Vibration mode identification and coupling assessment with the Mindlin plate equations and measurements in a quartz crystal plate

In the design of quartz crystal resonators, it is always required to have complete information about vibration couplings in the vicinity of thickness-shear mode for the suppression of spurious modes which always cause performance degradation as one of the most serious challenges in product development. The couplings of vibration modes are caused by the finite size of plates and material constants, while the effect is shown as occurrences of vibration near the primary resonance of thickness-shear vibration. The analysis of high frequency vibrations of quartz crystal plates has been done with the Mindlin plate theory with accurate solutions of the frequency and mode shapes for the identification of couplings of modes in relatively complicated frequency spectra. We have been using with the first-order Mindlin plate equations based on straight-crested wave assumptions for the frequency and mode shape solutions in both plane directions. We found the majority of spurious modes in the neighborhood of thickness-shear mode have been identified, then we can predict their movements with variations of plate and electrodes configurations and temperature changes. By measuring strong couplings of vibration modes in the vicinity of thickness-shear frequency of a crystal plate, we can obtain complete frequency spectra within the given band. Our calculations from a finite set of vibrations modes from truncated Mindlin plate equations are demonstrating the effectiveness in the analysis because the major couplings have been predicted in good agreement with measurements. It offers a promising approach for the improvement of current design procedure based on empirical knowledge.