We investigate the dynamics and global stability of a beam-based electrostatic microactuator, which is modeled as a first-order approximation of a reduced-order model (ROM) derived using the differential quadrature method (DQM). We show that the ROM dynamics is qualitatively similar to that of a higher-order approximation. We simulate the occurrence of dynamic pull-in for excitations near the first primary resonance using the finite difference method (FDM) and long-time integration. Limit-cycle solutions are obtained using the FDM, the generated frequency- and force-response curves exhibit cyclic-fold, saddle-node, and period-doubling bifurcations. We verify that symmetry breaking is not likely to occur because the orbit is already asymmetric. We identify the basin of attraction of bounded motions using various approximation levels. The simulations reveal that the erosion of the basin of attraction depends heavily on the amplitude and frequency of the AC voltage. We show that smoothness of the boundary of the basin of attraction can be lost and replaced by fractal tongues, which dramatically increase the sensitivity of the microbeam to initial conditions. According to these simulations, the locations of the two fixed points are likely to be disturbed.
Abdel-Rahman, E.M., Younis, M.I., and Nayfeh, A.H., 2005, "Finite-amplitude motions of beam resonators and their stability," Journal of Computational and Theoretical Nanoscience1, 1-7.
2.
Bochobza-Degani, O., Elata, D., and Nemirovsky, Y., 2002, "An efficient DIPIE algorithm for CAD of electrostatically actuated MEMS devices," Journal of Microelectromechnical Systems11, 612-620.
3.
De, S.K. and Aluru, N.R., 2004, "Full-lagrangian schemes for dynamic analysis of electrostatic MEMS," Journal of Microelectromechnical Systems13, 737-758.
4.
De, S.K. and Aluru, N.R., 2006, "Complex nonlinear oscillations in electrostatically actuated microstructures," Journal of Microelectromechnical Systems15, 355-369.
5.
Elata, D. and Bamberger, H., 2006, "On the dynamic pull-in of electrostatic actuators with multiple degrees of freedom and multiple voltage sources," Journal of Microelectromechnical Systems15, 131-140.
6.
Fargas-Marquès, A., Costa Castelló, R., and Shkel, A.M., 2005, "Modelling the electrostatic actuation of MEMS: state of the art 2005,"Technical Report IOC-DT-P-2005-18 , Universitat Polytèchnica de Catalunya.
7.
Fargas-Marquès, A. and Shkel, A.M., 2005, "On electrostatic actuation beyond snapping condition ," in Proceedings of the IEEE Sensors 05, Irvine, CA, pp. 600-603.
8.
Gui, C., Legtenberg, R., Tilmans, H.A.C., Fluitman, J.H.J., and Elwenspoek, M., 1998, "Nonlinearity and hysteresis of resonant strain gauges ," Journal of Micromechanics and Microengineering7, 122-127.
9.
Gupta, R.K., Hung, E.S., Yang, Y.J., Ananthasuresh, G.K., and Senturia, S.D., 1996, "Pull-in dynamics of electrostatically actuated beams ," in Technical Digest Solid State Sensor and Actuator Workshop, Hilton Head, SC, pp. 1-2.
10.
Lenci, S. and Rega, G., 2006, "Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam," Journal of Micromechanics and Microengineering16, 390-401.
11.
Li, G. and Aluru, N.R., 2002, "Dynamic analysis of electrostatic MEMS by meshless methods ," in Technical Proceedings of the International Conference on Modeling and Simulation of Microsystems, Nanotech 2002, San Juan, Puerto Rico, April 21-25, Vol. 1, pp. 356-359.
12.
Liqin, L., Gang, T.Y., and Zhiqiang, W., 2005, "Nonlinear dynamics of microelectromechanical systems ," Journal of Vibration and Control12, 57-65.
13.
Liu, S., Davidson, A., and Lin, Q., 2004, "Simulation studies on nonlinear dynamics and chaos in a MEMS cantilever control system," Journal of Micromechanics and Microengineering14, 1064-1073.
14.
