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«J NASA TECHNICAL TRANSLATION NASA TT F-14,576 A THEORY OF CALCULATION OF FLUTTER VIBRATIONS IN SUBSONIC FLOWS W. Roth c o P -* Translation of: "Eine Theorie zur Berechnung von Flatterschwingungen bei Unterschallstroemungen," Acta Meehaniaa, Vol. 6, pp. 22-41, 1968. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D.C. 20546 SEPTEMBER 1972 J NASA TT F-14,576 A THEORY OF CALCULATION OF FLUTTER VIBRATIONS IN SUBSONIC FLOWS 1 W. Roth ABSTRACT. Calculation of a critical flutter velocity for a membrane in a nonconservative system. It is shown that the damping forces have a destabilizing effect and that a dis- continuity appears in the stability criterion which depends on the damping coefficient and is characteristic of noncon- servative systems. By means of this theory, the flutter of weather vanes or sails, as well as the related traveling transverse waves, can be explained. I . Introduction . /22_* the calculation of the flutter oscillations of a thin profile or a plate without flow may be accomplished using the singularity method by means of a fictional vortex layer that changes with time. The calculation is relatively complex and vague; it eventually leads to the problem of the solution of an integral equation. Kuessner [1], Schwarz. [2] and Soehngen [3] performed calculations of flutter oscillations in 1936-1940. The calculation would be much simpler if there were a pressure law that would describe the reaction of the flowing medium on a plate element as an explicit function of the deformation of the elements. Such a pressure law would have the advantage that the problem of calculating the flutter oscillations would be separated from concepts of aerodynamics and become a pure problem of technical oscilla- tion theory. Now such a pressure law is available for supersonic flows, /23 provided by Ashley and Zartarian [4] according to the "Piston Theory." One is therefore led to wonder whether or not such a pressure law could be obtained for incompressible flows as well. x The present paper was delivered in abbreviated form by the author at the GAMM Congress in Zuerich (1967), under the title "A Contribution to the Calculation of Flutter Oscillations." *Numbers in the margin indicate pagination in the foreign text. 1 II. Derivation of the Pressure Law . In the following discussion, it will be assumed that we are dealing with incompressible flow which is essentially friction-free. The flow around an oscillating plate will be compared with the flow inside an oscillating tube through which a medium is flowing! Such tubes have recently been studied in connection with their stability [5] , and the results of the calculations are also in agreement with the phenomona that occur in technology. In order to understand the forces that are created at the tube wall by the flow velocity v of the medium in the tube and those causing the movement of the tube, see Figure 1. A medium element with relative velocity v^ = v is moving through the oscillating tube element with a deflection z(x,t). In the kinematic sense, the tube element constitutes a vehicular element, toward which a medium element moves with relative'velocity v . A familiar principle of kinematics [6], however, says in connection with such a relative movement toward a vehicle that the absolute acceleration is composed additively of three components, so that ', * - V +b t + v where b f is the vehicular acceleration, b is the relative acceleration, and b is the Coriolis acceleration. Since in the present case the motion of the c : tube elements occurs primarily in the z-direction, b f = z . The relative acceleration at constant r consists of the relative centripetal acceleration b =v 2 z . The Coriolis acceleration is, calculated as follows /24 r xx | b c * 2 (uj >i v r ), i I so that with an angular velocity of the vehicle to = z we will have b = 2 v z . If we let q equal the force exerted by the tube wall on the c xt flowing medium, based on the unit length of the tube axis, the motion equation of the medium element in the direction of the instantaneous perpendicular to the trajectory plane will be q dx = y p (v2 Zxx ^ 2 v z xt + z tt ) d x, from which we obtain as the