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TECHNICAL NOTE FREE VIBRATIONS OF RECTANGULAR PLATES OF EXPONENTIALLY VARYING THICKNESS AND WITH TWO FREE EDGES P. M.

BELLES*,

M. J.

MAURIZI,*

P. A. A. LAURA+ and H. C.

SANZI:~

*Departamento de Ingenieria, Universidad Nacional del Sur, 8000- Bahia Blanca, Argentina: ~Institute of Applied Mechanics (CONICET), 81100 - Bahia Blanca, Argentina; SENACE, SA, 100l-Buenos Aires, Argentina Abstract--This note is concerned with the transverse vibrations of the structural system described in the title, By using the classical plate theory and employing the Rayleigh-Ritz method and a finite element algorithmic procedure it is shown that the fundamental frequency coefficients obtained are in very good agreement. The fundamental frequency coefficient is determined using the optimized Rayleigh-Ritz method and the results are in good engineering agreement with the wdues obtained by means of a finite clement code.

INTRODUCTION

PLArES of non-uniform thickness are commonly used in ocean engineering systems-ship structures, offshore platforms, etc. The present study deals with the determination of the fundamental frequency of vibration of rectangular plates of exponentially varying thickness clamped at two adjacent edges and free at the other two (see Fig. la). It is assumed that the plate thickness varies according to the functional relations l~(x.y) = h,, e ~'""e

(1)

~'';'

which for moderate values of the parameters ¢x and 13 constitutes a good approximation for the case of linear variation of the thickness (see Fig. lb). In view of expression (1) it is convenient to approximate the fundamental mode shape using exponential coordinate functions (Laura and Cortinez, 1989) of the form w ( x . . O ~- w . ( x a , )

= ~, (~ - e~,'.")~(~ - e ~ ' ' " ) ~

(2)

where ~ and ~ are parameters which allow for optimization of the eigenvalue when using the Rayleigh-Ritz method. The determined eigenvalues possess good engineering accuracy and the procedure is quite simple and straightforward. APPROXIMATE

SOLUTION

OF TttE

The maximum strain energy of the plate is given by U ...... = 2

(~ ;~x- + av:/ 405

PROBLEM

406

P.M.

~

(

BEILI:S et at

Free edge

ree edge

¸f,~ffffJfffffff.,

"

X

(a) Plate system

~, ~ = 0 . 2

~

c==0.5

(b) Variation of the thickness at y=O Fro. l, V i b r a t i n g s t r u c t u r n l s y s t e m ,

¢'~2"'' 2

-2(l-~x)[

1 ~

[OzW OaW - ( ~1 IlcLrdv 3x ~ ay:

,#xi~y: 11

(3)

"

where

D(x,y)

E h~(x,y )

= 12(1-~x 2) =

D,,-

D,,e

'"-''e

3~,,,/,

(4a)

Eh~ 12(1"~:)

(4b)

while the maximum kinetic energy is calculated by means of the expression T A B L E I.

COMPARISON OF F U N D A M E N [ A I

FREQUENC't ( O E I . F I ( I E N I S O f [ I l l S[RUf_71LIRAL SYS1EM SItO~VN IN F i t ; .

CORRESPONDING 1 0 r i l e CASE OF UNIFORM I I I I C K N E S S [ h ( x , y )

~

]1.,1

Finite e l e m e n t results

Size of mesh

a/b 2/3 1

1.5

Present study

Lcissa 11973)

5 x 5

4.985

--

~.999

6.942

6.718

11.4911

11.2211

....

5. 106

I0 x

Itl

6.867

2(I ~ 2O

1

407

Technical Note T A B L E 2.

F U N D A M E N I A L FREQUENCY COEFIqCIENI OF IHE SYSIEM SHOWN IN DIFFERENI VALUES OF o~ ,'~ND [~

FI(;. 1,

FOR

/ ptt,, ~ " = \; o,, ,~,,a

[3

a/b

O. I 0.2 O.5 O.I 0.2 0.5 0.1 0.2 0.5

2/3

1

1.5

O.1

4.738 4.479 3.860 6.493 6.269 5.764 1(}.661 10.503 l/). 173

¢,

0.2

4.668 4.402 3.763 6.269 6.032 5.495 10.078 9.905 9.536

h ( x , y ) We(x,y)dxdy .

T ...... = 2 pt°2

Substituting (2) in

c~

(3)

cx

0.5

4.521 4.238 3.546 5.7(~4 5.495 4.862 8.686 8.468 7.479

(5)

and (5) and requiring that

u ...... = r , .....

((,)

one finally obtains the desired fundamental frequency coefficient /oh!,

t~,, = ~ D,, m, ,a 2 = Q , , ( y , .Ye) . Since { ~

(7)

is an u p p e r bound, by requiring O~ ay~

= O~]~

= 0

~S)

a7~

one obtains an optimized value of the frequency coefficient. In o r d e r to simplify the procedure the p a r a m e t e r y~ was taken as equal to y : = y in the present study. It is important to point out that expression (2) satisfies only the essential b o u n d a r y conditions. NUMERICAL RESULTS Table 1 depicts fundamental frequency coefficients for the uniform thickness case o b t a i n e d using the p r o c e d u r e previously described and those available in the literature (Leissa, 1973). G o o d engineering agreement is observed for the values of the ratios a/b - 2/3, 1 and 1.5. For other values of a/b which differ considerably from unity the present p r o c e d u r e yields u p p e r bounds which are rather high. Also shown in Table 1 are fundamental eigenvalues d e t e r m i n e d using the well-known S A P 90 finite element code. It is observed that the results a p p r o a c h the exact value from below. Table 2 shows f u n d a m e n t a l frequency coefficients of the system under study for

408

P.M. B~IL~S ~,t ul.

different values of o~ a n d !3. U s i n g a (20 x 20) finite e l e m e n t mesh, a value of [ ~ equal to 6.415 was o b t a i n e d for the case of a square plate w h e r e ~x ~= [3 - !~. I. This result is a b o u t 1% lower t h a n the f r e q u e n c y coefficient shown in T a b l e 2 ( ~ , ~493). All c a l c u l a t i o n s were p e r f o r m e d for ~ ( P o i s s o n ' s tatio) equal to 0.31i. Acknowledgements--Mrs Patricia Belies wa,', supportcd b) ~ gram irom Comisi0n dc In~c~tigaci~ncs

Cientfficas de la Provincia de Buenos Aires (CI(')~ The prcscm study has been par'dally sp,)nsorcd b'~ CONICET Research and Development Program (PID 3|)0|)5/~8) REFERENCI!S V . H 1989. Use of exponential coordinate lunctions containin~ an ~q~fimizalion parameter when solving mechanical vibrations problem~. ,L ,S'oumt Vihr, 12, 52(~-522. L~fSSA, A.W. 1973. The free vibration of rectangular plates. J. Srmml Vihr. 31, 257-293

LAURA, P.A.A. and C O R ~ Z ,

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