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Application of the spline element method to analyze vibration of annular sector plates
Journal ofSound
and Vibration (1991) 149(3), 461-470
APPLICATION ANALYZE
OF THE
VIBRATION
SPLINE
ELEMENT
OF ANNULAR
METHOD
SECTOR
TO
PLATES
T. MIZUSAWA Department of Construction and Civil Engineering, Duido Institute of Technology, Hukusuicho-40, Minumi-ku, Nagoya 457, Japan (Received 3 July 1990, and in revised form 1 October 1990)
This paper deals with the vibrations of isotropic annular sector plates with arbitrary boundary conditions using the spline element method. To demonstrate the accuracy of the method, several examples are solved, and results are compared with those obtained by analytical methods and by other numerical methods. Good accuracy is obtained. Frequencies of annular sector plates with free edges and with point supports are presented.
1. INTRODUCTION of annular sector plates employed in engineering designs have been investigated by analytical methods [l-8] and by numerical methods such as the RayleighRitz method [9, lo], the finite strip method [ 1l-151, the integral equation method [ 16-181 and the finite element method [19-211. These problems have also been studied by experimental approaches [22-241. Recently, Irie er al. [25] analyzed the vibration of polar orthotropic annular sector plates by the Ritz method with quintic spline functions and circular beam functions as the displacement functions. They [26] also investigated cantilever annular sector plates using the Ritz method with a power series. Mukhopadhyay [27, 281 presented a finite strip-finite difference method to calculate the vibration of annular sector plates with arbitrary boundary conditions. Harik and Molaghasemi [29, 301 described an analytical finite strip method with Bessel functions for the vibration of annular sector plates. Singh and Dey [3 l] applied the variational finite difference method associated with an integralbased finite difference approach to analyze the vibration of annular sector plates. Geannakakes [32] presented a semi-analytical finite strip method mapping arbitrarily shaped plates into the natural co-ordinate plane using serendipity functions, and analyzed the vibration of annular sector plates. In this paper, the spline element method [33,34], which is a discrete Ritz method with B-spline functions, is used to analyze the vibration of isotropic annular sector plate on the basis of thin plate theory. The accuracy of the present method is discussed via comparison of the results calculated by anaytical methods and by numerical methods. Frequencies of annular sector plates with free edges and with point supports are presented.
The free vibrations
2. METHOD
OF ANALYSIS
The analysis is based on the spline element method, which can be regarded as an alternative form of the displacement-based finite element procedure; it has previously been used to analyze skew plate problems [33, 341. 461 0022-460X/91/
18046lf
10 %03.00/O
@ 1991 Academic Press Limited
462
T. MIZUSAWA
Figure
1. Annular
sector
plate and polar co-ordinate
systems.
The spline element method is used here to analyze the vibration of an isotropic annular sector plate with arbitrary boundary conditions, as shown in Figure 1. It is convenient to introduce the non-dimensional polar co-ordinates (6, 7) as follows: 9=0/+
and
77=
CrmRR,)IB9
(1)
where 4 is the sector angle, B = R, -Rip and R, and Ri are the outer and inner radii of the annular sector plate, respectively. The respective displacement functions are expressed in terms of two-way spline functions as
w(& 77)= w,(if> 7))= 1 p=,
c
CpqN,,k(~)Nq,k(d~
(2)
9=1
where ie = k + MB - 1, i, = k + M, - 1, N,,,k(&) and Nn,k( 7)) are the normalized B-spline functions, k - 1 is the degree of the B-spline functions, MB and M, are the numbers of elements in the, 8- and r-directions, respectively, and the Cmns are unknown parameters which can be determined by the minimum total potential energy theorem. The method of artificial springs [32] is applied to deal with arbitrary boundary conditions. Three types of artificial spring, (Y,p and -y, corresponding to the deflection W, the slope deflection d W/a0 and d W/h, respectively, are introduced at each edge of the plate. The energy contribution lJ, due to these springs is given by ~b=~[~~W2+(~/~2)(~w/~~)2+(~/~2)(~~/~~)2~~~=0
+~~~*+~~/~*~~~~/~5~*+~(Y/~*~~~~/~77~*~1~=1 +~~~2+~~l~2~~~~l~~~2+~Y/~2~~~~l~71~2~l~=0
+~~~*+~~/~*~~~~/~~~*+~r/~*~~~~/~r1~*~1,=,1.
(3)
The strain energy due to bending, UP and the kinetic energy, T,, of the annular sector plate are given by 4 :” [(a’w,ar’+(l,r) d W,ar+(1,r’) a* W,df3*)z 4 = (D/2) II0
VIBRATION
OF ANNULAR
SECTOR
PLATES
463
1 1 = (W2)WR2)
II0 0
[{a2w/av2+(i/A)
aW/a7)+(l/A2/42)
a2W/at2)’
-2(1--~)a~W/a~~{(1~A)aW/a~+(1/A~/4~)a~W/a~~) +2(1-
4UllAld.d
a’Wla7
aW%)*lA d? de,
at-U/A2/44
where A=(77+Ri/B)={77+1/(S_l)},S=R,/Ri,
and
W2r dr de = (ph/2)02t$B2
Tp = (ph/2)02
(4)
W’(7) + RI/B) dv d5,
(5)
in which v is the Poisson ratio, p is the density, h is the thickness, D is the flexural rigidity of the plate, r$ is the sector angle and o is the frequency (rad/s). The functional of the annular sector plate Z7 is expressed by substituting equation (1) into Z7 as follows:
+Wzq
+(PI~')N~~~+(YIB~)NO,~,O~~}I~=,I
- (ph,2)c$B2
; ; ; ; C,,,,C,,[Zllp,J~],
(6)
mnPq
where Niyp,, N
I$, and .Zii’iare defined as follows: ijkl mw
=d’Nm,kWdS
djNn,k($ldd
dkNp,kWdSk
d’Nq.k(n)/drt’,
1
1”
mp
=
I d’%,&)/&
diNp,&Vd5’
d5,
0
1
JZ=
I 0
diNn,/c(T)ldTi d’N,.k(~)ldt7’(77+RiIB)’ dV*
(7)
Here the definite integrals, Zz, and Jg in equations (7) are calculated numerically by the Gauss-Legende quadrature formulae. By using the principle of minimum potnetial energy, the coefficients {C,,,,} are determined as solutions of aZMC,,I
= 0,
(8)
which may be expressed in matrix form as
rKI{Cm”I= A2[W{Crn”~.
(9)
T.
