The flutter of thin, plane, elliptic panels

TECüliiSCKi: HOGESCHOOL DELFT VUEGTUIGBCUVVIX'NDE

2 5 J U l j ' i ~ CoANOTEl

THE COLLEGE OF AERONAUTICS

C R A N F I E L D

THE FLUTTER O F THIN, PLANE, E L L I P T I C PANELS

by

T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

NOTE NO. 136

THE FLUTTER OF THIN, PLANE, ELLIPTIC PANELS by

D. J. Johns, R. J . Davies and G. H, F . Nayler

CORRIGENDA

Page 4. Equation (4. 5). For (80/ITS) read ( ^^QJ

Page 9. Section 4 . 5 . 1 . Equation (4.21) should read:

7 Z Z

nCHNïSCüE HOGESCHOOL DELFT

Vli:CTU!CECUVVKUNDE

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NOTE NO. 136 J a n u a r y , 1963. T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D T h e F l u t t e r of T h i n , P l a n e , E l l i p t i c P a n e l s b y -D. J . J o h n s , M . S c , M . I . A . S . , R. J . D a v i e s , B . S c , D . C . A e . , and G. H. F . N a y l e r , D . C . A e . SUMMARY

T h e effect of v a r i o u s p a r a m e t e r s on the f l u t t e r of e l l i p t i c p a n e l s i s studied by s i m p l e t h e o r e t i c a l a n a l y s e s . F o r clamped edge p a n e l s , c o m p r e s s i v e m e m b r a n e s t r e s s e s a r e shown to have d e s t a b i l i s i n g effects. S t r u c t u r a l d a m p i n g i s , in g e n e r a l , d e s t a b i l i s i n g , except for v e r y low v a l u e s of the r a t i o of s t r u c t u r a l and a e r o d y n a m i c d a m p i n g c o e f f i c i e n t s .

The addition of a c o n c e n t r a t e d m a s s m a y a l s o have a d e s t a b i l i s i n g effect,

d e p e n d i n g upon the position of the added m a s s . The effect of sweepback is s t a b i l i s i n g o r d e s t a b i l i s i n g , depending on w h e t h e r the p a n e l a s p e c t r a t i o i s l e s s t h a n , o r

g r e a t e r t h a n , unity. The effect of v a r i a t i o n s in the edge conditions i s shown for the c a s e of c i r c u l a r p a n e l s w h e r e the difference between s i m p l y s u p p o r t e d and fully c l a m p e d e d g e s c o r r e s p o n d s to a d i f f e r e n c e of 20% in c r i t i c a l panel t h i c k n e s s .

CONTENTS

Page Summary

Notation

Introduction 1 Statement of the Problem 1

Method of Analysis 2 Analyses for Elliptic Clamped Edge Panels 3

4 . 1 . Simple Two-Mode Solution 3 4 . 2 . Effect of Structural Damping 4 4 . 3 . Effect of Membrane Stresses 7 4 . 4 . Effect of a Concentrated Mass 8

4 . 5 . Effect of Sweepback 9 Analysis for Circular Panels with Variable

Edge Restraint 11 Limitations in the Previous Analyses 13

6 . 1 . Use of Static Deflection Modes 13 6.2. Assumption of Only Two Degrees of Freedom 13

Conclusions 14 References 15 Table 1. Ratio of Critical Mach Numbers for

Clamped Edge Elliptic Panels 16 Table 2. Buckling Coefficient for Clamped Edge

Elliptic Panels. 16 Table 3. Effect of Concentrated Mass Position on

the Flutter of a Clamped Edge Panel 16 Table 4. Effect of Variations in Concentrated Mass

on the Flutter of a Circular Clajnped Edge

Panel. 17 Table 5. Effect of Sweepback on Flutter Mach

Number for a Clamped Elliptic Aluminium

Panel at Sea Level 17 Table 6. Flutter P a r a m e t e r s for Circular and

Two-Dimensional Panels 17 Figures

NOTATION

speed of sound in undisturbed medium modal coefficients defined in equation 3 . 2 . semi-axes of elliptic panel in x and y directions edge restraint parameters defined in equation 5.2. panel chord

local flexural stiffness, Eh /12(1 - i^^)

Young's Modulus of elasticity of panel material

external force per unit length acting in the midplane of the panel, tensile force positive

critical compressive external force per unit length values of F in X and y directions

structural damping coefficient

matrix elements defined in equation 3.4. thickness of panel

coefficients in assumed modes for elliptic panel (equations 4 . 1 , 4, 21)

o

H value of H for circular panel, —^

^ a

i ^rï

k reduced frequency, cu)/2U K flutter parameter, -rrpr x

M L ) _ 2 4

3m u^ c L edge-fixity parameter (Section 5) m panel m a s s per unit surface a r e a

M magnitude of concentrated m a s s on panel M Mach number

p. . net perturbation p r e s s u r e , positive downwards 'x, t^

p p r e s s u r e in undisturbed supersonic flow p. p r e s s u r e on lower panel surface

q dynamic p r e s s u r e , 2 i° U

r , s coefficients defined after equation 4.4 a , _n a, B , . c D E F F _c ^ x g a: _n b Ba F y G mn h H n

Notation (Continued) R, t

s,

S'

u

0 T T> X , y , z X , , y ,

z

^(x.y) n(x,y) a y 6 X 13 \ V p

polar co-ordinates in circular panel time

coefficients defined after equation 4. 7 coefficients defined after equation 5.6 velocity of supersonic flow

coefficient defined after equation 4.15 co-ordinates defined in Fig. 1.

co-ordinates of position of concentrated mass M vertical displacement of panel = Z. . e

flutter mode shape for elliptic panel nth mode shape for elliptic panel

edge restraint coefficients defined in equation 5.4

concentrated mass parameter, M/wabm.

