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 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 which reduces the memory requirements and computational time significantly is employed to calculate the piezoelectric resonant frequencies. 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 square and circular plates are calculated. A predominant third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the third order thickness shear displacement and the zeroth-, first-, and second-order displacements 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.