Finite element analysis of nonlinear thickness-shear vibrations of AT-cut quartz crystal plates

The nonlinear finite element analysis is performed with the nonlinear Mindlin plate theory, which is further simplified to have only the thickness-shear and flexural modes to reduce the complicated couplings due to the consideration of material nonlinearity and higher-order strain components. The 2D nonlinear equations with two variables are implemented so the problem will have a smaller size in comparison with the 3D approach. General procedure of nonlinear finite element analysis based on the iterative method is implemented. The finite element program is in parallel implementation with advanced features such as the sparse matrix handling and linear algebra library in public domain. The program is developed and tested on a Linux cluster to enable fast solution of large scale problems. The solutions are given in displacements to make comparison with known linear solutions for verification and validation, but they can also be used for forced vibrations and future calculation of resonator electrical properties. In addition to essential results of finite element analysis in terms of vibration frequency and displacement solutions, we can extend the program to analyze the resonator behavior under driving voltage to explain many important behaviors like derive-level dependence and other nonlinear properties. These analytical capabilities will expand current features of finite element program and provide efficient tools for the nonlinear studies of quartz crystal resonators. Noting the finite element analysis of quartz crystal resonators has been making great contributions to the design and improvement with the fast shrinkage of resonator size and raised precision requirements, the full advantage of the finite element analysis can be taken if electrical parameters and performance behavior can be predicted with the improved analytical model and consideration of nonlinear material properties and field coupling. The current approach based on the nonlinear theory will meet these objectives since the advantage of the finite element analysis on parallel platforms have been well understood and widely implemented.