Exact solutions for free vibrations of rectangular thin plates on the Winkler foundation

The exact solutions of free vibrations of rectangular thin plates on the Winkler foundation are solved by using the method of separation of variables, and the effects of foundation on frequencies are analyzed. It was considered that there were not the closed form exact solutions for the plates with boundary conditions CCCC, SCCC and SSCC (S denotes simple support, and C denotes clamp). The exact eigenfunctions in the method of saperation of variables are directly solved from the governing differential equation, this is different from that in the inverse method, then the coefficients of the exact eigenfunctions and the exact eigenvalue equations are determined using boundary contions. The numerical results agree with those in literatures and FEM, this validates the correctness of the present method.