Symplectic elastic dynamical method for the in-plane free vibration of plane problem with two opposite edges simply-supported

In Hamiltonian symplectic dual system, the general method for solving exact solutions of free vibrations of rectangular plane problem with two opposite edges simply-supported are given directly by solving the Hamiltonian canonical equations using separation-of-variables method, and the analytical expressions of mode function and frequency equation of rectangular plane problem with two opposite edges simply-supported and the other two opposite edges clamped are obtained. The corresponding relationships of the two spatial eigenvalues with natural frequency are derived and discussed, and the generalized mode function vectors are obtained by using the conjugate symplectic ortho-normalization relationship of eigenvector. Based on boundary conditions, the exact frequency equations, two spatial eigenvalues and coefficients of generalized mode function are determined. Finally, present method is compared with the method based on Lagrange system, the correctness and universality of present method are validated.