# Full text of "DTIC ADA284539: Investigation of the Stability of the Vertical Position of a Gyroscope of Variable Mass. USSR"

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Best Available Copy \o OTS: 60-11,U68 AD-A284 539 JPRS: 2305 5 April i960 00 or INVESTIGATION OF THE STABILITY OF Bffi VERTICAL POSinON OF A GYROSCOPi: OF VARIABLE MASS ^ DTIC I^ELECTEf^ ^ AUG 02 19941 I - USSR - by V.S, Novoselov LIBRARY COPY APR 21 1960 o Thi3 do<^eni has been approved tor public release and sole; its distribution rs unliaited 94-23587 illlllllll'llilil Distributed by: FFICE OF TECHNICAL SERVICES , S. DEPAP.IMaTX OF COMMERCE WASHINGTON 2 $, D. C, U. S. JOINT PUBLICATIONS RESEARCH SERVICE 205 east U2nd SIREBT, SUITE 300 NEW YORK 17 , N, Y. 94 7 26 0 55 DTie QUALITY IJISPECTED S J?a3: 2305 CSO: 3367-1'1/C INVESTIGATION OF THE STABILITY OF THE VERTICAL POSITION OF A GYROSCOPE OF VARIABLE MASS / Following la a translation oF an article written by V* 3. Novoselov in Vestnlk Lenin- p:pad3kop:o Universlteta (Hem Id of Leningrad University, Ho, 5er Mat, Mekh 1 Astron, No. 4, 1959, pages 121-129. j This article Is devoted to a purely theoretical study of the stability of the vertical positinn of a gyroscope of a variable mass, the particles of which may possess an inter¬ nal motion. Technical features are not considered, Basic results of this article were presented in the author's the- sls / 1 1. Let UB obtain the "apex” equation CzJ of a gy¬ roscope oi’ variable mass. By this term we understand a body of variable mass that has kinetic symmetry and a stationary point. Ne shall also assume that a process of ejection but no accretion of particles might be taking place. The direc¬ tions of the principal axes of the body are assumed to be fi¬ xed. The formula of the variation of the kinetic moment of a body of a rarlable mass witn respect to a stationary point has the forraj ( 1 , 1 ) for the general case, allowing for the internal movement of particles / 3_/. _ In this formula Aqj-^Crk is the aforementioned kinetic moment expressed in the principal axes with the ortho- nomeal vectors i, jf* A and C are the moments of inertia; p, q, r the projections of the angular velocity. Further: D/Dt is the symbol of the derivative, with respect to time, which is computed on the as_sumptlon that all point masses of the rigid body are fixed. L and are the moments of the acting forces and of forces resulting from the variability of Codes y i/or / the total mass. Let u3 aae v for the velocity of the,end-point of the rector & which corresponds to the gyroscope's apex. We find{ + ( 1 . 2 ) Consequently, the kinetic moment shall be written in the foxrm: J^A(kxv)-^Cri. ( 1 , 3 ) on the basis of the formula (1*3) the equation (1,1) assumes the forms A{%xv)^CrS-^CfV^»t + K . (1,4) The moments of the active forces shall be composed of the moment of the force of gravity and the moments of the re¬ sistances to the gyroscope's motion, which will include the damping forces of the ejected particles. On neglecting fric¬ tion in the suspension and using formula (1,2) we may assume: —(••'>) Here h is the distance of the gyroscope's center of gravity from the point of suspension. The quantity h will oe greater than zero, equal to zero, or smaller than zero, depending oh whether the center of gravity is above the sta¬ tionary point, coincides with it, or is below it. V) is a po¬ sitive function of time, and vp a positive constant. (The quantities v^ and vg stand for resistance and damping). For a rapid rotation the resistance V 2 r must be replaced by the experimentally determined function f(r) of the resistance. It is not too difficult to realize a construction of a gyroscope, for which the relative kinetic moment of parti¬ cles moving in the gyroscope will be equal.to zero. In the case of such a construction the moment M / "bj would be /written in the form ( 1 . 