Equations And Solutions Of Nonlinear High Frequency Vibrations Of Piezoelectric Anisotropic Plates

Thickness-shear vibration of a quartz crystal plate is one of the most widely used functioning modes of piezoelectric resonators with broad applications for frequency control and detection. The classical plate theories, including Mindlin and Lee plate theories, are linear systems and have been successfully used for linear analysis of high frequency vibrations of piezoelectric plates and yielded important results for the design of piezoelectric resonators. With the growing demands for higher and more precise frequency and miniaturization of piezoelectric structures, the nonlinear phenomena have emerged and eventually caused frequency instability of resonators. In order to study the frequency shifts due to the nonlinear effects, we have established nonlinear two-dimensional equations of high frequency vibrations of quartz crystal plate with the consideration of kinematic and material nonlinearities. We further employed analytical methods to transform and solve these complicated nonlinear equations and investigated the different effects of nonlinearities of quartz crystal resonators on its frequency and corresponding circuit parameters.We have utilized the linear third-order Mindlin plate equations with natural correction and obtained exact cut-off frequencies of the fundamental and third-order thickness-shear modes as the foundation of nonlinear analysis. In order to study the couplings and interactions between thickness-shear and spurious modes, we have calculated frequency spectra in the vicinity of fundamental thickness-shear and third-order overtone modes. From frequency spectra, we have found that the coupling between thickness-shear and flexural modes depends strongly on the aspect ratios (length/thickness) and frequency spectra is the most basic tool to obtain optimal aspect ratios for decreasing and avoiding this coupling.The nonlinear equation of simple thickness-shear vibrations of an infinite and isotropic plate has been established and transformed into a nonlinear ordinary differential equation by Galerkin approximation. Then we solved this nonlinear equation by the perturbation method and homotopy analysis method, respectively. The amplitude-frequency relation we obtained showed the nonlinear frequency not only depends on amplitude but also related to the thickness of plate. Compared with results from perturbation method, our analytical solutions converge faster while homotopy analysis method provides an auxiliary parameter to control convergence region and speed.The nonlinear two-dimensional equations of high frequency vibrations of a quartz crystal plate for strongly coupled thickness-shear and flexural modes with the consideration of kinematic and material nonlinearities have been established. These equations are too complicated to solve directly. Based on the assumption of long thickness-shear waves and thickness-shear approximation, the nonlinear equation of thickness-shear modes has been transformed into an ordinary differential equation depends on time by Galerkin approximation. The amplitude-frequency relations of thickness-shear vibration have been obtained by perturbation and homotopy analysis methods. Numerical results showed neither kinematic nor material nonlinearities are the main factors of frequency shifts which suggest us to examine the effect of electrical field.The two-dimensional equations of high frequency vibration of a quartz crystal plate under a strong electric field have been established. Also based on the assumption of long thickness-shear waves and thickness-shear approximation, we utilized Galerkin approximation to transform the nonlinear partial differential equation of thickness-shear vibrations into an ordinary differential equation depending only on time. By successive approximation method, we obtained electrical current frequency-response relation and plotted nonlinear frequency-response curves for different amplitude ratios and driving voltages. Numerical results showed that frequency shift caused by the electrical field is significant and should be treated as the primary cause of frequency instability. We further computed some circuit parameters of AT-cut quartz crystal plates as simple resonators, and investigated the effect of nonlinearities on circuit parameters. The strongly coupled equations of vibrations of the thickness-shear and flexural modes have all been transformed into a system of nonlinear ordinary equations. By successive approximation method, we obtained frequency-response relations with the consideration of strong coupling between vibrations modes.The two-dimensional equations of high frequency vibrations of a quartz crystal plate with the consideration of material and kinematic nonlinearities we established and solved lay a firm theoretical foundation for nonlinear analysis of quartz crystal resonators, which also can be used for the finite element analysis and further analytical studies of nonlinear effects of bias fields on quartz crystal resonators. The results we obtained provide guidance for the precise analysis and design of quartz crystal resonators.