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SYMMETRIC AND ANTISYMMETRIC NORMAL MODES OF A CANTILEVER RECTANGULAR PLATE: EFFECT OF POISSON'S RATIO AND A CONCENTRATED MASS
Journal of Sound and Vibration (1996) 195(1), 142–148
SYMMETRIC AND ANTISYMMETRIC NORMAL MODES OF A CANTILEVER RECTANGULAR PLATE: EFFECT OF POISSON’S RATIO AND A CONCENTRATED MASS R. E. R P. A. A. L* Department of Engineering, Universidad Nacional del Sur and Institute of Applied Mechanics (CONICET), 8000—Bahi´ a Blanca, Argentina (Received 17 November 1995)
1. In spite of the fact that several well known authors have tackled the problem of transverse vibrations of a rectangular, cantilever plate† there is a lack of information regarding the influence of Poisson’s ratio upon the values of the frequency coefficients corresponding to symmetric and antisymmetric modes of vibration (Figure 1(a)). The present study deals with the first 10 normal modes of vibration (5 symmetric and 5 antisymmetric modes). The analysis is performed using the very accurate finite element algorithmic procedure due to Bogner et al. [2] for values of Poisson’s ratio, n, ranging from 0 to 0·5. In view of the practical implications of the problem the second part of this study depicts frequency coefficients as a function of the parameter (concentrated mass)/(total plate mass) in the case where the concentrated mass is placed at the center of the completely free edge, Figure 1(b). The effect of Poisson’s ratio is also investigated. 2. The present study uses the conforming rectangular element of 16 degrees-of-freedom developed by Bogner, Fox and Schmit [6] which yields excellent accuracy. One hundred elements with 121 nodes and a total of 420 degrees-of-freedom were used for the calculations presented in this paper. 3. Table 1 depicts natural frequency coefficients Vi = z(rh/D)vi a 2 corresponding to symmetric and antisymmetric modes for n = 1/3. The agreement with the results recently obtained by Gorman [7]‡ is remarkabley good. Interestingly, Gorman obtained the first four eigenvalues corresponding to symmetric and antisymmetric modes but in accordance with the results depicted in Table 1 the fifth natural frequency coefficient of a symmetric mode is lower than the fourth of an antisymmetric model for a/b = 0·5, 0·75, . . . 2. Table 2 presents natural frequency coefficients Vi for the first five symmetric and antisymmetric modes of cantilever plates for n ranging from 0 to 0·5. As expected, for n = 0 * Corresponding author. † A thorough treatment is contained in Leissa’s classical treatise [1]. Apparently the first numerical results, for the case of a square cantilever plate and n = 0·30, were obtained by D. Young [2]. Very accurate results have been obtained rather recently [3–5]. ‡ Gorman’s study [7] deals also with the isotropic plate, as a particular case.
142 0022–460X/96/310142 + 07 $18.00/0
7 1996 Academic Press Limited
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Figure 1. Vibrating structural system under study: (a) lower symmetric (S) and antisymmetric (A) modes; (b) cantilever plate with a concentrated mass M, m = M/Mplate .
the fundamental frequency coefficient V1 coincides with the value corresponding to a cantilever beam. On the other hand, as n increases the values of Vi always decrease monotonically, exceptions are the third frequency of the symmetric modes for a/b = 0·75 T 1 Frequency coefficients Vi = z(rh/D)vi a 2 of a cantilever plate for Poisson’s ratio equal to 1/3 (1) Symmetric modes, (2) antisymmetric modes
a/b
n = 1/3 ZXXXCXXXV 1 2
a/b
n = 1/3 ZXXXCXXXV 1 2
0·5
3·487 10·03 21·79 31·07 33·88
5·279 18·84 24·56 42·67 52·43
0·75
3·473 16·69 22·02 41·55 60·84
1·25
3·448 21·31 38·93 60·63 68·24
9·896 34·53 73·87 99·04 126·2
1·50
3·438 21·32 53·05 61·55 84·44
2·00
3·420 21·28 59·75 92·69 117·7
14·50 47·32 91·24 151·1 228·1
6·805 26·86 38·17 63·39 66·90 11·43 38·70 79·29 135·3 143·7
a/b
n = 1/3 ZXXXCXXXV 1 2
1·00
3·460 21·09 27·06 53·53 61·12
1·75
3·429 21·31 59·40 72·85 102·9
8·356 30·55 63·62 70·65 92·21 12·96 42·98 85·10 144·0 190·5
2·00
1·75
1·50
1·25
1·00
0·75
0·5
a/b
3·516 11·21 22·03 34·01 34·75 3·516 18·50 22·04 45·50 61·70 3·516 22·03 28·35 58·98 61·70 3·516 22·03 40·92 61·70 74·45 3·516 22·03 56·27 61·70 92·05 3·516 22·03 61·70 74·42 111·9 3·516 22·03 61·70 95·36 120·9
5·916 20·46 25·45 46·14 54·04 7·788 29·19 39·64 68·41 69·10 9·676 33·70 66·52 73·38 98·25 11·56 38·65 79·34 101·1 134·3 13·44 43·83 86·09 143·5 145·7 15·33 49·15 93·46 153·9 193·6 17·22 54·55 101·3 162·9 242·5
