Non-linear vibrations of a shallow cylindrical panel on a non-linear elastic foundation

Journal of Sound and Vibration (1979) 66(4), 507-512

NON-LINEAR

VIBRATIONS

OF A SHALLOW

PANEL ON A NON-LINEAR

CYLINDRICAL

ELASTIC FOUNDATION

C. MASSALAS AND N. KAFOUSIAS Department of Mechanics, University of’loannina, loannina, Greece (Received 22 December 1978)

The intluence of large amplitude on the free vibrations of a long shallow cylindrical panel with straight edges clamped and resting on a non-linear elastic foundation is investigated. The equilibrium of the panel when subjected to an external pressure is also studied. 1. INTRODUCTION Many studies of the large amplitude flexural vibrations of plates and shells have been reported in the literature in recent years. Extensive literature surveys on the topic of non-

linear shell vibrations have been given by Leissa and Kadi [l], by Mayers and Wrenn [2] and by El-Zaouk and Dym [3]. Reissner [4], using shallow shell theory, analyzed the non-linear motions of cylindrically curved shell panels with simply supported edges supported by shear diaphragms. A variational procedure and a perturbation technique were used to arrive at the relationship between the non-linear frequency and the amplitude. Leissa and Kadi [l] broke new ground by investigating the effects of curvature on the frequencies of free vibration. Ramachandran and Murthy [S] studied the non-linear vibrations and “snap-through” phenomenon of simply supported cylindrical panels made of orthotropic material and resting on an elastic Winkler foundation. The present paper is concerned with the non-linear vibrations of a relatively long panel of circular curvature with straight edges clamped and resting on a non-linear elastic foundation. The stability problem when the panel is subject to an external pressure is also considered. 2. PROBLEM

FORMULATION

In the present analysis it is assumed that the panel dimension in the direction of the generator (the length) is considerably larger than the dimension of the arc (the width); this makes it possible to consider the deformed curved surface as cylindrical, like the static shape. The system of initial non-linear differential equations (with tangential inertial neglected) for shallow shells is (D/h) v*P*w = w,,, @,‘y,+ w,yy@,,,,- 2w,,y @,,xy+ (I/R) +,,xx- PW,,,+ (P/h)> (I/E) PI+@

(I)

= ~5, - w,,, w,yy- (NW,,,,

(2) where w is the deflection, Q,is the stress function, x and y are the initial position co-ordinates, R is the panel radius, h is the panel thickness, D is the flexural rigidity, E is the modulus of elasticity, p is the mass density, P is the external load, t is time and ( ),, = a( )/ax. (A list of symbols

is given in the Appendix.) 507

0022460X/79/200507 + 06 %02.00/O

0 1979Academic Press Inc. (London)

Limited

508

C. MASSALAS

AND

N. KAFOUSIAS

The panel is clamped on fixed supports located along the straight edges and the external load P is given by P=

q-k,w-k3w3,

(3)

where q is a uniformly distributed pressure acting on the convex (outer) face, and k, and k, are foundation parameters. The stresses in the middle surface, acting along the width, are assumed to be constant for all points of the panel and equal to B,. The stresses zix (i = x, y) are considered as equal to zero [6]. On the basis of the assumptions made relative to the pattern of deformation of the panel, equation (1) can be written in the form DW&YYY - ha, wJY - (h/R) CJ~- q + k, w + k, w3 + FIW,~~= 0,

(4)

where F??= ph. The compatibility equation drops out, since the deflections are independent of x, because of the assumption about the pattern of deformation. In the case under discussion the boundary conditions are at y = 0, L, w = II = w,, = 0, (5) where L is the arc length.

3. STABILITY

OF PANEL

The stability problem is described by equation (4) when the inertia term is omitted. The deflection function w, to a first approximation, is assumed to be given by the expression w = f( 1- cos 1, y),

1, = 2x/L,

which satisfies the end conditions (5). The Bubnov-Galerkin

(6)

equation is

L

X(1-cos;Liy)dy = 0, I0 where X is determined from equation (4) and is equal to

(7)

X = Dw,~~,,~- CJ,,hw,,, - o,(h/R) i- k, w + k, w3 - q.

(8)

Substituting expressions (8) and (6) into equation (7) and carrying out the integrations gives q = D(A;‘/2)f - by h {-(A:/2)f‘+

l/R} + Sk,

f + +k, f 3.

(9)

The unknown CT,, is to be determined from the expression for the strain E,,: that is, a, = u., - (w/R) + +w;,

(10)

Ey = (a,/E)( 1- v2)

(11)

Since equation (10) can be written in the form

vg = Performing the integration conditions for u gives

(12)

in expression (12) and taking into account

the boundary (13)

cr = {E/(1 -v2)) 1(Z/4)f2 -(W)fJ. Introducing expression (13) into equation (9) gives

&$+;k, >f+

&;;A:f2

+ (Ah;

+;k3)f3.