non-dimensional co-ordinates of concentrated m a s s M

edge restraintparameter

frequency ratio, w^/u ; angular co-ordinate in circular panel relative to flow direction

angle of sweepback

panel-air mass ratio, m/pc Poisson's ratio

density of undisturbed supersonic stream

in vacuo frequency associated with modal shape Z flutter frequency

1

-1. Introduction

Up to the p r e s e n t t i m e , m o s t plane panel flutter a n a l y s e s have dealt with r e c t a n g u l a r p a n e l s having v a r i o u s edge conditions. In Ref. 1, J o h n s applied l i n e a r piston t h e o r y in an a n a l y s i s of a clamped edge e l l i p t i c panel. T h i s investigation h a s been extended in the p r e s e n t study to d e t e r m i n e the effects of the following p a r a m e t e r s on the flutter of unbuckled e l l i p t i c p a n e l s :

-(a) S t r u c t u r a l damping (b) M e m b r a n e s t r e s s e s (c) C o n c e n t r a t e d m a s s e s (d) Sweepback

(e) Edge r e s t r a i n t

T h i s Note i s b a s e d , in p a r t , on unpublished t h e s e s submitted by the two j u n i o r a u t h o r s in p a r t i a l fulfillment of the r e q u i r e m e n t s for a Diploma of the College of A e r o n a u t i c s . The work was s u p e r v i s e d by the s e n i o r a u t h o r , M r . D. J . J o h n s , who p r e p a r e d t h i s Note.

2. S t a t e m e n t of the P r o b l e m

A flat panel of uniform t h i c k n e s s , li, i s mounted in a rigid w a l l , with its

u p p e r s u r f a c e exposed to a s u p e r s o n i c flow of velocity U, p r e s s u r e p „ and d e n s i t y p ( F i g . 1). The l o w e r s u r f a c e of the p a n e l i s exposed to s t i l l a i r , p r e s s u r e p . , whose a c o u s t i c action i s i g n o r e d . The p a n e l i s a s s u m e d to be executing s i m p l e h a r m o n i c motion in the d i r e c t i o n n o r m a l to the plane of the w a l l , with frequency w and I n f i n i t e s i m a l l y s m a l l a m p l i t u d e .

A s s u m i n g s m a l l deflection t h i n - p l a t e t h e o r y , the differential equation of m o t i o n for such a panel i s , in g e n e r a l ,

D 7 * Z H- H E ! ? . F 9 l § _ - F ^ + ( P ^ - p . ) + p, ,.w 0 (2.1)

dt' "" 8 / y 8 y ' ' ^''•y'*^

iut w h e r e the d i s p l a c e m e n t ol the panel it. = z,, .e ifhere the d i s p l a c e m e n t of the panel Z = Z . .e

It will be a s s u m e d that t h e r e i s no s t a t i c p r e s s u r e differential a c r o s s the p a n e l ; i - e . p^ - p. = 0.

F i r s t o r d e r p i s t o n t h e o r y g i v e s , p , ^?' p a ^ ( U - — + —-) •^ ( x , y , t) "^ 9x 9t

.pa(jj~+wZ)e'''\ (2.2)

"= 8x

In Ref. 2 it w a s suggested that the a e r o d y n a m i c d a m p i n g t e r m should be multiplied by two to allow for the effects of the q u i e s c e n t a i r b e n e a t h the p a n e l . T h i s a s s u m p t i o n will not be made h e r e . It should, of c o u r s e , be r e a l i s e d that piston t h e o r y does not p e r m i t the p o s s i b i l i t y of negative a e r o d y n a m i c d a m p i n g .

2

-3. Method of Analysis

We may rewrite equation 2.1 as

. . « • • nTT ,

DV*Z - mu^Z - F Z - F Z + - ^ (UZ' + iwZ) - 0 (3.1)

X y M

where primes indicate differentiation with respect to x and dots denote differentiation with respect to y, and

V* Z - Z"" + 2Z' + Z

The flutter mode shape Z, , can be approximated by a linear combination of the form ^^'^'

Z, , - \ ' a Z , , (3.2) (x.y) // ^ r(xy)

r»l

where the coefficients a may represent complex amplitudes. Substitution of equation 3. 2 into equation 3.1 yields

.[

N \ ' a I D(Z"" + 2Z" + Z ) - mtJ^Z - F . Z " - F . Z / r { _ r r r r x r y r

+ S(UZ' + iuZ ) 1» 0 . (3.3)

M r ï* J

From equation 3.3, N simultaneous equations are obtained by the Galerkin process. The nth equation is obtained by multiplying equation 3.3 by Z and integrating over the area of the panel. The N equations may be written in matrix form

[°»>n]['n] • ["]• '"'

and the flutter conditions are determined fromi the characteristic equation

G 1 = 0 . (3.5) mn I

It is usual in panel flutter analyses to assume that the modes of displacement Z are the mode shapes for the unstressed panel vibrating in vacuo. The computation of these shapes for elliptic plates is rather tedious and, in the following analyses, modes are assumed which correspond to a uniformly distributed load and to a linearly varying load proportional to x . The accuracy of this assumption will be discussed later.

3

-4. A n a l y s e s for E l l i p t i c Clamped Edge P a n e l s 4 . 1 . Simple Two-Mode Solution

The flutter mode shape i s a p p r o x i m a t e d by

Z Z Z , . = a r - i + a, TT <4.1) ( x . y ) 1 H^ 2 H^ w h e r e Z^' 2 2 a^ b^ Z^» x | — H , ' H » 2 r 3 3 2 -\

[_ a b a b J

r

5 _ ^

1 ^ 2 n

^^^

L a b a b J

the r a n g e s of the i n t e g r a t i o n s in applying the G a l e r k i n p r o c e s s a r e

- b j l - ^ < y < -bJTTx

a -a < X < +a .