0·0 ZXXXCXXXV 1 2 3·514 10·92 22·02 33·28 34·60 3·512 18·11 22·04 44·51 61·64 3·511 21·96 27·98 57·61 61·67 3·511 21·98 40·48 61·63 72·77 3·510 21·98 55·70 61·72 90·04 3·509 21·98 61·52 74·00 109·6 3·508 21·97 61·55 94·85 120·7
5·745 20·13 25·23 45·27 53·79 7·524 28·67 39·33 67·40 68·43 9·320 32·95 66·12 72·68 96·70 11·11 37·63 78·20 100·8 132·4 12·89 42·53 84·57 142·4 144·9 14·68 47·56 91·51 151·9 193·2 16·47 52·67 98·85 160·4 239·9
0·1 ZXXXCXXXV 1 2 3·506 10·58 21·96 32·42 34·37 3·501 17·59 22·04 43·36 61·44 3·497 21·72 27·59 56·02 61·54 3·493 21·79 39·91 61·39 70·93 3·490 21·79 54·79 61·72 87·81 3·487 21·79 60·93 73·55 107·0 3·484 21·78 61·05 94·11 120·0
5·558 19·68 24·98 44·26 53·37 7·236 28·01 38·93 65·86 67·91 8·933 32·04 65·36 71·88 94·95 10·62 36·44 76·67 100·2 130·0 12·30 41·04 82·64 140·0 144·5 13·98 45·77 89·12 149·2 192·4 15·67 50·56 95·97 157·1 235·9
0·2 ZXXXCXXXV 1 2 3·493 10·18 21·84 31·43 34·03 3·482 16·94 22·03 42·04 61·04 3·471 21·29 27·20 54·19 61·26 3·462 21·46 39·20 60·88 68·94 3·454 21·47 53·54 61·62 85·33 3·446 21·46 59·88 73·04 104·0 3·440 21·44 60·15 93·11 118·5
5·352 19·08 24·67 43·09 52·71 6·566 27·18 38·39 64·06 67·20 8·508 30·96 64·15 70·98 92·94 10·09 35·05 74·67 99·39 127·3 11·66 39·33 80·23 136·6 144·0 13·23 43·73 86·21 145·5 191·1 14·80 48·18 92·53 152·8 230·4
0·3 ZXXXCXXXV 1 2 3·473 9·707 21·64 30·30 33·52 3·451 16·15 21·97 40·50 60·32 3·432 20·61 26·78 52·10 60·76 3·414 20·95 38·31 59·98 66·82 3·398 20·97 51·92 61·32 82·56 3·385 20·95 58·26 72·40 100·7 3·372 20·92 58·75 91·71 116·0
5·124 18·31 24·29 41·74 51·73 6·566 26·14 37·67 61·98 66·17 8·037 29·66 62·33 69·93 90·64 9·497 33·42 72·06 98·15 124·0 10·95 37·35 77·19 132·2 143·0 12·40 41·40 82·65 140·6 189·0 13·86 45·49 88·40 147·3 223·0
0·4 ZXXXCXXXV 1 2 3·444 9·141 21·31 29·02 32·76 3·407 15·19 21·80 38·74 59·06 3·374 19·66 26·28 49·73 59·88 3·345 20·21 37·15 58·51 64·60 3·319 20·26 49·80 60·72 79·45 3·296 20·23 55·96 71·49 96·95 3·276 20·18 56·70 89·69 112·6
4·867 17·32 23·78 40·21 50·24 6·169 24·84 36·66 59·64 64·64 7·508 28·09 59·73 68·63 87·99 8·840 31·48 68·69 96·19 120·2 10·16 35·04 73·38 126·2 141·3 11·49 38·70 78·27 134·3 185·0 12·81 42·41 83·40 140·3 213·2
0·5 ZXXXCXXXV 1 2
T 2 Frequency coefficients Vi of a cantilever plate for Poisson’s ratio equal to 0, 0·1, · · · 0·5· (1) Symmetric modes, (2) antisymmetric modes
144
a/b
2·00
1·75
1·50
1·25
1·00
0·75
0·5
3·516 11·21 22·04 34·01 34·75 3·516 18·50 22·04 45·50 61·70 3·516 22·03 28·35 58·98 61·70 3·516 22·03 40·92 61·70 74·45 3·516 22·03 56·27 61·70 92·05 3·516 22·03 61·70 74·42 111·9 3·516 22·03 61·70 95·36 120·9
5·916 20·46 25·45 46·14 54·04 7·788 29·19 39·64 68·41 69·10 9·676 33·70 66·52 73·38 98·25 11·56 38·65 79·34 101·1 134·3 13·44 43·83 86·09 143·5 145·7 15·33 49·15 93·45 153·9 193·6 17·22 54·55 101·3 162·9 242·5
3·531 10·98 22·13 33·44 34·77 3·530 18·20 22·15 44·73 61·96 3·529 22·07 28·12 57·90 61·98 3·528 22·09 40·68 61·94 73·14 3·527 22·09 55·98 62·03 90·50 3·527 22·09 61·83 74·37 110·2 3·526 22·08 61·86 95·33 121·3
5·774 20·23 25·36 45·50 54·06 7·562 28·82 39·53 67·74 68·78 9·367 33·11 66·45 73·05 97·19 11·16 37·82 78·60 101·3 133·0 12·96 42·75 85·00 143·1 145·7 14·76 47·80 91·97 152·7 194·1 16·56 52·93 99·35 161·2 241·1
0·0 0·1 ZXXXCXXXV ZXXXCXXXV 1 2 1 2 3·579 10·80 22·41 33·09 35·08 3·574 17·95 22·50 44·26 62·71 3·569 22·17 28·16 57·18 62·81 3·565 22·24 40·74 62·65 72·40 3·562 22·24 55·92 63·00 89·62 3·558 22·24 62·19 75·06 109·2 3·555 22·23 62·31 96·05 122·5
5·672 20·08 25·49 45·18 54·47 7·385 28·59 39·73 67·22 69·31 9·117 32·70 66·71 73·36 96·90 10·84 37·19 78·25 102·3 132·7 12·55 41·89 84·35 142·9 147·5 14·27 46·71 90·96 152·2 196·4 15·99 51·60 97·95 160·3 240·7
0·2 ZXXXCXXXV 1 2 3·662 10·67 22·90 32·95 35·67 3·650 17·76 23·10 44·07 63·99 3·639 22·31 28·51 56·81 64·22 3·629 22·50 41·10 63·82 72·27 3·620 22·50 56·13 64·60 89·45 3·613 22·49 62·77 76·57 109·0 3·606 22·47 63·06 97·60 124·2
5·610 20·00 25·87 45·18 55·26 7·252 28·50 40·24 67·15 70·44 8·918 32·45 67·24 74·41 97·43 10·57 36·74 78·27 104·2 133·4 12·22 41·23 84·10 143·3 150·9 13·87 45·84 90·37 152·5 200·4 15·52 50·51 97·00 160·2 241·5
0·3 ZXXXCXXXV 1 2 3·789 10·59 23·62 33·06 36·58 3·766 17·62 23·97 44·19 65·81 3·744 22·49 29·22 56·85 66·29 3·725 22·86 41·80 65·44 72·91 3·708 22·88 56·65 66·91 90·08 3·693 22·86 63·57 79·00 109·8 3·679 22·82 64·10 100·1 126·5