reaction of the flowing medium against the tube wall, the line load q as follows:,,. . y F (v 2 z +2 vz . + z..) XX xt tf (II. 1) Figure 1. Kinetic Values For a Tube Element with Flow Through It. In this pressure law, y through the internal cross-section of the tube according to the filamentary flow theory applied here is a known mass of the flowing medium directly in- volved in the transverse oscillations of the tube, based on the unit length of the tube axis. As far as the tube is concerned, there is no objection to using (II. 1), at least when a very thin tube is employed, for which the ratio r/L composed of the tube radius j of the internal cross section and the tube length 1 is much less than; 1. A theoretical expansion of (II. 1) occurs which is valid with respect to technical applications as well, if the right-hand side of (II. 1) is increased by a damping factor, so that we have M F (V Z x* + 2 V Z xt + Z tt> + 6 V (H - 2) Such an assumption of damping with damping factor <5 Q is necessary in the sense of modern stability theory according to Lyaponov [7], and to a certain degree is even necessary, as we shall see later on in the discussion of the results. The pressure law (II. 2) can be applied to a plate around which a medium is flowing if q and y are based on a unit area of the plate and weight per unit area u r is assumed known. The pressure law given by the "Piston Theory" for r supersonic flows, in comparison with (II. 2), is 6 1 v z x + 6 V (II. 3) where 6, <=6 n = <p /c represents a coefficient determined by the medium; 1 °° °° here < is the polytropic exponent, p ro and c w are the pressure and speed of sound in an undisturbed medium. It is advantageous for further calculations, however, to ensure a discussion of 4 the .results that is as^omplete as possible, so that 6 f 6.. In (II. 3) the deflection of the plate element is 'contained 1 in only one term with the derivative zr, the second term in this instance also; representis damping. • i III. Calculation of the Critical Velocity for a Membrane with Rectangular Boundaries . We shall now use pressure laws (II - 2) and (II. 3), for comparison of the results they yeild, in the case of a membrane (Figure 2) with rectangular boundaries subjected to flow in the x-c.irection at velocity v. The membrane is assumed to be stressed in the x- anc y-directions by forces S. and S based on the unit length, and these forces are further assumed to be positive, pressure forces. Since the edges of the membrane are assumed fixed, the desired solution z (x, y, t} for the mqvements of the membrane must satisfy the boundary conditions z (x, 0, t)_-.p ...: z (x, d, t) « In addition, in the sense of the stabi! disturbances ... z.ttCy. t) = o__ _ V ^ n (III.l) z ;(c, y, t) =0. j ity study at time t = 0, the initial z (x, y, 0) = £ (x, y) | z (x, y, 0) - * (x,y) (III. 2) \ may be given for the initial deflectiori and the initial velocity, where the functions <j> (x, y) and $ (x, y) are rar.dom within rather broad limits. If the pressure law (II. 3) is initially ajplied to the membrane, we will obtain the partial differential equation S. Z + S Z + 1 xx 2 yy W Z tt + & 1 v z x + 6 Z t (III. 3) as a motion equation, in which y is 1;he weight per unit area of the membrane;. It is not necessary to assume an additional external damping in this equation j which is proportional to the deflectiori velocity z ; instead, this can be j /25 thought of as included in the damping factor 6 Q since 6j t 6 Q and the two numbers may therefore be thought of as independent of each other. With the 7rmy rm _i9-z ^ (III. 4) u(x, t) sin —r- (m = !> l > 6 > ■••> y expression z = uix, tj sin -5 fill 3] is changed to the partial differential equation S l U xx + MP U tt + 6 1 V U x + 6 U t " S 2 ~^~ U = °" Lon This equation can then be solved by means of the separation expressi. ixx igt (III. 5) u = e where, with maintenance of the boundary conditions in x according to (III.l) and summation over all existing particular integrals the following solution can be obtained: ,V' r lib <» i » > (III. 6) Here c and c, are freely selectable constants, and the characteristic ' Imn 2mn exponents of the time are therefore ,> i \v V* * s '. I • T '"l s ; ' s '' f •""' | 2 (II 1. 