Journal of sound and Vibration (1980) 70(2), 221-229

NATURAL FREQUENCIES OF SIMPLY SUPPORTED CIRCULAR PLATES

A. W. LEISSA AND Y. NARITA~

Department of Engineering Mechanics, Ohio State University, Columbus, Ohio 43210, U.S.A.

(Received 5 July 1979, and in revised form 4 December 1979)

Although the problem of finding the natural frequencies of free vibration of a simply supported circular plate has a straightforward solution, very few numerical results are available in the literature. In the present work accurate (six significant figure) non- dimensional frequency parameters (A ) are given for all values of n + s 6 10, where n and s are the numbers of nodal diameters and internal nodal circles, respectively, and for Poissons ratios 0, 0.1, . . . , 0.5. Simplified formulas for determining additional values of A 2 for large s are derived by the use of asymptotic expansions.

1. INTRODUCTION

The free vibrations of circular plates have been of practical and academic interest for at least a century and a half [ 1,23. A thorough summary of the previously published literature through the year 1965 [3] revealed that a reasonable number of numerical results had been obtained for the two cases when the plate boundary is either clamped or free, but that few results were available for the simply supported boundary. A subsequent literature survey [4], as well as a further search of the literature by the present authors, turned up only one other significant publication on the simply supported case, namely, that of Pardoen [5], wherein 12 frequencies previously given in reference [3] for a Poissons ratio of 0.3 were compared against results obtained from a finite element method. In contrast is the recent work by Itao and Crandall[6] who published the lowest 701 eigenvalues (non-dimen- sional frequency parameters) and corresponding eigenvectors (defining the mode shapes of free vibration) for the completely free plate having a Poissons ratio of O-33.

The purpose of the present work is to determine accurate and extensive numerical results for the case of simply supported edges, which are useful to both designers and researchers, and to examine carefully what forms the solutions take when large numbers of nodal circles and/or nodal diameters are present.

2. THE FREQUENCY EQUATION

The solution of the equation of motion of classical plate theory for the case of free vibrations is well known (cf. [3]):

W,, (r, 0) = [A,, J, (kr) + GI,, (kr)] cos no, (1)

where W,, is the deflected shape of the vibrating plate, generally a function of both polar co-ordinates, r and 8 (see Figure l), J,(kr) and I,(kr) are ordinary and modified Bessel functions of the first kind, respectively, n is an integer, and k is related to the radian

t On leave from Hokkaido University, Sapporo, Japan.

221 0022-460X/80/100221 + 10 %02.00/O @ 1980 Academic Press Inc. (London) Limited

222 A. W. LEISSA AND Y. NARITA

iIlE++c Figure 1. Simply supported circular plate.

frequency, w, by

k4 = pw ID, (2)

in which p is the mass density per unit area of the plate and D is the flexural rigidity,

D=Eh3/12(1-V2) (3)

with E the Youngs modulus, h the plate thickness and v Poissons ratio. The ratios of the constant coefficients A, and C, (amplitude ratios) are determined from the boundary conditions.

From a physical viewpoint, the simply supported boundary conditions can represent either circular knife edge supports or hinges. Or they can be closely duplicated by cutting circular grooves of sufficient depth into both lateral surfaces of a larger plate. Mathema- tically, the boundary conditions state that the deflection and radial bending moment at a fixed radius (a) are zero: i.e.,

W(a, 9) = 0, a2w 1 aw 1 a2w

Wa,8)=--D 2+v ;ar+Tiae [ ( >1 =o. (4) ,=(1 Substituting equation (1) into equations (4), and using certain well-known relationships (cf., [3]) which relate the derivatives of Bessel functions to higher order functions, yields finally the frequency equation

Jn+l(A) I L+dA)_ 2A J,(A) I,(A) l-v

where A = ka. The amplitude ratios G/A, which are needed to determine the eigenfunctions (mode

shapes) are readily obtained from the first of equations (4):

G/An=-J,(A)/L(A). (6)

FREQUENCIES OF S.S. CIRCULAR PLATES 223

3. NUMERICAL RESULTS

The roots of equation (5), A, are the eigenvalues (or non-dimensional frequency parameters) of free vibration, where

A 2 = wa dp/D. (71

A double-precision computer program was written and utilized to extract the roots. Numerical results are summarized in Tables 1 through 6 for the full range of possible Poissons ratios, v = 0, 0~1,0~2,0~3,0~4 and O-5. Results are given for all values of n and s having n +s s 10, where n denotes the number of nodal diameters and s the number of interior nodal circles. Frequency parameters are given with six significant figures, which should be accurate due to the double precision (sixteen significant figure) arithmetic used both to sum the infinite series required for the Bessel functions and to evaluate equation (5) by the root-finding procedure.