The r e s u l t a n t flutter d e t e r m i n a n t (equation 3.5) h a s the t e r m s G,, « ^ + (pUiy - Mmu*)/5MH^ G „ » a*D + a''(pUi4J - Mmw^)/60MH^ (4.2) G » P U * / 1 0 M H 1 2 2 G j , " -puVlOMH^

To include h y s t e r e t i c s t r u c t u r a l d a m p i n g it i s only n e c e s s a r y to f a c t o r the stiffness t e r m s by (1 + ig), w h e r e g i s the s t r u c t u r a l damping coefficient. If g i s a s s u m e d

constant for the m o d e s c o n s i d e r e d , D i s r e p l a c e d by D(l + ig).

The solution of the flutter d e t e r m i n a n t for z e r o s t r u c t u r a l danaping y i e l d s the s t a b i l i t y b o u n d a r y (Ref. 1)

i-, = ("l - r / 7 5 s ' ' ' | r 4 M * k * / ( 3 - k ' ) l , (4.3) and the flutter frequency,

„=J1

50SD _ (4 4)

where r = 8H^ H^/3 and s - (2H^ + 9Hg)/45 are functions of the panel geometry, and M is the panel-air m a s s ratio, m / p c .

If aerodynamic damping effects are now ignored, which may be justified for zero structural damping as in the case of rectangular panels (Ref. 1), equations 4.3 and 4.4 reduce to

2qc' /MD - (8OV379) X

^ - 7 ]

39 + 14 2 - + 3 ^ . ( 4 . 5 )

V 2

Replacing M by M - 1 corresponds to using Ackeret-loading and should give better results for a wider range of supersonic Mach numbers.

Using equation 4 . 5 , Fig. 2 presents the variation of the parameter 2qc'/MD with panel eccentricity — . It can be concluded that the most critical value of h '^ a

— is for the two-dimensional panel (7- * 0) and it is interesting to note that the ratio of the critical thickness ratios for infinite and unit aspect ratio is 1.13, which agrees well with the ratio obtained by Hedgepeth (Ref. 3) for rectangula'^ simply supported panels. These results show that at high Mach numbers the effect of aspect ratio, in the range unity to infinity, on the critical thickness ratio is relatively small, both for rectangular and elliptic panels.

In Ref. 4, Fung has pointed out that the results shown in Fig. 2 become increasingly inaccurate as the aspect ratio d e c r e a s e s , because the aerodynamic tip effect ignored in the simple analysis becomes increasingly more important. Hence, the right hand side of the curve in Fig. 2 must be viewed with caution.

Fig. 3 gives the variation of the critical panel thickness ratio with Mach

number for the two dimensional (a/b = 0) and circular (a/b * 1) clamped edge panels, at sea level. Fig. 3 has been derived from equation 4. 5 with M replaced by ^ M*- 1 . 4 . 2 . Effect of Structural Damping

The effect of structural damping on panel flutter has received very little t r e a t -m e n t in the literature and there has been so-me doubt as to how such effects should

be included. In Ref. 5 it was decided to incorporate into the analysis viscous damping, which is proportional to, and in phase with, the velocity of the panel oscillations. The alternative approach would have been to use hysteretic damping involving a complex stiffness t e r m , which is proportional to the panel displacement and in phase with the velocity of the panel motion. It was reasoned that since the latter approach is only strictly applicable to simple harmonic motion it could lead to difficulties when complex frequencies are considered. Even so, it was shown that the addition of viscous structural damping could be destabilising. In another paper, (Ref. 6), it was shown that the effect of hysteretic structural damping could be stabilising or destabilising depending upon whether the Mach number of the

flow was less than, or g r e a t e r than, '^2. This dependence on M » / 2 " is apparently a consequence of the aerodynamic assumptions which in Ref. 6 included damping t e r m s based on linearised unsteady aerodynamic theory. Johns and P a r k s (Ref. 7), showed for a two-dimensional panel with simply supported edges that hysteretic structural damping could combine with aerodynamic damping to be destabilising. The analysis of Ref. 7 assumed linear piston theory for the aerodynamic forces.

fGCHN'iSCHE HOGESCHOOL DELFT

VUZGTUICECUVVKUNDE EIBLlOrüEEK

In t h i s s e c t i o n , the a p p r o a c h of Ref. 7 will be applied to the p r o b l e m of the clamped edge e l l i p t i c p a n e l .

The inclusion of h y s t e r e t i c s t r u c t u r a l d a m p i n g modifies the stiffness t e r m s (D) by a factor (1 + ig), w h e r e g i s the s t r u c t u r a l damping coefficient. F o r the two mode a n a l y s i s , the flutter d e t e r m i n a n t y i e l d s eventually the following two e q u a t i o n s , 27 i * r * 2 ~ l 2 2 r 2 R ,R> 2-1 ^ K 0 = - 0 s - e T + i + g 0 e s + e - T + (-) , (4.6) ( 4 . 7 ) R I T - 2S0^ T0''- 2 1 T^ n»,r 2 , „ 2 2 2 q c 8 D w h e r e K » 8Ma„/3iuu, c =• :r—- x "^ ^ ML) „ 2

s - mj2H^

T « 1 + S R « p a / m u oc' 1 0 = frequency r a t i o = w^/u 3mcj" c 40H D and u)^ = fundamental frequency = I —

The p a r a m e t e r R / g i s , effectively, the r a t i o of a e r o d y n a m i c and s t r u c t u r a l d a m p i n g coefficients and f r o m equation 4. 7 the value of the frequency r a t i o , 0, c o r r e s p o n d i n g to t h i s p a r a m e t e r m a y be d e t e r m i n e d for any e l l i p t i c p a n e l . The c r i t i c a l Mach n u m b e r , a s given by K, m a y then be found from equation 4. 6.

