Supersonic flutter of cylindrical shells

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Michfel de ^^^n^^{% ^i^U

THE COLLEGE OF AERONAUTICS

CRANFIELD

SUPERSONIC FLUTTER OF CYLINDRICAL SHELLS

by

T H E C O L L E G E O F A E F O N A U T I C S C E A N F I E L D S u p e r s o n i c F l u t t e r of C y l i n d r i c a l Shells b y -D. J . J o h n s , M . S c . ( E n g . ) , M. I. A. S. SUMMARY The g e n e r a l t h e o r i e s of thin e l a s t i c c y l i n d r i c a l s h e l l s a s d e r i v e d by Love and Novozhilov a r e c o m p a r e d and then u s e d in a s i m p l e b i n a r y flutter a n a l y s i s which p e r m i t s the e x i s t e n c e of both axial and c i r c u m -f e r e n t i a l waves o-f d e -f o r m a t i o n . L i n e a r piston t h e o r y h a s been used and the r e s u l t s obtained indicate that the a x i - s y i n m e t r i c mode of d e f o r m a t i o n i s the m o s t c r i t i c a l . C o m p a r i s o n s a r e then made with

other published r e s u l t s and a p p a r e n t i n c o n s i s t e n c i e s in t h o s e p a p e r s

a r e found to a r i s e from c e r t a i n a s s u m p t i o n s m a d e in the d e f o r m a t i o n equations u s e d . In a f u r t h e r a x i s y m m e t r i c mode a n a l y s i s the use of a t r a v e l l i n g wave form of r a d i a l deflection is shown to give s i m i l a r r e s u l t s a s standing wave f o r m s when applied to a s h e l l of finite length,

LIST O F CONTENTS ^ a g e S u m m a r y Notation 1. Introduction 1 2. G e n e r a l T h e o r y of C y l i n d r i c a l Shells 2 3. Standing Wave F l u t t e r A n a l y s i s I 5 3 . 1 . D i s c u s s i o n of r e s u l t s 8 4 . Standing Wave F l u t t e r Analysis II 9

5. T r a v e l l i n g Wave F l u t t e r A n a l y s i s : 10 Infinite Length C y l i n d e r

6. T r a v e l l i n g Wave F l u t t e r A n a l y s i s : 12 F i n i t e Length C y l i n d e r

7. Effect of Axial Stiffening 14

8. Conclusions 14 9. E e f e r e n c e s 15

a Radius of c y l i n d e r

A, B , C Modal coefficients in e x p r e s s i o n s for u , v , w c Speed of sound in a i r

D = E h ' / 1 2 ( 1 -v^ F l e x u r a l r i g i d i t y / u n i t width of shell E Young's Modulus

F Function defined after equation 3.6 h Shell t h i c k n e s s

i r-i

L Shell length between s u p p o r t s m N u m b e r of a x i a l half waves M Mach No.

n N u m b e r of c i r c u m f e r e n t i a l w a v e s

p Wave n u m b e r of axial w a v e s in infinitely long c y l i n d e r q Dynamic p r e s s u r e

r , s N u m b e r of axial half waves in a s s u m e d m o d e s

u , v , w Axial, t a n g e n t i a l and r a d i a l d i s p l a c e m e n t s of the middle s u r f a c e of the s h e l l (Sign Convention a s in Ref. 13) U Velocity of Supersonic Flow

V F l u t t e r speed p a r a m e t e r from Ref. 6 = falir^d -v^ X Axial c o - o r d i n a t e m e a s u r e d in a s t r e a m w i s e d i r e c t i o n

Notation (Continued)

a Shell bending parameter = h^/12a^

^ (M^ - 1)^

\ rmra other suffices used are r and s

m L

V Poissons ratio

p Density of supersonic flow

(T Shell m.ass per unit surface area

6 Parameter defined in equation 6.6

<p Circumferential shell co-ordinate

f Flutter speed parameter for shell with n circumferential

[ 2ql/^ waves \= MD /

Q Flutter frequency of oscillation

1. Introduction

Investigations into the panel flutter of cylindrical shells with the wind direction along the cylinder axis fall into two distinct categories; relating to unstiffened, infinitely long, cylinders and cylinders stiffened by rings and longerons.

The former problem has been examined in Refs. 1 - 3 and, although these various approaches a r e quite dissimilar, the conclusion was reached that infinite cylinders are extremely suscep+ible to panel flutter. Miles (Ref. 1) showed that a type of travelling wave with a wavelength which is small compared with the radius of the cylinder, and without a node around the circumference, is the most critical type of instability.

Stepanov (Ref. 3) showed a similar result using piston theory for the aerodynamic forces, but Miles in another paper (Ref. 4) questions the validity of Stepanov's work.

