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 circular annular plates of cylindrical anisotropy and non-uniform thickness

Determination of eigenvalues in a class of doubly connected regions in problems governed by Helmholtz's equation

Natural frequencies of a timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the other

Circular plates of variable thickness and elastically restrained against rotation along the edge resting on a Winkler‐Pasternak medium are addressed. Free and forced vibrations are studied using a very simple polynomial coordinate function and the Ritz method. In the case of simply supported and clamped plates of uniform thickness subjected to static loading the results are in very good agreement with values available in the open literature. The entire algorithmic procedure can be efficiently handled using a microcomputer.

R.H. Gutierrez

Papers

Vibrations of circular annular plates of cylindrical anisotropy and non-uniform thickness

Determination of eigenvalues in a class of doubly connected regions in problems governed by Helmholtz's equation

Natural frequencies of a timoshenko beam of non-uniform cross-section elastically restrained at one end and guided at the other

Vibrating Nonuniform Plates on Elastic Foundation

Free vibrations of rectangular plates elastically restrained against rotation with circular or square free openings