flutter of rectangular panel supersonic nasa report.pdf

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N A S A T E C H N I C L N O T E

FLUTTER OF BUCKLED,SIMPLY SUPPORTED,

RECTANGULAR PANELS AT SUPERSONIC SPEEDS



TECH LIBRARY KAFB,NY

FL UT TE R OF BUCKLED, SIMPLY SUPPORTED, RECTANGULAR PANELS

AT SUPERSONIC SPEEDS

By Robert W. Fralich and John A. McEl man

Langley Research Center

Langley Station, Hampton, Va.

NA TIONA L AERONAUTICS AND SPACE ADMINISTRATION

For sa le by the C lear inghouse fo r Federa l Sc ien t i f i c and Techn ica l n fo rmo t ion

Spr ing f ie ld V i rg in ia 22151 - CFSTI pr i ce 3 .0 0



FLUTTER O F BUCKLED, SIMPLY SUPPORTED, RECTANGULAR PANELS

AT SUPERSONIC SPEEDS

By Robert W Fral ich and Jo hn A. McElman

Langley Research Center

SUMMARY

A theoretical flutter analysis is presented for buckled, simply supported panels

subjected to supersonic flow over one surface. The analysis employs the Von Karman

large-deflection plate theory and linearized static aerodynamic strip theory. A Galerkin

procedure using four static mode shapes s employed to determine a set of differential

equations which is progr amed on an analog computer. The character of the output of the

analog is used to determine the flutter speed. Results are obtained for pane ls with r at io s

of length in the streamwise direction to length in the cross-flow direction equal to 1/2

and 1 for three specified in-plane edge-loading conditions. An as se ssme nt of effects of

cross-flow coupling of the modes is made by compar ison of the results with those obtained

when cross-flow coupling between the modes is neglected.

INTRODUCTION

Panel flutter has been encountered in the operation of a ir cra ft and mis si le s and hasbecome an important consideration in the design of struc tures for such vehi cles. The

panel flutter problem is influenced by many factors, such as aerodynamic effects, effects

of boundary conditions and midplane compressive loads, and length-width ratio. In addi-

tion, i f the midplane compressive stresses are of sufficient magnitude to cause buckling,

the flutter problem is further complicat ed. These various aspects of the flutt er probl em

are discussed, for example, in references 1 to 10; some of the earlier investigations are

listed in reference 1. Reference 1 con sid ers the fl utt er of buckled, simply supported

panels. In reference 1 he mechanism for flutter is shown to be a str eam wi se coupling

between the modes. The purpose of th e present analys is is to obtain a more accurate

solu tion to this parti cular probl em y investigating cross-flow coupling between the

modes. Cross-flow coupling is a phenomenon which does not occur in the small-

defle ction flu tter analy sis of uriouckled, simply supported panels.

The results presented in reference1we re obtained from a two-mode Galerkin solu-

tion, and rigorous analytical met hods were used to determine the panel stabilit y. In the

present analysis a general Galerkin solution is obta ined and the stab ility of the panel is

determined for a four-mode solutio n by use of an analog computer. Poi nts on the flutter



boundaries have been determined for three in-plane edge-loading conditions for ratios

of the length in the streamwise direction to the length in the cross-flow direction equal

to 1/2 and 1.

SYMBOLS

coefficients in displacement expressions (eqs. (9))

a length of pla tentreamwiseirection

bidth of platenross-flowirection

Cmnmplitudeoefficientsorateralisplacement

D flexuraligidity, Eh312(1 - p2)

E Young sodulus

Fmn,Gm,,Hmn Fourieroefficientsseeqs. (12))

hlate

Mach

pa ra me te rs defined in equations (21)

m,nnumber of half-waves n treamwise nd ross-flowdirections, espectively

Nx,Ny,Nxy midplane stressresul tants

Nx, y, q nondimensionalmidplane stres s esul tants ; --a2D r2D

and -Nxa 2 N y a 2 Nqa2

re sp ec tive ly 7T2D

Px2yaver age n-pl ane edge oadsperunit ength,positive

-P x 3 y nondimensionaln-plane dgeoads er nitength;

respectively

in compression

2



integers

dynamic pressure,3pV 2

time

in-plane displacements, positive in x- and y-directions, respec tively

nondimensional in-plane displacements, positive in x- and y-directions;

ha u and -ha2 v, respectivelyT2D a2Db

free-stream velocity

lateral deflection, positive in z-direction

nondimensional lateral deflection, positive in z-direction, w

rectangular Cartesian coordinate s (see fig. 1)

