Applied Acoustics 19 (1986) 17-24

Vibrations of Thin Elastic Plates with Point Supports: A Comparative Study

J. C. Utjes,* P. A. A. Laura,# G. S~nchez Sarmiento* and R. Gelos~

*Empresa Nuclear Argentina de Centrales El6ctricas SA, LN Alem 712, 1001, Buenos Aires (Argentina)

tlnstitute of Applied Mechanics, 8111, Puerto Belgrano Naval Base (Argentina)

SUniversidad Tecnolbgica Nacional, Facultad Regional Bahia Blanca (Argentina)

(Received: 13 September, 1984)

S UMMARY

Point-supported structural elements are commonly found in several fields of engineering:from civil and naval engineering applications to electronic systems (mainly point-supported printed circuit boards). The present paper deals with vibrating plates of different shapes and different support arrangements. Natural frequencies are determined using a finite element algorithm and compared with results available in the literature where possible. It is also shown that in the case of clamped and simply supported plates of regular polygonal shape with a central clamping support simple polynomial expressions yield accurate values for the fundamental frequency coefficient when the Ritz method is used.

INTRODUCTION

A recently published paper 1 dealt with the determination of fundamental frequency coefficients for the rectangular plates shown in Fig. 1. It stated:l , . . . i t is a rather unfortunate fact that no general or definite conclusions can be drawn about the absolute accuracy of the eigenvalues...'.

17

Applied Acoustics 0003-682X/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain

18 J. C. Utjes et al.

Support F ~ F

ss ~ c

(A) (BI

a) Rectangular Plates With Two Adjacent and SS-C and a Corner Support-

F

C x

(c)

Edges SS, C

Yl Y~ Point Support I SS / C /

+ ss

~. I b- SS x C

b) Rectangular, Circular and Regular Polygonal Plates With Central Support-

Fig. 1. Structural elements considered in the present investigation.

The present paper deals with the determination of the first three frequencies of vibration of the plates shown in Fig. l(a), making use of a finite element algorithm.2

Furthermore, eigenvalues for circular and regular polygonal plates are also determined when central point supports are present in the structural system (Fig. l(b)). It is assumed that these point supports restrict displacement and rotation in the case of square, pentagonal, hexagonal and heptagonal plates.

DETERMINATION OF NATURAL FREQUENCIES BY THE 'SAPIV' ALGORITHM 2

Figure 2 shows the element distribution for rectangular and square plates (Cases A, B and C of Fig. l(a)).

Vibrations of thin elastic plates with point supports 19

- - - ~ - X - - - D . - X

Square Configuration : 100 elements and 121 nodes.

Rectangular Configuration: 72 elements and 01 nodes.

Fig. 2. Application of the finite element method.

Each node possesses three degrees of freedom: transverse displacement and two rotations of axes parallel to the plate edges.

Isoparametric, quadrangular, four-node elements were used and the nets were generated automatically using the KUBIK program. 3

In order to obtain an idea of the accuracy of the procedure it was decided to determine the fundamental frequency coefficient of simply supported and clamped square plates (Table 1). Very good engineering agreement is achieved.

TABLE 1 Fundamental Frequency Coefficients for Simply Supported and

Clamped Square Plates

Fundamental Jrequenc y coefficient

x/ph/D o~1 la 2

Relative error 'SAPIV" Exact (%)

Simply supported 19.79 2n 2 0.25 Clamped 36.32 35.987 0-9

20 J.C. Utjes et al.

ANALYSIS OF RECTANGULAR, CIRCULAR AND REGULAR POLYGONAL PLATES WITH CENTRAL SUPPORTS

The case of circular plates (simply supported and clamped) has been treated by the Ritz method using polynomial coordinate functions following the procedure explained in ref. 1 for the case of rectangular plates with a corner support. As shown in Table 3, the fundamental eigenvalues are extremely high when compared with results available in the literature.

When dealing with polygonal plates, the analytical determinations followed the procedure explained in ref. 4 (the polynomial coordinate functions are multiplied by (x 2 + y2) in order to ensure null displacement and slope at the central support).

