Abstract
Nonlinear harmonic vibration of a micro-electro-mechanical beam is investigated, and the micro-actuator, which is considered in this study, is a special kind of electrostatic symmetric actuators. A fully clamped micro-beam with a uniform thickness is modeled as an electrostatic micro-actuator with two symmetric potential walls. The nonlinear forced vibration of the micro-beam is analyzed, and the non-dimensional governing equation of motion, using the Galerkin method, is developed. Higher-order nonlinear terms in the equation of motion are taken into account for the first time, and the perturbation method is utilized regarding these terms and hence, all the resonant cases have been considered. The multiple scales method is employed to solve the nonlinear equations, and therefore, the problem does not deal with the large deformations. The primary and secondary resonance conditions are determined, and the corresponding secular terms in each case are recognized. Harmonic responses are obtained for different cases of resonance, and eventually, the stable and unstable portions of the responses are identified. A parametric sensitivity study is carried out to examine the effects of different parameters on the amplitude–frequency characteristic equations.
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References
Qian, Y.H., Ren, D.X., Lai, S.K., Chen, S.M.: Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam. Commun. Nonlinear Sci. Numer. Simul. 17, 1947–1955 (2012)
Abdel-Rahman, E.M., Younis, M.I., Nayfeh, A.H.: Characterization of the mechanical behavior of an electrically actuated microbeam. J. Micromech. Microeng. 12, 759–766 (2002)
Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. J. 31, 91–117 (2003)
Kuang, J.-H., Chen, C.J.: Dynamic characteristics of shaped micro-actuators solved using the differential quadrature method. J. Micromech. Microeng. 14, 647–655 (2004)
Nayfeh, A.H., Younis, M.I.: Dynamics of MEMS resonators under superharmonic and subharmonic excitations. J. Micromech. Microeng. 15, 1840–1847 (2005)
Rhoads, J.F., Shaw, S.W., Turner, K.L.: The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation. J. Micromech. Microeng. 16, 890–899 (2006)
Younis, M.I., Alsaleem, F., Jordy, D.: The response of clamped–clamped microbeams under mechanical shock. Int. J. Non-linear Mech. 42, 643–657 (2007)
Batra, R.C., Porfiri, M., Spinello, D.: Electromechanical model of electrically actuated narrow microbeams. J. Microelectromech. Syst. 15, 1175–1189 (2006)
Batra, R.C., Porfiri, M., Spinello, D.: Vibrations of narrow microbeams predeformed by an electric field. J. Sound Vib. 309, 600–612 (2008)
Younesian, D., Askari, H., Saadatnia, Z.: Frequency analysis of strongly nonlinear generalized Duffing oscillators using He’s frequency–amplitude formulation and He’s energy balance method. Comput. Math. Appl. 59, 3222–3228 (2010)
Younesian, D., Askari, H., Saadatnia, Z.: Analytical approximate solutions for the generalized nonlinear oscillator. Appl. Anal. 91, 965–977 (2012)
Younesian, D., Askari, H., Saadatnia, Z., Yildirim, A.: Periodic solutions for the generalized nonlinear oscillators containing fraction order elastic force. Int. J. Nonlinear Sci. Numer. Simul. 11, 1027–1032 (2010)
Kuang, J.H., Chen, C.J.: Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators. Math. Comput. Model. 41, 1479–1491 (2005)
He, X.J., Wu, Q., Wang, Y., Song, M.-X., Yin, J.H.: Numerical simulation and analysis of electrically actuated microbeam-based MEMS capacitive switch. Microsyst. Technol. 15, 301–307 (2009)
Xia, W., Wang, L., Yin, L.: Nonlinear non-classical microscale beams: static bending, post buckling and free vibration. Int. J. Eng. Sci. 48, 2044–2053 (2010)
Mojahedi, M., Moghimi-Zand, M., Ahmadian, M.T., Babaei, M.: Analytic solutions to the oscillatory behavior and primary resonance of electrostatically actuated microbridges. Int. J. Struct. Stab. Dyn. 11, 119–1137 (2011)
Fu, Y., Zhang, J., Wan, L.: Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Curr. Appl. Phys. 11, 482–485 (2011)
Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity theory with surface energy. Arch. Appl. Mech. 82, 363–376 (2012)
