Abstract
As a preliminary attempt for the study on nonlinear vibrations of a finite crystal plate, the thickness-shear mode of an infinite and isotropic plate is investigated. By including nonlinear constitutive relations and strain components, we have established nonlinear equations of thickness-shear vibrations. Through further assuming the mode shape of linear vibrations, we utilized the standard Galerkin approximation to obtain a nonlinear ordinary differential equation depending only on time. We solved this nonlinear equation and obtained its amplitude–frequency relation by the homotopy analysis method (HAM). The accuracy of the present results is shown by comparison between our results and the perturbation method. Numerical results show that the homotopy analysis solutions can be adjusted to improve the accuracy. These equations and results are useful in verifying the available methods and improving our further solution strategy for the coupled nonlinear vibrations of finite piezoelectric plates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mindlin, R.D.: Thickness-shear and flexural vibrations of crystal plates. J. Appl. Phys. 22(3), 316–323 (1951)
Tiersten, H.F.: Linear Piezoelectric Plate Vibrations. Plenum, New York (1969)
Yang, J.S.: Analysis of ceramic thickness shear piezoelectric gyroscopes. J. Acoust. Soc. Am. 102(6), 3542–3548 (1997)
Reed, C.E., Kanazawa, K.K., Kaufman, J.H.: Physical description of a viscoelastically loaded AT-cut quartz resonator. J. Appl. Phys. 68(5), 1993–2001 (1990)
Mindlin, R.D. (Yang, J.S., editor): An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. World Scientific, Singapore (2006)
Wang, J., Yang, J.S.: Higher-order theories of piezoelectric plates and applications. Appl. Mech. Rev. 53(4), 87–99 (2000)
Wang, J., Zhao, W.H.: The determination of the optimal length of crystal blanks in quartz crystal resonators. IEEE Trans. Ultras. Ferroelectr. Freq. Control 52(10), 2023–2030 (2005)
Wang, J., Yong, Y.-K., Imai, T.: Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory. Int. J. Solids Struct. 36(13), 2303–2319 (1999)
Lee, P.C.Y., Yong, Y.-K.: Frequency–temperature behavior of thickness vibrations of double rotated quartz plates affected by plate dimensions and orientations. J. Appl. Phys. 60(7), 2327–2342 (1986)
Yang, J.S.: Two-dimensional equations for electroelastic plates with relatively large in-plane shear deformation and nonlinear mode coupling in resonant piezoelectric devices. Acta Mech. 196(1–2), 103–111 (2008)
Wang, J., Wu, R.X., Du, J.K., Huang, D.J.: Nonlinear Mindlin plate equations for the thickness-shear vibrations of crystal plates. In: Proceeding of the 2008 Symposium on Piezoelectricity, Acoustic Waves and Device Applications, pp. 87–92 (2008)
Yang, J. S., Guo, S.H., Effects of nonlinear elastic constants on electromechanical coupling factors. IEEE Trans. Ultras. Ferroelectr. Freq. Control 52(12), 2303–2305 (2005)
Yang, Z.T., Hu, Y.T., Wang, J., Yang, J.S.: Nonlinear coupling between thickness-shear and thickness-stretch modes in a rotated Y-cut quartz resonator. IEEE Trans. Ultras. Ferroelectr. Freq. Control 56(1), 220–224 (2009)
Wang, J., Wu, R. X., Yong, Y.-K., Du, J. K., et al.: An analysis of vibrations of quartz crystal plates with nonlinear Mindlin plate equations. In: Proceeding of the Joint Conference of 2009 IEEE International Frequency Control Symposium and the European Frequency and Time Forum, pp. 450–454 (2009)
Patel, M.S., Yong, Y.-K., Tanaka, M.: Drive level dependency in quartz resonators. Int. J. Solids Struct. 46(9), 1856–1871 (2009)
Yang, J.S.: Equations for the extension and flexure of electroelastic plates under strong electric fields. Int. J. Solids Struct. 36, 3171–3192 (1999)
Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton (2003)
Wang, J., Chen, J.K., Liao, S.J.: An explicit solution of the large deformation of a cantilever beam under point load at the free tip. J. Comput. Appl. Math. 202(2), 320–330 (2008)
Gao, L.M., Wang, J., Zhong, Z., Du, J.K.: An analysis of surface acoustic wave propagation in functionally graded plates with homotopy analysis method. Acta Mech. 208, 249–258 (2009)
Xu, H., Cang, J.: Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett., A 372, 1250–1255 (2008)
Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett., A 360, 109–113 (2006)
Wu, Y.Y., Cheung, K.F.: Homotopy solution for nonlinear differential equations in wave propagation problems. Wave Motion 46, 1–14 (2009)
Hiki, Y., Higher order elastic constants of solids. Ann. Rev. Mat. Sci. 11, 51–73 (1981)
Meurer, T., Qu, J., Jacobs, L.J.: Wave propagation in nonlinear and hysteretic media—a numerical study. Int. J. Solids Struct. 39, 5589–5614 (2002)
Abd-alla, A., Maugin, G.A.: Nonlinear phenomena in magnetostrictive elastic resonators. Int. J. Eng. Sci. 27(12), 1613–1619 (1989)
Wang, J., Wu, R.X., Du, J.K.: The nonlinear thickness-shear vibrations of an infinite and isotropic elastic plate. In: Proceedings of the Joint Conference of the 2009 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications and 2009 China Symposium on Frequency Control Technology, pp. 365–369 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, R., Wang, J., Du, J. et al. Solutions of nonlinear thickness-shear vibrations of an infinite isotropic plate with the homotopy analysis method. Numer Algor 59, 213–226 (2012). https://doi.org/10.1007/s11075-011-9485-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-011-9485-2