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Homoclinic bifurcation and chaos control in MEMS resonators. (English) Zbl 1228.70015

Summary: The chaotic dynamics of a micromechanical resonator with electrostatic forces on both sides are investigated. Using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters. Detailed numerical studies including basin of attraction, and bifurcation diagram confirm the analytical prediction and reveal the effect of parametric excitation amplitude on the system transition to chaos. The main result of this paper indicates that it is possible to reduce the electrostatically induced homoclinic and heteroclinic chaos for a range of values of the amplitude and the frequency of the parametric excitation. Different active controllers are applied to suppress the vibration of the micromechanical resonator system. Moreover, a time-varying stiffness is introduced to control the chaotic motion of the considered system. The techniques of phase portraits, time history, and Poincaré maps are applied to analyze the periodic and chaotic motions.

MSC:

70Q05 Control of mechanical systems
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
37N35 Dynamical systems in control
70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
70K50 Bifurcations and instability for nonlinear problems in mechanics
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[1] S. Shaw, K. Turner, J. Rhoads, R. Baskaranm, Parametrically excited MEMS-based filters. in Proc. IUTAM Symp. Caotic Dyn, Control Syst. Process. 122 (2003) 37-146.; S. Shaw, K. Turner, J. Rhoads, R. Baskaranm, Parametrically excited MEMS-based filters. in Proc. IUTAM Symp. Caotic Dyn, Control Syst. Process. 122 (2003) 37-146.
[2] Rhoads, J.; Shaw, S.; Turner, K.; Baskaran, R., Tunable MEMS¯lters that exploit parametric resonance, J. Vib. Acoust., 127, 423-430 (2005)
[3] Zhang, W.; Baskaran, R.; Turner, K., Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor, Sens. Actuat. A Phys., 102, 139-150 (2002)
[4] Requa, M.; Turner, K., Electromechanically driven and sensed parametric resonance in silicon microcantilevers, Appl. Phys. Lett., 88, 263508 (2006)
[5] Turner, K.; Miller, S.; Hartwell, P.; MacDonald, N.; Strogatz, S.; Adams, S., Five parametric resonances in a microelectromechanical system, Nature, 396, 149-152 (1998)
[6] Rhoads, J.; Shaw, S.; Turner, K.; Moehlis, J.; DeMartini, B.; Zhang, W., Generalized parametric resonance in electrostaticallyactuated microelectromechanical oscillators, J. Sound Vib., 296, 797-829 (2006)
[7] Rhoads, J.; Shaw, S.; Turner, K., The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation, J. Micromech. Microeng., 16, 890-899 (2006)
[8] B. DeMartini, J. Moehlis, K. Turner, J. Rhoads, S. Shaw, W. Zhang, Modeling of parametrically excited microelectromechanical oscillator dynamics with application to filtering. In Proc. IEEE Sens. Conf., Irvine. CA 345-348 (2005).; B. DeMartini, J. Moehlis, K. Turner, J. Rhoads, S. Shaw, W. Zhang, Modeling of parametrically excited microelectromechanical oscillator dynamics with application to filtering. In Proc. IEEE Sens. Conf., Irvine. CA 345-348 (2005).
[9] Abraham, G.; Chatterjee, A., Approximate asymptotics for a nonlinear Mathieu equation using harmonic balance based averaging, Nonlinear Dyn., 31, 347-365 (2003) · Zbl 1062.70597
[10] Chavarette, F. R.; Balthazar, J. M.; Felix, J. L. P.; Ra¯kov, M., A reducing of a chaotic movement to a periodic orbit, of a micro-electro-mechanical system, by using an optimal linear control design, Commun. Nonlinear Sci. Numer. Simulat., 14, 1844-1853 (2009)
[11] requa, M. V.; Turner, K. L., Precise frequency estimation in a microelectromechanical parametric resonator, Appl. Phys. Lett., 90, 17, 173508 (2007)
[12] Nichol, J. M.; Hemesath, E. R.; Lauhon, L. J.; Budakian, R., Controlling the nonlinearity of silicon nanowire resonators using active feedback, Appl. Phys. Lett., 95, 123116 (2009)
[13] david, G.; Nerea, O.; Lluis, P.; Antonio, L., Study of intermodulation in RF MEMS variable capacitors, IEEE Transactions on Microwave Theory and Techniques., 54, 3, 1120-1130 (2006)
