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Natural frequencies of rectangular plates with free edges
Journal of Sound and Vibration (1986) 105(3), 451-459
NATURAL FREQUENCIES OF RECTANGULAR PLATES WITH FREE EDGES T. MIZUSAWA Department of Construction Engineering, Daido Institute of Technology, Hakusuicho-40, Minami-ku, Nagoya, Japan (Received 28 November 1984, and in revised form 16 April 1985) This note presents vibration analysis of isotropic rectangular plates with free edges by the Rayleigh-Ritz method with B-spline functions. To show the accuracy of the present method, the results are compared with existing results based on other numerical methods and found to be in good agreement. Accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions. The effects of Poisson’s ratio on natural frequencies of square plates with free edges are also investigated.
1. INTRODUCTION [l-4] exists for the free vibrations of rectangular plates with free edges. Except for cases with two opposite edges simply supported, exact solutions do not exist, as mentioned in Leissa’s work [5]; therefore a variety of numerical methods has been used to analyze vibrations of rectangular plates with free edges. However the proper selection of the displacement functions is essential to obtain stable and accurate results for rectangular plates with free edges [6]. Warburton [7] presented an approximate solution for rectangular plates with combinations of free edges, obtained by using the Rayleigh-Ritz method with beam functions. Leissa [5] obtained comprehensive and accurate analytical results for vibration of rectangular plates in 21 distinct cases which involve all possible combinations of classical boundary conditions, employing the Ritz method with beam functions. Natural frequency parameters of rectangular plates with free edges evidently are influenced by Poisson’s ratio. In this note, vibrations of isotropic rectangular plates with free edges are analyzed by using the Ritz discretization procedure with B-spline functions [8,9] and results are compared with those obtained by other numerical methods. To show more accurate results for rectangular plates with free edges, the first five frequencies of plates with different aspect ratios and combinations of free edges have been calculated. The influence of Poisson’s ratios on the frequencies has also been investigated. A vast literature
2. METHOD OF ANALYSIS
The analysis is based on the spline element method, which discretization procedure with B-spline functions; it has been used skew plate problems [8,9]. The present method may be regarded of the displacement formulation of the finite element procedure total potential energy theorem is used to develop the relationship parameters and the applied loading. 451
0022-460X/86/060451 + 10$03.00/O
@ 1986Academic
is the Rayleigh-Ritz previously to analyze as an alternative form in that the minimum between the unknown
Press Inc. (London)
Limited
452
T. MIZUSAWA
1
Clomped
I
’ Simply
b
1 edges I
edge’ C
supported.
s
Free edge: F
Figure
1. Rectangular
plate and symbols
for boundary
conditions.
In this note, the spline element method is used to analyze vibrations of isotropic rectangular plates with free edges, as shown in Figure 1. The assumed displacement functions for a finite element take the form of two-way spline functions and are given by w=
; ; GJCl,,(~)~“,,(Y) m=, n=l
(1)
in which ix = k + M, - 2, i,, = k+ M,, - 2, and N,,+(X) and N,,,(y) are the normalized B-spline functions. In this formulation, k - 1 is the degree of the B-spline functions, M,
TABLE
1
Convergence study of frequency parameters, n*, of rectangular plate with SSCF;
Aspect ratio
Degree of spline
A=a/b
k-l
0.4
Leissa’s
1.0
Leissa’s
2.0
Number of nodes
Mx=My
Y = O-3
Modes Order of matrix?
\
r 1st
2nd
3rd
4th
5th
3
3 5 9 11 13 exact values [5]
25 49 121 169 225
10.204 10.190 10.189 10.189 10.189 10.189
13.630 13.606 13.604 13,604 13,604 13.604
20.516 20.116 20.098 20.097 20.097 20.097
39.701 30.643 29.626 29.623 29.622 29,622
42.811 39.675 39.639 39.639 39.638 39.638
3
3 5 9 11 13 exact values [5]
25 49 121 169 225
12.564 12.681 12.687 12.687 12.687 12.687
32.298 33.024 33.063 33.064 33.065 33.065
41,234 41.685 41.701 41.702 41.702 41.702
59,220 62.719 63.002 63.011 63.012 63.015
70.818 72.395 72.392 72.395 72.396 72.398
3
25 49 121 169 225
21.503 22.750 22.812 22.814 22.815
46.760 50.273 50.730 50.742 50.746
92.255 98.873 98.749 98.765 98.771
111.27 99.954 99.750 99.765 99.770
3 5 9 11 13
tTheorderofthematrixisgivenby(k+M,-2)x(k+M,-2).