7) These exponents describe the behavior of equation (III. 6) with time. In order to develop the solution z (x, y, t) to satisfy random initial disturbances (III. 2) according to (III. 6), all particular integrals are required, and hence the exponents (III. 7) must be discussed for all combinations of whole- -number values of m and n. It is advantageous in conjunction with further calculation to consider separately the two different cases 6 Q = and 5 Q > with arbitrary 6 > 0. If we initially assume that 5 Q = 0, i.e., if we calculate without damping, it becomes evident that instability is avoided either because no oscillations developed that become apparent with time or (what amounts to the same thing) exponents iS 12mn exhibit no positive real P arts > if rTTT jn Sj < and S 2 < ( TII - 8 J t , > n However, if we assume that S < and thereby for all values of 6 > 0. However, .„ sta bility imposes the assume damping, the retirement for avordance ^ t b 1 P ^ condition tm.S). ^though stabiUty » now ssured and^ ^ solution (HI. 6, exists which decreases wrth Um C - ^ reoul res validity for all combinations numbers^ ^ » ^ ^ ^ „ avoid instability. Prom the physical Stan porn t ^ ^ really not very convincing that crrterton (III.8, do s n vel ociUes V, and that there is consequent y ^ > ^° J J different ^ nutter oscillations of the ^ ra c h ^ ^ ^^ movement conditions are stable or unstab ^ ^ of the magnrtude of the flow veloc. y v ™ - _ ^^ Bolotin [8]. When bending resrstance » lne ^ [4] , [9] . t he velocity v of the medium can influence the stabrlrty , ftt ?l to the membrane leads to a solution The application of pressure ia. (1.2) ^^ for the z (x,y,t) which is different from (III. 6 , and motion equation for the membrane which is to differential equation with (III. 9) (III. 10) of the flow medium ,->•? 1-hp tlOW mcuxum „ - u + y is the sum of the weight per unit area y p of the Id the weigher »« area , p of the membrane „* the assumpt^" ., which sattsfres the boundary cond-ons o . ^ . (III.l), we obtain from (III. 9) the parx 71 II). \ 1 „ u = 0, which can be solved by means of the separation expression (III. 5) or directly by means of the expression h (x, t) - F (a .»■ + bt) sin --"-'- (» = 1, 2, 3. . . .)• Here, F (ax + bt) constitutes a transverse wave moving through the membrane in the x-direction at a velocity For the function F, we can find the conventional second-order differential equation ' ^ ,, -;r tt .+-;/ w - ;w;x**' + f '/«''" 'V Hi '' J i ' H J. with the argument u = ax + bt and •V* - 'Vf "" 'V • (II 1. 12) The exponential expression F = exp iru gives the characteristic exponents for the time s*\ \ . 1/ V , *■•• i*m» . i:''** 'Vl-- 1 ,'"-! 2 1. (HI. 13) t r i and the solution of (III .9) then has the form - , _•'?■-<,-„„ . --"''--"-(*-*.,, (IH-14) or <*2 w « ^ HI - .-.111 I I ■ I /I* M 1 » 1 = (•»•..'/.') =--- £ 2> u r • s " 1 -/ ' (III. 15) in - i ii i <•,»,» *'l »>„("■'• i l>l) l^nn, F 2m A"-*' bt)) with constants c^ and c^ which are again random. These constants can be established so that the initial disturbances (III. 2) are satisfied. The «= b f.y + btl and F (ax + bt) alone do not satisfy the transverse waves F lmn (ax + Dtj ana r 2mn icia > /28 ^nx c *~~ , rv v t~\ since although the factor sin boundary conditions in X for z (x.y.tj, since i appears in the solutions of (111.14) and (111.15). these waves are distorted uuasilocally or locally and the result is that the edges of *• -*'«™ regain at rest and do not undergo deflection. Solutions (III.6) and (III 41 as well as (III. 15), however, display exponential dependence on the wave oims in x Such wave forms were also given by Sparenberg [10] for an infinitely long membrane subjected to flow. Different results were obtained here, however, with respect to behavior with time. In order to determine the behavior of the solutions with time, the exponents („!..« must be discus by analogy with the above. It turns out that instability with vanishing dating 5 =0 can be avoided if ' S t < and .Sj* < » or N, = and S,** < (III. 16) or S t > o and V* < and A\* > « or \ r r. s .y~ <v ^ p..'