New Natural Frequency Studies of Orthotropic Plates by Adopting a Two-Dimensional Modified Fourier Series Method

Abstract

The free vibration behavior of orthotropic thin plates, which are clamped at three edges and free at one edge, is a matter of great concern in the engineering field. Various numerical/approximate approaches have been proposed for the present problem; however, lack precise analytic benchmark solutions are lacking in the literature. In the present study, we propose a modified two-dimensional Fourier series method to effectively handle free vibration problems of plates under various edge conditions. In the given solution, the adopted trial function automatically satisfies several boundary conditions. After imposing Stoke’s transformation in the trial function and letting it satisfy the remaining boundary conditions, we can change the present plate problem into calculating several systems of linear algebra equations which are easily handled. The present method can be regarded as an easily implemented, rational, and rigorous approach, as it can exactly satisfy both the governing equation and the associated edge conditions. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through the utilization of different types of Fourier series, and it can be extended for the further dynamic/static analysis of plates under different shear deformation theories. Finally, all the novel analytical solutions are confirmed to be sufficiently accurate since they match well with the FEM results. The new analytic solution obtained may serve as a benchmark for validating other numerical and approximate methods.