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