The question as to whether travelling wave instability based on linearised aerodynamic theory is significant for panels of finite length depends essentially on the wavelengths at which instability is predicted, and on whether an unstable travelling wave would experience sufficient growth before reaching the downstream support where it presumably would be reflected to form a standing wave. A method of dealing with the

ring-stiffened cylinder starting from a travelling wave analysis is presented in Fef. 2 but no numerical results are given. Other analyses which deal specifically with standing waves on a ring-stiffened cylinder of finite length are presented in Refs. 5 - 6 . These papers have been examined in considerable detail by Fung in Ref. 7 which contains a most valuable review of the entire panel flutter problem. It is stated in Ref. 7 that a major point of controversy concerns the choice of the appropriately simplified equations governing the deformations of the thin walled elastic shell, and related to this is the choice of flutter mode and the question as to how the critical flutter speed depends upon the number of circumferential and axial waves in that mode.

To avoid computational difficulties in using the general theory for cylindrical shells, which reduce to three simultaneous equations in t e r m s of the a x i a l , tangential and radial deformation components (u, v,w) recourse may be had to the well known Flugge's equations for thin cylinders (Ref. 8) which give way successively to the

Donnells equations (Ref, 9), the shallow-shell equations (Ref. 10)

2

-Flugge's equations were used in Ref, 2 as were also Donnell's equations, but no numerical results were giviin for the flutter of ring-stiffened cylinders, Donnell's equations were used in Ref. 6 and the existence of a minimum flutter speed was demonstrated for a flutter mode containing both axial and circumferential waves. This result is rather surprising since

it can be inferred from Refs, 3 and 5, which both used Goldenveiser's equation, that the most critical flutter mode is a x i - s y m m e t r i c ,

The accuracy of the Goldenveiser equation is suspect however since it neglects axial bending stiffness of the shell, whereas Donnell's equation is inaccurate when the condition n^ > > 1 is violated, i, e, n must be greater than 3,

Thus it can be seen that there is a need to derive more general flutter analyses for ring-stiffened cylinders of finite length, without the limiting assumptions of previous papers, and it is the purpose of this present paper to start from the general theories of cylindrical shells developed by Love and Novozhilov and to determine the critical flutter speed as a function of the numbers of axial and circumferential waves. An attempt will also be made to apply a travelling wave analysis to a cylinder of finite length in the manner outlined in Ref. 2. Linear piston theory will be used in the analyses and only loadings normal to the shell surface will be considered. Although the analyses will only strictly apply to thin shells with no axial stiffening i. e. no longerons, the effects of such stiffeneing will be discussed briefly.

2. General Theory of Cylindrical Shells

To establish the differential equations for the displacements, u, V and w, which define the deformation of a shell the following procedure is followed. The equations of equilibrium of forces and moments (six in all) are derived for an element of the thin shell in t e r m s of the deformations of the element. These general equations were obtained by Love (Ref. 12) and are presented by Timoshenko (Ref. 13), If the ssumption is then made that the membrane forces in the shell are much smaller than the critical values required to produce lateral buckling of the shell the following three equations can eventually be obtained (Fef, 13 eq, 303).

2a

gfu _^ 1

9 x2 9Ï1 . 1 -V 2 a' 9x9(^! - a^ 9x2 9 w 9(/) 9 0'

+

1 +y 2a

}L+ 1 i X + a

9 v 9 x 9^ a 9 w g x = o ( 2 . 1 ) (1-^) + a 9 (ƒ.' • g ' w 9 3^90 a^ 9 (^^ 9x=

+

g^w a^90^ O ( 2 . 2 ) a 9u gx a 9v d<^ + a 9 ' v

<^ - "' "aJa* ^ , %

W gV 2 9 V 9 X?» • ^ g i ? ? : 2 + a^d<^" _ Z ( l - y ^ ) E h ( 2 . 3 ) w h e r e o = h^/12a^" .

It should be e m p h a s i s e d t h a t , a c c o r d i n g t o Novozhilov (Ref. 14) the development given by Love i s not free f r o m i n a d e q u a c i e s with r e g a r d to c e r t a i n s m a l l t e r m s , s o m e of which a r e r e t a i n e d , and o t h e r s , which a r e of the s a m e o r d e r of m a g n i t u d e , a r e r e j e c t e d . T h i s point i s s u g g e s t e d a l s o by the lack of s y m m e t r y of the t e r m s in equations 2 . 1 - 2 , 3 . If the method formulated by Novozhilov is followed the equations obtained a r e ,

g x ^ 2a'^ g^u 1 +v 9V 2 a dxd^i V a ^ = 0 gx ( 2 . 4 ) 1 + v 2a

£u_

gx 9(/)

+

gV gx^ ^ 1 gV a^ 9(^^ 2(1 - v ) d% g^v 9x^ êd<i)^ a 9w g ^ + a (2 -v) gx?9(/) a^g(/>' ( 2 . 5 ) V a + a g u 9 x a ' a ' g V gx'*

+

gv Ckj} + a 2 9 ' g x^9(^'

+

(2 -v) 9 V a2g<f/ d\ 9^v 2 a J.3 9x^<i^ di^^^ Z ( l -v^ E h

+ 3 .

