A symplectic eigenfunction expansion approach for free vibration solutions of rectangular Kirchhoff plates

Abstract

The eigenfunction system of the Hamiltonian operator appearing in the free vibration of rectangular Kirchhoff plates with two opposite edges simply supported is studied. The governing differential equations for free vibration of rectangular Kirchhoff plates is rewritten as a Hamiltonian system based on the known results, and the associated Hamiltonian operator is obtained. Then, in the sense of Cauchy's principal value, the completeness of the symplectic eigenfunction system is proved. It offers a theoretical guarantee of the feasibility of variable separation methods based on the Hamiltonian system for the problem. The exact general solution for the corresponding Hamiltonian system of the problem is given by the symplectic eigenfunction expansion method. The general solution is more simple and convenient than the existing result. Examples are given to illustrate that, combining the general solution with the corresponding boundary conditions, the frequency equations and the transverse displacement functions for Lévy-type plates can be directly derived. Furthermore, the boundary conditions for the problem, which can be solved by this approach, are indicated.