Computation of modal stress resultants for completely free vibrating plates by LSFD method

When the Ritz method, the Galerkin's method and the finite element method are adopted for the vibration analysis of thin plates, the natural frequencies and mode shapes can normally be obtained accurately. However, the corresponding modal stress resultants usually violate the natural boundary conditions at the free edges and contain erroneous oscillations. Therefore, the accuracy of modal stress resultants obtained by such methods is uncertain. In this study, a meshfree least squares-based finite difference method (LSFD) is proposed for evaluating the vibration solutions of completely free plates. Examples treated include circular plates, elliptical plates, lifting-tab shaped plates and 45° right triangular plates. It will be shown that the LSFD method not only furnishes accurate natural frequencies, but also yields excellent modal stress resultants that satisfy the natural boundary conditions of the free edges and are smooth in their distribution over the plate domain.