1300 AIAA JOURNAL VOL. 28, NO. 7

Dynamic Stability of a Rectangular Plate with Four Free EdgesSubjected to a Follower Force

Ken Higuchi* and Earl H. DowelljDuke University, Durham, North Carolina 27706

The dynamic stability of a flexible rectangular plate, such as a plate-like large space structure, is analyzed. Oneof the four free edges of the plate is subjected to a tangential follower force. The plate shows both flutter anddivergence instabilities. The flutter load and divergence load are obtained for various slenderness ratios of therectangular plate by means of modal analysis. In the calculation, many weak instabilities were found; a weakinstability sometimes corresponds to a critical load which is significantly smaller than the strong flutter load. Theflutter mechanism was studied in depth because the behavior of flutter is fairly complicated with respect tovariations in slenderness ratio. Moreover, it was determined that a smaller slenderness ratio is generally desirableto realize a high acceleration of the plate.

Nomenclature= length of the plate, (see Fig. 1)= width of the plate, (see Fig. 1): bending rigidity of the plate: element of nondimensional load matrix: element of stiffness matrix: element of nondimensional stiffness matrix= element of mass matrix= element of nondimensional mass matrix= surface density= nondimensional follower force, [see Eq. (37)]= nondimensional divergence load: nondimensional flutter load= generalized force, [see Eq. (12)]= element of generalized force, [see Eq. (11)]= follower force per unit width: kinetic energy= time= strain energy= generalized coordinate, [see Eq. (5)]: see Eq. (15)

Received March 7, 1988; presented as Paper 88-2412 at the AIAA/ASME/ASCE/AHS 29th Structure, Structural Dynamics and Materi-als Conference, Williamsburg, VA, April 18-20, 1988; revision re-ceived Nov. 28, 1988. Copyright 1988 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

*Visiting Scholar at Duke University on leave from Tokyo DenkiUniversity, Japan. Member AIAA.

fProfessor and Dean, Department of Mechanical Engineering andMaterials Sciences. Fellow AIAA.

JNote to the Reader: The terms "flutter" or "divergence" areused here to denote a dynamic instability or a static instability of aflexible structure under the action of a nonconservative force. In aero-space engineering literature, the source of the nonconservative force isoften an aerodynamic flow. Here, the source is a follower force. Forthose readers familiar with the aerodynamic flutter and divergenceliterature but not the follower force literature, it may be noted that thebasic dynamics and statics are similar. The flutter mechanism in fol-lower force problems is almost invariably one of frequency coales-cence, analogous to bending-torsion wing flutter in an aerodynamicflow. This is because the follower force itself is usually a static, ratherthan a time dependent, force. If the follower force was time depen-dent, other forms of flutter may also arise. The reader may wish toconsult a standard reference for further details, for example, "A Mod-ern Course in Aeroelasticity", edited by Sitjhoff and Noorhoff, 2ndEdition, by Dowell, Curtiss, Scanlan and Sisto, 1989.

w(x,y,t)Xm(x)

x,yYn(y)Yn(ri)

a.fim, $n

v2( )5SmiJnj6 WNCX = a/bv6

= deflection= coordinate function in the direction of x= w-th free- free beam function in the direction

o f ?= coordinates, (see Fig. 1)= coordinate function in the direction of y= n-th free-free beam function in the direction

of 17= nondimensional acceleration, [see Eq. (39)]= eigenvalues of a free-free beam, [see Eqs. (22)

and (23), (26) and (27)]= a2( )/dx2 + a2( )/dy2= variation= Dirac's delta function= Kronecker's deltas= virtual work= slenderness ratio= Poisson's ratio= direction parameter of the follower force,

(see Fig. 1)= frequency, [see Eq. (15)]= nondimensional frequency, [see Eq. (36)]= imaginary part of ft= real part of ft=x/a=y/b=d()/dt= derivative with respect to a spatial coordinate

I. Introduction

LARGE space structures will be constructed for low Earthorbit and then will be moved, for example, to a geosta-tionary point. A plate-like large space structure, such as asolar power station, may undergo dynamic instabilities be-cause of its low rigidity when it is thrusted by follower forces.Investigated here is the dynamic instability of a rectangularplate that has four free edges and is subjected to a followerforce on one edge.