If, i n i t i a l l y , g i s a s s u m e d z e r o , solution of equations 4. 7 and 4 . 6 gives r e s p e c t i v e l y

0 = V~27T (4.8) ^ K ' ' = T ' ' - 4 S + 2 T . R ' . (4.9)

T h e s e e q u a t i o n s yield the r e s u l t s p r e v i o u s l y obtained in Section 4 . 1 v i z . equations 4 . 4 and 4 . 3 r e s p e c t i v e l y . Since R i s a l w a y s s m a l l one can conclude that for negligible a e r o d y n a m i c d a m p i n g , i . e . R = 0, equation 4 . 9 gives

2 7 2 2

^ K = T - 4S . (4.10) If, h o w e v e r , R i s i n i t i a l l y a s s u m e d to be z e r o in equations 4 . 7 and 4 . 6 , one o b t a i n s ,

r e s p e c t i v e l y ,

0 = V T / 2 S (4.11) and f^ K* = ^ F T ' ' - 4 S 1 + 4g''s. (4.12)

6

-Hence if g is now assumed to be zero, equation 4.12 reduces to give the result,

^ K ' = ^ [ T ' - 4 S ] , (4.13)

27 ^^2 4S 4 -p

which is completely dissimilar to equation 4,10.

Similar results to this were derived in Ref. 7 for the simply supported, two dimensional panel and a fuller discussion of the implications of such results is given there. For the purposes of this section it suffices only to say that equation 4.13 represents a lower critical Mach number than given by equation 4.10, the extent of the effect being indicated by the quantity MS . The value of this ratio for various elliptic planforms is given in Table 1, where the suffixes gR and Rg indicate the order in which these quantities are allowed to tend to z e r o .

Equation 4. 7 can be rearranged to read

0' = I + I I (T - 2S0') (4.14)

which for very small values of the ratio g/R gives

T R

Jl('-f')

(4.15)

Substituting these values into equation 4.6 yields, after some manipulation, 2 7 2 2 2

=r- K = T - 4S + 2TR

4

[-il(-fO]'

which should be compared with equation 4 . 9 ,

The square bracketed t e r m indicates that the effect of structural damping for small values of g/R is indeed stabilising but since R is always small and the term in g/R is also small, the overall effect is probably of no practical importance.

It can thus be shown from equation 4. 6 and 4. 7 that, except for very snaall values of the ratio g/R, the effect of structural damping is destabilising. For a two dimensional panel and with R = .088, the corresponding results for g = 0 and g = 0.05 are 0 = .485 and 0 = .6 respectively and the ratio of critical Mach numbers is

.^ Ï '—- = 0.9. The effect of structural damping is more destabilising as R -. 0 i . e . as altitude increases and the most extreme effect is evidenced by the results of equations 4.10 and 4 . 1 3 , which were discussed above.

7

-4 . 3 . Effect of M e m b r a n e S t r e s s e s

The p r e s e n c e of m i d - p l a n e s t r e s s e s in an a c t u a l panel i s i n e v i t a b l e , s i n c e v e r y l i t t l e of an a i r c r a f t s t r u c t u r e i s not l o a d c a r r y i n g . The inclusion of m i d -plane s t r e s s e s in two d i m e n s i o n a l a n a l y s e s i s c o m m o n , e . g . R e f s . 1, 3 and 7, but t h e r e a r e few s i m i l a r t h r e e - d i m e n s i o n a l a n a l y s e s . H o w e v e r , Hedgepeth (Ref. 3) h a s examined the flutter of a r e c t a n g u l a r s i m p l y supported panel of finite s i z e with m i d - p l a n e s t r e s s e s in both d i r e c t i o n s and he concluded that only the m i d - p l a n e s t r e s s e s in the d i r e c t i o n of the airflow w e r e i m p o r t a n t . The t r a n s v e r s e s t r e s s e s had no effect on the c r i t i c a l Mach n u m b e r .

In t h i s Section a t h r e e d i m e n s i o n a l a n a l y s i s i s p r e s e n t e d for the clamped edge e l l i p t i c p l a t e a s s u m i n g a constant load p e r unit length in both m u t u a l l y p e r p e n d i c u l a r d i r e c t i o n s i . e . F = F = F i n equations 3 . 1 and 3 . 3 .

X y

The flutter d e t e r m i n a n t which r e s u l t s h a s additional t e r m s in the e x p r e s s i o n s in G :

for G and G ( s e e equation 4. 2), viz

and in G. F a 15 H a b

b]

Solution of the flutter d e t e r m i n a n t for z e r o s t r u c t u r a l damping y i e l d s the

following e x p r e s s i o n s when a e r o d y n a m i c damping i s i g n o r e d , which m a y be c o m p a r e d with e q u a t i o n s 4 . 5 and 4 . 4 , 2qc='/MD = ( w = w h e r e 8 0 / 3 , 39 + 14 -2 + 3 - ^ + ^ ^ (13 + b b lOD

^]

150sD + Fv m 1 / 2 3 11

—^ + ~T ) i s a function of the panel g e o m e t r y .