464
MIZUSAWA
Here A is a frequency parameter defined by B’wm. The order of the matrices is given by (k + Me - 1) x (k + M, - l), in which k - 1 is the degree of the spline functions, and MB and kf, are the numbers of mesh divisions in the analytical region in the & and r-directions, respectively. [K] and [M] are stiffness and mass matrices obtained from equations (4) and (5). 3. NUMERICAL EXAMPLES AND DISCUSSION The vibrations of isotropic annular sector plates with various boundary conditions are analyzed and the accuracy of the present method for these problems is demonstrated by comparison with the existing results calculated by analytical and numerical methods. In all calculations, the degree of the B-spline functions of k - 1 = 4 and the mesh divisions of MB = M, = 12 were used. In Figure 2, the definitions of the boundary conditions are presented. The symbolism SC-SF, for example, identifies an annular sector plate with the edges 5 = O,t = 1, ~7= 0 and ~7= 1 having simply supported, clamped, simply supported and free boundary conditions, respectively.
-,
Figure 2. Definition of the boundary conditions. -, F-free edge.
TABLE
C-clamped
edge; ==,
S-simply
supported edge;
1
Convergence study of frequency parameters, A, of a clamped annular sector plate; R,l Ri = 2.0, ~ = 30”
Modes Degrees of spline, k-l
3
4
M,=M,
1st
2nd
3rd
4th
5th
4x4 6x6 8x8
48.27 48.08 48.05
86.70 85.91 85.62
107.6 105.0 104.5
154.8 146.2 143.5
245.5 152.5 151.6
10x 10 12x12
48.05 48.04
85.56 85.54
104.4 104.4
143.0 142.9
151.4 151.3
4x4 6x6 8x8 10x 10 12 x 12
46.07 48.01 48.04 48.04 48.04
83.12 85.43 85.51 85.52 85.52
100.6 104.4 104.4 104.4 104.4
134.0 142.1 142.7 142.8 142.8
143.0 151.2 151.3 151.3 151.3
4x4 6x6 8x8 10x10 12x12
46.77 47.98 48.03 48.04 48.04
82.76 85.17 85.47 85.50 85.52
101.0 103.9 104.3 104.4 104.4
138.0 142.6 142.6 142.7 142.8
140.4 150.0 151.0 151.2 151.3
VIBRATION
OF ANNULAR
TABLE
Frequency
SECTOR
465
PLATES
2
for annular sector plates with several boundar?, parameter, A = wB2Jm, conditions; Y = 0.0, 4 = 90”, k - 1 = 4 and M, = M, = 12 Modes 1st
2nd
3rd
4th
5th
Methods
10.09 10.57
10.68 12.48
11.66 15.67
13.04 20.13
14.82 25.85
Present Present
2-o
11.80 11.79 11.66
17.08 17.07 16.73
25.81 25.81 24.57
37.68 37.69 34.06
41.68 41.43 40.31
Present Reference [27] Reference [ 3 l]
4.0
15.97 15.94 15.70
31.93 31.90 31.05
47.89 47.64 46.25
54.94 54.93 51.75
69.96 69.72 67.27
Present Reference [27] Reference [ 3 1 ]
22.46 22.70
22.74 23.83
23.27 25.91
24.00 29.14
25.09 33.71
Present Present
2.0
23.33 23.00 22.40
26.88 26.56 25.79
33.87 33.57 32.01
44.65 44.38 40.61
58.96 61.29 50.48
Present Reference [27] Reference [ 3 1]
4.0
25.80 25.43 24.78
40.20 39.86 38.59
64.30 63.84 59.71
66.64 64.87 61.35
84.77 83.18 78.92
Present Reference [ 271 Reference [3 11
RJR, (a) SS-SS 1.25 1.5
(b) SS-CC 1.25 1.5
(c) SS-FF 1.25 1.5
0.1384 0.4395
0.7207 2.263
1.693 4,526
2.0
1.160 1.185 1.145
5.709 5.834 5.500
8.493 8,769 8.391
4.0
3,185 3.294 3.127
2.375 5.253
3.050 8.915
Present Present
12.68 13.02 11.76
17.44 17.96 16.98
Present Reference [ 271 Reference [ 3 1 ] Present Reference [27] Reference [ 3 11
13.55 14.02 13.02
18.87 19.37 18.52
28.88 30.03 26.69
41.85 42.95 40.07
22.49 22.80
22.86 24.20
23.51 26.76
24.44 30.62
25.75 35.93
Present Present
2.0
23.80 23.84 23.52 23.83
28.73 28.79 28.50 28.69
37.63 37.52 37.32 38.00
50.33 50.43 49.80 52.51
63.30 63.53 61.74 63.37
Present Reference [ 1 l] Reference [27] Reference [32]
4.0
29.58 29.60 29.26 29.62 29.64
49.55 49.60 49.28 49.48 49.95
69.54 69.68 68.16 69.58 70.37
77.09 77.37 76.51 78.78 78.75
94.66 94.89 93.74 94.51 95.85
Present Reference Reference Reference Reference
(d) CC-CC 1.25 1.5
[ 1 l] [27] [32] [ 131
T. MIZUSAWA
466
In Table 1 is shown the convergence study of the the present method for a clamped annular sector plate ( C$= 30”, R,/ Ri = 2.0) by changing the number of mesh divisions, MB = M,, and the degrees of the spline functions, k - 1. Good convergence is obtained by increasing the number of mesh divisions. In Table 2 is shown the comparison of the first five frequencies parameters, A = wB*m of annular sector plates having various boundary conditions with C#J = 90” and Y= 0.0. The radii ratio, R,/ Ri, varies from 1.25 to 4.0. The results obtained by the semi-analytical method [27] and the variational finite difference method [31] are listed. In the case of an all-clamped annular sector plate, in addition to these values, the solutions
TABLE
Frequency
parameter, A = wB*m, for annular sector plates with several boundary conditions; v=O*3, k-1=4 R,/R,=2*0 andM*=M,=12
Modes
Sector angle 4 (degrees)
Boundary conditions
30
45
60
90
3
r 1st
2nd
3rd
4th
5th
Methods
SS-FF
11.77 11.77 11.77
28.87 28.87 28.87
44.03 44.03 48.72
50.51 50.55 50.53
82.13 80.41 -
Present Reference [ 71 Reference [ 321
ss-ss
25.86
57.16
69.60
106.8
109.7
Present
ss-cc
33.90
75.11
76,46
121.8
135.3
Present
cc-cc
48.04
85.52
142.8
151.3
Present
SS-FF
5.267 5.267 5.268
104.4
16.68 16.69 16.68
20.40 20.40 22.59
36.60 36.66 36.61
44.03 44.03 -
Present Reference [ 71 Reference [ 321
ss-ss
