Frequency shift of a crystal quartz resonator in thickness-shear modes induced by an array of hemispherical material units

A two-dimensional model was established to study the dynamic characteristics of a quartz crystal resonator with the upper surface covered by an array of hemispherical material units. A frequency-dependent equivalent mass ratio was proposed to simulate the effect of the covered units on frequency shift of the resonator system. It was found that the equivalent mass ratio alternately becomes positive or negative with change of shear modulus and radius of each material unit, which indicates that the equivalent mass ratio is strongly related to the vibration mode of the covered loadings. The further numerical results show the cyclical feature in the relationship of frequency shift and shear modulus/radius as expected. The solutions are useful in the analysis of frequency stability of quartz resonators and acoustic wave sensors.