Zhao, J., Zhou, S., Wang, B., Wang, X.: Nonlinear microbeam model based on strain gradient theory. Appl. Math. Model. 36, 2674–2686 (2012)
Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. 50, 256–267 (2012)
Wang, Y.G., Lin, W.H., Feng, Z.J., Li, X.M.: Characterization of extensional multi-layer microbeams in pull-in phenomenon and vibrations. Int. J. Mech. Sci. 54, 225–233 (2012)
Hassanpour, P.A., Esmailzadeh, E., Cleghorn, W.L., Mills, J.K.: Experimental measurement of resonance frequencies of asymmetric micro-bridge resonators. J. Intell. Mater. Syst. Struct. 22, 127–136 (2011)
Hassanpour, P.A., Cleghorn, W.L., Esmailzadeh, E., Mills, J.K.: Nonlinear vibration of micromachined asymmetric resonators. J. Sound Vib. 329, 2547–2564 (2010)
Hassanpour, P.A., Cleghorn, W.L., Esmailzadeh, E., Mills, J.K.: Vibration analysis of micro-machined beam-type resonators. J. Sound Vib. 308, 287–301 (2007)
Abdel-Rahman, E.M., Nayfeh, A.H.: Secondary resonances of electrically actuated resonant microsensors. J. Micromech. Microeng. 13, 491–501 (2003)
Esmailzadeh, E., Nakhaie-Jazar, G.: Periodic solution of a Mathieu–Duffing type equation. Int. J. Non-linear Mech. 32(5), 905–912 (1997)
Esmailzadeh, E., Nakhaie-Jazar, G., Mehri, B.: Existence of periodic solution for beams with harmonically variable length. J Vib. Acoust. 119(3), 485–488 (1997)
Esmailzadeh, E., Mehri, B., Nakhaie-Jazar, G.: Periodic solution of a second order, autonomous, nonlinear system. Nonlinear Dyn. J. 10(4), 307–316 (1996)
Younesian, D., Esmailzadeh, E.: Non-linear vibration of variable speed rotating viscoelastic beams. Nonlinear Dyn. J. 60(1), 193–205 (2010)
Younesian, D., Esmailzadeh, E., Sedaghati, R.: Existence of periodic solutions for the generalized form of Mathieu equation. Nonlinear Dyn. J. 39(4), 335–348 (2005)
Krylov, S., Maimon, R.: Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force. J. Vib. Acoust. 126, 332–342 (2004)
Krylov, S., Harari, I., Cohen, Y.: Stabilization of electrostatically actuated microstructures using parametric excitation. J. Micromech. Microeng. 15, 1188–1204 (2005)
Krylov, S., Ilic, B., Lulinsky, S.: Bistability of curved microbeams actuated by fringing electrostatic fields. Nonlinear Dyn. J. 66, 403–426 (2011)
Rezazadeh, G., Madinei, H., Shabani, R.: Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method. Appl. Math. Model. 36, 430–443 (2012)
Ekici, H., Boyaci, H.: Effects of non-ideal boundary conditions on vibrations of microbeams. J. Vib. Control 13(9–10), 1369–1378 (2007)
Al Saleem, F.M., Younis, M.I.: Theoretical and experimental investigation of dynamic instabilities in electrostatic MEMS. Proceedings of the XIth International Congress and Exposition, 2–5 June (2008) Orlando, FL, USA
Elnaggar, A.M., El-Bassiouny, A.F., Mosa, A.: Harmonic and sub-harmonic resonance of MEMS subjected to a weakly non-linear parametric and external excitations. Int. J. Appl. Math. Res. 2(2), 252–263 (2013)
Alsaleem, F.M., Younis, M.I., Ouakad, H.M.: On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonator. J. Micromech. Microeng. 19(4), 045013 (2009)
Alsaleem, F.M., Younis, M.I., Ruzziconi, L.: An experimental and theoretical investigation of dynamic pull-in in MEMS resonators actuated electrostatically. J. Microelectromech. Syst. 19(4), 794–806 (2010)
Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn. J. 48(1–2), 153–163 (2007)
Fargas-Marques, A., Casals-Terré, J., Shkel, A.M.: Resonant pull-in condition in parallel-plate electrostatic actuators. J. Microelectromech. Syst. 16(5), 1044–1053 (2007)
De, S.K., Aluru, N.R.: Full-Lagrangian schemes for dynamic analysis of electrostatic MEMS. J. Microelectromech. Syst. 13(5), 737–758 (2004)
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Appendix
Appendix
The equation including the different secondary resonance conditions:
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Younesian, D., Sadri, M. & Esmailzadeh, E. Primary and secondary resonance analyses of clamped–clamped micro-beams. Nonlinear Dyn 76, 1867–1884 (2014). https://doi.org/10.1007/s11071-014-1254-z
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DOI: https://doi.org/10.1007/s11071-014-1254-z