Highly accurate closed-form solutions for the free in-plane vibration of rectangular plates with arbitrary homogeneous boundary conditions

In this paper, highly accurate closed-form solutions for the free in-plane vibration of rectangular isotropic plates with arbitrary homogeneous boundary conditions are achieved by an iterative separation-of-variable method, and all solutions are presented in an elegant explicit form. For plates with at least a pair of opposite edges simply supported, the present solutions are the exact solutions. For other homogeneous boundary conditions, the accurate closed-form solutions agree well with the results of the fine finite element method. This work shows that the iterative separation-of-variables method is a powerful method to find the closed-form eigensolutions of partial differential equations. The present solutions can be taken as the benchmarks for the validation of numerical methods, a guide in parametric design of structure and the basis of constructing new numerical methods.