For rectangular thick plates with non-Levy boundary conditions, it is important to explore analytical free vibration solutions because the classical inverse and semi-inverse exact solution methods are not applicable to this category of problems. This work is to develop an extended separation-of-variable (SOV) method to find closed-form analytical solutions for the free vibration of rectangular Mindlin plates with arbitrary homogeneous boundary conditions. In the extended SOV method, characteristic differential equations and boundary conditions in two directions are obtained by employing the Rayleigh principle and the assumption that the mode functions are in the SOV form, and two transcendental eigenvalue equations are achieved through boundary conditions. But these two eigenvalue equations cannot be solved simultaneously since there are two equations and only the natural frequency is the unknown variable. Considering this, the second assumption in this method is that the natural frequencies corresponding to two-direction mode functions are independent of each other in the mathematical sense, thus there are two unknowns in two transcendental eigenvalue equations, and the closed-form solutions for plates with arbitrary boundary conditions can be obtained non-iteratively. From the physical sense, the natural frequencies pertaining to different direction mode functions should be the same, and this conclusion is validated analytically and numerically. The present natural frequencies and mode shapes agree well with those obtained by other analytical and numerical methods. Especially, for the plates with at least two opposite sides simply supported, the present solutions are exact.
Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells
