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Mechanical behaviors of electrostatic microresonators with initial offset imperfection: qualitative analysis via time-varying capacitors

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Abstract

This paper investigates analytically and numerically the effect of initial offset imperfection on the mechanical behaviors of microbeam-based resonators. Symmetry breaking of DC actuation, due to different initial offset distances of microbeam to lower and upper electrodes, is concerned. For qualitative analysis, time-varying capacitors are introduced and a lumped parameter model, considering nonlinear electrostatic force and midplane stretching of microbeam, is adopted to examine the system statics and dynamics. The Method of Multiple Scales (MMS) is applied to determine the primary resonance solution under small vibration assumption. Meanwhile, the Finite Difference Method (FDM) combined with Floquet theory is utilized to generate frequency response curves for medium- and large-amplitude vibration simulations. Static bifurcation, phase portrait and Hamiltonian function are firstly investigated to examine the system inherent behaviors. Besides, basins of attraction are briefly depicted to grasp the effects of initial offset and AC excitation on the system global dynamics. Then, variation of equivalent natural frequency versus DC voltage is analyzed. Results show that initial offset may induce complex frequency rebound phenomenon as well as a separate frequency branch under secondary pull-in condition. In what follows, emergences of softening, linear and hardening vibration are classified through discussing a key parameter obtained from the frequency response equation. New linear behavior induced by initial offset imperfection is found, which exhibits much higher sensitivity to DC voltage. Medium- and large-amplitude in-well motions are also investigated, indicating the existence of alternations of softening and hardening behaviors. Finally, lumped parameters are deduced via Galerkin procedure, and case studies are provided to illustrate the effectiveness of the whole analysis.

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References

  1. Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, NewYork (2011)

    Book  Google Scholar 

  2. Lee, K.B.: Principles of Microelectromechanical Systems. Wiley, Hoboken (2011)

    Book  Google Scholar 

  3. Liu, C.: Foundations of MEMS. China Machine Press, Beijing (2008)

    Google Scholar 

  4. Hu, Y.C., Chang, C.M., Huang, S.C.: Some design considerations on the electrostatically actuated microstructures. Sens. Actuators A Phys. 112, 155–161 (2004)

    Article  Google Scholar 

  5. Abdel-Rahman, E.M., Nayfeh, A.H.: Secondary resonances of electrically actuated resonant microsensors. J. Micromech. Microeng. 13, 491–501 (2003)

    Article  Google Scholar 

  6. Najar, F., Choura, S., Abdel-Rahman, E.M., El-Borgi, S., Nayfeh, A.H.: Dynamic analysis of variable-geometry electrostatic microactuators. J. Micromech. Microeng. 16, 2449–2457 (2006)

    Article  MATH  Google Scholar 

  7. Hammad, B.K., Abdel-Rahman, E.M., Nayfeh, A.H.: Modeling and analysis of electrostatic MEMS filters. Nonlinear Dyn. 60, 385–401 (2010)

    Article  MATH  Google Scholar 

  8. Li, Y., Fan, S.C., Guo, Z.S., Li, J., Cao, L.: Study of dynamic characteristics of resonators for MEMS resonant vibratory gyroscopes. Microsyst. Technol. 18, 639–647 (2012)

    Article  Google Scholar 

  9. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn. 48, 153–163 (2007)

    Article  MATH  Google Scholar 

  10. Zhang, W.M., Yan, H., Peng, Z.K., Meng, G.: Electrostatic pull-in instability in MEMS/NEMS: a review. Sens. Actuators A Phys. 214, 187–218 (2014)

    Article  Google Scholar 

  11. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Reduced-order models for MEMS applications. Nonlinear Dyn. 41, 211–236 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nayfeh, A.H., Ouakad, H.M., Najar, F., Choura, S., Abdel-Rahman, E.M.: Nonlinear dynamics of a resonant gas sensor. Nonlinear Dyn. 59, 607–618 (2010)

    Article  MATH  Google Scholar 

  13. Elshurafa, A.M., Khirallah, K., Tawfik, H.H., Emira, A.: Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators. J. Microelectromech. Syst. 20, 943–958 (2011)