5·590 19·97 26·51 45·55 56·44 7·164 28·53 41·10 67·63 72·20 8·769 32·36 68·01 76·30 98·90 10·36 36·46 78·62 107·1 135·3 11·95 40·75 84·23 144·2 156·0 13·53 45·17 90·18 153·4 206·2 15·12 49·64 96·45 160·8 243·3
0·4 ZXXXCXXXV 1 2 3·977 10·55 24·61 33·51 37·83 3·934 17·53 25·18 44·73 68·19 3·896 22·71 30·35 57·42 69·14 3·862 23·34 42·90 67·56 74·60 3·832 23·39 57·51 70·11 91·74 3·806 23·36 64·62 82·55 111·9 3·782 23·31 65·47 103·6 130·0
5·620 20·00 27·46 46·43 58·02 7·124 28·68 42·33 68·86 74·64 8·670 32·44 68·97 79·25 101·6 10·21 36·35 79·32 111·1 138·8 11·74 40·46 84·73 145·8 163·2 13·26 44·69 90·38 155·1 213·7 14·79 48·97 96·30 162·0 246·2
0·5 ZXXXCXXXV 1 2
T 3 2 Modified frequency coefficients V* i = Vi z1 − n of a cantilever plate for Poisson’s ratio equal to 0, 0·1, . . . 0·5. (1) Symmetric modes, (2) antisymmetric modes
145
a/b
2·00
1·75
1·50
1·25
1·00
0·75
0·50
2·945 9·489 19·04 27·41 34·65 2·960 15·23 20·67 37·74 55·84 2·965 18·48 26·37 48·01 60·61 2·966 19·02 36·65 54·55 69·38 2·967 19·18 47·46 59·11 83·39 2·967 19·25 52·72 70·24 98·55 2·967 19·29 54·11 85·97 109·8
1·938 8·042 17·38 26·20 34·64 1·989 12·27 20·39 34·81 54·48 2·004 15·05 25·75 44·05 60·56 2·010 16·06 33·97 51·76 68·45 2·013 16·44 42·00 58·85 80·56 2·014 16·62 46·76 69·22 94·42 2·015 16·72 48·86 81·63 107·2
0·0 ZXXXCXXXV 0·1 0·5 2·937 9·192 19·09 26·71 34·48 2·954 14·72 20·89 36·77 55·78 2·958 18·12 26·31 46·99 60·48 2·960 18·83 36·15 54·38 68·23 2·960 19·05 46·40 59·67 81·53 2·960 19·14 51·82 70·40 96·63 2·960 19·19 53·51 85·17 109·4
1·926 7·797 17·41 25·60 34·48 1·981 11·87 20·61 33·94 54·29 1·997 14·69 25·80 43·21 60·39 2·004 15·81 33·58 51·62 67·49 2·007 16·26 41·15 59·39 78·90 2·008 16·48 45·87 69·52 92·83 2·009 16·60 48·15 81·02 106·9
0·1 ZXXXCXXXV 0·1 0·5 2·925 8·855 19·12 25·88 34·23 2·940 14·14 21·09 35·65 55·61 2·943 17·65 26·21 45·78 60·32 2·943 18·53 35·53 53·99 67·04 2·941 18·80 45·16 60·07 79·50 2·940 18·92 50·63 70·40 94·47 2·938 18·98 52·60 84·06 108·6
1·908 7·523 17·40 24·90 34·22 1·966 11·42 20·81 32·92 53·97 1·983 14·24 25·80 42·19 60·18 1·989 15·47 33·07 51·29 66·48 1·992 15·99 40·13 59·79 77·06 1·992 16·25 44·75 69·62 91·01 1·992 16·39 47·17 80·11 106·3
0·2 ZXXXCXXXV 0·1 0·5 2·906 8·471 19·11 24·91 33·85 2·918 13·49 21·25 34·36 55·25 2·917 17·04 26·07 44·36 60·10 2·913 18·09 34·75 53·34 65·80 2·908 18·43 43·69 60·31 77·26 2·903 18·57 49·10 70·19 92·08 2·898 18·64 51·31 82·56 107·4
1·885 7·215 17·34 24·09 33·83 1·944 10·91 20·97 31·73 53·44 1·961 13·71 25·71 40·99 59·92 1·966 15·03 32·42 50·75 65·40 1·966 15·61 38·90 60·03 75·01 1·965 15·90 43·37 69·45 88·98 1·963 16·06 45·88 78·80 105·4
0·3 ZXXXCXXXV 0·1 0·5 2·879 8·032 19·06 23·79 33·29 2·886 12·76 21·35 32·87 54·55 2·879 16·28 25·83 42·73 59·76 2·868 17·50 33·77 52·42 64·46 2·858 17·90 41·93 60·34 74·77 2·848 18·07 47·18 69·63 89·49 2·838 18·14 49·59 80·51 106·0
1·854 6·868 17·19 23·15 33·27 1·914 10·34 21·09 30·34 52·57 1·929 13·07 25·52 39·59 59·55 1·931 14·47 31·55 49·99 64·17 1·928 15·11 37·42 60·09 72·69 1·924 15·43 41·66 68·89 86·77 1·919 15·60 44·22 76·94 104·3
0·4 ZXXXCXXXV 0·1 0·5 2·843 7·527 18·90 22·49 32·46 2·840 11·91 21·37 31·14 53·22 2·823 15·36 25·44 40·88 59·18 2·804 16·72 32·50 51·22 62·86 2·785 17·19 39·79 60·09 71·93 2·768 17·38 44·77 68·52 86·73 2·753 17·46 47·31 77·68 104·4
1·814 6·473 16·92 22·09 32·44 1·874 9·691 21·15 28·70 51·13 1·884 12·32 25·14 38·00 58·98 1·881 13·77 30·41 49·02 62·65 1·874 14·46 35·60 59·89 69·99 1·866 14·80 39·55 67·70 84·44 1·857 14·99 42·09 74·32 103·0
0·5 ZXXXCXXXV 0·1 0·5
T 4 Frequency coefficients Vi of a cantilever plate with a concentrated mass M at (a, b/2) as a function of n. (a) m = M/Mplate ; (b) symmetric modes only
146
2·00
1·75
1·50
1·25
1·00
0·75
0·50
a/b
2·945 9·489 19·04 27·41 34·65 2·960 15·23 20·67 37·74 55·84 2·965 18·48 26·37 48·01 60·61 2·966 19·02 36·65 54·55 69·38 2·967 19·18 47·46 59·11 83·39 2·967 19·25 52·72 70·24 98·55 2·967 19·29 54·11 85·97 109·8
1·938 8·042 17·38 26·20 34·64 1·989 12·27 20·39 34·81 54·48 2·004 15·05 25·75 44·05 60·56 2·010 16·06 33·97 51·76 68·45 2·013 16·44 42·00 58·85 80·56 2·014 16·62 46·76 69·22 94·42 2·015 16·72 48·86 81·63 107·2
0·0 ZXXXCXXXV 0·1 0·5 2·952 9·238 19·19 26·84 34·66 2·969 14·79 21·00 36·95 56·06 2·973 18·21 26·44 47·23 60·79 2·975 18·93 36·33 54·65 68·58 2·975 19·14 46·64 59·97 81·94 2·975 19·24 52·08 70·75 97·12 2·975 19·29 53·78 85·59 109·9
1·935 7·837 17·50 25·73 34·65 1·990 11·93 20·72 34·11 54·57 2·007 14·76 25·93 43·43 60·69 2·014 15·89 33·75 51·88 67·83 2·017 16·35 41·35 59·69 79·29 2·018 16·56 46·10 69·87 93·29 2·019 16·68 48·40 81·43 107·5
