- Email: [email protected]
Vibrations of a rectangular plate of non-uniform thickness partially embedded in a Winkler medium
Journal of Sound and Vibration (1995) 185(5), 910–914
VIBRATIONS OF A RECTANGULAR PLATE OF NON-UNIFORM THICKNESS PARTIALLY EMBEDDED IN A WINKLER MEDIUM R. H. G P. A. A. L Institute of Applied Mechanics (CONICET) and Department of Engineering, Universidad Nacional del Sur 8000 Bahi´ a Blanca, Argentina H. C. S Facultad Regional Haedo (Universidad Tecnolo´gica Nacional) Haedo, Buenos Aires Province, Argentina G. E INTI. P.O. Box 157, San Martin, Buenos Aires Province, Argentina (Received 16 September 1994, and in final form 5 December 1994)
1. This study deals with the determination of the fundamental frequency of vibration of a clamped rectangular plate partially embedded in a Winkler type foundation; see Figure 1. The plate thickness varies according to the expression h(x¯ )=h(0){a(x¯ /a)+1},
aq0.
(1)
Accordingly the flexural rigidity is given by D(x¯ )=D(0){a(x¯ /a)+1)3.
(2)
The Winkler-type foundation is defined by (see Figure 1) k(x¯ , y¯ )=
6
k,
(x¯ , y¯ ) $ P1
0,
(x¯ , y¯ ) $ P−P1
7
.
(3)
The fundamental eigenvalue is obtained (a) by using the differential quadrature (DQ) technique [1–5], and (b) by the finite element method. The solution of this problem has not appeared in the open literature. 2. In terms of the dimensionless variables x=x¯ /a and y=y¯ /b the problem under consideration is governed by the differential system g 2(Wx 4+2l 2Wx 2y 2+l 4Wy 4 )+6ag(Wx 3+l 2Wxy 2 )+6a 2(Wx 2+l 2nWy 2 ) +(k'/g)W−V 2W=0,
(4)
W(0, y)=W(x, 0)=W(1, y)=W(x, 1)=0,
(5a)
Wx (0, y)=Wy (x, 0)=Wx (1, y)=Wy (x, 1)=0,
(5b)
910 0022–460X/95/350910+05 $12.00/0
7 1995 Academic Press Limited
911 y b
P v U
P1 0
P
a
u
x
h (0)
Figure 1. The vibrating mechanical system under study.
where l=a/b,
g(x)=ax+1,
V 2=rh(0)a 4v 2/D(0),
k'=ka 4/D(0), 0,
(see equation (3)). With the notation and approach of Bert and coworkers [2, 5], application of the differential quadrature method yields the following linear system of equations: N−1
N−1
s A2k 1 Wk 1 j=0,
s A2k 2 Wik 2=0,
j=3,..., N−1;
k 1=2
i=2,..., N−2;
k 2=2
N−1
s A(N−1)k 1 Wk 1 j=0,
j=2,..., N−2;
k 1=2
N−1
s A(N−1)k 2 Wik 2=0,
i=3,..., N−1;
k 2=2
$
N−1
N−1 N−1
N−1
k 1=2
k 1=2 k 2=2
k 2=2
gi2 s Dik 1 Wk 1 j+l 2 s s Bik 1 Bjk 2 Wk 1 k 2+l 4 s Djk 2 Wik 2
$ $
N−1
N−1 N−1
k 1=2
k 1=2 k 2=2
+6agi s Cik 1 Wk 1 j+l 2 s s Aik 1 Bjk 2 Wk 1 Wk 1 k 2 N−1
N−1
k 1=2
k 2=2
%
+6a 2 s Bik 1 Wk 1 j+l 2n s Bjk 2 Wik 2 +
%
%
k' W −V 2Wij=0, gi ij
i, j=3,..., N−2. (6)
12
3
4...
N–2 N–1 N
0 δ = 10–4
–4
δ = 10
Figure 2. The partition of the interval [0, 1] in the x and y directions when implementing the DQ approach.
912
z y
x
Figure 3. A typical modal shape when the system vibrates in its fundamental mode (l=1·50; u=v=0·6).
Here gi=g(xi ). The non-triviality condition yields a secular determinant the lowest root of which is the fundamental frequency coefficient of the system. The partition of the interval [0, 1] is denoted in Figure 2. For the cases u=v=0·2 and 0·6, the parameter N was taken equal to 9, while when considering the situations u=v=0·4 and 0·8, N was chosen equal to 11. 3. Since the structural system and its fundamental mode is symmetric with respect to y¯=b/2 (see Figure 1), one half of the plate was modelled by 450 rectangular elements. The results were obtained by using a standard finite element code. A typical fundamental modal shape obtained is depicted in Figure 3. 4. All calculations have been performed with n (the Poisson ratio)=0·30. In Table 1 are depicted fundamental frequency coefficients determined by means of the DQ method in the case of a plate of uniform thickness (a=0).
T 1 Fundamental frequency coefficient, V1 , of a clamped rectangular plate of uniform thickness partially embedded in a Winkler type foundation (the fundamental eigenvalue has been determined by using the DQ method) a=0, u=v 0·20 0·40 0·60 0·80
l=1 ZXXXXCXXXXV k'=320 800 1600 36·937 38·838 39·550 40·203
38·260 42·713 44·287 45·785
40·336 48·425 51·129 53·817
l=1·5 l=2 ZXXXXCXXXXV ZXXXXXCXXXXXV 320 800 1600 320 800 1600 61·359 62·508 63·239 63·392
62·161 64·984 66·736 67·071
63·462 68·888 72·183 72·791
98·725 99·423 99·733 100·009
99·215 100·983 101·698 102·380
100·020 103·519 105·763 106·216
913
T 2 Fundamental frequency coefficient of a clamped rectangular plate of linearly varying thickness (a=0·20); (1) results obtained by using the DQ method; (2) finite element results a=0·2 u=v
l=1 ZXXXXCXXXXV k'=320 800 1600
l=1·5 l=2 ZXXXXCXXXXV ZXXXXXCXXXXXV 320 800 1600 320 800 1600
0·2 (1) (2)
40·309 40·26
41·414 41·39
43·170 43·19
67·188 67·06
67·848 67·74
68·927 68·85
108·122 107·98
108·520 108·40
109·173 109·08
0·4 (1) (2)
41·898 41·76
45·200 44·95
50·182 49·74
68·133 67·96
70·203 69·96
73·506 73·13
108·681 108·50
109·969 109·80
112·072 111·80
0·6 (1) (2)
42·508 42·74
46·572 47·21
52·595 53·82
68·531 68·57
71·122 71·44
75·218 75·97
108·963 108·90
110·596 110·75
113·256 113·72
0·8 (1) (2)
43·063 43·00
47·876 47·80
54·968 54·87
68·868 68·74
71·980 71·82
76·886 76·74
109·160 109·00
111·155 111·00
114·402 114·24
Tables 2 and 3 are for plates of non-uniform thickness (a=0·2 and 0·4, respectively). The eigenvalues obtained by means of the DQ technique are compared with those determined by means of the finite element algorithmic procedure. One concludes immediately that an excellent agreement is obtained for all the situations considered. Analysis of the tables reveals that, in general, the effect of doubling u is larger than that of doubling k'. It is also seen that the effect of the taper parameter, a, is relatively small for the range of values of a considered. The present investigation also shows that the DQ method can be easily implemented when the embedding medium properties vary in a discontinuous fashion.
The present study has been sponsored by CONICET Research and Development Program (PID-BID 003-92). The authors are indebted to the reviewer of this paper and to Professor P. E. Doak for their valuable criticism.
T 3 Fundamental frequency coefficient of a clamped rectangular plate of linearly varying thickness (a=0·40); (1) results obtained by using the DQ method; (2) finite element results a=0·4 u=v
l=1 ZXXXXCXXXXV k'=320 800 1600
l=1·5 l=2 ZXXXXCXXXXV ZXXXXXCXXXXXV 320 800 1600 320 800 1600
0·2 (1) (2)
43·592 43·53
44·523 44·44
46·016 46·03
72·772 72·62
73·317 73·19
74·213 74·12
116·894 116·71
117·210 117·04
117·731 117·60
0·4 (1) (2)
44·938 44·81
47·780 47·55
52·141 51·75
73·557 73·40
75·307 75·07
78·123 77·78
117·331 117·10
118·400 118·20
120·152 119·90
0·6 (1) (2)
45·476 45·66
48·996 49·55
54·318 55·42
73·920 73·92
76·128 76·38
79·655 80·31
117·611 117·50
118·988 119·09
121·238 121·64
0·8 (1) (2)
45·962 45·90
50·167 50·09
56·481 56·38
74·210 74·08
76·898 76·76
81·177 81·02
117·775 117·60
119·501 119·30
122·322 122·10
914
1. R. B and J. C 1971 Journal of Mathematical Analysis and Applications 34, 235–238. Differential quadrature and long-term integration. 2. C. W. B, S. K. J and A. G. S 1989 Computational Mechanics 5, 217–226. Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature. 3. S. K. J, C. W. B and A. G. S 1989 International Journal of Numerical Methods in Engineering 28, 561–577. Application of differential quadrature to static analysis of structural components. 4. A. G. S, S. K. J and C. W. B 1988 Thin-walled Structures 6, 51–62. Nonlinear bending analysis of thin circular plates by differential quadrature. 5. A. R. K, J. F and C. W. B 1992 Journal of Engineering Mechanics 118, 1221–1238. Fundamental frequency of tapered plates by differential quadrature method.
