Free transverse vibration analysis of general polygonal plate with elastically restrained inclined edges

Abstract

In this study, a semi-analytical solution for the free transverse vibration of polygonal isotropic thin plates with arbitrary shapes and elastically restrained edges is obtained. All edges including inclined edges are elastically restrained by setting two groups of boundary springs along each edge of the polygonal plate to simulate the boundary forces and moments. Specifically, the complex surface integrals over the irregular plate area are simplified into a summation of plate boundary points by introducing the divergence theorem. The Chebyshev polynomial series are adopted to expand the displacement of the polygonal plate. Then, the Ritz method is performed to obtain the natural frequencies and mode shapes of several polygonal plates, such as triangular plates, quadrilateral plates, hexagonal plates, and irregular polygonal plates. An experiment study is performed and the validated comparisons between the current results and the reference results demonstrate that the proposed method can obtain accurate eigenpairs for general polygonal plates. The present method offers a unified procedure to address the vibration of polygonal plates, in which the change in the geometrical shape of plates can be readily converted with the increase or decrease in the corresponding boundary points. The results show that when the geometry and boundary restraints of a general polygonal plate are axisymmetric, the vibration modes will also present the feature of axial symmetry. Furthermore, cutting off a small part of the plate shows little influence on the modal shape whereas the influence on natural frequencies depends on the boundary conditions.