A complete symplectic approach for a class of partial differential equations arising from the elasticity

摘要

The symplectic approach is used to establish the unified framework for solving governing equations of some thin plate problems in elasticity. By introducing appropriate functions, the given partial differential equation is first transferred into a separable Hamiltonian system. The completeness of the system of generalized eigenvectors of corresponding Hamiltonian operator matrix is proved, which serves as the theoretical foundation of symplectic approach. Utilizing the expansion theorems, the general solutions of the boundary value problem for partial differential equation under consideration are obtained. Moreover, the bending, buckling, and free vibration problems of fully clamped rectangular thin plates governed by the given partial differential equation are solved analytically by the technique of superposition. Numerical results for bending, buckling, and free vibration plates are presented to demonstrate the availability and validity of the approach by comparison with those available in the literatures.