The most complete numerical results available heretofore were those published by Gontkevich [7] and Wah [8] for v = 0.3, which were reported in reference [3]. These are presented in Table 7, together with the relevant data taken from Table 4 rounded off to the same number of significant figures to make comparison easier. It is seen that significant inaccuracy exists in the previous data, probably due to inaccuracy in computing the Bessel functions. It should be pointed out, however, that Wahs work was aimed at demonstrating the effects of in-plane forces upon the vibration frequencies of a simply supported circular plate, although the frequency equation he presented [8] reduces straightforwardly to equation (5) when the in-plane force is zero. In a relatively recent work by Pardoen [5] in the 12 values given in Table 7 were also presented, and these agree exactly with the present results, not only for the five significant figures listed in Table 7, but for all six significant figures listed in Table 4 for these 12 values.

Comparison of the results in Tables 1 through 6 shows that the effect of Poissons ratio upon the frequency parameter A * = wa *&@ is significant only for the lowest frequencies. This effect can be seen clearly in Table 8 wherein the ratio of A * (0*5)/A*(O) is given for selected values of n and S, where A*(0.5) and A*(O) are the values of A2 for Y = 0.5 and 0, respectively. The variation of A * with Y was discussed by Jacquot and Lindsay [9] for the lowest frequency, axisymmetric mode. However, if the ratios of the frequencies them- selves are compared, the effects of v are seen to be more pronounced, as is also evident in Table 8. The circumferential stiffening is particularly important in the axisymmetric mode where, as it is seen, the frequency can differas much as 35% for different Y.

Finally, some additional values of wa*Jp/~ for v = O-3 are given in Table 9 for the purpose of future measurement of the accuracy of extrapolation of the tables.

4. OTHER VALUES OF FREQUENCY PARAMETERS

Tables 1 through 6 each contain 66 values of A *, which should be sufficient for most practical needs for vibration frequencies. however, in choosing to present frequency results for n +s s 10, only the lowest 27 values can be found for each value of Y.

To obtain additional values of frequency parameters not given in Tables 1 through 6, one may calculate additional roots of the frequency equation (5). The standard compu- tational procedure for obtaining Bessel functions of integer order is to sum their series representations:

J (*)= f (-mv2Y+2 n

r=O r!(n +r)! ' * (h/2)n+2r

L(A)= c ,=o r!(n + r)! (8)

TA

BLE

1

Freq

uenc

y pa

ram

eter

w

a T

D

for

v =

0

n

s 0 1

2 3 4 ii

7 ; 10

0 1

2 3

4 5

6 7

8 9

10

4.44

361

13*5

013

25.2

446

39.6

031

56.4

964

75.8

642

97-6

602

121.

848

148.

399

177.

288

208.

494

29.3

638

48.1

361

69.7

816

94.2

183

121.

375

151.

194

183.

625

218-

630

256.

173

269.

225

73.8

229

102.

445

133.

973

168.

353

205.

531

245.

460

288.

097

333.

405

381.

351

137.

995

176.

481

217.

884

262.

168

309.

291

359,

217

411.

908

467.

331

221.

897

270.

250

321.

526

375.

698

432.

735

492.

607

555.

281

448,

909

325.

534

383.

755

516,

999

444,

903

588,

017

661.

948

508.

956

575.

892

738.

77 1

64

5.68

1

592.

022

669-

980

750.

869

834.

675

754.

873

937.

462

1035

.16

842.

701

933.

459

1139

.79

TA

BLE

2

Freq

uenc

y pa

ram

eter

w

a C

D

for

v =

0.1

n

S 0

1 2

3 4

5 6

7 8

9 10

0 4.

6192

3 1

29.4

850

z 73

.935

3 13

8.10

3 4

222.

003

5 32

5.64

0 6

449.

014

7 59

2.12

6

i 75

4.97

7 93

7.56

5 10

11

39.8

9

13.6

384

48.2

522

102.

556

176.

588

270-

356

383.

860

517,

103

670.

084

842,

804

1035

.26

25.3

707

69.8

949

134.

083

217.

991

321-

631

445.

007

588-

121

750.

972

933.

562

39.7

236

56.6

134

75.9

788

97.7

731

121.

960

148.

510

177.

398

208.

603

94.3

298

121.

485

151.

303

183.

734

218,

737

256.

280

296.

332

168.

461

205.

639

245.

567

288.

203

333.

511

381.

456

262.

274

309.

397

359.

322

412.

013

467.

436

375.

803

432.

841

492.

711

555.

385

509.

061

575.

996

645.

786

662.

052

738.

875

834.

778

TA

BLE

3

Freq

uenc

y pa

ram

eter

w

a 2 J

;;Tis

for

v =

0.2

s 0

1 2

3 4

5 6

7 8

9 10

0 4.

7825

8 13

-770

5 25

.493

5 39

.841

6 56

.728

4 76

-091

8 97

.884

5 12

2.07

0 14

8.61

9 17

7.50

6 20

8.71

1 1

29.6

037

48.3

665

70.0

067

94.4

400

121-

594

151.

411

183.

841

218.

844

2.56

386

296.

438

i 13

8.21

1 74

.046

4 10

2665

17

6.69

5 21

8,09

7 13

4.19

1 26

2.38

0 16

8.56

8 20

5.74

5 30

9.50

3 24

5.67

3 35

9.42

7 41

2.11

7 28

8.30

9 46

7.54

0 33

3.61

6 38

1.56

1

4 22

2.11

0 27

0.46

1 32

1,73

6 37

5.90

8 43

2.94

5 49

2.81

5 55

5.48

8 5

325.

745

383.

965

445,

112

509.

165

576.

099

645.

889

6 44

9.11

8 5

17.2

07

588.

225

662,

155

738.

978

7 59

2.23

0 67

0.18

8 75

1.07

6 83

4.88

2 8

755.

079

842.

907

933.

665

9 93

7.66

8 10

35.3

7 10

11

40*0

0

TA

BLE

4

Freq

uenc

y pa

ram

eter

w

adp

/D

for

v =

O-3

tl

s 0

1 2

3 4

5 6

7 8

9 10

0 4.

9351

5 13

.898

2 25

.613

3 39

.957

3 56

.841

6 76

.203

1

97.9

945

122.

179

148.

727

177.

614

208.

818

1 29

:720

0 48

.478

9 70

.117

0 94

.549

0 12

1.70

2 15

1.51

8 18

3.94

8 21

8.95

0 25

6.49

2 29

6.54

3 32

13

8,31

8 74

.156

0 10

2,77

3 17

6.80

1 21

8.20

2 13

4.29

8 26

2.48

5 16

8.67

5 30

9.60

8 20

5,85

1 24

5.77

8 35

9.53

2 41

2.22

1 28

8.41

4 46

7.64

4 33

3.72

1 38

1.66

6

4 22

2.21

5 27

0.56

6 32

1.84

1 37

6.01

2 43

3.04

8 49

2.91

9 55

5.59

2 iz

44

9.22

2 32

5,84

9 38

4.06

9 51

7.31

0 44

5.21

5 58

8.32

8 66

2.25

8 50

9.26

8 57

6.20

2 73

9.08

1 64

5.99

2

z 59

2.33

2 75

5.18

2 67

0.29

0 84

3.00

9 93

3.76

8 75

1.17

9 83

4.98

4

9 93

7.77

1 10

3-s-

47

10

1140

*10

TABL

E 5

Freq

uenc

y pa

ram

eter

w

a&

@

for

v =

O-4

n

s 0

1 2

3 4

5 6

7 8

9 10

0 : 3 : 6 7 8 9 10

5.07

817

29.8

339

74.2

644

138.

424

222.