The procedure is quite simple and straightforward but too lengthy to be included here.

NUMERICAL RESULTS--DISCUSSION

Table 2 presents a comparison of results for the configurations shown in Fig. 1. The following conclusions can be drawn from Table 2.

(1) The results obtained by Cox 5 (Methodology (1): Finite Differences) are consistently lower than the values obtained by other researchers (Cox dealt exclusively with square plates 5).

(2) Polynomial coordinate functions, coupled with the Ritz method,1 yield the highest values for all cases considered. However, the results obtained for square plates possess reasonable agreement with the values obtained by means of other methods. Since a one-term approximation allows for the determination of a simple frequency equation, one may conclude that this procedure is ideal, from a practical viewpoint, when the plates analysed (Fig. 1) are square or almost square, since other complicating factors such as orthotropy, in-plane forces, concentrated masses, etc., can be easily taken into account.

(3) The results obtained by means of SAPIV are presumably very accurate. They are in excellent agreement with the results already available. 6

(4) Results obtained using SAPIV and those calculated by means of

Vibrations of thin elastic plates with point supports 21

TABLE 2 F u n d a m e n t a l F r e q u e n c y C o e f f i c i e n t s : C o m p a r i s o n o f R e s u l t s a n d F r e q u e n c i e s

C o r r e s p o n d i n g t o T w o H i g h e r M o d e s

[(1) F i n i t e d i f f e r e n c e s ; (2) m o d a l c o n s t r a i n t m e t h o d ; (3) p o l y n o m i a l s - - R i t z m e t h o d ; (4)

f i n i t e s t r i p m e t h o d ]

a/b Mode Method

r ¥

t b I

' , _ _ _ e . . . .

#=0"33 1~=0"30 IX=0"33 #=0"30

I

1

1 I I

/ / / / / / / / / V / / / / /

p=0"33 ~=0"30

1.0 1 S A P I V 9 . 5 9 9 9 . 7 0 0 15-22 15 .380 11-90 12-030

(1) - - 8"998 - - 13.683 - - - -

(2) 11.02 - - 17.42 - - 13.28 - -

(3) - - 10 ' 49 - - 19"57 - - 14"00

(4) - - 9"61 - - 15.17 - - - -

2 S A P I V 16"81 16"81 23"29 23"29 20"69 - -

(4) - - 17 .32 - - 23"93 - - - -

3 S A P I V 3 0 . 4 4 30 .44 39 .29 39 .29 - - - -

(4) - - 30 .60 - - 39 .40 34 .87 - -

0 .5 1 S A P I V 3-923 3 . 9 6 4 6 .465 6 .533 4 -997 5 .050

(i) . . . . . . (2) 3-95 - - 7.11 - - 5 .58 - -

(3) . . . . . .

2 S A P I V 10.25 10.25 14.12 - - 11.43 11.43

3 S A P I V 14-38 14.38 19,44 - - 15.88 15-88

2 1 S A P I V 15 .692 15-856 25"86 2 6 . 1 3 2 2 2 . 3 5 6 2 2 ' 5 9 0

(1) . . . . . . (2) 15.81 - - 28 -44 - - 22 .33 - -

(3) . . . . . .

2 S A P I V 4 1 . 0 0 4 1 . 0 0 56 .48 - - 4 8 ' 9 6 4 8 . 9 6

3 S A P I V 57 .52 57"52 77"76 - - 74 .20 74 .20

the modal constraint s method differ by at least 10 ~ except for Cases A (a/b = 0.5 or 2) and C (a/b = 2), where agreement is remarkably good.

(5) A very minor variation in the frequency coefficients is observed when the Poisson ratio varies from 0.33 to 0.30 (use of 'SAPIV' indicates a 1 ~o variation of the fundamental frequency corresponding to a 10 variation in the Poisson ratio).