17.09 17.09
37.75 37.74
47.40 47.40
69.60 69.61
70.90 70.90
Present Reference [ 71
ss-cc
26.89 26.91
44.70 44.69
67.37 74.03
76.45 77.34
86.60 88.32
Present Reference [7]
cc-cc
31.39
56.85
70.22
94.54
96.73
Present
11.77 11.77 11.87
11.87 11.89 13.06
25.51 25.51 30.73
28.87 28.87 -
Present Reference [ 71 Reference [32]
SS-FF
2,856 2.856 2.857
ss-ss
13.99
25.86
44.00
44.75
57.16
Present
ss-cc
24.74
33.89
51.44
64.79
75.08
Present
cc-cc
26.53
39.85
61.49
65.96
79.38
Present
7.779
11.77 11.77 15.01
16.68 -
-
Present Reference [ 31 Reference [32]
SS-FF
ss-ss
1.068 1.068 1.069 11.77
5.267 5.267 5.858
7.780 7,779
17.09
25.86
37.75
41.59
Present
44.65
58.96
Present
50.33 52.51 50.42
63.30 63.37 63.30
Present Reference [32] Reference [ 321
ss-cc
23.33
26.88
33.87
cc-cc
23.80 23.83 23.80
28.73 28.69 28.77
37.63 38.00 37.69
VIBRATION
OF
ANNULAR
SECTOR
467
PLATES
calculated by the finite strip method [ll, 131 and the semi-analytical finite strip method [32] are also shown. It is seen that the values calculated by the present method show good agreement for all the cases. In Table 3 are shown the frequency parameters A, of annular sector plates (Y = 0.3, R,/ Ri = 2.0) with several boundary conditions. The sector angle I$ varies from 30” to 90”. The results are compared with those obtained by the analytical method [3, 71 and the finite strip method [32]. In the case of annular sector plates with simply supported straight edges, excellent agreement is shown compared with the analytical values. In Table 4 are shown the first five frequency parameters, A, of annular sector plates ( v = 0.3 and R,/ Ri = 2.0) with several clamped edges and free edges. In Table 5 are shown the frequency parameters, A, of cantilever annular sector plates ( v = 0.3 and R,/ Ri = 2.0) for the different sector angles. In this case, the results for 4 = 90” are compared with the numerical solutions calculated by the Ritz method [26] and the finite element method [24], and with the experimental results [24]. Good agreement is obtained. Lastly, the frequency parameters of annular sector plates with rigid point supports are shown in Table 6. The arrangement of the point supports is depicted in Figure 3. The frequencies of the plates are influenced by the sector angles and the positions of point supports. 4. CONCLUSIONS The spline element method has been applied to analyze vibrations of isotropic annular sector plates with various boundary conditions. The accuracy of the method has been demonstrated in the comparative studies between the present results and the existing values calculated by analytical and numerical methods.
TABLE
4
Frequency parameter, A = wB2m, for annular sector plates with free edges; v = 0.3, k- 1~4, R,/Ri*2*0 and MB= M, = 12; (a) one free edge; (b) two free edges
Modes -__
Sector angle, 4 (degrees)
Boundary conditions
30
CC-FC
1st
2nd
3rd
4th
46.75 25.12 25.06
69.57 49.91 52.45
99.17 65.77 64.29
104.4 87.17 95.20
149..3 110.6 108.7
CC-CF CF-CC
16.92 I.435 22.37
33.73 18.04 28.33
36.32 26.82 41.91
57.62 33.98 60.85
60.64 2.45 63.38
30
CC-FF FF-CC CF-FC CF-FC
25.12 21.94 13.04 6.649
48.34 28.36 31.84 23.30
65.77 56.63 49.46 28.94
70.54 60.64 69.22 57.05
100.4 70.22 76.94 63.74
60
CC-FF FF-CC CF-FC CF-CF
6.926 21.82 6.905 3.707
17.09 23.25 18.31 8.384
17.96 30.08 25.21 18.55
33.82 43.84 37.16 22.09
34.20 60.17 39.50 30.27
5th
(a)
CC-CF CF-CC 60
(b)
CC-FC
TABLE
5
A = wB2m, for cantilever annular sector plates; R,/Ri=2~0andMO=M,=12
Frequency parameter,
v = 0.3, k - 1 = 4,
Modes
Sector angle, $I (degrees)
Boundary conditions
30
45
60
90
p 1st
2nd
3rd
4th
5th
CF-FF
5.292
13.14
26.57
32.68
5345
FF-CF
2.884
19.91
29.28
31.74
FF-FC
4.232
23.03
41.22
51.98
CF-FF
2.480
7.686
13.14
26.82
28.35
FF-CF
2.968
5.316
14.06
21.10
25.26
FF-FC
4.250
23.02
28.63
33.12
7.286 14.31
10.82
CF-FF
1449
5.147
8.121
17.57
20.97
FF-CF
3.034
4.387
9.289
18.99
20.80
FF-FC
4.262
8.982
23.11
29.21
CF-FF
0.6930
2.587
4.600
8.426
12.41
FF-CF
3.112 3.122
3.604 3.604
5.713 5.696
10.02 9.948
16.72 16.59
Reference
[26]
FEM
3.114
3.662
5.814
9.876
15.94
Reference
[ 241
Experiment
3.076
3.593
5.730
10.09
16.82
Reference
[ 241
FF-FC
4.272
7,018
21.58
22.99
TABLE
Frequency parameters,
Sector angle, qb (degrees)
Method
19.55
12.63
6
A = oB2m, of point supported annular sector plaes; v = 0.3, k-1=4, R,/Ri=24andMO=M,=12 Modes
Boundary conditionst
1st
2nd
3rd
4th
5th
30
4 points A 6 points B 6 points C 8 points D
8.012 11.51 9.614 22.79
16.16 25.38 18.88 36.66
18.88 38.35 36.11 38.35
36.03 44.70 42.43 56.48
42.43 56.48 49.25 67.38
45
4 points 6 points 6 points 8 points
A B C D
4.729 5.231 9.372 14.84
12.45 16.15 13.65 19.80
13.65 19.80 23.95 29.14
22.23 35.83 26.69 38.24
23.95 38.24 30.68 47.20
60
4 points A 6 points B 6 points C 8 points D
2.720 2.847 8.671 11.02
9.900 11.64 9.900 11.64
10.75 11.76 17.08 21.40
16.14 24.7 1 19.34 26.99
17.08 26.99 22.26 35.93
90
4 points 6 points 6 points 8 points
1.048 1.066 5.003 5.249
5.003 5.249 6.147 6.564
7.718 7.772 12.94 14.38
10.33 11.68 13.87 16.41
12.94 16.41 14.59 20.09
t Boundary
conditions
A B C D
are defined
in Figure 3.
