- Email: [email protected]
TRANSVERSE VIBRATIONS OF ISOTROPIC, THIN CIRCULAR PLATES WITH A RECTANGULARLY ORTHOTROPIC CIRCULAR CORE
Journal of Sound and
TRANSVERSE VIBRATIONS OF ISOTROPIC, THIN CIRCULAR PLATES WITH A RECTANGULARLY ORTHOTROPIC CIRCULAR CORE R. H. GUTIERREZ, P. A. A. LAURA AND R. E. ROSSI Department of Engineering, CONICE¹, ;niversidad Nacional del Sur, BahnHa Blanca 8000, Argentina (Received 24 September 2001)
1. INTRODUCTION
The present note is an outgrowth of previous publications [1}4] and deals with the determination of the fundamental frequency of transverse vibration of the system as shown in Figure 1. For the sake of generality, it is assumed that a point mass M is rigidly attached to the plate center. The optimized Rayleigh}Ritz method is employed to determine the fundamental eigenvalue and an independent solution is obtained using the "nite element procedure [5]. Good engineering agreement is observed between both sets of results.
2. APPROXIMATE SOLUTION
In the case of normal modes of vibration the dynamic behavior of the system is governed by the functional:
. [(=V #=W )!2(1!)(=V =W !=VW)] dxN dyN
J(=)"D
. (D=V #2D=V =W #D =W #4DI =VW ) dxN dy
#
!h
. = dxN dyN !M=(0, 0)
(1)
and appropriate boundary conditions. Substituting: D "D /D, D "D /D, D "D /D and xN "ax, yN "ay into equation (1) one I I obtains
. [(=V #=W )!2(1!)(=V =W !=VW)] dx dy
a J(=)" D
. (D=V #2D=V =W #D =W #4DI =VW ) dx dy
#
0022-460X/02/$35.00
2002 Elsevier Science Ltd. All rights reserved.
196
LETTERS TO THE EDITOR
Figure 1. Vibrating mechanical system under study (note: PM "isotropic portion, PM "orthotropic core; PM "PM PM ).
!
= dx dy#=(0, 0) ,
.
(2)
where "(ha/D), "M/M ; M is the plate mass. N N In order to apply the Rayleigh}Ritz method one makes , =(x, y):= " C (x, y) ? H H H and applying Ritz minimization condition one obtains a J , " 2D C G H
.
( # )( # ) HV HW GV GW
!(1!)( # !2 ) HW GV HV GW HVW GVW
(3)
197
LETTERS TO THE EDITOR
TABLE 1 Fundamental frequency coe.cients "(h/D aR
SS
(A)
(B)
(C)
(D)
(E)
r "0 @
0)2
0)4
0)6
0)8
0
4)93(1) (4)94)(2) 3)76 (3)76) 3)15 (3)13)
4)52 (4)83) 3)42 (3)62) 2)86 (3)00)
4)27 (4)57) 3)24 (3)42) 2)70 (2)83)
4)13 (4)31) 3)14 (3)24) 2)63 (2)69)
4)03 (4)11) 3)08 (3)11) 2)58 (2)58)
9)58 (9)93) 6)51 (6)46) 5)23 (5)10)
9)39 (9)60) 6)38 (6)28) 5)13 (4)95)
9)20 (9)37) 6)21 (6)12) 4)98 (4)82)
8)79 (9)02) 5)92 (5)89) 4)75 (4)65)
0)2 0)4 0
C
0)2 0)4
10)20 (10)22) 7)00 (6)87) 5)64 (5)47)
Note: One quarter of the plate domain was subdivided into: (A) 5069 elements, (B) 5066, (C) 5076, (D) 5081, (E) 5094 respectively. RComparison of analytical and "nite element values: (1) Optimized Rayleigh}Ritz method. (2) Finite element predictions.
. [D HV GV #D (HW GV #HV GW )
#
#D #4D ] dx dy I HVW GVW HW GW !
.
dx dy# (0) (0) H G H G
C "0. H
(4)
In the present study, N"2 and " rN# r#1,
" rN># r#r,
r"(x#y,
(5)
where the 's and 's are obtained substituting each co-ordinate function in the boundary conditions for the cases of simply supported and clamped edges respectively. Once the fundamental eigenvalue is determined one minimizes it with respect to &&p'' (see equations (5)).
3. NUMERICAL RESULTS
Calculations of fundamental frequency coe$cients were performed for the following constitutive characteristics and geometric parameters: ""0)30, D /D"4/5, D /D "1/2, D /D "1/3; r "b/a"0, 0)2, 0)4, 0)6, 0)8; "0, 0)2, 0)4. I @ Table 1 depicts values of "((h/D) a obtained by means of the optimized Rayleigh}Ritz method and the "nite element technique for simply supported and clamped plates. The agreement is good from an engineering viewpoint. Some analytical results are
198
LETTERS TO THE EDITOR
lower than the "nite element values. However, since the latter are presumably very accurate in view of the large number of elements used, this fact may be due to a slight numerical instability when making use of the optimized variational approach. The model presented is also a "rst order approximation for the situation where an isotropic circular plate has been damaged in a central portion and acquires orthotropic characteristics.
ACKNOWLEDGMENTS
The present study has been sponsored by CONICET Research and Development Program and by SecretarmH a General de Ciencia y TecnologmH a of Universidad Nacional del Sur.
