Thickness-shear vibration of crystal plates under time-dependent biasing deformations

Incremental vibration of a crystal plate raider time-harmonic biasing deformations is studied using the equations for small fields superposed on finite biasing fields. It is shown that the incremental thickness-shear vibration of certain crystal plates, including Y-cut quartz and langasite plates, is governed by an equation with time dependent coefficients. The equation is reduced to the well known Mathieu equation. Approximate analytical solutions are obtained. In particular, the case when the frequency of the biasing deformations is much lower than the frequency of the incremental thickness-shear vibration is examined in detail. The thickness-shear vibration is shown to be both frequency and amplitude modulated, with the frequency modulation as a first-order effect and the amplitude modulation a second-order effect. The results are of fundamental importance to the study of the vibration sensitivity of crystal resonators.