VIBRATION OF ANNULAR
(a)
SECTOR
PLATES
469
(b)
Cc)
(d)
Figure 3. Depiction of point supported annular sector plates and the arrangement of point supports.(a) four points A; (b)six points B; (c) six points C; (d) eight points D.
Good agreement has been shown. The frequency parameters of annular sector plates with free edges and with point supports, which can be difficult to solve analytically, have also been presented. The present method can be used, without difficulty, to analyze annular sector plates with a variable thickness or with stiffeners.
REFERENCES 1. M. BEN-AMOZ 1959 Journalof Applied Mechanics 26,136-137. Note on deflections and flexural vibrations of clamped sectorial plates. 2. H. YONEZAWA 1962 Journal of Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers 88, l-21. Moments and free vibrations in curved girder bridges. and V. X. KUNUKKASSERIL 1973Journal ofSound and Vibration 30, .3 . R. RAMAKRISHNAN 127-129. Free vibration of annular sector plates. 4. C. RUBIN 1975Journal of Sound and Vibration 39,523-526. Nodal circles and natural frequencies for the isotropic wedge. 1975 Journal of Sound and Vibration 41,503-505. 5. A. P. BHATTACHARYA and K. N. BHOWMIC Free vibration of a sectorial plate. 6. V. X. KUNUKKASSERIL and R. RAMAKRISHNAN 1975Earthquake Engineering and Structural Dynamics 3, 217-232. Dynamic response of circular bridge decks. and V. X. KUNUKKASSERIL 1976Journal ofSound and Vibration 44, 7. R. RAMAKRISHNAN 209-221. Free vibration of stiffened circular bridge decks. 8. R. RAMAKRISHNAN ~~~V.X.KUNUKKASSERIL~~~~ EarthquakeEngineeringandStructural Dynamics 5, 377-394. Response of circular bridge decks to moving vehicles. 9. G. K. RAMAIAH and K. VIJAYAKUMAR 1974Journal of Sound and Vibration 34,53-61. Natural frequencies of circumferentially trancated sector plates with simply supported straight edges. 10. G. K. RAMAIAH 1980Journal of Sound and Vibration 70, 589-596. Flexural vibrations of polar orthotropic sector plates with simply supported straight edges. 11. Y. K. CHEUNG and M. S.CHEUNG 1971Journal of Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers 97, 391-411. Flexural vibration of the rectangular and other polygonal plates. 12. P. R. BENSON and E.HINTON 1976International Journalfor Numerical Methods in Engineering 10, 665-678. A thick finite strip solution for static, free vibration and stability problems. 13. M. S. CHEUNG and M. Y. T. CHAN 1981 Computers and Structures 14, 79-88. Static and dynamic analysis of thin and thick sectorial plates by the finite strip method. 1984Computers and Structures 18, 27-32. Dynamic 14. S. S. DEY and N. BALASUBRAMANIAN response of orthotropic curved bridge decks due to moving loads.
470
T. MIZUSAWA
15. T. MIZUSAWA Journal ofSoundand Vibration (to appear). Vibration of thick annular sector plates using semi-analytical methods. 16. R. S. SRINIVASAN and V. THIRUVENKATACHARI 1985 Computers and Structures 21,395-403. Static and dynamic analysis of stiffened plated. 17. R. S. SRINIVASAN and V. THIRUVENKATACHARI 1985 Journal of Sound and Vibration 101, 193-201. Free vibration of transverse isotropic annular sector Mindlin plates. 18. R. S. SRINIVASAN and V. THIRUVENKATACHARI 1986 Journal ofSound and Vibration 109, 89-96. Free vibration analysis of laminated annular sector plates. 19. P. K. DAS 1971Ph.D. Thesis of University of Pennsylvania. Coupled vibration of a horizontally curve bridge subjected to simulated highway loadings. 20. P. GURUSWAMY and T. Y. YANG 1979 Journal of Sound and Vibration 62, 505-516. A sector element for dynamic analysis of thick plates. 21. M. N. BAPU RAO, P. GURUSWAMY and K. S. SAMPATH KUMARAN 1977 International Journal of Nuclear Engineering and Design 41, 247-255. Finite element analysis of thick annular and sector plates. 22. M. D. WALLER 1952 Proceedings ofthe Royal Society (London) Series A 211,265-276. Vibrations of free plates: line symmetry; corresponding modes. 23. K. MARUYAMA and 0. ICHINOMIYA 1981 Journal of Sound and Vibration 74, 565-573. Experimental investigation of free vibrations of clamped sector plates. 24. M. SWAMINADHAM, J. DANELSKI and 0. MAHRENHOLTZ 1984 Journal of Sound and Vibration 95, 333-340. Free vibration analysis of annular sector plates by holographic experiments. 25. T. IRIE, G. YAMADA and F. ITO 1979 Journal of Sound and Vibration 67,89-100. Free vibration of polar-orthotropic sector plates. 26. T. IRIE, K. TANAKA and G. YAMADA 1988 Journal of Sound and Vibration 122, 69-78. Free vibration of a cantilever annular sector plate with curved radial edges. 27. M. MUKHOPADHYAY 1979 JoumalofSound and Vibration 63,87-95. A semi-analytical solution for free vibration of annular sector plates. 28. M. MUKHOPADHYAY 1982 Journal of Sound and Vibration 80, 275-279. Free vibration of annular sector plates with edges possessing different degrees of rotational restraint. 29. I. E. HARIK and H. R. MOLAGHASEMI 1989 Journal of Engineering Mechanics, Proceedings of American Society of Civil Engineers 115, 2709-2722. Anlytical solution of free vibration of sector plates. 30. I. E. HARIK and H. R. MOLAGHASEMI 1990 Journal of Sound and Vibration 138, 524-528. Vibration of sector plates on elastic foundations. 31. J. P. SINGH and S. S. DEY 1990 Journal of Sound and Vibration 136, 91-104. Vibrational finite difference method for free vibration of sector plates. 32. G. N. GEANNAKAKES 1990 Journal of Sound and Vibration 137, 283-303. Vibration analysis of arbitrarily shaped plates using beam characteristic orthogonal polynomials in the semianalytical finite strip method. 33. T. MIZUSAWA, T. KAJITA and M. NARUOKA 1979 Journal of Sound and Vibration 62,301-308. Vibration of skew plates by using B-spline functions. 34. T. MIZUSAWA and T. KAJITA 1987 Journal of Sound and Vibration 115,243-251. Vibration of skew plates resting on point supports.
and Vibration (1991) 149(3), 461-470
APPLICATION ANALYZE
OF THE
VIBRATION
SPLINE
ELEMENT
OF ANNULAR
METHOD
SECTOR
TO
PLATES
T. MIZUSAWA Department of Construction and Civil Engineering, Duido Institute of Technology, Hukusuicho-40, Minumi-ku, Nagoya 457, Japan (Received 3 July 1990, and in revised form 1 October 1990)
This paper deals with the vibrations of isotropic annular sector plates with arbitrary boundary conditions using the spline element method. To demonstrate the accuracy of the method, several examples are solved, and results are compared with those obtained by analytical methods and by other numerical methods. Good accuracy is obtained. Frequencies of annular sector plates with free edges and with point supports are presented.