(14)

PANEL

ON A NON-LINEAR

509

FOUNDATION

cm-

-100

Figure

Upon introduction

-

1. Influence

of k: on q*. 6, = 0.032: 6, = 31; k; = 0.

of the dimensionless parameters 6, = h/R,

5 =flh,

6, = L/h,

k; = {h3k3(l -v2)/E}(L/h)4,

kT = h{k,(l-v2)/E)(L/h)4, q* = (q/E)(L/h)4(1

-v’),

(15)

equation (14) takes the form q* =.q*(Q

= (+c”+@d;+$kT)<

- 37~‘6,6;5~

+ (2n4+yk;)t3.

(16)

Curves for q* as a function of 5 are given in Figures 1 and 2 for 6, = O-032,6, = 31 and different values of k7f and kz. For k: = k$ = 0 and 6, = 0.032 the “snap-through” phenomenon enters into the picture for 20 < 6, < 20.1. On the other hand for 6, = 0.032, 6, = 31, kf = 0 and kf small the curves q * = q*(t) have the usual loop shape. With increasing k: they go higher and become more level. At k: > 270 the zone of negative rigidity disappears. At this point the phenomenon of “snap-through” becomes impossible. For 6, = 0.032,6, = 31, k: = 100 the influence of the hardening non-linear foundation (kf$ # 0) on the equilibrium curves of the panel is shown in Figure 2. At k: > 30.1 the “snap-through” phenomenon disappears. For kz < 0 (softening foundation) the zone of negative rigidity is increasing.

Figure 2. Influence of k: on q*. S, = 0.032: 6, = 31: k: = 100.

510

C. MASSALAS AND N. KAFOUSIAS

4. VIBRATION OF PANEL In this problem the vibrational mode shape is taken as

w(y,t) =fw(l-cos~,Y),

(17)

where f(t) is an unknown function of time. The Bubnov-Galerkin J,, + &f-

W1f2

+ (@Z&1 +%sz)f3

equation gives

= 0,

(18)

where 4 E,

=

h/R < 1,

E2

=

kk,

3iii

I E

2:

E

rxl =---l-v22m

2E h 31-v2R2iii

(+&+-__

1,

+

=

f m=O

i: n=O

m’

11 a3 = ykl. 12m

A;R

Q = m->

According to Lindstedt’s method [7] and a multiparameter seeks a periodic solution of equation (18) in the form f

+5

(19)

perturbation

technique one

m,n = 0,1,2 ,...,

Mf,,,

(20)

it being required that each f,, (m, n = 0, 1,2, . . .) be periodic. The non-linear terms can be expected to alter the period by amounts of order .si and s2, This is equivalent to the statement that the frequency of the periodic oscillation depends on .sr and s2. Hence one can assume that Q=f

f

0;

m=o n=O

w 00 = *02

%l” 9

(21)

where the frequency Sz is as yet unknown. Upon making use of the transformation equation (18) becomes Q=f"+CO;f-ol,&,f2+(CC2E,+~,E2)f3

=

r = Or

0,

(22)

where ( )’= d( )/dr. Substituting the expressions (20) and (21) into equation (18) and equating like powers of sr and E= gives Ey&;:f;

+fo

=

0,

w;:f;o

+f10

=

-(2%o/~o)f,"

+

(d4)f,'

EyE2:f;)1

+fo1

=

-(2~o,/~o)f6

-

(~,l~it)fo39

E:.E;:f;o

+f2()

=

-(~10/~0)2fd' +

+fo2

=

-(~ol/~o)=fo

%&2:f;l

+f11

=

-(2~,o/aJf61

-

-

-

(2~o,l~o)f;, -

(2@2o/~o)f6

(3~2l~;)fo'fio~

(2~o,/~o)f,", -

(2c+$)f,f,,

(~,/d)fo">

(2~,o/~o)f;,

(%/&wJf10

&y&f;2

+

-

-

(2~02bo)f; -

(3a2/4)fii2f01

(2Wll/~O)f6' -

-

(3~3/~~)fo"f&~ -

2~,0(~01/49fd

(3~3/~i?)fo2flO~

(23)

The initial conditions are assumed to be fk.

=

0

at

z = 0,

m,n = 0,1,2 ,...,

which can be done by including a phase angle in r, because the system is autonomous.

(24)

PANEL ON A NON-LINEAR FOUNDATION

511

7-ll--l

20

40

60

80

n/tAl, Figure 3. R/w, as a function of A. s1 = 004; s2 = 0.05.

