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The Nonlinear Thickness-shear Vibrations of Quartz Crystal Plates under a Strong Electric Field

With higher frequency and miniaturization of piezoelectric resonators, many nonlinear phenomena such as derive level dependency (DLD) and activity dip have emerged and needed to be systematically studied. Our earlier studies shown neither kinematic nor material nonlinearities are the main factors of frequency shifts and performance fluctuation of quartz crystal resonator. With the consideration of material and kinematic nonlinearities, a nonlinear system of two-dimensional equations for the coupled thickness-shear and flexural vibrations of piezoelectric plates is established by expanding the mechanical displacements and the electrical potential into power series in the plate thickness coordinate. The nonlinear equation of thickness-shear vibrations in a strong electric field have been solved by combination of the Galerkin approximation and successive approximation method. Through these effects, we can obtain nth-order electrical current amplitude-frequency relation. We found the higher-order solutions only improve accuracy of solutions slightly while it is extremely complicated to solve directly. The first-order electrical current amplitude-frequency relation is accurate enough to give some nonlinear characteristics of thickness-shear vibration of quartz crystal plate.