Transverse vibrations of an elliptic plate with variable thickness

Journal ofSound

and Vibration (1985) 99(3), 379-391

TRANSVERSE VIBRATIONS OF AN ELLIFTIC PLATE WITH VARIABLE THICJKNESS B. SINGH Department

of Mathematics,

(Received

AND

D. K. TYAGI

University of Roorkee, Roorkee 247667, India

1 February 1984, and in revised form 20 June 1984)

Gale&in’s method has been used to obtain the frequencies of symmetric transverse vibrations of an elliptic plate with clamped edges and variable thickness. The procedure can be used to generate a sequence of approximations which may be truncated when the required number of frequencies have converged to the desired accuracy. The numerical values of the first four frequencies are reported correct to five significant digits for various values of the taper parameter and the ratio of the semi-axes. Mode shapes and the nodal ellipses have also been worked out for the second and third modes. The results for an elliptic plate with uniform thickness, circular plate with parabolically varying thickness and a circular plate with uniform thickness have been obtained as special cases.

1. INTRODUCTION

AND BASIC EQUATIONS

A lot of literature is available on the transverse vibrations of plates of various shapes, thickness varying according to different laws and different conditions satisfied at the boundary. Some important references are the publications by Leissa [ 11, Timoshenko and Woinowsky [2], and Soedel [3]. Unfortunately, very little has been reported for elliptic plates even for the simplest case of uniform thickness and clamped edges. Leissa [l] and McNitt [4] have given very rough estimates of the first two frequencies only. The aim of this paper is to obtain the frequencies of symmetric transverse vibrations of an elliptic plate with clamped edges and variable thickness. The procedure can be used to obtain a sequence of approximations by Gale&in’s method and this is continued until one finds that the required number of frequencies have converged to the desired accuracy. Here the results are reported for the first four frequencies correct to five significant digits for various values of the taper parameter and the ratio of semiaxes. The results for (i) an elliptic plate with uniform thickness (ii) a circular plate with parabolically varying thickness and (iii) a circular plate with uniform thickness have been obtained as special cases. The mode shapes and the nodal ellipses have also been numerically worked out for second and third modes. The basic equation governing the transverse vibration of a plate with variable thickness is [l, 21 V2(DV2w) - (1 - v)[D,,w,

-2&w,,+

D,w,,]+

$rw,, = 0

in R,

(1)

where R h, t, and

is the closed domain of the xy-plane occupied by the plate. As usual, w, v, 7, V2 are respectively the displacement, Poisson ratio, density (mass per unit volume), the variable thickness, time and the two-dimensional Laplace operator. The flexural rigidity D appearing in equation (1) is defined by D = Eh3/[12( 1 - v2)], 379 0022-460X/85/070379+ 13

%03.00/O

@ 1985 Academic

(2) Press Inc. (London)

Limited

B. SINGH

380

AND D. K. TYAGI

where E is the Young’s modulus of the material of the plate. For a plate with clamped edges the boundary conditions are w = 0,

awlan = 0

on dR for all r,

(3)

where n is the normal to the boundary. The variable thickness is taken to be of the form h(x, Y) = &f(x,

Y),

(4)

where a is a characteristic length, and h, is the non-dimensional thickness at a standard point, say (x,,, yO), which may be taken as the origin. Thus f(x,, y,,) = 1. For discussing vibrations one seeks a solution of the form w(x, y, t) = W(x, y) = W(x, y) exp (iwt).

(5)

Substituting equations (4) and (5) in equation (1) gives V’(FV2W)-(~-Y)(F~~W~--~F~~W~~+F~W~~)-A~~W/~~=O where F = f 3 and h is a non-dimensional

(7)

conditions on W are aW/an = 0.

w=o, The boundary

(6)

parameter defined by

A2=12(1-v2).ya2w2/(Eh;). The boundary

inR,

(8)

of the elliptic plate is defined by (see Figure 1) x2/a2+y2/b2=

1.