. On the other hand, if we assume doping with 6 Q > 0, in order to avoid in- stability or implementation of stability we will have the required (III. 17) We can enjoy only a slight degree of conviction in accepting crlt n C K J in contrast to (111.17) because of its inhomogeneous structure which includes numerous possibilities and because it also partly contains the *"» *1 which cannot be explained physically. Hence, fro, the physical and t «h ni» standpoint, it is also unacceptable because it was developed without onsidera tion of damping and damping forces are always present in reality Th. stability theory of Lyapunov offers the clue that the criterion (II obtained with consideration of dating may be considered correct. Criterion rill 16) can only guarantee avoidance of instability but cannot guarantee • u „c a - n was assumed in establishing establishment of stability, inasmuch as 6 Q - was assum this criterion. However, according to Lyapunov, this is precisely the critical case, in which the theory predicts that the smallest nonlineanties /29 a A in the present linearized calculation, can lead to which were disregarded in the piesen uncer tain which motion ,•■.-*. ti- i<; therefore, in fact, unceitaxi i::;i:rr;:i ;u — « ^-^^cizr the motive power produced by the stream pressure « , ^ ^ s „ess on the membrane in the x-direct.on «"'"£'" ' md then allo ws calculation of the critical flow velocity v R ,,...-, , - Sl . (III. 18) Th e doping factor «„ U no longer contained in th, criterion,^ criterion [IU .16, is not always obtained for S „ C J my th ere is a discontinuity wrth respect to am -^ ^ ^ration criteria. Since criterron (I II. 17) C indicating that the f damping, the present result ^ * ^ , destabilil ing damping that exists when pressure law (II. 2) » oscil lations eff ect on the oscillating membrane and ensures that flutter develop for all flow velocities V > V instability phenomena, caused by critical --ciUes^can also develop ln the case of rudders or single-bladed »,« * < mad e by „eidenha«r [11]. - such a system, « * °" iMl the critical velocity depends on the darning factor « velocity given by criterion (III. IS) is -depen en ^ ^^ with respect to the dam p f r J^ ^ ^ ^ ^.^ degrees of freedom. Ziegler L i J reS ultant unsteadiness in the stability criterion for a aouoie v IV. Determination of the Weight Per Unit Area or Pressure Layer . The calculation, as performed thus far, is based on a known weight per unit area y of the flowing medium. The critical flow velocity v k according to (V.10) is dependent on y and the value of y p is likewise dependent on the form of oscillation of the membrane. In the following, we shall describe a calculation of y„ which is based on the singularity method. According to r Birnbaum [13] and Glauert [14], the flow around a thin profile (Figure 3) can be obtained through interaction with a vortex layer of intensity k(x). The induced velocity v (x) at point x, caused by the totality of interactions k(x J ) at points x 1 , is given by the Biot-Savart law l '»W=^J -(7' -'*)-■ x' « o 77777777 h Figure 2. Membrane Plate With Rec- tangular Boundaries Subject to Flow. The expression k (.»•) = 2 v\A ctg J + £ A n sin n &\ n - 1 (IV. 1) for the vortex application k(x), with substitutions x = j (1 - cos©) and X 1 = J (1 - COS<f>) (0 < < tt, < <j> < *) with consideration of integral values rr -t ■ ,, In =- 0, 1, 2, . . .) C eon u (p dip J {eon ip - cos (-)) . ^ 1 r„ M , -f and the flow condition, becomes V dz 'dx A + ]£ A H conn (-). (IV. 2) Figure 3. Flow Around a Thin Profile With Application of a Vortex Layer. 10 v.njs and wt obtain A _, .s ■ ,,^-r.--\ : ^\.* ip. •' •■' - 1 /' ''" Ai-I A -'{''," cos i' ( ' f ,,< "'- r, J ,U ( '°- A " » J "' <-■> u (i it The lift, based on the unit length of the profile depth, is calculated according to the Kutta-Joukowski lift fo»»la with consideration of the corre.pondxng sign determination which applies here and the density p of the medium at d P/dx q = - vpk(x). If we equate this lift to the static pressure components , , f TT ?1 hv the relative centripetal acceleration developed in the pressure law (II. 2) by the reiauv (Figure 1), and use (IV. 1), we willhaye and thus obtain for the weight per unit .area__ ... Thus the value obtained'^ this' fashion for „ F can also be used for the terms in motion equation (III. 9) in