138.34 128.90 132.12 132.22 132.24
FREQUENCIES
OF RECTANGULAR
453
PLATES
and M,, are the number of nodes in the x- and y-directions, respectively and the Cmn’s are unknown parameters. The details of the analysis formulation have been carried out by using the procedure described in reference [8]. 3. NUMERICAL
RESULTS AND DISCUSSSIONS
In Figure 1, the definitions of the boundary conditions are presented. The symbolism SCSF, for example, identifies a rectangular plate with the edges x = 0, x = a, y = 0, y = b having simply supported, clamped, simply supported, and free boundary conditions, respectively (see Figure 1). The order of the matrices obtained is given by (k+ M, -2) x (k + MY - 2). In the calculations of several examples, k - 1 = 3 and &f, = M,, = 13 are used. To demonstrate the accuracy of the present method, a few example problems were solved to obtain results for comparison with those obtained by other numerical approaches. Table 1 shows the influence of the number of nodes on the convergence of natural of rectangular plates with SSCF for different frequency parameters, n* = wa’~, aspect ratios and numbers of nodes. Here k - 1 = 3 and v = 0.3 are assumed. Stable convergence was obtained with increase in the number of nod&. Results are also compared with Leissa’s exact values. All show a good agreement. Table 2(a) shows the first five frequency parameters, n* = wa*m, of rectangular plates with one free edge for aspect ratios A = a/b of 1-O and 1.5. Here D is the flexural TABLE
2
Comparison of the frequency parameters, n*, for rectangularplates and M, = MY = 13
(v = O-3); k - 1 = 3
(a) one free edge Aspect ratio; A = a/b Boundary conditions
(
1.0
1.5
A
Modes
r
*
\
CFCC
1 2 3 4 5
23.71 39.77 62.68 76.54 80.05
(24.02) (70.04) (63.49) (76.76) (80.71)
50.69 65.26 98.98 137.3 154.3
(51.78) (66.21) (99.82) (140.4) (155.0)
CFCS
1 2 3 4 5
17.47 35.93 51.60 70.82 74.28
(17.62) (36.05) (52.07) (71.19) (74.35)
36.18 54.06 91.21 112.9 131.8
(36.65) (54.45) (91.58) (114.3) (132.9)
SFCC
1 2 3 4 5
23.20 35.36 62.37 66.58 76.89
(23.46) (35.61) (63.13) (66.81) (77.50)
SO.43 62.05 90.70 137.2 140.3
(51.42) (62.93) (91.54) (140.2) (141.1)
SFCS
1 2 3 4 5
16.73 31.03 51.20 63.93 67.34
(16.87) (31.14) (51.63) (64.04) (67.65)
35.73 50.14 82.22 112.7 129.2
(36.15) (50.51) (82.58) (114.1) (130.3)
) are results obtained
by Leissa [5] using the Ritz method with beam functions.
454
T.
MIZUSAWA
TABLE 2 (cont.) (b) TWO free edges Aspect ratio: A = a/b Boundary conditions
1.0 Modes
1.5
y
CFCF
1 2 3 4 5
6.883 23.10 26.56 41.34 62.54
(6.942) (24.03) (26.68) (47.79) (63.04)
11.11 29-59 51.98 67.50 76.40
(11.22) (29.90) (52.61) (68.09) (77.04)
CFSF
1 2 3 4 5 . 1 2 3 4 5
5,338 19.03 24.61 42-95 52.71
(5.364) (19.17) (24.77) (43.19) (53.00)
6.910 27.14 38.38 63.90 67.16
(6.931) (27.29) (38.59) (64.25) (67.47)
22.03 26.05 43.20 60.72 66.25
(22.27) (26.53) (43.66) (61.47) (67.55)
[22.17]* [26.40]* [43.6]* [61-2]* [67+2]*
49.30 52-93 69.64 103.8 135.7
(50.21) (54.70) (71.33) (105.3) (138.5)
1 2 3 4 5
15.15 20.46 39.58 49.21 55.90
15.29 20.67 39.78 49.73 56.62
[ 15.191** [20-57]** [39.72]** [49.52]** [56.32]**
34.10 39.65 59.45 96.48 110.8
34.52 40.39 60.15 97.18 112.1
FFCC
FFCS
( ) are results obtained by Leissa [S] using the Ritz method with beam functions, [ ]* are values calculated by Claassen and Thome [IO] utilizing the series method and [ ]** are solutions obtained by Fujii and Hoshino using the mixed approach combined with spline functions 111 J. (c) cantilever plates
Aspect ratio; A = a/b Boundary conditions CFFF
1.0 Modes 1 2 3 4 5
1.5
I 3.467 (3.492) 8462 (8.525) 21.19 (21.43) 27.18 (27.33) 30.77 (31.11)
[3.431]* [20-87]* [26-50]*
[3.473]** [21-30]** [27.29]**
3.452 11.63 2144 39.23 53.54
(3.477) (11.68) (21.62) (39.49) (53.88)
( ) are results obtained by Leissa [5] using the Ritz method with beam functions, [ ]* are lower bounds calculated by Bazley et al. [12] utilizing the Rayleigh-Ritz method with beam functions and [ 1**are upper bounds obtained by Sigillito [ 131 using the Ritz method with products of beam functions and Legendre functions.
rigidity, h is the thickness and p is the density of the plate, and w is the frequency (rad/s). The values obtained by Leissa [5] using the Ritz method with 36 terms containing the products of beam functions are also shown in comparison with the present results. It is evident that the results obtained by the present method are in good agreement with Leissa’s solutions, within an accuracy of 5%.