1. INTRODUCTION of annular sector plates employed in engineering designs have been investigated by analytical methods [l-8] and by numerical methods such as the RayleighRitz method [9, lo], the finite strip method [ 1l-151, the integral equation method [ 16-181 and the finite element method [19-211. These problems have also been studied by experimental approaches [22-241. Recently, Irie er al. [25] analyzed the vibration of polar orthotropic annular sector plates by the Ritz method with quintic spline functions and circular beam functions as the displacement functions. They [26] also investigated cantilever annular sector plates using the Ritz method with a power series. Mukhopadhyay [27, 281 presented a finite strip-finite difference method to calculate the vibration of annular sector plates with arbitrary boundary conditions. Harik and Molaghasemi [29, 301 described an analytical finite strip method with Bessel functions for the vibration of annular sector plates. Singh and Dey [3 l] applied the variational finite difference method associated with an integralbased finite difference approach to analyze the vibration of annular sector plates. Geannakakes [32] presented a semi-analytical finite strip method mapping arbitrarily shaped plates into the natural co-ordinate plane using serendipity functions, and analyzed the vibration of annular sector plates. In this paper, the spline element method [33,34], which is a discrete Ritz method with B-spline functions, is used to analyze the vibration of isotropic annular sector plate on the basis of thin plate theory. The accuracy of the present method is discussed via comparison of the results calculated by anaytical methods and by numerical methods. Frequencies of annular sector plates with free edges and with point supports are presented.
The free vibrations
2. METHOD
OF ANALYSIS
The analysis is based on the spline element method, which can be regarded as an alternative form of the displacement-based finite element procedure; it has previously been used to analyze skew plate problems [33, 341. 461 0022-460X/91/
18046lf
10 %03.00/O
@ 1991 Academic Press Limited
462
T. MIZUSAWA
Figure
1. Annular
sector
plate and polar co-ordinate
systems.
The spline element method is used here to analyze the vibration of an isotropic annular sector plate with arbitrary boundary conditions, as shown in Figure 1. It is convenient to introduce the non-dimensional polar co-ordinates (6, 7) as follows: 9=0/+
and
77=
CrmRR,)IB9
(1)
where 4 is the sector angle, B = R, -Rip and R, and Ri are the outer and inner radii of the annular sector plate, respectively. The respective displacement functions are expressed in terms of two-way spline functions as
w(& 77)= w,(if> 7))= 1 p=,
c
CpqN,,k(~)Nq,k(d~
(2)
9=1
where ie = k + MB - 1, i, = k + M, - 1, N,,,k(&) and Nn,k( 7)) are the normalized B-spline functions, k - 1 is the degree of the B-spline functions, MB and M, are the numbers of elements in the, 8- and r-directions, respectively, and the Cmns are unknown parameters which can be determined by the minimum total potential energy theorem. The method of artificial springs [32] is applied to deal with arbitrary boundary conditions. Three types of artificial spring, (Y,p and -y, corresponding to the deflection W, the slope deflection d W/a0 and d W/h, respectively, are introduced at each edge of the plate. The energy contribution lJ, due to these springs is given by ~b=~[~~W2+(~/~2)(~w/~~)2+(~/~2)(~~/~~)2~~~=0
+~~~*+~~/~*~~~~/~5~*+~(Y/~*~~~~/~77~*~1~=1 +~~~2+~~l~2~~~~l~~~2+~Y/~2~~~~l~71~2~l~=0
+~~~*+~~/~*~~~~/~~~*+~r/~*~~~~/~r1~*~1,=,1.
(3)
The strain energy due to bending, UP and the kinetic energy, T,, of the annular sector plate are given by 4 :” [(a’w,ar’+(l,r) d W,ar+(1,r’) a* W,df3*)z 4 = (D/2) II0
VIBRATION
OF ANNULAR
SECTOR
PLATES
463
1 1 = (W2)WR2)
II0 0
[{a2w/av2+(i/A)
aW/a7)+(l/A2/42)
a2W/at2)’
-2(1--~)a~W/a~~{(1~A)aW/a~+(1/A~/4~)a~W/a~~) +2(1-
4UllAld.d
a’Wla7
aW%)*lA d? de,
at-U/A2/44
where A=(77+Ri/B)={77+1/(S_l)},S=R,/Ri,
and
W2r dr de = (ph/2)02t$B2
Tp = (ph/2)02
(4)
W’(7) + RI/B) dv d5,
(5)
in which v is the Poisson ratio, p is the density, h is the thickness, D is the flexural rigidity of the plate, r$ is the sector angle and o is the frequency (rad/s). The functional of the annular sector plate Z7 is expressed by substituting equation (1) into Z7 as follows:
+Wzq
+(PI~')N~~~+(YIB~)NO,~,O~~}I~=,I
- (ph,2)c$B2
; ; ; ; C,,,,C,,[Zllp,J~],
(6)
mnPq
where Niyp,, N
I$, and .Zii’iare defined as follows: ijkl mw
=d’Nm,kWdS
djNn,k($ldd
dkNp,kWdSk
d’Nq.k(n)/drt’,
1
1”
mp
=
I d’%,&)/&
diNp,&Vd5’
d5,
0
1
JZ=
I 0
diNn,/c(T)ldTi d’N,.k(~)ldt7’(77+RiIB)’ dV*
(7)
Here the definite integrals, Zz, and Jg in equations (7) are calculated numerically by the Gauss-Legende quadrature formulae. By using the principle of minimum potnetial energy, the coefficients {C,,,,} are determined as solutions of aZMC,,I
= 0,
(8)
which may be expressed in matrix form as
rKI{Cm”I= A2[W{Crn”~.
(9)
T.