( 2 . 6 )

4

The only changes in t h e s e equations c o m p a r e d with equations 2 . 1 -2. 3 o c c u r in the s q u a r e b r a c k e t e d a t e r m s in equation 2 . 2 . By i n s p e c t i o n of the t e r m s in equations 2, 2 and 2. 5 it can be shown that the s q u a r e b r a c k e t e d a t e r m s in v a r e negligible, and Vlasov (Fef. 15) h a s s u g g e s t e d that all the s q u a r e b r a c k e t e d a t e r m s in equations 2 . 2 2,6 a r e s m a l l q u a n t i t i e s of the s a m e o r d e r a s t h o s e which w e r e d i s -r e g a -r d e d in the d e -r i v a t i o n of equations 2, 2 - 2 . 6 , An o -r d e -r of magnitude c o m p a r i s o n of all the t e r m s in equations 2, 2 - 2,6 does not support t h i s s u g g e s t i o n but if the a p p r o p r i a t e t e r m s a r e neglected the following set of equations a r e obtained from the Love and Novozhilov d e v e l o p m e n t s , g^u 1 -V 9^1 ^ 1 +y g^v V dw + + ^ "*" ^ d V _ v_ dw - 0 (2 7) g x^ 2a^ 9<i^ 2a Q^Q^ a 9x 1 + 1 ^ g^u l_-_v g^v _!_ 1 9^v _ 1 9w _ Q (2 8) 2a dxd(l> 2 ~ 3^2 a^ Q ^^ \2 ^ V 9 ] i - _ L 9 v . + Z _ + a ' / ^ a ^ 9'* w ^ 2 g-^w _j, 9^w - a gx " a ' 9<^ a^ V d >ê dx^d<\>^ a^g<^^ = Z<1 - ^^') ( 2 . 9 ) E h

It can e a s i l y be shown that for the c a s e of axi - s y m m e t r i c d e f o r m a t i o n s the above equations r e d u c e to the s i m p l e r e s u l t

j3d_lw ^ mnv_ ^ 2 • (2 JO) d X a

3. Standing Wave F l u t t e r A n a l y s i s I

T h i s f i r s t a n a l y s i s i s c o n c e r n e d with flutter of a c y l i n d r i c a l s h e l l of length L , in which the simplified equations 2. 7 - 2 . 9 will be u s e d . The flutter mode a s s u m e d c o r r e s p o n d s to that of a s i m p l y - s u p p o r t e d s h e l l and is defined by the e x p r e s s i o n s

u = U , ,v e (x,(^) ifit V _(X.(^) iQt and w = W _(x,(^) iQt w h e r e U = V and W r—J : 1 \ X ) ) A ^ n C O s n < ^ c o s -f-X -f-X = V ^ " ^ m n «i^ ""^ sin - ^ r-^ r-i \ n ^ ( 3 . 1 ) ( 3 . 2 ) ( 3 . 3 . )

When t h e s e e x p r e s s i o n s a r e s u b s t i t u t e d into equations 2. 7 and 2 . 8 one o b t a i n s , after c o n s i d e r a b l e m a n i p u l a t i o n , m n rnn m 2 2 (1 - v)n + ( 2 - v+v^) \^n -2* k^ 1 - y 2 < ^ m + " ' ) < ^ m + ^ T r - n ) . and B mn = C mn n n^ + ( 2 + V) \ m (X^ + n M , m ' and inclusion of t h e s e e x p r e s s i o n s into equation 2.9 r e s u l t s in

) V^C c o s rut sin \ mn \ X m a ( X ' ^ + n^)+ X ^ l -v^ Z a^(l -v) e"^"* = E h (X^^fn^) ( 3 . 4 )