REFERENCES 1. P. A. A. LAURA, C. A. ROSSIT, D. R. AVALOS and H. A. LARRONDO 2000 Journal of Sound and
TRANSVERSE VIBRATIONS OF ISOTROPIC, THIN CIRCULAR PLATES WITH A RECTANGULARLY ORTHOTROPIC CIRCULAR CORE R. H. GUTIERREZ, P. A. A. LAURA AND R. E. ROSSI Department of Engineering, CONICE¹, ;niversidad Nacional del Sur, BahnHa Blanca 8000, Argentina (Received 24 September 2001)
1. INTRODUCTION
The present note is an outgrowth of previous publications [1}4] and deals with the determination of the fundamental frequency of transverse vibration of the system as shown in Figure 1. For the sake of generality, it is assumed that a point mass M is rigidly attached to the plate center. The optimized Rayleigh}Ritz method is employed to determine the fundamental eigenvalue and an independent solution is obtained using the "nite element procedure [5]. Good engineering agreement is observed between both sets of results.
2. APPROXIMATE SOLUTION
In the case of normal modes of vibration the dynamic behavior of the system is governed by the functional:
. [(=V #=W )!2(1!)(=V =W !=VW)] dxN dyN
J(=)"D
. (D=V #2D=V =W #D =W #4DI =VW ) dxN dy
#
!h
. = dxN dyN !M=(0, 0)
(1)
and appropriate boundary conditions. Substituting: D "D /D, D "D /D, D "D /D and xN "ax, yN "ay into equation (1) one I I obtains
. [(=V #=W )!2(1!)(=V =W !=VW)] dx dy
a J(=)" D
. (D=V #2D=V =W #D =W #4DI =VW ) dx dy
#
0022-460X/02/$35.00
2002 Elsevier Science Ltd. All rights reserved.
196
LETTERS TO THE EDITOR
Figure 1. Vibrating mechanical system under study (note: PM "isotropic portion, PM "orthotropic core; PM "PM PM ).
!
= dx dy#=(0, 0) ,
.
(2)
where "(ha/D), "M/M ; M is the plate mass. N N In order to apply the Rayleigh}Ritz method one makes , =(x, y):= " C (x, y) ? H H H and applying Ritz minimization condition one obtains a J , " 2D C G H
.
( # )( # ) HV HW GV GW
!(1!)( # !2 ) HW GV HV GW HVW GVW
(3)
197
LETTERS TO THE EDITOR
TABLE 1 Fundamental frequency coe.cients "(h/D aR
SS
(A)
(B)
(C)
(D)
(E)
r "0 @
0)2
0)4
0)6
0)8
0
4)93(1) (4)94)(2) 3)76 (3)76) 3)15 (3)13)
4)52 (4)83) 3)42 (3)62) 2)86 (3)00)
4)27 (4)57) 3)24 (3)42) 2)70 (2)83)
4)13 (4)31) 3)14 (3)24) 2)63 (2)69)
4)03 (4)11) 3)08 (3)11) 2)58 (2)58)
9)58 (9)93) 6)51 (6)46) 5)23 (5)10)
9)39 (9)60) 6)38 (6)28) 5)13 (4)95)
9)20 (9)37) 6)21 (6)12) 4)98 (4)82)
8)79 (9)02) 5)92 (5)89) 4)75 (4)65)
0)2 0)4 0
C
0)2 0)4
10)20 (10)22) 7)00 (6)87) 5)64 (5)47)
Note: One quarter of the plate domain was subdivided into: (A) 5069 elements, (B) 5066, (C) 5076, (D) 5081, (E) 5094 respectively. RComparison of analytical and "nite element values: (1) Optimized Rayleigh}Ritz method. (2) Finite element predictions.
. [D HV GV #D (HW GV #HV GW )
#
#D #4D ] dx dy I HVW GVW HW GW !
.
dx dy# (0) (0) H G H G
C "0. H
(4)
In the present study, N"2 and " rN# r#1,
" rN># r#r,
r"(x#y,
(5)
where the 's and 's are obtained substituting each co-ordinate function in the boundary conditions for the cases of simply supported and clamped edges respectively. Once the fundamental eigenvalue is determined one minimizes it with respect to &&p'' (see equations (5)).
3. NUMERICAL RESULTS
Calculations of fundamental frequency coe$cients were performed for the following constitutive characteristics and geometric parameters: ""0)30, D /D"4/5, D /D "1/2, D /D "1/3; r "b/a"0, 0)2, 0)4, 0)6, 0)8; "0, 0)2, 0)4. I @ Table 1 depicts values of "((h/D) a obtained by means of the optimized Rayleigh}Ritz method and the "nite element technique for simply supported and clamped plates. The agreement is good from an engineering viewpoint. Some analytical results are
198
LETTERS TO THE EDITOR
lower than the "nite element values. However, since the latter are presumably very accurate in view of the large number of elements used, this fact may be due to a slight numerical instability when making use of the optimized variational approach. The model presented is also a "rst order approximation for the situation where an isotropic circular plate has been damaged in a central portion and acquires orthotropic characteristics.
ACKNOWLEDGMENTS
The present study has been sponsored by CONICET Research and Development Program and by SecretarmH a General de Ciencia y TecnologmH a of Universidad Nacional del Sur.
REFERENCES 1. P. A. A. LAURA, C. A. ROSSIT, D. R. AVALOS and H. A. LARRONDO 2000 Journal of Sound and