(9)

Y

Figure 1. Plate geometry

Introducing

the variable u defined by u = x2/a2+y2/b2-

1,

(10)

one finds that u varies from -1 at the centre of the plate to zero at the boundary. f(x, y), which controls the variation of thickness, is now assumed to be of the form f(x,y)=l-p(x2/a2+y2/b2)=1-P(l+u),

(11)

where p is a parameter known as the taper parameter. Note that equation (11) amounts to taking the same thickness along an ellipse u = constant. Further, in the special case of a circular plate f(x, y) reduces to f(x, Y) = 1 -Pr2/a2,

This corresponds

r2=x2+y2.

to the axisymmetric case of parabolic variation.

(12)

VARIABLE

THICKNESS

2. METHOD

ELLIPTIC

381

PLATE

OF SOLUTION

Since it is not possible to solve the problem exactly in closed form, the well-known approximate method due to Galerkin has been used, in which where one obtains a sequence of approximations { WN}zC1, where W, = f

Ajuj+‘,

Nal.

(13)

j=l

Note that for each integer N 2 1 W, satisfies the boundary conditions (8). The trial functions in W, are not the most general ones. The dependence of W, on u means only that the displacement is assumed to be the same along concentric ellipse u = const. This is particularly appropriate for symmetric vibrations analogous to these symmetric vibrations of a circular plate for which the displacement depends on radial distance only. Galerkin’s equations are

E,u

i+’dx dy = 0,

i= l(l)N,

(14)

R

where EN is the residual obtained by substituting expression for EN can be written as EN = i

ek(akw/auk)

equation

(13) in equation

- A*f(W/a”),

(15)

k=l

e~=-96~fz(~+$~+16f3[3($+$)+--$-p(l+u)],

a4

e,=l6($+$r,

2

K =L+L+b4

a4

(16)

2v

L=l+l+-

3a2b2’

(6). The

b4

a2b2’

(17)

After substituting the expression for EN in equation (14), one faces the tedious task of integration, and grouping the terms suitably to put the result in a simple non-dimensional form. Furtunately it is not difficult to derive the formula xpy4ur dx dy = (-l)‘aP+‘bq

(18)

+‘FFi;ri/p,

R

where p, q are even integers and p, q, r > -1. This helps in expressing all the integrals involved in a closed form. We omit these details here and only write the final result, c” (a,-h2bti)Aj=0,

i=l(l)N,

j=l

(19)

where a0 and b, by a,={12(i+l)(j+l)/(i+j+l)(i+j+2)(i+j+3)(i+j+4)} x(-1)‘+‘[/3L,(i+j+3-4~)(i+j+4)+6/3*) +{ijK,/(i+j-l)(i+j)}{(i+j+2-9P)(i+j+3)+36P2)(i+j+4)-60P3}]

(20)

382

B. SINGH

AND

D. K. TYAGI

b,={(i+j+4-/3)/(i+j+3)(i+j+4)}(-l)’+’,

(21)

(22,231 It is important to note that aV and b, are symmetric in the indices i and j. Solution of (19) is a standard eigenvalue problem. It leads to a characteristic equation

equation

involving

a polynomial

of degree hiN’,i=l(l)N,

N

in A’. The

N

values

whereh$N’
TABLE

J

i b/a

by

ifi
(24)

1

Values of AiN’, i=l,2,3,4;