(4.17) (4.18)

As a check on the a c c u r a c y of the solution, the buckling condition for the panel can be obtained f r o m the e x p r e s s i o n for G^ ^ and c o m p a r e d with m o r e e x a c t s o l u t i o n s . T h u s , in the a b s e n c e of i n e r t i a and a e r o d y n a m i c f o r c e s , the buckling force F i s given by, o r F a c F b c 3 + 2 ^ + 3 - 4 b b D 3 + 2 -g + 3 « T / a r 2 - 1

b J / b |_ b J

b b

In Table 2, the v a r i a t i o n of the buckling p a r a m e t e r F a*

c

D with panel e c c e n t r i c i t y , T h i s shows that the a / b , i s shown and c o m p a r e d with the r e s u l t s of R e f s . 8, 9.

e n e r g y method used in Ref. 8 i s only 0.7 / o in e r r o r for a c i r c u l a r p a n e l , c o m p a r e d with the e x a c t solution of Ref. 9. The p r e s e n t a n a l y s i s gives a c o r r e s p o n d i n g e r r o r of 9 /o.

8

-Inspection of equation 4.17 shows that the stabilising effect of the tensile membrane stresses is greatest for a/b = 0 i.e. the two-dimensional panel. Also, as a/b - x , i.e. when the stresses are mainly transverse to the air flow, the stabilising effect is insignificant. The latter result agrees with the conclusions of Ref. 3 that forces in the y direction have no effect on the stability boundaries for rectangular simply supported panels. For a/b = 0, the reduction in critical panel thickness/chord ratio for Fa*/D = 10 compared with Fa*/D = 0 is 9.2 /o. 4.4. Effect of a Concentrated Mass

The addition of a concentrated mass to the panel is of interest analytically because

(a) it may be necessary experimentally to mount a strain gauge or other transducer to the panel, and the effect of this must be known,

(b) the concentrated mass may have a stabilising effect analogous to the mass balancing of control surfaces.

The analysis follows that of Section 3, except that equation 3. 3 is modified by the addition of a concentrated mass of magnitude M at a point on the panel having co-ordinates (x,, y,). The flutter determinant obtained in Section 4.1 (i.e. equation 4. 2) has the following additional terms in each of the expressions

- (j^/9m0/H ,

- u'^/Sm^a^ ri" /H^ , - w'/Sm^a/j/H , - uVm^arj/H^ .

In the above ternas, M = 0mira.h defines the term /9 ,

x^ y _

n = — and ^ = ~ defines the co-ordinates of M a b m n viz. in G 11 in G 22 inG^, inG,,

and

^ =

Iv^^e -ly .

Solution of the flutter determinant for zero structural damping yields the following result for the flutter frequency

0)"= 300sD/l 2m + 5/9m^(l + 12 7J*) j . (4.19) If aerodynamic damping is also neglected, the critical dynamic pressure parameter

can be expressed by

9

-The above r e s u l t s a r e s t r i c t l y t r u e only for an additional m a s s s y s t e m which i s s y m m e t r i c a l about the x - a x i s but not about the y a x i s , i . e . M / 2 should be c o n c e n t r a t e d at 77, ± g . F o r a m a s s s y s t e m which i s s y m m e t r i c a l about the y a x i s a l s o , the additional t e r m s in G^ ^ and G^^ would be the s a m e a s above but t h o s e in G and G would be z e r o , i . e . for M / 4 c o n c e n t r a t e d at ± n , ± E .

1 2 2 1 ' •• t>

F r o m the above r e s u l t s it can be shown that the addition of a c o n c e n t r a t e d m a s s on the y - a x i s i s always s t a b i l i s i n g and i s m o s t effective when added at the c e n t r e of the p a n e l . F o r e x a m p l e , for ^ = 1 the r e s u l t obtained from equation 4. 20, and to be c o m p a r e d with equation 4 . 5 , i s

2 q c ' = 8 0 ^

MD 9

Y-f 33 + 13-1+6-*] . (4,21)

T a b l e 3 p r e s e n t s the r e s u l t s for a c i r c u l a r panel and a t w o - d i m e n s i o n a l panel with a c o n c e n t r a t e d m a s s , ( ^ = 1), situated at v a r i o u s points along the x a x i s . It can be s e e n that a m a r k e d i n c r e a s e in s t a b i l i t y o c c u r s when TJ = 0 and t h i s

d e c r e a s e s a s T? i n c r e a s e s until, for V = 0.6 and 0.8, t h e r e i s , in fact, a reduction in s t a b i l i t y . H o w e v e r , if the added m a s s had b e e n utilised to double the p a n e l t h i c k n e s s , the value of the p a r a m e t e r 2 q c ' / M D , in t e r m s of the o r i g i n a l t h i c k n e s s , would be 6,880 for the c i r c u l a r panel and 4,800 for the two d i m e n s i o n a l p a n e l . T h u s it can be i n f e r r e d that the addition of a c o n c e n t r a t e d m a s s a s a naeans of s t a b i l i s a t i o n i s not a s effective a s i n c r e a s i n g the panel t h i c k n e s s .

T a b l e 4 p r e s e n t s c o r r e s p o n d i n g r e s u l t s for a c i r c u l a r panel with a c o n c e n t r a t e d m a s s at the c e n t r e of the p a n e l , showing the effects of v a r i a t i o n s i n ^ .

4 . 5 . Effect of Sweepback 4 . 5 . 1 . T h r e e Mode Solution

F o r the e l l i p t i c panel shown in F i g . 1, the a i r flow v e l o c i t y , U, h a s components UcosX and UsinX in the x and y d i r e c t i o n s r e s p e c t i v e l y , when the panel y a x i s i s swept by an angle X . The effect of the sweepback i s a l s o introduced by a s s u m i n g that the f l u t t e r mode shape i s now a p p r o x i m a t e d by (cf. equation 4 . 1 )

^(xy) = ^ Ï T + a , ~ ^ c o s X + a 3 ^ ^ s i n X (4.21)

. ^' " 2 w h e r e Z .