6mn

x

xc r

P

= a/b

5 rl

P

7

mass density of plate material

Kronecker delta, equals 1 f m = n, equals 0 i f m st n

speed parameter,1 ~ ~ ~ 3

37r4pD

flutter speed parameter

Poisson’s ratio

nondimensional oordinates;x/a and y / a , respectively

free-stream density of fluid

nondimensional ime, Ea2 Y

3



When subsc ripts . 5 q and T follow a comma, they indicate partial differentia-

tionwithrespect o [, q and T , respectively.Dotsoversymbolsdenotederivatives

with respect o 7

STATEMENT OF PROBLEM

The configuration analyzed in this report is the simply supported, flat, rectangular

panel shown in fig ure 1. The panel has a constant thickness h with air flowing over

the top surface at a Mach number M. No flow of

air beneath the panel is considered. Average

in-planeedge oads Px and PY perunit ength

(positive in compression) are specified at the

kT boundaries. No in-p lanehearingorcesre

C l t t t l + t

c

applied to the plate.

METHOD O F SOLUTION

The present analysis employs the large-

X [ deflection pl ate theory of Von Karman and linear-

ized static aerodynamic strip theory. This aero-

J

dynamic approximation has previously been shown

to yield accurate flutter boundaries for Machnum-

bers great er than abou t 1.6. The resulting equa-

tions are analyzed in the appendixby means of a

7h

Figure 1.- Rectangularanelndoordinate Galerkinpro ced ure which util izes he doublysystem.

in fin it e set of static buckling modes. This pro-

cedure yields a doubly infinite set of second-order non linear ordina ry differential equa-

tions for the time-dependent amplitude coefficients cmn.These equations can be

reduced to those for various approximate analyses that utilizea fini te number of sta ti c

buckling modes. The set of four equations, obtained from an approxima te analysis that

use s four of th e stat ic modes, is analyzed by mea ns of an analog computer in or de r to

find a f lutter speed parameter her. The modes considered have amplitude coefficients

Cmn, where m = 1 and 2 and n = 1 and 2 a r e the number of half waves in the strea m-

wise and cross-flow directions, respectively.

In the analog analysis, the character of the ti me hi sto rie s of the amplitude coeffi-

cients is observed while the initial static buckling condition is alt ere d by gradually

increasing the speed parameter to a given evel. Analog tr ac es which illu strat e the

method of dete rmin ing flutter are given in figu re 2. For lev els of below the cri ti cal

value X, (fig. 2(a)), the amplitude coefficients do not build up with time and the motion

4



I

2 - :

c12 0

7

atStablemotion. A < her)

c21

A

bt Unstablemotion. A > Acr)

Figure 2.- Variation of amplitudecoefficientswith ime.

is consideredstable.For evels of X above hecritical lutterspeed X,, (fig.2(b)),

the amplitude coefficients show a dr as ti c buildup with time . The charac ter of the motion

is observ ed for differ ent evels of X until the critical flutter speed parameter Xcr isdetermined. The value of hcr for the case illustrated in figures 2 a) and 2(b) is indi-

cated by the tick mark . Values of Xc r are determined for three in-plane loading con-

ditions for two value s of a/b, the ratio of leng th in the stream wise direc tion to width in

the cross-flow direction.

RESULTS ND DISCUSSION

Results for the flutter boundaries obtained from the present analysis are givenby

the circles in f igures 3 to 5. Flut ter bound ariesA

= Acr) are plotted in these figuresas a function of in-plane edge oading for a/b of 1 / 2 and 1. Flutter occurs above

these bound aries and stable motion is obtained below the bounda ries. Resu lts for speci-

fied values of i? ith i?, 0, forspecified Py with i? 0, and forspecified

P = Py are shown in fig ure s 3, 4, and 5, respectively. Also shown in these figures

are the flu tter bo undari es prese nted in r eferen ce which we re obtained by a rigorous

stability analysis using just two modes. The curve labeled n = 1 is the flutter

-

5



”- modes buckled)

o 4 modes presentnalysis

0 2 4 6 8 1 0 1 2

A

A

0 5 10 15 20 25 30

-PX

b) v = 1.

Fig ure 3.- Flut terboundaries for pan_els with

streamwlse ompressiveoad P Py = 0.

- - _ modes unbuckled) ref )

modes buckled) io 4 modes presentnalysis 1

n=l

0 0

111110 10 20 30 40 50

-P

Figure 4.- Flutterboundaries fcr panelswith

cross-flow ompressive loadPy. P = 0;

v = 1.

boundary obtained by consi dering only the C11

and C21 modes,whereas heone abeledn = 2

utilizeshe C12 andC22modes.

