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. •All homogeneous boundary conditions are considered.•All eigensolutions are accurate and in an explicit closed form.•The present solutions agree well with the results of FEM.