An estimate of the second-order in-plane acceleration sensitivity of a Y-cut Quartz thickness-shear resonator

We perform a theoretical analysis of the secondorder in-plane acceleration sensitivity of a Y-cut quartz thickness-shear mode resonator. The second-order nonlinear theory of elasticity for anisotropic crystals is used to determine the biasing fields in the resonator under in-plane acceleration. The acceleration-induced frequency shift is determined from a perturbation analysis based on the plate equations for small-amplitude vibrations superposed on a finite bias. We show that, whereas the first-order acceleration-induced frequency shift is zero for a structurally symmetric resonator under in-plane acceleration, the second-order frequency shift is nonzero and is quadratic in the acceleration. As the fourth-order nonlinear elastic constants of quartz have never been measured, we can only estimate the magnitude of the second-order frequency shift. For a particular case of interest, we find Δω/ω₀~10^-18, 10^-16, and 10^-14 when the acceleration is 1, 10, and 100 g, respectively.