Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang, China.
IEEE Trans Ultrason Ferroelectr Freq Control. 2012 Jan;59(1):30-9. doi: 10.1109/TUFFC.2012.2153.
We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. 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.
我们研究了考虑运动学和材料非线性的石英晶体板的厚度剪切和弯曲模态的非线性振动，采用非线性 Mindlin 板方程。通过采用非线性本构关系和近似方法来跟随 Mindlin，建立了强耦合厚度剪切和弯曲模态的非线性 Mindlin 板方程。基于长厚度剪切波近似，并借助相应的线性解，通过 Galerkin 和同伦分析方法的组合，求解了石英晶体板厚度剪切振动的非线性方程。我们得到的振幅频率关系表明，厚度剪切振动的非线性频率取决于振动幅度、厚度和板的长度，这与线性情况有显著的不同。本研究的数值结果还表明，在我们观察到的石英晶体谐振器中，运动学和材料非线性都不是频率偏移和性能波动的主要因素。这些努力将为进一步研究偏置场下石英板的非线性效应提供适用的求解技术，以便对石英晶体谐振器进行精确的分析和设计。
