Thickness-shear and thickness-twist vibrations of rectangular quartz crystal plates with nonuniform thickness

References

• H.F. Tiersten, Thickness vibrations of piezoelectric plates, J. Acoust. Soc. Am., vol. 35, pp. 53–58, 1963.
   Web of Science ®Google Scholar
• H.F. Tiersten, A corrected modal representation of thickness vibrations in quartz plates and its influence on the transversely varying case, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 50, pp. 1436–1443, 2003.
   PubMed Web of Science ®Google Scholar
• H.F. Tiersten and R.C. Smythe, An analysis of contoured crystal resonators operating in overtones of coupled thickness shear and thickness twist, J. Acoust. Soc. Am., vol. 65, pp. 1455–1460, 1979.
   Web of Science ®Google Scholar
• A.V. Apostolov and S.H. Slavov, Frequency spectrum and modes of vibration in circular, convex AT-cut bevelled-design quartz resonators, Appl. Phys. A, vol. 29, pp. 33–37, 1982.
   Web of Science ®Google Scholar
• D. Salt, Quartz Crystal Devices, Van Nostrand Reinhold, Wokingham, Berkshire, UK, p. 55.
   Google Scholar
• S.H. Slavov, Modes of vibration, motion inductance, and resonance interval of circular, convex AT-cut bevelled design trapped energy quartz resonators, Appl. Phys. A, vol. 40, pp. 59–65, 1986.
   Web of Science ®Google Scholar
• S.H. Slavov, Equivalent resonance radius of contoured AT-cut quartz resonators, Appl. Phys. A, vol. 43, pp. 111–116, 1987.
   Web of Science ®Google Scholar
• S.H. Slavov and D.G. Ouroushev, Investigation of harmonic modes of vibration in spherically contoured AT-cut quartz resonators by the degenerate hypergeometric function, J. Phys. D: Appl. Phys., vol. 23, pp. 434–438, 1990.
   Web of Science ®Google Scholar
• H.F. Tiersten and Y.S. Zhou, Transversely varying thickness modes in quartz resonators with beveled cylindrical edges, J. Appl. Phys., vol. 76, pp. 7201–7208, 1994.
   Web of Science ®Google Scholar
• J.S. Yang and H.F. Tiersten, Effects of free edge on the vibration characteristics of a contoured, beveled quartz resonator, Int. J. Appl. Electromagnet. Mech., vol. 46, pp. 649–661, 2014.
   Web of Science ®Google Scholar
• H.F. Tiersten, B.J. Lwo, and B. Dulmet, Transversely varying thickness modes in trapped energy resonators with shallow and beveled contours, J. Appl. Phys., vol. 80, pp. 1037–1046, 1996.
   Web of Science ®Google Scholar
• J.J. Shi, C.Y. Fan, M.H. Zhao, and J.S. Yang, Trapped thickness-shear modes in a contoured, partially electroded AT-cut quartz resonator, Eur. Phys. J. Appl. Phys., vol. 69, art. ID 10302, 2015.
   Web of Science ®Google Scholar
• H.F. Tiersten, Analysis of intermodulation in thickness-shear and trapped energy resonators, J. Acoust. Soc. Am., vol. 57, pp. 667–681, 1975.
   Web of Science ®Google Scholar
• H.F. Tiersten, Analysis of trapped-energy resonators operating in overtones of coupled thickness shear and thickness twist, J. Acoust. Soc. Am., vol. 59, pp. 879–888, 1976.
   Web of Science ®Google Scholar
• H.F. Tiersten, Analysis of overtone modes in monolithic crystal filters, J. Acoust. Soc. Am., vol. 62, pp. 1424–1430, 1977.
   Web of Science ®Google Scholar
• H.F. Tiersten and R.D. Mindlin, Forced vibrations of piezoelectric crystal plates, Q. Appl. Math., vol. 20, pp. 107–119, 1962.
   Google Scholar
• R.D. Mindlin, High frequency vibrations of piezoelectric crystal plates, Int. J. Solids Struct., vol. 8, pp. 895–906, 1972.
   Google Scholar
• N. Bugdayci and D.B. Bogy, A two-dimensional theory for piezoelectric layers used in electro-mechanical transducers, Int. J. Solids Struct., vol. 17, pp. 1159–1178, 1981.
   Web of Science ®Google Scholar
• P.C.Y. Lee, S. Syngellakis, and J.P. Hou, A two-dimensional theory for high-frequency vibrations of piezoelectric crystal plates with or without electrodes, J. Appl. Phys., vol. 61, pp. 1249–1262, 1987.
   Web of Science ®Google Scholar
• P.C.Y. Lee, J.D. Yu, and W.S. Lin, A new two-dimensional theory for vibrations of piezoelectric crystal plates with electroded faces, J. Appl. Phys., vol. 83, pp. 1213–1223, 1998.
   Web of Science ®Google Scholar
• Z.N. Zhao, Z.H. Qian, B. Wang, and J.S. Yang, Thickness-shear and thickness-twist modes in an AT-cut quartz acoustic wave filter, Ultrasonics, vol. 58, pp. 1–5, 2015.
   PubMed Web of Science ®Google Scholar
• H.J. He, J.S. Yang, and J.A. Kosinski, Scalar differential equation for slowly-varying thickness-shear modes in AT-cut quartz resonators with surface impedance for acoustic wave sensor application, IEEE Sens. J., vol. 13, pp. 4349–4355, 2013.
   Web of Science ®Google Scholar
• P. Li, F. Jin, and J.S. Yang, Thickness-shear vibration of an AT-cut quartz resonator with a hyperbolic contour, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 59, pp. 1006–1012, 2012.
   PubMed Web of Science ®Google Scholar
• H.F. Tiersten and R.C. Smythe, Coupled thickness-shear and thickness-twist resonances in unelectroded rectangular and circular AT-cut quartz plates, Proceedings of the 35th Annual Frequency Control Symposium, pp. 230–236, May 27–29, Philadelphia, PA, 1981.
   Google Scholar
• C.J. Wilson, Vibration modes of AT-cut convex quartz resonators, J. Phys. D: Appl. Phys., vol. 7, pp. 2449–2455, 1974.
   Web of Science ®Google Scholar
• R. Bechmann, Elastic and piezoelectric constants of alpha-quartz, Phys. Rev., vol. 110, pp. 1060–1061, 1958.
   Web of Science ®Google Scholar