Potential theory of in-plane vibrations of rectangular and circular plates

Basic equations of in-plane plate vibrations are specified. Governing differential equations of motion are solved analytically introducing displacement potentials and employing the method of separation of variables. Frequency equations for a rectangular plate with two opposite simply supported edges and two remaining edges clamped, free and combined clamped-free are derived. Three types of solutions are possible, depending on values of involved function arguments. For the purpose of vibration analysis of a circular plate, differential equations of motion of a rectangular plate are transformed. Concerning a circular plate, circumferential variation of displacement potentials is assumed in the form of trigonometric series, while variation in radial direction is obtained by solving Bessel’s differential equations. Frequency equations for clamped and free plate edges are given, and the same procedure is applied for the annular plate. Application of the developed theory is illustrated in cases of a rectangular plate simply supported at two opposite edges, clamped, free and combined clamped-free at two remained edges. Vibrations of clamped and free circular plates are also analyzed as well as of a clamped-free annular plate. In all the considered cases, analytical values are compared with FEM results.