Accurate free vibration analysis of clamped orthotropic plates by the method of superposition

Journal of Sound and Vibration (1990) 140(3), 391-411

ACCURATE

FREE

VIBRATION

ORTHOTROPIC

PLATES

ANALYSIS BY THE

OF CLAMPED METHOD

OF SUPERPOSITION D. J. Department

of Mechanical

(Received

Engineering,

GORMAN

The University of Ottawa, Ottawa,

Canada, K 1 N 6N5

18 April 1989, and in revised form 9 October 1989)

The Superposition Method is exploited for the first time to analyze the free vibration frequencies and mode shapes of fully clamped rectangular orthotropic plates. The method is found to work extremely well and excellent agreement is obtained when comparison is made between computed results and earlier reliable published data. Simple modifications required to adapt the method to orthotropic plate problems are described in detail. Accurate eigenvalues are tabulated for a wide range of plate geometries and vibration mode families with various levels of orthotropy.

1. INTRODUCTION Numerous publications have appeared in the literature in recent years related to the free vibration analysis of orthotropic plates. In a fairly recent paper by Marangoni et al. [l] a review of much of this literature is presented. In particular, reference is made to the work of Dickinson [2] which will be referred to at a later stage in this paper. lt is pointed out that the most popular method for analyzing rectangular plate problems which do not lend themselves to a single L&y type solution, i.e., problems involving plates which do not have at least one pair of opposite edges simply supported, is found to be the Rayleigh-Ritz energy method. It is characteristic of this method that one must choose a set of functions to represent the shape of the deformed plate and it is here, as pointed out by Marangoni et al., that researchers have followed quite different paths. Almost all of these approaches have taken their roots in earlier studies related to isotropic plates. Marangoni et al. have, in fact, utilized the Rayleigh-Ritz method to establish upper bounds for the free vibration frequencies of fully clamped plates and they have utilized the method of Bazely and Fox [3] to establish lower limits. Dickinson employed the sine series solution developed by Dill and Pister. The superposition method as developed by the author has now gained a strong position as a contender among techniques for the free vibration analysis of isotropic plates of rectangular, triangular and other geometries. The effects of point supports, elastic edge support, etc., are easily incorporated into the solution [4,5]. The method is quite unique in that no shape functions need be presupposed for the deformed plate, and the governing differential equation is satisfied exactly throughout the plate. Satisfaction of the boundary conditions depends on the number of terms utilized in the series expansions and any arbitrary degree of exactitude in the computed eigenvalues can be obtained by increasing this number of terms. Convergence is found to be rapid. It is natural to raise the question of whether or not this method would be suitable for the analysis of orthotropic rectangular plates. In this paper the method is applied in an 391 002:2-460X/90/150391

+21 $03.00/O

@ 1990 Academic

Press Limited

392

D. J. GORMAN

orderly and comprehensive manner to the fully clamped plate. It will be seen that the method works extremely well, and excellent agreement is obtained when comparison is made between computed results and the limited reliable data available in the literature. Accurate eigenvalues are computed and tabulated for a wide range of plate geometry and several levels of plate orthotropy. The objective of this paper is two-fold. It is desired to provide the reader with a clear understanding of how the method is applied to orthotropic plates and, also, it is desired to provide the designer with a reasonably wide list of accurate tabulated eigenvalues. These eigenvalues should also be of interest to other researchers who may wish to compare their computed results with those tabulated here. 2. MATHEMATICAL PROCEDURE TO THE GOVERNING DIFFERENTIAL EQUATION 2.1. SOLUTIONS All plates studied in this paper possess what is sometimes called “special orthotropy”: i.e., the principal directions of orthotropy coincide with the rectangular plate co-ordinate axis. This has been the type of orthotropy assumed in most of the preceding vibration studies. It has been pointed out by Leissa [6] that only three flexural rigidity parameters are required to characterize such plates and, in fact, it follows that only two parameters involving ratios of the above parameters are required in order to carry out a free vibration analysis. With symbols as described in the Appendix, one may write the plate governing differential equation as [6],

d4W(X, Y, t) +

D

x

3X4

2H

d4W(X, Y, t) + ax2ay2

a4w(x, Y, t)

D

Y

aY4

+P

a2w(x, Y, t) at2

Separating the time and space variables and non-dimensionalizing plate lateral displacement, one obtains

=

0.

(1)

the equation for

a”w(S,7) w4 In the analysis to follow the variables S and n are separated by expressing W([, q) in the form proposed by L&y as W(& a) =

$

m=1,2

(3)

Y,(n) sin mr5.

Substituting equation (3) into equation (2) gives (4)

Y!&7)+%Y~(71)+%Ym(7))=o, where the Roman superscripts indicate the order of differentiation co-ordinate 7. The coefficients CY,and a2 are given by (Y,= -2DHY+‘(m7r)’

and

with respect to the

a2= DXY~4{(m~)4-h4}.

(5)

With the plate parameters DHX and DHY as specified (see the Appendix) the quantities (Y, and a2 are available for conducting the vibration study. It is at this stage that the analysis differs slightly from that described for isotropic plates [4]. The four roots of the indicial equation obtained from equation (4) are *{(-a,f@=G$2}1’2.

(6)

CLAMPED

ORTHOTROPIC

PLATE

FREE

393

VIBRATION

First, consider the case when a: - 4a 22 0. It can easily be shown that this relationship is always satisfied for isotropic plates (DHY = DXY = 1). With z = a: - 4az, the solution for equation (4) is [4], for A+ a1 3 0, Y,,,( T,I)= A,,, sinh Pm77+ B cash Pm7 + C,,, sin ymv + Q,, cos ‘y,,,~

(7)

and, for &+ CY,~0, Y,( 7) = A, sinh Pm7 + B, cash Pm7 + C,,, sinh Y,,,v+ 0,

cash y,,,ym77,

(8)

where, & = i{&- a,}, and & = ${A+ a,}, or -f{&+ a,}, whichever is positive, and A,,,, B,, etc., are constants to be determined. Equations (7) and (8) are already well known solutions employed in connection with isotropic plate analysis. Next to be considered is the situation, which can be encountered in orthotropic plate analysis, when ai- 4~~s 0. The solution to equation (4) can now be formulated as follows. With the parameter z as defined above and the further parameters z, = --$a,, z2 := f&Y, z3 = tan - ’(zJz,), and z,= (z:+z:)“~, one can establish the quantities R = z4 sin (z,/2)

and

s =

z4 cos

(zJ2).

(9)

The solution for equation (4) then becomes Y,,,(71)= A, sin Rv sinh ST + B,,, sin Rv cash Sr] + C,,, cos Rv sinh ST + D, cos Rv cash ST.

(10)

It is the complementing of equations (7) and (8) with equation (10) that permits the superposition method to be extended to the analysis of orthotropic plates. 2.2..