For a free-free beam (missile) subjected to an end thrust, thedynamic instability has been studied by several authors.1'7Beal,1 using a Galerkin method, extensively investigated theproblem including the effect of a control and a pulsatingthrust. In later studies, Wu,3 and Peters and Wu4 stated thatthe second lowest branch of the eigenvalue curves determinesthe critical load. They demonstrated this numerically by usinga finite-element method based upon an adjoint formulation.They further studied the mode shapes analytically by using

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asymptotic expansions. Matsumoto and Mote,5 Park andMote,6 and Park7 considered a control system in order toincrease the stability of the beam. The history of research ona free-free beam with distributed parameters subjected to anend thrust is discussed in Ref. 4. A free-free beam with non-conservative loading on both ends has also been studied.8'9

Celep10'11 studied a circular plate with all free edges under-going axisymmetric and asymmetric deformations and sub-jected to a nonconservative edge load. Of course, by defini-tion, a circular plate is limited to a single slenderness ratio. Fora rectangular plate, there are also some studies,12'17 thoughnone for all free edges. A cantilever plate12 and a cantileverplate with simply supported two opposite sides13'14 subjectedto nonconservative edge loading on free edges were consid-ered. Leipholz and Pfendt17 investigated the stability formula-tion of a rectangular plate with various edge conditions sub-jected to follower forces uniformly distributed over thesurface of the plate, by using an extended Galerkin equation.These studies are helpful in understanding the complicatedcharacteristics of dynamic stability of plates under nonconser-vative forces. But none of their examples includes the casewhere all edges are free.

In this study, instability phenomena of the completely freeedged plate thrusted by a compressive tangential force uni-formly distributed on one edge are examined by means ofmodal analysis. Flutter load and divergence load are obtainedfor various slenderness ratios. And along with the change ofthe slenderness ratio, the mechanism of flutter is discussed.The effect of the true damping ratio on the flutter load is alsoinvestigated. Finally, the optimum slenderness ratio, wherethe largest acceleration can be realized without any instability,is determined.

II. FormulationHamilton's Principle

Hamilton's Principle can be written as

\t](5T - 6U + 5WNC)dt = 0 (1)where kinetic energy T potential energy U and the virtual workdone by nonconservative forces dWNC are written as followsfor a uniform, isotropic plate subjected to a follower force(Fig. 1).

2(1 - 3xdy

dxdy

dw _

1302 K. HIGUCHI AND E. H. DOWELL AIAA JOURNAL

Coordinate FunctionsOne may choose the mode functions of a free-free beam as

the coordinate functions Xm(x) and Yn(y).19'20 Here, let usintroduce the nondimensional coordinates as follows:

= x/a (0 < < 1)

=y/b ( 0 = 0.3).

20 30 40 50 60LOAD Q

Fig. 3 Eigenvalue curves (A = 0.3, jf = 0.3).

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350

300

250

0

2000

FLU

TT

ER

8~~ (3

i0

50

j \ | | -~ i

\ LOWEST '. QF ,

W^^SYMMETRIC MODE

\

\

\ SECOND LOWEST: QF>2\ ANTISYMMETRIC MODE

W _ O- - . _ . / . .O xvj ~~O ' ' ^ ^~" ~~~ C

\-\- V =0.3

9 16 25 36 49 64, -. __ LJ I 1 1 1 1 1 M 1 I

10 20 30 40

MATRIX SIZE

50 60

Fig. 4 Convergence of flutter load with number of modes used in theanalysis.

HI. ResultsEigenvalue Curves

Examples of the eigenvalue curves are shown in Figs. 2 and3. Figure 2 is for the case of square plate, i.e., X = 1. Flutteroccurs when two real frequencies merge into a pair of complexfrequencies. Divergence occurs when a real frequency becomeszero. These phenomena are clearly shown in Fig. 3 for the caseof A = 0.3. In this figure, QD^ and QDi2 denote divergenceloads, QF,2 and QFA show strong flutter loads, and QFl andQ/r3 show weak flutter instabilities. The figure shows that QF5corresponds to flutter of a much higher (frequency) mode.The weak instability QF.I, not the strong flutter load QF2 fora lower mode nor the divergence load QD>1, is the critical loadin this case.

There are altogether 64 frequencies in the model used forthe calculations. The lowest three frequencies of the freerectangular plate for 9 = 0 are always equal to zero, corre-sponding to rigid body modes. It is assumed that the rigidbody modes can be controlled properly and do not cause anydestructive behavior of the structure. So, the rigid body modesare not illustrated in the eigenvalue curves.

Figure 4 shows the convergence of flutter load for a squareplate with respect to the number of modes applied. Taking upto the eighth mode in each direction, which means a 64 x 64matrix size, appears to be enough to ensure convergence.

I4O

'DIVERGENCE POINT_

FLUTTER POINT

0.125 0.25 0 . 5 I 2 4 8

SLENDERNESS RATIO \

Fig. 5 Flutter load and divergence load vs slenderness ratio.

