Free vibration of thick rectangular orthotropic plates with clamped edges- using asymptotic analysis of infinite systems

Abstract

Exact solutions for free vibration of thick rectangular orthotropic plates when their all edges are clamped are sought through asymptotic analysis of infinite systems without resorting to the usual truncation of series solution. The use of modified trigonometric functions made it possible to obtain a general solution for the problem which has the same form for all four cases of symmetry of the quarter plate. Thus, an infinite system of linear algebraic equations is derived for the unknown coefficients of the series representing the solution for each case. This is in sharp contrast to previous publications based on series-solution which does not allow the satisfaction of the quasi-regularity condition of the corresponding infinite system, and therefore, the method used earlier, was not amenable to asymptotic solution of the infinite system. In this investigation, the quasi-regularity of the infinite system is proved, but importantly, an algorithm for determining the natural frequencies of the plate based on the theorem of the existence of the solution for the quasi-regular system is presented. The asymptotic behaviour of the non-trivial solution of the homogeneous quasi-regular infinite system is ascertained by generalising the asymptotic law of Koialovich which essentially led to the development of the algorithm. Numerical examples are given with significant conclusions drawn.