# Free vibration characteristics analysis of rectangular plate with rectangular opening based on Fourier series method

Chinese Journal of Ship Research - - 郑超凡等：任意边界及耦合条件下的多跨梁结构振动特性 -

WANG Minhao1，LI Kai2，QIU Yongkang1，LI Tianyun1，3，4，ZHU Xiang1，3 1 School of Naval Architecture and Ocean Engineering，Huazhong University of Science and Technology， Wuhan 430074，China 2 China Ship Development and Design Center，Wuhan 430064，China 3 Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics，Wuhan 430074，China 4 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration，Shanghai 200240，China Abstract：Plate structures with openings are common in many engineering structures. The study of the vibration characteristics of such structures is directly related to the vibration reduction, noise reduction and stability analysis of an overall structure. This paper conducts research into the free vibration characteristics of a thin elastic plate with a rectangular opening parallel to the plate in an arbitrary position. We use the improved Fourier series to represent the displacement tolerance function of the rectangular plate with an opening. We can divide the plate into an eight zone plate to simplify the calculation. We then use linear springs, which are uniformly distributed along the boundary, to simulate the classical boundary conditions and the boundary conditions of the boundaries between the regions. According to the energy functional and variational method, we can obtain the overall energy functional. We can also obtain the generalized eigenvalue matrix equation by studying the extremum of the unknown improved Fourier series expansion coefficients. We can then obtain the natural frequencies and corresponding vibration modes of the rectangular plate with an opening by solving the equation. We then compare the calculated results with the finite element method to verify the accuracy and effectiveness of the method proposed in this paper. Finally, we research the influence of the boundary condition, opening size and opening position on the vibration characteristics of a plate with an opening. This provides a theoretical reference for practical engineering application. Key words：arbitrary opening position rectangular plate with an opening； improved Fourier series ； method；energy variational method