    Article  Google Scholar 

  14. Younesian, D., Sadri, M., Esmailzadeh, E.: Primary and secondary resonance analyses of clamped–clamped micro-beams. Nonlinear Dyn. 76, 1867–1884 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xu, T.T., Younis, M.I.: Nonlinear dynamics of carbon nanotubes under large electrostatic force. J. Comput. Nonlinear Dyn. 11, 021009 (2016)

    Article  Google Scholar 

  16. Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.H.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 15, 672–680 (2003)

    Article  Google Scholar 

  17. Mestrom, R.M.C., Fey, R.H.B., Phan, K.L., Nijmeijer, H.: Simulations and experiments of hardening and softening resonances in a clamped–clamped beam MEMS resonator. Sens. Actuators A Phys. 162, 225–234 (2010)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Alkharabsheh, S.A., Younis, M.I.: Statics and dynamics of MEMS arches under axial forces. J. Vib. Acoust. 135, 021007 (2013)

    Article  Google Scholar 

  20. Krylov, S., Harari, I., Cohen, Y.: Stabilization of electrostatically actuated microstructures using parametric excitation. J. Micromech. Microeng. 15, 1188–1204 (2005)

    Article  Google Scholar 

  21. Mobki, H., Rezazadeh, G., Sadeghi, M., Vakili-Tahami, F., Seyyed-Fakhrabadi, M.M.: A comprehensive study of stability in an electro-statically actuated micro-beam. Int. J. Non-linear Mech. 48, 78–85 (2013)

    Article  Google Scholar 

  22. Stanciulescu, I., Mitchell, T., Chandra, Y., Eason, T., Spottswood, M.: A lower bound on snap-through instability of curved beams under thermomechanical loads. Int. J. Non-Linear Mech. 47, 561–575 (2012)

    Article  Google Scholar 

  23. Younis, M.I., Ouakad, H.M., Alsaleem, F.M., Miles, R., Cui, W.L.: Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation. J. Microelectromech. Syst. 19, 647–656 (2010)

    Article  Google Scholar 

  24. Mora, K., Gottlieb, O.: Parametric excitation of a microbeam-string with asymmetric electrodes: multimode dynamics and the effect of nonlinear damping. J. Vib. Acoust. 139, 040903 (2017)

    Article  Google Scholar 

  25. Krylov, S., Ilic, B.R., Schreiber, D., Seretensky, S., Craighead, H.: The pull-in behavior of electrostatically actuated bistable microstructures. J. Micromech. Microeng. 18, 055026 (2008)

    Article  Google Scholar 

  26. Tajaddodianfar, F., Pishkenari, H.N., Yazdi, M.R.H.: Prediction of chaos in electrostatically actuated arch micro-nano resonators: analytical approach. Commun. Nonlinear Sci. Numer. Simul. 30, 182–195 (2016)

    Article  MathSciNet  Google Scholar 

  27. Medina, L., Gilat, R., Llic, B.R., Krylov, S.: Experimental dynamic trapping of electrostatically actuated bistable micro-beams. Appl. Phys. Lett. 108, 073503 (2016)

    Article  Google Scholar 

  28. Zhang, Y., Wang, Y.S., Li, Z.H., Huang, Y.B., Li, D.H.: Snap-through and pull-in instabilities of an arch-shaped beam under an electrostatic loading. J. Microelectromech. Syst. 16, 684–693 (2007)

    Article  Google Scholar 

  29. Chen, X., Meguid, S.A.: On the parameters which govern the symmetric snap-through buckling behavior of an initially curved microbeam. Int. J. Solids Struct. 66, 77–87 (2015)

    Article  Google Scholar 

  30. Ouakad, H.M.: Electrostatic fringing-fields effects on the structural behavior of MEMS shallow arches. Microsyst. Technol. 1–9 (2016). https://doi.org/10.1007/s00542-016-2985-1

  31. Jia, X.L., Yang, J., Kitipornchai, S., Lim, C.W.: Pull-in instability and free vibration of electrically actuated poly-SiGe graded micro-beams with a curved ground electrode. Appl. Math. Model. 36, 1875–1884 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ouakad, H.M., Younis, M.I.: The dynamic behavior of MEMS arch resonators actuated electrically. Int. J. Non-linear Mech. 45, 704–713 (2010)

