Free in-plane vibration of rectangular plates

Based on the Kantorovich-Krylov variational method (Kantorovich; L. V, and Krylov, V.I., Approximate Methods of Higher Analysis, Noordhoff International, Groningen, The Netherlands, 1964, pp. 241-357), we analytically solve for modal frequency and displacement mode shapes of in-plane vibration of rectangular plates with free and clamped boundary conditions. Free and clamped boundary conditions do not allow closed-form solutions from partial differential equations. We develop analytical expressions of plate mode shapes consisting of a linear combination of progressive waves with unknown wave amplitudes. We then use these to calculate the modal frequencies and the wave amplitudes with analytical mode shape expressions. An iteration scheme is presented to calculate efficiently the natural frequency and corresponding mode shape functions based on the Kantorovich-Krylov method. Three configurations are considered: a plate with four edges clamped, a plate with three edges clamped and one edge free, and a plate with two parallel edges clamped and the other two edges free. The first six natural frequencies predicted by our approach are validated using NASTRAN and other analyses from the literature. Improvements in accuracy of the first six predicted natural frequencies were achieved using the Kantorovich-Krylov method when compared against other results in the literature.