Thickness-shear and thickness-twist vibrations of an AT-cut quartz mesa resonator

We study thickness-shear and thickness-twist vibrations of an AT-cut quartz plate mesa resonator with stepped thickness. The equations of anisotropic elasticity are used with the omission of the small elastic constant c(56). An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and energy trapping are calculated and examined. The solution shows that a mesa resonator exhibits strong energy trapping of thickness-shear and thickness-twist modes, and that the trapping is sensitive to some of the structural parameters of the resonator.