(6) F rom the analysis of Tables 3 and 4 one concludes that the ratio of the fundamental frequency of a plate with central clamped support to the

22 J . C . Utjes et al.

TABLE 3 Fundamental Frequency Coefficient of Simply Supported and Clamped Plates with a

Central Clamping Support

Shape Boundary x / ph/ D o31 , a e condition

Present Present sludy study

(SAP1V) (Ritz polynomial)

Leissa 8 Blevins 9

Square

Square

Rectangular (a/b = 0.50)

Idem

Circular a: outer radius

Circular Pentagonal a:

side of the regular polygon

Pentagonal Hexagonal Hexagonal Heptagonal Heptagonal

Simply supported 50.18 90 52.60 (49.65)

Clamped 75-28 95 (74-49)

Simply supported 20.10 (19.89)

Clamped 32.84 . . . . . (32-49)

Simply supported - - 18 14.80

Clamped -- 29 22.70 Simply supported 30.76 35 - -

Clamped 45-54 49 - - Simply supported 20.40 23 . . . . . . Clamped 29.59 30 Simply supported 14.86 16 -- Clamped 20.98 21 --

Values in parentheses have been calculated for/1 = 0.33; for the remainder/~ = 0.30.

fundamental frequency of a plate without central support 7 is higher for simply supported plates than for rigidly clamped plates. In fact, this ratio is equal to 2.5 for a square plate, 2.9 for a circular plate and about 2.7 for pentagonal, hexagonal and heptagonal plates, while for clamped plates it is equal to 2 for a square plate, 2.2 for a circular plate and approximately 2.3 for pentagonal, hexagonal and heptagonal plates.

(7) In the case of plates of regular polygonal shape with central supports, the agreement between analytical and finite element results can be considered as reasonably good, especially for hexagonal and heptagonal plates.

Vibrations of thin elastic plates with point supports

TABLE 4 Fundamental Frequency Coefficients of Simply Supported and Clamped Plates of Regular Polygonal Shape: Comparison of Results

23

Order of the polygon

Values of f]l 1 = x/Ph/D °911 a2 (a: side of the polygon)

Simply supported Clamped

ReJl 4 SAP1V Ref 4 SAPIV

4 19.74 19.79 36.1 36.32 (2~Z 2) (35"987)

5 11.00 11.23 19.95 20"06 6 7.15 7-54 12.85 12.93 7 5.06 5"68 9"06 923

( ): exact value.

A C K N O W L E D G E M E N T S

The present investigation has been partially sponsored by Comisi6n de Investigaciones Cientificas (Buenos Aires Province).

R E F E R E N C E S

1. J. G. M. Kerstens, P. A. A. Laura, R. O. Grossi and L. Ercoli, Vibrations of rectangular plates with point supports: comparison of results, Journal of Sound and Vibration, 89 (1983), pp. 291-3.

2. K. J. Bathe, E. L. Wilson and F. E. Peterson, SAPIV: A structural analysis program for static and dynamic response of linear systems, University of California, College of Engineering, Berkeley, California, Report No. EERC 73-11 (1973).

3. S. Pissanetzky, KUBIK: An automatic three-dimensional finite element mesh generator, International Journal for Numerical Methods in Engineering, 17 (1981), pp. 255-69.

4. P. A. A. Laura, L. E. Luisoni and G. Sfinchez Sarmiento, A method for the determination of the fundamental frequency of orthotropic plates of polygonal boundary shape, Journal of Sound and Vibration, 70 (1980), pp. 77-84.

5. H. L. Cox, Vibration of certain square plates having similar adjacent edges, Quarterly Journal of Mechanics and Applied Mathematics, VIII (1955), pp. 454-6.

24 J.C. Utjes et al.

6. S. C. Fan and Y. K. Cheung, Flexural free vibrations of rectangular plates with complex support conditions, Journal of Sound and Vibration, 93 (1984), pp. 81-94.

7. P.A.A. Laura and R. H. Gutierrez, Fundamental frequency of vibration of clamped plates of arbitrary shape subjected to a hydrostatic state of in-plane stress, Journal of Sound and Vibration, 48 (1976), pp. 327-32.

8. A. W. Leissa, Vibration of Plates, NASA SP 160, 1969. 9. R. D. Blevins, Formulas jbr Natural Frequency and Mode Shape, Van

Nostrand Reinhold Company, New York, 1979.