3 19

32

5.95

3 44

9.32

5 59

2.43

5 75

5,28

4 93

7.87

3 11

40.2

0

14.0

215

48.5

896

102.

880

176.

906

270.

670

384.

172

517.

413

670.

393

843.

112

1035

*57

25.7

301

70.2

259

134.

404

218.

307

321.

944

4453

19

58

8.43

1 75

1.28

1 93

3.87

0

40-0

707

94.6

566

168.

780

262.

589

376.

116

509.

371

662.

361

835.

086

56-9

528

76.3

128

98.1

030

122.

287

148.

834

177.

720

208.

924

121.

809

151.

624

184,

053

219.

056

256.

597

296.

647

205.

956

245.

883

288.

5 18

33

3.82

5 38

1.77

0 30

9.71

1 35

9.63

5 41

2.32

5 46

7.74

8 43

3,15

2 49

3.02

2 55

5695

57

6.30

5 64

6.09

5 73

9.18

3 TABL

ET

Freq

uenc

y pa

ram

eter

w

a d

z fo

r Y

= 0

.5

n

s 0

1 2

3 4

5 6

7 8

9 10

0 5.

2126

5 14

.140

7 25

.844

1 40

.181

9 57

.062

2 76

.420

9 98

.210

2 12

2.39

3 14

8.94

0 17

7.82

5 20

9.02

9 :

29.9

456

74.3

713

102.

986

48.6

985

134.

509

70.3

334

168-

885

94.7

631

206.

060

121.

915

245.

987

151.

729

288.

622

184.

158

333.

928

219.

160

256.

701

381.

872

296.

75

1

3 13

8,52

9 17

7,01

1 21

8-41

1 26

2-69

3 30

9.81

4 35

9.73

8 41

2.42

8 46

7.85

0 4

222.

423

270.

774

322-

048

376.

218

433-

255

493.

125

555.

798

ii 44

9.42

8 32

6.05

6 38

4.27

5 51

7.51

6 44

5.42

58

8.53

3 1

662.

463

509,

473

576.

408

739.

285

646.

185

;: 59

2.53

8 75

5.38

6 67

0.49

5 84

3.21

4 75

1.38

6 93

3.97

2 83

5.18

8

9 93

7.97

5 10

35.6

7 10

11

40.3

0

FREQUENCIES OF S.S. CIRCULAR PLATES

TABLE 7

Comparison of frequency parameters wa2JrD with those of Gontkeuich [7] and Wah [S] for v = O-3

227

II s Reference [7] Reference [8] Present work

0 0 1 2 3

1 0 1 2 3

2 0 1 2 3

4.977 4.94 4.935 29.76 29.72 29.72 74.20 74.15 74.16

138.34 - 138.32

13.94 13.47 13.90 48.51 47.89 48.48

102.80 103.43 102.77 176.84 - 176.80

25.65 25.60 25.61 70.14 68.89 70.12

134.33 134.56 134.30 218.24 - 218.20

TABLE 8

Ratios of frequency parameters and frequencies having Y = 0.5 and v = 0

A 2(o*5) w(O*5) n s

A W) 43

0 0 1.17307 1.35454

0 1 1.01981 1.17758 0 2 1.00743 1.16328 0 10 1.00045 1.15522

1 0 1.04736 1.20934 2 0 1.02375 1.18212

10 0 1.00257 1.15470

TABLE 9

Some additional values of A2 = oa2Jp/D for Y = 0.3

n

s 0 5 10 15 20

0 - - - 399.061 645.319 5 - - 1036.90 - -

10

4249.03 2447.82

1707.16 2346.42 - -

;i - - - - - - - -

228 A. W. LEISSA AND Y. NARITA

However, for large A these series converge slowly. In this case it may be desirable to represent the Bessel functions by their asymptotic expansions [lo, 111:

J,(A) = J2/?rA(P,, cos C& -Q, sin &),

I,(A) = {eAlJ2d}(P?i - Qfi 1 (94

where

p,=l_R',2'+~lf)_R(n6'+...,p~=l+Rlf)+Rlf)+R(n6)+...,

Q,=R',"_RIT'+R',5'-...,Qg=R(,)+R(,3)+RJIS)+..., Pb)

with O), (3, (3),. . . being superscripts, and where

4 = A - (n/4)(2n + l), R, = (4n2- 1)/8A, (lOa, b)

R(i+)

= R(i) [4n2-(2i + l>'l 8A(i+l)

, i = 1,2,3 . . . . (1Od

It is seen that the terms R, in the asymptotic expansions should be terminated when (2i + 1) becomes greater than 2n in equation (10~).

