Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Two distinct types of simple support boundary conditions are formulated for rectangular plates undergoing in-plane free vibration. Each type of simple support is shown to be analogous to the well known simple support edge conditions encountered in the free transverse vibration analysis of rectangular plates. It is shown that exact solutions may be obtained for plate free vibration eigenvalues and mode shapes when two opposite plate edges are given either type of simple support, the other two edges being given any combination of classical edge conditions. A complete analysis is provided for plate in-plane vibration with pairs of clamped or free edge conditions imposed on the non-simply supported boundaries and exact eigenvalues are tabulated for a large range of plate aspect ratios. This appears to be the first thorough study of these in-plane vibration problems with exact solutions.