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FUNDAMENTAL FREQUENCY OF TRANSVERSE VIBRATION OF ORTHOTROPIC PLATES OF REGULAR POLYGONAL SHAPE CARRYING A CONCENTRATED MASS
Journal of Sound and Vibration (1999) 221(1), 175–179 Article No. jsvi.1998.1841, available online at http://www.idealibrary.com.on
FUNDAMENTAL FREQUENCY OF TRANSVERSE VIBRATION OF ORTHOTROPIC PLATES OF REGULAR POLYGONAL SHAPE CARRYING A CONCENTRATED MASS P. A. A. L R. H. G Institute of Applied Mechanics (CONICET) and Department of Engineering, Universidad Nacional del Sur 8000, Bahı´ a Blanca, Argentina (Received 17 September 1998)
1. The present study deals with the solution of the title problem using a conformal mapping approach coupled with the optimized Rayleigh–Ritz method [1, 2]. By conformally transforming the given shape in the z-plane onto a unit circle in the z-plane it is possible to construct co-ordinate functions which satisfy the essential boundary conditions in the case of simply supported and clamped plates. For the sake of simplicity, the azimuthal variation in the z-plane is disregarded and the following co-ordinate functions are used. (1) Simply supported plates: W 2 Wa = C1 (1 − rp) + Ca (1 − r p + 1) + C2 (1 − rp + 2),
z = r eiU.
(1)
(2) Clamped plates: W 2 Wa = C1 (1 − r p)2 + C2 (1 − r p + 1)2 + C3 (1 − rp + 2)2
(2)
where p is Rayleigh’s optimization parameter [2]. It should be pointed out that orthotropic plates are commonly used in engineering practice, e.g., printed circuit boards used in electronics applications. 2. Following Lekhnitskii’s standard notation [3] one expresses the governing functional in the form J(W) =
gg
2 (D1 Wx2 2 + 2D1 n2 Wx 2 Wy 2 + D2 Wy22 + 4Dk Wxy ) dx dy − rhv2
p
gg
W2 dx dy − Mv 2W2(0, 0),
(3)
p
where it has been assumed that the concentrated mass M is rigidly attached at the center of the plate. 0022–460X/99/110175 + 05 $30.00/0
7 1999 Academic Press
176
In the case of regular polygons of degree ‘‘s’’ the mapping function is given by [1] z = As ap F(z) = AS ap
g
z
0
dz , (1 + zs)2/s
(4)
where ap is the apothem of the polygon. Defining now 1 e−2Ui , 4 F'2(z)
U1 + V1 i =
U2 + V2 i =
1 F0(z) −Ui e , 2 F'3(z)
(5)
and substituting equations (4) and (5) into equation (3) one obtains the transformed energy functional in the form
As2 ap2 J(Wa ) = D1
gg 8& 00
Wa −
2
1
c
− 8n2
+
&
D2 −2 D1
+ 16
−
$0
War 2 −
Dk D1
1
War U1 − War U2 r
War 2 −
$0
As2 p 16tg s 4
00
1
War 1 U1 − War U2 + 2 r
War 2 −
+
1
n2 2
0
War 2 +
1
War 1 U1 − War U2 + 2 r
1
War V1 − War V2 r
$ gg
V 2 As2
%
2
War r =F'(z)=2
War 2 +
%9
1
War r
=F'(z)=4
2
'
War r =F'(z)=2
War 2 +
2
2
=F'(z)=2r dr du
Wr2 =F'(z)=2r dr du + mstg
c
'
2
%
p 2 W , s (0)
(6)
where the fact that Wa , defined in equations (1) and (2), does not contain the azimuthal variable u has been taken into account, and m = M/Mp , Mp = plate mass, tgp/s = tan p/s, and V12 = (rha4/D1 )v12 . 3. Tables 1 and 2 depict values of the fundamental frequency coefficient V1 in the case of isotropic, simply supported and clamped plates, respectively. Reasonably
177
y
a
M
x
ap
M
Figure 1. Orthotropic plate of regular polygonal shape carrying a central, concentrated mass, M.
good agreement with values available in the literature, for the case where m = 0, is obtained. Pentagonal, exagonal and heptagonal plates are considered in the present investigation.
