The effect of time-delayed feedback controller on an electrically actuated resonator. (English) Zbl 1281.93047
Summary: This paper presents a study of the effect of a time-delayed feedback controller on the dynamics of a Microelectromechanical systems (MEMS) capacitor actuated as a resonator by DC and AC voltage loads. A linearization analysis is conducted to determine the stability chart of the linearized system equations as a function of the time delay period and the controller gain. Then the method of multiple-scales is applied to determine the response and stability of the system for small vibration amplitude and voltage loads. It is shown that negative time-delay feedback control gain can lead to unstable responses, even if AC voltage is relatively small compared to the DC voltage. On the other hand, positive time delay can considerably strengthen the system stability even in fractal domains. We also show how the controller can be used to control damping in MEMS, increasing or decreasing, by tuning the gain amplitude and delay period. Agreements among the results of a shooting technique, long-time integration, basin of attraction analysis with the perturbation method are achieved.
MSC:
| 93B52 | Feedback control |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74G10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics |
References:
| [1] | Hu, H., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, Berlin (2002) · Zbl 1035.93002 · doi:10.1007/978-3-662-05030-9 |
| [2] | Balachandran, B., Kalmár-Nagy, T., Gilsinn, D.: Delay Differential Equations: Recent Advances and New Directions. Springer, New York (2009) · Zbl 1162.34003 |
| [3] | Erneux, T., Kalmar-Nagy, T.: Nonlinear stability of a delayed feedback controlled container crane. J. Vib. Control 13, 603-616 (2007) · Zbl 1182.74122 · doi:10.1177/1077546307074245 |
| [4] | Kalmar-Nagy, T., Stepan, G., Moon, F.C.: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121-142 (2001) · Zbl 1005.70019 · doi:10.1023/A:1012990608060 |
| [5] | Kurdi, M.H., Haftka, R.T., Schmitz, T.L., Mann, B.P.: A robust semi-analytical method for calculating the response sensitivity of a time delay system. J. Vib. Acoust. 130, 064504 (2008). doi:10.1115/1.2981093 · doi:10.1115/1.2981093 |
| [6] | Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421-428 (1992) · doi:10.1016/0375-9601(92)90745-8 |
| [7] | Pyragas, K., Pyragas, V., Benner, H.: Delayed feedback control of dynamical systems at a subcritical Hopf bifurcation. Phys. Rev. E 70, 056222 (2004). doi:10.1103/PhysRevE.70.056222 · Zbl 1254.34083 · doi:10.1103/PhysRevE.70.056222 |
| [8] | Masoud, Z.N., Daqaq, M.F., Nayfeh, N.A.: Pendulation reduction on small ship-mounted telescopic cranes. J. Vib. Control 10(8), 1167-1179 (2004) · doi:10.1177/1077546304043890 |
| [9] | Nayfeh, A.H., Nayfeh, N.A.: Time-delay feedback control of lathe cutting tools. J. Vib. Control 18, 1106-1115 (2012) · doi:10.1177/1077546311410763 |
| [10] | Nakajima, H., Ueda, Y.: Half-period delayed feedback control for dynamical systems with symmetries. Phys. Rev. E 58, 1757-1763 (1998) · doi:10.1103/PhysRevE.58.1757 |
| [11] | Yamasue, K., Hikihara, T.: Persistence of chaos in a time-delayed-feedback controlled Duffing system. Phys. Rev. E 73, 036209 (2006). doi:10.1103/PhysRevE.73.036209 · doi:10.1103/PhysRevE.73.036209 |
| [12] | Yamasue, K., Hikihara, T.: Control of microcantilevers in dynamic force microscopy using time delayed feedback. Rev. Sci. Instrum. 77, 053703 (2006). doi:10.1063/1.2200747 · doi:10.1063/1.2200747 |
| [13] | Hikihara, T., Kawagoshi, T.: An experimental study on stabilization of unstable periodic motion in magneto-elastic chaos. Phys. Lett. A 211, 29-36 (1996) · doi:10.1016/0375-9601(95)00925-6 |
| [14] | Hu, H.Y., Dowell, E.H., Virgin, L.N.: Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dyn. 15, 311-327 (1998) · Zbl 0906.34052 · doi:10.1023/A:1008278526811 |
| [15] | Wang, Z.H., Hu, H.Y.: Stability switches of time-delayed dynamic systems with unknown parameters. J. Sound Vib. 233, 215-233 (2000) · Zbl 1237.93159 · doi:10.1006/jsvi.1999.2817 |
| [16] | Wang, H.L., Hu, H.Y., Wang, Z.H.: Global dynamics of a Duffing oscillator with delayed displacement feedback. Int. J. Bifurc. Chaos 14, 2753-2775 (2004) · Zbl 1075.34077 · doi:10.1142/S0218127404010990 |
| [17] | Wang, H.L., Hu, H.Y.: Bifurcation analysis of a delayed dynamic system via method of multiple scales and shooting technique. Int. J. Bifurc. Chaos 15, 425-450 (2005) · Zbl 1080.34056 · doi:10.1142/S0218127405012326 |