the sense of an approximation theory for describing the flutter oscillations. Since Vf can also be interpreted as the weight per unit mass of a flowing medium for a tube with curvature ,U „ ith flow through it and having a rectangular cross section of width equal to 1, division by the density p of the medium gives the t he cross section, so that Vp - ph. This height h can be represented thickness or the height of a pressure layer, in which the pressure buildup on the surface of the membrane subjected to flow is created by the flowing of the medium at the velocity v, governed by the centrifugal forces of the medium elements in the vicinity of the bent membrane. The flow around the Lbrane can therefore be expressed by the totality of an infinite num ber "ubes having cross sectional height h and infinitesimal cross sectional width a ranged close together and parallel to the X-axis. Hence «»•=«« Less of the oscillating membrane with flow over it is attributed primary to the interaction between the membrane and the pressure layer or the weight per unit area Vf by analogy with the tube with flow through it. The flow 11 around the .emhrane plate corresponds to flow through a tuhe with resistance E I - 0, or flow throu 8 h a hose [IS], The weight per unit area fro, (IV.3J or the thickness h'of the pressure layer will generally vary wrth the profile coordinate x pr e. Hence, it is appropriate withtn the of an approximate method, such as is developed here to introduce an average value for „ p which is still .ore exact instead of Vf and likewise introduce an average alue for the pressure layer thickness h as weli. n order t ohtain initially an estimate of the order of magnitude of ,.„ and to ascerta.n the influence of constant curvature of the profile with a bulge f on the weight per unit area, we will assume for the circuLr profile accordtng Figure 4 / ,, .> /i\ (IV. 4) to be the result of the calculation. We 8f which has the curvature z 11 = p-> „f ja -nr-n-7^4...). Hence, will have A Q = 0, A 1 = 4 - and A n - (n - 2,3, 4,... J [IV. 5) ,,,,. :r.= ,., r sin (-), as the average weight per unit .area y p we will have . /f „„ = ; (,„,/,•=-■: 'jr. ■ 0,785 e r (IV. 6) as the average value over the up^rdly convex profile with profile depth c. The average thickness of the pressure layer is therefore /, = I r : ■: 0,7S5 c. (IV. 7) qn(1 h is accomplished with particular simplicity in this Determination of y and h is accompli*" .,.,„,. for the case, since in (iv's) the curvature that appears in the denominator for profile (IV. 4) is constant over the entire depth of the profile. Fieure 4. Circular Profile With Flow Around It as the Lowest Possible Form of Oscillation of the Membrane. /32 12 The lowest form of oscillation of the membrane plate can be approximated by the circular pattern of the deflections according to (IV. 4). Higher oscillation forms can be approximated by lining up such segments of a circle with alternating signs, so that a periodic wave "train results, which correspond to a sinusoidal pattern for the wave profile at higher wave numbers. This /33 approximation would have the advantage that in (IV. 3) the curvature z" would be constant for each segment and therefore taking the average over y p would also be simple from the calculating standpoint. In the meantime, however, on the assumption of such a wave profile, the curvature over the base c of the profile would behave in an unsteady fashion, and the calculation would prove to be quite tedious as far as determining the deflections z according to Fourier from the curve of the curvatures^ is concerned. Hence, for the profile ^/HiupA-D-V-, (A.=-l,2,"a,...r (IV. 8) which describes a sinusoidal half-arc over the base c with A = 1 and which can then be compared with the circular profile (IV. 4), after which the weight per unit area y p can be determined. For values X > 1 (IV. 8) describes a profile with 2\ - 1 half-waves. If we ;recall that the relationship cos (.«• sin y) = /„ (.<0 4 2 £ / 2 „ (x) coh 2 w y ( IV . 9) exists, where the I 2n (x) are the Bessei functions of the first type and the second order, we will then obtain for the profile according to (IV. 8) as a function of the coordinate using (IV. 9) ,= -/(_ 1)4/. (A*- -jM + 2£ /,.