T 1 Frequency coefficients of simply supported isotropic plates of regular polygonal shape (n = 0·30) s
m=0
0·10
0·20
0·30
0·40
m = 0 [4]
5 6 7
11·00 6·96 4·97
9·28 5·90 4·22
8·14 5·19 3·72
7·33 4·68 3·35
6·72 4·30 3·08
11·01 7·15 5·06
T 2 Frequency coefficients of clamped isotropic plates of regular polygonal shape s
m=0
0·10
0·20
0·30
0·40
m = 0 [5]
5 6 7
19·24 12·56 8·91
15·11 9·91 7·05
12·76 8·39 5·97
11·23 7·39 5·27
10·13 6·68 4·76
19·71 12·81 9·08
178
T 3 Frequency coefficients of simply supported orthotropic plates of regular polygonal shape (n2 = 0·30) Dk /D1
s
m=0
0·10
0·20
0·30
0·40
m = 0 [6]
4
0·85
5 6 7
17·08 10·73 7·68
14·41 9·10 6·51
12·64 8·01 5·73
11·38 7·23 5·17
10·42 6·63 4·75
17·03 11·70 –
1
0·85
5 6 7
12·49 7·78 5·50
10·54 6·60 4·67
9·24 5·81 4·12
8·32 5·24 3·71
7·62 4·81 3·41
12·54 8·48 –
1
0·10
5 6 7
10·17 6·50 4·68
8·58 5·51 3·97
7·53 4·85 3·49
6·78 4·37 3·15
6·21 4·01 2·89
10·45 7·16 –
0·25
0·10
5 6 7
8·33 5·35 3·84
7·04 4·54 3·26
6·17 3·99 2·87
5·56 3·60 2·59
5·10 3·30 2·37
8·65 5·94 –
D2 /D1
Tables 3 and 4 present values of V1 for simply supported and clamped orthotropic plates. The following orthotropic parameters have been considered: D2 /D1 = 4, Dk /D1 = 0·85, n2 = 0·30; D2 /D1 = 1, Dk /D1 = 0·85, n2 = 0·30; D2 /D1 =1, Dk /D1 = 0·10, n2 = 0·30; D2 /D1 = 0·25, Dk /D1 = 0·10, n2 = 0·30. In the case of bare plates (m = 0) the eigenvalues have been compared with those determined in reference [6]. T 4 Frequency coefficients of clamped orthotropic plates of regular polygonal shape (n2 = 0·30) Dk /D1
s
m=0
0·10
0·20
0·30
0·40
m = 0 [6]
4
0·85
5 6 7
29·77 19·42 13·79
23·37 15·31 10·90
19·73 12·96 9·23
17·36 11·41 8·14
15·67 10·31 7·36
30·46 20·18 14·82
1
0·85
5 6 7
21·69 14·13 9·99
17·02 11·14 7·90
14·36 9·43 6·69
12·63 8·30 5·90
11·40 7·50 5·33
22·09 14·52 10·62
1
0·10
5 6 7
17·90 11·70 8·32
14·06 9·23 6·58
11·88 7·82 5·58
10·45 6·89 4·92
9·44 6·22 4·45
18·66 12·25 9·05
0·25
0·10
5 6 7
14·71 9·62 6·83
11·56 7·59 5·41
9·77 6·43 4·58
8·59 5·67 4·04
7·76 5·12 3·65
15·38 10·18 7·41
D2 /D1
179
Present results are, in general, somewhat lower. Possibly the values determined in reference [6] are rather high upper bounds since a single approximating function was used. The present study has been sponsored by CONICET Research and Development Program, Secretarı´ a General de Ciencia y Tecnologia of Universidad Nacional del Sur and Comisio´n de Investigaciones Cientificas (Buenos Aires Province). 1. R. S and P. A. A. L 1991 Conformal Mapping: Methods and Applications. Elsevier: Amsterdam. 2. P. A. A. L 1995 Ocean Engineering 22, 235–250. Optimization of variational methods. 3. S. G. L 1968 Anisotropic Plates. New York: Gordon and Breach Science Publishers. 4. P. A. S, R. P and P. A. A. L 1967 Journal of the Acoustical Society of America 42, 806–809. Application of complex variable theory to the determination of the fundamental frequency of vibrating plates. 5. P. A. A. L and R. H. G 1976 Journal of Sound and Vibration 48, 327–332. Fundamental frequency of vibration of clamped plates of arbitrary shape subjected to a hydrostatic state of in-plane stress. 6. P. A. A. L, L. E. L and G. S´ S 1980 Journal of Sound and Vibration 70, 77–84. A method for the determination of the fundamental frequency of orthotropic plates of polygonal boundary shape.