10

The r e s u l t a n t flutter d e t e r m i n a n t h a s the t e r m s .2 11 G 1 G G G G a i 1 3 8D ( pUiu - Mmu) 3 \ 5MH 1 2 ^ 2/ pUiw - Mmw'

^ ° ^ ^ V 60MH,

, 2 ^ , 2 / pUiu) - Mmu' b D + b \ 6OMH3 P U ' c o s ' X / l O M H j -pU''cos''>./10MH pU sin X/IOMH, •pU' sin*X/10MH cos*X sin ^ > 8 3 G. 3 8 0 . ( 4 . 2 3 )

A g e n e r a l solution of the flutter d e t e r m i n a n t to give a closed f o r m r e s u l t i s not p o s s i b l e s o a p a r t i c u l a r solution h a s been obtained for the following v a l u e s of the v a r i o u s p a r a m e t e r s (in l b . s l u g . f t . units),

D m 67.5 0.0432 0,75 p a l 2967 p a . 2.64 T h e s e v a l u e s c o r r e s p o n d to an a l u m i n i u m e l l i p t i c panel at s e a l e v e l having a p a n e l t h i c k n e s s / c h o r d r a t i o h / c = 4 x 1 0 " 3 . T a b l e 5 p r e s e n t s the r e s u l t s of t h i s a n a l y s i s , giving v a l u e s of flutter frequency, u , and c r i t i c a l Mach n u m b e r , M^, for different a n g l e s of s w e e p b a c k , c o m p a r e d with a t w o - m o d e a n a l y s i s . T h o s e r e s u l t s for X = 0 and 90 could be obtained d i r e c t l y from equations 4 . 4 and 4 . 5: t h e r e would be slight d i f f e r e n c e s in the v a l u e s for M^, so obtained b e c a u s e of the neglect of a e r o d y n a m i c d a m p i n g in the equation 4 . 5 .

1 1

-4 . 5 . 2 . Two Mode Solution

An alternative approach to the problem is to assume that the flutter mode shape is approximated by

Z/ > = a rj^ + a ( rj^ cosX + r-^ sinX ) (4.24) (x.y) ^ H 2 \H^ Hj )

The four t e r m s in the flutter determinant can now be written directly from the expressions in equation 4 . 2 3 , viz :

(4.25)

Using the parameters given above in Section 4 , 5 , 1 . , Table 5 presents the results of this analysis, compared with those from the three mode solution. The agreement between the two solutions is good and in both cases the variation of M with sweepback follows a reasonable trend.

5. Analysis for Circular Panels with Variable Edge Restraint

The analyses in the previous section were for elliptic clamped edge panels and it is desirable that corresponding investigations be made for panels with other forms of edge restraint. However, simple analytical expressions for either the vibration modes, or the p r e s s u r e modes as used in Section 4, are not available for general elliptic panels and the present analysis is restricted therefore to circular panels for v/hich the required p r e s s u r e modes are known,

G,,

G„

\^ \ s = = = = G,i Ga2 G « Gai

^G„

+ G,3 + G3,

Thus the assumed flutter mode shape is

a Z + a Z (5.1)

1 1 2 2

„here Z. • [, - ïllji ] [B, - ïi±^ ]

•^

> (5.2)

These modes correspond to uniform and linearly varying p r e s s u r e loadings respectively, where the t e r m s B^ and B^ are given by (Ref. 10) :

-1 + ^ -1 + -1 V V

B, = — and B = — for simply supported edges, 1 + * 1 +

-V -V and B = B = 1 for fully clamped edges,

12

-F o r v a l u e s of edge r e s t r a i n t i n t e r m e d i a t e between s i m p l y supported and fully clEunped, the a p p r o p r i a t e boundary conditions to be satisfied at the edge of the p l a t e a r e 0 . o ( a'^z

aR

_ £ az _y_ a^z \ az « "" R aR ^ T,'^ a f l W ^ 8R w h e r e L i s an e l a s t i c constant for the support at the e d g e , i . e .

L = 0 for s i m p l y supported e d g e s L = oe for fully clamped edges

L > 0 for i n t e r m e d i a t e , e l a s t i c , edge r e s t r a i n t .

T h u s it can be i n f e r r e d from s i m i l a r a n a l y s e s in R e f s . 11 and 12 that m o r e g e n e r a l e x p r e s s i o n s for B and B^ a r e B. 1 + 5K 1 +K and B 1 + 7K 1 +3K where K = 1/ \v + — j .

T h u s the v a l u e s of L quoted above would c o r r e s p o n d to

K = 1/v for s i m p l y supported e d g e s K = 0 for fully clamped e d g e s

0 < K < 1/v for e l a s t i c a l l y r e s t r a i n e d e d g e s .

U s i n g the above m o d e s in equation 3 . 3 , and applying the G a l e r k i n p r o c e s s , y i e l d s the following t e r m s in the flutter d e t e r m i n a n t (neglecting m e m b r a n e s t r e s s e s )

w h e r e and G 11 G . . G 1 a a 6 y X * H = = = = = = = = _ 8D 3 • " -^ D . y + - G m 2 1

14<^1

14<^1

1 + 2(B^ 1 + 3(B^

-l4(B,

8 { pUiu) - Mmu* \ 5MH /pUiw - Mmu'* \

\ 60MH J

lOMHa • ** - 1) - 1 ) + | ( B , - I ) ' ' • 1 ) - l ) + | ( B ^ - D ' - 1 ) + | ( B , - 1) + . 6

r

( 5 . 3 ) > (5.4) 1 ) ( B ^ - 1 )