FREQUENCIES
OF RECTANGULAR
PLATES
455
Table 2(b) shows the first five frequency parameters of rectangular plates with two free edges. In the cases of square plates, Leissa’s results [5] calculated by the Rayleigh-Ritz method with beam functions, results obtained by Claassen and Thorne [lo] using the series method, and solutions calculated by Fujii and Hoshino [ 111 utilizing mixed approaches combined with spline functions are also shown. It is evident that good agreement is obtained, the present results being slightly less exact than those calculated by Leissa, by Claassen and Thorne and by Fujii and Hoshino. Frequency parameters of cantilever rectangular plates are also shown in Table 2(c). To show the accuracy of the present method, the present results are compared with Leissa’s values, lower bound solutions obtained by Bazley et al. [ 121 using the RayleighRitz method with beam functions and upper bounds of the solutions calculated by Sigillito [13] utilizing the Ritz method with products of beam functions and Legendre functions. It can be seen that the present results are always between upper bounds and lower bounds. TABLE
3
Frequency parameters, n*, of rectangular plates (u = 0.3); Aspect Boundary conditions
ratio:
k - 1 = 3 and M, = MY = 13 A
I Modes
0.5
0.75
1.25
1.0
SFSS
1 2 3 4 5
4.034 11.68 18.82 24.01 27.76
(a) one free 11.68 22.70 27.76 41.20 24.01 41.17 59.07 51.63 61.86
CFSS
1 2 3 4 5
5.704 12.69 24.69 24.94 33.06
8.570 24.69 28.41 45.75 52.07
SFCS
1 2 3 4 5
5.142 14.06 19.33 27.66 29.30
SFCC
1 2 3 4 5
CFCS
CFCC
1.5
1.75
2.0
edge 17.25 33.92 63.29 68.23 81.61
24.01 41.17 75.82 90.29 108.9
31.99 49.55 84.58 122.2 138.4
41.20 59.07 94.48 148.5 159.1
12.69 33.06 41.70 63.01 72.40
18.06 38.85 63.68 78.29 85.02
24.69 45.75 85.40 90.61 111.9
32.58 53.80 93.69 122.5 143.7
41.70 63.01 103.2 159.3 162.4
10.02 24.24 29.58 45.38 57.68
16.73 31.03 51.20 63.93 67.34
25.30 39.66 72,12 78.92 95.27
35.73 50.14 82.22 112.7 129.2
48.02 62.48 94.23 145.6 152.6
62.1.6 76.69 108.2 159.1 198.2
6.592 16.77 19.94 31.08 31.63
13.55 26.23 35.85 50.28 58.78
23.20 35.36 62.37 66.58 76.89
35.50 47.31 77.19 96.22 111.0
50.43 62.05 90-70 137.2 140.3
67.92 79.57 107.1 155.3 185.1
87.89 99.87 126.5 173.1 239.6
1 2 3 4 5
6.564 14.92 25.35 28.26 34.42
11.01 29.67 30.18 49.63 60.35
17.47 35.93 51.60 70.82 74.28
25.87 44.05 79.18 81.78 98.27
36.18 54.06 91.21 112.9 131.8
48.38 65.98 102.6 152.5 159.6
6244 79.83 115.9 172.2 198.3
1 2 3 4 5
7.764 17.51 25.84 32.16 35.99
14.30 31.39 36.3 1 54.20 69.30
23.71 39.77 62.68 76.54 80.05
35.87 51.07 86.31 96.41 113.5
50.69 65.26 98.98 137.3 154.3
68.09 82.33 114.6 168.4 185.2
87.99 102.3 133.3 185.4 239.6
7.302
T. MIZUSAWA
456
TABLE 3 (cont.)
Boundary conditions
Aspect ratio: A
r Modes
, 0.5
0.75
1.0
1.25
1.5
1.75
2.0
(b) two free edges FFSS
1 2 3 4 5
2.378 6.881 9.63 1 16.13 21.82
5.388 11.10 21.82 29.21 30.91
9.631 16.13 36.73 38.95 46.74
15.11 22.14 43.66 61.00 68.98
21.82 29.21 51.65 87.99 90.80
29.77 37.39 60.67 100.4 120.0
38.95 46.74 70.74 111.0 156.8
FFSC
1 2 3 4 5
3.762 7.811 12.25 18.18 25.67
8.502 13.38 27.68 32.26 34.18
15.15 20.46 39.58 49-27 55.90
23.68 29.19 48.66 76.98 83.34
34.10 39.65 59.45 96.48 110.8
46.39 51.87 71.97 108.9 150.5
60.53 65.86 86.24 123.1 178.9
FFCC
1 2 3 4 5
5.503 8.963 15.17 20.53 27.34
12.40 16.31 33.92 34.20 39.87
22.03 26.05 43.20 60.72 66.25
34.34 38.25 55.11 90.30 94.61
49.30 52.93 69.64 103.8 135.7
66.84 70.07 86.79 120-l 173.7
86.91 89.63 106.6 139.1 191.5
SFSF
1 2 3 4 5
1.661 6.344 14.69 16.29 22.27
2.518 11.39 17.10 29.19 30.78
3.367 17.32 19.29 38.21 51.04
4.202 19.65 27.14 48.40 53.83
5.024 21.46 37.53 55.22 60.74
5.837 23.37 49.61 57,72 74.04
6.644 25.38 58.74 65.18 89.10
CFSF
1 2 3 4 5
4.005 7.800 15.76 22.35 27.56
4.613 12.60 23.20 31.07 34.75
5.338 19.03 24.61 42.95 52.71
6.115 25.04 28.50 52.97 64.73
6.910 27,14 38.38 63.90 67.16
7.707 29.00 50.52 68.11 78.15
8.502 30.94 64.14 70.90 92.90
CFCF
1 2 3 4 5
4.245 9.011 18.23 22.52 28.55
5.356 15.62 23.63 36.03 38.04
6.883 23.70 26.56 47.34 62.54
8.797 26.92 37.47 60.04 66.02
11.11 29.59 51.98 67.50 76.40
13.83 32.62 67.56 72.23 94.01
1 2 3 4 5
3.476 5.272 10.04 18.87 21.62
3.473 6.857 16.82 21.96 26.88
3.452 11.63 2144 39.23 53.54
3445 13.21 2144 43.65 59.82
(c) cantilever CFFF
16.98 36.04 72.92 90.06 114.2
plates
3.467 8.462 21.19 27.18 30.77
3,459 10.05 21.41 34.92 39.19
3.439 14.79 21.42 48-12 60.11
To show more accurate frequencies of rectangular plates with free edges, the first five parameters of rectangular plates with one free edge, two free edges, and three
frequency
free edges are tabulated in Tables 3(a), 3(b) and 3(c), respectively. from 0.5 to 2.0 and v = 0.3 was used.