6

-F o r the flutter p r o b l e m the effective, e x t e r n a l , l a t e r a l p r e s s u r e Z contains a e r o d y n a m i c t e r m s and o s c i l l a t o r y i n e r t i a t e r m s p r o p o r t i o n a l to the s h e l l m a s s . If l i n e a r piston t h e o r y i s used to d e r i v e the a e r o d y n a m i c f o r c e s , equation 3 , 4 b e c o m e s , for the mn th m o d e , a(X^ +n^)'*+ X'* (1 -v^)'] X x COS Tie s i n — E h a ^ d - i ^ m m ' ( X^ 4- n M .2 X £U _ 2 1 c o s n^ COS M a X X m. pJJiQ M a n J c o s n^ sin X X m _{( 3 . 5 )}

v/here U and M a r e the velocity and Mach N u m b e r of the s u p e r s o n i c flow p a s s i n g in an axial d i r e c t i o n along the e x t e r n a l s u r f a c e of the s h e l l . If only two d e g r e e s of freedom a r e c o n s i d e r e d m = r and m = s with n fixed in e a c h , then by applying the G a l e r k i n p r o c e s s t o equation 3, 5 the following f l u t t e r d e t e r m i n a n t i s obtained,

Plf 4 r s LM s ' - r Eh w h e r e F ^ = ^ a ^ ( l - ^ ^ pU 4 r s LM • s^ - r '

F +

'I''' -

aQ^

s M ' a ( x ; + n ' ) ' + ^ ; ( l -v") ( X ^ + n M 0 . ( 3 . 6 ) , with a s i m i l a r

e x p r e s s i o n for F g , and the condition that ( r + s) m u s t be odd a p p l i e s . When the d e t e r m i n a n t is expanded, the i m a g i n a r y and r e a l t e r m s can be s e p a r a t e d giving e x p r e s s i o n s for the flutter frequency and speed r e s p e c t i v e l y , 2a 0^ = F „ + F

f P_21^\ f ^rs

The t e r m \ L M / Vr^- s^ ( F _ F ) % 4 ( - ^ ^ ' ' ^ ^ r ^ s ' ^ * \ M ( 3 . 7 ) ( 3 . 8 )

M r e p r e s e n t s the effects of a e r o d y n a m i c damping which c a n p r o b a b l y be neglected a s was justified in Ref. 16.

If t h i s i s d o n e , equation 3. 8 b e c o m e s after s o m e manipulation, with r > s f 2qL n '' MD L V / r ^ - s ' a 8 r s ( V ' - X | ) ( 2 n 4 - X ^ r + ' i ) + 1 - V' a

K

X' » , 2 l ^ ^ n +Xj,

„=.x^

When n = 0 the r e s u l t is obtained for a x i - s y m m e t r i c f l u t t e r , v i z .

( 3 . 9 )

o

/ 2 2 . 2 , 2 , 2 v

TT^ ( r - s ) ( r + s ) ( 3 . 1 0 )

and equation 3,9 may be r e w r i t t e n a s 4 n

+

LA / r ^ - s^ 8 r s n^+X: 2 n M ^ r - ^ s ) +

K

n ^ + X LZÜ: a ( 3 . 1 1 )

It was pointed out in Fef, 16 that if a e r o d y n a m i c damping can be neglected then " s t a t i c " a e r o d y n a m i c f o r c e s given by A c k e r e t a r e m o r e a c c u r a t e than t h o s e given by piston tl^eory. The modification involves r e p l a c i n g M in i/r by ^ » (M^ - 1)2 and should l o w e r the Mach No, r a n g e of applicability of equations 3,9 - 3 . 1 1 .

8

-3. 1. D i s c u s s i o n of r e s u l t s

It i s s e e n that the m o s t c r i t i c a l value of i^jj i s fQ s i n c e the second t e r m on the right hand side of equation 3.11 is always p o s i t i v e , and hence the m o s t c r i t i c a l flutter mode is a x i - s y m m e t r i c , The m o s t c r i t i c a l combination of m o d e s r and s giving the lowest flutter speed was investigated in Ref. 17, which used equation 2. 10 to define the e l a s t i c d e f o r m a t i o n of the s h e l l , and found to be r = 2 and s = 1 whence f^ - 274.

It can a l s o be s e e n that the second t e r m on the right hand side of equation 3.11 h a s a m i n i m u m value at s o m e value of n o t h e r than z e r o . By equating d i^^ ^o z e r o and a s s u m i n g that n^ » X^ the

d n "

following e x p r e s s i o n i s obtained

6 / 2 , 2 V 77^ a ^ / 1 - y ^ \ / o 1 o \

n = ( r + s ) 5 - ( 1 ( 3 . 12) 2

L a

Shulman (Ref, 6) d e m o n s t r a t e d the e x i s t e n c e of a m i n i m u m flutter speed for n ^ 0 using D o n n e l l ' s equations and l i n e a r piston t h e o r y in an a n a l y s i s employing 8 m o d e s . F o r a s h e l l having the p r o p e r t i e s — = and •— = 6 Shulman quoted a c r i t i c a l value of n = 8 a 200 a