464
MIZUSAWA
Here A is a frequency parameter defined by B’wm. The order of the matrices is given by (k + Me - 1) x (k + M, - l), in which k - 1 is the degree of the spline functions, and MB and kf, are the numbers of mesh divisions in the analytical region in the & and r-directions, respectively. [K] and [M] are stiffness and mass matrices obtained from equations (4) and (5). 3. NUMERICAL EXAMPLES AND DISCUSSION The vibrations of isotropic annular sector plates with various boundary conditions are analyzed and the accuracy of the present method for these problems is demonstrated by comparison with the existing results calculated by analytical and numerical methods. In all calculations, the degree of the B-spline functions of k - 1 = 4 and the mesh divisions of MB = M, = 12 were used. In Figure 2, the definitions of the boundary conditions are presented. The symbolism SC-SF, for example, identifies an annular sector plate with the edges 5 = O,t = 1, ~7= 0 and ~7= 1 having simply supported, clamped, simply supported and free boundary conditions, respectively.
-,
Figure 2. Definition of the boundary conditions. -, F-free edge.
TABLE
C-clamped
edge; ==,
S-simply
supported edge;
1
Convergence study of frequency parameters, A, of a clamped annular sector plate; R,l Ri = 2.0, ~ = 30”
Modes Degrees of spline, k-l
3
4
M,=M,
1st
2nd
3rd
4th
5th
4x4 6x6 8x8
48.27 48.08 48.05
86.70 85.91 85.62
107.6 105.0 104.5
154.8 146.2 143.5
245.5 152.5 151.6
10x 10 12x12
48.05 48.04
85.56 85.54
104.4 104.4
143.0 142.9
151.4 151.3
4x4 6x6 8x8 10x 10 12 x 12
46.07 48.01 48.04 48.04 48.04
83.12 85.43 85.51 85.52 85.52
100.6 104.4 104.4 104.4 104.4
134.0 142.1 142.7 142.8 142.8
143.0 151.2 151.3 151.3 151.3
4x4 6x6 8x8 10x10 12x12
46.77 47.98 48.03 48.04 48.04
82.76 85.17 85.47 85.50 85.52
101.0 103.9 104.3 104.4 104.4
138.0 142.6 142.6 142.7 142.8
140.4 150.0 151.0 151.2 151.3
VIBRATION
OF ANNULAR
TABLE
Frequency
SECTOR
465
PLATES
2
for annular sector plates with several boundar?, parameter, A = wB2Jm, conditions; Y = 0.0, 4 = 90”, k - 1 = 4 and M, = M, = 12 Modes 1st
2nd
3rd
4th
5th
Methods
10.09 10.57
10.68 12.48
11.66 15.67
13.04 20.13
14.82 25.85
Present Present
2-o
11.80 11.79 11.66
17.08 17.07 16.73
25.81 25.81 24.57
37.68 37.69 34.06
41.68 41.43 40.31
Present Reference [27] Reference [ 3 l]
4.0
15.97 15.94 15.70
31.93 31.90 31.05
47.89 47.64 46.25
54.94 54.93 51.75
69.96 69.72 67.27
Present Reference [27] Reference [ 3 1 ]
22.46 22.70
22.74 23.83
23.27 25.91
24.00 29.14
25.09 33.71
Present Present
2.0
23.33 23.00 22.40
26.88 26.56 25.79
33.87 33.57 32.01
44.65 44.38 40.61
58.96 61.29 50.48
Present Reference [27] Reference [ 3 1]
4.0
25.80 25.43 24.78
40.20 39.86 38.59
64.30 63.84 59.71
66.64 64.87 61.35
84.77 83.18 78.92
Present Reference [ 271 Reference [3 11
RJR, (a) SS-SS 1.25 1.5
(b) SS-CC 1.25 1.5
(c) SS-FF 1.25 1.5
0.1384 0.4395
0.7207 2.263
1.693 4,526
2.0
1.160 1.185 1.145
5.709 5.834 5.500
8.493 8,769 8.391
4.0
3,185 3.294 3.127
2.375 5.253
3.050 8.915
Present Present
12.68 13.02 11.76
17.44 17.96 16.98
Present Reference [ 271 Reference [ 3 1 ] Present Reference [27] Reference [ 3 11
13.55 14.02 13.02
18.87 19.37 18.52
28.88 30.03 26.69
41.85 42.95 40.07
22.49 22.80
22.86 24.20
23.51 26.76
24.44 30.62
25.75 35.93
Present Present
2.0
23.80 23.84 23.52 23.83
28.73 28.79 28.50 28.69
37.63 37.52 37.32 38.00
50.33 50.43 49.80 52.51
63.30 63.53 61.74 63.37
Present Reference [ 1 l] Reference [27] Reference [32]
4.0
29.58 29.60 29.26 29.62 29.64
49.55 49.60 49.28 49.48 49.95
69.54 69.68 68.16 69.58 70.37
77.09 77.37 76.51 78.78 78.75
94.66 94.89 93.74 94.51 95.85
Present Reference Reference Reference Reference
(d) CC-CC 1.25 1.5
[ 1 l] [27] [32] [ 131
T. MIZUSAWA
466
In Table 1 is shown the convergence study of the the present method for a clamped annular sector plate ( C$= 30”, R,/ Ri = 2.0) by changing the number of mesh divisions, MB = M,, and the degrees of the spline functions, k - 1. Good convergence is obtained by increasing the number of mesh divisions. In Table 2 is shown the comparison of the first five frequencies parameters, A = wB*m of annular sector plates having various boundary conditions with C#J = 90” and Y= 0.0. The radii ratio, R,/ Ri, varies from 1.25 to 4.0. The results obtained by the semi-analytical method [27] and the variational finite difference method [31] are listed. In the case of an all-clamped annular sector plate, in addition to these values, the solutions
TABLE
Frequency
parameter, A = wB*m, for annular sector plates with several boundary conditions; v=O*3, k-1=4 R,/R,=2*0 andM*=M,=12
Modes
Sector angle 4 (degrees)
Boundary conditions
30
45
60
90
3
r 1st
2nd
3rd
4th
5th
Methods
SS-FF
11.77 11.77 11.77
28.87 28.87 28.87
44.03 44.03 48.72
50.51 50.55 50.53
82.13 80.41 -
Present Reference [ 71 Reference [ 321
ss-ss
25.86
57.16
69.60
106.8
109.7
Present
ss-cc
33.90
75.11
76,46
121.8
135.3
Present
cc-cc
48.04
85.52
142.8
151.3
Present
SS-FF
5.267 5.267 5.268
104.4
16.68 16.69 16.68
20.40 20.40 22.59
36.60 36.66 36.61
44.03 44.03 -
Present Reference [ 71 Reference [ 321
ss-ss
