Electrically forced thickness-shear vibration of a crystal plate under time-dependent biasing deformations

Electrically forced thickness-shear vibrations of a Y-cut quartz plate resonator under time-dependent extensional biasing vibrations are studied using the two-dimensional equations for small deformations superposed on finite biasing deformations in electroelastic plates. It is shown that the thickness-shear vibration is governed by the well known Mathieu equation with a time-dependent coefficient and a driving term. Two approximate analytical solutions are obtained. One may be called a quasi-static solution which is valid when the frequency of the biasing extensional deformation is much lower than the frequency of the thickness-shear vibration. The other is a first-order perturbation solution valid when the biasing deformations are infinitesimal. Both solutions show that the static and motional capacitances of the resonator become time-dependent with a frequency the same as that of the biasing deformation.