Thickness-shear vibration of an AT-cut quartz resonator with a hyperbolic contour

A theoretical analysis is performed on thicknessshear vibrations of a contoured AT-cut quartz resonator with a hyperbolic thickness variation using the Legendre equation and hypergeometric function. Based on the solution, resonant frequencies and modes are calculated. Strong energy trapping of the modes is observed. The effects of the parameters of the hyperbolic contour on resonant frequencies and modes are examined. A comparison with the conventional contoured resonator in the literature with a quadratic thickness variation is made. The behaviors of the two types of resonators are qualitatively similar, with small but significant quantitative differences.

[1]  Jiashi Yang,et al.  An analysis of partially electroded, contoured, quartz resonators with beveled cylindrical edges , 2007, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[2]  Jiashi Yang,et al.  Shear horizontal vibrations of a piezoelectric/ferroelectric wedge , 2004 .

[3]  R. D. Mindlin,et al.  Thickness‐Shear and Flexural Vibrations of Contoured Crystal Plates , 1954 .

[4]  H. Tiersten,et al.  Transversely varying thickness modes in quartz resonators with beveled cylindrical edges , 1994 .

[5]  H. F. Tiersten,et al.  An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation , 1986 .

[6]  J. Bleustein.  Thickness-twist and face-shear vibrations of a contoured crystal plate , 1966 .

[7]  P. C. Lee,et al.  Vibrations of Contoured and Partially Plated, Contoured, Rectangular, AT‐Cut Quartz Plates , 1969 .

[8]  Y. Watanabe,et al.  Analysis of 3-D vibrations of rectangular AT-cut quartz plates with a bi-mesa structure , 2001, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[9]  J. Wang,et al.  Piezoelectrically forced thickness‐shear and flexural vibrations of contoured quartz resonators , 1996 .

[10]  C. J. Wilson.  Vibration modes of AT-cut convex quartz resonators , 1974 .

[11]  Ji Wang,et al.  The effect of cubically varying contours on the thickness-shear and flexural vibrations of quartz plates , 1996, 1996 IEEE Ultrasonics Symposium. Proceedings.

[12]  D. Vangheluwe.  The Frequency and Motional Capacitance of Partial Contoured Crystal Resonators , 1980 .

[13]  H. Tiersten.  The influence of an asymmetric air gap on the vibration characteristics of quartz trapped energy resonators , 1995, Proceedings of the 1995 IEEE International Frequency Control Symposium (49th Annual Symposium).

[14]  Ji Wang,et al.  THICKNESS-SHEAR AND FLEXURAL VIBRATIONS OF LINEARLY CONTOURED CRYSTAL STRIPS WITH MULTIPRECISION COMPUTATION , 1999 .

[15]  H. F. Tiersten,et al.  An analysis of contoured crystal resonators operating in overtones of coupled thickness shear and thickness twist , 1979 .

[16]  H. Tiersten,et al.  On the Effect of Edge Boundaries on Nearly Trapped Vibrations of Contoured Rectangular Quartz Resonators , 1986, IEEE 1986 Ultrasonics Symposium.

[17]  Jiashi Yang,et al.  Vibrations of an isotropic elastic wedge , 2005, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[18]  H. Tiersten,et al.  An analysis of contoured quartz resonators with beveled cylindrical edges , 1995, Proceedings of the 1995 IEEE International Frequency Control Symposium (49th Annual Symposium).

[19]  H. Tiersten,et al.  Transversely varying thickness modes in trapped energy resonators with shallow and beveled contours , 1996 .

[20]  Pcy Lee,et al.  Thickness‐shear and flexural vibrations of contoured crystal strip resonators , 1996 .