Journal of Sound and Vibration (1992) 152(1), 149-155 TRANSVERSE VIBRATION OF SIMPLY SUPPORTED ELLIPTICAL AND CIRCULAR PLATES USING BOUNDARY CHARACTERISTIC ORTHOGONAL POLYNOMIALS IN TWO VARIABLES B. SINGH Department of Mathematics, University of Roorkee, Roorkee, India AND S. CHAKRAVERTY Computer Centre, Central Building Research Institute, Roorkee, India (Received 26 September 1990, and in final form 6 December 1990) The first four frequencies and mode shapes have been computed for elliptical and circular plates of uniform thickness with simply supported edges by employing orthogonal polyno- mials in the Rayleigh-Ritz method. In general, this is found to give better results than traditional methods. Successive approximations have been worked out to ensure conver- gence. Comparisons have been made with existing results in some cases. Calculations have been done for several values of b/a; i.e., the ratio of the semi-major and semi-minor axes. Mode shapes have been used to obtain three-dimensional plots of plates in displaced configurations. 1. INTRODUCTION In recent years orthogonal polynomials have been widely used in solving the problem of vibration of plates of varying shapes and under a variety of boundary conditions. Some important references in this connection are those of Bhat [1, 2], Bhat et al. [3], Laura et al. [4], Liew et al. [5] and Dickinson and Blasio [6]. The authors [7] have also applied these polynomials to the vibration problems of elliptical and circular plates with clamped boundaries. The present paper is an effort to extend the same method to elliptical and circular plates with simply supported boundaries. The main attractions of the orthogonal polynomials are the simplification of the eigenvalue problem and the better convergence characteristics. The three basic steps in the method are (a) to generate the desired ortho- gonal polynomials, (b) to use them in approximating the displacement which is to be substituted in Rayleigh's quotient and minimized as a function of parameters, and, finally, (c) to solve the resulting eigenvalue problem for frequencies and mode shapes. 2. GENERATION OF ORTHOGONAL POLYNOMIALS We follow the well-known Gram-Schmidt process to generate a set of orthogonal poly- nomials as has been done by Bhat [1] and Liew et al. [5] for a rectangular plate. For an 149 0022-460X/92/010149 + 07 $03.00/0 © 1992AcademicPress Limited