FUNDAMENTAL FREQUENCY OF TRANSVERSE VIBRATION OF ORTHOTROPIC PLATES OF REGULAR POLYGONAL SHAPE CARRYING A CONCENTRATED MASS P. A. A. L R. H. G Institute of Applied Mechanics (CONICET) and Department of Engineering, Universidad Nacional del Sur 8000, Bahı´ a Blanca, Argentina (Received 17 September 1998)
1. The present study deals with the solution of the title problem using a conformal mapping approach coupled with the optimized Rayleigh–Ritz method [1, 2]. By conformally transforming the given shape in the z-plane onto a unit circle in the z-plane it is possible to construct co-ordinate functions which satisfy the essential boundary conditions in the case of simply supported and clamped plates. For the sake of simplicity, the azimuthal variation in the z-plane is disregarded and the following co-ordinate functions are used. (1) Simply supported plates: W 2 Wa = C1 (1 − rp) + Ca (1 − r p + 1) + C2 (1 − rp + 2),
z = r eiU.
(1)
(2) Clamped plates: W 2 Wa = C1 (1 − r p)2 + C2 (1 − r p + 1)2 + C3 (1 − rp + 2)2
(2)
where p is Rayleigh’s optimization parameter [2]. It should be pointed out that orthotropic plates are commonly used in engineering practice, e.g., printed circuit boards used in electronics applications. 2. Following Lekhnitskii’s standard notation [3] one expresses the governing functional in the form J(W) =
gg
2 (D1 Wx2 2 + 2D1 n2 Wx 2 Wy 2 + D2 Wy22 + 4Dk Wxy ) dx dy − rhv2
p
gg
W2 dx dy − Mv 2W2(0, 0),
(3)
p
where it has been assumed that the concentrated mass M is rigidly attached at the center of the plate. 0022–460X/99/110175 + 05 $30.00/0
7 1999 Academic Press
176
In the case of regular polygons of degree ‘‘s’’ the mapping function is given by [1] z = As ap F(z) = AS ap
g
z
0
dz , (1 + zs)2/s
(4)
where ap is the apothem of the polygon. Defining now 1 e−2Ui , 4 F'2(z)
U1 + V1 i =
U2 + V2 i =
1 F0(z) −Ui e , 2 F'3(z)
(5)
and substituting equations (4) and (5) into equation (3) one obtains the transformed energy functional in the form
As2 ap2 J(Wa ) = D1
gg 8& 00
Wa −
2
1
c
− 8n2
+
&
D2 −2 D1
+ 16
−
$0
War 2 −
Dk D1
1
War U1 − War U2 r
War 2 −
$0
As2 p 16tg s 4
00
1
War 1 U1 − War U2 + 2 r
War 2 −
+
1
n2 2
0
War 2 +
1
War 1 U1 − War U2 + 2 r
1
War V1 − War V2 r
$ gg
V 2 As2
%
2
War r =F'(z)=2
War 2 +
%9
1
War r
=F'(z)=4
2
'
War r =F'(z)=2
War 2 +
2
2
=F'(z)=2r dr du
Wr2 =F'(z)=2r dr du + mstg
c
'
2
%
p 2 W , s (0)
(6)
where the fact that Wa , defined in equations (1) and (2), does not contain the azimuthal variable u has been taken into account, and m = M/Mp , Mp = plate mass, tgp/s = tan p/s, and V12 = (rha4/D1 )v12 . 3. Tables 1 and 2 depict values of the fundamental frequency coefficient V1 in the case of isotropic, simply supported and clamped plates, respectively. Reasonably
177
y
a
M
x
ap
M
Figure 1. Orthotropic plate of regular polygonal shape carrying a central, concentrated mass, M.
good agreement with values available in the literature, for the case where m = 0, is obtained. Pentagonal, exagonal and heptagonal plates are considered in the present investigation.
