Identifying all combinations of boundary conditions for in-plane vibration of isotropic and anisotropic rectangular plates

This work presents a combined study of an effective vibration analysis and a counting theory to identify all combinations of boundary conditions for in-plane vibration of isotropic, specially orthotropic and symmetrically laminated rectangular plates. First, a method of analysis is proposed to obtain the in-plane natural frequencies of plates under any in-plane boundary conditions of free, clamp and two types of simple supports, and this method makes it possible to calculate the frequencies of rectangular plates subject to 256(=4 powered by 4) sets of boundary conditions. Secondly, Polya counting theory is introduced to determine theoretically the total number of distinct combinations of plate boundary conditions, and to reduce the 256 sets into the essentially identical subsets of frequencies. In numerical experiments, all sets of natural frequencies are calculated for the rectangular plates with different aspect ratio, material property and lamination, and are sorted into the classes with identical sets of frequencies. The distinct combinations are thus obtained for in-plane vibration of the plates, and it is shown that the number of combinations from the experiment exactly agrees with prediction by Polya counting theory.