To utilize equations (9) and (10) in the frequency equation (5), one observes that

I P,+i sin (A - 7r/4) + Qntl cos (A - 7r/4)

P,, cos (A - 7r/4) - Q, sin (A - r/4) n even J&= J,(A)

-P,+I cos (A - r/4) + Qn+i sin (A - 7r/4) P, sin (A -n/4) + Q, cos (A -v/4)

, n odd

I,+i(A)/I,(A) = (P;+i - Q:+I)/@: - QXL

so that equation (5) can be written in the form

tan (A - 1rj4) = S,/ T,,,

where PC+, -Q:+,

p _Q* n

T, =Pn+i-On P,*+I - a:+~+ 2AQ,

P:-Q; l-v

S*=1_ 2AQ,+p

?I+1 _Q P:+,-Q:+I

V n P:-Q;

Tn = Q,+i+Pn P,*,I - Q:+I 2APn --

P:-Q; l-v i

(11)

n even

n odd

For very large A and relatively small values of n, equations (9) and (10) yield

P,=PX =Pnsl=P;+l = 1, Q, = Qt = (4n2 - 1)/U

Q ?I+1 = a;+, ={4(n+1)2-1)/8A,(P~+i-Q~+~)l(P~-Q~)=1,

and the frequency equation (11) then simplifies to

8~/(3-4~+4n*), n even (3-4v+4n2)/8A,n odd

(12)

FREQUENCIES OF S.S. CIRCULAR PLATES 229

TABLE 10

Percent error in using equation (12) to calculate frequency parameters A 2 = wa 2dp/D for Y = 0.3

n

s 0 2 4 10

Equations (12) are relatively simple transcendental forms for the frequency equation. For fixed values of Y and n, the right-hand sides are a straight line and a hyperbola which intersect the tangent function at values of A separated by w as A increases. Examples showing the percent error in using the approximate frequency equations (12) to determine A 2, compared with the accurate values listed in Tables 4 and 9, are presented in Table 10 for v = 0.3. In this table a positive error indicates that the approximate values are too large. It is seen that equation (12) is useful only for the smaller values of n and larger values of s.

Crequency parameters for larger n can be obtained from the tables by extrapolation. 1

REFERENCES

1. 2.

3. 4.

5.

6.

7.

8.

9.

10. 11.

E. F. F. CHLADNI 1803 Die Akustik. Leipzig. S. D. POISSQN 1829 Memoires de IAcademie Royales des Sciences de llnstitutde la France, ser. 2 8, 357. LEquilibre et le mouvement des corps Clastiques. A. W. LEISSA 1969 NASA SP- 1960. Vibration of plates. A. W. LEISSA 1977 Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations: classical theory. G. C. PARDOEN 1978 Computers and Structures 9, 89-85. Asymmetric vibration and stability of circular plates. K. ITAO and S. H. CRANDALL (to appear) Journal ofApplied Mechanics. Natural modes and natural frequencies of uniform, circular, free-free plates. V. S. GONTKEVICH 1964 Natural Vibrations of Plates and Shells (in Russian, A. P. Filippov, editor). Kiev: Naukova Dumka. T. WAH 1962 Journal of the Acoustical Society of America 34,275281. Vibration of circular plates. R. G. JACQUOT and J. E. LINDSAY 1977 Journal of Sound and Vibration 52,603-605. On the influence of Poissons ratio on circular plate natural frequencies. N. MCLACHLAN 1948 Bessel Functions for Engineers. London: Oxford University Presss. M. ABRAMOWITZ and I. A. STEGUN 1964 Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series 55.

0 3.204 0.966 7.84 -11*210 2 0.044 0.045 - - 5 0.004 0,006 - 3.626

10 0.001 - - 0.990