The aspect ratio varies
For isotropic materials, Poisson’ratio (v) can vary between 0 and O-5. It has been seen that frequency parameters depend on v if one or more of the edges of the plate is free. The effects of Poisson’s ratio on the frequencies of square plates with free edges are shown in Tables 4(a), 4(b) and 4(c). Poisson’s ratio values of v = O-0, O-15, 0.3 and 0.5
FREQUENCIES
OF RECTANGULAR TABLE
457
PLATES
4
Frequency parameters, n*, as functions of v for square plates; k - 1 = 3 and
Poisson’s Boundary conditions
ratio: v
r Modes
0.15
0.0
0.3
0.5
(a) one free edge SFSS
1 2 3 4 5
12.23 29.05 41.93 60.77 63.11
12.03 28.44 41.73 60.07 62.50
11.68 27,76 41.20 59.07 61.86
10.87 26.71 39.51 57.18 60.98
CFSS
1 2 3 4 5
1344 34.46 42.56 64.99 73.61
13.14 33.80 42.31 64.16 73.01
12.69 33.06 41.70 63.01 72.40
11.69 31.98 39.85 60.93 71.55
SFCS
1 2 3 4 5
17.22 32.30 51.97 65.23 69.01
17.05 31.71 51.76 64.60 68.33
16.73 31.03 51.20 63.93 67.34
15.96 29.97 49.43 63.01 65.50
SFCC
1 2 3 4 5
23.65 36.57 63.25 67.90 78.52
23.49 36.01 62.99 67.26 77.85
23.20 35.36 62.37 66.58 76.89
22.45 34.33 60.45 65.62 75.12
CFCS
1 2 3 4 5
18.13 37-33 52.49 72.73 75,56
17.88 36.67 52.23 71.93 74.94
17.47 35.93 51.60 70.82 74.28
16.52 34.82 49.68 68.81 73.37
CFCC
1 2 3 4 5
24.32 41.13 63.66 77.84 81.94
24.10 40.50 63.36 77.20 81.15
23.71 39.77 62.68 76.54 80.05
22.83 38.67 60.61 75.61 78.08
(b) two free edges FFSS
1 2 3 4 5
9.870 17.88 39.23 39-48 48.91
9.816 17.12 38.04 39.36 48.12
9.631 16.13 36.73 38.95 46.74
9.079 14.35 34.78 37.52 43.48
FFSC
1 2 3 4 5
15.39 22.06 42.10 49.86 58.04
15.33 21.37 40.91 49.72 57.27
15.15 20.46 39.58 49.27 55.90
14.60 18.75 37.58 47.78 52.63
FFCC
1 2
22.28 27.51
22.21 26.89
22.03 26.05
21.49 2446
458
T. MIZUSAWA TABLE
Boundary conditions
4 (cont.) Poisson’s ratio: v
r Modes
0.0
0.15
0.3
0.5
3 4 5
4569 61.41 68.46
44.52 61.23 67.65
43.20 60.72 66.25
41.20 59.08 62.91
SFSF
1 2 3 4 5
3.956 19.31 19.39 41.10 53.40
3,681 18.43 19.41 39.82 5244
3.367 17.32 19.29 38.21 51.04
2,872 15.32 18.76 35.45 47.84
CFSF
1 2 3 4 5
5.902 20.43 25.40 46.04 54.01
5.639 19.88 25.05 44.67 53.59
5.338 19.03 24.61 42.95 52.71
4.858 17.27 23.71 40.05 50.23
CFCF
1 2 3 4 5
7442 25.91 26.64 50.61 65.05
7.183 24.93 26.68 49.15 64.13
6.883 23.70 26.56 47.34 62.54
6.392 21.52 25.97 44.33 58.93
3.467 8.462 21.19 27.18 30.77
3.373 7.47 1 19.56 26.28 27.89
(c) cantilever plates CFFF
3.509 9.635 21.99 28.30 33.57
3.499 9.086 21.80 27.75 32.35
were used in the calculations. It can be seen from the tables that Poisson’s ratio has an obvious effect on the frequency parameters, and that the frequencies are also influenced by the number of free edges and by their arrangement.
4. CONCLUSIONS
In this note, free vibrations of isotropic rectangular plates with free edges has been discussed. Good accuracy of the present method is shown in comparisons with results obtained by other numerical methods. The effect of Poisson’s ratio on the frequencies of the plates has been investigated. It is found that frequencies of the plates with free edges are significantly influenced by Poisson’s ratio.
ACKNOWLEDGMENT The results presented in this paper were obtained
Grant No. 59750362, supported the Japanese Government.
in the course of a research programme, by the Ministry of Education, Science and Culture of
FREQUENCIES
OF RECTANGULAR
PLATES
459
REFERENCES 1. A. W. LEISSA 1969 NASA SP-160. Vibration of plates. 2. A. W. LEISSA 1977 The Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations: classical theory. 3. A. W. LEISSA 1981 7%e Shock and Vibration Digest 13, 11-22. Plate vibration research, 1976-1980: classical theory. 4. D. J. GORMAN 1982 Free Vibration Analysis OfRectangular Plates. Amsterdam: Elsevier, North Holland. 5. A. W. LEISSA 1973 Journal ofSound and Vibration 31,257-293. The free vibration of rectangular plates. 6. S. F. BASSILY and S. M. DICKINSON 1975 Journal of Applied Mechanics 42, 858-864. On the use of beam functions for problems of plates involving free edges. 7. G. B. WARBURTON 1954 Proceedings of the Institudon of Mechanical Engineers, Series A, 168, 371-384. The vibration of rectangular plates. 8. T. MIZUSAWA, T. KA.IITA and M. NARUOKA 1979 Journal of Sound and Vibration 62,301-308. Vibration of skew plates by using B-spline functions. 9. T. MIZUSAWA, T. KA.IITA and M. NARUOKA 1980 Journal of Sound and Vibration 73,575584. Analysis of skew plate problems with various constraints. 10. R. W. CLAASSEN and C. J. THORNE 1960 U.S. Naval Test Station, China Lake, California, NOTS Tech. Pub. 2379, NAVWEPS Report 7016. Transverse vibrations of thin rectangular isotropic plates. 11. F. FUJII and T. HOSHINO 1983 Journal of Sound and Vibration 87, 525-534. Discrete and non-discrete mixed methods applied to eigenvalue problems of plates. 12. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 Applied Physics Laboratory, The Johns Hopkins University, Technical Memorandum TG-705. Upper and lower bounds for frequencies of rectangular cantilever plates. 13. V. G. SIGILLITO 1965 Applied Physics Laboratory, The Johns Hopkins University, Engineering Memorandum EM-4012. Improved upper bounds for frequencies of rectangular free and cantilever plates.