4

and a m i n i m u m value of f = 7, 85 x 10 , If the above s h e l l

p a r a m e t e r s a r e s u b s t i t u t e d into equation 3, 12 c o r r e s p o n d i n g v a l u e s of n = 9 and ^f^^ = 5 x 10"* (when r = 2, s = 1) a r e obtained which a r e in r e a s o n a b l e a g r e e m e n t with S h u l m a n ' s r e s u l t s . Table 1 gives n u m e r i c a l v a l u e s of f for the above shell_using equation 3 , 1 1 and a l s o given a r e the c o r r e s p o n d i n g v a l u e s of V with S h u l m a n ' s v a l u e s for c o m p a r i s o n . The a g r e e m e n t i s good and the r e s u l t s indicate m o s t s t r o n g l y the fact that a x i - s y m m e t r i c flutter i s the m o s t c r i t i c a l m o d e . (Note V = ira/-n^{l -v^)).

If the f i r s t t e r m on the right hand s i d e of equation 3 . 9 i s n e g l e c t e d - which i s equivalent t o neglecting the a t e r m o r the bending stiffness of the s h e l l - and if it i s a s s u m e d that n » X^ , the following r e s u l t is obtained.

12(1 -i;^ / a Y , ^

The above a s s u m p t i o n , r e g a r d i n g the neglect of the bending s t i f f n e s s , is the s a m e a s that m a d e in the G o l d e n v e i s e r ' s equation which was used

that '^n "^ '''o ^ ° ^ l a r g e n. T h i s i s obviously false since from equation 3 . 1 1 and Table 1 it i s s e e n that f^ » ^ITQ for l a r g e n. 4 . Standing Wave F l u t t e r A n a l y s i s II

In Section 3 t h e simplified e l a s t i c equations 2.7 - 2.9 w e r e u s e d and useful r e s u l t s have been obtained, but a f u r t h e r a n a l y s i s i s w a r r a n t e d to justify the o m i s s i o n of the v a r i o u s s q u a r e b r a c k e t e d

a t e r m s in e(^uations 2 . 1 - 2 . 6 . Since the neglect of the t e r m in v

in equations 2. 2 and 2. 5 can be e a s i l y justified the a n a l y s i s of Section 3 h a s been r e p e a t e d , including in equations 2. 7 - 2.9 only the s y m m e t r i c t e r m s due to Novozhilov, i . e .

a (2 -v) g y d \ v

a^

ge-in equation 2. 8

and _a (2 -v) g" V g^ V in equation 2. 9

Following the p r o c e d u r e outlined in Section 3 the v a r i o u s c o r r e s p o n d i n g r e s u l t s a r e , m n m n \l(l-v)n''+(2-v + v V ^ \^ - 2v X^^^ + 2a(l + v) 2(X^^ + nM^ {X^^+ ^ ^ n ^ ) '

[1

(2 -t^) V + n 2 ' 1 \\_ + 1 - ^ rf < " ' m Kl-v)

mn

C^^ n|n^+(2+.)X^^ + 2 4 ( 2 - v ) X ^ + n « J [ ^ + i ^ r f J

•\2 , 2 X + n m and F „ b e c o m e s Eh a^(l-y^ a I (X^ + n^)^- 2n"

'(2 -v)l^^xf\

(2+y)X^^ + n2] j+X^d -v'^

if t e r m s in a a r e n e g l e c t e d . (X'^ + n^ )

10

-The corresponding result to equation 3,11 is

^n = ^o \ T ) ^ 4 ,. LY,/:-^ -^ _{' r - s} r s 2nMx:. - Xf) - 2n^ , . . , 2 (4 -y^X^ + 4n^?^ + n (4 -v^)\% + 4n2 A| + n-^-] / 1 _ ^2 + ( n 2 + x | )2 a

(n^

+

x ; )

X^ r X^ s .(n^+X^J^ (n^+X^)^ (4.1) and it can be seen immediately that the additional t e r m s do not

affect the original conclusion that n = 0 gives the most critical mode. Also as n -» ex the additional t e r m s vanish and the general statement may be made that the neglect of the square bracketed a t e r m s in Section 3 was justified. To support this statement Shulman's shell p a r a m e t e r s have been substituted into equation 4.1 and Table 1 shows the small changes in the values of I/TJ^ caused by the additional t e r m s ,

