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Acta Mechanica Solida Sinica

, Volume 26, Issue 3, pp 245–254 | Cite as

Thickness-Shear and Thickness-Twist Vibrations of Circular AT-Cut Quartz Resonators

  • Huijing He
  • Jiashi Yang
  • Qing Jiang
Article

Abstract

Exact solutions for free vibration frequencies and modes are obtained for thickness-shear and thickness-twist vibrations of unelectroded circular AT-cut quartz plates governed by the two-dimensional scalar differential equation derived by Tiersten and Smythe. Comparisons are made with experimental results and the widely-used perturbation solution by Tiersten and Smythe under the assumption of weak in-plane anisotropy. Our solution is found to be much closer to the experimental results than the perturbation solution. For the frequency of the fundamental thickness-shear mode, the error of the perturbation method is 0.4549%, significant in resonator applications.

Key Words

plate linear vibration resonance 

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References

  1. 1.
    Koga, I., Thickness vibrations of piezoelectric oscillating crystals. Physics, 1932, 3(2): 70–80.CrossRefGoogle Scholar
  2. 2.
    Tiersten, H.F., Thickness vibrations of piezoelectric plates. Journal of the Acoustical Society of America, 1963, 35(1): 53–58.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Mindlin, R.D., High frequency vibrations of crystal plates. Quarterly of Applied Mathematics, 1961, 19(1): 51–61.CrossRefGoogle Scholar
  4. 4.
    Tiersten, H.F. and Mindlin, R.D., Forced vibrations of piezoelectric crystal plates. Quarterly of Applied Mathematics, 1962, 20(2): 107–119.CrossRefGoogle Scholar
  5. 5.
    Mindlin, R.D., High frequency vibrations of piezoelectric crystal plates. International Journal of Solids and Structures, 1972, 8(7): 895–906.CrossRefGoogle Scholar
  6. 6.
    Mindlin, R.D. and Lee, P.C.Y., Thickness-shear and flexural vibrations of partially plated, crystal plates. International Journal of Solids and Structures, 1966, 2(1): 125–139.CrossRefGoogle Scholar
  7. 7.
    Wang, J. and Zhao, W.H., The determination of the optimal length of crystal blanks in quartz crystal resonators. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2005, 52(11): 2023–2030.CrossRefGoogle Scholar
  8. 8.
    Wang, J., Zhao, W.H. and Du, J.K., The determination of electrical parameters of quartz crystal resonators with the consideration of dissipation. Ultrasonics, 2006, 44(Suppl.1): E869–E873.CrossRefGoogle Scholar
  9. 9.
    Zhang, C.L., Chen, W.Q. and Yang, J.S., Electrically forced vibration of a rectangular piezoelectric plate of monoclinic crystals. International Journal of Applied Electromagnetics and mechanics, 2009, 31(4): 207–218.Google Scholar
  10. 10.
    Wang, J.N., Hu, Y.T. and Yang, J.S., Frequency spectra of at-cut quartz plates with electrodes of unequal thickness. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2010, 57(5): 1146–1151.CrossRefGoogle Scholar
  11. 11.
    Chen, G.J., Wu, R.X., Wang, J., Du, J.K. and Yang, J.S., Five-mode frequency spectra of x3-dependent modes in AT-cut quartz resonators. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2012, 59(4): 811–816.CrossRefGoogle Scholar
  12. 12.
    Wang, J. and Yang, J.S., Higher-order theories of piezoelectric plates and applications. Applied Mechanics Reviews, 2000, 53(4): 87–99.CrossRefGoogle Scholar
  13. 13.
    Tiersten, H.F. and Smythe, R.C., Coupled thickness-shear and thickness-twist vibrations of unelectroded AT-cut quartz plates. Journal of the Acoustical Society of America, 1985, 78(5): 1684–1689.CrossRefGoogle Scholar
  14. 14.
    Stevens, D.S. and Tiersten, H.F., An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation. Journal of the Acoustical Society of America, 1986, 79(6): 1811–1826.CrossRefGoogle Scholar
  15. 15.
    EerNisse, E.P., Analysis of thickness modes of contoured, doubly rotated, quartz resonators. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2001, 48(5): 1351–1361.CrossRefGoogle Scholar