13

-If h y s t e r e t i c s t r u c t u r a l damping i s a s s u m e d , a s in Section 4 . 2 , the following two equations a r e obtained from the flutter d e t e r m i n a n t

f l ^ ' K % ^ - - [ 0 V - 0 ^ ' + l] +g^0==[0^S'+0 2.T'+(5)^] (5

L T'

R T T ' - 2 S ' 0 ^ " g 1 T ' 0 ' - 2 w h e r e S ' 5) (5.6) 9 6y 2 • a x T ' = 1 + S ' 0 = frequency r a t i o 1 u

and u) = fundamental frequency = J —— -r

E q u a t i o n s 5.5 and 5.6 a r e s i m i l a r to equations 4 . 6 and 4 . 7 , and hence a s i m i l a r a n a l y s i s can be m a d e h e r e of the effects of s t r u c t u r a l and a e r o d y n a m i c d a m p i n g on the flutter of clamped and s i m p l y supported c i r c u l a r p a n e l s . The p r i n c i p a l findings of t h i s s e c t i o n a r e s u m m a r i s e d in T a b l e 6, t o g e t h e r with o t h e r c o m p a r a b l e data from e a r l i e r s e c t i o n s and from R e f s . 1, 11 and 12.

6, L i m i t a t i o n s in the P r e v i o u s A n a l y s e s 6 . 1 . U s e of Static Deflection Modes

In the p r e v i o u s a n a l y s e s , s t a t i c deflection mode s h a p e s w e r e used b e c a u s e the 'in v a c u o ' v i b r a t i o n a l mode s h a p e s for g e n e r a l elliptic p a n e l s could not be r e p r e s e n t e d in a convenient a n a l y t i c f o r m . H o w e v e r , a n a l y s e s in Ref. 11 for c i r c u l a r p a n e l s have indicated that v e r y l i t t l e difference in the r e s u l t s i s obtained when v i b r a t i o n a l m o d e s h a p e s a r e u s e d . When v i b r a t i o n a l m o d e s a r e u s e d , the flutter d e t e r m i n a n t h a s to be d e t e r m i n e d s e p a r a t e l y for e a c h chosen value of edge fixity coefficient K , s i n c e K does not e n t e r into the deflection function d i r e c t l y . T h u s it i s suggested that s t a t i c deflection mode s h a p e s a r e m o r e convenient to use and do not i n t r o d u c e any significant e r r o r s .

The flutter d e t e r m i n a n t obtained in Ref. 11 for the c i r c u l a r panel using

v i b r a t i o n a l mode s h a p e s had t e r m s G^^n which, when c o m p a r e d to t h o s e in equation 4 . 2 , w e r e found to a g r e e t o within t 2 p e r cent. The calculated flutter p a r a m e t e r s a r e c o m p a r e d in T a b l e 6 with t h o s e d e t e r m i n e d f r o m s t a t i c deflection mode a n a l y s e s , and t h o s e from Ref. 1, a s s u m i n g z e r o s t r u c t u r a l d a m p i n g .

6 . 2 . A s s u m p t i o n of Only Two D e g r e e s of F r e e d o m

It i s of i n t e r e s t to d e t e r m i n e how the inclusion of m o r e d e g r e e s of f r e e d o m in the panel f l u t t e r p r o b l e m affects the r e s u l t s obtained, i . e . w h e t h e r o r not the solution c o n v e r g e s a s the n u m b e r of t e r m s in the flutter mode i n c r e a s e s , and, if it d o e s c o n v e r g e , how r e a l i s t i c i s the two m o d e a p p r o x i m a t i o n . A n a l y s e s of t h i s type have been m a d e in Ref. 11 for a clamped edge c i r c u l a r panel using a flutter mode of the f o r m :

14

-Z = (1 - R^)^ A + A Rcosö + A R%os^0 + A R \ O S " 0

L o 1 2 n J The first and second terms represent the static deflection shapes assumed in equation 4.1 and the other t e r m s represent mode shapes which have nodal lines perpendicular to the flow direction and situated symmetrically about the centre.

The results obtained indicate that the flutter parameter K was larger for the three-mode than for the two-mode approximation by about 25 per cent but that the use of further t e r m s produced only a very small change in K, Further, the four t e r m approximation gave somewhat lower, and the five term approximation somewhat higher, results than the preceding approximations. It was also noticed that for the range of Fa „ , K no longer showed a linear dependence on Fa - contrary

D ^ D to the result shown in equation 4.14, However, in a more recent paper, Lee (Ref. 13)

has investigated the same convergence problem by using a slightly different approximation, viz.

Z = (1 - R^)*rA + A RCOS0 +

A,R''COS20

+ A R"cosn0 |

The only difference is in replacing cos 0 by cos n0, which is the same for the vibration mode in vacuo. Calculations using this approximation were carried out and, for comparison, analyses using three vibration modes in vacuo were made. The same rapid convergence of the approximate modes approach was observed, as in Refs. 11 and 12, but it was found that K varied linearly with respect to the midplane load parameter.

To summarise, therefore, the two mode results presented earlier may be expected to be conservative for design purposes on the basis of the resuls of Refs. 11, 12 and 13.

