Numerical analysis of third harmonic overtone of thickness-shear vibrations in SC-Cut quartz resonators

Research on algorithms, strategies and problems associated with the numerical modeling of high frequency piezoelectric resonators was performed. The research was applied principally to SC-cut quartz crystal resonators vibrating at the third overtone of thickness shear mode and to piezoelectric laminated plates vibrating at the fundamental thickness shear mode. Finite elements using high frequency piezoelectric plate equations of motion were implemented in computer codes. Scientific visualization techniques of high frequency modes of vibration were performed. Algorithms for efficient storage of mass and piezoelectric stiffness matrices were proposed. Algorithms for the calculation of eigenpairs in the piezoelectric eigenvalue matrix problem were proposed. These algorithms reduced the memory requirement and computational time for large scale piezoelectric eigenvalue matrix problem by approximately two orders of magnitude over the current methods where the electrical degrees of freedom were separated from the mechanical degrees of freedom in the global piezoelectric stiffness matrix. The proposed method interleaved the electrical and mechanical degrees of freedom.