Symplectic analytical solution for forced vibration of a multilayer plate system

The classical laminate and lattice sandwich plate structure can be simplified into a multilayer plate system, wherein the plate components of the system are continuously joined along the transverse direction by elastic layers and can have different combinations of boundary conditions. A symplectic analytical wave propagation approach is developed for the forced vibration of a system of multiple elastically connected thin plates considering the Kirchhoff thin plate theory. The proposed method overcomes the limitation of the traditional analytical method, wherein the exact vibration field function only exists for a system with all edges of the plate components simply supported. First, the coupled partial differential equations governing the vibration of the multi-plate system are decoupled using a technique based on matrix theory; for decoupled equations, a general “vibration” state is innovatively introduced into the symplectic dual system. Next, the general “vibration” state can be analytically described in symplectic space by solving the symplectic eigenproblem and utilizing wave propagation theory. Finally, by using these analytical wave shapes and satisfying the physical boundary conditions of the system, the forced responses can be analytically calculated. In the numerical examples, the forced transverse vibrations of the double- and three-plate systems are investigated, and the cases with various combinations of boundary conditions are considered. The effectiveness of the present method is validated by comparing the present results with the results from the literature and those calculated using the finite element method. The influence of the elastic layer stiffness and number of plate components on the vibration is also investigated.