    Article  Google Scholar 

  33. Das, K., Batra, R.C.: Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches. Smart Mater. Struct. 18, 115008 (2009)

    Article  Google Scholar 

  34. Krylov, S., Dick, N.: Dynamic stability of electrostatically actuated initially curved shallow micro beams. Contin. Mech. Therm. 22, 445–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Farokhi, H., Ghayesh, M.H., Hussain, S.: Pull-in characteristics of electrically actuated MEMS arches. Mech. Mach. Theory 98, 133–150 (2016)

    Article  Google Scholar 

  36. Ouakad, H.M.: An electrostatically actuated MEMS arch band-pass filter. Shock Vib. 20, 809–819 (2013)

    Article  Google Scholar 

  37. Ramini, A.H., Hennawi, Q.M., Younis, M.I.: Theoretical and experimental investigation of the nonlinear behavior of an electrostatically actuated in-plane MEMS arch. J. Microelectromech. Syst. 25, 570–578 (2016)

    Article  Google Scholar 

  38. Ghayesh, M.H., Farokhi, H., Alici, G.: Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes. Int. J. Mech. Sci. 103, 247–264 (2015)

    Article  Google Scholar 

  39. Ruzziconi, L., Lenci, S., Younis, M.I.: An imperfect microbeam under an axial load and electric excitation: nonlinear phenomena and dynamical integrity. Int. J. Bifurc. Chaos 23, 1350026 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ruzziconi, L., Younis, M.I., Lenci, S.: An efficient reduced-order model to investigate the behavior of an imperfect microbeam under axial load and electric excitation. J. Comput. Nonlinear Dyn. 8, 011014 (2013)

    Article  MATH  Google Scholar 

  41. Ramini, A., Bellaredj, M.L.F., Al Hafiz, M.A., Younis, M.I.: Experimental investigation of snap-through motion of in-plane MEMS shallow arches under electrostatic excitation. J. Micromech. Microeng. 26, 015012 (2015)

    Article  Google Scholar 

  42. Rahim, F.A., Younis, M.I.: Control of bouncing in MEMS switches using double electrodes. Math. Probl. Eng. 2016, 3479752 (2016)

    MathSciNet  Google Scholar 

  43. Krylov, S.: Parametric excitation and stabilization of electrostatically actuated microstructures. Int. J. Multiscale Comput. 6, 563–584 (2009)

    Article  Google Scholar 

  44. Han, J.X., Zhang, Q.C., Wang, W.: Static bifurcation and primary resonance analysis of a MEMS resonator actuated by two symmetrical electrodes. Nonlinear Dyn. 80, 1585–1599 (2015)

    Article  Google Scholar 

  45. Han, J.X., Zhang, Q.C., Wang, W.: Design considerations on large amplitude vibration of a doubly clamped microresonator with two symmetrically located electrodes. Commun. Nonlinear Sci. Numer. Simul. 22, 492–510 (2015)

    Article  Google Scholar 

  46. Mestrom, R.M.C., Fey, R.H.B., van Beek, J.T.M., Phan, K.L., Nijmeijer, H.: Modelling the dynamics of a MEMS resonator: simulations and experiments. Sens. Actuators A Phys. 142, 306–315 (2008)

    Article  Google Scholar 

  47. Luo, A.C.J., Wang, F.Y.: Chaotic motion in a microelectro-mechanical system with non-linearity from capacitors. Commun. Nonlinear Sci. Numer. Simul. 7, 31–49 (2002)

    Article  Google Scholar 

  48. Lenci, S., Rega, G.: Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam. J. Micromech. Microeng. 16, 390–401 (2006)

    Article  Google Scholar 

  49. Lakrad, F., Belhaq, M.: Suppression of pull-in instability in MEMS using a high-frequency actuation. Commun. Nonlinear Sci. Numer. Simul. 15, 3640–3646 (2010)

    Article  Google Scholar 

  50. Haghighi, H.S., Markazi, A.H.D.: Chaos prediction and control in MEMS resonators. Commun. Nonlinear Sci. Numer. Simul. 15, 3091–3099 (2010)

