The Finite Element Analysis Of High Frequency Vibrations Of SC-cut Quartz Crystal Plates With Mindlin Plate Theory

Quartz crystal resonators are widely used in various electronic devices, for the role offrequency control and detection in circuits. SC-cut quartz crystal resonators are mainly used inthe field of high precision devices, because of its superior stability. For this reason, the finiteelement method is used in the analysis of SC-cut quartz crystal resonators. Natural andsymmetric methods were used in the correction of Mindlin’s first-order and third-order plateequations. Material, mass, and stiffness effects are considered in the correction. Free vibrationsof SC-cut quartz crystal plates and forced vibrations of SC-cut quartz crystal resonators are studiedwith the corrected Mindlin plate equations.We started with the basic theory of wave propagation of piezoelectric solids, deduced thefundamental equations of SC-cut quartz crystal plates with Mindlin plate theory, and obtained theexact frequency of thickness-shear vibrations of infinite SC-cut quartz crystal plates. Then thefirst-order and third-order Mindlin plate equations were corrected with natural and symmetrictechniques, and then we used exact frequency equation of electroded SC-cut quartz crystal plates toobtain the exact frequencies of electroded SC-cut quartz crystal plates. With the exact frequencies,we corrected Mindlin’s first-order and third-order equations, mass and stiffness effects of gold,silver and aluminium were considered in the correction. After obtained the correction factors, wefitted them functions by polynomials of mass ratio. With corrected equations, we analyzed andobtained frequency spectra and frequencies of thickness-shear of SC-cut quartz crystal plates.Finally, forced vibrations of SC-cut quartz crystal resonators were analyzed with piezoelectriceffect considered in Mindlin plate equations, and we obtained the capacitance ratio curves atdifferent driving frequencies.