Thickness modes of vibration of contoured SBTC-cut quartz plates

High-performance thickness-shear quartz resonators use trapped energy modes of vibration since these modes exhibit reduced mounting losses and possible reflections of anharmonic overtones from the electrode or plate boundaries. Trapped energy modes of vibration are obtained by appropriate contouring or beveling of the plate surfaces. The effect of contouring on energy trapping depends strongly on the crystalline orientation and the selected thickness harmonic mode of vibration. The earlier work of D. S. Stevens and H. F. Tiersten (1986) on the analysis of the SC-cut contoured quartz resonators for the C-mode of vibration is extended to obtain results for any of the three thickness (A-, B-, or C-) modes of vibration of any doubly-rotated cut of quartz