INTRODUCTION
The concept of differential quadrature method introduced by Bellman and Casti (1971) and Bellman et al. (1972) has been used extensively 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 thus has become an alternative to the existing methods. Chen and Zhong (1997) pointed out that the differential quadrature method and the 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. One of the fields among which can find extensive applications of the differential quadrature method is structural mechanics. Civan (1994) solved multivariable mathematical models using differential quadrature method and differential cubature method. Malik and Bert (1996) implemented the differential quadrature method multiple boundary conditions to free vibration of plates. In order to derive the weighting coefficients in a more explicit way, a generalized differential quadrature method was proposed by Quan and Chang (1989a, b). They presented that the generalized differential quadrature method is rather convenient in comparison with the differential quadrature method. Hua and Lam (1998) pointed out that generalized differential quadrature method is no restriction on the distribution and the number of discrete grid points used in the approximation process and the weighting coefficients are determined by using a simple recurrence relationship instead of by solving a set of linear algebraic equations. Dynamic characteristics of annular plates are of considerable importance in many designs. Azimi (1988a) solved free vibration of circular plates with elastic edge supports using the receptance method. Azimi (1988b) solved free vibration of circular plates with elastic or rigid interior support. Wang et al. (1995) studied free vibration analysis of circular annular plates with nonuniform thickness by the differential quadrature method. In this work, the generalized differential quadrature method is employed to formulate the discrete eigenvalue problems of annular plates. In the generalized differential quadrature method, two points, separated by a very small distance, δ are placed at each boundary point. If δ is used, the solution may show oscillation. δ influences the accuracy of the solution seriously. In this study a different approach is proposed for applications of the boundary conditions using the modified generalized differential quadrature method. The integrity and computational efficiency of the modified generalized differential quadrature method in this problem will be demonstrated through a series of case studies.
THE GENERALIZED DIFFERENTIAL QUADRATURE METHOD
For a function f(r), generalized differential quadrature method approximation for the m order derivative at the ith sampling point is given by
where N is the number of sampling points, r_{i} the location of ith
sampling points in the domain, f(r_{i}) the functional values at these
points and
the weighting coefficients of m order differentiation attached to these
functional values. The advantages of the generalized differential quadrature
method over the differential quadrature method are that the general form of
the test function is taken to be Lagrangian interpolation polynomial in order
to determine the weighting coefficients
instead of solving a set of linear algebraic equations. The test function is
where
Other types of polynomials could be used as well. However, by using the Lagrangian interpolation polynomial, the numerical ill conditions in calculating the weighting matrix would be overcome. Introducing Eq. 2 into Eq. 1, one derives that
and
Once the sampling points are selected, the weighting matrix can easily be obtained from Eq. 3 and 4. For the second order and higher order derivatives, the weighting coefficients are obtained by using the following simple recurrence relation
It is emphasized that the number of the test functions must be greater than the highest order of derivative in the governing equations.
CHOICE OF THE SAMPLING POINTS
The selection of locations of the sampling points plays a significant role in the accuracy of the solution of differential equations. For a domain separate into by N points, one could obtain equally spaced sampling points. Inner points are
At each end, two points, separated from each other by a small distance δ, should be placed and boundary conditions applied at both points. Boundary points are
where, δ is small distance, b is the inner radius of the annular plate
and a is the outer radius of the annular plate. Using equally spaced points
can be considered to be a convenient and an easy selection method. For a domain
separate into by N points, one could obtain a more accurate solution by choosing
a set of unequally spaced sampling points. Bert and Malik (1996) pointed out
simple and good choice can be the ChebyshevGaussLobatto points. Following
nonuniform grid spacing gives better and more reliable results. Inner points
are
boundary points are
VIBRATION ANALYSIS
The axisymetric motion of an annular plate is governed by
where, D is the flexural rigidity, h is the thickness of the annular plate,
w = W sin(ωt), ω is the natural frequency of the annular plate, E
is Young’s modulus of the material of the annular plate and ρ is the
density of the material of the annular plate. Equation 17
can be transformed into the following equation
where,
At a clamped boundary, the transverse deflection of the plate is zero:
at a free boundary, the condition of zero normal moment can be reduced to
The condition of zero effective shear force at a free boundary is given by
The generalized differential quadrature method is applied and Eq.
1 substituted into Eq. 1822. Equations
1822 can be rewritten as the following equations:
where,
By solving the eigenvalue problem of Eq. 23, one will obtain
the eigenvalues.
A DIFFERENT APPROACH FOR VARIOUS BOUNDARY CONDITIONS
If δ is too small, oscillation of solution may occur. In this study a different approach is proposed for application of the boundary conditions. The selection of locations of the sampling points plays a significant role in the accuracy of the solution of differential equations. For a domain separate into by N points, one could obtain unequally spaced sampling points. Inner points are
Different formulas are no two points at each of the ends and the boundary conditions applied at each of the points. Boundary points are
Equations 1822 can be rewritten as the
following equations:
where,
One will obtain the eigenvalues by solving the eigenvalue problem of Eq.
39. Computer programs are coded and developed in MATLAB to find the numerical
results that are based on the modified generalized differential quadrature method.
RESULTS AND DISCUSSION
Figure 1 shows the nondimensional frequencies
of the clampedfree annular plate for b/a = 0.1. The present method gives the
most accurate results for the fundamental frequency and shows good convergence.
The numerical results solved using the modified generalized differential quadrature
method show better than the results calculated using the generalized differential
quadrature method. The numerical results are greatly improved using the modified
generalized differential quadrature method. If δ is too small, the polynomial
solution may show oscillations. One can observe that the present method renders
very accurate numerical results even in the 7 sampling points case. One can
numerically show convergence when the sampling points increase using the modified
generalized differential quadrature method. Numerical results solved using the
generalized differential quadrature method with, δ = 10^{6}, 5,
11, 13 and 24 sample points have ill conditions. Figure 2
plots the nondimensional frequencies
of the clampedfree annular plate for b/a = 0.2. The present method gives the
most accurate results for the fundamental frequency and shows good convergence.
The results solved using the present method show better than the results solved
using the small distance δ for the clampedfree annular plates.

Fig. 1: 
Nondimensional frequencies
of the clampedfree annular plate for b/a = 0.1 

Fig. 2: 
Nondimensional frequencies
of the clampedfree annular plate for b/a = 0.2 
One can observe that the present method presents very accurate numerical results
even in the 7 sampling points case. One cannot numerically show convergence
when the sampling points increase using the generalized differential quadrature
method. Numerical results calculated using the generalized differential quadrature
method with, δ = 10^{6}, 21 and 24 sample points have ill conditions.
CONCLUSIONS
In this study the modified generalized differential quadrature method is presented using a new differential quadrature formulation. A numerical solution for vibration of an annular plate is developed based on the implementation of modified generalized differential quadrature method. The results from several case studies validate the applicability of the method for solving such an engineering problem. The method provides accurate results with relatively minimal computational and modeling efforts. The modified generalized differential quadrature method did not use the δ method. It is concluded that the demonstrated accuracy and simplicity of the modified generalized differential quadrature method make it a good candidate for modeling more complicated cases of vibrations of annular plates. Results indicated that the modified generalized differential quadrature method accompanies appropriate boundary conditions is feasible and valid in formulating the eigenvalue problems of the annular plates.