17.09 17.09
37.75 37.74
47.40 47.40
69.60 69.61
70.90 70.90
Present Reference [ 71
ss-cc
26.89 26.91
44.70 44.69
67.37 74.03
76.45 77.34
86.60 88.32
Present Reference [7]
cc-cc
31.39
56.85
70.22
94.54
96.73
Present
11.77 11.77 11.87
11.87 11.89 13.06
25.51 25.51 30.73
28.87 28.87 -
Present Reference [ 71 Reference [32]
SS-FF
2,856 2.856 2.857
ss-ss
13.99
25.86
44.00
44.75
57.16
Present
ss-cc
24.74
33.89
51.44
64.79
75.08
Present
cc-cc
26.53
39.85
61.49
65.96
79.38
Present
7.779
11.77 11.77 15.01
16.68 -
-
Present Reference [ 31 Reference [32]
SS-FF
ss-ss
1.068 1.068 1.069 11.77
5.267 5.267 5.858
7.780 7,779
17.09
25.86
37.75
41.59
Present
44.65
58.96
Present
50.33 52.51 50.42
63.30 63.37 63.30
Present Reference [32] Reference [ 321
ss-cc
23.33
26.88
33.87
cc-cc
23.80 23.83 23.80
28.73 28.69 28.77
37.63 38.00 37.69
VIBRATION
OF
ANNULAR
SECTOR
467
PLATES
calculated by the finite strip method [ll, 131 and the semi-analytical finite strip method [32] are also shown. It is seen that the values calculated by the present method show good agreement for all the cases. In Table 3 are shown the frequency parameters A, of annular sector plates (Y = 0.3, R,/ Ri = 2.0) with several boundary conditions. The sector angle I$ varies from 30” to 90”. The results are compared with those obtained by the analytical method [3, 71 and the finite strip method [32]. In the case of annular sector plates with simply supported straight edges, excellent agreement is shown compared with the analytical values. In Table 4 are shown the first five frequency parameters, A, of annular sector plates ( v = 0.3 and R,/ Ri = 2.0) with several clamped edges and free edges. In Table 5 are shown the frequency parameters, A, of cantilever annular sector plates ( v = 0.3 and R,/ Ri = 2.0) for the different sector angles. In this case, the results for 4 = 90” are compared with the numerical solutions calculated by the Ritz method [26] and the finite element method [24], and with the experimental results [24]. Good agreement is obtained. Lastly, the frequency parameters of annular sector plates with rigid point supports are shown in Table 6. The arrangement of the point supports is depicted in Figure 3. The frequencies of the plates are influenced by the sector angles and the positions of point supports. 4. CONCLUSIONS The spline element method has been applied to analyze vibrations of isotropic annular sector plates with various boundary conditions. The accuracy of the method has been demonstrated in the comparative studies between the present results and the existing values calculated by analytical and numerical methods.
TABLE
4
Frequency parameter, A = wB2m, for annular sector plates with free edges; v = 0.3, k- 1~4, R,/Ri*2*0 and MB= M, = 12; (a) one free edge; (b) two free edges
Modes -__
Sector angle, 4 (degrees)
Boundary conditions
30
CC-FC
1st
2nd
3rd
4th
46.75 25.12 25.06
69.57 49.91 52.45
99.17 65.77 64.29
104.4 87.17 95.20
149..3 110.6 108.7
CC-CF CF-CC
16.92 I.435 22.37
33.73 18.04 28.33
36.32 26.82 41.91
57.62 33.98 60.85
60.64 2.45 63.38
30
CC-FF FF-CC CF-FC CF-FC
25.12 21.94 13.04 6.649
48.34 28.36 31.84 23.30
65.77 56.63 49.46 28.94
70.54 60.64 69.22 57.05
100.4 70.22 76.94 63.74
60
CC-FF FF-CC CF-FC CF-CF
6.926 21.82 6.905 3.707
17.09 23.25 18.31 8.384
17.96 30.08 25.21 18.55
33.82 43.84 37.16 22.09
34.20 60.17 39.50 30.27
5th
(a)
CC-CF CF-CC 60
(b)
CC-FC
TABLE
5
A = wB2m, for cantilever annular sector plates; R,/Ri=2~0andMO=M,=12
Frequency parameter,
v = 0.3, k - 1 = 4,
Modes
Sector angle, $I (degrees)
Boundary conditions
30
45
60
90
p 1st
2nd
3rd
4th
5th
CF-FF
5.292
13.14
26.57
32.68
5345
FF-CF
2.884
19.91
29.28
31.74
FF-FC
4.232
23.03
41.22
51.98
CF-FF
2.480
7.686
13.14
26.82
28.35
FF-CF
2.968
5.316
14.06
21.10
25.26
FF-FC
4.250
23.02
28.63
33.12
7.286 14.31
10.82
CF-FF
1449
5.147
8.121
17.57
20.97
FF-CF
3.034
4.387
9.289
18.99
20.80
FF-FC
4.262
8.982
23.11
29.21
CF-FF
0.6930
2.587
4.600
8.426
12.41
FF-CF
3.112 3.122
3.604 3.604
5.713 5.696
10.02 9.948
16.72 16.59
Reference
[26]
FEM
3.114
3.662
5.814
9.876
15.94
Reference
[ 241
Experiment
3.076
3.593
5.730
10.09
16.82
Reference
[ 241
FF-FC
4.272
7,018
21.58
22.99
TABLE
Frequency parameters,
Sector angle, qb (degrees)
Method
19.55
12.63
6
A = oB2m, of point supported annular sector plaes; v = 0.3, k-1=4, R,/Ri=24andMO=M,=12 Modes
Boundary conditionst
1st
2nd
3rd
4th
5th
30
4 points A 6 points B 6 points C 8 points D
8.012 11.51 9.614 22.79
16.16 25.38 18.88 36.66
18.88 38.35 36.11 38.35
36.03 44.70 42.43 56.48
42.43 56.48 49.25 67.38
45
4 points 6 points 6 points 8 points
A B C D
4.729 5.231 9.372 14.84
12.45 16.15 13.65 19.80
13.65 19.80 23.95 29.14
22.23 35.83 26.69 38.24
23.95 38.24 30.68 47.20
60
4 points A 6 points B 6 points C 8 points D
2.720 2.847 8.671 11.02
9.900 11.64 9.900 11.64
10.75 11.76 17.08 21.40
16.14 24.7 1 19.34 26.99
17.08 26.99 22.26 35.93
90
4 points 6 points 6 points 8 points
1.048 1.066 5.003 5.249
5.003 5.249 6.147 6.564
7.718 7.772 12.94 14.38
10.33 11.68 13.87 16.41
12.94 16.41 14.59 20.09
t Boundary
conditions
A B C D
are defined
in Figure 3.