Luo, A.C.J. and Wang, F.Y., 2004, "Nonlinear dynamics of a micro-electro-mechanical system with time-varying capacitors," Journal of Vibration and Acoustics126, 77-83.
15.
Moon, F.C., 1992, Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers, Wiley, New York.
16.
Najar, F., Choura, S., Abdel-Rahman, E.M., El-Borgi, S., and Nayfeh, A.H., 2006, "Dynamic analysis of variable-geometry electrostatic microactuators ," Journal of Micromechanics and Microengineering16, 2449-2457.
17.
Najar, F., Choura, S., El-Borgi, S., Abdel-Rahman, E.M., and Nayfeh, A.H., 2005, "Modeling and design of variable-geometry electrostatic microactuators," Journal of Micromechanics and Microengineering15, 419- 429.
18.
Najar, F., Nayfeh, A.H., Abdel-Rahman, E.M., Choura, S., and El-Borgi, S., 2009, "Nonlinear analysis of MEMS electrostatic microactuators: primary and secondary resonances of the first mode,"Journal of Vibration and Control, accepted.
19.
Nathanson, H.C., Newell, W.E., Wickstrom, R.A., and Davis, J.R., 1967, "The resonant gate transistor," IEEE Transactions on Electron Devices14, 117-133.
20.
Nayfeh, A.H. and Balachandran, B., 1995, Applied Nonlinear Dynamics, Wiley, New York.
21.
Nayfeh, A.H. and Mook, D.T., 1979, Nonlinear Oscillations, Wiley, New York.
22.
Nayfeh, A.H. and Younis, M.I., 2005, "Dynamics of MEMS resonators under superharmonic and subharmonic excitations," Journal of Micromechanics and Microengineering15, 1840-1847.
23.
Nayfeh, A.H., Younis, M.I., and Abdel-Rahman, E.M. , 2007, "Dynamic pull-in phenomenon in MEMS resonators," Nonlinear Dynamics48, 153-163.
24.
Nielson, G.N. and Barbastathis, G., 2006, "Dynamic Pull-In of Parallel-plate and torsional electrostatic MEMS actuators," Journal of Microelectromechnical Systems15, 811-821.
25.
Rega, G. and Lenci, S., 2005, "Identifying, evaluating, and controlling dynamical integrity measures in nonlinear mechanical oscillators," Nonlinear Analysis63, 902-914.
26.
Rhoads, J.F., Shaw, S.W., and Turner, K.L., 2006, "The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation," Journal of Micromechanics and Microengineering16, 890-899.
27.
Seeger, J.I. and Boser, B.E., 2003, "Negative capacitance for control of gap-closing electrostatic actuators," in Proceedings of the 12th International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers 03), Boston, MA, pp. 484-487.
28.
Soliman, M.S. and Thompson, J.M.T., 1992, "Global dynamics underlying sharp basin erosion in nonlinear driven oscillator," Physical Review A45, 3425-3436.
29.
Tadayon, M.A., Rajaei, M., Sayyaadi, H., Nakhaie Jazar, G., and Alasty, A., 2006, "Nonlinear dynamics of microresonators," Internatonal MEMS Conference, Journal of Physics: Conference Series34, 961-966.
30.
Veijola, T., Mattila, T., Jaakkola, O., Kihamäki, J., Lamminmäki, T., Oja, A., Ruokonen, K., Seppä, H., Seppälä, P., and Tittonen, I., 2000, "Large-displacement modeling and simulation of micromechanical electrostatically driven resonators using the harmonic balance method," in Proceedings of the International Microwave Symposium , Boston, MA, June 11-16, pp. 99-102.
31.
Wang, Y.C., Adams, S.G., Thorp, J.S., MacDonald, N.C., Hartwell, P., and Bertsch, F., 1998, "Chaos in MEMS, parameter estimation and its potential application," IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications45, 1013-1020.
32.
Younis, M.I., 2001, "Investigation of the mechanical behavior of microbeam-based MEMS devices," MSc Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
33.
Younis, M.I. and Nayfeh, A.H., 2003, "A study of the nonlinear response of a resonant microbeam to an electric actuation," Nonlinear Dynamics31, 91-117.