6 ) Here V is the loss per second of the relative kineticfo¬ ment of the particles across the gyroscope's surface; is the loss per second of the transferable kinetic moment of the particles through the same surface. Assuming the process of the ejection of particles to be symmetric we shall have: where K is a known function of time. And we shall have further: ... 2 - ol -^OiXi) + C>i: (1.7) M.H) The syn?'^''! &nd.^, in the formula (1,8) are the known oositlv ttlona of time, corraapondln^^ to the mo¬ ments Gi iner . of loss of mass ^er second, Vie shall con¬ sider gyroscopes which Ay ^ 1^1/ ^ I Cf • The func¬ tions of time K, A'-y as well as A', C, h, ra and , can be determined experimentally upon the Btand._^ Takia^ the scalar and vector products of k and the equation (1,4), and teklrxf, Into consideration formulas (1,5) — (l»'"j)j obtain ^ 4*^1^^ 4 V, A)v (1.10) The equation (1,9) shall be called ''equation of the a- pex" as used by Yu, A, Krutkov, 2, Equation (1,9) has the Solution (1.11) For a very rapid rotation of the gyroscope, when the resistance may no longer be considered linear, the equation of rotation about the instrument's own axis can be integrated approximately. It is obvious that equation (1,10) has a particular solution __ _ v-ssO. (1,11-9 where a is the unit vector of the vertical axis, vie shall study the stability of the vertical position (1,12), For this we shall have equations by the first appro¬ ximation. Let k = a + R, we will consider the fixed system of the coordinates (x, y, z), with origin o at the end-point of the vector a. The axes ox, oy shall be horizontal, while oz is directed vertically downwards, V/e shall have the formulas v^R'y A?(l (1,10) On projecting, the equation (1,10) onto tne axes ox and oy by the first approximation with respect to quantities X and z and their derivatives, we obtain: * -T-*—Py-f fljf. j y^pjf -iy+ «y. I (l.l-V A A Cr ,• wiiece . P i Y *• Ih +''i + ai'e some bounded functions of time. i'^or the projection z ws have the formula z * 1 — V\ - Therefore, it follows from stability with respect to X and y and their derivatives, and of their derivative func¬ tions, that motion in the z direction is also stable. 2 Multiplying the first equation of the system (1,13) by X and the second by y we obta^, on adding, i V'-.~.2t(i* + y*)-i(x»+y’), (2.1) where we introduce the notation V^->+y‘-a(x’-ry*) (2.2) Let the center of gravity be located below the sta- tlnnarv^ point; then^<0 and consequently «<0. The function V shall be definite positive, since^*, /A/ and ^>0* Let OC no’w be a mon^otonlcaliy Increasing function. In that ease the derivative -^Kis definite negative, as follows from for¬ mula (2,1). For sufficiently small initial conditions, this derivative will remain negative definite if computed using the non-linear equations of the gyroscopic motion, because terms of the second and of higher orders cannot affect the sign in tne ri<i:ht member of the relation (2,1). Hence, by the iundamental theorem of A. 'A. Lyapunov's second method l_ 4_J7 vje conclude that tne vertical position of the gyros¬ cope of variable mass is asymptotically stable in the case considered. Let us suppose that the center of gravity coincides with the stationary point, i,e. Using the same theorem of Lyapunpv we shall prove asymptotic stability with respect to X and y. Since these latter are small for small initial conditions, we can use for their evaluation the equations 4 oftha firct approximation (1,13), wlilch In this case admit" tna l:iX.o J-C.1 jS)^ (2.3) Denoting by the symbol the averaging we find for the case when oL^C from the equations (1,13) A — ^ ~ V* ( j : _ -“?