DEVELOPMENT

OF EIGENVALUE

MATRIX

The eigenvalue matrix is constructed in a manner almost identical to that described earlier for the fully clamped plate [4]. Only a brief description will be provided here for the sake of completeness. It will be appreciated that there are three families of vibration modes for the fully clamped plate. These are modes fully symmetric about the plate central co-ordinate axis, modes fully antisymmetric about the same axis, and modes which are symmetric with respect to one axis and antisymmetric with respect to the other. Only a limited amount of discussion pertaining to the analysis of the fully symmetric modes will provided here. The analysis is carried out by means of the building blocks of Figure 1. In view of the symmetry only one quarter of the plate is analyzed as shown on the left hand side of the figure. Two small circles adjacent to a plate segment boundary indicate slip shear conditions: i.e., vertical edge reaction and slope taken normal to the boundary are everywhere zero.

b

Figure I. Building orthotropic plate.

blocks

utilized

in the free vibration

analysis

of the fully symmetric

modes of the clamped

394 The

D. J. GORMAN

solution for the first building block is taken in the form

F

W,(& 17)=

mvT

Y,(s)

COS-Tj-.

(11)

m=1,3.5

The edge of the block, 5 = 1, is simply supported and therefore equation (11) satisfies exactly the boundary conditions along this edge and the edge opposite it, as required of L&J~ type solutions. The edge, r] = 1, of the first building block has lateral displacement forbidden along it. It is subjected to a distributed dimensionless harmonic bending moment of circular frequency U, with a distributed amplitude given by Mb* -= a4

E, cos y.

;

(12)

m=l,3,5

The solution for the iunctions Y,,,(n) of equation (11) will be given by equations (7), (8) or (lo), depending on the values of (Y,and a2 of equation (5) where the quantity rnr is now replaced by mr/2. For the fully symmetric mode analysis discussed here all terms which are antisymmetric about the 5‘axis will be deleted. There remains only the task of enforcing the boundary conditions along the driven edge. This has been discussed in numerous publications. Formulation of the building block solution as discussed in reference [7] is the most desirable since it permits supression of overflow and underflow problems previously encountered in the computations. The expression for Y,(n) of equation (7) is written as

Y,(q) = Em &,m

cashPmq+

e

sinh P,,,

cosYm77

13m

sin yrn I ’

(13)

with a similar expression for the solution of equation (8). The solution for equation (10) takes the form Y,,,= E,{8,,,

sin Rq sinh Sn + &,,, cos Rq cash Sn}.

(14)

These solutions are obtained upon enforcing the boundary conditions at the plate driven edge. The second building block of Figure 1 has boundary conditions that are identical to those of the first block but lying along different edges. Because of symmetry the solution for the second is easily extracted from that obtained from the first. Construction of the eigenvalue matrix is achieved through following standard procedures [4]. The contributions of each building block toward slope, along the edges, n = 1 and 5 = 1, are expanded in trigonometric series as utilized in equation (11). It is required that the net slope contributions from the superimposed pair of blocks should be zero. This is enforced by requiring the net Fourier coefficient of each term in the above series to equal zero. The result is a set of 2K simultaneous homogeneous algebraic equations constraining homogeneous algebraic equations constraining the coefficients, E,, etc., where K is the number of terms utilized in the building block solutions. Eigenvalues are obtained by seeking out those values of A2 which cause the determinant of the coefficient matrix of this set of equations to vanish. Mode shapes are obtained by setting one of the driving coefficients equal to unity and solving for the remaining coefficients. The free vibration mode shapes can then be obtained. Finally, it will be appreciated that pairs of building blocks for analyzing the other two families of modes discussed above are easily obtained through slight modifications to the building blocks of Figure 1. For the fully antisymmetric modes, for example, one requires

CLAMPED

simple support conditions, and II axes.

ORTHOTROPIC

PLATE

FREE

395

VIBRATION

rather than slip shear conditions, to be satisfied along the 5

3. PRESENTATION OF COMPUTED RESULTS Before discussing the tabulated eigenvalues it is appropriate to study comparisons between data computed by the superposition method and earlier published data. We begin by examining the results of Table 1. Here we have a tabulation of computed fundamental mode eigenvalues for fully clamped square plates with various combinations of orthotropic parameters. The data of Dickinson and Marangoni were extracted from the paper of Marangoni [ 11. The results of Marangoni were obtained by the Rayleigh-Ritz method. Those of Dickinson were obtained by the series solution method of Dill and Pister. It is seen that there is remarkable agreement between all of the computed results. This agreement is even more impressive when one reflects on the vast differences in the Rayleigh-Ritz method and the superposition method employed to gain these results. It adds substantially to the confidence which one may place in the data. In Table 2 the first five computed eigenvalues for a square orthotropic plate are tabulated. They are from the same sources as those of Table 1. Again, agreement is seen to be excellent for all five modes. While no delineation between the various types of modes was listed in the other sources, such delineation was immediately obvious when the superposition method was employed. The pairs of letters in the column to the right of the table indicate the mode types. The letters S and A indicate symmetric and antisymmetric:, respectively. The first mode is, of course, fully symmetric about the plate central axis. The second mode is symmetric about the x (or 5) central axis and antisymmetric about the y (or 7) axis. The third mode is antisymmetric about the x axis and symmetric about the y axis. This order is to be expected for these two later modes in view of the given stiffness parameters. Had these parameters been equal to 1-O (the isotropic case) the eigenvalues for the above two modes would have been identical. TABLE

Comparison of computedfundamental

DH.X

DHY

2.0 2.0

2.0

2.3 1.0

1.0 0.5 0.5

Dickinson

Marangoni

9.925

et al.

Superposition

method

9.925

9.927

11.41 13.89 10.60

11.41 13.89 10.60

TABLE

Comparison

1

mode eigenvalues A2 for a clamped square orthotropic plate

11.41 13.89 10.60

2

of computed eigenvalues A2 for a clamped square plate (DHX

= I/ 1.543,

DHY = l/4+310) -Mode ---

1 2 3 4 5

Dickinson 11.87 19.51 28.58 33.28 33~88

Marangoni 11.87 19.50 28.58 33.28 33.87

et al.

Superposition 11.87 19.51 28.58 33.28 33.88

method

Mode type s-s S-A A-S s-s A-A

D. J. GORMAN

396

It is always prudent to make some data comparisons for non-square plates in order to increase confidence in the analysis. Such a comparison is made in Table 3 with results extracted from the work of Marangoni et al. [ 11. Their data comes from the lower portions of the third and fifth segments of their Table 4 (a/b = 1.5 and 2.0). Again, agreement is seen to be very good, with their upper limits being identical or very close to values computed by the present method. TABLE

3

Comparison of computed eigenvalues A2 for clamped orthotropic rectangularplates of various aspect ratios

Marangoni et al. \

@ = (b/a)