NATURAL FREQUENCIES OF RECTANGULAR PLATES WITH FREE EDGES T. MIZUSAWA Department of Construction Engineering, Daido Institute of Technology, Hakusuicho-40, Minami-ku, Nagoya, Japan (Received 28 November 1984, and in revised form 16 April 1985) This note presents vibration analysis of isotropic rectangular plates with free edges by the Rayleigh-Ritz method with B-spline functions. To show the accuracy of the present method, the results are compared with existing results based on other numerical methods and found to be in good agreement. Accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions. The effects of Poisson’s ratio on natural frequencies of square plates with free edges are also investigated.
1. INTRODUCTION [l-4] exists for the free vibrations of rectangular plates with free edges. Except for cases with two opposite edges simply supported, exact solutions do not exist, as mentioned in Leissa’s work [5]; therefore a variety of numerical methods has been used to analyze vibrations of rectangular plates with free edges. However the proper selection of the displacement functions is essential to obtain stable and accurate results for rectangular plates with free edges [6]. Warburton [7] presented an approximate solution for rectangular plates with combinations of free edges, obtained by using the Rayleigh-Ritz method with beam functions. Leissa [5] obtained comprehensive and accurate analytical results for vibration of rectangular plates in 21 distinct cases which involve all possible combinations of classical boundary conditions, employing the Ritz method with beam functions. Natural frequency parameters of rectangular plates with free edges evidently are influenced by Poisson’s ratio. In this note, vibrations of isotropic rectangular plates with free edges are analyzed by using the Ritz discretization procedure with B-spline functions [8,9] and results are compared with those obtained by other numerical methods. To show more accurate results for rectangular plates with free edges, the first five frequencies of plates with different aspect ratios and combinations of free edges have been calculated. The influence of Poisson’s ratios on the frequencies has also been investigated. A vast literature
2. METHOD OF ANALYSIS
The analysis is based on the spline element method, which discretization procedure with B-spline functions; it has been used skew plate problems [8,9]. The present method may be regarded of the displacement formulation of the finite element procedure total potential energy theorem is used to develop the relationship parameters and the applied loading. 451
0022-460X/86/060451 + 10$03.00/O
@ 1986Academic
is the Rayleigh-Ritz previously to analyze as an alternative form in that the minimum between the unknown
Press Inc. (London)
Limited
452
T. MIZUSAWA
1
Clomped
I
’ Simply
b
1 edges I
edge’ C
supported.
s
Free edge: F
Figure
1. Rectangular
plate and symbols
for boundary
conditions.
In this note, the spline element method is used to analyze vibrations of isotropic rectangular plates with free edges, as shown in Figure 1. The assumed displacement functions for a finite element take the form of two-way spline functions and are given by w=
; ; GJCl,,(~)~“,,(Y) m=, n=l
(1)
in which ix = k + M, - 2, i,, = k+ M,, - 2, and N,,+(X) and N,,,(y) are the normalized B-spline functions. In this formulation, k - 1 is the degree of the B-spline functions, M,
TABLE
1
Convergence study of frequency parameters, n*, of rectangular plate with SSCF;
Aspect ratio
Degree of spline
A=a/b
k-l
0.4
Leissa’s
1.0
Leissa’s
2.0
Number of nodes
Mx=My
Y = O-3
Modes Order of matrix?
\
r 1st
2nd
3rd
4th
5th
3
3 5 9 11 13 exact values [5]
25 49 121 169 225
10.204 10.190 10.189 10.189 10.189 10.189
13.630 13.606 13.604 13,604 13,604 13.604
20.516 20.116 20.098 20.097 20.097 20.097
39.701 30.643 29.626 29.623 29.622 29,622
42.811 39.675 39.639 39.639 39.638 39.638
3
3 5 9 11 13 exact values [5]
25 49 121 169 225
12.564 12.681 12.687 12.687 12.687 12.687
32.298 33.024 33.063 33.064 33.065 33.065
41,234 41.685 41.701 41.702 41.702 41.702
59,220 62.719 63.002 63.011 63.012 63.015
70.818 72.395 72.392 72.395 72.396 72.398
3
25 49 121 169 225
21.503 22.750 22.812 22.814 22.815
46.760 50.273 50.730 50.742 50.746
92.255 98.873 98.749 98.765 98.771
111.27 99.954 99.750 99.765 99.770
3 5 9 11 13
tTheorderofthematrixisgivenby(k+M,-2)x(k+M,-2).
138.34 128.90 132.12 132.22 132.24
FREQUENCIES
OF RECTANGULAR
453
PLATES
and M,, are the number of nodes in the x- and y-directions, respectively and the Cmn’s are unknown parameters. The details of the analysis formulation have been carried out by using the procedure described in reference [8]. 3. NUMERICAL
RESULTS AND DISCUSSSIONS
In Figure 1, the definitions of the boundary conditions are presented. The symbolism SCSF, for example, identifies a rectangular plate with the edges x = 0, x = a, y = 0, y = b having simply supported, clamped, simply supported, and free boundary conditions, respectively (see Figure 1). The order of the matrices obtained is given by (k+ M, -2) x (k + MY - 2). In the calculations of several examples, k - 1 = 3 and &f, = M,, = 13 are used. To demonstrate the accuracy of the present method, a few example problems were solved to obtain results for comparison with those obtained by other numerical approaches. Table 1 shows the influence of the number of nodes on the convergence of natural of rectangular plates with SSCF for different frequency parameters, n* = wa’~, aspect ratios and numbers of nodes. Here k - 1 = 3 and v = 0.3 are assumed. Stable convergence was obtained with increase in the number of nod&. Results are also compared with Leissa’s exact values. All show a good agreement. Table 2(a) shows the first five frequency parameters, n* = wa*m, of rectangular plates with one free edge for aspect ratios A = a/b of 1-O and 1.5. Here D is the flexural TABLE
2
Comparison of the frequency parameters, n*, for rectangularplates and M, = MY = 13
(v = O-3); k - 1 = 3
(a) one free edge Aspect ratio; A = a/b Boundary conditions
(
1.0
1.5
A
Modes
r
*
\
CFCC
1 2 3 4 5
23.71 39.77 62.68 76.54 80.05
(24.02) (70.04) (63.49) (76.76) (80.71)
50.69 65.26 98.98 137.3 154.3
(51.78) (66.21) (99.82) (140.4) (155.0)
CFCS
1 2 3 4 5
17.47 35.93 51.60 70.82 74.28
(17.62) (36.05) (52.07) (71.19) (74.35)
36.18 54.06 91.21 112.9 131.8
(36.65) (54.45) (91.58) (114.3) (132.9)
SFCC
1 2 3 4 5
23.20 35.36 62.37 66.58 76.89
(23.46) (35.61) (63.13) (66.81) (77.50)
SO.43 62.05 90.70 137.2 140.3
(51.42) (62.93) (91.54) (140.2) (141.1)
SFCS
1 2 3 4 5
16.73 31.03 51.20 63.93 67.34
(16.87) (31.14) (51.63) (64.04) (67.65)
35.73 50.14 82.22 112.7 129.2
(36.15) (50.51) (82.58) (114.1) (130.3)
) are results obtained
by Leissa [5] using the Ritz method with beam functions.
