New analytic free vibration solutions of orthotropic rectangular plates by a novel symplectic approach

Abstract

This paper aims at analytically solving the free vibration problems of orthotropic rectangular plates, which have found broad applications as a class of crucial aerostructures. A novel symplectic superposition method is presented to give analytic solutions of plates without two opposite edges simply supported (i.e., non-Levy-type plates). Since these solutions are very difficult to obtain within the classical Lagrangian system framework, the new approach proceeds in the Hamiltonian system, in physics, and in the symplectic space, in mathematics. The validity of separation of variables and symplectic eigenexpansion guarantees the realization of new analytic solutions in a rigorous step-by-step way, which cannot be achieved by some classical analytic methods where the predetermination of solution forms is usually inevitable. Representative challenging problems, including those incorporating two adjacent free edges (e.g., cantilever and free plates), are successfully solved and validated by the refined finite element modeling. Comprehensive natural frequency and mode shape results show the accuracy of the present solutions. The generality of the symplectic superposition method enables one to pursue more analytic solutions of similar problems that are governed by higher-order partial differential equations.