Frequencies and energies of vibrations for a honeycomb sandwich plate with negative Possion's ratio are investigated. Based on Reddy's third-order shear deformation plate theory, von Karman type nonlinear theory and Hamilton theory, the equations of motion for the plate are obtained, free vibrations and vibrations with the damping and in-plane force of the plate are considered, respectively. The changes of real parts of the eigenvalues with the damping and in-plane forces of the plate are obtained. In order to obtain the analytic solutions of the frequencies and energies of the plate, lateral free vibrations are considered by using the Hamilton energy method, and we compare the results with that of the previous free vibrations. The potential energies and the total energies variation with parameters of the plate are given. The research results from the two methods show that the change trends of the frequencies and energy of the plate are exactly consistent, but the values of natural frequencies and potential energies are lager in lateral vibrations.