454
T.
MIZUSAWA
TABLE 2 (cont.) (b) TWO free edges Aspect ratio: A = a/b Boundary conditions
1.0 Modes
1.5
y
CFCF
1 2 3 4 5
6.883 23.10 26.56 41.34 62.54
(6.942) (24.03) (26.68) (47.79) (63.04)
11.11 29-59 51.98 67.50 76.40
(11.22) (29.90) (52.61) (68.09) (77.04)
CFSF
1 2 3 4 5 . 1 2 3 4 5
5,338 19.03 24.61 42-95 52.71
(5.364) (19.17) (24.77) (43.19) (53.00)
6.910 27.14 38.38 63.90 67.16
(6.931) (27.29) (38.59) (64.25) (67.47)
22.03 26.05 43.20 60.72 66.25
(22.27) (26.53) (43.66) (61.47) (67.55)
[22.17]* [26.40]* [43.6]* [61-2]* [67+2]*
49.30 52-93 69.64 103.8 135.7
(50.21) (54.70) (71.33) (105.3) (138.5)
1 2 3 4 5
15.15 20.46 39.58 49.21 55.90
15.29 20.67 39.78 49.73 56.62
[ 15.191** [20-57]** [39.72]** [49.52]** [56.32]**
34.10 39.65 59.45 96.48 110.8
34.52 40.39 60.15 97.18 112.1
FFCC
FFCS
( ) are results obtained by Leissa [S] using the Ritz method with beam functions, [ ]* are values calculated by Claassen and Thome [IO] utilizing the series method and [ ]** are solutions obtained by Fujii and Hoshino using the mixed approach combined with spline functions 111 J. (c) cantilever plates
Aspect ratio; A = a/b Boundary conditions CFFF
1.0 Modes 1 2 3 4 5
1.5
I 3.467 (3.492) 8462 (8.525) 21.19 (21.43) 27.18 (27.33) 30.77 (31.11)
[3.431]* [20-87]* [26-50]*
[3.473]** [21-30]** [27.29]**
3.452 11.63 2144 39.23 53.54
(3.477) (11.68) (21.62) (39.49) (53.88)
( ) are results obtained by Leissa [5] using the Ritz method with beam functions, [ ]* are lower bounds calculated by Bazley et al. [12] utilizing the Rayleigh-Ritz method with beam functions and [ 1**are upper bounds obtained by Sigillito [ 131 using the Ritz method with products of beam functions and Legendre functions.
rigidity, h is the thickness and p is the density of the plate, and w is the frequency (rad/s). The values obtained by Leissa [5] using the Ritz method with 36 terms containing the products of beam functions are also shown in comparison with the present results. It is evident that the results obtained by the present method are in good agreement with Leissa’s solutions, within an accuracy of 5%.