DHX

DHY

Mode

Upper limit

Lower limit

1.5 1.5 2.0 2.0 1.5 1.5 2.0 2.0

l/2*0 l/2*0 l/2*0 l/2*0 ,1-o 1.0 1.0 1.0

1.0 1.0 1.0 1.0 l/2*0 l/2*0 l/2.0 l/2.0

1 2 1 2 1 2 1 2

A* = 6,201 A2 = 11.84 A*=5.876 A2 = 9.727 A*=7.201 h*=8.825 A2 = 6.308 A* = 6.255

6.191 5.876 7.186 6.299 -

Superposition Method 6.201 11.84 5.848 9.731 7.201 8.826 6.308 6.257

It should be pointed out that all of the data computed here was prepared with a view to obtaining four significant digits. Preliminary computer runs were performed with K, the total number of terms utilized in the series, equal to eight. Convergence tests indicated that this was almost sufficient to give the required accuracy. Nevertheless, 14 terms were utilized in all final passes. Experience indicates that this should be quite adequate for four digit accuracy. We now examine the computed and stored eigenvalues of this paper. They are stored in Tables 4-7. The first four eigenvalues are made available for all plate configurations and orthotropic stiffness ratios studied. Plate aspect ratios are varied from 1 to 3 for the fully symmetric and fully antisymmetric mode data. It will be appreciated that studies for these families of modes with 4 less than 1 would be redundant. For the symmetricantisymmetric modes the aspect ratio is varied from l/3 to 3. A major decision which has to be made in any study of this type concerns the range of stiffness parameters for which data is to be provided. It is obvious that the range of these parameters must be restricted in order to keep storage requirements within acceptable limits. In this paper it was decided to provide data for stiffness parameters DHX, DHY and their inverses, taking on the values 1, l-5 and 2.0. It is hoped that this range of data will prove useful to the designer, recognizing of course, that interpolation and extrapolation will have to be relied upon to obtain frequencies which lie between tabulated values or outside their range. We look next at some plotted results which help indicate how much confidence can be placed in data obtained through interpolation. This data is found in Figure 2. The eigenvalues for the first four fully symmetric modes of a clamped square plate are plotted continuously as a function of the stiffness ratio DHX as it varies from l/2 to 2.0. The stiffness ratio DHY is held constant at 1.0. Heavy dots interrupting these curves indicate eigenvalues listed in the tables provided.

1 2 3 4

6.401 13.11 26.38 31-10

2-00 6.063 lOa 18.38 30.74

2.50

8.454 26.65 33.27 54.11

1.25

1.50

5.807 7.923 12-98 20,97

2.50

6.087 10.30 18.97 30.79

2.00 5.885 8.398 13.74 21.86

2.50

7.450 19.96 32.29 42-73

1.50 6-540 13.41 16.14 31.35

2.00 6.161 10.45 18.54 30.31

2.50

= l-5, DHX = 2.0)

6.591 14-71 30.54 31.28

(DHY

7.174 19.32 31.79 42.32

8.390 27-97 32.78 52.12

4

1.25

l-00

5.967 9.701 18.12 30.62

2.00

= 1.5, DHX = 1.0)

6.390 13-97 29-64 30.96

l-50

t#~ (DHY

6.905 18.52 31.34 40.92

10.45 35.15 39.07 67.08

7.259 19.95 31.86 43.92

1.50

C#J(DHY = l-0, DHX = l/1*5)

8-248 27.01 32.67 51.19

8.027 27-14 32.09 48-21

1.00

A2 = wa’dm,

(DHY = 1.0, DHX = 1.5)

l-25

10.28 34.26 40.15 63.83

1 2 3 4

1.25

4

eigenvalues,

1.00

1.00

Mode

Computed

TABLE

4

5.971 8.880 14.45 22.59

3.00.

5.785 7.432 10.99 16.51

3.00

5.731 7.060 10.32 15.67

3.00

5.899 8.534 14.12 22.60

3*00

I 2 3 4

1 2 3 4

Mode

modes of the clamped plate

7.577 23.10 31.95 46.02

1.00

9-476 33.90 34.07 60.08

l-00

6.602 16.29 31.06 35.98

7.495 2.368 31-63 44.41

6.211 11-18 20.95 30.95

2.50

6.201 12.43 25.84 30.78

l-50

5.876 8.859 15.95 26.95

2.00

5.755 7.412 11.59 18.39

2.50

= l-0, DHX = l/2*0)

6.647 14.77 30.26 31.39

2.00

6.691 16.12 31.28 34.74

1.25

1.50,

7.318 11.96 22.85 31.07

2-00

6.025 9.481 16.32 26.43

2.50

6.277 12.45 25.18 30-93

l-50

5.927 9.020 15.84 26.25

2.00

5.798 7.584 11.69 18.17

2-50

= 1.5, DHX = l/1.5)

7.034 17,54 31.79 37-14 C#J(DHY

7.841 23.28 32.54 49.62

1.25

qb (DHY = 1.5, DHX = 1.5)

l-25

l-00

l-50 7.731 22.78 32.38 48.28

(DHY

8.955 30.98 3344 56.44

11.41 35.50 46.22 71.55 4

1.25

1.00

c$ (DHY = l-0, DHX = 2.0)

for the first four fully symmetric

5.722 6.877 9.550 13.89

3-00

5-879 8.190 12.84 19.79

3-00

5.697 6.725 9.371 13-87

3.00

5.997 9.301 15-96 25.86

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1.25

1.25

1.25

1.25

9.831 7.894 33.24 26.17 39.30 32.00 59.05 47.38

1.00

5.741 7.141 IO*51 16,OO

2.50

6.219 11.62 22.62 30.91

1.50

6.004 9.162 15.19 23.99

2.50

5.906 8.658 14.57 23.56

2.00 5.780 7.408 10.99 16.59

2.50

DHX=1/1+5)

6.274 11.34 20.85 31.06

2.00

6.981 19-1s 31.39 4144

l-50 6.227 12.40 25.49 30-84

2.00 5.948 9.495 17.54 29,81

2.50

d, (DHY = l/1.5, DHX = 1.0)

7.340 6.579 20.78 14.77 31.88 30.87 44.87 31.25

I*00

6.917 16.20 31.75 33.24

1.50

cp (DHY=2.0,

9.046 7.627 3044 21.18 33.84 32.47 58.09 45.77

1.00

5.846 8.302 14.02 22-92

2.00

cf, (DYH = 2.0, DHX = 1.5)

6.114 11.15 22.00 30.76

1.50

(DHY = 1.5, DHX = l/2*0)

7.135 6.453 20.21 14.25 31.53 30.24 42.63 31.02

1.00

4

5,818 8.048 13.35 21.70

3-00

5.717 6,782 9-139 12.90

3.00

5,868 8.010 12.15 18.23

3.00

5-690 6.581 8.736 12.36

3.00

1 2 3 4

--

1 2 3 4

: 3 4

1 2 3 4

TABLE b-continued.

Il.37 34.65 47.93 69.03

1.00

6.056 9,826 17.37 28.54

2.00 5.871 8.158 12.84 19.89

2.50

7.301 18.39 32.24 38.19

1.50

6.483 12.67 23.82 31.33

2.00

6.135 10.06 17.20 27.46

2.50

1.25

1.50 8.817 7.577 31.78 23.10 32.85 31.95 53.91 46.02

1.25

ci, (DHY=l/l.S,

6.069 10.45 X9.81 30.74

I.50

5.734 7.001 9.930 14.65

2.50

6.520 15.65 30.94 31.14

2.00

6.117 10.92 21.09 30.79

2.50

DHX=l.S)

5.830 8.007 12.94 20.62

2.00

(DHY = 2.0, DHX = l/2+0)

8.188 24-20 33.18 52.67

1.25

4

6.507 13.64 27.37 31.25

l-50

d, (DHY = 2.0, DHX = 2.0)

7,018 19.62 31.71 37.56

1.25

6.946 6.348 18.23 13.11 31.48 26.91 40.15 30.99

1.00

9.925 34.87 35.05 64.71

l*OO

8.068 25.10 32.68 50.59

1.00

c$ (DHY = 2.0, DHX = 1.0)

