Symplectic superposition method for the free-vibrating problem of sigmoid functionally graded material rectangular thin plates clamped at four edges

The symplectic superposition method (SSM) is implemented to investigate the free vibration problem of sigmoid functionally graded material (FGM) rectangular thin plates clamped at four edges. Firstly, the vibration equation of the sigmoid FGM rectangular thin plate is transformed into Hamiltonian canonical equations. Then based on the analysis of boundary conditions of the plate, the vibration problem of the sigmoid FGM rectangular thin plate clamped at four edges is decomposed as two sub-problems simply supported on two opposite edges. And then based on the symplectic elasticity method (SEM), the series expansion solutions for the two sub-vibrating problems are obtained. Afterward, by superposing the series expansion solutions for the two sub-vibrating problems, a symplectic superposition solution of the freely vibrating sigmoid FGM rectangular thin plates clamped at four edges is obtained. Finally, we apply the obtained symplectic superposition solution to research the frequencies and vibration mode functions of some specific sigmoid FGM rectangular thin plates, and the impact of the aspect ratio a/b and material gradient index k on the vibration frequencies of the sigmoid FGM rectangular thin plates is studied.