Solution of natural characteristics of a hard-coating plate based on Lindstedt–Poincaré perturbation method and its valedictions by FEM and measurement

The solution of natural characteristics of a hard-coating cantilever plate, which is made of a layer of anisotropic hard-coating material and the substrate of isotropic metal, is investigated based on the theory of nonlinear elasticity of plate. The hard-coating plate is regarded as a nonlinear elastic plate, which is made of equivalent nonlinear material. Nonlinear governing differential equations of the hard-coating plate are built based on the theory of nonlinear elasticity of plate and transformed from nonlinear partial differential equations into nonlinear ordinary differential ones by Galerkin method. These equations are solved by Lindstedt–Poincaré perturbation method (small parameter method), and the approximate analytic solutions of natural characteristics are obtained. To test the validity and computational accuracy of approximate analytic method, comparison and validation of these solutions are proceeded by FE method and experimental test. The results indicate that, the natural frequencies obtained by approximate analytical method of hard-coating plate natural characteristics, which is based on Lindstedt–Poincaré perturbation method, and by FE method are extremely close to the experimental test results, the relative errors are less than 6 %. However, the calculated natural frequencies obtained by linear analytical method have a comparatively big difference after the third order.