(**- -J) <-1>»cu,2»6>]. The flow condition (IV. 2) then becomes , . i / i ■* \ Hin 2 n (-) A- 1 If we use the relationship ^l?_!i®. = 2 V cos (2 i - 1) 6>, (n = 1, 2, 3, . . .) i-l 13 we will then have v, (-r) clz v dx 00 00 = (- 1)' 16 I £ £ **P [ X n - t) (~ l ) p P cos ( 2 m-\)G m — 1 p — m = (- 1)* 16 -{- £ «2m-i cos (2 m - 1) <9 w- 1 with no "2W.-1 = ^ 7 2 p (A 7T - J I (_ ]) P/J . /34 If we compare this result with the flow condition (IV. 2), we will obtain the coefficients of the vortex interaction with A = 0, A = (n = 2,4,6,...) and " m = Vr^,) / z " < /j: = (~ 1) A 2 n -/, (2 A - 1)» For calculating the average weight per unit area p_ we must now define an r average value !» = (- 1)» 16 A a 2m ^ = (_ l)M6 { £ /„ [A * - -J-] (- i), p . 10) ^ = (- 1)» 16 A fl^.j = (_ i)M6 A 2 7 2p (A * - J] (- i), p . P -„, l j (IV. 1( (n= 1, 3, 5, ...) for (IV. 3), in which the integration extends over the central portion of the average half-wave of the wave profile (IV. 8) from the point x = C ^ X " ^ to the point x - . UA . .. . Here j. ^ = -^v.-^r~- represents the arithmetic mean over the same region of the profile (IV. 8) It is attained in such a fashion that the curvature of the profile (IV. 3) behaves constantly approximately like a circular wave profile. Further calculation then gives 1 Q C l' F 2 (/,-«,)? A x (2e + sin 2 f) -f (IV. 11) * - z "J 14 for the profile (IV. 8) or !<r = " n (ik'-i) j«, (2f + «in 2t) - V i_-J)".^» ii ((2 m - 1) Bin (2 /« - 1) e co» £ - cos (2 m - 1) >' sin f ) 2, "i {M - 1) u (IV. 12) The angle Tis the complementary angle of the angles e, and e, associated /35 with boundaries Xj^ and x 2> sins = - ( 2T-"I)" between which the relationships &1 = Ji" E ' °2 " 2 ' * ^ for x = 1 and x -_£(l- cosGO hold. The numerical evaluatxon of (IV.12) a Teads'to e = f and then to the disappearance of the entire sum expressxon x„ (IV. 11) and (IV.12), so that ~ f:= ___ 4cffli remains, for this lowest half-wave number, we obtain the value of coefficients /,",.= 0.800 e r (IV. 13) a to a = - 0.223, so that and for the thickness of the pressure layer we will have the value A = 0,890 r (IV. 14) The differences between the numerical values (IV.6) and (IV.7) for the icu rofile and the values (1V.13) and (IV.14) for the s nuso.dal of.le relatively sHg- according to the method used for ^- ^ ~ which represents a certain confirmation of the accuracy of thx yp ° approximation, figure S shows the manner in which*, » ^^ k«^c of the profile. In this diagram, the curve or y p /P h-iaher half-wave numbers of tne pruiJ-xc. according to (IV.!,) and therefore that of h/c over the higher half-wave numbers X is shown. Figure 5. Dimensionless Weight Per "nit Area I /pc or Dimensionless Pressure Layer Thxckness h/c as the Function of the Wave Factor X of the Oscillating Membrane. 15 ■„,. definition used for calculation of the average -eight pen una volume „ „ pressure .oyer thtckness h fp W d in C«V.10, contains a certaan degr f arbitrariness As indicated by a comparison of numerical values wath he r. t— — «- -— - v the r - n r t/:: : . " , average (IV.10) does make sense from a mechanical standpoant. If » ».. y t al po nt of profile 3 - \ on the middle half-wave for eva uata g ow pro ess around the wave profile due to the pressure distrabutto and perform the calculations for this point, for which the curvature of the profile z „ = ( _ iy . 1*'- (2 A - 1)» /36 is y according to (IV. 3), we will have F It is, however ;t2 _ _ _e_ c _.._ r fll - «. + «,-, -«,+ •• •]• K _ „ 3 + «.-«,+ ...] = -! '«*« [A» - 2 J ( l + 2 f ) and therefore the formula /'#■ = -„• (2 A- 11" 2rf 2,4< l 2 J ( - n wbich is much simplier in contrast to (1V.11, and (IV.12,. i. «^-»^» calculation of the weight per unit area or pressure layer thacaness. T, alculated for „ in this fashion can also he used as representataves :;: r « *.