Hamiltonian system-based analytic modeling of the free rectangular thin plates’ free vibration

•Symplectic superposition method is applied to free vibration of foundation plates.•Analytic free vibration solutions of rectangular thin foundation plates are obtained.•Predetermination of the solution form is avoided in the present solution approach.

Analytic free vibration solutions of free rectangular thin plates with or without an elastic foundation are obtained by using an up-to-date Hamiltonian system-based symplectic superposition method. Such boundary value problems are known to be very difficult and they were generally solved by the approximate/numerical methods. The present analytic solutions are expected to be the benchmarks for future verification of other methods. The advantage of the method is that the solution procedure is conducted in the symplectic space, where the symplectic eigen expansion is valid and the predetermination of the solution form is avoided. This significantly extends the approach to the analytic solutions of similar problems.