Weak InstabilitiesDivergence load and flutter load for various slenderness

ratios of the rectangular plate are shown in Fig. 5. Divergenceloads for symmetric and antisymmetric modes show smoothbehavior as the slenderness ratio changes. On the other hand,flutter loads sometimes change abruptly as the slendernessratio changes. The flutter mechanism is very sensitive to achange of slenderness ratio. This is because there exist manyweak instabilities due to higher modes and there may be achange in the most critical flutter mode due to a change inslenderness ratio. And these weak instabilities sometimes de-termine the critical load for a given slenderness ratio as shownin Fig. 5, and more clearly as previously shown in Fig. 3.

However, it is possible to identify smooth envelopes offlutter loads for similar eigenmodes. Flutter points that arenot connected by envelope curves in Fig. 5 correspond toextremely weak instabilities or weak instabilities of muchhigher modes. An envelope curve, written as A in Fig. 5,changes abruptly at its left end into an envelope curve, written

o:cj

0.15

O.I

0.05

O

X = 0.9

"SYMMETRIC MODE "ANTISYMMETRIC MODE-

i i i i Hfc

' X = l.25

V,B' :SYMMETRIC MODE

cj

o

oI O.05I R

0.125 0.25 0 . 5 I 2 4

SLENDERNESS RATIO

16

Fig. 9 Flutter acceleration and divergence acceleration vs slendernessratio.

quency for mode D due to an increase of slenderness ratio hasa limit, Fig. 7. However, the nondimensional frequency formode C is almost linear with respect to an increase of slender-ness ratio. Therefore, the difference between the two frequen-cies for modes C and D becomes smaller as the slendernessratio increases. Thus, with increasing slenderness ratio, thetwo modes can merge more easily into an instability. How-ever, the intensity of the instability becomes weaker; this weakinstability disappears at the intersection of the two modes Cand D.

DampingAs shown in Fig. 5, the lowest instability load may change

abruptly due to a change of slenderness ratio if there is nodamping. In the presence of structural damping, some of theweak instabilities disappear. A perturbation analysis showsthat the true damping ratio OI/^R, is approximately equivalentto one-half of a structural damping coefficient g.21 It is ex-pected, therefore, from Fig. 6 that a weak instability whoseintensity is smaller than a certain level of true damping ratiowill not occur for sufficiently large g. The lowest flutter loadand divergence load are shown in Fig. 8 for three valuesof Q,/QR or g/2: 0, 0.01, and 0.05. In the case of Qj/0R > 0.05,the flutter load variation with slenderness ratio is compara-tively smooth. Moreover, there is no significant change in thenature of this variation if the threshold of $VQR is increasedbeyond 0.05.

Acceleration LimitsIn the application of these results to the optimum design of

a large space structure, the best choice of slenderness ratio fora rectangular plate will be desired. This choice may be defined,for example, as the maximum in-plane rigid body accelerationof the plate for which no instability occurs for a given totalarea of a rectangular plate. Toward this end, based upon thecondition that the area is held constant, the nondimensionalacceleration ce is defined here as

qb/mab(39)

where qb/(mab) is the actual acceleration of the plate, Q is thenondimensional load, and X is the slenderness ratio. The low-est flutter acceleration and the lowest divergence accelerationfor various slenderness ratios are shown in Fig. 9. Thethreshold of 1VQR for flutter in this figure is 0.05. It is seenthat a smaller slenderness ratio is generally desirable and thatthere exist maximal values around X = 0.4 and 0.6. However,the rapid change of the results of flutter limit and divergencelimit in that slenderness ratio range, 0.4 to 0.6, may requirespecial attention in actual applications.

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IV. Concluding RemarksFlutter and divergence loads of flexible rectangular plates

with free edges subjected to a follower force were obtained forvarious slenderness ratios. The flutter mechanism was studiedclosely because the behavior of flutter is fairly complicated.Many weak instabilities were found. The weak instabilitysometimes determines the critical load, which is often signifi-cantly smaller than the strong flutter load or the divergenceload. Therefore, critical loads show a rapid variation with achange of slenderness ratio for zero structural damping. How-ever, some instabilities do not occur if damping exists.

A slenderness ratio near 0.4 or 0.6 is desirable from theviewpoint of obtaining a large in-plane rigid body accelerationcaused by the thrust without flutter or divergence. However,the rapid variation of the minimum acceleration at whichflutter occurs in this slenderness ratio range may suggest thatcaution is needed in using these results.

AcknowledgmentThis work was supported, in part, by Army Research Office

Contract DAAL03-87-K-0023. Gary Anderson is the Techni-cal Monitor.