*.»" «- » lu .. *« -now from (I v. 15 , ^ I d ewise those for pressure layer thickness with h- and carry ou the aLulation for the odd numerical values X - 1,3,5,7 and >*'™££^ unit area and pressure layer thickness according hoth to (W.12) C™. deviation hetween the numerical valuea occurs in the case of X - 1^ **« . since the factor of ,-, or h* calculated according to t e samp = , e f . c n f rxsrilation of the membrane plate is located nv 151 for these lowest forms of osciiation vx „ :„ the factor ( 1V,, and t .V.7, for the eircula rly curve d eontour and th e factor CIV.l, and C.V.14, for the sinusoidal a f-wave the valu a calculated in simplified fashaon according to ( V 5). ( smaller than those from (IV.12), are also acceptable. T*e 16 i « lie slightly below the curve *„ the case of even X-values, lie siignw in those ways, xn the o ^ ^ ^ ^ in the case plotted for odd X-vaU.es. ™= r * t the point x . c relative of ov en x-values their downwash of the prof ^ ^ ^ ^ ^ „ the bulge of the profile at porn X - P^ ^^ ^ ^ . odd values of X. Numerically, these responding sUght ly s nusoldal pattern of the curv en r ^ ^ — "** ^""^^rhryl «1 undlteoted. nnall, these Figure 5). are so — 1 1 tl»« they c representati ons of the average discrepancies are a conscience ° ^ ™ b ? combining several half- values, which can, however, he •^»^ lly ^ . sm00thing of the — of the profile in the avera ging^ ^J ^ f ^ „ _ curves results. For practical requir m ^ ^ suf£ice; since it is better to Keep , numer U, ^^ ^^ small, there is a safety factor provided xn calcula hlc; frlQC | _°^i 3 I.„ 0,16187 0,15005 0,08084 0,05827 0,05488 0,04373 0,04144 V Prospects. ' t rt . .ritical flow velocities v of the membrane The emulation of the cr t * ^ ^ ^ according to («".» » - ° "^ „, velocity Vfc is obtained for which we have determined. The lowest half . wa ve as a deflection th. lowest form of oscillation of the membrane with for which (according to IV.13) „ F - 0.890 PC so that irTi24-' s ''- i,o« I (V.l) ■ „n higher wave numbers for the membrane. High er critical velocities are given.gh « ^^ ^ ^ ^^ ^ z^7T£~^*» •■;» «~ ai r° tiv - i2) cTable5, so that Bt := j,r_"o,173 ^ = *.**\ -V (V.2) 17 pe £r T Fi r: 5 , r a" — : - - -» — - - — the radical in (V.1J ana iv.zj s theorV with the aid appropriate critical velocities is — >^ , "^ "tafcen. In the of Figure 5, from which the factors of the Vf /pc values c case of even half-wave numhers, the calcuiation was not performed. U ^ corresponding to (IV. 8). Instead,™, v crit erion (IV. 1) give the factors in front of the radical sign m - a. «, « ^ or aiso (1V,> -ith suffic ent accu - • ^; ^ ^ half „„ aves l U; Vecause one can calculate fro, it the lowest .deed POSS.lv critical velocity v, and achieve .utility of the membrane for all velocities V below v,. according (V.l). Recent lv, Thwaites [16]. Melsen [17], and Hey.at, and Zierep [18] rrr. rr:"r:::"';,::.;::'-r= : .— , fashion, i.e., witnou have critlcal tion that resting critical wave shapes of the sal resu ^ - — -;• F Tv ™ tc:rrr:-ti. ., ». ^. Factors Factors 2 T he kinetic stability criteria for such sails wit g^ r for branes free are, however, the same as ^ criteria ci ^ ^ ^ solutlon with laterally fastened edges This follows ^ ^.^ £unction (III 4) for the motion equation (lli.yj instead of the sine function in y. th. first line and factors taken from [16] and [17] on the second line ft that even at five naif-widths the difference between the numerical u f 2. 7 3 and 2 .4S is still ve ry sli g ht constitutes a con -• ^ •w.a in this naoer In mechanics, it amounts to the agree the theory described an this paper. ^ nent of the static with the kinetac stability criteria 7 the present case, the situation is somewhat more complex, .inc. th In the present da ™r,i nE in the motion equation inclusion of the supporting member and the damping „„ 9) gives a correct result only in conjunction with the proper pre ( I] The calculated results indicate that the pressure law (1.2) i P i ly correct. It is not without interest from the mathematical a a oint that even with a more general pressure la. which is com, . * r tt ?■) and fll 3) the calculation again yields criterion additively from (II.