The non-linear thickness-shear vibrations of quartz crystal plates under an electric field

•Established nonlinear equations of electrostatic plates under an electric field.•Both kinematic and material nonlinearities are considered.•Coupled equations for flexural and thickness-shear vibrations are extracted.•Nonlinear vibration frequencies of different amplitude ratios and electric fields are presented.•Electric fields have stronger effect on vibration frequencies of quartz crystal resonators.

With the consideration of material and kinematic non-linearities, a non-linear system of two-dimensional equations for the strongly coupled thickness-shear and flexural vibrations of electroelastic plates is established by expanding mechanical displacements and electric potential into power series in the plate thickness coordinate and integrating over the thickness. Since the non-linear equations are too complicated to be solved directly by known methods, we utilized the Galerkin approximation to convert the non-linear equation of thickness-shear vibrations into an ordinary differential equation depending only on time by assuming the mode shape of linear vibrations. This non-linear forced vibration equation has been solved by the successive approximation method and we plotted frequency–response curves with different amplitude ratios and electrical voltages. Numerical results showed that the electric field has a more significant effect on vibration frequency compared with other known factors.