Vibration sensitivity of a Y-cut quartz thickness-shear resonator

Incremental thickness-shear (TSh) vibrations of a Y-cut quartz crystal plate under time-harmonic biasing extensional deformations are studied using the two-dimensional equations for small fields superposed on finite biasing fields in an electroelastic plate. The incremental TSh vibrations are governed by the well known Mathieu´s equation with a time-dependent coefficient. Both free and electrically forced vibrations are studied when the frequency of the biasing deformation is much lower than that of the incremental TSh vibration. Numerical results show that the incremental TSh free vibration mode is both frequency and amplitude modulated. The forced vibration solutions show that both the static and motional capacitances become time-dependent due to the time-harmonic biasing deformation.