B

are denoted

N=lO

1.0

0.8

0.6

0.4

0.2

-0.8

1 2 3 4

15.815 523873 114.04 199.50

20.501 68.549 147.86 258.67

31.450 105.22 226.98 397.11

64.266 215.16 464.26 812.32

244.74 819.89 1769.4 3096.1

-0.6

1 2 3 4

14.428 49.771 108.21 189.90

18.703 64.528 140.30 246.22

28.696 99.050 215.39 378.00

58.651 202.57 440.57 773.24

223.39 771.95 1679.2 2947.3

-0.4

1 2 3 4

13.032 46.569 102.15 179.88

16.895 60.378 13244

25.927 92.686 203.33

233.23

358.07

53.005 189.57 415.93 732.48

201.93 722.48 1585.3 2792.0

-0.2

1 2 3 4

11.629 43.246 95.805 169.35

15.076 56.071 124.22 219.58

23.141 86.081 190.71 337.65

47.324 176.08 390.13 689.12

180.34 671.13 1487.0 2628.8

0.0

1 2 3 4

104.216 39.771 89.104 158.18

13.246 51.567 115.53 205.10

20.337 79.173 117.38 314.90

41.605 161.97 362.88 644.22

158.59 617.41 1383.3 2455.7

0.2

1 2 3 4

8.7923 36.096 81.936 146.18

11.401 46.804 106.24 189.53

17.511 71.868 163.12 291.01

35.841 147.05 333.74 595.37

136.68 560.60 1272.2 2269.5

0.4

1 2 3 4

7.3561 32.141 74.116 133.00

9.5402 41.677 96.102 172.45

14.659 64-004 147.57 264.79

30.023 130.98 301.94 541.74

114.55 49944 1151.1 2065.2

0.6

1 2 3 4

5.9024 27.749 65.286 118.00

7.6563 35.984 84.654 153.01

11.771 55.271 130~00 234.94

24.128 113.14 266.02 480.72

92.125 431.49 1014.3 1832.7

0.8

1 2 3 4

4.4082 22.511 54.527 99.542

5.7193 29.193 70.706 129.07

8.7995 44.854 108.59 198.20

18.055 91.852 222.25 405.58

68.992 350.42 847.51 1546.3

VARIABLE

THICKNESS

ELLIPTIC

383

PLATE

to the fundamental mode, the next, AiN’, to The smallest value, viz. AiN), corresponds the second mode, AiN’ to the third mode, and so on. The major problem with Galerkin’s method is that although the lower frequencies are reasonably accurate, the higher ones are not so. But as N is increased to obtain the successive approximations, more and more of the lower frequencies converge. Thus the process may be truncated when the required number of frequencies have converged to the desired accuracy. We have carried out computations up to N = 10 for various values of p and the ratio ‘b/u. It is found that in all cases considered the first four of five frequencies converged to at least five significant digits. As /3 + 1, the convergence becomes slow. However, we have not encountered any difficulties up to /3 = 0.9. Comparisons have been made with results available in special cases. Table 1 gives the first four frequencies for ~=0.3, b/a =0*2(0*2)1*0 and p = -0.8(0.2)0+8. All the computations were carried on the DEC-2050 system at the University of Roorkee Regional Computer Centre. The CPU time for one set of values of V, b/a, /3 and N = l(l)10 is roughly 0.9 s.

3. SPECIAL 3.1.

ELLIPTIC

Putting

PLATE

WITH

UNIFORM

p = 0 in equations

CASES

THICKNESS

(20) and (21) simplifies

the expressions

for aij and b, to

b,=(-l)“‘/(i+j+3).

a,={12&ij(i+1)(j+1)/(i+j-1)(i+j)(i+j+1)}(-1)’+’,

(25) Leissa [l] and McNitt [4] have given results for N = 2 only, but their result for the second harmonic is not correct; actually for N = 2, the quadratic equation in A* is found to be (h2/12K0)2-306(A2/12Ko)/5+189=0, which gives A2=36K,,(51f2*)/5= 39.145 K,, 695.25 K,,, as compared with the Leissa and McNitt values of 39.218 & and 129.18 K,,. We have computed results for various values of b/u, which give K,, from equation (22). In the special case of a circular plate of uniform thickness K0 = 8/3. This case is separately discussed below. Table 2 gives the values of AiN’ for N = l( l)lO, i = l(1) N, b/a = 0.5, and Y = 0.3. This table clearly shows how the various frequencies converge. Up to N = 10 first four values have converted to five significant digits.