FREQUENCIES
OF RECTANGULAR
PLATES
455
Table 2(b) shows the first five frequency parameters of rectangular plates with two free edges. In the cases of square plates, Leissa’s results [5] calculated by the Rayleigh-Ritz method with beam functions, results obtained by Claassen and Thorne [lo] using the series method, and solutions calculated by Fujii and Hoshino [ 111 utilizing mixed approaches combined with spline functions are also shown. It is evident that good agreement is obtained, the present results being slightly less exact than those calculated by Leissa, by Claassen and Thorne and by Fujii and Hoshino. Frequency parameters of cantilever rectangular plates are also shown in Table 2(c). To show the accuracy of the present method, the present results are compared with Leissa’s values, lower bound solutions obtained by Bazley et al. [ 121 using the RayleighRitz method with beam functions and upper bounds of the solutions calculated by Sigillito [13] utilizing the Ritz method with products of beam functions and Legendre functions. It can be seen that the present results are always between upper bounds and lower bounds. TABLE
3
Frequency parameters, n*, of rectangular plates (u = 0.3); Aspect Boundary conditions
ratio:
k - 1 = 3 and M, = MY = 13 A
I Modes
0.5
0.75
1.25
1.0
SFSS
1 2 3 4 5
4.034 11.68 18.82 24.01 27.76
(a) one free 11.68 22.70 27.76 41.20 24.01 41.17 59.07 51.63 61.86
CFSS
1 2 3 4 5
5.704 12.69 24.69 24.94 33.06
8.570 24.69 28.41 45.75 52.07
SFCS
1 2 3 4 5
5.142 14.06 19.33 27.66 29.30
SFCC
1 2 3 4 5
CFCS
CFCC
1.5
1.75
2.0
edge 17.25 33.92 63.29 68.23 81.61
24.01 41.17 75.82 90.29 108.9
31.99 49.55 84.58 122.2 138.4
41.20 59.07 94.48 148.5 159.1
12.69 33.06 41.70 63.01 72.40
18.06 38.85 63.68 78.29 85.02
24.69 45.75 85.40 90.61 111.9
32.58 53.80 93.69 122.5 143.7
41.70 63.01 103.2 159.3 162.4
10.02 24.24 29.58 45.38 57.68
16.73 31.03 51.20 63.93 67.34
25.30 39.66 72,12 78.92 95.27
35.73 50.14 82.22 112.7 129.2
48.02 62.48 94.23 145.6 152.6
62.1.6 76.69 108.2 159.1 198.2
6.592 16.77 19.94 31.08 31.63
13.55 26.23 35.85 50.28 58.78
23.20 35.36 62.37 66.58 76.89
35.50 47.31 77.19 96.22 111.0
50.43 62.05 90-70 137.2 140.3
67.92 79.57 107.1 155.3 185.1
87.89 99.87 126.5 173.1 239.6
1 2 3 4 5
6.564 14.92 25.35 28.26 34.42
11.01 29.67 30.18 49.63 60.35
17.47 35.93 51.60 70.82 74.28
25.87 44.05 79.18 81.78 98.27
36.18 54.06 91.21 112.9 131.8
48.38 65.98 102.6 152.5 159.6
6244 79.83 115.9 172.2 198.3
1 2 3 4 5
7.764 17.51 25.84 32.16 35.99
14.30 31.39 36.3 1 54.20 69.30
23.71 39.77 62.68 76.54 80.05
35.87 51.07 86.31 96.41 113.5
50.69 65.26 98.98 137.3 154.3
68.09 82.33 114.6 168.4 185.2
87.99 102.3 133.3 185.4 239.6
7.302
T. MIZUSAWA
456
TABLE 3 (cont.)
Boundary conditions
Aspect ratio: A
r Modes
, 0.5
0.75
1.0
1.25
1.5
1.75
2.0
(b) two free edges FFSS
1 2 3 4 5
2.378 6.881 9.63 1 16.13 21.82
5.388 11.10 21.82 29.21 30.91
9.631 16.13 36.73 38.95 46.74
15.11 22.14 43.66 61.00 68.98
21.82 29.21 51.65 87.99 90.80
29.77 37.39 60.67 100.4 120.0
38.95 46.74 70.74 111.0 156.8
FFSC
1 2 3 4 5
3.762 7.811 12.25 18.18 25.67
8.502 13.38 27.68 32.26 34.18
15.15 20.46 39.58 49-27 55.90
23.68 29.19 48.66 76.98 83.34
34.10 39.65 59.45 96.48 110.8
46.39 51.87 71.97 108.9 150.5
60.53 65.86 86.24 123.1 178.9
FFCC
1 2 3 4 5
5.503 8.963 15.17 20.53 27.34
12.40 16.31 33.92 34.20 39.87
22.03 26.05 43.20 60.72 66.25
34.34 38.25 55.11 90.30 94.61
49.30 52.93 69.64 103.8 135.7
66.84 70.07 86.79 120-l 173.7
86.91 89.63 106.6 139.1 191.5
SFSF
1 2 3 4 5
1.661 6.344 14.69 16.29 22.27
2.518 11.39 17.10 29.19 30.78
3.367 17.32 19.29 38.21 51.04
4.202 19.65 27.14 48.40 53.83
5.024 21.46 37.53 55.22 60.74
5.837 23.37 49.61 57,72 74.04
6.644 25.38 58.74 65.18 89.10
CFSF
1 2 3 4 5
4.005 7.800 15.76 22.35 27.56
4.613 12.60 23.20 31.07 34.75
5.338 19.03 24.61 42.95 52.71
6.115 25.04 28.50 52.97 64.73
6.910 27,14 38.38 63.90 67.16
7.707 29.00 50.52 68.11 78.15
8.502 30.94 64.14 70.90 92.90
CFCF
1 2 3 4 5
4.245 9.011 18.23 22.52 28.55
5.356 15.62 23.63 36.03 38.04
6.883 23.70 26.56 47.34 62.54
8.797 26.92 37.47 60.04 66.02
11.11 29.59 51.98 67.50 76.40
13.83 32.62 67.56 72.23 94.01
1 2 3 4 5
3.476 5.272 10.04 18.87 21.62
3.473 6.857 16.82 21.96 26.88
3.452 11.63 2144 39.23 53.54
3445 13.21 2144 43.65 59.82
(c) cantilever CFFF
16.98 36.04 72.92 90.06 114.2
plates
3.467 8.462 21.19 27.18 30.77
3,459 10.05 21.41 34.92 39.19
3.439 14.79 21.42 48-12 60.11
To show more accurate frequencies of rectangular plates with free edges, the first five parameters of rectangular plates with one free edge, two free edges, and three
frequency
free edges are tabulated in Tables 3(a), 3(b) and 3(c), respectively. from 0.5 to 2.0 and v = 0.3 was used.
