Analytical solutions based on Hamiltonian system for the free vibration of polyline thin plates

It is challenging to establish analytical solutions for conventional inverse and semi-inverse approaches because of the intricacy of standard mathematical calculations. In this study, the symplectic superposition approach is used to provide the analytical solution for the free vibration of a thin plate with a polyline shape. The polyline plate is separated into four sub-plates based on the superposition principle, and then the problem is solved by using the symplectic eigen-expansion method. Multiple sets of constants dependent on boundary conditions can be found by applying a specific mechanical quantity to the edge of each sub-plate. The four sub-plate solutions can be combined to provide the final analytical solution. By contrasting the numerical results obtained under various boundary restrictions, the correctness and effectiveness of the method are confirmed. The frequency values decrease with increasing a 2 /a 1 due to the decrease of structure’s stiffness. The method proposed in this work can be extended to solve the vibration of anisotropic thin plate with other boundary conditions and irregular plates.