The symplectic superposition method for the free vibration problem of fully free sigmoid functionally graded materials rectangular thin plates on two-parameter elastic foundations

The symplectic superposition method (SSM) is employed to investigate the free vibration problem of sigmoid functionally graded materials (S-FGMs) rectangular thin plates with all sides free (FFFF) on two elastic foundations (Pasternak foundation and Winkler foundation). First, the initial vibration problem is decomposed into two sub-vibration problems. After that, both sub-problems are independently solved using the Hamiltonian system-based variable separation method. Next, the series expansion solution for the initial vibration problem is obtained using the SSM. Lastly, the accuracy of the solution obtained by the SSM is confirmed using a specific instance of the S-FGMs rectangular thin plate. Additionally, the effects of the aspect ratio, material gradient index, and elastic foundation parameters on the thin plate’s vibration frequencies are studied.