284
F. Lin et al.
numerical solutions of vibration analysis of structures. Kitipornchai et al. [2] used
DQM to discretize space domain of partial differential motion equations for studying
nonlinear vibration of edge cracked functionally graded Timoshenko beams. Similarly, Ke et al. [3] observed nonlinear free vibration of embedded double–walled
carbon nanotubes based on nonlocal Timoshenko beam theory by the same way.
Fung [4] performed DQM to discretize time domain of differential motion equations
to solve first–order initial value problems. Furthermore, Peng et al. [5] applied the
time–domain DQM to survey forced vibration analysis of beams. For the mechanical
analysis of plate, Wang [6] used DQM to discretize two–dimensional space domain of
partial differential motion equations for studying linear static analysis, elastoplastic
buckling analysis, and linear buckling analysis of thin plates, ARANI and JAFARI [7]
for Nonlinear vibration analysis of laminated composite Mindlin micro/nano–plates
resting on orthotropic Pasternak medium, and Safarpour et al. [8] for static and free
vibration analysis of graphene platelets. For the statically forced and free vibration
of plates, however, there are DQM to only discretize two–dimensional space domain
of partial differential motion equations and for the dynamically forced vibration no
literature available on basis of the time–domain DQM.
In this paper, a study of the forced vibration of simply supported plates subjected
to a transverse uniform dynamic load is presented on the basis of the time–domain
DQM. Applying DQM both in space domain and time domain, the partial differential
governing equations and initial–boundary conditions are discretized into linear algebraic equations for studying the forced vibration of simply supported plates subjected
to a transverse uniform load.
23.2 Solution Technique
Equation (23.1) defines the governing equation [9] of simply supported plates
subjected to a transverse uniform load q:
∇
2
∇
2
w +
¯
m
D
∂
2
w
∂t 2 =
q
D
(23.1)
where ∇
2
∇
2 is double Laplacian Operator, w transverse deflection, ¯
m mass per unit
volume, D bending stiffness, t time, and some of them are defined by Eqs. (23.2a–d):
∇
2
∇
2
=
∂
4
∂ x 4 + 2
∂
4
∂ x 2 ∂ y 2 +
∂
4
∂ y 4 , w = w(x, y, t), D =
Eδ
3
12(1 − μ 2 )
,
q = q(x, y, t) (23.2a–d)
in which E is elastic modulus, δ thickness, and μ Poisson’s ratio.
For simply supported plates, the boundary conditions are described by
Eqs. (23.3a–d):
F. Lin et al.
numerical solutions of vibration analysis of structures. Kitipornchai et al. [2] used
DQM to discretize space domain of partial differential motion equations for studying
nonlinear vibration of edge cracked functionally graded Timoshenko beams. Similarly, Ke et al. [3] observed nonlinear free vibration of embedded double–walled
carbon nanotubes based on nonlocal Timoshenko beam theory by the same way.
Fung [4] performed DQM to discretize time domain of differential motion equations
to solve first–order initial value problems. Furthermore, Peng et al. [5] applied the
time–domain DQM to survey forced vibration analysis of beams. For the mechanical
analysis of plate, Wang [6] used DQM to discretize two–dimensional space domain of
partial differential motion equations for studying linear static analysis, elastoplastic
buckling analysis, and linear buckling analysis of thin plates, ARANI and JAFARI [7]
for Nonlinear vibration analysis of laminated composite Mindlin micro/nano–plates
resting on orthotropic Pasternak medium, and Safarpour et al. [8] for static and free
vibration analysis of graphene platelets. For the statically forced and free vibration
of plates, however, there are DQM to only discretize two–dimensional space domain
of partial differential motion equations and for the dynamically forced vibration no
literature available on basis of the time–domain DQM.
In this paper, a study of the forced vibration of simply supported plates subjected
to a transverse uniform dynamic load is presented on the basis of the time–domain
DQM. Applying DQM both in space domain and time domain, the partial differential
governing equations and initial–boundary conditions are discretized into linear algebraic equations for studying the forced vibration of simply supported plates subjected
to a transverse uniform load.
23.2 Solution Technique
Equation (23.1) defines the governing equation [9] of simply supported plates
subjected to a transverse uniform load q:
∇
2
∇
2
w +
¯
m
D
∂
2
w
∂t 2 =
q
D
(23.1)
where ∇
2
∇
2 is double Laplacian Operator, w transverse deflection, ¯
m mass per unit
volume, D bending stiffness, t time, and some of them are defined by Eqs. (23.2a–d):
∇
2
∇
2
=
∂
4
∂ x 4 + 2
∂
4
∂ x 2 ∂ y 2 +
∂
4
∂ y 4 , w = w(x, y, t), D =
Eδ
3
12(1 − μ 2 )
,
q = q(x, y, t) (23.2a–d)
in which E is elastic modulus, δ thickness, and μ Poisson’s ratio.
For simply supported plates, the boundary conditions are described by
Eqs. (23.3a–d):