    Article  Google Scholar 

  51. Ibrahim, M.I., Younis, M.I.: The dynamic response of electrostatically driven resonators under mechanical shock. J. Micromech. Microeng. 20, 025006 (2010)

    Article  Google Scholar 

  52. Tusset, A.M., Balthazar, J.M., Bassinello, D.G., Pontes, B.R., Felix, J.L.P.: Statements on chaos control designs, including a fractional order dynamical system, applied to a “MEMS” comb-drive actuator. Nonlinear Dyn. 69, 1837–1857 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Chen, C.P., Hu, H.T., Dai, L.M.: Nonlinear behavior and characterization of a piezoelectric laminated microbeam system. Commun. Nonlinear Sci. Numer. Simul. 18, 1304–1315 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  54. Tajaddodianfar, F., Pishkenari, H.N., Yazdi, M.R.H., Miandoab, E.M.: On the dynamics of bistable micro/nano resonators: analytical solution and nonlinear behavior. Commun. Nonlinear Sci. Numer. Simul. 20, 1078–1089 (2015)

    Article  MathSciNet  Google Scholar 

  55. Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N., Tajaddodianfar, F.: Study of nonlinear dynamics and chaos in MEMS/NEMS resonators. Commun. Nonlinear Sci. Numer. Simul. 22, 611–622 (2015)

    Article  Google Scholar 

  56. Shao, S., Masri, K.M., Younis, M.I.: The effect of time-delayed feedback controller on an electrically actuated resonator. Nonlinear Dyn. 74, 257–270 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)

    MATH  Google Scholar 

  58. Alkharabsheh, S.A., Younis, M.I.: Dynamics of MEMS arches of flexible supports. J. Microelectromech. Syst. 22, 216–224 (2013)

    Article  Google Scholar 

  59. 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, 794–806 (2010)

    Article  Google Scholar 

  60. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  61. Masri, K.M., Shao, S., Younis, M.I.: Delayed feedback controller for microelectromechanical systems resonators undergoing large motion. J. Vib. Control. 21, 2604–2615 (2015)

    Article  Google Scholar 

  62. Najar, F.: Static and Dynamic Behaviors of MEMS Microactuators. University of Waterloo, Waterloo (2008)

    Google Scholar 

  63. Kacem, N., Hentz, S., Pinto, D., Reig, B., Nguyen, V.: Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors. Nanotechnology 20, 275501 (2009)

    Article  Google Scholar 

  64. Azizi, S., Ghazavi, M.R., Khadem, S.E., Rezazadeh, G., Cetinkaya, C.: Application of piezoelectric actuation to regularize the chaotic response of an electrostatically actuated micro-beam. Nonlinear Dyn. 73, 853–867 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant no. 11702192, 11372210, 11772218, 51405343, 11602169), Tianjin Research Program of Application Foundation and Advanced Technology (Grant no. 15JCQNJC05000, 16JCQNJC04700), Innovation Team Training Plan of Tianjin Universities and colleges (Grant no. TD12-5043), Tianjin Science and Technology Planning Project (Grant no. 15ZXZNGX00220) and the Scientific Research Foundation of Tianjin University of Technology and Education (Grant no. KYQD1701, KYQD16009) and Scientific Research Program of Tianjin Education Committee (Grant no. JWK1602).

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Authors and Affiliations

  1. Tianjin Key Laboratory of High Speed Cutting and Precision Machining, School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin, 300222, China

    Jianxin Han, Houjun Qi, Gang Jin & Baizhou Li

  2. Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, School of Mechanical Engineering, Tianjin University of Technology, Tianjin, 300384, China

    Jingjing Feng

  3. Tianjin Key Laboratory of Nonlinear Dynamics and Control, School of Mechanical Engineering, Tianjin University, Tianjin, 300350, China

    Qichang Zhang

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  1. Jianxin Han
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Han, J., Qi, H., Jin, G. et al. Mechanical behaviors of electrostatic microresonators with initial offset imperfection: qualitative analysis via time-varying capacitors. Nonlinear Dyn 91, 269–295 (2018). https://doi.org/10.1007/s11071-017-3868-4

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