The eigenvalue problem of the Hamiltonian operator associated with the free vibration of a moderately thick rectangular plate was investigated.First,the eigenfunctions of the operator with the mixed boundary conditions of the generalized displacements and generalized internal forces were solved directly.Then,the completeness of the eigenfunctinons was proved, demonstrating the feasibility of using separation of variables to solve the problems.Finally,the general solution was obtained by using the proved expansion theorem,and a concrete numerical example was given.