Explicit Analytical Solution for Free Vibration of Restrained Orthotropic Plates

An approximate explicit analytical solution to the free vibration behavior of orthotropic plates with all edges rotationally restrained against rotation based on the deformation shape function and the Galerkin method is presented. The deformation shape function along each direction is uniquely constructed through a weighted combination of vibration characteristic functions of simply supported-clamped beam and modified vibration characteristic functions of clamped-simply supported beam and clamped-clamped beam to meet the elastically restrained opposite edges against different rotation; whereas to satisfy the elastically restrained opposite edges against the same rotation, the deformation shape function along each direction is chosen by combining the vibration characteristic function of simply supported beam and the modified vibration characteristic function of clamped-clamped beam. It is revealed that the present approximate explicit analytical solution can avoid possible numerical instabilities in the method based on the classical vibration characteristic function of beam and is effective and accurate for performing the free vibration analysis of orthotropic plates with all the opposite edges arbitrarily elastically restrained against rotation.