TABLE VuhesofA~N),N=1(1)10,i=1(1)N,/3=

1 2 3 4 5 6 7 8 9 10

28.048 27.746 27-743 27.743 27.743 27.743 27.743 27.743 27.743 27,743

0,

2

3

4

116.93 108.41 108.01 108.01 108.01 108.01 108.01 108.01 108.01

297.62 247.56 242.25 241.99 241.99 241 a99 241.99 241.99

621.07 459.83 432.69 429.74 429.58 429.58 429.58

5

2

b/a = O-5 (ellipticplate 6

1152.7 771.53 1969.0 687.69 1216.4 672.49 1023.9 670.88 975.07 670.78 966.51

with uniform thickness)

7

8

9

10

3157.8 1832.1 1463.9 1348.4

4818.4 2659.6 2033.2

7061.3 3743.8

10009.0

384 3.2.

B. SINGH CIRCULAR

PLATE

WITH

VARIABLE

AND

D. K. TYAGI

THICKNESS

For this case one puts a = b, K0 = 8/3 and L,, = 2( 1 + Y) in equation now varies according to the law

(20). The thickness

h = ah,( 1 -@‘/a’). Soni [5] has given some results has computed the values of

for this case, obtained

(26) by series solution.

Instead

of A he

p = Ah&/i?,

(27)

with h,, = 0.2. In Table 3, a comparison of our results with his is shown. The values in the fundamental mode almost tally. The difference increases in the second and third modes. Note that all our values are correct to the last digit stated.

TABLE

Values of p = Ah,/fi,

3

h,, = 0.2, b/a = 1 (circular plate with variable thickness)

Fundamental

Second

I

, Present results r

P

A

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7

15.1224 13.7310 12.3317 10.9235 9.50548 8.07595 6.63202 5.16409

3.3.

CIRCULAR

Putting

Soni CL

, CL 0.873 1 0.7928 0.7120 0.6307 0.5488 0.4663 0.3829 0.2981

PLATE

0.8730 0.7927 0.7119 0.6306 0.5488 0.4662 0.3829 0.2981

WITH

Present results w A CL 51.3333 48.1833 44,9242 41.5301 37.9627 34.1610 30.0152 25.2850

UNIFORM

2.9637 2.7819 2.5937 2.3977 2.1918 1.9723 1.7329 1.4598

r

Soni I*

Third h Present results

F A 111.156 105.214 99.0173 92.5050 85.5877 78.1241 69.8624 60.2572

2.9634 2.7816 2.5935 2.3975 2.1916 1.9721 1.7328 1.4020

CL 6.4176 6.0745 5.7168 5.3408 4.9414 4.5105 4.0335 3.4789

, Soni p 6.4174 6.073 1 5.7165 5.3402 4.9409 4.5112 4.0329 3.4760

THICKNESS

a = b and K0 = 813, from equations

(25) one gets b, = (-l)““/(i+j+3).

a,={32ij(i+1)(j+1)/(i+j-1)(i+j)(i+j+1)}(-1)’+’,

(28)

This case has been discussed at length in the literature, the analytic solution being given in terms of Bessel functions [l, 31. This involves determination of zeros of expressions involving Bessel functions. The present method requires much less computation. Table 4 shows the various convergences up to N = 10. It can be seen that the first five frequencies have converged up to the last digit. For improving results for stiil higher modes N will have to be increased beyond 10.

4. MODE SHAPES AND NODAL

LINES

After having determined the frequencies, the mode shapes Aj, j = l(l)N can be computed from equations (19) for a known value of A. Thus for the nth mode and Nth order approximation one has ; j=l

[aG-(A’,N’)2bii]A;;‘,=0,

lcn
(29)

VARIABLE

THICKNESS TABLE

Values of AIN’, N=

Ni

1

2

ELLIPTIC

385

PLATE

4

i= 1 ( 1 ) N, p = 0, b/a = 1.0 (circular plate with uniform thickness)

