An analytical method for nonlinear and nonhomogeneous boundary value problems of plates

Nonhomogeneous and nonlinear boundaries are difficult for dynamic studies of plates. The widely used modal expansion method only fits ideal boundary conditions. In this paper, an approximate analytic method for unideal boundary conditions is presented. The method treats boundary conditions as generalized governing equations and introduces corresponding generalized variables. All boundary conditions have equal status with the governing equation of the plate, and they can be involved in the solving process all the time. This ensures the accuracy of the proposed method. The proposed solution consists of a traditional modal expansion part and a correctional part. The first part satisfies derived linear and homogeneous boundary conditions and reveals natural characters. It ensures the convergence efficiency of the proposed method. The second part replenishes the modal expansion solution to make the whole solution satisfy initial nonlinear and nonhomogeneous boundaries. After the governing equation of the plate is projected into modal space based on the whole solution, a set of ordinary differential equations about modal variables and generalized variables is produced. Together with initial boundaries, the number of equations matches the number of undetermined variables. They are solved by the harmonic balance method. This ensures the proposed method can deal with strong nonlinearities. Three examples are presented to discuss the convergence and accuracy of the new method. They are responses subjected to the lateral basement excitation, the moment excitation, and the nonlinear boundary conditions. The differential quadrature element method is carried out to verify the proposed method. The comparison shows the new method has good accuracy in dealing with strong nonlinear and nonhomogeneous boundary conditions of rectangular plates. It further develops an analytical method for plate and helps investigation on it to have broader application.