5. Travelling Wave Flutter Analysis: Infinite Length Cylinder Miles (Fef, 1) showed that an axisymmetric flutter mode was the most critical for infinitely long shells, using a travelling wave analysis, and a similar result was shown in Sections 3 and 4 for finite length shells with a standing wave analysis. In this section therefore, only a simplified axi-symmetric analysis will be

presented using equation 2,10 reproduced below,

D d^w dx^ Ehw a Z (2.10) If it is assumed that the cylinder wall may deform into any number of sinusoidal waves of any wavelength and constant amplitude along its length, and that the motion is simple harmonic in time, then the radial deflection of the wall maybe written

w Re Ce -ip (x

nt

( 5 . 1 . ) where C is the complex amplitude of the motion and p is the real

Substiuting for Z and w into equation 2 . 1 0 y i e l d s the r e s u l t , E h , pu'^ . PU M ^ M Dp4 + £ ^ = + £ L i p . ^ in + a n ^ ( 5 . 2 ) By s e p a r a t i n g r e a l and i m a g i n e r y t e r m s one o b t a i n s , U = J I , ( 5 . 3 ) P . 4 ^ E h a" and Dp"* + - ^ = CT Q^ , ( 5 . 4 )

and s u b s t i t u t i n g for Q into equation 5.4 gives

u'

Dp^ + Eh/a^ p^ _{a .} (5.5)

Eh The m i n i m u m value of U o c c u r s when p^ » —

a"D

1

and U ^ . =. Eh^ /oa [3(1 - v V ' . ( 5 . 6 ) niin L

T h i s r e s u l t a g r e e s with that obtained by Stepanov (Ref. 3), who a l s o used piston t h e o r y , from a r a t h e r different method of a n a l y s i s . It should however be pointed out that the a s s u m e d deformation of equation 5.1 h a s the f o r m of a wave t r a v e l l i n g in the positive x d i r e c t i o n with v e l o c i t y _Q. , hence t h e flow of velocity U outside the v i b r a t i n g c y l i n d e r i s equivalent t o a flow of velocity U - - ^

P outside the " s t a t i o n a r y " c y l i n d e r . However it i s s e e n that ( U - -0- j is z e r o from equation 5. 3 and we have an a p p a r e n t

i n c o n s i s t e n c y s i n c e for piston t h e o r y to be valid U » c , the s p e e d of sound. T h i s h a s been pointed out r e c e n t l y in Ref. 4 and i s a c o n s e q u e n c e of l i m i t a t i o n s i n h e r e n t in the u s e of piston t h e o r y when applied to t r a v e l l i n g wave m o t i o n s ,

12

-6, T r a v e l l i n g Wave F l u t t e r A n a l y s i s ; F i n i t e Length C y l i n d e r In Ref, 2 a method of a n a l y s i s was p r o p o s e d for extending the t r a v e l l i n g wave a n a l y s i s to a c y l i n d e r of finite length. That method will be applied h e r e to the p r o b l e m of a x i - s y m m e t r i c flutter only, using equation 2 . 1 0 , The configuration to be c o n s i d e r e d c o n s i s t s of

an unstiffened c y l i n d e r with added rigid r i n g s t i f f e n e r s which p r e v e n t r a d i a l deflections at the l o c a t i o n s x = - jL(j =• 0, 1, 2, ). As in s e c t i o n s 3 and 4 t h e s e s t i f f e n e r s , whose p o s i t i o n s define the length of the s h e l l b a y s , a r e a s s u m e d not to i n t e r f e r e with the e x t e r n a l flow of a i r , and the a e r o d y n a m i c f o r c e s a r e given by l i n e a r piston t h e o r y .

The r a d i a l deflection of the wall m a y be w r i t t e n

CO

- i m77x iQi

w = Re ) C ^ e L e ( 6 . 1 )

oc

provided that the coefficienteC satisfy the c o n s t r a i n i n g r e l a t i o n s , which c o r r e s p o n d to z e r o deflection at the r i n g l o c a t i o n s

0 0

y

C j ^ = 0 ( 6 . 2 )

m = - « ( m odd)

y Cjj^ - 0 ( 6 . 3 ) m = -cc(m even)

It i s a s s u m e d in equation 6 . 1 that w i s p e r i o d i c o v e r two bay lengths in the x d i r e c t i o n and, consequently equation 6. 3 c o r r e s p o n d s to

motion which is i d e n t i c a l in each bay, w h e r e a s equation 6. 2 c o r r e s p o n d s to motion having the s a m e amplitude from bay t o bay but with a l t e r n a t i n g d i r e c t i o n .