^J ana U 1 -^ . aaai ' . trt ; c witerion is concerned, it is . As far as the derivation of this criterion Zrl « have inclusion of the inertial term in the motion equation of ~a kv t-hP foriolis acceleration 2 v z t - the membrane caused by the torioi^ a xt The flutter oscillations can be seen particularly well in ^ <"*°J •bhln or flags Figure 6 shows a photograph of a ribbon in a wind tunne ribbons or flags, ngu n.ttern which changes with nne can clearly see the instantaneous sinusoidal pattern, Une can cxcm./ j^~ ri~, ■ TVii <; wave :;;:: r;;^ ""::: velocity. „*. * k according „ ,,,, n cttt ill the simple relationship we will obtain from (III. 11 J tne simpi ■ r H * (V.3) V -- /39 „hich expresses the wave velocity « as a function of the flow velocity^^ -: rMii::::;; r;::::i:::-5 ~ — - • - — r v < v k (subcritical veloci ty v through the membrane and " ' \T:l X:^2» o run llll state) the transverse waves travel 3The author would like to fxpresa his |^* c ^/Sf ^rt 1 had t beS°c^pleted, S^rLaMinglhlfcoSari-retween-umrerical values to be made subsequently. ■ vTv.j ' w» 'g .i>wii». W;-jij .-up. i u i- ' jy 1 '"."-wm ma- nn,i-t.i| i iMH a (P|ii;uwi H 'I Figure 6. Photograph of a Fluttering Cloth in a Wind Tunnel In the first case, nothing can be seen jof the flutter, since the slightest disturbances are immediately damped. Ijn the second case, the travel of the transverse waves can be seen clearly. However, in the fluttering membrane the same types of phenomona canbe^eeT^ as in an oscillating drive belt, which served in [15] as a special case of a tube with flow through it. In particular, the test in the wind tunnel also revealed that as the flow velocity v increases the wave velocity w likewise increases and according to (V.3) vw+v. Thus, in addition to the good agreement of the aboye described numerical values, it is a phenomenological confirmation of the theory given above. In addition, the sails of sailboats moving with the wind show transverse oscillations of the type under discussion. These flutter oscillations are not caused by some disturbing objects in the airstream, such as the mast supporting the sail of a sailboat, which could create vortices and thus stimulate the flutter oscillations, but they arise solely from the supply of energy in the airstream. An energy balance based on the fluttering system composed of the membrane is not possible, since any amount of energy can be drawn from a medium flowing with a constant flow velocity v. This is a typical feature of nonconservative systems, to which the membrane plate with flow over it must be assigned and which leads to the described lack of steadiness in the derivation of the stability criterion. /40 20 - PFERENCES!. : [I] Kuessner, H. G., "Concluding Report on Nonstationary Lift of Wings," Luftfahrtforschung, Vol. 13, pp. 410, 1936, Kuessner, H. G. and L. Schwarz, "The Oscillating Wing with Aerodynamical ly Balanced Rudder," Luftfahrtforschung, Vol. 17, pp. 337, 1940. [2] Schwarz, L., "Calculation of Pressure Distribution of a Harmonically Deformed Airfoil in a Flow," Luftfahrtforschung, Vol. 17, pp. 379, 1940. [3] Soehngen, H,, "Determination of Lift Distribution for Random Nonstationary Movements (Plane Problem) ," Luftfahrtforschung, Vol . 17, pp. 401, 1940. r [4] Ashley, H. and G.Zartarian, "Piston Theory -- A New Aerodynamic Tool | for the Aeroelastician," Journal of the Aeronautical Sciences, Vol. 23, ; pp. 1109, 1956. ■! ■ ■ . . 1 [5] Ashley, H. and G. Haviland, "Bending Vibrations of a Pipe Line Containing Flowing Fluid," Journ. Appl. Meah. , Vol. 17, pp. 229, :1950. Housner, G. W. , "Bending V^bratio^s of, u a ( Pipe Line Containing Flowing Fluid," Journ. Appl. Mech. , Vol. 19, pp. 205, 1952. 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Glauert, H. , "The Elements oiP Aerofoil ' and Airscrew Theory," S. ,87, Cambridge, University Press, 19^9. . Roth, W., "Transverse Oscillations of Cords with Flow Through Them," Z$P, Vol. 16, pp. 201, t ms'/ a f tiVYe ' I Thwaites, B., "The Aerodynamic Theory of Sails. I. Two-Dimensiional Sails," Proa. Roy. Soa. London $ev. A., Nr. 261, S. 402, 1961. : Nielsen, J. N., "Theory of Flexible Aerodynamic Surfaces," Journ, Appl. Meoh. 3 Vol. 30, pp. 435, 1963. Heynatz, J. T., and J. Zierep, "The Slightly Pitched Sail at Supersonic, Hypersonic, and Sonic Velocity, 1 ! Aoia Mechanica, Vol. 3, pp. 278, 1967, Coveh Pa< -Translated for the National Aeronautics Contract No. NASw-2037 by Techtran Corp< Maryland 21061; translator, Williap J 30 L 'iO i- source and Space d, rat ion, F Grimes , N . I . L Administration- ur^der - .0. Box 729, Glen Bumie, 4'j HtV, ! i 22 KOH)..;n Odd