VIBRATION OF ANNULAR
(a)
SECTOR
PLATES
469
(b)
Cc)
(d)
Figure 3. Depiction of point supported annular sector plates and the arrangement of point supports.(a) four points A; (b)six points B; (c) six points C; (d) eight points D.
Good agreement has been shown. The frequency parameters of annular sector plates with free edges and with point supports, which can be difficult to solve analytically, have also been presented. The present method can be used, without difficulty, to analyze annular sector plates with a variable thickness or with stiffeners.
REFERENCES 1. M. BEN-AMOZ 1959 Journalof Applied Mechanics 26,136-137. Note on deflections and flexural vibrations of clamped sectorial plates. 2. H. YONEZAWA 1962 Journal of Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers 88, l-21. Moments and free vibrations in curved girder bridges. and V. X. KUNUKKASSERIL 1973Journal ofSound and Vibration 30, .3 . R. RAMAKRISHNAN 127-129. Free vibration of annular sector plates. 4. C. RUBIN 1975Journal of Sound and Vibration 39,523-526. Nodal circles and natural frequencies for the isotropic wedge. 1975 Journal of Sound and Vibration 41,503-505. 5. A. P. BHATTACHARYA and K. N. BHOWMIC Free vibration of a sectorial plate. 6. V. X. KUNUKKASSERIL and R. RAMAKRISHNAN 1975Earthquake Engineering and Structural Dynamics 3, 217-232. Dynamic response of circular bridge decks. and V. X. KUNUKKASSERIL 1976Journal ofSound and Vibration 44, 7. R. RAMAKRISHNAN 209-221. Free vibration of stiffened circular bridge decks. 8. R. RAMAKRISHNAN ~~~V.X.KUNUKKASSERIL~~~~ EarthquakeEngineeringandStructural Dynamics 5, 377-394. Response of circular bridge decks to moving vehicles. 9. G. K. RAMAIAH and K. VIJAYAKUMAR 1974Journal of Sound and Vibration 34,53-61. Natural frequencies of circumferentially trancated sector plates with simply supported straight edges. 10. G. K. RAMAIAH 1980Journal of Sound and Vibration 70, 589-596. Flexural vibrations of polar orthotropic sector plates with simply supported straight edges. 11. Y. K. CHEUNG and M. S.CHEUNG 1971Journal of Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers 97, 391-411. Flexural vibration of the rectangular and other polygonal plates. 12. P. R. BENSON and E.HINTON 1976International Journalfor Numerical Methods in Engineering 10, 665-678. A thick finite strip solution for static, free vibration and stability problems. 13. M. S. CHEUNG and M. Y. T. CHAN 1981 Computers and Structures 14, 79-88. Static and dynamic analysis of thin and thick sectorial plates by the finite strip method. 1984Computers and Structures 18, 27-32. Dynamic 14. S. S. DEY and N. BALASUBRAMANIAN response of orthotropic curved bridge decks due to moving loads.
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T. MIZUSAWA
15. T. MIZUSAWA Journal ofSoundand Vibration (to appear). Vibration of thick annular sector plates using semi-analytical methods. 16. R. S. SRINIVASAN and V. THIRUVENKATACHARI 1985 Computers and Structures 21,395-403. Static and dynamic analysis of stiffened plated. 17. R. S. SRINIVASAN and V. THIRUVENKATACHARI 1985 Journal of Sound and Vibration 101, 193-201. Free vibration of transverse isotropic annular sector Mindlin plates. 18. R. S. SRINIVASAN and V. THIRUVENKATACHARI 1986 Journal ofSound and Vibration 109, 89-96. Free vibration analysis of laminated annular sector plates. 19. P. K. DAS 1971Ph.D. Thesis of University of Pennsylvania. Coupled vibration of a horizontally curve bridge subjected to simulated highway loadings. 20. P. GURUSWAMY and T. Y. YANG 1979 Journal of Sound and Vibration 62, 505-516. A sector element for dynamic analysis of thick plates. 21. M. N. BAPU RAO, P. GURUSWAMY and K. S. SAMPATH KUMARAN 1977 International Journal of Nuclear Engineering and Design 41, 247-255. Finite element analysis of thick annular and sector plates. 22. M. D. WALLER 1952 Proceedings ofthe Royal Society (London) Series A 211,265-276. Vibrations of free plates: line symmetry; corresponding modes. 23. K. MARUYAMA and 0. ICHINOMIYA 1981 Journal of Sound and Vibration 74, 565-573. Experimental investigation of free vibrations of clamped sector plates. 24. M. SWAMINADHAM, J. DANELSKI and 0. MAHRENHOLTZ 1984 Journal of Sound and Vibration 95, 333-340. Free vibration analysis of annular sector plates by holographic experiments. 25. T. IRIE, G. YAMADA and F. ITO 1979 Journal of Sound and Vibration 67,89-100. Free vibration of polar-orthotropic sector plates. 26. T. IRIE, K. TANAKA and G. YAMADA 1988 Journal of Sound and Vibration 122, 69-78. Free vibration of a cantilever annular sector plate with curved radial edges. 27. M. MUKHOPADHYAY 1979 JoumalofSound and Vibration 63,87-95. A semi-analytical solution for free vibration of annular sector plates. 28. M. MUKHOPADHYAY 1982 Journal of Sound and Vibration 80, 275-279. Free vibration of annular sector plates with edges possessing different degrees of rotational restraint. 29. I. E. HARIK and H. R. MOLAGHASEMI 1989 Journal of Engineering Mechanics, Proceedings of American Society of Civil Engineers 115, 2709-2722. Anlytical solution of free vibration of sector plates. 30. I. E. HARIK and H. R. MOLAGHASEMI 1990 Journal of Sound and Vibration 138, 524-528. Vibration of sector plates on elastic foundations. 31. J. P. SINGH and S. S. DEY 1990 Journal of Sound and Vibration 136, 91-104. Vibrational finite difference method for free vibration of sector plates. 32. G. N. GEANNAKAKES 1990 Journal of Sound and Vibration 137, 283-303. Vibration analysis of arbitrarily shaped plates using beam characteristic orthogonal polynomials in the semianalytical finite strip method. 33. T. MIZUSAWA, T. KAJITA and M. NARUOKA 1979 Journal of Sound and Vibration 62,301-308. Vibration of skew plates by using B-spline functions. 34. T. MIZUSAWA and T. KAJITA 1987 Journal of Sound and Vibration 115,243-251. Vibration of skew plates resting on point supports.