l(l)lO,

3

4

5

6

7

8

9

10

1

10,328

2

10.217

3

10.216

39.921

4

10.216

39.773

91.158

5

10.216

39.771

89.204

169.32

6

10.216

39.771

89.106

159.33

284.10

725.05

7

10.216

39.771

89.104

158.24

253.23

447.91

8

10.216

39.771

89.104

158.19

247.63

377.01

674.63

9

10.216

39.771

89.104

158.18

247.04

359.05

539.05

979.36

2600.2

10

10.216

39.771

89.104

158.18

247.01

355.90

496.53

748.67

1378.6

43.058 109.59 228.70 424.45 1162.8 1774.3 3685.7

where A:(T)) are the mode shapes corresponding to hLN’. As N is increased the mode shapes converge, as can be seen from Table 5 where results are given for the second mode (n = 2) with /I = 0, b/a = 1 and N = 2(1)10. The normalization is done by taking the highest component as unity. The nodal lines, which will be concentric ellipses, are then obtained from equation (13) by solving WN = 0. One obvious solution is u = 0 which corresponds to the clamped boundary. Taking out u2 as a common factor, one obtains the other solutions from c” A;;;

nj-l = 0,

2sncN.

(30)

j=l

TABLE

Mode shapes A;;: 1 2 3

O-83936 -0.45358

2

5

(second mode, /3 = 0, b/a = 1 .O) 3

4

5

6

1 *OOooo 0.32039

1~00000 0.09821

4

o-95191

0.46325

5

1~00000

O-27094

-0.58156

0.24521

-0.34866

6

1~00000

0.29917

-0.45029

0.50543

-0.10827

7

1~00000

O-29690

-0.46476

0.46315

-0.17187

0.03509

8

1~00000

0.29704

-0.46364

0.46763

-0.16216

0.04675

9 10

1~00000

0.29703

-0.46370

0.4673

-0.16308

o-04517

1~00000

O-29703

-0.46370

0.46733

-0.16302

o-04531

9

z’l;i

2

-0.83936

0.40080

3

-0.85247

0.38410

4

-0.85604

0.37942

5

-0.85635

0.37901

6

-0.85636

0.37900

-0.85636

0.37900

7

-0.01381

8

-0.00651

0.00185

9

-0.00810

0*00100

-0~00019

10

-0.00789

0.00117

-0*00011

10

1

0.08236

G1

7

8

1~00000

0~00002

-0.85636

0.37900

-0.85636

0.37900

-0.85636

0.37900

B. SINGH AND D. K. TYAGI

386

From physical considerations this equation will have n - 1 real roots between -1 and 0. These values are denoted by u iyij , k = l(1) n - 1. The nodal ellipses are then given by (x2/u2) + (y2/b2) = 1+ upon)), So, in non-dimensional (b/a), where

form, the semi-axes

k=l(l)n-1.

of the nodal

ellipses

z$$, = Jiz$$

(31) will be z$$,), and z@)

(32)

The numerical values of ukyd, and z&), for the second mode (n = 2), p = 0, b/a = 1, N = 2( 1)lO are also given in Table 5. The fast convergence is immediately evident. The value O-37900 agrees with that given by Leissa [l], who has taken it from Gontkevich [6]. For brevity not all the results computed showing the influences of various parameters and the orders of approximation are presented. The values of ziy;il) from the tenth approximation, i.e., N = 10, and for the second and third modes, i.e., n = 2 and 3, which immediately determine the semi-axes of the nodal ellipses, for p = -0*8(0*2)0*8 are shown in Table 6, and the corresponding nodal ellipses, including u = 0 (boundary), are shown in Figure 5 of the next section. A three-dimensional view of the second and third modes along with the nodal curves is given in Figure 6 of the next section, for p = 0 and b/a = O-5. TABLE

6

Values of z for second and third modes (10)

P

zw)

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

0.34911 0.35528 0.36218 0.36999 0.37900 0.38966 0.40277 o-41994 O-44560

(10) zl(3)

0.54559 0.55334 0.56201 0.57188 0.58330 0.59689 O-61370 0.63586 0.66923

0.23127 0.23608 0.24149 0.24766 0.25483 0.26339 0.27402 0.28811 0.30942

For Y = O-3, the values of K,, and Lo are almost the same, and so the effect of K,, (or b/a) on ati reduces to just a constant factor. This leads to a negligible effect of b/a on the mode shapes and therefore on u&)) and ziyi). In fact, if b/a is varied from 0.1 to 1.0, the first three places after the decimal point remain practically unchanged. For Y = l/3 the dependence on b/a will completely disappear. However, for values of v significantly different from l/3, the effect of b/a on mode shapes will not be so small.

