MECHANICS RESEARCH COMMUNICATIONS Voi.ii(6),397-400, 1984. Printed in the USA 0093-641.3/84 $3.00 + .00 Copyright (c) 1985 Pergamon Press Ltd

A NOTE ON VIBRATING CIRCULAR PLATES CARRYING CONCENTRATED MASSES

~ . A . A . L a u r a , P . A . L a u r a , G .Diez and V . H . C o r t i n e z I n s t i t u t e o f A p p l i e d M e c h a n i c s and U n i v e r s i d a d N a c i o n a l d e l S u r , 8111 P u e r t o B e l g r a n o Nava l B a s e , A r g e n t i n a

(Rec~ved 31 October 1984; accepted for prin~6 December 1984]

I n t r o d u c t i o n

T h i s n o t e d e a l s w i t h t h e a p p r o x i m a t e d e t e r m i n a t i o n o f t h e f u n d a - m e n t a l frequency of vibration of circular plates elastically restrained against rotation and carrying a concentrated mass at its center. In the case of clamped plates the results are in excellent agreement with values predicted by the exact solution. The present algorithmic procedure allows for the calculation of the fundamental frequency of vibration for any value of the flexibility coefficient in a very simple yet quite accurate fashion.

Variational Solution

When t h e p l a t e e x e c u t e s n o r m a l m o d e s , i t s b e h a v i o r i s d e s c r i b e d by t h e d i f f e r e n t i a l s y s t e m

DV4W(r) - ph~2W(r) - M w 2 6 ( x ) 6 ( y ) W ( r ) = 0 ( l a )

W(a) = 0 ( l b )

dW (a) = - ,D(d2W + ~ ( , e l d r d r 2 r d r r=a

w h e r e ¢: f l e x i b i l i t y c o e f f i c i e n t d e f i n e d by e q . ( l c ) .

An e x a c t s o l u t i o n o f (1) i s c e r t a i n l y p o s s i b l e [1] b u t i n g e n e -

r a l r a t h e r i n c o n v e n i e n t f rom t h e p o i n t o f v i e w o f t h e d e s i g n e r

o f m e c h a n i c a l s y s t e m s .

I t w i l l be shown h e r e t h a t a v e r y p r a c t i c a l s o l u t i o n c a n be a t -

~C)7

398 P.A.A. LAURA, P.A. LAURA, G. DIEZ, and VoH. CORTINEZ

tained using Galerkin's method and minimizing then the fundamen-

tal frequency coefficient with respect to a parameter V which is

introduced as an exponent in the polynomial coordinate function

which approximates the modal shape [2,3].

Making

W(r) ~ Wa(r) = AI(~rY+Br2+I) (2)

where in order to satisfy (Ib) and (Ic):

1 2 1 1 + { ' ( 1 + p ) ]

a Y y[1+~'(y-1+~)]-211+¢~'(1+#)]

B _ y [ 1+~ ' ( y - 1 + # ) ]

2 a y I l + q b ' ( y - l + # ) ] - 2 [ l + @ ' ( l + p ) ]

~D a

and substituting in (la) one obtains an "error" or "residual

function" c(r). Galerkin's method requires now that e(r) be or-

thogonal with respect to Wa(r) over the entire domain.

Once this straightforward integration is performed one obtains:

= ~ a 2 _- ~oo ~ ° ' )oo

= _ay (y-- 2)2 ( a a2Y/y__z_ 1 + 2 BaY__~+2/y + 2a Y/_____yy- 2)

a 2 a2Y/y+1 + B2a4/3 + I +4~BaY+2/y+4 + 4aaY/y+2 + Ba 2 +

where (Mp = p~a2h) (3)

S ince (3) i s an upper bound of the e x a c t f r e q u e n c y c o e f f i c i e n t ,

by r e q u i r i n g :

oo _ 0 (4)

one obtains an optimum value of ~oo for the coordinate function

selected as approximate modal shape. Since ~oo is a rather com-

plicated function of y it is more expedient to obtain (~oo)min

numerically assigning values to the parameter y.

VIBRATING PLATES WITH CONCENTRATED MASSES 399

Numerical Results

Table I shows a comparlson of results between exact frequency

coefficients and those obtained in the present investigation in

the case of rigidly clamped plates (~=0). The agreement is excel

lent for all the situations considered.

Table 2 depicts frequency coefficients available in the open li-

terature for simply supported plates [4] and eigenvalues deter-

mined in the present study (~=0.30). Results obtained in the in-

vestigation reported here are considerably lower and accordin-

gly, more accurate.

An inherent advantage of the present approach is its simplicity

and the fact that it can be implemented on a microcomputer or

even a pocket programmable calculator.

TABLE 1 - (Rigidly Clamped Plate)

"M/M p

0 0.05 0.10 0.20 0.50 1 . 0 0

~oo

[1]

1 0 . 2 1 4 9 . 0 1 2 8 .111 7 .00 5 .00 3 . 7 5

P r e s e n t S t u d y

10 .22 9 .01 8 .11 6 . 8 7 5 .02 3 .75

Y

3.3 2.9 2.7 2.4 2.2 2.10

TABLE 2 - (Simply Supported Plate)

M/M P

0 0.10 0.20 0.50 1 . 0 0

~oo

[4]

4.935

3.767 2.945 2.291

P r e s e n t Study

4 . 9 3 4 . 2 3 3 . 7 5 2 .92 2 .25

3 .4 2 .9 2 .6 2 .6 2 .2

400 P.A.A. LAURA, P.A. LAURA, G. DIEZ, and V.H. CORTINEZ

References

I. A.W.Leissa, NASA SP-160. Vibration of Plates (1969). 2. R.Schmidt, J.A.M., 49, 639 (1982). 3. C.W.Bert, Industrial Math. 34(I)65 (1984). 4. P.A.A.Laura, A.Arias and L.E.Luisoni, J.Sound Vib. 45(2)298

(1976).