5.927 9.020 15.84 26.25

3.00 -

5.687 6.506 8.400 11.53

3.00

5.956 8.658 13.63 20.76

3.00

5-778 7.300 10.45 15-26

3.00

1 2 3 4

1 2 3 4

1 2 3 4

9.240 3244 36.69 54.20

la0

9.350 33.01 35.93 56.48

12.36 35.01 54.61 73.84

6.821 12.18 24.11 30.97

2.50

5.919 9.633 18.48 30.53

2.00 5.773 7.798 13-04 21-48

2.50

7.880 25.86 32.04 47.49

1.50 6.635 16.05 31.17 34.91

2.00 6-169 11.70 23.49 30.80

2-50

= 1/2-O, DHX = 1-O)

6.326 14.13 30.70 30.82

l-50

= l/ 1.5, DHX = l/2*0)

6.800 16.59 31.43 35.55

2.00

DHX=24)

7.503 24-34 31.50 43.90

I.25

6.709 17.77 31.05 38.87

I.50 6.079 11~50 23-68 30.65

2*00 5.856 8.856 16.23 27-72

2.50

C/J (DHY = 112.0, DHX = l/l-S)

1.25

l-00

l-SO

8.129 26.45 32.51 so-17

d, (DHY

6.843 18-95 31.12 39.75

8.002 28.09 31.77 46.94

f

l-25

C$J(DHY

9.651 33.67 36.53 59.72

1.25

l-00

12.71 36.00 55~22 77.71

la0

C#J (DHY=lj1*5,

5-757 7.578 12.35 20-11

3-00

5.954 9.480 17.40 2544

3-00

5.707 6.934 10.25 15.87

3aO

6.034 9.895 17.99 30.11

3-00 1

1 2 3 4

1 2 3 4

2 3 4

7.211 21.63 31.42 42.45

8.656 32.20 32.33 51-30

8,478 31.85 31.96 49.32

la0

13.89 36.46 62.95 80.30

7.075 21.28 31.18 40.92

l-25

c$ (DHY

10.30 33.89 41.34 62.80

1.25

1.50

5.832 S-403 14.70 24.58

2.50

5.983 10.09 19.47 30.61

2.50

6.946 18.23 31.48 40.15

2.00

6.348 13.11 26.91 30.99

2.50

DHX = 2-O)

6.308 13.51 28.71 30.86

2.00

DHX = 1.0)

6.024 10.64 21.08 30.63

2*00

DHX=l/l.S)

6.449 15.64 30.84 34.90

1.50

5.962 IO-35 20.70 30.54

2*00

5.792 8.166 14.34 24.18

2.50

= 112.0, DHX = l/2.0)

8.504 29.67 32.63 51.97

1.50

= l/2.0,

7-201 21.38 31.45 42.54

(DHY

8.292 29.53 32.11 49.33

LO.60 33.49 44.73 62.81 d,

1.25

l-00

la0

6.552 1598 31.01 35.29

1.50

ct, (DHY = l/2.0,

1.25

(DHY=l/l.S,

1.00

4

5.716 7-13s 11a 17.64

3.00

6-069 10.45 19-81 30.74

3-00

5.836 8.392 14.58 24.29

3-00

5.744 7.324 11.38 18.03

3.00

1 2 3 4

2 3 4

1

1 2 3 4

1 2 3 4

l+IO

20.30 36.47 54*41 63-97

23.83 SO-38 57.32 81-14

21-57 16-12 31.28 34.74

25.52 23.10 31.95 46.02

32.54 68.42 73.64 114.1

18-83 28.84 45.14 53.49

2.00 17.52 23.76 34.04 48.30

2.50

19.53 12-45 25.18 30.93

l-50 17.61 9-020 15.84 26.25

2-00

2.50

16.77 7.584 11.69 18-17

2-50

16-39 19.59 25.49 34.28

DHX=l-0)

17.02 22.43 32.30 46-63

2.00

26.27 52.92 61.73 91-94

l-25

22-88 41-65 58*08 70-08

l-50 19.55 30.35 46.65 54.48

2-00 18.02 25-06 35.71 49.80

2-50

i#J (LmY = l-5, Lmx = 2-O)

1-2s

l*OO

l-00

IS-57 29.20 47-85 52.94

l-50

r#J (DHY=l-5,

1-2s

F

21-85 40.15 56.43 69.48

1.50

= l-0, DHX = 1.5)

TABLE 5

17.21 22.15 29.70 39.63

3-00

16.34 6.877 9.550 13.89

3-00

16.06 18.16 22.02 27.84

3-00

16.84 21-08 28.11 37.92

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

22.67 44-53 56.80 77.59

l+O

29.25 64.19 64.45 101.9

1GO

22.04 44.34 55.58 74.59

l-00

34.94 69.71 84.20 121.2

l-00 23.60 4544 58-43 79.64

l-50 19.83 32.05 51-28 54.62

2*00 18.16 28.92 38.23 52.89

2.50

17-84 26.45 42.18 52.21

1.50

16-64 20.91 29.03 41.15

2.00

16-15 18.64 23.40 30.70

2.50

19-74 33.19 54.17 55.70

1.25

18.61 27.43 41.19 53-39

2.00

18.27 27.25 42.57 52.81

l-50

16.91 21-63 29.86 41.63

2.00

1.5, DHX=

21.28 36.94 56.17 61-27 d, (DHY=

24.04 46.55 59.01 81.50

l-50

16.34 19.21 24.42 31.53

2.50

l/l+)

17.41 23.05 31.92 43.88

2-50

(DWY = 1.5, DHX = 1.5)

1.25

4

19-20 32.53 53.34 55.97

1.25

t$ (DHY = l-0, DHX = l/2.0)

27.52 59.03 62.36 95.69

1.25

f$ (DHY = 1.0, DHX = 2.0)

for the first four fully antisymmetric modes of the clamped plate

a, (DHY = 1.0, DHX = l/1.5)

25.07 51.76 59.50 86.59

l-25

c#J (DHY

1.00

31.25 65.34 73.40 108.0

,

Computed eigenvalues, A2 = oa’m,

16.04 17.96 21.32 26.22

3.00

16.78 20.69 26.88 35.23

3.00

15.91 17.52 20.58 25.33

3.00

17.28 22.64 31.18 42.82

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

29.19 61.70 70.21 99.38

1dM

22.06 41.29 56.53 72.32

1.00

28-18 59-26 63.78 98.77

1-00

21.10 39.63 55.17 69.96

1.00 16.55 20.27 27.01 36.90

2.00

18.11 26.21 39.66 52.74

1.50 16.86 21.22 28.56 39.88

2.00 17.86 25.37 31.09 38.13

16.31 19.01 23.59 30.06

23.42 48.81 56.99 79.73

l-25

2.00 17.99 26.71 42-21 5244

1.50

20.55 37.48 54.62 66.19

16.95 22.11 31.52 45.24

2.50 16.43 19.79 25.98 35.18

3.00

3.00

16.75 20.48 26.23 33.80

3.00

15.89 17.36 20.02 24.00

3.00

2.50

l/1*5)

17.35 22-68 30.79 41.49

2.50

(DHY = l/1*5, DHX = 1.0)