In using equation 6. 1 in equation 2 . 1 0 allowance m u s t be m a d e for the r e a c t i o n f o r c e s e x e r t e d on the c y l i n d e r wall by the r i n g

s t i f f e n e r s at the end of e a c h b a y . If t h i s is done and if equation 2 . 1 0 , 1 m'/rx

suitably modified, is multiplied by g—^ and i n t e g r a t e d o v e r 2 bay lengths of the c y l i n d e r the r e s u l t a n t e x p r e s s i o n s obtained from the c o n s t r a i n i n g r e l a t i o n s of equations 6. 2 and 6. 3 a r e

DO

y - i . = 0 (6.4)

( m odd)m=--o<: ^

V.

J _ » 0 (6.5) / Or m = - a ( m even)

w h e r e » „ , ' ^ ^ < "Jn " " ' > + 4 ^ < - ^ ' " 1 ^ ' <«•«•>

and

.'^

, iuB^ .

f> /o

.

L a

A t t e m p t s have been m a d e with equations 6 . 4 and 6 . 5 to obtain c l o s e d f o r m s o l u t i o n s for the flutter frequency and velocity using i n c r e a s i n g l y h i g h e r a p p r o x i m a t i o n s in the s u m m a t i o n s but only the t h r e e t e r m a p p r o x i m a t i o n to equation 6. 5 was s a t i s f a c t o r y in t h i s r e s p e c t . It i s worth noting that v e r y l a r g e v a l u e s of m should not influence e i t h e r equation 6 . 4 o r 6 . 5 significantly, s i n c e then

—— -» 0 a s m - oc. The t h r e e - t e r m a p p r o x i m a t i o n to equation

^ m 1 1 1

6 . 5 m a y be w r i t t e n a s + + =» 0,

% ®2 ®-2

and s u b s t i t u t i o n of t h e a p p r o p r i a t e f o r m s of equation 6 . 6 y i e l d s , eventually, the following e x p r e s s i o n s for flutter frequency and s p e e d ,

| a t ? = 1 ^ , 1 4 4 ( 6 . 8 ) MD VY2

The fact that t h i s r e s u l t i s m o r e c o n s e r v a t i v e than the c o r r e s p o n d i n g r e s u l t {fQ = 274) of Section 3 i s not s u r p r i s i n g and it would be expected that a s m o r e t e r m s a r e included in the a p p r o x i m a t i o n to e i t h e r

equation 6 . 4 o r 6 . 5 a l e s s c o n s e r v a t i v e r e s u l t would be obtained. T h i s d e c r e a s e in c o n s e r v a t i s m with a l a r g e r n u m b e r of m o d e s w a s noted in a n a l y s e s of r e c t a n g u l a r p a n e l s in R e f e r e n c e s 16 and 18.

Thus s t a r t i n g from the t r a v e l l i n g wave form of equation 6 . 1 it i s s e e n that the inclusion of boundary conditions at the r i n g s t i f f e n e r s , i . e . z e r o r a d i a l deflection at x » * j L . h a s led to a f o r m of solution c o m p a r a b l e with that obtained from the standing wave a n a l y s i s of Section 3 . Hence t h e r e i s no i n t r i n s i c difference in the two t y p e s of solution and p r e s u m a b l y the a s s u m p t i o n of a d o w n s t r e a m support for the s h e l l h a s led to the t r a v e l l i n g wave being reflected t o f o r m a standing w a v e .

14

-7. Effect of Axial Stiffening

The previous analyses have been for thin cylindrical shells with no additional axial stiffening, and the inclusion of such effects would, in general, necessitate modifications to the elastic and inertia t e r m s . Since the addition of heavy stringers would seem to preclude the possibility of axi-symmetric flutter - which would

require the stringers to bend, it would appear that a more critical flutter mode might then exist for some value of n when the s t r i n g e r s lie along circumferential nodal lines and hence do not bend. It is suggested therefore that equation 3.9 be used in this case and the value of 2n assumed ( i . e . number of nodes) should correspond to the number of stringers or a multiple of it whichever is the most critical.

8, Conclusions

By starting from the general theory of deformation for thin cylindrical shells a simple result has been obtained (Equation

3.9) from a binary flutter analysis for a shell of finite length which indicates that the axi-symmetric flutter mode is the most critical. This equation reduces, after further simplifying assumptions, into forms comparable with earlier papers using Donnell's equation (Fef. 6) and Goldenveiser's equation (Ref. 5), and shows how certain apparent inconsistencies in these papers have been derived.

In further axi-symmetric flutter mode analyses the use of a travelling wave form of radial deflection has been shown to give similar results as standing wave forms for the finite length shell, and to give a very simple, but suspect, result for the infinitely long shell.

Although the analyses were for shells with no axial stiffening in the form of s t r i n g e r s , suggestions were made as to how such effects might be considered.

