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.
Similar content being viewed by others
References
Ai S. (2003). Multi-bump solutions to Carrier’s problem. J. Math. Anal. Appl. 277, 405–422
Ai S., and Hastings S.P. (2002). A shooting approach to layers and chaos in a forced Duffing equation. J. Diff. Eqns. 185, 389–436
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, 262–272
Bao M., and Wang W. (1996). Future of microelectromechanical systems (MEMS). Sensor Actuator A 56, 135–141
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.
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, 1511–1514
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.
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, 622–625
Hastings S.P., and McLeod J.B. (1991). On the periodic solutions of a forced second-order equation. J. Nonlin. Sci. 1, 225–245
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, 91–101
Hirsch M.W., Smale S., and Devaney R.L. (2004). Diffenential equations, dynamical systems, and an introduction to chaos, 2nd edn. Elsevier/Academic Press, Amsterdam
Kim P., and Lieber C.M. (1999). Nanotube nanotweezers. Science 286, 2148–2150
Muldavin, J. B., and Rebeiz, G. M. (1999). 30 GHz Tuned MEMS Switches. IEEE MTT-S Digest, pp. 1511–1518.
Nathanson H.C., Newell W.E., Wickstrom R.A., and Davis J.R. (1967). The resonant gate transistor. IEEE Tran. on Elec. Dev. 14, 117–133
Poncharal P., Wang Z.L., Ugarte D., and de Heer W.A. (1999). Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 283, 1513–1516
Saif M.T.A., Alaca B.E., and Sehitoglu H. (1999). Analytical modeling of electrostatic membrane actuator micro pumps. IEEE J. Microelectromech. Syst. 8, 335–344
Seeger, J. I., and Boser, B. E. (1999). Dynamics and Control of Parallel-Plate Actuators Beyond the Electrostatic Instability, Transducers ’99, pp. 474–477.
Shi F., Ramesh P., and Murkherjee S. (1995). Simulation methods for micro- electro-mechanical structures (MEMS) with application to a microtweezer. Comput. Struct. 56, 769–783
Tilmans H.A.C., Elwenspoek M., and Fluitman J.H.J. (1992). Micro resonant force gauges. Sensor Actuator A 30, 35–53
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ai, S., Pelesko, J.A. Dynamics of a Canonical Electrostatic MEMS/NEMS System. J Dyn Diff Equat 20, 609–641 (2008). https://doi.org/10.1007/s10884-007-9094-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-007-9094-x