Frequency Shift Of A Quartz Resonator Under The Thickness-shear Modes Caused By Surface Micro-beam Arrays

Two-dimensional models about the attached substance or structure to study the dynamic characteristics of a compound quartz crystal resonator(QCR) system are established in this paper, while the compound system consists of a quartz crystal resonator and arrays of micro-beams. The governing equations of micro-beams are derived from the Euler-Bernoulli beam theory, Timoshenko beam theory and the modified Euler beam theory when taking into account the first-order strain gradient. The influence of physical and geometrical parameters on frequency shift of QCR system is studied by using the established model. Some useful results are obtained, and finally the physical implication is analyzed in detail.Numerical results show that the resonance frequency of the composite QCR system becomes positive or negative with gradually change of physical and geometrical parameters of each beam unit. And jump point will occur between two adjacent cycles. By further solving the natural frequency of the jump points of the attached beams we can find that, the natural frequency of the beams is just equal to the vibration frequency of QCR, which implies resonance. Besides, it can be also found that, when the elastic modules of the attached micro beams become large enough, the periodic character disappears and the attached layer only decreases the frequency. The reason for this phenomenon is that, when fixed the other parameters and the elastic modules increases to a certain degree, the natural frequency of the micro beams is positively greater than the vibration frequency of the resonator, and the resonance can not occur. In this case, the attached micro beam arrays can be equivalent to a rigid quality layers. So, the resonant frequency of the system is always decreased and the solution satisfies the Sauerbrey equation. This solution examines the validity of the model established in this paper. And in high-order vibration modes, the result solved by Sauerbrey model will no longer be correct. So, this is a supplement to Sauerbrey model. Then we analyze the effect of shear deformation on frequency shift between surface micro beams and quartz resonator. Shear deformation has certain influence on frequency shift by compared with the previous results, and the influence is more obvious in the higher order vibration modes. The strain gradient theory is a newly-developed theory to explain the size effect phenomenon of materials in the micro scales. The effects of first-order strain gradient on the vibration characteristics between surface micro beams and quartz resonator is discussed in the end of the paper. It can be found from the results that the strain gradient leads to a little left or right shift in the frequency shift curves, and a stronger strain gradient produces a larger left or right shift. The reason for these changes is that the strain gradient effect leads to an increase in beam bending stiffness.Through the analysis of this paper, we have a better understanding about the vibration characteristics of the quartz resonator. And the obtained results have important significance in resonator design and applications.