Closed-form solutions for the free vibrations of three-dimensional orthotropic rectangular plates

In the vibration analysis of plates, the three-dimensional (3-D) elasticity theory provides more accurate modes and frequencies than all the two-dimensional (2-D) plate theories. In this work, the extended separation-of-variable method proposed by the present authors (Xing Y, Wang Z. An extended separation-of-variable method for the free vibration of orthotropic rectangular thin plates. Int J Mech Sci 2020;182:105739) is developed to solve the vibrating problem of orthotropic plates based on the 3-D elasticity theory. In this method, the mode functions are in an explicit separation-of-variable form, and the frequencies corresponding to eigenfunctions in three spatial directions are mathematically independent of each other. The closed-form analytical eigensolutions satisfy the Rayleigh's principle exactly. This method can deal with all homogeneous boundary conditions. For plates with at least two pairs of parallel surfaces simply supported, the eigensolutions satisfy the differential governing equation exactly as the well-known Levy types of exact solutions. Numerical experiments validate the accuracy of the present solutions. The effects of plate thickness, shear correctness factor and material properties on the accuracy of the 2-D plate theories are investigated.