0·1 ZXXXCXXXV 0·1 0·5 2·985 9·037 19·51 26·41 34·93 3·001 14·44 21·53 36·38 56·75 3·004 18·01 26·76 46·73 61·57 3·003 18·91 36·26 55·10 68·43 3·002 19·19 46·09 61·31 81·14 3·000 19·31 51·67 71·85 96·42 2·998 19·37 53·68 85·80 110·8
1·948 7·678 17·76 25·41 34·92 2·006 11·66 21·24 33·60 55·09 2·024 14·53 26·33 43·06 61·42 2·030 15·79 33·76 52·35 67·85 2·033 16·32 40·95 61·02 78·65 2·033 16·58 45·67 71·05 92·88 2·033 16·72 48·15 81·76 108·5
0·2 ZXXXCXXXV 0·1 0·5 3·046 8·880 20·04 26·12 35·48 3·059 14·15 22·27 36·01 57·91 3·058 17·86 27·33 46·50 63·00 3·054 18·97 36·43 55·92 68·98 3·049 19·32 45·80 63·22 80·99 3·043 19·47 51·47 73·58 96·53 3·038 19·54 53·79 86·55 112·6
1·976 7·564 18·17 25·25 35·47 2·038 11·44 21·99 33·26 56·02 2·056 14·37 26·96 42·97 62·81 2·061 15·76 33·98 53·20 68·56 2·061 16·37 40·78 62·93 78·63 2·060 16·67 45·46 72·80 93·27 2·058 16·83 48·10 82·60 110·5
0·3 ZXXXCXXXV 0·1 0·5 3·142 8·764 20·79 25·96 36·32 3·149 13·92 23·30 35·86 59·52 3·141 17·77 28·18 46·62 65·20 3·130 19·09 36·85 57·19 70·33 3·118 19·53 45·75 65·83 81·58 3·107 19·71 51·48 75·98 97·64 3·097 19·79 54·10 87·85 115·6
2·023 7·494 18·76 25·26 36·30 2·089 11·29 23·01 33·10 57·36 2·105 14·27 27·84 43·20 64·97 2·107 15·79 34·43 54·55 70·01 2·104 16·49 40·83 65·56 79·31 2·099 16·83 45·45 75·16 94·67 2·094 17·02 48·24 83·95 113·8
0·4 ZXXXCXXXV 0·1 0·5 3·283 8·692 21·83 25·97 37·49 3·280 13·75 24·67 35·96 61·45 3·260 17·73 29·37 47·20 68·34 3·238 19·31 37·53 59·14 72·59 3·216 19·85 45·95 69·38 83·05 3·197 20·06 51·70 79·12 100·1 3·179 20·16 54·63 89·69 120·5
2·095 7·474 19·53 25·51 37·46 2·164 11·19 24·42 33·14 59·04 2·176 14·23 29·03 43·88 68·10 2·172 15·90 35·11 56·60 72·34 2·164 16·69 41·10 69·15 80·82 2·154 17·09 45·66 78·17 97·51 2·145 17·30 48·60 85·82 119·0
0·5 ZXXXCXXXV 0·1 0·5
T 5 Modified frequency coefficients V* i = Vi /z1 − n of a cantilever plate with a concentrated mass M at (a, b/2) as a function of n 2
147
148
and the fourth frequency for a/b = 1·5 which show a very slight increase for 0 E n E 0·20 and then decrease monotonically. In order to assess the influence of n it was considered to be of interest to obtain the 2 values of V* i = Vi /z1 − n and to present them in Table 3. It is now observed that V* i increases as a function of n whilst the remaining V* i decrease in some cases and increase in others. If one considers the third and fourth frequencies of the symmetric modes for a/b = 0·75 and 1·5, respectively, one notices that now both increase monotonically as a function of n. Table 4 shows the influence of a concentrated mass M rigidly attached to the plate at (a, b/2) upon the frequencies of symmetric modes of the structural element for n = 0, 0·1, . . . 0·5. It is again observed that the values of V1 decrease monotonically as n increases from 0 to 0·5 but the variation is, in general, slightly larger percentage-wise than in the case when no mass is present (for instance, when a/b = 1, the fundamental frequency decreases by 5% and 6% for m = 0·1 and 0·5, respectively, when n varies from 0 to 0·5, while V1 decreases by 4% when no mass is attached to the plate). 2 Table 5 depicts values of V* i = Vi /z1 − n for the situations treated in Table 4. It is again observed that V* i increases monotonically as n increases, for a fixed value of m except for m = 0·5 and a/b = 0·5 when n increases from 0 to 0·10. The remaining V* i decrease in some cases and increase in others. ACKNOWLEDGMENTS
The present study has been sponsored by Secretarı´ a General de Ciencia y Tecnologı´ a of Universidad Nacional del Sur and by PID-BID 003/92 and PID 30010/92 (CONICET). 1. A. W. L 1969 NASA SP-160. Vibration of plates. 2. D. Y 1950 Journal of Applied Mechanics 17, 448–453. Vibration of rectangular plates by the Ritz method. 3. A. W. L 1973 Journal of Sound and Vibration 31, 257–293. The free vibration of rectangular plates. 4. S. F. B and S. M. D 1975 Journal of Applied Mechanics 42, 858–864. On the use of beam function for problems of plates involving free edges. 5. R. B. B 1985 Journal of Sound and Vibration 102, 493–499. Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. 6. F. K. B, R. L. F and L. A. S J 1966 Matrix Methods in Structural Mechanics AFFDL-TR-66-80, 397–443. The generation of inter-element-compatible stiffness and mass matrices by the use of interpolation formulas. 7. D. J. G 1985 Journal of Sound and Vibration 181, 605–618. Accurate free vibration analysis of the orthotropic cantilever plate.