Th e res ul ts show effect s of cross-flow

coupling of the m odes on th e flu tter of simply

supported, rectangular panels in sup ersonic flow.Theoretically, this coupling does not exist on the

portion of the flutter boundary prior to buckling

(the dashed line porti ons of fi gs. 3 , 4, and 5 . One

result in the present four-mode analysis was

determined in the unbuckled range. (See fig. 4.)

Figure 4 confirms the absence of cross-flow

coupling in the unbuckled range and gives an

”modes unbuckled) f ref. 1- modes buckled)

o 4 modes presentnalysis )

l

0 4 8 12 16

a) v = 1/2.

4 1

0 4 8 126

b) v = 1.

Figure 5.- Flutterboundaries orpanels

with equal streamwi ie and_ cross-flow

compres sive loads. P = Py.

6



indication of the validity of the procedu re used in determi ning the flutte r crit erion . On

the portion of the flutter boundary or buckled panels, cross-flow coupling of the modes

has an effect on the flu tt er bound ary for certa in values of a/b and in-plane loading

conditions. For he two square panels under streamwise loading and under biaxial

load ing (figs. 3(b) and 5@ the effects are negligible. However, fo r th e pa ne ls shown in

figu res 3(a), 4, and 5(a), effects of cross -flow coupling on the flut ter boundary are sig-

nificant. For hese cases (except for a reg ion of fig. 3(a)), the effect of cross -flow

coupling is to lower the flutter boundaries.

The result s show that the effects of cross-flow coupling can be important in deter-

mining flutter boundaries for buckled panels. In th i s analy sis no investigation is made

of the convergence of the Galerk in solu tion . To do this more mod es would have to be

considered. In order to facil itat e such a study, the modal solution that includes all the

static buckling modes is presented in the appendix.

CONCLUDINGREMARKS

A supersonic flutter analysis is presented for simply supported, rectangular panels

subjected to specified in-plane compressive edge loads. A Galerkin procedure that

uti liz es t he doubly inf init e set of the static buckl ing modes yields a doubly infinite set of

differential equations that can be reduced to any desi red finite number of equations. The

equations have been programed on an analog computer for a stability analysis for a four-

mode solution that exhibits both stream wise and cross-flow coupling of the modes. The

ch ar ac te r of the output of the analog computer is then used to determine the critical

flutter condition. Num erical results are presented for panels with ra ti os of length in thestreamwise direction to length in the cross-flow direction equal to 1/2 and 1. The fol-

lowing specified in-plane boundary edge conditions are consider ed: (a) streamwise com-

pr es si ve loading only, (b) cross -flow compressive loading only, and (c) equal streamw ise

and cross-flow compressive loading. The results show that the effec ts of cross-flow

coupling are important in determining flutter boundaries for buckled panels since the

flutter speed can be appreciably reduced from the value determined without cross-flow

coupling.

Langley Research Center,

National Aeronautics and Space Administration,

Langley Station, Hampton, Va., July 6, 1967,

126-14-02-24-23.

7



APPENDIX

ANALYSIS

The nonlinear differential equations expressing the equilibrium of an aerodynami-

cally loaded oscillating panel based onVon Karman large-deflection plate theory areobtained in nondimensional form in reference 1. These equations can be written as

follows:

where

in which i, 7, and W are nondimensionaldisplacements, N,, N y, and NxY are non-

dimensional str ess esul tan ts, [ and q are nondimensional x and coordinates,

is nondimensional ime, IJ is Poisson s atio,and h is thespeedparameter.

The boundary conditions to be satisfied by W are the simple-support conditions

Th e boundaryconditions obesatisfied by and V are those for uniform displace-

ment of e ach edge in the plane of the pla te

8



APPENDIX

and those for zero in-plane shear stress at the edge of the pl ate

Appropriate boundary c.onditions for the edge loadsare

and Px and Py a r e nondimensional n-planeedge oadsperunit ength.

The boundary conditions (eqs. (5), 6), and (7)) can be satisfied if the displacements

and ? arewrit ten as

i i = A g + A 1 t + 1 A s inmntcos

m =l n=O

V = BO + B1q +b

m=O n=l

and if the normal displacement W is written as

The nonlin ear terms on the right-hand side of equations 1) and (2) can also be

expanded in Fourier series as follows:



APPENDIX

where

where

6 = 0 (m f n)

6 = 1 (m = n)

When equations (9) and (11) are substi tuted into equations 1) and (2) the coefficients

Am, andBmn be determinedner ms of Fmn, Gmn, and Hmn as follows:

Thedisplacements and ? a r e now known in er ms of Ao, AI, Bo, B1,

Fmn, Gmn, and Hmn. The stress resultants are obtained in terms of the coeffi-

cients A i , B1, Fmn, Gmn, and Hmn by subst itut ion romequations (9), ( l l) , and

(13) intoequation (4). In or de r for Rx and INy tosatisfy heboundaryconditions

(eq. 8 ) ) , theconstantsA1andB1musthave hevalues .