19.45 31.43 51.06 54.05

1.25

DHX=

C#J(DHY=2.0,

2.00 18.49 26.69 39.06 63.33

1.50

20.97 35.21 56.02 56-72

23.50 43.70 58.75 74.36

l-25

4

16.11 18.34 22.38 28.40

2.50

Q, (DHY = 2.0, DHX = 1.5)

18.76 29.79 48-89 51.16

1.50

= 1.5, DHX = l/2.0)

17.60 24.84 37.70 52.12

(DHY

l-25

4

1 2 3 4

-

1 2 3 4

2 3 4

1

1 2 3 4

19.40 29.46 44.20 54.51

22.50 39.60 57.90 64.77

17.48 23.99 35.24 51.17

1.50

1.25 26.51 58.70 60.20 90.98

34.01 66.79 85.21 116.4

17.95 24.61 34.37 46.98

2.50

16.09 18.18 21.85 27.17

2.50

2.00 19.15 30.81 50.47 53.63

1.50 22.67 44.53 56.80 71.59

17.66 24.76 36.99 52.24

2.50

DHX = 1.5)

16.51 19.93 25.93 34.58

2.00

DHX=1/2.0)

2.00

C#J(DHY = l/1.5,

18.53 28.31 44.94 53.06

1.25

16.74 20.57 26.72 35.10

2.50

DHX=2.0)

17-53 23.57 33.18 46.27

2.00

DHX =l.O)

I.50

(DHY=2.0,

19.31 30.15 47.25 54.08

1.50

cj (DHYz2.0,

25.61 49.58 61.39 85.28

1.25

4

21.17 36.85 55.97 61.48

1.25

1.00

20.61 36.62 54.96 63.16

1.00

31-26 67.61 67.87 110.2

1.00

24.70 49.29 59.54 85-71

1.00

c#a (DHY=2.0,

16.91 21.63 29.86 41.63

3.00

15.87 17.28 19.73 23.30

3.00

17.17 21.90 28.92 37.93

3.00

16.32 18.69 23.22 29.05

3-00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

26.98 58.77 64-83 90.87

1-w

27.87 60.84 64.92 95-14

36.55 67-18 95.57 121.2

21-87 44.86 55.07 73.59

1.25

t#~ (DHY

1.25

l-50 20.23 34.41 54.79 57.54

2.00 18.53 27.15 41.72 52.98

2.50

DHX=2-0)

16.76 21.83 31.83 46.79

2-00 16.20 19.07 24.84 33.85

2.50

19.45 32-66 53-76 55.29

2.00 17.80 25.72 39-71 52.30

2.50

19.42 34-38 53.28 61+KI

1.50 17-33 24.66 38-68 51-69

2*00 16.52 20.66 28.91 41.45

2.50

= 1/2-O, DHX = l/1.5)

23.45 48.51 57.15 79-98

1.50

= 1/2-O, DHX = l-5)

18.17 28.69 48.13 52.35

1.50

= l/1.5, DHX = l/2-0)

24.61 50.59 58.91 84.80

t#~ (DHY

19.84 36.25 53.59 65.16

1.25

1.00

23.36 50.89 56.15 78.70

l-00

29.27 63.22 67.15 101-O

38.22 71.52 97.91 125.3

C#J(DHY

1.25

1.00

t#, (DHY=l/l-5,

16.14 18.73 23.98 32.20

3-00

16.99 22-17 31.51 45.03

3.00

15.93 17.74 21-38 27.20

3-00

17.38 23.33 33.28 47-18

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

TABLE 5-continued.

24-61 56.60 56.79 82.58

l-00

41.23 73,24 109.9 126.3

l-00

34.17 62.70 78.60 105.5

1.00

2546 57.95 58.17 86.15

l-00

1.25 1.50 16.46 20.13 27.26 38.04

2.50

20.45 39-62 53.84 68.46

1.25

,#I (DHY

30.91 64.05 74-38 106.0

1.25

1.50 18-21 28.14 45.06 52.52

2.00

17.05 22.82 33.66 49.53

2.50

20.61 36.62 54-96 63.16

2.00

18.53 28.31 44.94 53.06

2.50

18.50 30.77 52.47 53.41

1.50

16.87 22.71 34.39 51.26

2.00

16-25 19.49 26.21 36.73

2.50

= 112.0,DHX = l/2*0)

25.56 55.25 59.36 87.70

1.50

= 1/2-O, DHX = 2.0)

21.13 40.64 54.86 71.41

c$ (DHY

24.45 53.80 57.45 82.90

l-25

c#J (DHY = 1/2-o, DHX = 1-O)

17.18 23.57 35-63 51.63

2.00

l/l-5, DHX = l/l+)

19.00 31.90 53.11 54.82

(DHY=

21-10 40.88 54-75 71.22

4

15.96 17-96 22.16 28.95

3-00

17.48 23.99 35.24 51.17

3.00

16.48 20.18 27.25 37.87

3.00

16.10 18.45 23.02 30.10

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

20.31 50.36 54-91 88.49

1.00

18.04 37.36 52.50 71.91

1.00

17.43 34.88 51.76 67.03

1.00

19.64 49.26 53.91 83.83

1.00

l-50

16.17 22.87 37.54 50.70

18.41 37.86 53.04 71.48

l-25

1.50

15-94 20.12 28.40 40.68

2.50

15.97 20.34 28.97 41.77

2.00

2.50

15.76 18.47 23.87 31.95

2.50

15-65 17-67 22.18 29.53

DHX=l.O)

15.81 19-19 26.80 38.92

2.00

17.43 31.05 52.06 54.65

l-50 16.50 24.23 37.83 51.12

2.00 16-09 21.06 29-95 42.32

2.50

= 1.5, DHX = 2.0)

1646 24.57 40.28 51.02

C#J(DHY

16.98 28.99 51.52 51.87

1.25

16.28 23.03 36.21 50.85

2x)0

= l-0, DHX = l/1.5)

C#J(DHY=1.5,

16.58 26.92 48.85 51.06

1.50

17.06 29.60 51.60 5346

(DHY

1.25

4

l-I.92 36.42 52.39 70.89

1.25

TABLE 6

15.87 19.33 25.62 3437

3.00

15.65 17.49 21-18 26.71

3.00

15.58 16.91 19.85 24.68

3.00

15.77 18.60 24.25 32.67

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

-

18-67 41.05 53.17 76.71

20.86 55.15 56.13 92.00

17.21 31.82 51.67 60.69

1.00

19.21 44-36 53.72 80.64

1.00

16.30 24.57 43.03 50.79

16.95 31.19 51.32 61.28

16.48 25.31 43.34 51.01

l-25

1.50

16.11 21.45 31.52 46.05

2.50

15.71 18.33 24.47 34.60

2.00

15.59 17.14 20-71 26.73

2.50

1.50

15.92 19.81 27.09 37.51

2.50

16.12 21.96 34.13 50.67

1.50

15.79 18.85 25.30 35.31

2.00

15.65 17.51 21.45 27.60

2.50

= 1.5, DHX = l/1.5)

16.23 22.37 33.70 49.89

2+MI

= 1.5, DHX = 1.5)

16.95 28.00 48.02 51.55 C#J(DHY

17.71 33.73 52.29 62.47

l-25

16-55 25.04 40.80 51.14

2.00

DHX=2.0)

= 1.0, DHX = l/2.0)