SYMMETRIC AND ANTISYMMETRIC NORMAL MODES OF A CANTILEVER RECTANGULAR PLATE: EFFECT OF POISSON’S RATIO AND A CONCENTRATED MASS R. E. R P. A. A. L* Department of Engineering, Universidad Nacional del Sur and Institute of Applied Mechanics (CONICET), 8000—Bahi´ a Blanca, Argentina (Received 17 November 1995)
1. In spite of the fact that several well known authors have tackled the problem of transverse vibrations of a rectangular, cantilever plate† there is a lack of information regarding the influence of Poisson’s ratio upon the values of the frequency coefficients corresponding to symmetric and antisymmetric modes of vibration (Figure 1(a)). The present study deals with the first 10 normal modes of vibration (5 symmetric and 5 antisymmetric modes). The analysis is performed using the very accurate finite element algorithmic procedure due to Bogner et al. [2] for values of Poisson’s ratio, n, ranging from 0 to 0·5. In view of the practical implications of the problem the second part of this study depicts frequency coefficients as a function of the parameter (concentrated mass)/(total plate mass) in the case where the concentrated mass is placed at the center of the completely free edge, Figure 1(b). The effect of Poisson’s ratio is also investigated. 2. The present study uses the conforming rectangular element of 16 degrees-of-freedom developed by Bogner, Fox and Schmit [6] which yields excellent accuracy. One hundred elements with 121 nodes and a total of 420 degrees-of-freedom were used for the calculations presented in this paper. 3. Table 1 depicts natural frequency coefficients Vi = z(rh/D)vi a 2 corresponding to symmetric and antisymmetric modes for n = 1/3. The agreement with the results recently obtained by Gorman [7]‡ is remarkabley good. Interestingly, Gorman obtained the first four eigenvalues corresponding to symmetric and antisymmetric modes but in accordance with the results depicted in Table 1 the fifth natural frequency coefficient of a symmetric mode is lower than the fourth of an antisymmetric model for a/b = 0·5, 0·75, . . . 2. Table 2 presents natural frequency coefficients Vi for the first five symmetric and antisymmetric modes of cantilever plates for n ranging from 0 to 0·5. As expected, for n = 0 * Corresponding author. † A thorough treatment is contained in Leissa’s classical treatise [1]. Apparently the first numerical results, for the case of a square cantilever plate and n = 0·30, were obtained by D. Young [2]. Very accurate results have been obtained rather recently [3–5]. ‡ Gorman’s study [7] deals also with the isotropic plate, as a particular case.
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Figure 1. Vibrating structural system under study: (a) lower symmetric (S) and antisymmetric (A) modes; (b) cantilever plate with a concentrated mass M, m = M/Mplate .
the fundamental frequency coefficient V1 coincides with the value corresponding to a cantilever beam. On the other hand, as n increases the values of Vi always decrease monotonically, exceptions are the third frequency of the symmetric modes for a/b = 0·75 T 1 Frequency coefficients Vi = z(rh/D)vi a 2 of a cantilever plate for Poisson’s ratio equal to 1/3 (1) Symmetric modes, (2) antisymmetric modes
a/b
n = 1/3 ZXXXCXXXV 1 2
a/b
n = 1/3 ZXXXCXXXV 1 2
0·5
3·487 10·03 21·79 31·07 33·88
5·279 18·84 24·56 42·67 52·43
0·75
3·473 16·69 22·02 41·55 60·84
1·25
3·448 21·31 38·93 60·63 68·24
9·896 34·53 73·87 99·04 126·2
1·50
3·438 21·32 53·05 61·55 84·44
2·00
3·420 21·28 59·75 92·69 117·7
14·50 47·32 91·24 151·1 228·1
6·805 26·86 38·17 63·39 66·90 11·43 38·70 79·29 135·3 143·7
a/b
n = 1/3 ZXXXCXXXV 1 2
1·00
3·460 21·09 27·06 53·53 61·12
1·75
3·429 21·31 59·40 72·85 102·9
8·356 30·55 63·62 70·65 92·21 12·96 42·98 85·10 144·0 190·5
2·00
1·75
1·50
1·25
1·00
0·75
0·5
a/b
3·516 11·21 22·03 34·01 34·75 3·516 18·50 22·04 45·50 61·70 3·516 22·03 28·35 58·98 61·70 3·516 22·03 40·92 61·70 74·45 3·516 22·03 56·27 61·70 92·05 3·516 22·03 61·70 74·42 111·9 3·516 22·03 61·70 95·36 120·9
5·916 20·46 25·45 46·14 54·04 7·788 29·19 39·64 68·41 69·10 9·676 33·70 66·52 73·38 98·25 11·56 38·65 79·34 101·1 134·3 13·44 43·83 86·09 143·5 145·7 15·33 49·15 93·46 153·9 193·6 17·22 54·55 101·3 162·9 242·5
0·0 ZXXXCXXXV 1 2 3·514 10·92 22·02 33·28 34·60 3·512 18·11 22·04 44·51 61·64 3·511 21·96 27·98 57·61 61·67 3·511 21·98 40·48 61·63 72·77 3·510 21·98 55·70 61·72 90·04 3·509 21·98 61·52 74·00 109·6 3·508 21·97 61·55 94·85 120·7
