Journal of Dynamics and Differential Equations

, Volume 20, Issue 3, pp 609–641 | Cite as

Dynamics of a Canonical Electrostatic MEMS/NEMS System


The mass-spring model of electrostatically actuated microelectromechanical systems (MEMS) or nanoelectromechanical systems (NEMS) is pervasive in the MEMS and NEMS literature. Nonetheless a rigorous analysis of this model does not exist. Here periodic solutions of the canonical mass-spring model in the viscosity dominated time harmonic regime are studied. Ranges of the dimensionless average applied voltage and dimensionless frequency of voltage variation are delineated such that periodic solutions exist. Parameter ranges where such solutions fail to exist are identified; this provides a dynamic analog to the static “pull-in” instability well known to MEMS/NEMS researchers.


MEMS nanotechnology electrostatics periodic solutions saddle-node bifurcation shooting method 

Mathematics Subject Classification (1991)

34C15 34C60 70K40 


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  1. 1.
    Ai S. (2003). Multi-bump solutions to Carrier’s problem. J. Math. Anal. Appl. 277, 405–422MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ai S., and Hastings S.P. (2002). A shooting approach to layers and chaos in a forced Duffing equation. J. Diff. Eqns. 185, 389–436MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Anderson M.J., Hill J.A., Fortunko C.M., Dogan N.S., and Moore R.D. (1995). BroadBand electrostatic transducers: modeling and experiments. J. Acoust. Soc. Am. 97, 262–272CrossRefGoogle Scholar
  4. 4.
    Bao M., and Wang W. (1996). Future of microelectromechanical systems (MEMS). Sensor Actuator A 56, 135–141CrossRefGoogle Scholar
  5. 5.
    Camon, H., Larnaudie, F., Rivoirard, F., and B. Jammes, B. (1999). Analytical simulation of a 1D single crystal electrostatic micromirror. Proc. Model. Simul. Microsyst., pp. 628–631.Google Scholar
  6. 6.
    Chan E.K., Kan E.C., Dutton R.W., and Pinsky M.P. (1997). Nonlinear dynamic modeling of micromachined microwave switches. IEEE MTT-S Digest 3, 1511–1514Google Scholar
  7. 7.
    Chu, P. B., and Pister, K. S. J. (1994). Analysis of closed-loop Control of parallel-plate electrostatic microgrippers. Proc. IEEE Int. Conf. Robotics and Automation, pp. 820–825.Google Scholar
  8. 8.
    R. Gao R., Wang Z.L., Bai Z., de Heer W.A., Dai L., and Gao M. (2000). Nanomechanics of individual carbon nanotubes from pyrolytically grown arrays. Phys. Rev. Lett. 85, 622–625CrossRefGoogle Scholar
  9. 9.
    Hastings S.P., and McLeod J.B. (1991). On the periodic solutions of a forced second-order equation. J. Nonlin. Sci. 1, 225–245MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Horenstein M, Bifano T., Pappas S., Perreault J., and Krishnamoorthy-Mali R. (1999). Real time optical correction using electrostatically actuated MEMS device. J. Electrostatics 46, 91–101CrossRefGoogle Scholar
  11. 11.
    Hirsch M.W., Smale S., and Devaney R.L. (2004). Diffenential equations, dynamical systems, and an introduction to chaos, 2nd edn. Elsevier/Academic Press, AmsterdamGoogle Scholar
  12. 12.
    Kim P., and Lieber C.M. (1999). Nanotube nanotweezers. Science 286, 2148–2150CrossRefGoogle Scholar
  13. 13.
    Muldavin, J. B., and Rebeiz, G. M. (1999). 30 GHz Tuned MEMS Switches. IEEE MTT-S Digest, pp. 1511–1518.Google Scholar
  14. 14.
    Nathanson H.C., Newell W.E., Wickstrom R.A., and Davis J.R. (1967). The resonant gate transistor. IEEE Tran. on Elec. Dev. 14, 117–133CrossRefGoogle Scholar
  15. 15.
    Poncharal P., Wang Z.L., Ugarte D., and de Heer W.A. (1999). Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 283, 1513–1516CrossRefGoogle Scholar
  16. 16.
    Saif M.T.A., Alaca B.E., and Sehitoglu H. (1999). Analytical modeling of electrostatic membrane actuator micro pumps. IEEE J. Microelectromech. Syst. 8, 335–344CrossRefGoogle Scholar
  17. 17.
    Seeger, J. I., and Boser, B. E. (1999). Dynamics and Control of Parallel-Plate Actuators Beyond the Electrostatic Instability, Transducers ’99, pp. 474–477.Google Scholar
  18. 18.
    Shi F., Ramesh P., and Murkherjee S. (1995). Simulation methods for micro- electro-mechanical structures (MEMS) with application to a microtweezer. Comput. Struct. 56, 769–783CrossRefGoogle Scholar
  19. 19.
    Tilmans H.A.C., Elwenspoek M., and Fluitman J.H.J. (1992). Micro resonant force gauges. Sensor Actuator A 30, 35–53CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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