5. DISCUSSION From Table 1 and Figures 2-4, in which A has been plotted against p and b/a for the fundamental, second and third modes, one can make the following observations. (i) As /3 is increased with b/a fixed, h decreases for all modes. For the fundamental and second modes the graphs are almost straight lines. This is not so for the third and higher modes. (ii) As b/a increases with /3 fixed, again A decreases for all modes. Initially h decreases at a faster rate for all /3. This rate of decrease becomes very small as b/a approaches unity.

VARIABLE

THICKNESS

ELLIPTIC

240

387

PLATE

(b)

150 1 120 x 90 60

b/o -

Figure 2. Fundamental

mode (a) A vs. /3 for various

b/a;

(b) A vs. b/a

for various p.

B. SINGH

388

AND

D.

K. TYAGI

500-

i x 300-

100-

06 08 1.0

-0.8

-0.4

I 04

I 0.0

P-

(b)

Figure 3. Second

mode (a) A vs. f3 for various

b/a

; (b) A vs. b/a for various @

VARIABLE

THICKNESS

ELLIPTIC

389

PLATE

1800 (a)

1500-

lOoot A

(b)

3

I

02

I

I

0.6

0.4

I

08

l-0

b/o-

Figure4.

Third mode (a) A vs. 0 for various b/a; (b) A vs. b/a

for various 0,

390

B. SINGH

AND

D. K. TYAGI

(iii) The effect of the parameter ho (the non-dimensional thickness at the standard point which is taken as the origin here) has not been explicitly studied, as it has been included in A. This was done in keeping with the definitions used by other authors: e.g., Leissa. However, if one is interested in also finding the effect of !I,-,,a new parameter such as p defined in equation (27) and used by Soni [S], or p = Aho, may be considered. Since it will only be a constant multiple of A, it is not of much interest. Second /3=-o.*

Third

@

@

-0.6

@

@

-0.4

@

@J

Figure 5. Nodal

(b)

Figure 6. Mode shapes;

curves (b/a

= 0.5).

hbdoi curves

/3 = 0, b/n = O-5. (a) Second mode; (b) third mode.

VARIABLE THICKNESS ELLIF’TIC PLATE

391

From Tables 5 and 6, and Figures 5 and 6, one can draw the following conclusions. (iv) As p is increased from -0.8 to 0.8 the nodal ellipses increase in size for all modes and b/a. (v) The “second mode” here is second only in the framework of the constraints adopted. This may not be so for a general choice of trial functions, in which case other frequencies and nodal curves will also be present. (iv) The trial functions chosen may not be so good for elliptic plates of large aspect ratio, because in that case the nodal lines may not exactly be concentric ellipses. For elliptic plates that are perturbations of a circular plate the assumption may be quite good but may rapidly deteriorate as the aspect ratio of the plate increases. ACKNOWLEDGMENTS

We are grateful to Professor C. Prasad for suggesting the problem and helpful discussions. Thanks are also due to Dr Jia Lal, Sri P. K. Agarwal and Sri A. K. Gupta for helping in computations. We are also thankful to the referee for making valuable suggestions, particularly regarding the mode shapes, and for pointing out the limitations of the trial functions. REFERENCES 1. A. W. LEISSA 1969 NASA SP-160. Vibration of plates. 2. S. TIMOSHENKO and K. S. WOINOWSKY 1959 77ieory ofPlates and Shells. New York: McGrawHill. 3. W. S~EDEL 1981 Vibrations ofShells and Plates. New ‘York and Basel: Marcel Dekker, Inc. 4. R. P. MCNIIT 1962 Journal of Aerospace Science 29, 1124-1125. Free vibrations of a clamped elliptic plate. 5. S. R. SONI 1972 Ph.D. Thesis, University of Roorkee. Vibrations of plates and shells of variable thickness. 6. V. S. GONTKEVICH 1964 Natural Vibrations of Phtes and Shells (A. P. Filippov editor). Kiev: Nauk. Dumka.