The nonlinear thickness-shear vibrations of an infinite and isotropic elastic plate

Thickness-shear vibrations of a plate is one of the most widely used functioning modes of quartz crystal resonators. For an analysis of vibrations, the Mindlin and Lee plate theories based on the displacement expansion of the thickness coordinate have been used as the linear theories. However, due to lacking of available method and complexity of the problem, the nonlinear thickness-shear vibrations have been rarely studied with analytical methods. As a preliminary step for the research on nonlinear vibrations in a finite crystal plate, nonlinear thickness-shear vibrations of an infinite and isotropic elastic plate are studied. First, using the Galerkin approximation and forcing the weighted error to vanish, we have obtained a nonlinear ordinary differential equation depending on time. By assuming corresponding solution and neglecting the high-order nonlinear terms, the amplitude-frequency relation of the nonlinear vibrations is obtained. In order to verify the accuracy of our study, we have also employed the perturbation method to solve this ordinary differential equation and obtained the second-order amplitude-frequency relation. These equations and results are useful in verifying the available methods and improving our solution techniques for the coupled nonlinear vibrations of finite piezoelectric plates.