VIBRATIONS OF A RECTANGULAR PLATE OF NON-UNIFORM THICKNESS PARTIALLY EMBEDDED IN A WINKLER MEDIUM R. H. G P. A. A. L Institute of Applied Mechanics (CONICET) and Department of Engineering, Universidad Nacional del Sur 8000 Bahi´ a Blanca, Argentina H. C. S Facultad Regional Haedo (Universidad Tecnolo´gica Nacional) Haedo, Buenos Aires Province, Argentina G. E INTI. P.O. Box 157, San Martin, Buenos Aires Province, Argentina (Received 16 September 1994, and in final form 5 December 1994)
1. This study deals with the determination of the fundamental frequency of vibration of a clamped rectangular plate partially embedded in a Winkler type foundation; see Figure 1. The plate thickness varies according to the expression h(x¯ )=h(0){a(x¯ /a)+1},
aq0.
(1)
Accordingly the flexural rigidity is given by D(x¯ )=D(0){a(x¯ /a)+1)3.
(2)
The Winkler-type foundation is defined by (see Figure 1) k(x¯ , y¯ )=
6
k,
(x¯ , y¯ ) $ P1
0,
(x¯ , y¯ ) $ P−P1
7
.
(3)
The fundamental eigenvalue is obtained (a) by using the differential quadrature (DQ) technique [1–5], and (b) by the finite element method. The solution of this problem has not appeared in the open literature. 2. In terms of the dimensionless variables x=x¯ /a and y=y¯ /b the problem under consideration is governed by the differential system g 2(Wx 4+2l 2Wx 2y 2+l 4Wy 4 )+6ag(Wx 3+l 2Wxy 2 )+6a 2(Wx 2+l 2nWy 2 ) +(k'/g)W−V 2W=0,
(4)
W(0, y)=W(x, 0)=W(1, y)=W(x, 1)=0,
(5a)
Wx (0, y)=Wy (x, 0)=Wx (1, y)=Wy (x, 1)=0,
(5b)
910 0022–460X/95/350910+05 $12.00/0
7 1995 Academic Press Limited
911 y b
P v U
P1 0
P
a
u
x
h (0)
Figure 1. The vibrating mechanical system under study.
where l=a/b,
g(x)=ax+1,
V 2=rh(0)a 4v 2/D(0),
k'=ka 4/D(0), 0,
(see equation (3)). With the notation and approach of Bert and coworkers [2, 5], application of the differential quadrature method yields the following linear system of equations: N−1
N−1
s A2k 1 Wk 1 j=0,
s A2k 2 Wik 2=0,
j=3,..., N−1;
k 1=2
i=2,..., N−2;
k 2=2
N−1
s A(N−1)k 1 Wk 1 j=0,
j=2,..., N−2;
k 1=2
N−1
s A(N−1)k 2 Wik 2=0,
i=3,..., N−1;
k 2=2
$
N−1
N−1 N−1
N−1
k 1=2
k 1=2 k 2=2
k 2=2
gi2 s Dik 1 Wk 1 j+l 2 s s Bik 1 Bjk 2 Wk 1 k 2+l 4 s Djk 2 Wik 2
$ $
N−1
N−1 N−1
k 1=2
k 1=2 k 2=2
+6agi s Cik 1 Wk 1 j+l 2 s s Aik 1 Bjk 2 Wk 1 Wk 1 k 2 N−1
N−1
k 1=2
k 2=2
%
+6a 2 s Bik 1 Wk 1 j+l 2n s Bjk 2 Wik 2 +
%
%
k' W −V 2Wij=0, gi ij
i, j=3,..., N−2. (6)
12
3
4...
N–2 N–1 N
0 δ = 10–4
–4
δ = 10
Figure 2. The partition of the interval [0, 1] in the x and y directions when implementing the DQ approach.
912
z y
x
Figure 3. A typical modal shape when the system vibrates in its fundamental mode (l=1·50; u=v=0·6).
Here gi=g(xi ). The non-triviality condition yields a secular determinant the lowest root of which is the fundamental frequency coefficient of the system. The partition of the interval [0, 1] is denoted in Figure 2. For the cases u=v=0·2 and 0·6, the parameter N was taken equal to 9, while when considering the situations u=v=0·4 and 0·8, N was chosen equal to 11. 3. Since the structural system and its fundamental mode is symmetric with respect to y¯=b/2 (see Figure 1), one half of the plate was modelled by 450 rectangular elements. The results were obtained by using a standard finite element code. A typical fundamental modal shape obtained is depicted in Figure 3. 4. All calculations have been performed with n (the Poisson ratio)=0·30. In Table 1 are depicted fundamental frequency coefficients determined by means of the DQ method in the case of a plate of uniform thickness (a=0).