•(>- Vo). y— y« «• p* (jf ~ x^) - f* (y — y^). Therefore^ for small velocities, the coordinates will suffer negligibly from the initial values, 2. If the process of variation in mass occurs only in a finite time interval tJ a-nd should the resistance of the air to the rotation of the gyroscope about its own axis be negligible, the coefficients factors of the system (1,13) will be constant for t > T, The characteristic equation shall be written as: X-fr-l-2|>^ 4-2»-I* ?•) i* - «**0. In order to satisfy Hurwitz's conditions It is both sufficient and necessary'that In our case < 0, Therefore, if at the instant t the quantity h Is negative,, the vertical position of the gyroscope shall be stable. The nature of the gyroscope'^ motion can be examined by the method of B, 7, Bulgakov / 5_J7, Let the coefficients In the equations of the system (1,13) approach finite limits as t infinity. Assuming vg = 0, we obtain a boundary system with constant coefficients. If the limit _i3 less than zero, then by the theorem of K, P, Persidskiy vertical position of the gyroscope is also stable In this case, 3 1, Should internal motion of the particles in the gy¬ roscope be absent, as for example in the case of we shall have a purely superficial burning away, v'e shall neglect the air resistance. Then as a conse¬ quence of formulas (1,11) and (1,13) we shall find: 5 (3.1) ^ rf/, f- I > • pi + ay. I (3.2) 2* As a preliminary we will prove a theorem refer¬ ring to the approximate solution of a system of linear dif¬ ferential eauatlona with variable coefficients. Let us suppose we have a linear system: - <i« 1.2,. ., (.1.S) /-« where are continuous functions of time, and where /a^j/ < a for'O <1 t < infinity. V/e shall consider an arbitrary time interval ^0, T_7 subdivide It by points X — X ® parts so that <, 4 ^ — ■= 'C. X write the approximating system with the notation: dt • 2 All a,y(0) npH 0<r</,. aQitf) • npM 4 * 4 ) npH • (3.4) (3.'}) The theorem consists in the following: for any arbl- trarilv small £. > 0 and an arbitrarily large T 0 we can find a" sufficiently small ■t:> 0, so that for -//ir 0 ^ t ^ T we shall have: The continuous solution of the system {3»^) with the Initial conditions yj,© will be unique, Now we shall use the method of successive approximations for computing yi, taking XjL as the zero zpproxlmatlon. By virtue of the system (3»^) we have: yi^yiv-irJ^o'uyjdf 6 Thus we arrive at the following result: y, ^ X (y?* ~ *')’ AkI The functions a^j are continuous, therefc^e they shall he uniforialy continuous In the closed interval Gon- seouently, a value of /C can be found for which This gives the estimate (:5,6). It must be mentioned that this estimate shall be uniform relative to T as varies. 3. Let us examine the stability of the zero solution of the system (3,2), This system Is linear and the theorem Just proven applies. Let us consider the approximating system; where fi<'and /i*are constants obtained by the approximation rule oi' (3#5)« Settins? + (i» t ^ t ) , we reduce the ayaten (3,7) to a single equation S A- /f'u — a'tt - ‘t .-or the jth interval we have Let us set *“yf +-a;«y*0. Uj^rVjt 1%^ ♦ ( 3 . 1 ", which yields the equation for the variable Wj, r'ormulas (3,10) and (3,11) show that the motion of a point having the coordinates X| and can be represented as a set of relative motion in accordance with equation (3,11) and translatory motion of uniform rotation with an angular velocity /3j/A. V7e shall note that the stability of the sequence of functions u^, Implies the stability of the zero solution of system (3,7), since by formula (3,10) we have A solution in the variables u and d must be continu¬ ous, hence by virtue of (3,12) we have the following transi¬ tion conditions from the interval j to the Interval J + 1: (3 1?. where is the modulus of the complex number uj ; la th‘" aa xj u' velocity of relative motion or u derivative wliu ri.iocr t to -iniG w:.oi’e ars^ument in the complex number ; anu hj Si I — (l ^, Let us suppose that- tat . Denoting by the quantity Indicated on the interval, we obtain tne fol¬ lowing obvious integrals of tne equation (3,11); //^-consl, ^3 _- ‘^onsl. ( 3 . 