石英晶体板非线性厚度剪切振动的二维有限元分析
An Analysis of Nonlinear Thickness-shear Vibrations of Quartz Crystal Plates by Two-Dimensional Finite Element Method
摘要：
本文用二维有限元法对AT切石英晶片的高频厚度剪切振动进行了非线性分析。我们扩展了运动学和本构非线性Mindlin板方程，然后将其截断为一阶方程作为近似，这是后来用于制定所有零阶和一阶位移和电势的非线性有限元分析。采用一阶谐波近似建立了系统的矩阵运动方程，并采用直接迭代法求解了广义非线性特征系统。得到了主曲线和相应的振型，并进行了分析。非线性有限元程序是在原有的线性有限元程序的基础上开发的，可用于小型化石英晶体谐振器设计过程中的非线性特性预测。
英文摘要：
A nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut quartz crystal plates is presented with the two-dimensional finite element method. We expanded both kinematic and constitutive nonlinear Mindlin plate equations and then truncated them to the first-order equations as an approximation, which is used later for the formulation of nonlinear finite element analysis with all zeroth- and first-order displacements and electric potentials. The matrix equation of motion is established with the first-order harmonic approximation and the generalized nonlinear eigensystem is solved by a direct iterative procedure. A backbone curve and corresponding mode shapes are obtained and analyzed. The nonlinear finite element program is developed based on earlier linear edition and can be utilized to predict nonlinear characteristics of miniaturized quartz crystal resonators in the design process.
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