T 1 Frequency coefficients of simply supported isotropic plates of regular polygonal shape (n = 0·30) s
m=0
0·10
0·20
0·30
0·40
m = 0 [4]
5 6 7
11·00 6·96 4·97
9·28 5·90 4·22
8·14 5·19 3·72
7·33 4·68 3·35
6·72 4·30 3·08
11·01 7·15 5·06
T 2 Frequency coefficients of clamped isotropic plates of regular polygonal shape s
m=0
0·10
0·20
0·30
0·40
m = 0 [5]
5 6 7
19·24 12·56 8·91
15·11 9·91 7·05
12·76 8·39 5·97
11·23 7·39 5·27
10·13 6·68 4·76
19·71 12·81 9·08
178
T 3 Frequency coefficients of simply supported orthotropic plates of regular polygonal shape (n2 = 0·30) Dk /D1
s
m=0
0·10
0·20
0·30
0·40
m = 0 [6]
4
0·85
5 6 7
17·08 10·73 7·68
14·41 9·10 6·51
12·64 8·01 5·73
11·38 7·23 5·17
10·42 6·63 4·75
17·03 11·70 –
1
0·85
5 6 7
12·49 7·78 5·50
10·54 6·60 4·67
9·24 5·81 4·12
8·32 5·24 3·71
7·62 4·81 3·41
12·54 8·48 –
1
0·10
5 6 7
10·17 6·50 4·68
8·58 5·51 3·97
7·53 4·85 3·49
6·78 4·37 3·15
6·21 4·01 2·89
10·45 7·16 –
0·25
0·10
5 6 7
8·33 5·35 3·84
7·04 4·54 3·26
6·17 3·99 2·87
5·56 3·60 2·59
5·10 3·30 2·37
8·65 5·94 –
D2 /D1
Tables 3 and 4 present values of V1 for simply supported and clamped orthotropic plates. The following orthotropic parameters have been considered: D2 /D1 = 4, Dk /D1 = 0·85, n2 = 0·30; D2 /D1 = 1, Dk /D1 = 0·85, n2 = 0·30; D2 /D1 =1, Dk /D1 = 0·10, n2 = 0·30; D2 /D1 = 0·25, Dk /D1 = 0·10, n2 = 0·30. In the case of bare plates (m = 0) the eigenvalues have been compared with those determined in reference [6]. T 4 Frequency coefficients of clamped orthotropic plates of regular polygonal shape (n2 = 0·30) Dk /D1
s
m=0
0·10
0·20
0·30
0·40
m = 0 [6]
4
0·85
5 6 7
29·77 19·42 13·79
23·37 15·31 10·90
19·73 12·96 9·23
17·36 11·41 8·14
15·67 10·31 7·36
30·46 20·18 14·82
1
0·85
5 6 7
21·69 14·13 9·99
17·02 11·14 7·90
14·36 9·43 6·69
12·63 8·30 5·90
11·40 7·50 5·33
22·09 14·52 10·62
1
0·10
5 6 7
17·90 11·70 8·32
14·06 9·23 6·58
11·88 7·82 5·58
10·45 6·89 4·92
9·44 6·22 4·45
18·66 12·25 9·05
0·25
0·10
5 6 7
14·71 9·62 6·83
11·56 7·59 5·41
9·77 6·43 4·58
8·59 5·67 4·04
7·76 5·12 3·65
15·38 10·18 7·41
D2 /D1
179
Present results are, in general, somewhat lower. Possibly the values determined in reference [6] are rather high upper bounds since a single approximating function was used. The present study has been sponsored by CONICET Research and Development Program, Secretarı´ a General de Ciencia y Tecnologia of Universidad Nacional del Sur and Comisio´n de Investigaciones Cientificas (Buenos Aires Province). 1. R. S and P. A. A. L 1991 Conformal Mapping: Methods and Applications. Elsevier: Amsterdam. 2. P. A. A. L 1995 Ocean Engineering 22, 235–250. Optimization of variational methods. 3. S. G. L 1968 Anisotropic Plates. New York: Gordon and Breach Science Publishers. 4. P. A. S, R. P and P. A. A. L 1967 Journal of the Acoustical Society of America 42, 806–809. Application of complex variable theory to the determination of the fundamental frequency of vibrating plates. 5. P. A. A. L and R. H. G 1976 Journal of Sound and Vibration 48, 327–332. Fundamental frequency of vibration of clamped plates of arbitrary shape subjected to a hydrostatic state of in-plane stress. 6. P. A. A. L, L. E. L and G. S´ S 1980 Journal of Sound and Vibration 70, 77–84. A method for the determination of the fundamental frequency of orthotropic plates of polygonal boundary shape.