  16. 16.
    Tiersten, H.F. and Smythe, R.C., An analysis of contoured crystal resonators operating in overtones of coupled thickness shear and thickness twist. Journal of the Acoustical Society of America, 1979, 65(6): 1455–1460.CrossRefGoogle Scholar
  17. 17.
    Stevens, D.S., Tiersten, H.F. and Sinha, B.K., Temperature dependence of the resonance frequency of electroded contoured AT-cut quartz crystal resonators. Journal of Applied Physics, 1983, 54(4): 1704–1716.CrossRefGoogle Scholar
  18. 18.
    Tiersten, H.F. and Shick, D.V., On the normal acceleration sensitivity of contoured quartz resonators rigidly supported along rectangular edges. Journal of Applied Physics, 1990, 67(1): 60–67.CrossRefGoogle Scholar
  19. 19.
    Tiersten, H.F. and Zhou, Y.S., On the normal acceleration sensitivity of contoured quartz resonators with the mode shape displaced with respect to rectangular supports. Journal of Applied Physics, 1991, 69(5): 2862–2970.CrossRefGoogle Scholar
  20. 20.
    Tiersten, H.F. and Zhou, Y.S., On the normal acceleration sensitivity of contoured quartz resonators stiffened by quartz cover plates supported by clips. Journal of Applied Physics, 1992, 72(4): 1244–1254.CrossRefGoogle Scholar
  21. 21.
    Tiersten, H.F. and Zhou, Y.S., On the in-plane acceleration sensitivity of contoured quartz resonators supported along rectangular edges. Journal of Applied Physics, 1991, 70(9): 4708–4714.CrossRefGoogle Scholar
  22. 22.
    Tiersten, H.F. and Zhou, Y.S., The increase in the in-plane acceleration sensitivity of the plano-convex resonator resulting from its thickness asymmetry. Journal of Applied Physics, 1992, 71(10): 4684–4692.CrossRefGoogle Scholar
  23. 23.
    Zhou, Y.S. and Tiersten, H.F., In-plane acceleration sensitivity of contoured quartz resonators stiffened by quartz cover plates supported by clips. Journal of Applied Physics, 1993, 74(12): 7067–7077.CrossRefGoogle Scholar
  24. 24.
    Tiersten, H.F. and Zhou, Y.S., Transversely varying thickness modes in quartz resonators with beveled cylindrical edges. Journal of Applied Physics, 1994, 76(11): 7201–7208.CrossRefGoogle Scholar
  25. 25.
    Yang, J.S. and Tiersten, H.F., An analysis of contoured quartz resonators with beveled cylindrical edges. In: Proceedings of the IEEE International Frequency Control Symposium 1995, Institute of Electrical and Electronics Engineers, 1995: 727–739.Google Scholar
  26. 26.
    Yang, J.S. and Tiersten, H.F., The influence of the free edge on the vibration characteristics of a contoured, beveled cylindrical quartz resonator. In: Proceedings of the IEEE International Frequency Control Symposium 1996, Institute of Electrical and Electronics Engineers, 1996: 657–664.Google Scholar
  27. 27.
    Huang, L.D., Tiersten, H.F. and Yang, J.S., An analysis of contoured quartz resonator with beveled cylindrical edges using the correct variation of thickness. In: Proceedings of the IEEE International Frequency Control Symposium 1997, Institute of Electrical and Electronics Engineers, 1997: 668–676.Google Scholar
  28. 28.
    Yang, J.S., An analysis of partially electroded, contoured quartz resonators with beveled cylindrical edges. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2007, 54(11): 2407–2409.CrossRefGoogle Scholar
  29. 29.
    McLachlan, N.W., Theory and Application of Mathieu Functions. Oxford: Oxford Press, 1951.zbMATHGoogle Scholar
  30. 30.
    Tiersten, H.F., Linear Piezoelectric Plate Vibrations. New York: Plenum, 1969.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2013

Authors and Affiliations

  1. 1.Department of Mechanical and Materials EngineeringUniversity of Nebraska-LincolnLincolnUSA
  2. 2.School of EngineeringSun Yat-Sen UniversityGuangzhouChina
  3. 3.Guangzhou Ande Biotechnology Co. Ltd.GuangzhouChina

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