15.99 21.26 33.44 50.52

c,f~(DHY

l-25

1.00

1.50 17.57 32.96 52.13 61.00

$J (DHY

1.25

(DHY=l.O,

l-00

4

D,, f&r the first four modes symmetric about the 5 axis and antisymmetric the q axis of the clamped plate (4 2 1 .O)

g5 (DHY = 1.0, DHX = 1.5)

Computed eigenvalues, A2 = wa’Jp/

15.57 16.83 19.46 23.58

3&-.I_

15.76 18.43 23.51 30.79

T 3.00

15.54 16.55 18.85 22.73

3.00

15.89 19.54 26-52 36.62

3.00

about

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

18.79 46-47 52-82 77.17

1.00

17.10 30.17 51.62 55-35

1.00

18.98 41-68 53.61 79.11

1.00

16.78 28.64 51.25 53.17

1.00

15.95 20.53 30.57 46.20

15.59 17.02 20.12 25.14

2-50

15-91 19-64 26-41 35-82

2.50

16.09 21.48 32.28 48.36

1.50 15-78 18.67 24.52 33.35

2-00 15.64 17.43 21.06 26.59

2.50

=2.0, DHX = l/1*5)

16.21 22.03 32-37 46.77

2.00

DHX=1.5)

15.70 18.05 23.24 31.56

2.00

17.33 33-97 51.68 66.25

1.25

16-64 27.49 50.41 51.11

1.50 16.03 21-51 33.63 50.57

2.00 15.79 19.02 26.27 37.85

2.50

Cp (DHY = l/1-5, DHX = 1.5)

16.43 2446 40.30 50.99

l-25

S#J(DHY

16.89 27.16 45-06 51.52

1.50

(DHY=2*0,

17.60 32-29 52.24 57-71

l-25

4

16.22 23.25 38.35 50.75

1.50

(DHY = 1.5, DHX = 1/2-O)

1.25

4

15.66 17-78 22.51 30.21

3.00

15.57 16.79 19.25 23.00

3.00

15.76 18.34 23.12 29.79

3-00

15.53 16.49 18.53 21.84

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

20.26 54.17 55.80 87.62

1.00

16.69 27.26 48.60 51.21

1.00

20.02 47.22 54.77 86.64

1.00

17.88 35.25 52-42 66.80

1.00

TABLE 6-continued.

16-41 23.93 37.93 51.01

1.50 15.95 20.08 27.94 39.29

2.00

2.50 15.75 18.35 23.36 30.63

DHX=l-0)

18.21 40.10 52.51 73.03

1.25

16.07 20.85 29.13 40.32

2.50

15.69 17.91 22.60 29.93

2.00

15.59 16-95 19.81 24.30

2.50

17.21 31.82 51.67 60.69

l-50

16.33 23.96 39.66 50.88

2.00

15.97 20.58 30,24 45.00

2.50

l/1.5, DHX = 1.5)

15.92 20.15 29.02 42-63

l-50

C#I (DHY=

16.18 22.56 35.78 50.73

1.25

16.47 23.81 36.24 51.10

2.00

= 2.0, DHX = l/2.0)

17.35 30.03 51.17 52.03

C#I (DHY

18.27 36-15 52.98 65.94

1.50

(DHY = 2.0, DHX = 2.0)

1.25

4

16.91 27.87 48.05 51.49

l-25

c$ (DHY=2-0,

15.79 18.85 25.30 35.31

3.00

15-53 16.45 18.37 21.37

3.00

15.87 19.21 25.15 33-18

3.00

15.64 17.43 20.89 25-95

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

18.04 42.73 52.00 71.50

1.00

20.84 54.43 61.64 91.49

1.00

17.19 34.67 51.41 65.02

l-00

21.63 55.51 63.75 96.93

1.00

(DHY

15.60 17.32 21.56 28.94

2.50

16.85 31.22 51.17 62.30

l-25

16.31 25.38 46.30 50,76

1.50 15.86 20.18 30.83 48.14

2.00 15.68 18.13 24.42 34.66

2.50

= l/2.0, DHX = l/1*5)

15.99 21.01 31.96 48.94

2.50

c$ (DHY

!6.39 24.85 42.83 50.90

2.00

17.36 33.89 51.73 66.49

1.50

= l/2.0, DHX = l-5)

15.73 18.72 26.19 38.71

2.00

18.48 43.47 52.63 74.97

1.25

1.50

16.04 22-31 37.33 50.54

C#J(DHY

16.40 26.41 49.22 50.83

1.25

(DHY

= l/l-5, DHX = l/2.0)

16.15 22.02 33.72 50.72

2-50

4

16.63 26.18 44-87 51.18

2*00

17.77 35.61 52.22 69.41

1.50

= l/l-5, DHX = 2,O)

19.04 45.40 53.33 79.21

1.25

4

15.59 17.15 20.99 27.71

3.00

15.80 19.08 26.31 37.75

3.00

15.54 16.64 19.31 24.01

3.00

15.91 19.85 27.80 39.74

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

17.42 37.82 51.50 66.78

1.00

22.36 55.84 70.56 101.6

l-00

19.21 51.15 52.99 80.11

1.00

17.74 39.01 51.88 69.31

1.00 15.67 17.90 23.24 32.20

2.50

1.50

15.80 19.33 27.60 40.98

2.50

16.50 28.13 50.87 54,71

1.25

(DHY

16.09 23.30 40.85 50.56

1.50

15.75 19.10 27.81 42.42

2.00

15.61 17.49 22.37 30.00

2.50

= l/2.0, DHX = l/2.0)

16.18 22.56 35.78 50.73

2.50

4

16.69 27.26 48.60 51.21

2.00 17.96 38.07 52.30 71.16

1.50

= l/2.0, DHX = 2.0)

16.07 22.17 36.13 50.59

2.00

19.39 49.35 53.49 81.60

1.25

15.83 19.70 28.89 43.71

2*00

= l/2.0, DHX = l-0)

16.74 29.09 51.15 55.60

C#I (DHY

17.52 36.63 51.76 67.68

1.25

1.50

= 111.5, DHX = l/1.5)

16.24 24.16 42.15 50.73

c$ (DHY

16.72 29.15 51.11 56.10

l-25

C#J(DHY

15.55 16.73 19.75 25.22

3.00

15.92 20.15 29.02 42.63

3.00

15.67 17.95 23.27 32.12

3.00

15.58 17.03 20.43 26.24

3.00

1 2 3 4

1 2 3 4

19.64 49.26 53.91 83.53

1 2 3 4

20.31 50.37 54.91 88-49

18.04 37.36 52.50 71.91

1.00

17.43 34.88 51.76 67.03

1.00

1.00

Mode 8,326 13.12 21.38 32.99

9.347 17.26 30.49 39.79

9.901 24.96 29.94 45.01

1.50

2.50 5*856 10.59 18.82 25-92

2.00 6.871 14.80 26.72 28.12

7,265 15.48 27.52 28.88

2.00 6-153 11.19 19.65 26.45

2.50

= l-5, DHX = 1.0)

9.391 24.25 28.60 41.50

(DHY

2.50

2.00

= 1.0, DHX = X/1*5)

1.50

(DHY

11.81 26.57 42.13 50.48

1.50

(DHY = 1.0, DHX = l-5)