5·745 20·13 25·23 45·27 53·79 7·524 28·67 39·33 67·40 68·43 9·320 32·95 66·12 72·68 96·70 11·11 37·63 78·20 100·8 132·4 12·89 42·53 84·57 142·4 144·9 14·68 47·56 91·51 151·9 193·2 16·47 52·67 98·85 160·4 239·9
0·1 ZXXXCXXXV 1 2 3·506 10·58 21·96 32·42 34·37 3·501 17·59 22·04 43·36 61·44 3·497 21·72 27·59 56·02 61·54 3·493 21·79 39·91 61·39 70·93 3·490 21·79 54·79 61·72 87·81 3·487 21·79 60·93 73·55 107·0 3·484 21·78 61·05 94·11 120·0
5·558 19·68 24·98 44·26 53·37 7·236 28·01 38·93 65·86 67·91 8·933 32·04 65·36 71·88 94·95 10·62 36·44 76·67 100·2 130·0 12·30 41·04 82·64 140·0 144·5 13·98 45·77 89·12 149·2 192·4 15·67 50·56 95·97 157·1 235·9
0·2 ZXXXCXXXV 1 2 3·493 10·18 21·84 31·43 34·03 3·482 16·94 22·03 42·04 61·04 3·471 21·29 27·20 54·19 61·26 3·462 21·46 39·20 60·88 68·94 3·454 21·47 53·54 61·62 85·33 3·446 21·46 59·88 73·04 104·0 3·440 21·44 60·15 93·11 118·5
5·352 19·08 24·67 43·09 52·71 6·566 27·18 38·39 64·06 67·20 8·508 30·96 64·15 70·98 92·94 10·09 35·05 74·67 99·39 127·3 11·66 39·33 80·23 136·6 144·0 13·23 43·73 86·21 145·5 191·1 14·80 48·18 92·53 152·8 230·4
0·3 ZXXXCXXXV 1 2 3·473 9·707 21·64 30·30 33·52 3·451 16·15 21·97 40·50 60·32 3·432 20·61 26·78 52·10 60·76 3·414 20·95 38·31 59·98 66·82 3·398 20·97 51·92 61·32 82·56 3·385 20·95 58·26 72·40 100·7 3·372 20·92 58·75 91·71 116·0
5·124 18·31 24·29 41·74 51·73 6·566 26·14 37·67 61·98 66·17 8·037 29·66 62·33 69·93 90·64 9·497 33·42 72·06 98·15 124·0 10·95 37·35 77·19 132·2 143·0 12·40 41·40 82·65 140·6 189·0 13·86 45·49 88·40 147·3 223·0
0·4 ZXXXCXXXV 1 2 3·444 9·141 21·31 29·02 32·76 3·407 15·19 21·80 38·74 59·06 3·374 19·66 26·28 49·73 59·88 3·345 20·21 37·15 58·51 64·60 3·319 20·26 49·80 60·72 79·45 3·296 20·23 55·96 71·49 96·95 3·276 20·18 56·70 89·69 112·6
4·867 17·32 23·78 40·21 50·24 6·169 24·84 36·66 59·64 64·64 7·508 28·09 59·73 68·63 87·99 8·840 31·48 68·69 96·19 120·2 10·16 35·04 73·38 126·2 141·3 11·49 38·70 78·27 134·3 185·0 12·81 42·41 83·40 140·3 213·2
0·5 ZXXXCXXXV 1 2
T 2 Frequency coefficients Vi of a cantilever plate for Poisson’s ratio equal to 0, 0·1, · · · 0·5· (1) Symmetric modes, (2) antisymmetric modes
144
a/b
2·00
1·75
1·50
1·25
1·00
0·75
0·5
3·516 11·21 22·04 34·01 34·75 3·516 18·50 22·04 45·50 61·70 3·516 22·03 28·35 58·98 61·70 3·516 22·03 40·92 61·70 74·45 3·516 22·03 56·27 61·70 92·05 3·516 22·03 61·70 74·42 111·9 3·516 22·03 61·70 95·36 120·9
5·916 20·46 25·45 46·14 54·04 7·788 29·19 39·64 68·41 69·10 9·676 33·70 66·52 73·38 98·25 11·56 38·65 79·34 101·1 134·3 13·44 43·83 86·09 143·5 145·7 15·33 49·15 93·45 153·9 193·6 17·22 54·55 101·3 162·9 242·5
3·531 10·98 22·13 33·44 34·77 3·530 18·20 22·15 44·73 61·96 3·529 22·07 28·12 57·90 61·98 3·528 22·09 40·68 61·94 73·14 3·527 22·09 55·98 62·03 90·50 3·527 22·09 61·83 74·37 110·2 3·526 22·08 61·86 95·33 121·3
5·774 20·23 25·36 45·50 54·06 7·562 28·82 39·53 67·74 68·78 9·367 33·11 66·45 73·05 97·19 11·16 37·82 78·60 101·3 133·0 12·96 42·75 85·00 143·1 145·7 14·76 47·80 91·97 152·7 194·1 16·56 52·93 99·35 161·2 241·1
0·0 0·1 ZXXXCXXXV ZXXXCXXXV 1 2 1 2 3·579 10·80 22·41 33·09 35·08 3·574 17·95 22·50 44·26 62·71 3·569 22·17 28·16 57·18 62·81 3·565 22·24 40·74 62·65 72·40 3·562 22·24 55·92 63·00 89·62 3·558 22·24 62·19 75·06 109·2 3·555 22·23 62·31 96·05 122·5
5·672 20·08 25·49 45·18 54·47 7·385 28·59 39·73 67·22 69·31 9·117 32·70 66·71 73·36 96·90 10·84 37·19 78·25 102·3 132·7 12·55 41·89 84·35 142·9 147·5 14·27 46·71 90·96 152·2 196·4 15·99 51·60 97·95 160·3 240·7
0·2 ZXXXCXXXV 1 2 3·662 10·67 22·90 32·95 35·67 3·650 17·76 23·10 44·07 63·99 3·639 22·31 28·51 56·81 64·22 3·629 22·50 41·10 63·82 72·27 3·620 22·50 56·13 64·60 89·45 3·613 22·49 62·77 76·57 109·0 3·606 22·47 63·06 97·60 124·2
5·610 20·00 25·87 45·18 55·26 7·252 28·50 40·24 67·15 70·44 8·918 32·45 67·24 74·41 97·43 10·57 36·74 78·27 104·2 133·4 12·22 41·23 84·10 143·3 150·9 13·87 45·84 90·37 152·5 200·4 15·52 50·51 97·00 160·2 241·5
0·3 ZXXXCXXXV 1 2 3·789 10·59 23·62 33·06 36·58 3·766 17·62 23·97 44·19 65·81 3·744 22·49 29·22 56·85 66·29 3·725 22·86 41·80 65·44 72·91 3·708 22·88 56·65 66·91 90·08 3·693 22·86 63·57 79·00 109·8 3·679 22·82 64·10 100·1 126·5
5·590 19·97 26·51 45·55 56·44 7·164 28·53 41·10 67·63 72·20 8·769 32·36 68·01 76·30 98·90 10·36 36·46 78·62 107·1 135·3 11·95 40·75 84·23 144·2 156·0 13·53 45·17 90·18 153·4 206·2 15·12 49·64 96·45 160·8 243·3