[14] Lorenz, E., Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141 (1963) · Zbl 1417.37129
[15] M. Basso, L. Giarre, M. Dahleh, I. Mezi’c, Numerical analysis of complex dynamics in atomic force microscopes, in Proc. IEEE Int. Conf. Control Appl., Trieste, Italy. 1-4, 1026-1030 (1998).; M. Basso, L. Giarre, M. Dahleh, I. Mezi’c, Numerical analysis of complex dynamics in atomic force microscopes, in Proc. IEEE Int. Conf. Control Appl., Trieste, Italy. 1-4, 1026-1030 (1998).
[16] Ashhab, M.; Salapaka, M.; Dahleh, M.; Mezi’c, I., Melnikov-based dynamical analysis of microcantilevers in scanning probe microscopy, Nonlinear Dyn., 20, 197-220 (1999) · Zbl 0964.74041
[17] Ashhab, M.; Salapaka, M.; Dahleh, M.; Mezi’c, I., Dynamical analysis and control of microcantilevers, Automatica, 35, 663-1670 (1999) · Zbl 0941.93044
[18] Hu, S.; Raman, A., Chaos in atomic force microscopy, Phys. Rev. Lett., 96, 036 107 (2006)
[19] Jamitzky, F.; Stark, M.; Bunk, W.; Heckl, W.; Stark, R., Chaos in dynamic atomic force microscopy, Nanotechnology, 17, S213-S220 (2006)
[20] Wang, Y.; Adams, S.; Thorp, J.; MacDonald, N.; Hartwell, P.; Bertsch, F., Chaos in MEMS, parameter estimation and its potential application, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 45, 1013-1020 (1998)
[21] Liu, S.; Davidson, A.; Lin, Q., Simulation studies on nonlinear dynamics and chaos in a MEMS cantilever control system, J. Micromech. Microeng., 14, 1064-1073 (2004)
[22] Luo, A.; Wang, F., Nonlinear dynamics of a micro-electro-mechanical system with time-varying capacitors, J. Vib. Acoust., 126, 77-83 (2004)
[23] De, S. K.; Aluru, N., Complex oscillations and chaos in electrostatic microelectromechanical systems under superharmonic excitations, Phys. Rev. Lett., 94, 204 101 (2005)
[24] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems Bifurcations of Vector Fields (1983), Springer- Verlag: Springer- Verlag New York · Zbl 0515.34001
[25] Siewe Siewe, M.; Moukam Kakermi, F. M.; Tchawoua, C.; Woafo, P., Nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well \(Φ^6\)-Rayleigh oscillator combined to parametric excitations, J. Comput. Nonlin. Dyn, 1, 196-204 (2006)
[26] Melnikov, V. K., Trans. Moscow Math. Soc., 12, 1-57 (1963)
[27] Haghighi, H. S.; Markazi, A. H.D., Chaos prediction and control in MEMS resonators, Commun. Nonlinear Sci. Numer. Simulat., 10, 3091-3099 (2010)
[28] Chacon, R., Comparison between parametric excitation and additional forcing terms as chaos-suppressing perturbations, Phys. Lett. A, 247, 431-436 (1998)
[29] A. H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational experimental Methods. Wiley-VCH Verlag Gmbh and Co. KGaA (2004).; A. H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational experimental Methods. Wiley-VCH Verlag Gmbh and Co. KGaA (2004). · Zbl 0848.34001
[30] Kao, Y. H.; Wang, C. S., Analog study of structures in a Van der Pol oscillator with nonlinear restoring force, Phys. Rev. E, 48, 2514 (1993)
[31] Gilboa, G.; Sochen, H.; Zeevi, Y. Y., Image sharpening by flow based on triple well potentials, J Math Imag. Vision., 20, 121-131 (2004) · Zbl 1366.94051
[32] Wagner, C.; Kiefhaber, T., Intermediates ca accelerate protein folding, Proc. Natl. Acad. Sci. U.S.A., 96, 6716-6721 (1999)
[33] Zhujun, J.; Ruiqi, W., Complex dynamics in Duffing system with two external forcings, Chaos Soliton. Fract., 23, 399-411 (2005) · Zbl 1077.34048
[34] Li, G. X.; Moon, F. C., Criteria for chaos of a three-well potential oscillator with homoclinic and heteroclinic orbits, J. Sound Vib., 136, 17-34 (1990) · Zbl 1235.74096
[35] Chac´on, R.; Bejarano, J. D., Homoclinic and hetroclinic chaos in a triple-well oscillator, J. Sound Vib., 186, 269-278 (1995) · Zbl 1049.70644
[36] Awrejcewicz, J.; Dzyubak, O.; Dzyubak, L., Chaos in three-well potential, Mech. Res. Commun., 31, 287-294 (2004) · Zbl 1082.70010
[37] V. Chua, Cubic-Quintic Duffing Oscillators. College of Engineering School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta Georgia. 12 (2003).; V. Chua, Cubic-Quintic Duffing Oscillators. College of Engineering School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta Georgia. 12 (2003).
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