Frequency Shift Of A Quartz Crystal Resonator Induced By An Attached Two-dimensional Structure Layer

We establish a two-dimensional model about the attached structure configuration tostudy the dynamic characteristics of a compound QCR system, while the QCR systemconsists of a QCR and a layer of attached structure units. The attached units are simplifiedinto hemispherical bodies or frustum-shaped cones, respectively. Using the establishedmodel, we study the influence of shear modulus and configuration dimension onfrequency shift of QCR system. A frequency-dependent equivalent mass ratio is putforward, which may be either positive or negative. The physical implication of thepositive/negative definiteness of equivalent mass ratio is investigated in detail.Numerical results show that the equivalent mass alternately become positive or negativewith gradually change of shear modulus and radius of each material unit. It is also foundthat there exists jump transition between two adjacent cycles. Further analysis indicatesthat one natural frequency of the attached unit is just equal to the vibration frequency ofQCR, which implies resonance. When the shear stress and the displacement of QCRsurface are with the same phases, the equivalent mass ratio is positive, that is, the attachedlayer increases the system inertia and reduces the resonance frequency of the QCR system.On the other hand, when the shear stress and the displacement of QCR surface are with theoppisite phases, the equivalent mass ratio is negetave, that is, the attached layer decreasesthe system inertia and increases the resonance frequency of QCR system. In addition, it isalso found that when the shear modules of the attached layer become larger enough, theattached layer decreases the frequency only. This is because for a given size of attachedunits, the fundamental natural frequency of each unit is positively greater than thevibration frequency of the resonator when the shear modulus increases to a certain degree.Under that case, resonance does not occur so that the equivalent mass ratio presentspositive effect. Thus, the resonant frequency of the QCR system is always decreased andclose to the solution of Sauerbrey equation.The solutions are useful in design of QCR system and analysis of frequency stability ofboth quartz resonators and acoustic wave sensors.