References!Beal, T. R., "Dynamic Stability of a Flexible Missile under Con-

stant and Pulsating Thrust," AIAA Journal, Vol. 3, No. 3, 1965, pp.486-494.

2Walter, W. W., and Anderson, G. L., "Stability of a System ofThree Degrees of Freedom Subjected to a Circulatory Force," Journalof Sound and Vibration, Vol. 45, No. 1, 1976, pp. 105-114.

3Wu, J. J., "On Missile Stability," Journal of Sound and Vibra-tion, Vol. 49, No. 1, 1976, pp. 141-147.

4Peters, D. A., and Wu, J. J., "Asymptotic Solutions to a StabilityProblem," Journal of Sound and Vibration, Vol. 59, No. 4, 1978, pp.591-610.

5Matsumoto, G. Y., and Mote, C. D., Jr., "Time Delay Instabili-ties in Large Order Systems With Controlled Follower Forces,"Transactions of the ASME, Journal of Dynamic Systems, Measure-ments, and Control, Vol. 94, No. 4, 1972, pp. 330-334.

6Park, Y.P., and Mote, C. D., Jr., "The Maximum ControlledFollower Force on a Free-Free Beam Carrying a Concentrated Mass,"Journal of Sound and Vibration, Vol. 98, No. 2, 1985, pp. 247-256.

7Park, Y. P., "Dynamic Stability of a Free Timoshenko BeamUnder a Controlled Follower Force," Journal of Sound and Vibra-

tion, Vol. 113, No. 3, 1987, pp. 407-415.8Celep, Z., "On the Stability of a Discrete Model of the Free-Free

Beam Subjected to End-Loads," Journal of Sound and Vibration,Vol. 59, No. 1, 1978, pp. 153-157.

9Celep, Z., "On the Vibration and Stability of a Free-Free BeamSubjected to End-Loads," Journal of Sound and Vibration, Vol. 61,No. 3, 1978, pp. 375-381.

10Celep, Z., "Axially Symmetric Stability of a Completely FreeCircular Plate Subjected to a Non-Conservative Edge Load," Journalof Sound and Vibration, Vol. 65, No. 4, 1979, pp. 549-556.

HCelep, Z., "Vibration and Stability of a Free Circular PlateSubjected to a Non-Conservative Loading," Journal of Sound andVibration, Vol. 80, No. 3, 1982, pp. 421-432.

12Farshad, M., "Stability of Cantilever plates Subjected to BiaxialSubtangential Loading," Journal of Sound and Vibration, Vol. 58,No. 4, 1978, pp. 555-561.

13Culkowski, P. M., and Reismann, H., "Plate Buckling Due toFollower Edge Forces," Transactions of the ASME, Ser. E, Journalof Applied Mechanics, Vol. 44, 1977, pp. 768-769.

14Adali, S., "Stability of a Rectangular Plate Under Nonconserva-tive and Conservative Forces," International Journal of Solids andStructures, Vol. 18, No. 12, 1982, pp. 1043-1052.

15Leipholz, H. H. E., "Stability of a Rectangular Simply Sup-ported Plate Subjected to Nonincreasing Tangential FollowerForces," Transactions of the ASME, Ser. E, Journal of AppliedMechanics, Vol. 45, No. 3, 1978, pp. 223-224.

16Leipholz, H. H. E., and Pfendt, F., "On the Stability of Rectan-gular, Completely Supported Plates with Uncoupling Boundary Con-ditions Subjected to Uniformly Distributed Follower Forces," Com-puter Methods in Applied Mechanics and Engineering, Vol. 30, 1982,pp. 19-52.

17Leipholz, H. H. E., and Pfendt, F., "Application of ExtendedEquations of Galerkin to Stability problems of Rectangular plateswith Free Edges and Subjected to Uniformly Distributed FollowerForces," Computer Methods in Applied Mechanics and Engineering,Vol. 37, 1983, pp. 341-365.

18Dowell, E. H., "Aeroelasticity of plates and Shells," NoordhoffInternational publishing, Leyden, 1975, pp. 26-29.

19Felgar, R. P., Jr., "Formulas for Integrals Containing Character-istic Functions of a Vibrating Beam," The University of Texas publi-cation, Circular No. 14, 1950.

20Young, D., and Felgar, R. P., Jr., "Tables of CharacteristicFunctions Representing Normal Modes of Vibration of a Beam," TheUniversity of Texas publication, Engineering Research Series No. 44,1949.

21Dowell, E. H., Curtiss, H. C., Jr., Scanlan, R. H., and Sisto, F.,"A Modern Course in Aeroelasticity," Sijthoff & Noordhoff, Alphenaan den Rijn, 1978, pp. 104-110.

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