The differential quadrature method is a numerical solution technique for initial and/or boundary problems. It was developed by the late Richard Bellman and his associates in the early 70s and, since then, the technique has been successfully employed in a variety of problems in engineering and physical sciences. The method has been projected by its proponents as a potential alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. This paper presents a state-of-the-art review of the differential quadrature method, which should be of general interest to the computational mechanics community.

Differential Quadrature Method in Computational Mechanics: A Review

Plates are integral parts of most engineering structures and their vibration analysis is required for safe design. Vibration of Plates provides a comprehensive, self-contained introduction to vibration theory and analysis of two-dimensional plates. Reflecting the author's more than 15 years of original research on plate vibration, this book presents new methodologies and demonstrates their effectiveness by providing comprehensive results. The text also offers background information on vibration problems along with a discussion of various plate geometries and boundary conditions, including the new concepts of Boundary Characteristic Orthogonal Polynomials (BCOPs).

Vibration of Plates

Free vibrations of beams with general boundary conditions

The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem.

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Eigenvalues of the Laplacian in Two Dimensions

The free and forced vibration of a laminated functionally graded beam of variable thickness under thermally induced initial stresses is studied in this paper within the framework of Timoshenko beam theory. The beam consists of a homogeneous substrate and two inhomogeneous functionally graded layers whose material composition follows a power law distribution in the thickness direction in terms of the volume fractions of the material constituents. Both the axial and rotary inertia of the beam are considered in the present analysis. It is assumed that the beam may be clamped, hinged, or free at its ends and is subjected to one-dimensional steady heat conduction in the thickness direction before undergoing dynamic deformation. To include the effect of temperature change, the initial stress state is determined through a thermo-elastic analysis before the free and forced vibration analyses. The differential quadrature method that makes use of Lagrange interpolation polynomials is employed as a numerical solution tool to solve both the thermo-elastic equilibrium equation and dynamic equation. Numerical results are presented in both tabular and graphical forms for various laminated functionally graded beams, showing that vibration frequencies, mode shapes and dynamic response are significantly influenced by the thickness variation, temperature change, slenderness ratio, volume fraction index, the thickness of the functionally graded layer, and the end support conditions.

Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction

Vibrations of an elastically restrained cantilever beam of varying cross section with tip mass of finite length

The main advantages of the differential quadrature method are its inherent concep­ tual simplicity and the fact that easily programmable algorithmic expressions are obtained. It was developed by Bellman in the 1970s but only recently has been applied in the solution of technically important problems. Essentially, it consists of the ap­ proximate solution of the differential system by means of a polynomial-collocation approach at afinite number of points selected by the analyst. This article reports some numerical experiments on vibrating Timoshenko beams of nonuniform cross­ section. © 1993 John Wiley & Sons, Inc.

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Analysis of Vibrating Timoshenko Beams Using the Method of Differential Quadrature

The free vibrations of circular plates having flexible edge supports have been studied by several researchers for the restricted case when the supports are represented by springs having constant stiffness. In the present paper a general method is presented for dealing with supports having translational and rotational flexibilities which vary in an arbitrary manner around the boundary. It is shown that the varying stiffnesses can be represented as accurately as desired by expanding them into trigonometric series in the polar angle. The exact solution in polar coordinates of the differential equation of motion for the plate is then substituted into the elastic boundary conditions. The resulting infinite characteristic determinant is solved by successive truncation. As an example the case of a plate having a simply supported edge (infinite translational stiffness) with rotational stiffness varying according to L0 + L1 cos θ (L0 and L1 being constants) is considered. Numerical results are obtained by the meth...

R.H. Gutierrez

Papers

Vibrations of an elastically restrained cantilever beam of varying cross section with tip mass of finite length

Analysis of Vibrating Timoshenko Beams Using the Method of Differential Quadrature

Transverse vibrations of circular plates having nonuniform edge constraints

A note on transverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass at the other

A note on transverse vibrations of stiffened rectangular plates with edges elastically restrained against rotation