INTRODUCTION
Vibration is the most important modes of failure in plates with sprung mass and it plays a crucial role in engineering. Avalos et al. (1993) dealt with the solution of vibration by a simply mounted concentred mass using the wellknown normal mode. Xiang et al. (1997) used the Ritz method combined with a variation to solve vibration of rectangular mindlin plates resting on elastic edge supports. Laura and Grossi, (1981) calculated the fundamental frequency coefficient for a rectangular plate with edges elastically restrained against both translation and rotation using polynomial coordinate functions and the RayleighRitz=s method. Wu and Luo (1997) solved the problem of the natural frequencies and the corresponding mode shapes of a uniform rectangular flat plate carrying any number of point masses and translational springs using the analyticalandnumericalcombined method. Grossi and Nallim (1997) analyzed the problem the strain energy stored in rotational springs at the plate edges of nonuniform thickness using the RayleighRitz method with a polynomial expression as approximating function. Nicholson and Bergman (1986) used the Green=s function express the natural modes for the damped plateoscillator systems. Gorman (1997) solved the free vibration problem of shear deformable plates resting on uniform elastic foundations using the modified SuperposionGalerkin method.
This work focuses on the application of differential quadrature method to the vibration of plates with various boundary conditions and sprung masses. In the following section an overview of differential quadrature method to preset the computation of its weighting coefficients offered and discussed the selection problem. The integrity and computational efficiency of the method will be demonstrated through a series of case studies. Very few study in the literature have presented the vibration analysis of rectangular plates with various boundary conditions and sprung masses using the differential quadrature method.
THE DIFFERENTIAL QUADRATURE METHOD
With the increasing use of new fast and affordable computers, along with the
availability of various numerical methods, the solutions of several complicated
engineering problems have now become efficiently achievable. The finite differences
method, the finite element method and the boundary element method have been
used extensively for solving linear and nonlinear differential equations and
consequently there are several commercially developed software packages. The
development of new techniques from the standpoint of computational efficiency
and numerical accuracy is of primal interest. The differential quadrature method
is originally proposed by Bellman et al. (1972). Since it has been developed,
several researchers have applied the differential quadrature method to solve
a variety of problems in different fields of science and engineering. The differential
quadrature method has been shown to be a powerful contender in solving initial
and boundary value problems and become an alternative to the existing methods
such as the finite element method or the finite difference method. One of the
fields among which can find extensive applications of differential quadrature
method is structural mechanics. Civan (1994) solved multivariable mathematical
models using differential quadrature method and differential cubature method.
Han and Liew (1999) analyzed the axisymmetric free vibration of moderately thick
annular plates using the differential quadrature method. Chen and Zhong (1997)
pointed out that differential quadrature method and differential cubature method,
due to their global domain property, are more efficient for nonlinear problems
than the traditional numerical techniques such as finite element method and
finite difference method. The differential quadrature method is used analyze
the mechanical behavior of anisotropic plates and beams (Bert et al.,
1993).
The partial differential equation can be reduced to a set of algebraic equations using the differential quadrature method. Possible oscillations of numerical results arising from higher order polynomials can be avoided by using numerical interpolation methods. The differential quadrature method uses the basis of the Gauss method in deriving the derivative of a function. It follows that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of function values at some intermediate points in that variety. A differential quadrature approximation at the ith discrete point on a grid in the direction of xaxis is given by
A differential quadrature approximation at the ith discrete point on a grid in the direction of yaxis may be written as
where A_{Ij}^{(m)} and B_{ij}^{(m)} are the weighting coefficients.The test function can be written as
Substituting Eq. (3) to Eq. (1) and (2),
Eq. (1) and (2) are computed by
and
The higherorder derivates may be obtained using following equations
where
are the 1th, 2th, …, mth order weighting coefficient matrix in the direction
of xaxis, respectively.
where
are the 1th, 2th, …, mth order weighting coefficient matrix in the direction
of yaxis respectively. The above relation gives the higher order weighting
coefficient matrix based on the firstorder derivative weighting coefficients.
The above relations are not restricted to the choice of sampling points. It
is emphasized that the number of the test functions must be greater than the
highest order of derivative in the governing equations.
The selection of locations of the sampling points is important in ensuring the accuracy of the solution of differential equations. Using equally spaced points can be considered to be a convenient and an easy selection method. A more accurate solution could be obtained by choosing a set of unequally spaced sampling points for a domain separate into by N_{x} and N_{y} points. A simple and good choice can be the roots of shifted Chebyshev and Legendre points. The inner points are
in the direction of xaxis
in the direction of yaxis and boundary points are
in the direction of xaxis.
in the direction of yaxis. δ_{x} and δ_{y} are small distance, a is the length of the plate in the direction of xaxis and b is the length of the plate in the direction of yaxis.
TRANSVERSE VIBRATION OF A RECTANGULAR PLATE AND MASSSPRING SYSTEM
Figure 1 depicts the geometry of a plate with massspring system. The strain energy of the plate and massspring system is
The kinetic energy of the plate and massspring system is
where w is the deflection of the plate, M is the concentrated mass, k is the
spring constant, z is the sprung mass location, x_{0} is the location
of sprung mass in the direction of xaxis, y_{0} is the location of
sprung mass in the direction of yaxis, t is the time, D = Eh^{3}/(12(1v^{2}))
is the flexural rigidity, E is Young=s modulus, ρ is the density of the
plate material and h is the plate thickness. Substituting Eqs. (24) and (25)
into Hamilton=s equation, this leads to the equations of motion of the plate
with sprung mass as:

