Simulating Frequency Nonlinearities in Quartz Resonators at High Temperature and Pressure

In order to facilitate the design of quartz resonators as sensors for the oil and gas industry, the current work focuses on the development of a three-dimensional finite element model to calculate the frequency change of anisotropic quartz resonators associated with the application of temperature and pressure. In doing so, the simulation employs the incremental linear field equations for superimposed small vibrations onto nonlinear thermoelastic stressed media, as given by Lee and Yong [1]. This method involves solving geometric and material nonlinearities for the both the thermal stress and piezoelectric models in COMSOL. The thickness-shear mode frequency response of the model was benchmarked to experimental sensor data with temperature ranging from 50°C to 200°C (in 25°C increments) and pressure from 14 psi to 20,000 psi (with 2,000 psi increments, approximately). The normalized frequency response to the change in external pressure matched very well with experimental data at lower temperatures, and by the same token, the temperature-frequency response matched the experimental trend well for lower pressures. The study found, however, that applying high temperature and pressure simultaneously leads to considerable error in the frequency response. It is hypothesized that the increased error at extreme conditions is due to the current lack of certain material properties of quartz, known as the temperature derivatives of the third-order elastic coefficients.
Keywords: quartz, temperature, pressure, sensor, nonlinear