New Analytic Free Vibration Solutions of Free-Edge Orthotropic Moderately Thick Rectangular Plates by the Symplectic Superposition Method

The goal of this paper is to uniformly study the free vibration problems of free-edge (FFFF) orthotropic/isotropic rectangular plates with different aspect ratios and thickness-to-width ratios using the symplectic superposition method (SSM). The governing equations for the orthotropic moderately thick rectangular plate (MTRP) are first transformed into Hamiltonian canonical equations. Then, by analyzing the boundary conditions (BCs) of the plate, the vibration problem of the original FFFF orthotropic MTRP is decomposed into two sub-vibration problems with sliding supports on two opposite sides. After that, the general solutions for these two sub-vibration problems are obtained using the separation of variables method in the Hamiltonian framework. Then, based on the superposition method, the symplectic superposition solution for the original vibration problem is derived by superimposing the general solutions of these two sub-vibration problems. In examples, the symplectic superposition solution is applied to present the vibration frequencies and corresponding modes for orthotropic rectangular plates with different thickness-to-width ratios and aspect ratios. Additionally, the change rules of vibration frequencies with aspect ratios, thickness-to-width ratios, and elastic modulus ratios are analyzed. The SSM does not need to set any trial function in advance, its solving process can be achieved through step-by-step rigorous derivation, and this method has a wide range of applications. For example, this method can be used to study the buckling and vibration of plates with different materials and shapes under more complex boundary conditions.