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

This paper is concerned with analyzing the resonance frequency change of a quartz resonator under initial thermal and mechanical biases. the change in the resonance frequency of the resonator subjected to a pair of diametrical forces is called the forcefrequency effect and is quantified by force-frequency coefficient. the experimental data suggest that this coefficient may change due to homogenous thermal biases on the quartz crystal. in this article, we investigate the effect of initial homogenous thermal strains on the force-frequency effect. 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°c using the experimental results. the model leads to a better understanding of quartz crystal behavior which increases accuracy of pressure sensors in applications such as down-hole pressure measurement in the oil and gas industry.