Numerical algorithms and results for SC-cut quartz plates vibrating at the third harmonic overtone of thickness shear

Finite element matrix equations, derived from two-dimensional, piezoelectric high frequency plate theory proposed by Lee, Syngellakis, and Hou (1987), are solved to study the vibrational behavior of the third overtone of thickness shear in square and circular SC-cut quartz resonators. The mass-loading and electric effects of electrodes are included. A perturbation method is employed to calculate the piezoelectric resonant frequencies which reduces the memory requirements and computational time significantly. A new storage scheme is introduced which reduces memory requirements for mass matrix by about 90% over that of the envelope storage scheme. Substructure techniques are used in eigenvalue calculation to save storage. Resonant frequency and the mode shapes of the harmonic third overtone thickness shear vibrations for rectangular and circular plates are calculated. A pure third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the zeroth, first and second order displacements and the third order thickness shear displacement is noted. The magnitudes of the lower order displacements are found to be about two orders smaller than that of the third overtone thickness shear displacement