Mesh-free least-squares-based finite difference method for large-amplitude free vibration analysis of arbitrarily shaped thin plates

A mesh-free least-squares-based finite difference (LSFD) method is applied for solving large-amplitude free vibration problem of arbitrarily shaped thin plates. In this approximate numerical method, the spatial derivatives of a function at a point are expressed as weighted sums of the function values of a group of supporting points. This method can be used to solve strong form of partial differential equations (PDEs), and it is especially useful in solving problems with complex domain geometries due to its mesh-free and local approximation characteristics. In this study, the displacement components of thin plates are constructed from the product of a spatial function and a periodic temporal function. Consequently, the nonlinear PDE is reduced to an ordinary differential equation (ODE) in terms of the temporal function. The accuracy, simplicity and efficiency of this mesh-free method are demonstrated for plates with simple as well as complex shapes. The ODE solutions obtained allow one to investigate the effect of large deflection amplitude on the vibration frequencies or periods.