| [18] | Wang, H.L., Wang, Z.H., Hu, H.Y.: Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback. Acta Mech. Sin. 20, 426-434 (2004) · doi:10.1007/BF02489381 |
| [19] | Hu, H.Y.: Using delayed state feedback to stabilize periodic motions of an oscillator. J. Sound Vib. 275, 1009-1025 (2004) · Zbl 1236.93124 · doi:10.1016/j.jsv.2003.07.006 |
| [20] | Wang, Z.H., Hu, H.Y.: Stabilization of vibration systems via delayed state difference feedback. J. Sound Vib. 296, 117-129 (2006) · Zbl 1243.93094 · doi:10.1016/j.jsv.2006.02.013 |
| [21] | Hu, H.Y., Wang, Z.H.: Singular perturbation methods for nonlinear dynamic systems with time delays. Chaos Solitons Fractals 40, 13-27 (2009) · Zbl 1197.34113 · doi:10.1016/j.chaos.2007.07.048 |
| [22] | El-Bassiouny, A.F.: Fundamental and subharmonic resonances of harmonically oscillation with time delay state feedback. Shock Vib. 13, 65-83 (2006) |
| [23] | El-Bassiouny, A.F.: Vibration control of a cantilever beam with time delay state feedback. Z. Naturforsch. A, J. Phys. Sci. 61, 629-640 (2006) |
| [24] | El-Bassiouny, A.F., El-Kholy, S.: Resonances of a nonlinear single-degree-of-freedom system with time delay in linear feedback control. Z. Naturforsch. A, J. Phys. Sci. 65, 357-368 (2010) · Zbl 1379.70056 |
| [25] | Qaroush, Y., Daqaq, M.F.: Vibration mitigation in multi-degree-of-freedom structural systems using filter-augmented delayed-feedback algorithms. Smart Mater. Struct. 19, 085016 (2010). doi:10.1088/0964-1726/19/8/085016 · doi:10.1088/0964-1726/19/8/085016 |
| [26] | Daqaq, M.F., Alhazza, K.A., Qaroush, Y.: On primary resonances of weakly nonlinear delay systems with cubic nonlinearities. Nonlinear Dyn. 64, 253-277 (2011) · doi:10.1007/s11071-010-9859-3 |
| [27] | Erneux, T.: Strongly nonlinear oscillators subject to delay. J. Vib. Control 16, 1141-1149 (2010) · Zbl 1269.70033 · doi:10.1177/1077546309341130 |
| [28] | Rand, R.H., Suchorsky, M.K., Sah, S.M.: Using delay to quench undesirable vibrations. Nonlinear Dyn. 62, 407-416 (2010) · Zbl 1218.34085 · doi:10.1007/s11071-010-9727-1 |
| [29] | Hamdi, M., Belhaq, M.: Control of Bistability in a Delayed Duffing Oscillator. Adv. Acoust. Vib. 2012, 872498 (2012) |
| [30] | Nayfeh, A.H., Chin, C.M., Pratt, J.: Perturbation methods in nonlinear dynamics—applications to machining dynamics. J. Manuf. Sci. Eng. 119, 485-493 (1997) · doi:10.1115/1.2831178 |
| [31] | Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91-117 (2003) · Zbl 1047.74027 · doi:10.1023/A:1022103118330 |
| [32] | Alsaleem, F.M., Younis, M.I.: Integrity analysis of electrically actuated resonators with delayed feedback controller. J. Dyn. Syst. Meas. Control 133, 031011 (2011). doi:10.1115/1.4003262 · doi:10.1115/1.4003262 |
| [33] | Alsaleem, F.M., Younis, M.I.: Stabilization of electrostatic MEMS resonators using a delayed feedback controller. Smart Mater. Struct. 19, 035016 (2010). doi:10.1088/0964-1726/19/3/035016 · doi:10.1088/0964-1726/19/3/035016 |
| [34] | 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) · doi:10.1109/JMEMS.2010.2047846 |
| [35] | Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York (2011) · doi:10.1007/978-1-4419-6020-7 |
| [36] | Nayfeh, A.H.: The Method of Normal Forms. Wiley, New York (2011) · Zbl 1234.37003 · doi:10.1002/9783527635801 |
| [37] | Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) · Zbl 0449.34001 |
| [38] | Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995) · Zbl 0848.34001 · doi:10.1002/9783527617548 |
| [39] | Daqaq, M.F., Alhazza, K.A., Arafat, H.N.: Non-linear vibrations of cantilever beams with feedback delays. Int. J. Non-Linear Mech. 43, 962-978 (2008) · Zbl 1203.74060 · doi:10.1016/j.ijnonlinmec.2008.07.005 |
| [40] | Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn. 48, 153-163 (2007) · Zbl 1177.74191 · doi:10.1007/s11071-006-9079-z |
| [41] | Ruzziconi, L., Younis, M.I., Lenci, S.: An electrically actuated imperfect microbeam: dynamical integrity for interpreting and predicting the device response. Meccanica (2013). doi:10.1007/s11012-013-9707-x · Zbl 1293.74271 · doi:10.1007/s11012-013-9707-x |
| [42] | Ruzziconi, L., Lenci, S., Younis, M.I.: An imperfect microbeam under axial load and electric excitation: nonlinear phenomena and dynamical integrity. Int. J. Bifurc. Chaos 23(2), 1350026 (2013) (17 pages) · Zbl 1270.74115 · doi:10.1142/S0218127413500260 |
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