7. Conclusions

Theoretical analyses have shown the destabilising effects of compressive membrane s t r e s s e s and structural damping (in general)on the flutter of clamped edge elliptic panels. The addition of a concentrated mass to the panel has been investigated and the possibility of such an addition being destabilising shown. The effect of sweepback is stabilising or destabilising, depending on whether the panel aspect ratio — is less than, or greater than, unity; the predicted variation of critical Mach number with sweepback has been shown to agree well from two and three mode analyses,

Simple analyses have also shown that the effect of edge restraint on the flutter of circular panels is equivalent to a difference of 20 per cent in critical panel thickness between simply supported and fully clamped panels,

8. References

TECHNiSCHE flOGESCHOOL DELF

VLIwGTUlGECUV/KUNDE BIBLIOlHtEK 15 -1. Johns, D . J . 2. Houbolt,J.C. 3. Hedgepeth, J, M, 4. Fung, Y . C . Hedgepeth, J.M. Budiansky, B, Leonard, R. W . Nelson, H.C., Cunningham, H. J, 7, Johns, D,J., Parks, P. C, 8. Voinoisky-Krieger,S. 9. Bryan, G,H. 10. Timoshenko,S, 11. Goodman,L. E , , Rattaya, J, V, 12, Rattaya, J . V . 13, L e e , F , A .

Some panel flutter studies using piston theory. J . Aero/Space S c i . , Vol.25, 1958, pp 679-684. A study of several aerothermoelastic problems of aircraft structures in high speed flight.

Inst, Flug. und Leicht. Mitt, 5, 1958.

Flutter of rectangular simply supported panels at high supersonic speeds.

J . A e r o . S c i . , Vol.24, 1957, pp 563-573, Panel flutter.

AGARD Manual on Aeroelasticity, Part 5. (To be published).

Analysis of flutter in compressible flow of a panel on many supports.

J . A e r o . S c i . , Vol.21, 1954, pp 475-486. Theoretical investigation of flutter of two

dimensional flat panels with one surface exposed to supersonic potential flow.

NACA Tech. Note 3465, July 1955. (or NACA Rep. 1280), Effect of structural damping on panel flutter.

Aircraft Engineering, Vol.32, 1960, pp 304-308, The stability of a clamped edge elliptic plate under uniform compression.

J . Appl.Mech., Vol.4, 1937, pp A177-178.

On the stability of a plane plate under thrusts in its own plane, with applications to the buckling of the sides of a ship,

P r o c . Lond, Math. Soc. , Vol.22, Dec. 1890, p. 54. Theory of plates and shells.

New York, McGraw-Hill, 1940.

Supersonic speed flutter analysis of circular panels with edges elastically restrained against rotation, WADD Tech. Report 60-309, August, 1960. Flutter analysis of circular panels.

J,Aero/Space Sci. , Vol.29, 1962, pp 534-539. Comments on flutter analysis of circular panels. J.Aero/Space Sci, Vol,29, 1962, pp 1123-1124.

16 -a b M „ % -gR 0 .645 1/2 .668 1 .77 2 .925 OO .98 TABLE 1

Ratio of C r i t i c a l Mach N u m b e r s for Clamped Edge E l l i p t i c P a n e l s . M 1. e . M

M

gR 4S T '

1 ^

b Ref, 8 P r e s e n t A n a l y s i s Ref. 9 0 12.00 9.87 1/2 -11.80 ~ 1 14.79 16.00 14.68 2 44.08 47.20 " • 3 99.09 105.60 ~ 4 178.40 183.06 ~ 5 282.50 296.62 ~ I T A B L E 2

Buckling Coefficient for Clamped Edge E l l i p t i c P a n e l s . F a * / D

M a s s P o s i t i o n rj = F l u t t e r P a r a m e t e r ^ -( C i r c u l a r P a n e l ) 2 q c ' F l u t t e r P a r a m e t e r /v ( T w o - D i m e n s i o n a l Panel) 0 1830 1162 .2 1550 983 .6 776 496 .8 634 304 1.0 860 600 TABLE 3

Effect of C o n c e n t r a t e d M a s s P o s i t i o n on the F l u t t e r of a Clamped Edge P a n e l ( /9= 1 ; M a s s on x - a x i s )

17 -Mass P a r a m e t e r /3 Flutter P a r a m e t e r -T^T-MD 0 860 .2 1318 .4 1542 .6 1680 .8 1768 1.0 1830 TABLE 4

Effect of Variations in Concentrated Mass on the Flutter of a Circular Clamped Edge Panel. (Mass on Centre of Panel)

X 0 30 45 60 90 Three Mode u r a d / s e c 846 864 883 913 1059 Solution c 3.24 3.51 3.76 4.12 4.98

Two Mode Solution (J r a d / s e c 846 915 906 958 1059 M c 3.24 3.47 3.79 4.22 4.98 TABLE 5

Effect of Sweepback on Flutter Mach Number for a Clamped Elliptic Aluminium Panel at Sea Level, (h/c = 4 x 10 12 in. , b = 9 in.)

m u ' c ^ / D 2qc'/MD Simply Supported Circular 1775 (1648) 485 (468) Two-Dimensional ((832)) ((274)) Clamped Circular 4693 (4480) 860 (804) Two-Dimensional 2020 600 TABLE 6

Flutter P a r a m e t e r s for Circular and Two-Dimensional Panels The figures not in brackets correspond to

analyses using static deflection mode shapes in this Note. The figures in brackets

correspond to analyses using vibrational mode shapes, i . e . (( )) from Ref. 1 and

FIG.I. CO-ORDINATE SYSTEM FOR THE ELLIPTIC PANEL. 3 0 0 0 2 0 OO 1 0 0 0 PLcm 1 J

U^.-j ' /

/ _ _ _ ^ 1 1 I 1 1 0-5 l-O 1-5 ECCENTRICITY a/b. 2 0 2-5

FIG.2. VARIATION OF CRITICAL FUJTTEP PARAMETER WITH PANEL ECCENTRICITY.

r 8 X X. cc Z u X »- 2 _ t UJ z * / b = o -i 1 ! 1 1 t 1

j

^ FLOW f - 2 a . - H 2 3 MACH NO.

F i a a . VARIATION OF CRITICAL PANEL THICKNESS WITH MACH NO. AT SEA LEVEL