Fig. 1: 
A plate partially rotational spring supported 
At a simply supported or a clamped boundary, the transverse deflection of the plate is zero:
at a simply supported boundary, the condition of zero normal moment can be reduced to
in the direction of xaxis and
in the direction of yaxis. The condition of zero normal moment at a free boundary in the direction of xaxis is given by
where v is Poisson=s ratio. The condition of zero normal moment at a free boundary in the direction of yaxis is given by
The condition of zero effective shear force at a free boundary is given by
in the direction of xaxis and
in the direction of yaxis. The following form gives the condition of spring supported:
in the direction of xaxis and
in the direction of yaxis. k_{φ} is the torsion stiffness. Substituting
w = We^{iωt} and z = Ze^{iωt} into Eq.
(26) and (27), Eq. (26) and (27)
can be written as
where ω is the natural frequency. Substituting Eq. (1)
and (2) to Eq. (37) and (38),
leads to
The algorithmic procedure of the differential quadrature method leads to a simply supported or a clamped boundary, the transverse deflection of the plate at a simply supported boundary can be written as
The condition of zero normal moment can be reduced to the following discrete forms. For example, at the edge y = 0
in the direction of yaxis. The condition of zero normal moment at a free boundary can be reduced to the following discrete forms. For example, at the edge y = 0
The condition of zero effective shear force at a free boundary can be reduced to the following discrete forms. For example, at the edge y = 0
The following discrete form gives the condition of spring support. For example, at the edge y = 0
in the direction of yaxis. Eq. (39)(45)
can be rearranged in matrix form as
where K_{ij} is the stiffness matrix element, the subscript b and i
refer to the locations at the boundary and the interior regions, respectively.
The vector {W_{b}} and {W_{i}} are the normal deflection vectors
corresponding to the boundary and interior points. By substituting Eq.
(46) into a general eigenvalue form, Eq. (46) can be
expressed as
The eigenvalues will be obtained by solving the eigenvalue problem of Eq.
(47).
RESULTS AND DISCUSSION
Figure 2 shows the eigenvalue of the plates which are supported
as all of edges are simple support with a/b = 0.5 and the uniform equidistant
distribution of discrete grid points. The dimensionless natural frequency is
defined as .

Fig. 2: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 0.5 and the uniform equidistant
distribution of discrete grid points 

Fig. 3: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 0.5 with the roots of shifted
Chebyshev and Legendre points 
The other results in Figure 2 are cited from reference (Leissa,
1969). It can be seen that the numerical results agree with the data from theory
to within 5.38(%) when just 16x16 sample points are used. Figure
3 lists the eigenvalue of the plates which are supported as all of edges
are simple support with a/b = 0.5 with the roots of shifted Chebyshev and Legendre
points. It can be seen that the numerical results agree with the data from theory
to within 4.43(%) when just 10x10 sample points are used.

Fig. 4: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 0.8 and the uniform equidistant
distribution of discrete grid points 

Fig. 5: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 0.8 and the roots of shifted
Chebyshev and Legendre points 
Figure 4 shows the eigenvalue of the plates which are supported
as all of edges are simple support with a/b = 0.8 and the uniform equidistant
distribution of discrete grid points. It can be seen that the numerical results
agree with the data from theory to within 0.50(%) when just 16x16 sample points
are used. Figure 5 lists the eigenvalue of the plates which
are supported as all of edges are simple support with a/b = 0.8 with the roots
of shifted Chebyshev and Legendre points.

Fig. 6: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.0 and uniform equidistant
distribution of discrete grid points 

Fig. 7: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.0 and the roots of shifted
Chebyshev and Legendre points 
It can be seen that the numerical results agree with the data from theory to
within 0.01(%) when just 12x12 sample points are used.
Figure 6 shows the eigenvalue of the plates which are supported
as all of edges are simple support with a/b = 1.0 and the uniform equidistant
distribution of discrete grid points. It can be seen that the numerical results
agree with the data from theory to within 0.09(%) when just 16x16 sample points
are used.

Fig. 8: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.2 and the uniform equidistant
distribution of discrete grid points 

Fig. 9: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.2 and the roots of shifted
Chebyshev and Legendre points 
Figure 7 displays the eigenvalue of the plates which are
supported as all of edges are simple support with a/b = 1.0 with the roots of
shifted Chebyshev and Legendre points. It can be seen that the numerical results
agree with the data from theory to within 0.01(%) when just 12x12 sample points
are used. The computational time for using the differential quadrature method
with 8x8, 10x10, 12x12, 14x14, 15x15 and 16x16 sample points are 0.601, 1.423,
11.186, 27.059, 37.474 and 37.484 seconds, respectively.