The aspect ratio varies
For isotropic materials, Poisson’ratio (v) can vary between 0 and O-5. It has been seen that frequency parameters depend on v if one or more of the edges of the plate is free. The effects of Poisson’s ratio on the frequencies of square plates with free edges are shown in Tables 4(a), 4(b) and 4(c). Poisson’s ratio values of v = O-0, O-15, 0.3 and 0.5
FREQUENCIES
OF RECTANGULAR TABLE
457
PLATES
4
Frequency parameters, n*, as functions of v for square plates; k - 1 = 3 and
Poisson’s Boundary conditions
ratio: v
r Modes
0.15
0.0
0.3
0.5
(a) one free edge SFSS
1 2 3 4 5
12.23 29.05 41.93 60.77 63.11
12.03 28.44 41.73 60.07 62.50
11.68 27,76 41.20 59.07 61.86
10.87 26.71 39.51 57.18 60.98
CFSS
1 2 3 4 5
1344 34.46 42.56 64.99 73.61
13.14 33.80 42.31 64.16 73.01
12.69 33.06 41.70 63.01 72.40
11.69 31.98 39.85 60.93 71.55
SFCS
1 2 3 4 5
17.22 32.30 51.97 65.23 69.01
17.05 31.71 51.76 64.60 68.33
16.73 31.03 51.20 63.93 67.34
15.96 29.97 49.43 63.01 65.50
SFCC
1 2 3 4 5
23.65 36.57 63.25 67.90 78.52
23.49 36.01 62.99 67.26 77.85
23.20 35.36 62.37 66.58 76.89
22.45 34.33 60.45 65.62 75.12
CFCS
1 2 3 4 5
18.13 37-33 52.49 72.73 75,56
17.88 36.67 52.23 71.93 74.94
17.47 35.93 51.60 70.82 74.28
16.52 34.82 49.68 68.81 73.37
CFCC
1 2 3 4 5
24.32 41.13 63.66 77.84 81.94
24.10 40.50 63.36 77.20 81.15
23.71 39.77 62.68 76.54 80.05
22.83 38.67 60.61 75.61 78.08
(b) two free edges FFSS
1 2 3 4 5
9.870 17.88 39.23 39-48 48.91
9.816 17.12 38.04 39.36 48.12
9.631 16.13 36.73 38.95 46.74
9.079 14.35 34.78 37.52 43.48
FFSC
1 2 3 4 5
15.39 22.06 42.10 49.86 58.04
15.33 21.37 40.91 49.72 57.27
15.15 20.46 39.58 49.27 55.90
14.60 18.75 37.58 47.78 52.63
FFCC
1 2
22.28 27.51
22.21 26.89
22.03 26.05
21.49 2446
458
T. MIZUSAWA TABLE
Boundary conditions
4 (cont.) Poisson’s ratio: v
r Modes
0.0
0.15
0.3
0.5
3 4 5
4569 61.41 68.46
44.52 61.23 67.65
43.20 60.72 66.25
41.20 59.08 62.91
SFSF
1 2 3 4 5
3.956 19.31 19.39 41.10 53.40
3,681 18.43 19.41 39.82 5244
3.367 17.32 19.29 38.21 51.04
2,872 15.32 18.76 35.45 47.84
CFSF
1 2 3 4 5
5.902 20.43 25.40 46.04 54.01
5.639 19.88 25.05 44.67 53.59
5.338 19.03 24.61 42.95 52.71
4.858 17.27 23.71 40.05 50.23
CFCF
1 2 3 4 5
7442 25.91 26.64 50.61 65.05
7.183 24.93 26.68 49.15 64.13
6.883 23.70 26.56 47.34 62.54
6.392 21.52 25.97 44.33 58.93
3.467 8.462 21.19 27.18 30.77
3.373 7.47 1 19.56 26.28 27.89
(c) cantilever plates CFFF
3.509 9.635 21.99 28.30 33.57
3.499 9.086 21.80 27.75 32.35
were used in the calculations. It can be seen from the tables that Poisson’s ratio has an obvious effect on the frequency parameters, and that the frequencies are also influenced by the number of free edges and by their arrangement.
4. CONCLUSIONS
In this note, free vibrations of isotropic rectangular plates with free edges has been discussed. Good accuracy of the present method is shown in comparisons with results obtained by other numerical methods. The effect of Poisson’s ratio on the frequencies of the plates has been investigated. It is found that frequencies of the plates with free edges are significantly influenced by Poisson’s ratio.
ACKNOWLEDGMENT The results presented in this paper were obtained
Grant No. 59750362, supported the Japanese Government.
in the course of a research programme, by the Ministry of Education, Science and Culture of
FREQUENCIES
OF RECTANGULAR
PLATES
459
REFERENCES 1. A. W. LEISSA 1969 NASA SP-160. Vibration of plates. 2. A. W. LEISSA 1977 The Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations: classical theory. 3. A. W. LEISSA 1981 7%e Shock and Vibration Digest 13, 11-22. Plate vibration research, 1976-1980: classical theory. 4. D. J. GORMAN 1982 Free Vibration Analysis OfRectangular Plates. Amsterdam: Elsevier, North Holland. 5. A. W. LEISSA 1973 Journal ofSound and Vibration 31,257-293. The free vibration of rectangular plates. 6. S. F. BASSILY and S. M. DICKINSON 1975 Journal of Applied Mechanics 42, 858-864. On the use of beam functions for problems of plates involving free edges. 7. G. B. WARBURTON 1954 Proceedings of the Institudon of Mechanical Engineers, Series A, 168, 371-384. The vibration of rectangular plates. 8. T. MIZUSAWA, T. KA.IITA and M. NARUOKA 1979 Journal of Sound and Vibration 62,301-308. Vibration of skew plates by using B-spline functions. 9. T. MIZUSAWA, T. KA.IITA and M. NARUOKA 1980 Journal of Sound and Vibration 73,575584. Analysis of skew plate problems with various constraints. 10. R. W. CLAASSEN and C. J. THORNE 1960 U.S. Naval Test Station, China Lake, California, NOTS Tech. Pub. 2379, NAVWEPS Report 7016. Transverse vibrations of thin rectangular isotropic plates. 11. F. FUJII and T. HOSHINO 1983 Journal of Sound and Vibration 87, 525-534. Discrete and non-discrete mixed methods applied to eigenvalue problems of plates. 12. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 Applied Physics Laboratory, The Johns Hopkins University, Technical Memorandum TG-705. Upper and lower bounds for frequencies of rectangular cantilever plates. 13. V. G. SIGILLITO 1965 Applied Physics Laboratory, The Johns Hopkins University, Engineering Memorandum EM-4012. Improved upper bounds for frequencies of rectangular free and cantilever plates.