1 = 0 (=>- fa = 0 , <f>\ = 0 )! Indeed, then the equations of the (m)-constrained pendulum motion decouple to the Routh-Voss equations: fa: X x = m 2 l 2 [cosfa(d' , fa/df) — sinfa(dfa/dt) ] (kinetostatic) , (n) fa : d 2 fa/dt 2 + {g/l 2 ) sin (/> 2 = 0 {kinetic). (o) With the initial conditions at, say, t = 0: <p 2 {0) = fa = 0 and fa(0) = fa, equation (o) readily integrates, in well-known elementary ways, to (the energy equation) (fa) 2 = (faf ~ (2g// 2 )(l - cos fa), (p) in which case, (n) yields the constraint reaction in terms of the angle fa = faf) and its initial conditions Ai = ■ ■ ■ = m 2 [(2 - 3 cos fa)g - l 2 (fa) 2 ]sinfa = Ai (t,fa,fa). (q) Finally, since 6 , W R = [X l (df/dfa)\Sfa = R l 6fa = (Xil x cosfa)6fa = X l 6(l l sin fa) = A, 6y x (= 0), (r) the multiplier represents the (variable) horizontal force of reaction needed to preserve the constraint (m). [Other forms of (m) will result in different, but physically equiva¬ lent, forms of the multiplier. See also §3.7: Relaxation of Constraints.] WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS (ii) Similarly, if 0 2 acquires a prescribed motion, say <j) 2 = f{t) = known function of time, then, since in that case <50 2 = bf(t) = 0 [= (0) bfi + (1) <50 2 ], we must add a term A 2 - 0 = 0 to the right side of the 0]-equation, and a term A 2 • 1 to the right side of the 0 2 -equation [where A 2 = multiplier corresponding to the constraint f 2 = 0 2 — /(/) = 0 => 8f 2 = 0]. The rest of the calculations are left to the reader. Problem 3.5.6 Constrained Double Pendulum. Continuing from the preceding example, assume that we impose on our pendulum the constraint fi=y 2 = h sin 0 i + h sin 0 2 = 0 . (a) (i) Show that in this case, and for the special simplifying choice l x = l 2 = / [=> sin 0 i + sin 0 2 = 0 => 0 i + 0 2 = 0 ], the equations of motion reduce to (m x + m 2 )l 2 (d 2 cj) x /dt 2 ) — m 2 l 2 cos(2</) x ) (d 2 f x /dt 2 ) + ;w 2 / 2 sin( 20 1 )(c/ 0 1 /dt ) 2 + (m x + m 2 )g l sin 0! = Ai/cos0 1; (b) — m 2 l 2 (d 2 <f>\/dr) + m 2 l 2 cos( 20 i) (d 2 f x /dt 2 ) — m 2 l 2 sin( 20 1 )(c/ 0 1 /(f /) 2 - m 2 g I sin f x = Ai/cos 0 i. (c) (ii) From the above, deduce that [e.g. by adding (b) and (c) etc.]: A[ = (m 1 // 2 )(l/cos 0 1 )(c? 2 0 1 /r(t 2 ) + (m 1 g/ 2 )tan 0 1 . (d) Interpret the multiplier Aj physically. (iii) From the above, deduce that [e.g. by subtracting (b) and (c) from each other etc.]: (m!+4m 2 sin 2 0 i)(ii 2 0 i/fl!f 2 )+ 2 m 2 s'm(2(/)i) (dfi / dt) 2 + (m l +2 m 2 )(g/l) sin 0 ! = 0 ; (e) i.e., a single pendulum-like, reactionless (kinetic) and nonliner equation. Example 3.5.9 Small (Linearized) Oscillations of Double Pendulum. Continuing from the preceding example, let us study the small (linearized) amplitude/velocity/ acceleration oscillatory motions of our planar double mathematical pendulum about its equilibrium configuration 0 ! = 0 , 0 2 = 0 . There are two ways to proceed. Either (i) we keep up to quadratic terms in 0 1; 0 2 and their derivatives in T and V (or up to linear ones in the g’s) so that the corresponding Lagrangean equations end up linear in these functions; or (ii) we directly linearize the earlier-found equations of motion (for a more general