Fig. 10: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.5 and the uniform equidistant
distribution of discrete grid points 

Fig. 11: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.5 and the roots of shifted
Chebyshev and Legendre points 
However, a computational time over 68.659 seconds is required for using FEM
in the similar problem.
Figure 8 shows the eigenvalue of the plates which are supported
as all of edges are simple support with a/b = 1.2 and the uniform equidistant
distribution of discrete grid points. It can be seen that the numerical results
agree with the data from theory to within 0.10(%) when just 16x16 sample points
are used.

Fig. 12: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.8 and the uniform equidistant
distribution of discrete grid points 

Fig. 13: 
The natural frequencies of the plates which are supported
as all of edges are simple support with a/b = 1.8 and the roots of shifted
Chebyshev and Legendre points 
Figure 9 displays the eigenvalue of the plates which are
supported as all o f edges are simple support with a/b = 1.2 with the roots
of shifted Chebyshev and Legendre points. It can be seen that the numerical
results agree with the data from theory to within 0.01(%) when just 12x12 sample
points are used.

Fig. 14: 
The natural frequencies of the plates which are a springmass
system mounted on and supported by simple support with different M/M_{P} 
Figure 10 shows the eigenvalue of the plates which are supported
as all of edges are simple support with a/b = 1.5 and the uniform equidistant
distribution of discrete grid points. It can be seen that the numerical results
agree with the data from theory to within 7.08(%) when just 10x10 sample points
are used. Figure 11 plots the eigenvalue of the plates which
are supported as all of edges are simple support with a/b = 1.5 with the roots
of shifted Chebyshev and Legendre points. It can be seen that the numerical
results agree with the data from theory to within 3.76(%) when just 10x10 sample
points are used.
Figure 12 shows the eigenvalue of the plates which are supported
as all of edges are simple support with a/b = 1.8 and the uniform equidistant
distribution of discrete grid points. It can be seen that the numerical results
agree with the data from theory to within 4.09(%) when just 16x16 sample points
are used. Figure 13 displays the eigenvalue of the plates
which are supported as all of edges are simple support with a/b = 1.8 with the
roots of shifted Chebyshev and Legendre points. It can be seen that the numerical
results agree with the data from theory to within 0.30(%) when just 12x12 sample
points are used. It can be observed from Fig. 213
that the results solved using the roots of shifted Chebyshev and Legendre points
are more accurate than the results solved using the uniform equidistant distribution
of discrete grid points and the results solved using 10x10 grid points did not
agree with the reference data.

Fig. 15: 
The natural frequencies of the plates that are a springmass
system mounted on, free at all of edges, except simple supported with torsion
spring at half of edge with different k/k_{p} 
Figure 14 shows the natural frequencies of the plates with
a springmass system and simple supported at all of edge. The data used in this
analysis are as follows: a = 1.0 m, b = 1.0 m, h = 0.005 m, v = 0.3, ρh
= 39.25 kg m^{2}, E = 2.051x10^{11}N m^{2},D = Eh^{3}/[12(1v^{2})]
= 2.3478x10^{3} Nxm, M_{P} = ρhab = 2355 kg, k_{P}
= D/a^{2} = 5.8695x10^{2} N m^{1 }and k/k_{p}
= 0.2. The eigenvalue of the springmass system mounted on the plate is calculated
and this plate is simple supported. The results reveal that ω decrease
as M/M_{P} increase. The difference between ω solved using the
DQM and ω from reference paper is less than 0.44%.
Figure 15 plots the natural frequencies of the plates that
are a springmass system mounted on with free at all of edges, except simple
supported with torsion spring at half of edge. There is a sprung mass in the
central point of the plate. The data used in this analysis are as follows: 15x15
sample points, δ_{x} = 10^{5}, δ_{y} = 10^{5},
a = 1.0 m, b = 1.0 m, h = 0.005 m, v = 0.3, ρh = 39.25 kg m^{2},
E = 2.051x10^{11}N m^{2}, D = Eh^{3}/[12(1v^{2})]
= 2.3478x10^{3} NHm, M_{P} = ρhab = 2355 kg, k_{p}
= D/a^{2} = 5.8695x10^{2} N/m, k_{φ}a/D and M/M_{p}
= 0.1. It is worthy of mentioning that M_{p} is the total mass of plate
and k_{p} is the stiffness of the plate. The results indicate that the
magnitude of ω increases as k/k_{p} increases.
CONCLUSIONS
The differential quadrature method is shown a powerful means of obtaining accurate solutions to the problem of rectangular plates with various boundary conditions and sprung masses. The natural frequencies of the plates increase as the torsion spring stiffness increases. The natural frequencies of the plates with sprung mass decrease as the weights of the sprung masses increase. The investigation into the integrity of the various grid spacing schemes indicates that the use of unequally spaced grids in conjunction with the technique can produce the fastest convergence. The comparisons and numerical examples show the effectiveness of the differential quadrature method.