Hamiltonian System-Based Symplectic Framework for Analytical Vibration Analysis of Microplates

Abstract The classical elasticity is inadequate for the dynamic analysis of microplates due to the size effect. This study incorporates a higher-order strain gradient theory into the Hamiltonian system-based symplectic framework and derives new analytical solutions for the free vibration of microplates. The analytical solutions are obtained using rigorous mathematical techniques, including separation of variables, symplectic eigen expansion, and superposition, without relying on predetermined solution forms. Hence, they are not restricted to Lévy-type boundary conditions. Using these analytical solutions, we present comprehensive vibration results for microplates and perform detailed parametric studies to explore the impact of length scale parameters on the natural frequencies. Given the growing demand for microplates in advanced engineering applications, the obtained analytical solutions are expected to facilitate their design and performance optimization.