On the symplectic superposition method for analytic free vibration solutions of right triangular plates

Abstract

The analytic free vibration solutions of triangular plates are important for both rapid analyses and preliminary designs of similar structures. Due to the difficulty in solving the complex boundary value problems of the governing high-order partial differential equations, the current knowledge about the analytic solutions is limited. This study presents a first attempt to explore an up-to-date symplectic superposition method for analytic free vibration solutions of right triangular plates. Specifically, an original problem is regarded as the superposition of three fundamental subproblems of the corresponding rectangular plates that are solved by the symplectic eigenexpansion within the Hamiltonian-system framework, involving the coordinate transformation. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition. By comparison with the numerical results for the right triangular plates under six different combinations of clamped and simply supported boundary constraints, the fast convergence and high accuracy of the present approach are well confirmed. Within the current solution framework, the extension to the problems of more polygonal plates is possible.