9. R e f e r e n c e s 1. M i l e s , J . W . 2. L e o n a r d , R. W. , Hedgepeth, J . M . 3. 4. 5. 6. 7. 8. 9. 10. Stepanov, R . D . M i l e s , J . W . S t r a c k , S, L . , Holt, M. Shulman, Y. F u n g , Y . C . F l u g g e , W. Donnell, L . H . V l a s o v , V . S . Supersonic Panel F l u t t e r of a C y l i n d r i c a l Shell. J o u r . A e r o . S c i . P a r t I V o l . 2 4 , 1957, pp 107-118 Ti^art n V o l . 2 5 , 1958, pp 312-316 On P a n e l F l u t t e r and D i v e r g e n c e of Infinitely Long Unstiffened and Ring-Stiffened Thin Walled C i r c u l a r C y l i n d e r s .

NACA R e p o r t 1302, 1957.

On the F l u t t e r of C y l i n d r i c a l Shells and P a n e l s Moving in a Flow of G a s . ( T r a n s l a t i o n ) NACA TM. 1438, 1958. On Supersonic F l u t t e r of Long P a n e l s . J o u r . A e r o / S p a c e Sci. , V o l . 2 7 , 1960, pp 476. Supersonic P a n e l F l u t t e r of a C y l i n d r i c a l Shell of F i n i t e Length. IAS p a p e r N o . 6 0 - 2 2 , 1960. V i b r a t i o n and F l u t t e r of C y l i n d r i c a l and Conical S h e l l s . O . S . R . TR. 59-776, M . I . T. , 1958. P a n e l F l u t t e r , AGARD Manual on A e r o e l a s t i c i t y , 1959, Statik und Dynamik d e r Schalen.

J u l i u s S p r i n g e r , B e r l i n , 1939. Stability of Thin-Walled T u b e s Under T o r s i o n . NACA R e p o r t 479, 1933. B a s i c Differential E q u a t i o n s in G e n e r a l T h e o r y of E l a s t i c S h e l l s . NACA T M . 1 2 4 1 , 1951.

R e f e r e n c e s (Continued) 16 -11, 12, 13, 14, 15. 16. 17, 18. G o l d e n v e i s e r , A . L . L o v e , A . E . H . T i m o s h e n k o , S. H. W o i n o w s k y - K r i e g a r , S. Novozhilov, V. V. V l a s o v , V . Z . J o h n s , D , J . J o h n s , D . J . Hedgepeth, J . M . T h e o r y of E l a s t i c Thin S h e l l s . G . T . L . , Moscow, 1953. E l a s t i c i t y . 4th Edition C h a p . 24, 1927 pp. 515 T h e o r y of P l a t e s and S h e l l s ,

2nd E d i t i o n . McGraw Hill Book Co. , 1959, The t h e o r y of Thin Shells *

Noordhof f L t d . , 1959.

A G e n e r a l T h e o r y of S h e l l s . Moscow, 1949.

Some P a n e l F l u t t e r Studies Using P i s t o n T h e o r y . J o u r . A e r o . Sci. V o l . 2 5 , 1958, pp 6 7 9 - 6 8 4 . Supersonic F l u t t e r of a C y l i n d r i c a l P a n e l in an A x i s y m m e t r i c m o d e . J o u r . R o y . A e r o . S o c . V o l . 6 4 , 1960, pp 3 6 2 - 3 6 3 . F l u t t e r of R e c t a n g u l a r Simply Supported P a n e l s at High Supersonic S p e e d s . J o u r . A e r o . Sci. V o l . 2 4 , 1957, pp 5 6 3 - 5 7 3 .

TABLE 1 C r i t i c a l F l u t t e r Speed P a r a m e t e r s a 200 a 6 , r =» 2, s =" 1 ^Q » 274 n 1 2 3 4 5 6 7 8 9 10 11 12 -4 ^ X 10 n 3 V x 10 Equation 3.11 2411 450 117 4 1 . 5 1 8 . 7 1 0 . 1 6 . 8 5.42 5.05 5.20 5.66 6,36 562 105 2 7 . 3 9-67 4 , 3 6 2.36 1.58 1.26 1.17 1.21 1.33 1.49 V x 10"^ Ref. 6 -3 8 . 0 1 3 . 1 5.69 3.15 2.14 1.83 1,88 2,22 2.70 3.26 ^^ X l O " ' * n C o r r e c t i o n T e r m s in Equation 4 . 1 - 0 . 0 4 0 - 0 . 0 7 0 - 0 . 0 7 0 - 0 . 0 7 5 - 0 . 0 8 0 - 0 . 0 8 0 - 0 . 0 8 0 - 0 . 0 8 0 - 0 . 0 8 0 - 0 . 0 8 0