The solution of equations (23) under the conditions (24) is f. = A cosr,

A = constant.

(25)

Following the procedure of reference [7] one obtains oio/oo w&00

= &/&$A2,

= &Jw;)*A*

oo&o

- &(az/w;)%4”,

(26, 27)

= &/o$A*, wo2/00 = -&(cx&;)%~

ml 1/% = -i&/W;)(C13/W%‘43

-

i%("2/o~)(a3/w;)

(28, 29) (30)

A4.

Introducing the notation Cl = a,/~&

c2 = a&i&

c3 = L7,/0;

(31)

and inserting equations (26H30) into equation (21) gives an analytical expression for the fundamental frequency Sz of the panel: s1 =

o,[1+&&2A2+&2&3A2+&:{~c$42-$C;A4} -E:&c~A4-&lE2{~c1C3A3-#JC2c3A4}]+

The vibrational mode shape, to a first approximation,

(32)

0(&3).

is

w(y;t) = [A c0s(52t+~)-&,{~c,A2C0S(SZt+c#J)+&4*C0S2(Qt+~) -&C*A3COS3(SZt+~)}+E2&C3A3COS3(S2t+~)](1-COS/I~y)

+ O(2).

(33)

The variation of the frequency ratio Q/o, with amplitude A for large deflections (h = 0.1 m, = 0.05) is shown in Figures 3 and 4. From these figures one can observe that the non-linearity due to the

L = 2.5 m, E = 6 x lo5 t/m2, ti = 0.07 t/m2, v = 0.30, k: = 150, .sr = h/R, E* = k,/k,

I .6

I

I

I

I

I

I

I.2

0.4

/ 0.c

,_

1

l.co2

I

1.004

1.006 I.008

sl/w, Figure 4. ajo0 as a function of A. cl = 0, c2 = 0.05.

512

C. MASSALASAND N. KAFOUSIAS

foundation is always of a hardening type; on the other hand the type of the geometrical nonlinearity of the panel depends on the values of ci Cl].

5. CONCLUSIONS

The non-linear vibration and “snap-through” phenomenon of a relatively long shallow cylindrical panel resting on a non-linear foundation have been studied. The formulation of the problem was based on the assumption of constant stress cY at all points of the middle surface. This assumption is reasonably accurate for a simply supported panel [S, 81. The mathematical analysis is based on the Bubnov-Galerkin method and a multiparameter perturbation technique. For the fundamental frequency B of the non-linear vibration an analytical expression has been obtained.

REFERENCES

1. A. W. LENA and A. S. KADI 1971 Journal of Sound and Vibration 16,173-l 87. Curvature effects on shallow shell vibrations. 2. J. MAYER and B. WRENN 1967 Developments in Mechanics, Proceedings of the Tenth Midwestern Mechanics Conference 4,819. New York: Johnson Publishing Co. On the nonlinear free vibrations of thin cylindrical shells. 3. B. EL-ZAOUKand C. L. DYM 1973 Journal of Sound and Vibration 31,89-103. Non-linear vibrations of orthotropic doubly-curved shallow shells. 4. E. REIISSNER1955 The Ramo-Wooldridge Corporation Report AM 5-6. Non-linear effect in vibrations of cylindrical shells. 5. J. RAMACHANDRAN and P. A. K. MURTHY 1976 Journal of Sound and Vibration 41,495-500. Nonlinear vibrations of a shallow cylindrical panel on an elastic foundation. 6. A. WOLMIR1962 Biegsame Platten and Schalen. Berlin: VEB Verlag fur Bauwesen. 7. C. MASSALAS1977 Journalof Soundand Vibration 54,613-615. Fundamental frequency ofvibration of a beam on a non-linear elastic foundation. 8. B. KORBUT 1962 Proceedings of the 4th All-Union Conference on Shells and Plates, Erevan 5,24-31 October. On the stability of shallow cylindrical shells supported on the inner surface by an elastic foundation.

APPENDIX:

DEFINITION

initial co-ordinates of a middle surface point R radius of the middle surface of the panel h panel thickness L circumferential length of the panel v, w displacement components of the middle surface in the circumferential and radial directions respectively distributed external 4 uniformly load P mass density t time non-dimensional time X,Y

OF SYMBOLS

linear vibration frequency non-linear vibration frequency f amplitude of w displacement D Eh3/12(1 - v2) flexural rigidity E Young’s modulus Poisson’s ratio k,, k: foundation parameters dimensionless parameters i ;,I a( )/ax &I h/R < 1 < 1 -52 k,lk, coefftcients ai,cj,aj A initial amplitude V2V2 xz v4 biharmonic operator @ stress function 00

Q