A new analytical framework for free vibration solutions of rectangular plates with internal discontinuities

Abstract

In this study, a new analytical framework for free vibration solutions of rectangular plates with various internal discontinuities is developed. The discontinuities caused by a line hinge, a rigid line support, or a step change in thickness are under consideration. All such discontinuities are bypassed via the domain decomposition technique, which divides the entire plate into sub-plates, and then the symplectic superposition is leveraged to analytically address the free vibration of the sub-plates. Specifically, the governing free vibration equations are elegantly reformulated in the Hamiltonian system and rigorously solved by the three mathematical treatments, i.e., separation of variables, eigen expansion, and superposition. This innovative framework, developed through the integration of the domain decomposition and the symplectic superposition, is particularly notable for its ability to simplify the mathematical challenges in obtaining analytical solutions of plates with internal discontinuities. The obtained analytical solutions are well validated by other methods and are thus capable of serving as benchmarks for future studies. These solutions also reveal distinct vibration characteristics of plates with different internal discontinuities. In comparison to continuous plates, plates with a line hinge exhibit lower natural frequencies due to increased flexibility, whereas plates with a rigid line support demonstrate higher natural frequencies due to restricted deflection at the support location. Furthermore, the design of stepped plates with thickened sub-plates effectively suppresses vibration, hence achieving higher natural frequencies. In view of the wide usage of plate assemblies in engineering, the present framework is expected to facilitate their early-stage design and parametric optimization.