15mJ 37.07 44.74 69.86

1.25

12.19 2744 41.68 51.51

1.50 9.504 17.96 31.43 38.67

2.00 8.326 13.65 22.22 33.97

2.50

7.714 11.36 17.26 25.38

3.00

5.599 8.947 14.70 22.77

3.00

5.374 8.412 14.01 22.01

3.00 -

7.818 10.98 16.55 24.48

3.00

1 2 3 4

1 2 3 4

20.86 55.15 56.13 92-00

1 2 3 4

17.21 31.82 51.67 60.69

1.00

19.21 44.36 53.72 80.64

1.00

16.95 31.19 51.32 61.28

1.00

1.00

Mode

11.84 27.17 33.76 48.12

1.25

l.O/$

13.90 35.87 39.11 62-62

1.25

X*0/+

11.62 27.04 33.42 46.35

1.25

1*O/d,

15.71 37.42 51.10 72.00

1.25

X-O/+

6.257 14,26 23.23 27.62

8.826 23.75 25.02 37.62

8461 16.77 30.18 33.56

2.00

7.322 12.49 20.97 32.33

2.50

9.004 24.08 24.80 38.94

6.340 14.55 22.60 27.97

2.00

5.226 10.24 18.71 21.66

2.50

= l-5, DHX = 1,‘l.S) 1.50

(DHY

11-11 26.23 36.30 50.27

1.50

5.221 10.00 18.38 22-49

2.50

= 1.5, DHX = l-5)

2.00

1.50

(DHY

9.500 14.43 22.74 34.34

2.50

= 1.0, DHX = l/2.0)

10.55 18.58 31.83 45.87

13-04 27.87 4844 51.78 (DIfY

2.00

= 1.0, DHX = 2.0)

1.50

(DHY

4.68 1 7.979 13.74 21.17

3.00

6.741 10.23 16.03 24.11

3.00

4.734 7.797 1344 21.48

3.00

8.967 12.26 17.89 25.85

3.00

for the$rst four modes symmetric about the 5 axis and antisymmetric about the 7 axis of the clamped plate (4 6 1.0)

1*0/4J (DHY = 1.5, DHX = 2.0)

12.71 32.48 34.62 54.40

1.25

l.O/@

12.14 30.68 33.89 SO.23

1.25

1.0/d

14-48 36.13 44.57 66-24

1.25

l-O/#

Computed eigenvalues, A *2 = wb2m,

TABLE 7

2 3 4

1

1 2 3 4

2 3 4

1

1 2 3 4

8-801 22.65 23.99 37.59

l-50

1.25

13.72 35.06 42.63 61.09

18.79 46.47 52.82 77.17

11-16 25.54 40.72 49.33

l-50

4-693 9.729 18.22 18.81

6.763 12.15 20.75 28.58

2.50

4.880 10% 18.60 19.17

2.50

8.889 16.34 29.39 38.97

2-00 8.008 12.33 20.35 31.85

2.50

DHX = 1.0)

6.055 14.42 20.23 27.89

2-00

DHX=1/1.5)

7-978 16.50 29.94 30.04

2.00

l/1.5,

(DHY=2.0,

10.73 26.05 33aO 50.14

1.50

l.O/c#~ (DHY=

11.68 25.22 33-69 47.02

l-25

l-00

17.10 30.17 51.62 55.35

l-00

13.60 35.71 36.07 60.71

18.98 41.68 53.61 79.11

1.O/d

l-25

5.821 14.06 19.68 26.86

2-50

= 2-0, DHX = l-5)

8.519 21.76 23.63 35-52

2.00

= 1.5, DHX = l/2.0)

l-50

l*O/c#a (DHY

11.38 24.06 33.33 44-64

1.25

1.00

16.78 2864 51.25 53.17

1.00

l.O/c#~ (DHY

7.580 10.32 15.61 23.40

3.00

4.292 7.750 13.60 18.62

3.00

6.132 9.821 15.76 23.91

3.00

4.146 7448 13.23 18.36

3.00

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

20.26 54.17 55.80 87.62

l*OO

16-69 27.26 48.60 51.21

1.00

20.02 47.22 54.77 86.64

l-00

17.88 35.25 52.42 66.80

la0

l-50

15.28 36.50 51.70 70.92

1.25

1.014

11.26 22.42 33.27 43-74

1.25

8.929 17.64 31.22 34.50

2-00

= 2.0)

7.669 13.24 21-94 32.95

2.50

5.710 10.94 19.49 23.39

2.50

5.589 13.96 17.64 25.40

4.404 9-585 16.66 18.14

2.50

12.77 27.05 49.60 50.76

l-50

10.53 17.95 30.92 47.63

2.00

9.635 14al 21.96 33.39

2.50

= 111.5, DHX = 1.5)

8.358 19.93 23.56 34-41

2*00

= 2.0, DHX = l/2.0)

1.50

(DHY

?a0 6.891 15-29 24.59 28-76

(DHY

1l-74 27.21 37.85 51.36

l.O/#

9.623 24.83 27.28 43.25

1.50

1.0/c/1 (DHY

14-62 36.88 41-18 67.57

l-25

12.49 30.04 34.52 52.93

l-25

l*O/c#~ (DHY = 2.0, DHX = 1.0)

9.198 12-01 17.27 25.00

3.00

3.817 7.264 13.13 16.15

3-00

7.003 lo-87 16.92 25.13

3.00

5.110 8.639 14.50 22.63

3-00

1 2 3 4

1 2 3 4

1 2 3 4

16.69 37.89 59.38 72.29

21.63 55.51 63.75 96.93

18.04 42.13 s2*00 71.50

1.00

20.84 54.43 61.64 91.49

12.99 34.23 39.38 56.01

1.25

1.0/d,

16.04 36.85 57.95 71.14

1.25

6-8.56 14-54 27.70 27.79

2-00 5.924 10.39 18.61 27.10

2.50

13.67 27.51 51-03 56.08

I.50 11.60 18.61 31.32 49.38

2.00 lo-78 14.83 22.51 33.77

2.50

(DHY = l/2.0, DHX = I-5)

9.264 23-93 29-23 40-56

1.50

11*03 15.49 23-46 34.86

2.50

l/1*5, DHX = l/2+0)

11-95 19.43 32.37 50.53

2-00

10.45 24.70 37.78 48.29

1.50 8-255 15.51 28.52 36.37

2.00 7.428 11.54 19.49 30.96

2.50

(DHY = 112.0, DHX = l/1.5)

11.96 30.98 33.56 48.78

17.19 34.67 51.41 65.02

1.00

1.25

1.014

14.20 28.48 52.15 57.10

1.50

(DHY=

1.00

1.0/b,

1.25

la0

l*O/i#J (DHY = l/ 1.5, DHX = 2.0)

7.054 9.581 14.76 22.51

3.00

1040 12.96 17.95 25.49

3.00

5.499 8.289 13.74 21.68

3.00

10.57 13.49 18.78 26.50

3.00

1 2 3 4

1 2 3 4

-

TABLE ii-continued.

