On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space

Abstract

In this paper, we obtain accurate analytic free vibration solutions of rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected.