High-frequency vibrations of quartz crystals subject to initial thermo-mechanical bias

When a quartz resonator is exposed to an initial mechanical bias, its resonance frequency changes. This change in the resonance frequency is called the forcefrequency effect and isquantified by force frequency coefficient. The experimental data suggest that force frequency coefficient may change due to homogenous thermal biases on the quartzcrystal. In this article, we investigate the effect of initial homogeneous thermal strains on the force frequency effect. Then we derive an explicit formula that predicts the resonance frequency shift of the thickness shear mode of AT-cut quartz crystals. The mathematical model is validated on circular AT-cut quartz crystals at 78 using the experimental results. The model leads to a better understanding of quartz crystal behavior which increases accuracy of pressure sensors in applications suchas down-hole pressure measurement in the oil and gas industry.