The study of various vibration problems of elliptical and circular plates has a long history, and papers for vibration of the plates were widely reviewed by Leissa [1,2], and Yamada and Irie [3]. Thus, the papers are not mentioned individually here. Rather, considering the fact that most of the earlier papers treated isotropic and polar orthotropic plates, more recent works for rectilinear orthotropic plates with an elliptical shape are mentioned here.
Irie and Yamada [4] treated circular and elliptical annular plates using the Rayleigh-Ritz method with spline functions as the admissible functions, Tomar and Gupta [5] studied clamped elliptical plates using the Galerkin method with two terms of admissible functions, and Narita [6] analyzed vibration problem of free elliptical plates. Young and Dickinson [7] studied plates with curved edges using the Rayleigh-Ritz method with products of simple polynomials, which is basically the same approach as in the present paper. More recently, Chakraverty and Petyt [8] studied elliptical and circular plates with seven types of orthotropic material properties for all the classical free, simply supported and clamped boundary conditions using the Rayleigh-Ritz method with two-dimensional boundary characteristic orthogonal polynomials as the admissible functions. They presented an exhaustive graphical results of the first five frequencies for various aspect ratios. Chakraverty et al.
[9] also studied the orthotropic annular elliptic plates. Their study contains results for the first eight frequency parameters for various values of aspect ratios of the outer and inner ellipse.
In this letter, the free vibration problem of elliptical and circular plates is studied using the Rayleigh-Ritz method with products of simple polynomials as the admissible functions. The functions allow one to treat the free, simply supported and clamped boundary conditions simply taking the power of the starting polynomials as 0, 1 and 2, and the integral involved in the analysis is calculated simply by recurrence relationships. It may be worth noting that the generation of the function and the evaluation of the integral are very simple. The analysis is presented for rectilinear orthotropic material and some sample results are presented to demonstrate' the applicability and compared with existing results to show the accuracy. In addition, exhaustive numerical results are tabulated for isotropic plates.
Basically, the same method was used before by the author and some of the results for isotropic plates were presented in Korean [10]. The approach used is similar to that used by Leissa [11], who analyzed simply supported elliptical plates and by who treated rectangular and triangular plates. In particular, the method used by Narita [6]