0·4 ZXXXCXXXV 1 2 3·977 10·55 24·61 33·51 37·83 3·934 17·53 25·18 44·73 68·19 3·896 22·71 30·35 57·42 69·14 3·862 23·34 42·90 67·56 74·60 3·832 23·39 57·51 70·11 91·74 3·806 23·36 64·62 82·55 111·9 3·782 23·31 65·47 103·6 130·0
5·620 20·00 27·46 46·43 58·02 7·124 28·68 42·33 68·86 74·64 8·670 32·44 68·97 79·25 101·6 10·21 36·35 79·32 111·1 138·8 11·74 40·46 84·73 145·8 163·2 13·26 44·69 90·38 155·1 213·7 14·79 48·97 96·30 162·0 246·2
0·5 ZXXXCXXXV 1 2
T 3 2 Modified frequency coefficients V* i = Vi z1 − n of a cantilever plate for Poisson’s ratio equal to 0, 0·1, . . . 0·5. (1) Symmetric modes, (2) antisymmetric modes
145
a/b
2·00
1·75
1·50
1·25
1·00
0·75
0·50
2·945 9·489 19·04 27·41 34·65 2·960 15·23 20·67 37·74 55·84 2·965 18·48 26·37 48·01 60·61 2·966 19·02 36·65 54·55 69·38 2·967 19·18 47·46 59·11 83·39 2·967 19·25 52·72 70·24 98·55 2·967 19·29 54·11 85·97 109·8
1·938 8·042 17·38 26·20 34·64 1·989 12·27 20·39 34·81 54·48 2·004 15·05 25·75 44·05 60·56 2·010 16·06 33·97 51·76 68·45 2·013 16·44 42·00 58·85 80·56 2·014 16·62 46·76 69·22 94·42 2·015 16·72 48·86 81·63 107·2
0·0 ZXXXCXXXV 0·1 0·5 2·937 9·192 19·09 26·71 34·48 2·954 14·72 20·89 36·77 55·78 2·958 18·12 26·31 46·99 60·48 2·960 18·83 36·15 54·38 68·23 2·960 19·05 46·40 59·67 81·53 2·960 19·14 51·82 70·40 96·63 2·960 19·19 53·51 85·17 109·4
1·926 7·797 17·41 25·60 34·48 1·981 11·87 20·61 33·94 54·29 1·997 14·69 25·80 43·21 60·39 2·004 15·81 33·58 51·62 67·49 2·007 16·26 41·15 59·39 78·90 2·008 16·48 45·87 69·52 92·83 2·009 16·60 48·15 81·02 106·9
0·1 ZXXXCXXXV 0·1 0·5 2·925 8·855 19·12 25·88 34·23 2·940 14·14 21·09 35·65 55·61 2·943 17·65 26·21 45·78 60·32 2·943 18·53 35·53 53·99 67·04 2·941 18·80 45·16 60·07 79·50 2·940 18·92 50·63 70·40 94·47 2·938 18·98 52·60 84·06 108·6
1·908 7·523 17·40 24·90 34·22 1·966 11·42 20·81 32·92 53·97 1·983 14·24 25·80 42·19 60·18 1·989 15·47 33·07 51·29 66·48 1·992 15·99 40·13 59·79 77·06 1·992 16·25 44·75 69·62 91·01 1·992 16·39 47·17 80·11 106·3
0·2 ZXXXCXXXV 0·1 0·5 2·906 8·471 19·11 24·91 33·85 2·918 13·49 21·25 34·36 55·25 2·917 17·04 26·07 44·36 60·10 2·913 18·09 34·75 53·34 65·80 2·908 18·43 43·69 60·31 77·26 2·903 18·57 49·10 70·19 92·08 2·898 18·64 51·31 82·56 107·4
1·885 7·215 17·34 24·09 33·83 1·944 10·91 20·97 31·73 53·44 1·961 13·71 25·71 40·99 59·92 1·966 15·03 32·42 50·75 65·40 1·966 15·61 38·90 60·03 75·01 1·965 15·90 43·37 69·45 88·98 1·963 16·06 45·88 78·80 105·4
0·3 ZXXXCXXXV 0·1 0·5 2·879 8·032 19·06 23·79 33·29 2·886 12·76 21·35 32·87 54·55 2·879 16·28 25·83 42·73 59·76 2·868 17·50 33·77 52·42 64·46 2·858 17·90 41·93 60·34 74·77 2·848 18·07 47·18 69·63 89·49 2·838 18·14 49·59 80·51 106·0
1·854 6·868 17·19 23·15 33·27 1·914 10·34 21·09 30·34 52·57 1·929 13·07 25·52 39·59 59·55 1·931 14·47 31·55 49·99 64·17 1·928 15·11 37·42 60·09 72·69 1·924 15·43 41·66 68·89 86·77 1·919 15·60 44·22 76·94 104·3
0·4 ZXXXCXXXV 0·1 0·5 2·843 7·527 18·90 22·49 32·46 2·840 11·91 21·37 31·14 53·22 2·823 15·36 25·44 40·88 59·18 2·804 16·72 32·50 51·22 62·86 2·785 17·19 39·79 60·09 71·93 2·768 17·38 44·77 68·52 86·73 2·753 17·46 47·31 77·68 104·4
1·814 6·473 16·92 22·09 32·44 1·874 9·691 21·15 28·70 51·13 1·884 12·32 25·14 38·00 58·98 1·881 13·77 30·41 49·02 62·65 1·874 14·46 35·60 59·89 69·99 1·866 14·80 39·55 67·70 84·44 1·857 14·99 42·09 74·32 103·0
0·5 ZXXXCXXXV 0·1 0·5
T 4 Frequency coefficients Vi of a cantilever plate with a concentrated mass M at (a, b/2) as a function of n. (a) m = M/Mplate ; (b) symmetric modes only
146
2·00
1·75
1·50
1·25
1·00
0·75
0·50
a/b
2·945 9·489 19·04 27·41 34·65 2·960 15·23 20·67 37·74 55·84 2·965 18·48 26·37 48·01 60·61 2·966 19·02 36·65 54·55 69·38 2·967 19·18 47·46 59·11 83·39 2·967 19·25 52·72 70·24 98·55 2·967 19·29 54·11 85·97 109·8
1·938 8·042 17·38 26·20 34·64 1·989 12·27 20·39 34·81 54·48 2·004 15·05 25·75 44·05 60·56 2·010 16·06 33·97 51·76 68·45 2·013 16·44 42·00 58·85 80·56 2·014 16·62 46·76 69·22 94·42 2·015 16·72 48·86 81·63 107·2
0·0 ZXXXCXXXV 0·1 0·5 2·952 9·238 19·19 26·84 34·66 2·969 14·79 21·00 36·95 56·06 2·973 18·21 26·44 47·23 60·79 2·975 18·93 36·33 54·65 68·58 2·975 19·14 46·64 59·97 81·94 2·975 19·24 52·08 70·75 97·12 2·975 19·29 53·78 85·59 109·9
1·935 7·837 17·50 25·73 34·65 1·990 11·93 20·72 34·11 54·57 2·007 14·76 25·93 43·43 60·69 2·014 15·89 33·75 51·88 67·83 2·017 16·35 41·35 59·69 79·29 2·018 16·56 46·10 69·87 93·29 2·019 16·68 48·40 81·43 107·5
