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Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 1–15 | Cite as

Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay

  • Mustapha Hamdi
  • Mohamed BelhaqEmail author
Original Paper

Abstract

The effect of time-delayed feedback and fast harmonic excitation (FHE) on stationary periodic vibration and quasi-periodic responses in a parametric and self-excited weakly nonlinear oscillator is analyzed in this paper. The method of direct partition of motion and two stages of multiple scales analysis are conducted to obtain analytical approximation for quasi-periodic oscillation envelopes and frequency-locking area near primary resonance. A parameter study shows that, in the absence or the presence of high-frequency excitation, time-delayed feedback may reduce significantly the amplitude and the envelopes of quasi-periodic oscillations leading to a quasi synchronization of the response over the whole frequency range around the resonance. The results presented for the parameters tested agree well with results obtained by numerical simulation.

Keywords

Self-excitation Parametric excitation Time delay Frequency-locking Quasi-periodic vibration Control 

References

  1. 1.
    Tondl, A.: On the interaction between self-excited and parametric vibrations. In: Monographs and Memoranda, vol. 25. National Research Institute for Machine Design, Prague (1978) Google Scholar
  2. 2.
    Schmidt, G.: Interaction of Self-Excited Forced and Parametrically Excited Vibrations. In: The 9th International Conference on Nonlinear Oscillations, vol. 3. Application of the Theory of Nonlinear Oscillations. Naukowa, Dumka (1984) Google Scholar
  3. 3.
    Belhaq, M.: Numerical study for parametric excitation of differential equation near a 4-resonance. Mech. Res. Commun. 17(4), 199–206 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Szabelski, K., Warminski, J.: Self excited system vibrations with parametric and external excitations. J. Sound Vib. 187(4), 595–607 (1995) CrossRefGoogle Scholar
  5. 5.
    Szabelski, K., Warminski, J.: The nonlinear vibrations of parametrically self-excited system with two degrees of freedom under external excitation. Nonlinear Dyn. 14, 23–36 (1997) zbMATHCrossRefGoogle Scholar
  6. 6.
    Warminski, J.: Synchronisation effects and chaos in van der Pol–Mathieu oscillator. J. Theor. Appl. Mech. 39(4), 861–884 (2001) zbMATHGoogle Scholar
  7. 7.
    Warminski, J., Balthazar, J.M.: Vibrations of a parametrically and self-excited system with ideal and non-ideal energy sources. J. Braz. Soc. Mech. Sci. Eng. 25, 413–420 (2003) CrossRefGoogle Scholar
  8. 8.
    Belhaq, M., Fahsi, A.: Higher-order approximation of subharmonics close to strong resonances in the forced oscillators. Comput. Math. Appl. 33(8), 133–144 (1997) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Yano, S.: Analytic research on dynamic phenomena of parametrically and self-exited mechanical systems. Ing.-Arch. 57, 51–60 (1987) zbMATHCrossRefGoogle Scholar
  10. 10.
    Yano, S.: Considerations on self- and parametrically excited vibrational systems. Ing.-Arch. 59, 285–295 (1989) zbMATHCrossRefGoogle Scholar
  11. 11.
    Abouhazim, N., Belhaq, M., Lakrad, F.: Three-period quasi-periodic solutions in self-excited quasi-periodic Mathieu oscillator. Nonlinear Dyn. 39, 395–409 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Pandey, M., Rand, R.H., Zehnder, A.: Frequency locking in a forced Mathieu–van der Pol–Duffing system. Nonlinear Dyn. 54, 3–12 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Pandey, M., Rand, R.H., Zehnder, A.: Perturbation analysis of entrainment in a micromechanical limit cycle oscillator. Commun. Nonlinear Sci. Numer. Simul. 12, 291–301 (2007) CrossRefGoogle Scholar
  14. 14.
    Kalmar-Nagy, T., Stepan, J., Moon, F.C.: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121–142 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fahsi, A., Belhaq, M., Lakrad, F.: Suppression of hysteresis in a forced van der Pol–Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1609–1616 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Belhaq, M., Fahsi, A.: 2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator. Nonlinear Dyn. 53(1–2), 139–152 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Belhaq, M., Fahsi, A.: Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation. Nonlinear Dyn. 57, 275–287 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ji, J.C.: Nonresonant Hopf bifurcations of a controlled van der Pol–Duffing oscillator. J. Sound Vib. 297, 183–199 (2006) zbMATHCrossRefGoogle Scholar
  19. 19.
    Maccari, A.: Vibration amplitude control for a van der Pol Duffing oscillator with time delay. J. Sound Vib. 317, 20–29 (2008) CrossRefGoogle Scholar
  20. 20.
    Li, X.Y, Ji, J.C., Hansen, C.H., Tan, C.X.: The response of a Duffing–van der Pol oscillator under delayed feedback control. J. Sound Vib. 291, 644–655 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hamdi, M., Belhaq, M.: Self-excited vibration control for axially fast excited beam by a time delay state feedback. Chaos Solitons Fractals 41, 521–532 (2009) CrossRefGoogle Scholar
  22. 22.
    Belhaq, M., Sah, S.M.: Horizontal fast excitation in delayed van der Pol oscillator. Commun. Nonlinear Sci. Numer. Simul. 13, 1706–1713 (2008) CrossRefGoogle Scholar
  23. 23.
    Belhaq, M., Sah, S.M.: Fast parametrically excited van der Pol oscillator with time delay state feedback. Int. J. Non-Linear Mech. 43, 124–130 (2008) CrossRefGoogle Scholar
  24. 24.
    Suchorsky, M.K., Sah, S.M., Rand, R.H.: Using delay to quench undesirable vibrations. Nonlinear Dyn. 62, 407–416 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Sah, S.M., Belhaq, M.: Control of a delayed limit cycle using the tilt angle of a fast excitation. J. Vib. Control 17, 175–182 (2011) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979) zbMATHGoogle Scholar
  27. 27.
    Thomsen, J.J.: Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer, Berlin (2003) Google Scholar
  28. 28.
    Kuzmak, G.E.: Asymptotic solutions of nonlinear second-order differential equations with variable coefficients. J. Appl. Math. Mech. 23, 730–744 (1959) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Blekhman, I.I.: Vibrational Mechanics—Nonlinear Dynamic Effects, General Approach, Application. World Scientific, Singapore (2000) CrossRefGoogle Scholar
  30. 30.
    Niculescu, S.L.: Delay Effects on Stability: A Robust Control Approach. Springer, London (2001) zbMATHGoogle Scholar
  31. 31.
    Logemann, H., Rebarber, R.: The effect of small time-delays on closed-loop stability of boundary control systems. Math. Control Signals Syst. 9(2), 123–151 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Logemann, H., Townley, S.: The effect of small delays in the feedback loop on the stability of neutral systems. Syst. Control Lett. 27, 267–274 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    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) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Belhaq, M., Houssni, M.: Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dyn. 18, 1–24 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Rand, R.H., Guennoun, K., Belhaq, M.: 2:2:1 resonance in the quasi-periodic Mathieu equation. Nonlinear Dyn. 31, 187–193 (2003) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Shampine, L.F., Thompson, S.: Solving delay differential equations with dde23 (2000). PDF available on-line at http://www.radford.edu/~thompson/webddes/tutorial.pdf

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Laboratory of MechanicsUniversity Hassan II-Aïn ChockMaârif, CasablancaMorocco

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