10



APPENDIX

where, when use is made of equations (10) and (12),

F00 = 1 p2Cpq2

p=l q=l

The stress resultants become

-Nx = -Px +

bco s mnt; cos

n=1

Ny = -Py + GmO cos mnt; + m2Jmn2

m=1 m=l=l [ m 2 a i ]

sin mat; sin2 b

where

n

When use is made of equations (12) and 14), FOn, Gn,O, and Jmn become

(n = 1,2 , ., m

11



APPENDIX.

Substitution of equations 10)and (16) into equation f3). and.application of the Galerkin

procedure yields

The modal solution (eq. (19)) tha t util ize s the doubly infinite se t of sta tic mode

shapes reduce s to any desired approximate soluti on that uses a finite number of modes

simply by delet ing the undesired modes. In the present analysis numerical results are

found for a four-mode solution so that

and equation (19) becomes

1 2



APPENDIX

M3 = 3 + 4v2By - 1 + 4v2 2

M = 4Bx + 4v2Py - 16 1 + ~ 2 ) ~

q14. d6

P2 = k 16 + v4)

P3 = ~ l+ 16~4)

P4 = 1 + v4

Q = -4 1[ + v4) + 81v42 +

v4

1 + 42) 9+ 42)

s =LI 16 + v4) + 81v4

(4 + v q 2 (4 +v4vq2

T = 16

K = 16+ v4) +

[: v4 .]

81v4V

16 + v22 16+ 9v2)

N = - 1 + 16v4) + 81v4

1 + 1 6 ~ ~ ) ~9+ 16v2)

H=l +v 4+ 25v4 25v4

1 + 9v2)2 9 + v q 2

inwhich v = a/b. Inequation 21) thedoubledotsover hecoefficientsCmnrepre-

sent the second derivative with respect to T. Equations 21) are se t up on an analog com-

puter in order to determine the flutter speed = Xcr as a function of the in-plane edge

load s anda/b.

13



REFERENCES

1. Fra lic h, Rob ert W.: Postb uckling Effect s on the Flut ter of Simply Supported

Rectangular Panels at Supersonic Speeds. NASA TN D-1615 , 1963.

2 . Kobayashi, Shigeo,: Fl ut te r of Simply Supported Rectangular Panels in a Supersonic

Flow. Tr ans. Japan SOC.Aeron. Space Sci., vol. 5, no. 8, 1962, pp. 79- 89 .

3 . Kobayashi, Shigeo: Two-Dimens ionalPanelFlutter.Trans.Japan SOC.Aeron.

Space Sci., vol. 5, no. 8, 1962.

I. Simply Supported Panel, pp. 90- 102 .

11. Clamped Panel, pp. 103-118.

4 . Dixon, Sidney C.: Experimental Invest igat ion at Mach Number 3 . 0 of Effects of

Thermal Stress and Buckling on Flutter Characteristics of Flat Single-Bay Panels

of Length-Width Ratio 0.96 . NASA TN D-1485, 1962.5. Dowell, Earl H.: Nonlinear Oscillations of a Flutt ering Plate . AIAA Pa pe r

No. 6 6 - 7 9 , Jan. 1966.

6 . Shide ler, John L.; Dixon, Sidney C.; and Shore, Ch ar le s P.: Flutter at Mach 3 of

Therm ally Stress ed Panel s and Compariso n With Theory for Panel s With Edge

Rota tional Rest raint . NASA TN D-3498, 1966.

7 . Erickson, Lar ry L.: Super sonic Flut ter of Flat Rectangular Orthotropic Panels

Elast ically Rest rained Against Edge Rota tion . NASA TN D-3500, 1966.

8. Dixon, Sidney C.: Comparison of Panel Flu tte r Res ult s From App rox imate Aer o-dynam ic Theor y With Resul ts From Exact Invisc id Theor y and Experiment. NASA

TN D-3649 , 1966.

9 . Dugundji, John: Theore tic al Con sidera tions of Panel Flutter at High Supersonic

Mach Numbers. AIAA J., vol. 4 , no. 7, July 1966, pp. 1257-1266.

10. Ketter, D . J.: Fl ut te r of Flat, Rectangular, Orthotropic Panels. AZAA J., vol. 5,

no. 1, Jan. 1967, pp. 116-124.

14 NASA-Langley, 1968- 2 L- 040

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