Analytical solutions of nonlinear vibrations of thickness-shear and flexural modes of quartz plates

As the systematic study of nonlinear vibrations of quartz crystal resonators, we have established the nonlinear Mindlin plate equations which include the strongly coupled thickness-shear and flexural modes. Based on the simplification of these complicated nonlinear equations, equations of thickness-shear vibrations of quartz crystal plates have been solved by the combination of the Galerkin method and homotopy analysis method (HAM). The amplitude-frequency relation we have obtained shows that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude and thickness and length of plate, which is quite different from the linear cases. All these efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.