15 , where is the relative velocity on the j^h in¬ terval. 3y virtue of formulas (3,13) - (3,15) the following relation will hold; I Let us suppose 4r^ Is decreasing (not necessarllv ra¬ pidly), Then by virtue of the formulas (5,13) - (3,16) -f — b] -h ^ p*(t) 4* 13 , 17 ) The expression in square brackets In (3,17) equals: In it the index indicates the averaging In the Indicated ib-^ interval a(; and fi; are derivatives computed at the ins¬ tant Sinceis decreasing we can expect that for suffi¬ ciently high relative velocities and a sufficient momentum K, «<0. Consequently, for any value of the indices 1 and j we shall have 3rom the formulas (3,17) - (3,18) we obtain: //,>! < 4 2| Si + A«A^( HeCr+'f). ( 3 . 19 > in wiaich we use the notations; 9 W w |^ll9j *♦* y ‘V(r) arsupj Juppose the initial conditions are sufricleritl 7 small .hen lattin-^ MNiTl ('•i) we obtain tne inequality K ^ t-+ 'i^"P 5 )^ t ti\»> The estimate (.5,20) is independent or 1', consequently it is valid lor the interval 0 4t '2 <- Ini’lnlty. ihc case v;here./3 and Increase is also of interest. (The increase does not have to be fast). Cn account of the Intewrai (3,1-^), the expression -tc ^ Gj p ^ conM. Oj ■» c(>n«ii ri. 211 will ai'5o bo an Intej^ral of the equation (2,7) on tne time interval. Due to the formulas (3,13) - (3,l6) and (3,21), we ha ve where v:e introduce the notation llnca b2 is an Increasia'? function, we have *V ^ piT" ■*" (P/,i ““ ij’urther on we will have 10 Let us suppose that the relationship (I < is sa¬ tisfied for a given number q. For sufflolently high rela¬ tive velocities of the moving particles, is considerably larger than \dj > Consequently, a sufficiently large number P can be found with the property that - —J— jrom the above, for a sufficiently small X, and fci <» the inequality will be satisfied. -/e have proved that in the two cases discussed the stability of the vertical position of a gyroscope of varia¬ ble mass is valid by the first approximation. And we ob- talnel, in addition, the estimates for the changes of the variables. BIBLIOGRAPHY 1* Novoselov, V* S, Certain Problems of the Hechanics of Variable Hasses, Author's Abstracts Candidate's Dissertation. Leningrad State University imeni A, A. Zhdanov, 1952* 2, Krutkov, Yu, A. On the Equations of the "Tops” Apex, I. News of the Acad Soi USSR, Ser. Math* and Hatural Sciences, 1932, pages 489-501, 3, Novoselov, V. S. Certain Problems of the Mechanics of Variable Masses in Considering the Internal Motion of Particles, I. Herald of Leningrad State Univer¬ sity, No. 19, 1956. 4, Lyapunov, A. M, General Problem of the Stability of Mo¬ tion. Moscow State Publishing House of Technical and Theoretical Literature, 1950, 5, Bulgakov, b. V. Vibrations. Moscow, Gostekhlzdat, 1954. 6, Persidskiy, K, P. On the Characteristic Numbers of Dif¬ ferential Equations, News of the Acad, Sci, KazSSR, Ser. Math, and Mech., No. 42, vyp 1, 1947* Article received by the edi¬ tor on 6 February 1958, 2023 END 12 FOR KEASCIIS OF SPEED AND ECONOMY TK13 FEPORT FAS BEEN REPRODUCED ELECTRONICALLY DIRECTLY FROM %'R CONTR'vCTOR’S TYPESCRIPT TaLs pa'.-).!!cation uaz prepared uiider contract to tne UI'IITED STATES JOINT PUBLICATICRS RE3EARCN SERVICE, a federal r;cvernine:vc orcian .zat tcn established to ser.ice bNe translaticn and research needs of t’.ie ari'^us government departments.