Analysis of nonlinear oscillator arising in the microelectromechanical system by using the parameter expansion and equivalent linearization methods

  • Authors

    • D. V. Hieu Department of Mechanics, Thai Nguyen University of Technology
    • N. T. K. Thoa
    • L. Q. Duy
    2018-04-21
    https://doi.org/10.14419/ijet.v7i2.9191
  • Parameter Expansion Method, Equivalent Linearization Method, Nonlinear Oscillator, MEMS.
  • Abstract

    In this paper, the nonlinear oscillator arising in the microbeam-based micro-electromechanical system (MEMS) is described. The motion equation of a microbeam is simplified into an ordinary differential equation by using the Galerkin method. The nonlinear ordinary differential equation is solved by using two methods including the Parameter-Expansion and Equivalent Linearization Methods. To verify the accuracy of the present methods, illustrative examples are provided and compared with other analytical, exact and numerical solutions.

  • References

    1. [1] E. M. Abdel-Rahman, M.I. Younis, A.H. Nayfeh. “Characterization of themechanical behavior of an electrically actuated microbeamâ€. Journal of Micromechanics and Microengineering , Vol. 12 (2002), pp. 759-766. https://doi.org/10.1088/0960-1317/12/6/306.

      [2] J.H. Kuang, C.J. Chen. “Dynamic characteristics of shaped micro-actuators solved using the differential quadrature methodâ€. Journal of Micromechanics and Microengineering, Vol.14 (2004), pp. 647-655. https://doi.org/10.1088/0960-1317/14/4/028.

      [3] J.H. Kuang, C.J. Chen. “Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuatorsâ€. Mathematical and Computer Modelling, Vol. 41 (2005), pp. 1479-1491. https://doi.org/10.1016/j.mcm.2005.06.001.

      [4] M. I. Younis and A. H. Nayfeh. “A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuationâ€. Nonlinear Dynamics, Vol. 31 (2003), pp. 91–117. https://doi.org/10.1023/A:1022103118330.

      [5] Fu, Y., Zhang, J., and Wan, L. “Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)â€. Current Applied Physics, Vol.11 (3) (2011), pp.482-85. https://doi.org/10.1016/j.cap.2010.08.037.

      [6] Y.H. Qian, D.X. Ren, S.K. Lai, S.M. Chen. “Analytical approximations to nonlinear vibration of an electrostatically actuated microbeamâ€. Commun Nonlinear Sci Numer Simulat. Vol.17 (2012), pp. 1947–1955. https://doi.org/10.1016/j.cnsns.2011.09.018.

      [7] Mahdi Bayat, Mahmoud Bayat, Iman Pakar. “Nonlinear vibration of an electrostatically actuated microbeamâ€. Latin American Journal of Solids and Structures. Vol.11 (2014),pp. 534 – 544. https://doi.org/10.1590/S1679-78252014000300009.

      [8] J. H. He. “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineeringâ€. International Journal of Modern Physics B (IJMPB), Vol.22(21) (2008), pp. 3487–3578. https://doi.org/10.1142/S0217979208048668.

      [9] J. H. He and D. H. Shou. “Application of parameter-expanding method to strongly nonlinear oscillatorsâ€. International Journal of Nonlinear Sciences and Numerical Simulation, Vol.8 (2007), pp.121–124.

      [10] He J H. “Bookkeeping parameter in perturbation methodsâ€. Int J Non-linear Sci Numer Simul, (2001); pp.257–264. https://doi.org/10.1515/IJNSNS.2001.2.3.257.

      [11] M. Kaya and S. Altay Demirbag. “Application of parameter expansion method to the generalized nonlinear discontinuity equationâ€. Chaos, Solitons & Fractals, Vol. 42(4) (2009), pp.1967–1973. https://doi.org/10.1016/j.chaos.2009.03.143.

      [12] H. M. Liu. “Approximate period of nonlinear oscillators with discontinuities by modified lindstedt-poincare methodâ€. Chaos, Solitons & Fractals, Vol. 23(2) (2005), pp. 577–579. https://doi.org/10.1016/j.chaos.2004.05.004.

      [13] T. Ozis and A. Yildirim. “Determination of periodic solution for a u1/3 force by He’s modified lindstedt-poincaré methodâ€. Journal of Sound and Vibration, Vol. 301(1-2) (2007), pp. 415–419. https://doi.org/10.1016/j.jsv.2006.10.001.

      [14] S. Q. Wang and J. H. He. “Nonlinear oscillator with discontinuity by parameter-expansion methodâ€. Chaos, Solitons & Fractals, Vol. 35(4) (2008), pp. 688–691. https://doi.org/10.1016/j.chaos.2007.07.055.

      [15] L. Xu. “Application of he’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wireâ€. Physics Letters A, Vol. 368(3-4) (2007), pp.259–262. https://doi.org/10.1016/j.physleta.2007.04.004.

      [16] L. Xu. “Determination of limit cycle by he’s parameter-expanding method for strongly nonlinear oscillatorsâ€. Journal of Sound and Vibration, Vol. 302(1-2) (2007), pp.178–184. https://doi.org/10.1016/j.jsv.2006.11.011.

      [17] L. Xu. “He’s parameter-expanding methods for strongly nonlinear oscillatorsâ€. Journal of Computational and Applied Mathematics, Vol. 207(1) (2007), pp.148–154. https://doi.org/10.1016/j.cam.2006.07.020.

      [18] J.F. Rhoads, S.W. Shaw, K.L. Turner. “The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuationâ€. Journal of Micromechanics and Microengineering, Vol.16 (2006), pp. 890-899. https://doi.org/10.1088/0960-1317/16/5/003.

      [19] R.C. Batra, M. Porfiri, D. Spinello. “Electromechanical model of electrically actuated narrow microbeamsâ€. Journal of Microelectromechanical Systems, Vol.15 (2006), pp. 1175-1189. https://doi.org/10.1109/JMEMS.2006.880204.

      [20] Krylov, N., Bogoliubov N., Introduction to nonlinear mechanics, New York: Princenton University Press, (1943).

      [21] Caughey, T. K., “Equivalent linearization techniqueâ€, The Journal of the Acoustical Society of America, Vol. 35 (1959), 1706–1711. https://doi.org/10.1121/1.1918794.

      [22] Anh, N. D., Schiehlen, W., “New criterion for Gaussian equivalent linearizationâ€, Euror Journal of Mechanics- A/Solid, Vol. 16 (1997), 1025– 1039.

      [23] Anh, N. D., Di Paola, M., “Some extensions of Gaussian equivalent linearizationâ€, Proceedings of the International Conference on Nonlinear Stochastic Dynamics. pp. 5–16. Hanoi, Vietnam (1995).

  • Downloads

  • How to Cite

    Hieu, D. V., Thoa, N. T. K., & Duy, L. Q. (2018). Analysis of nonlinear oscillator arising in the microelectromechanical system by using the parameter expansion and equivalent linearization methods. International Journal of Engineering & Technology, 7(2), 597-604. https://doi.org/10.14419/ijet.v7i2.9191

    Received date: 2018-01-16

    Accepted date: 2018-04-06

    Published date: 2018-04-21