treatment of linearized motions, see §3.10). Let us begin with the first way; it is not hard to show that the earlier T , V (Q i 2 ) approximate to the homogeneous quadratic (linear ) forms: 2 T = ( m x + m 2 ) I f (fi)" + 2m 2 l i/ 2 0 i 02 + m 2 l 2 2 (ff) 2 (a) 2V = (m x + m 2 )gl x (f> 2 + m 2 gl 2 (j) 2 2 + constant terms , (b) Q\ = -(mi + m 2 )g /[ 0 i, Q 2 = —m 2 g l 2 f 2 . (c) WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS Then, with L = T — V = Lagrangean of the system, we easily obtain dL/df i = (mi -\- m 2 )l 2 fi + m 2 lil 2 ^> 2 , (d) (dL/dfi) = (nii + m 2 )l\~4>\ + m 2 lil 2 (f> 2 , (e) dL/dfi = -(mi +m 2 )gh<t>\ (= Q\)\ (f) dL/d (}>2 = m 2 l iL^i + m 2 ^i ^21 (g) (dL/dff) = m 2 l il 2 cji + m 2 l 2 2 (j> 2 , (h) dL/ d(j> 2 = —m 2 g l 2 cj) 2 (=Qi). (i) Therefore, Lagrange’s linearized (but still coupled!) equations are (nii + m 2 )li(d 2 (j)i/dt 2 ) + m 2 l 2 (d 2 (j) 2 /dt 2 ) + (m x + mfgfi = 0, (j) 1 1 (d~ fi / dt) + l 2 (d~cf> 2 /dr) + g <fi 2 = 0. (k) The reader can verify that (j, k) result by direct linearization of (k, 1) of the preceding example, respectively. Solution of System of Equations (j, k) As the theory of differential equations/linear vibration teaches us, the general solu¬ tion of this homogeneous system is a linear combination, or superposition, of the following harmonic motions (or modes)'. fi = A sin(wt + e) and cj) 2 = Bsin(ujt + e), (1) where A,B = mode amplitudes, to = mode frequency, and e = mode phase. Substitu¬ ting ( 1 ) into (j, k), we are readily led to the algebraic system for the mode amplitudes: [(nii + m i)(g ~ hu 2 )\A + (—m 2 l 2 tj?)B = 0, (m) (— liui 2 )A + (g — 1 2 cj 2 )B = 0. (n) The requirement for nontrivial A and B leads, in well-known ways, to the determi- nantal (secular) equation (mi +m 2 )(g- liu?) -fui 2 — / 7 ? 2 12^ g - hu 2 = 0 , (o) which, when expanded, becomes (milffuf - [(mi + m 2 )(li + l 2 )g\ur + (m x + m 2 )g 2 = 0 . (p) To simplify the algebra we, henceforth, assume that m x = m 2 = m and l\ = l 2 = l. Then (p) reduces to tv 4 - A(g/l)u? + 2 (g/l) 2 = 0, WWW.EBOOK777.COM (q) CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS and its positive roots can be easily shown to be w i = {P - (2) 1/2 ]0//)} 1/2 {lowerfrequency), (rl) w 2 = {P+(2 ) 1/2 ](g//)} 1/2 (> uj u higher frequency). (r2) For lu = u x , u> 2 , the amplitude ratios H^B/A = [l l u?/{g-l 2 u?)}=J/[{g/l)-J} [=n{u?)] (s) [obtained from (n), for /[ = l 2 ] are found to be Mi = B x /A x = [2 - (2) 1 / 2 ]/[(2 ) 1/2 - 1] = (2) 1 / 2 , (si) M 2 = B 2 /A 2 = -[2 + (2)‘/ 2 ]/[l + (2) 1 / 2 ] = —(2) 1 / 2 , (s2) that is, B x = {2) l ' 2 A x and B 2 = — (2)^ 2 A 2 , for any initial conditions, and therefore the general solution of (j, k) is — 4>\,l + ^1,2) = 4*2,\ T 4*2, (t) where 0i,i — A\ sin(c<Di/ + £i), 02,i — Mi^i sin(cDi? + £i), (tl) (f> 1,2 = ^2 sin(w 2 t + £ 2 ) , 0>,2 = M 2^2 sin(w 2 t + s 2 )- (t2) The above show that, for each frequency u) k {k = 1,2), the ratio of the correspond¬ ing mode amplitudes <j> x k and <j> 2k is constant ; that is, independent of the initial conditions 02,i / 0i,i — Mi — (2) 1 / 2 and 02, 2 /0i,2 — M2 — —(2)*^“- (t3) The remaining four constants A x , s x , and A 2 , e 2 are