0·1 ZXXXCXXXV 0·1 0·5 2·985 9·037 19·51 26·41 34·93 3·001 14·44 21·53 36·38 56·75 3·004 18·01 26·76 46·73 61·57 3·003 18·91 36·26 55·10 68·43 3·002 19·19 46·09 61·31 81·14 3·000 19·31 51·67 71·85 96·42 2·998 19·37 53·68 85·80 110·8
1·948 7·678 17·76 25·41 34·92 2·006 11·66 21·24 33·60 55·09 2·024 14·53 26·33 43·06 61·42 2·030 15·79 33·76 52·35 67·85 2·033 16·32 40·95 61·02 78·65 2·033 16·58 45·67 71·05 92·88 2·033 16·72 48·15 81·76 108·5
0·2 ZXXXCXXXV 0·1 0·5 3·046 8·880 20·04 26·12 35·48 3·059 14·15 22·27 36·01 57·91 3·058 17·86 27·33 46·50 63·00 3·054 18·97 36·43 55·92 68·98 3·049 19·32 45·80 63·22 80·99 3·043 19·47 51·47 73·58 96·53 3·038 19·54 53·79 86·55 112·6
1·976 7·564 18·17 25·25 35·47 2·038 11·44 21·99 33·26 56·02 2·056 14·37 26·96 42·97 62·81 2·061 15·76 33·98 53·20 68·56 2·061 16·37 40·78 62·93 78·63 2·060 16·67 45·46 72·80 93·27 2·058 16·83 48·10 82·60 110·5
0·3 ZXXXCXXXV 0·1 0·5 3·142 8·764 20·79 25·96 36·32 3·149 13·92 23·30 35·86 59·52 3·141 17·77 28·18 46·62 65·20 3·130 19·09 36·85 57·19 70·33 3·118 19·53 45·75 65·83 81·58 3·107 19·71 51·48 75·98 97·64 3·097 19·79 54·10 87·85 115·6
2·023 7·494 18·76 25·26 36·30 2·089 11·29 23·01 33·10 57·36 2·105 14·27 27·84 43·20 64·97 2·107 15·79 34·43 54·55 70·01 2·104 16·49 40·83 65·56 79·31 2·099 16·83 45·45 75·16 94·67 2·094 17·02 48·24 83·95 113·8
0·4 ZXXXCXXXV 0·1 0·5 3·283 8·692 21·83 25·97 37·49 3·280 13·75 24·67 35·96 61·45 3·260 17·73 29·37 47·20 68·34 3·238 19·31 37·53 59·14 72·59 3·216 19·85 45·95 69·38 83·05 3·197 20·06 51·70 79·12 100·1 3·179 20·16 54·63 89·69 120·5
2·095 7·474 19·53 25·51 37·46 2·164 11·19 24·42 33·14 59·04 2·176 14·23 29·03 43·88 68·10 2·172 15·90 35·11 56·60 72·34 2·164 16·69 41·10 69·15 80·82 2·154 17·09 45·66 78·17 97·51 2·145 17·30 48·60 85·82 119·0
0·5 ZXXXCXXXV 0·1 0·5
T 5 Modified frequency coefficients V* i = Vi /z1 − n of a cantilever plate with a concentrated mass M at (a, b/2) as a function of n 2
147
148
and the fourth frequency for a/b = 1·5 which show a very slight increase for 0 E n E 0·20 and then decrease monotonically. In order to assess the influence of n it was considered to be of interest to obtain the 2 values of V* i = Vi /z1 − n and to present them in Table 3. It is now observed that V* i increases as a function of n whilst the remaining V* i decrease in some cases and increase in others. If one considers the third and fourth frequencies of the symmetric modes for a/b = 0·75 and 1·5, respectively, one notices that now both increase monotonically as a function of n. Table 4 shows the influence of a concentrated mass M rigidly attached to the plate at (a, b/2) upon the frequencies of symmetric modes of the structural element for n = 0, 0·1, . . . 0·5. It is again observed that the values of V1 decrease monotonically as n increases from 0 to 0·5 but the variation is, in general, slightly larger percentage-wise than in the case when no mass is present (for instance, when a/b = 1, the fundamental frequency decreases by 5% and 6% for m = 0·1 and 0·5, respectively, when n varies from 0 to 0·5, while V1 decreases by 4% when no mass is attached to the plate). 2 Table 5 depicts values of V* i = Vi /z1 − n for the situations treated in Table 4. It is again observed that V* i increases monotonically as n increases, for a fixed value of m except for m = 0·5 and a/b = 0·5 when n increases from 0 to 0·10. The remaining V* i decrease in some cases and increase in others. ACKNOWLEDGMENTS
The present study has been sponsored by Secretarı´ a General de Ciencia y Tecnologı´ a of Universidad Nacional del Sur and by PID-BID 003/92 and PID 30010/92 (CONICET). 1. A. W. L 1969 NASA SP-160. Vibration of plates. 2. D. Y 1950 Journal of Applied Mechanics 17, 448–453. Vibration of rectangular plates by the Ritz method. 3. A. W. L 1973 Journal of Sound and Vibration 31, 257–293. The free vibration of rectangular plates. 4. S. F. B and S. M. D 1975 Journal of Applied Mechanics 42, 858–864. On the use of beam function for problems of plates involving free edges. 5. R. B. B 1985 Journal of Sound and Vibration 102, 493–499. Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. 6. F. K. B, R. L. F and L. A. S J 1966 Matrix Methods in Structural Mechanics AFFDL-TR-66-80, 397–443. The generation of inter-element-compatible stiffness and mass matrices by the use of interpolation formulas. 7. D. J. G 1985 Journal of Sound and Vibration 181, 605–618. Accurate free vibration analysis of the orthotropic cantilever plate.