1.00

17.42 37.82 Sl*SO 66.78

1.00

22.36 55.84 70.56 101.6

1.00

19.21 51.15 52.99 80.11

1.00

17-74 39.01 51-88 69.31

I

12.28 33.68 34.48 51.08

1.25

l-O/c$

17.62 38.34 66.63 72.56

1.25

l-O/4

l-50 6.689 11.08 19.21 30.78

2.50

9.680 24.10 32.91 43.29

1.50

2.00

12.37 16.48 24.14 35.34

2.50

7.406 14.82 27.93 31.57

2.00

6.552 10.77 18.83 30.37

2.50

DHX = 112.0)

13.20 20.24 32.89 50.89 = l/2.0,

15.23 29.05 52.50 64.61

1.50

8.925 12.96 20.75 32.12

2.50

DHX = 2.0)

9.731 16.82 29.67 44.45

(DHY = l/2.0,

11.84 25.86 45.99 49.51

2.00

= l/2*0, DHX = 1.0)

1.50

(DHY

2.00

DHX = l/1-5)

7.595 15.17 28.32 31.91

= l/1.5,

9.937 24-48 33.51 45.03

(DHY

(DHY

14.29 35-30 47.69 64.73

1.25

l.O/c$

12.57 34.06 35.31 53.20

1.25

1*0/6, .

6.170 8,750 14.03 21.87

3.00

11.96 14.61 19.61 27.11

3.00

8.550 11.06 16.11 23.75

3.00

6.270 9.017 14-39 22.26

3.00

CLAMPED

ORTHOTROPIC

PLATE

FREE

409

VIBRATION

D KC

65-

366 46-

6035-

2. 0 DHX

Figure 2. Plot of eigenvalues of the first four fully symmetric of orthotropic parameter DHX.

modes

of clamped

square

plate as functions

It is seen that for DHX greater than 1.0 the relationship between this parameter and the eigenvalues is quite linear. For values of DHX less than I.0 the plotted curves do not generate straight lines and interpolation using the tabulated data will not give as high a level of accuracy. Numerous families of curves similar to those of Figure 2 have been plotted for other ranges of the stiffness parameters. The results obtained invariably had a character quite similar to that of Figure 2. In fact, somewhat similar families of curves have been plotted in reference [l]. It appears that from the designers’ point of view fairly reasonable values for free vibration frequencies can be obtained by interpolation between the data points provided here, or extrapolating slightly beyond them. It was these observations that led to the decision to tabulate data over the ranges for which it is provided in this paper. Eigenvalues for the isotropic plate are tabulated in Table 8. They were obtained by setting the parameters DHX and DHY equal to unity. They are provided here for the convenience of analysts who may need them as limiting cases in order to perform interpolation.

4. SUMMARY AND CONCLUSIONS It is found that the superposition method works extremely well in free vibration analysis of clamped rectangular plates with special orthotropy. In fact, it is obvious that it will work just as well for all the other plates with combinations of classical boundary conditions, just as it has already worked so well for the isotropic plates [4]. The same can be said of orthotropic plates resting on point supports, elastic edge supports, etc. The eigenvalues tabulated herein represent, to the author’s knowledge, the most extensive compilation of accurate eigenvalues ever prepared for this problem. It is expected that they will play a strong role in meeting the needs of designers and will provide a

8

1 2 3 4 5

8.996 32.98 32.98 55*01 77.26

1 2 3 4 5

18.35 41.25 52.63 74.08 85.15

1.00

I.00

Mode

17.13 31.08 51.59 58.75 64.77

1.25

15.53 25.78 44.61 51.06 59.33

1.50 16.00 20.82 30.92 46.34 SO.56

2.00 15.77 1869 24.86 3444 47.38

2.50

modes)

2.50

QI(symmetric-antisymmetric

2.00 5.911 8.854 15.37 25.34 30.59

6~751 16.63 31.32 36.05 40.3 1

1.50

modes)

&144 11.19 21.81 30.81 35.59

7412 22.31 31.88 45.32 50.52

1.25

C#I (symmetric-symmetric

15.66 17.61 21.72 28.16 39.96

3.00

5.799 7,685 11-99 18.76 27.87

3.00

1 2 3 4 5

1 2 3 4 5

Mode

18.35 41.25 52.63 74.08 85.15

1.00

27.05 60.57 60.57 92.84 114.7

1.00 19.95 34.02 54.36 57.5 I 67.79

1.50 17.77 25.20 37.97 52.34 55.99

2+IO

13.13 34.80 36.89 57.18 69.41

1.25

10.43 25.20 34.66 48.30 49.14

1.50

7.956 15.83 29.09 32.59 39.87

2.00

~.O/C#J (symmetric-antisymmetric

22.34 43.25 56.50 76.39 71.51

1.25

6.952 11.67 19.94 31.56 31.69

2.50

modes)

16.85 21.36 29.23 40.49 51.45

2.50

modes)

6.465 9.523 15.08 23-03 31.22

3.00

16.38 19.38 24.65 32.27 42.23

3.00

except for symmetrk-

4 (antisymmetric-antisymmetric

Computed eigenualues for the jirst Jive modes of isotropic plate mode families, A 2 = oa*m, antisymmetric modes with (p < 1.0, where h”2 = wb*m

TABLE

5

E

8

I-

?

CLAMPED ORTHOTROPIC PLATE FREE VIBRATION

411

valuable reference against which other researchers can compare their results. The author recognizes that with the accelerated use of fiber reinforced composites in industry there may be need for design data well outside of the range reported. He is in possession of all of the necessary software and would be prepared to discuss how these needs could be met, through private communication. REFERENCES 1. R. D. MARANGONI, L. M. COOK and N. BASAVANHALLY 1978 International Journal ofsolids and Structures 14, 61 l-623. Upper and lower bounds to the natural frequencies of vibration of clamped rectangular orthotropic plates. 2. S. M. DICKINSON 1969 American Society ofMechanical Engineers, Journal of Applied Mechanics 36, 101-106. The flexural vibration of rectangular orthotropic plates. 3. N. W. BAZELY and D. W. FOX 1964 Applied Physics Laboratory, The Johns Hopkins University, Report T6-609. Methods for lower bounds to frequencies of continuous elastic systems. 4. D. J. GORMAN 1982 Free Vibration Analysis of Rectangular Plates. New York: Elsevier North Holland. 5. D. J. GORMAN 1989 American Society of Mechanical Engineers, Journal of Applied Mechanics 56, 893-899. A comprehensive study of the free vibration of rectangular plates resting on symmetrically distributed uniform elastic edge supports. 6. A. W. LEISSA 1978 Shock and Vibration Digest 10(12), 21-35. Recent research in plate vibrations 1973-76: complicating effects. 7. D. J. GORMAN 1988 Journal of Sound and Vibration 125, 281-290. Accurate free vibration analysis of rhombic plates with simply supported and fully clamped edge conditions.

APPENDIX: NOMENCLATURE dimensions of quarter plate in x and y directions, respectively a, b Q> D.y,H orthotropic plate flexural rigidity parameters DHX

DHY DXY K W

W x, P ” 4 P 5 :2 h*2

= H/D, = H/D, = D,/D, actual number of terms used in truncated series expansion plate lateral dispiacement amplitude of plate vibration co-ordinate axes, coincide with 6 and 77 axes, respectively circular frequency of plate vibration plate aspect ratio, equals b/a mass of plate per unit area dimensionless distance along x axis, equals x/a dimensionless distance along y axis, equals y/b = wa2m, eigenvalue = ob2m, alternate formulation of eigenvalue

of L&y type solutions