determined from the initial conditions. For example, if at t = 0 we choose = 0, fa = 0, and f 2 = f 0 , <fi 2 = 0, then, since <j) x — A j lo x cos(cc j t T £ x ) -f- A 2 uj 2 cos(c o 2 t T e 2 ) , (ul) <j) 2 = (2) A x uj x cos(uij t-fCj) — (2) *' ~ A 2 lu 2 cos(cc 2 t T c 2 ) , (n2) eqs. (t-t 2 ), the above, and the initial conditions lead to the following algebraic system: <t>\- 0 = Ai sin^ + A 2 sine 2 , (vl) </>o = (2 ) X/1 A { sin^ - (2) l//2 A 2 sin e 2 , (v2) 01 : 0 = A j UJ\ COS £\ ~\~ A 2 (jJ 2 cos e 2 , (v3) fi- 0 = ( 2 ) ^A\uj\ cos£i — ( 2 ) t~A 2 uj 2 cose 2 . (v4) From 0 => [( 2) 1/2 the last two equations, we readily conclude that cos e x = cos e 2 = e x = e 2 = 7 t/ 2 ; and so the first two reduce to A x + A 2 = 0 and A x — A 2 = /2]f 0 , and from these we easily find A x = [(2) 1/<2 /4 ]f 0 and A 2 = WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS LOWER Frequency ( CO p HIGHER Frequency (tip Figure 3.4 Angular modes of planar double pendulum, for its two frequencies: (a) lower frequency, (b) higher frequency. The amplitudes of <j >i t, <j>-\ 2 depend on the initial conditions. — [(2)'' 2 /4]^ 0 . Hence, the particular solution of our system (j, k), satisfying the earlier chosen initial conditions, is h = [ < / > o(2) 1/2 /4][cos(tui?) — cos(w 2 ?)], (wl) 02 = (0 o /2) [cos (uq t) F cos(w 2 t)]; (w2) where oj|, u 2 are given by (rl, 2). The relative modal contributions for each frequency are shown in fig. 3.4(a, b). Problem 3.5.7 Double Pendulum; Noninertial Coordinates. Consider the double pendulum of fig. 3.3. (i) Show that its (Lagrangean) equations of motion in the angles 0j(= 0p and 0 2 , under gravity, are [n%il F T m2(/1 - T 2/ \l 2 cos 9 2 F l 2 2 J\(d ^ 9 1 / dt 2 ^) F ni 2 l 2 (l \ cos 9 2 F l2)(d~9 2 /dt 2 ^j — ( m 2 l x l 2 sin 9 2 )(d9 2 /dt) 2 — (2m 2 /[/ 2 sin9 2 )(d9i/dt)(d9 2 /dt) F (mi F m 2 )l\gsm9\ F m 2 l 2 g %m(6 { F 0 2 ) = 0, (a) ( m 2 I 2 2 )(d 2 0 2 /dt 2 ) + m 2 l 2 (l\ cos 0 2 F l 2 )(d 2 9i/dt 2 ) F (m 2 l 1 l 2 sin9 2 )(d9 l /dt) 2 F m 2 l 2 gsin(9i F 9 2 ) = 0. (b) (ii) Obtain its equations of small motion; that is, linearize (a, b). (iii) What do (a, b) reduce to for /[ = 0, or / 2 = 0, before and after their linear¬ ization? Problem 3.5.8 Double Physical Pendulum. A rigid body / of mass M can rotate freely about a fixed and smooth vertical axis. A second rigid body II of mass m can rotate freely about a second smooth and also vertical axis that is fixed on body I (fig. 3.5). (i) Show that the (double) kinetic energy of this double planar “physical” pendu¬ lum is 2T = Aip 2 F2r<j>ij)F # 0 2 , WWW.EBOOK777.COM (4 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS where A = MK 2 + ma 2 , B = m(k 2 + b 2 ), T = mab cos(</> — ip) = mab cos (ip — </>) = in a b cos x (definition of angle x); K(k) = radius of gyration of /(//) about 0 (G^. (ii) Show that, in this (force-free) case, dT/dij> + dT/dip = = total angular momentum about G-axis = constant = c, (b) or (A + r){df/dt) + {B + r)(dip/dt) = c-, (bl) and 2T = E ( another constant). (c) (iii) Show that, with the help of x — ^ — e q. (bl) can be further transformed to (A + 2r + B){dip/dt) = c — (A + r)(dx/dt), or (A + 2r + B)[A(d<p/dt) + r{dip/dt)\ = {A + r)c+(AB - r 2 )(d X /dt). (d) (iv) With the help of this integral, show that the energy integral (c) can be rewritten as (dx/dt){A{d<p / dt) + r(dip/dt)\ + c(dip/dt) = E, or (, d X /dt) 2 (AB - r 2 ) + c 2 = (A + 2r + B)E. WWW.EBOOK777.COM (e) §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS (v) Finally, and recalling the F-definition, show that (e) transforms to ( d\/dt ) 2 = [(A + B + 2mabcosx)E — c 2 ]/[AB — ( mat ) 2 cos 2 x] = /(x)> (f) that is, the problem has been led to a quadrature. For further discussion of this famous problem, and of its many variations, see, for example (alphabetically): Marcolongo (1912, pp. 213-216), Schell (1880, pp. 549- 551), Thomson and Tait (1912, pp. 310, 324-325), Timoshenko and Young (1948, pp. 209-211, 215-216, 249-250, 276-278, 312-314). Problem 3.5.9 Double Physical Pendulum: Vertical Axes. Continuing from the preceding problem (penduli axes through O and O' vertical), obtain its Lagrangean equations of motion. What happens to these equations if the center of mass of the entire system I + II is at its maximum / minimum distance from 01 Assume that O , G, O' are collinear, and OG = h. HINT Introduce the new angular variables cj\ = (p and q 2 = 9 = ip — <p (= — x) = inclination of body II relative to / (positive counterclockwise). Then, T —> T(6;<p,9), and so on. Problem 3.5.10 Double Physical Pendulum: Horizontal Axes. Consider the pre¬ ceding double pendulum problem, but now with both axes through O and O' horizontal. In addition, assume that the mass center of body /, G, lies in the plane of the axes O and O' , and OG = h. Show that here T is the same, in form , as in the previous vertical axes case, but the potential of gravity forces, V, equals (exactly) V = — M ghcoscp — mg(a cos <p + b cos if) + constant , (a) and therefore the corresponding Lagrangean impressed forces are Qcj, = -dV/dcp = • • • and = -dV/dtp = ■ ■ ■. (b) Then write down Lagrange’s equations for q\= f and q 2 = tp. Problem 3.5.11 Double Physical Pendulum: Horizontal Axes; Small Oscillations. Continuing front the preceding problem (O and O' horizontal), show that for small oscillations about the vertical equilibrium position ip = 0, linearization of the exact equations leads to the coupled system (Mh + ma)[L(d 2 cp/dt 2 ) + grp] + m ab(d 2 ip/dt 2 ) = 0, a(d 2 (p/dt 2 ) + L'(d 2 ip/dt 2 ) + gip = 0 , (a) where L = (MK 2 + ma 2 )/(Mh +ma) and L'= (k 2 + b 2 )/b. (b) Interpret L and L' in terms ofsingle pendulum quantities. Then, assume as solutions of (a) cp = (p„ sin(uV + e) and ip = ip 0 sin(cut + e), (c) WWW.EBOOK777.COM 439 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS where 4> 0 ,ip 0 = angular amplitudes, e = initial phase, and u> = frequency, and show that the ur are real, positive, and unequal, and are the roots of (£2 — L)(Q — L 1 ) = (. ma 2 b)/(Mh + ma ), where Q = g/w 2 ; (d) say Q l < Q 2 ', and thus conclude that 0, < min (L,L r ) < ma x(L,L') < Q 2 ■ (e) Finally, show that f 0 /f 0 = a/(Q-L') = ---, (f) and, therefore, (i) for the smaller to (—> larger Q = Qf), • ip > 0 (i.e., in the slower mode, the angles have the same sign); while (ii) for the larger lo (—> smaller Q = Q { ), f-ip <0 (i.e., in the faster mode, the