T 1 Fundamental frequency coefficient, V1 , of a clamped rectangular plate of uniform thickness partially embedded in a Winkler type foundation (the fundamental eigenvalue has been determined by using the DQ method) a=0, u=v 0·20 0·40 0·60 0·80
l=1 ZXXXXCXXXXV k'=320 800 1600 36·937 38·838 39·550 40·203
38·260 42·713 44·287 45·785
40·336 48·425 51·129 53·817
l=1·5 l=2 ZXXXXCXXXXV ZXXXXXCXXXXXV 320 800 1600 320 800 1600 61·359 62·508 63·239 63·392
62·161 64·984 66·736 67·071
63·462 68·888 72·183 72·791
98·725 99·423 99·733 100·009
99·215 100·983 101·698 102·380
100·020 103·519 105·763 106·216
913
T 2 Fundamental frequency coefficient of a clamped rectangular plate of linearly varying thickness (a=0·20); (1) results obtained by using the DQ method; (2) finite element results a=0·2 u=v
l=1 ZXXXXCXXXXV k'=320 800 1600
l=1·5 l=2 ZXXXXCXXXXV ZXXXXXCXXXXXV 320 800 1600 320 800 1600
0·2 (1) (2)
40·309 40·26
41·414 41·39
43·170 43·19
67·188 67·06
67·848 67·74
68·927 68·85
108·122 107·98
108·520 108·40
109·173 109·08
0·4 (1) (2)
41·898 41·76
45·200 44·95
50·182 49·74
68·133 67·96
70·203 69·96
73·506 73·13
108·681 108·50
109·969 109·80
112·072 111·80
0·6 (1) (2)
42·508 42·74
46·572 47·21
52·595 53·82
68·531 68·57
71·122 71·44
75·218 75·97
108·963 108·90
110·596 110·75
113·256 113·72
0·8 (1) (2)
43·063 43·00
47·876 47·80
54·968 54·87
68·868 68·74
71·980 71·82
76·886 76·74
109·160 109·00
111·155 111·00
114·402 114·24
Tables 2 and 3 are for plates of non-uniform thickness (a=0·2 and 0·4, respectively). The eigenvalues obtained by means of the DQ technique are compared with those determined by means of the finite element algorithmic procedure. One concludes immediately that an excellent agreement is obtained for all the situations considered. Analysis of the tables reveals that, in general, the effect of doubling u is larger than that of doubling k'. It is also seen that the effect of the taper parameter, a, is relatively small for the range of values of a considered. The present investigation also shows that the DQ method can be easily implemented when the embedding medium properties vary in a discontinuous fashion.
The present study has been sponsored by CONICET Research and Development Program (PID-BID 003-92). The authors are indebted to the reviewer of this paper and to Professor P. E. Doak for their valuable criticism.
T 3 Fundamental frequency coefficient of a clamped rectangular plate of linearly varying thickness (a=0·40); (1) results obtained by using the DQ method; (2) finite element results a=0·4 u=v
l=1 ZXXXXCXXXXV k'=320 800 1600
l=1·5 l=2 ZXXXXCXXXXV ZXXXXXCXXXXXV 320 800 1600 320 800 1600
0·2 (1) (2)
43·592 43·53
44·523 44·44
46·016 46·03
72·772 72·62
73·317 73·19
74·213 74·12
116·894 116·71
117·210 117·04
117·731 117·60
0·4 (1) (2)
44·938 44·81
47·780 47·55
52·141 51·75
73·557 73·40
75·307 75·07
78·123 77·78
117·331 117·10
118·400 118·20
120·152 119·90
0·6 (1) (2)
45·476 45·66
48·996 49·55
54·318 55·42
73·920 73·92
76·128 76·38
79·655 80·31
117·611 117·50
118·988 119·09
121·238 121·64
0·8 (1) (2)
45·962 45·90
50·167 50·09
56·481 56·38
74·210 74·08
76·898 76·76
81·177 81·02
117·775 117·60
119·501 119·30
122·322 122·10
914
1. R. B and J. C 1971 Journal of Mathematical Analysis and Applications 34, 235–238. Differential quadrature and long-term integration. 2. C. W. B, S. K. J and A. G. S 1989 Computational Mechanics 5, 217–226. Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature. 3. S. K. J, C. W. B and A. G. S 1989 International Journal of Numerical Methods in Engineering 28, 561–577. Application of differential quadrature to static analysis of structural components. 4. A. G. S, S. K. J and C. W. B 1988 Thin-walled Structures 6, 51–62. Nonlinear bending analysis of thin circular plates by differential quadrature. 5. A. R. K, J. F and C. W. B 1992 Journal of Engineering Mechanics 118, 1221–1238. Fundamental frequency of tapered plates by differential quadrature method.