Resonant frequency function of thickness-shear vibrations of rectangular crystal plates

Abstract

The resonant frequencies of thickness-shear vibrations of quartz crystal plates in rectangular and circular shapes are always required in the design and manufacturing of quartz crystal resonators. As the size of quartz crystal resonators shrinks, for rectangular plates we must consider effects of both length and width for the precise calculation of resonant frequency. Starting from the three-dimensional equations of wave propagation in finite crystal plates and the general expression of vibration modes, we obtained the relations between frequency and wavenumbers. By satisfying the major boundary conditions of the dominant thickness-shear mode, three wavenumber solutions are obtained and the frequency equation is constructed. It is shown the resonant frequency of thickness-shear mode is a second-order polynomial of aspect ratios. This conforms to known results in the simplest form and is applicable to further analytical and experimental studies of the frequency equation of quartz crystal resonators.

Thickness-shear vibration analysis of circular quartz crystal plates by a differential quadrature hierarchical finite element method

“…Quartz is the most widely used crystal for resonators. A large portion of quartz crystal resonators, usually in circular or rectangular platform with all edges free [3], are operated with the so-called thickness-shear modes of a plate [4,5]. Theoretical modeling of crystal resonators using the equations of anisotropic elasticity or piezoelectricity is very complicated due to material anisotropy and electromechanical coupling [6].…”

“…Considerable efforts have been spent in analysis of piezoelectric resonator on these aspects since Mindlin [7,8]. Although accurate and reliable results have been obtained through the efforts of many researchers for several decades [3], with emerging problems and practical needs, the sizes of quartz crystal resonators are shrinking fast and their frequencies are increasing rapidly. Therefore, precise analysis of thickness-shear vibrations of crystal plates is even more important now.…”

“…Wang et al [23] and Yong et al [24] have applied FEA to thickness-shear vibration of circular crystal plates. However, the computational cost is very expensive for numerical procedures [3].…”

Використання Матрично-Операторного Методу Для Дослідження Товщинно-Зсувних Коливань Кварцових Резонаторів

“…У роботі [4] запропонована математична модель другого класу, визначена резонансна частота TSh коливань. У роботі [5], на відміну від [6], врахований вплив відносного масонавантаження та отримані співвідношення для резонансних частот коливання ПЕ, встановлений факт зменшення резонансної частоти TSh коливань в умовах збільшення відносного масонавантаження. У роботі автор [7], ґрунтуючись на лінійній апроксимації TSh коливань, запропонованій Міндліном, отримав рівняння для резонансної частоти у випадку квадратичного закону розподілу відносного масонавантаження уздовж довжини і ширини ПЕ.…”