angles have opposite signs). [For a discussion of the historically famous case of the nonringing, or ",silent ”, bell of Koln (Cologne), Germany (1876; bell + clapper = double pendulum), based on (a), see, for example, Flamel ([1922(b)] 1912, 1st ed., pp. 514 If.), Szabo (1977, pp. 89-90), Timoshenko and Young (1948, p. 278).] Problem 3.5.12 General Form of Lagrange’s Equations for a 2 DOF System. Consider a 2 DOF holonomic and scleronomic system; for example, a particle on a fixed surface, or the previous double pendulum, with (double) kinetic energy 2T = A(dx/dt)~ + 2r(dx/dt)(dy/dt) + B(dy/dtf , (a) and such that 6'W = X8x + Y5y, where A,B,r,X , Y, are functions of x, y. (i) Show that its Lagrangean equations of motion in q\= x and q 2 = y are A(d 2 x/dt 2 ) + r(d 2 y/dt 2 ) + (l/2)(dA/dx)(dx/dt) 2 + ( dA/dy)(dx/dt)(dy/dt ) + [dr/dy - (1/2 )(dB/dx)](dy/dt) 2 = X , (b) B(d 2 y / dt 2 ) + r(d 2 x/dt 2 ) + (1/2 ){dB/dy){dy/dt) 2 + (dB/dx)(dx/dt)(dy/dt) + [dr/dx — (1/2 )(dA/dy)](dx/dt) 2 = Y, (c) and ponder over the geometrical/kinematical/inertial meaning and origin of each of these terms. (ii) Show that these equations linearize to the (still coupled) system: A 0 {d 2 x/dt 1 ) + r 0 {d 2 y/dt 1 ) = {dX/dx) 0 x + {dX/dy) a y, (d) B 0 (d 2 y/dt 2 ) + r o {d 2 x/dt 2 ) = {dY/dx) a x + (dY/dy) a y, (e) where (.. ,) o = (...) evaluated at x,y = 0. Example 3.5.10 Lagrange’s Equations, 2 DOF: Elastic Pendulum, or Swinging Spring. Let us derive and discuss the equations of plane motion, under gravity, of a pendulum consisting of a heavy particle (or bob) of mass m suspended by a linearly elastic and massless spring of stiffness k (a positive constant) and unstretched (or natural) length b (fig. 3.6). This is a holonomic and scleronomic WWW.EBOOK777.COM §3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS o two DOF system; that is, n = 2, m = 0. With Lagrangean coordinates as the polar coordinates of the bob: cp = r, q x = <j>, its (double) kinetic energy is 2T = mv 2 = m(ds/dt ) 2 = m[(f) 2 + r 2 (</>) 2 ], (a) while the virtual work of its impressed forces, gravity and spring force, equals S'W = —k(r — b) 6r + (mg cos </) 6r — (mg sin (j>)(r 8(j>) = Q r 6r + Q^ 6(j> , (b) that is, Q r = -k(r-b) +mgcos<t>, Q <j> = -m gr sin cj>. (c) Alternatively, the potential energy of the system is V = (1/2 )k(r — b) 2 — mgr cos 0 = V(r, (j>), (d) and so the corresponding Lagrangean forces are Q,. = — dV/dr = ■ ■ ■, = — dV/d(j) = ■ ■ -, equations (c). We also notice that for r > b: Q rsprmg = — k(r — b) < 0, as it should; and analogously for r < b. Lagrange’s equations, then, are E r (T) = E r = Q r : (mr)'— mr(<j>) 2 =—k(r — b) + mgcostj), (e) E d T ) = E^> = Qf (mr 2 <j>y =-mgr sin fa (f) or, after some simplifications (since r ^ 0), d 2 r/dt 2 — r(d(j>/dt ) 2 = —(k/m)(r — b) + geos </, (g) r(d 2 (j)/dt 2 ) + 2(dr/dt)(dcf>/dt) = —gsincf). (h) The general solution of this nonlinear and coupled system is unknown, and so we will limit ourselves to some simple and physically motivated special solutions of it. WWW.EBOOK777.COM CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS (i) Equilibrium solution : Setting all (.../'-derivatives in (g, h) equal to zero, we find [wi