Skip to main content
SpringerLink
Log in

On the quasi-periodic response in the delayed forced Duffing oscillator

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We explore the quasi-periodic (QP) vibrations in a delayed Duffing equation submitted to periodic forcing. The second-step perturbation method is applied on the slow flow of the oscillator to derive the slow–slow flow near the primary resonance. The QP solution corresponding to the nontrivial equilibrium of the slow–slow flow as well as its modulation envelope is predicted analytically. The influence of different system parameters on the QP response is reported and discussed. The analytical results show that for weak nonlinearity and small damping large-amplitude QP vibration induced by destabilization of limit cycle via Neimark–Sacker bifurcation occurs in a broadband of the excitation frequency and in large range of delay parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Hu, H., Dowell, E.H., Virgin, L.N.: Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dyn. 15, 311–327 (1998)

    Article  MATH  Google Scholar 

  2. Lu, W.L.Y., Liu, Y.: Vibration control for the primary resonance of the Duffing oscillator by a time delay state feedback. Int. J. Nonlinear Sci. 8, 324–328 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Rusinek, R., Weremczuk, A., Warminski, J.: Regenerative model of cutting process with nonlinear duffing oscillator. Mech. Mech. Eng. 15, 129–143 (2011)

    Google Scholar 

  4. Rusinek, R., Weremczuk, A., Kecik, K., Warminski, J.: Dynamics of a time delayed Duffing oscillator. Int. J. Nonlinear Mech. 65, 98–106 (2014)

    Article  Google Scholar 

  5. Szabelski, K., Warminski, J.: Self excited system vibrations with parametric and external excitations. J. Sound Vib. 187, 595–607 (1995)

    Article  MATH  Google Scholar 

  6. Luongo, A., Zulli, D.: Parametric, external and self-excitation of a tower under turbulent wind flow. J. Sound Vib. 330, 3057–3069 (2011)

    Article  Google Scholar 

  7. Hamdi, M., Belhaq, M.: Quasi-periodic oscillation envelopes and frequency locking in excited nonlinear systems with time delay. Nonlinear Dyn. 73, 1–15 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kirrou, I., Mokni, L., Belhaq, M.: On the quasiperiodic galloping of a wind-excited tower. J. Sound Vib. 32, 4059–4066 (2013)

    Article  MATH  Google Scholar 

  9. Hamdi, M., Belhaq, M.: Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay gain. J. Vib. Control (2015). doi:10.1177/1077546315597821

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rand, R.H., Guennoun, K., Belhaq, M.: 2:2:1 Resonance in the quasi-periodic Mathieu equation. Nonlinear Dyn. 31, 187–193 (2003)

  12. Belhaq, M., Fahsi, A.: 2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator. Nonlinear Dyn. 53, 139–152 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Abouhazim, N., Belhaq, M., Lakrad, F.: Three-period quasi-periodic oscillations in a self-excited quasi-periodic Mathieu equation. Nonlinear Dyn. 39, 395–409 (2005)

  14. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscil. Wiley, New York (1979)

    Google Scholar 

  15. Shampine, L.F., Thompson, S.: Solving delay differential equations with dde23. http://www.radford.edu/thompson/webddes/tutorial (2000)

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mechanics, University Hassan II-Casablanca, Casablanca, Morocco

    Ilham Kirrou & Mohamed Belhaq

Authors
  1. Ilham Kirrou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohamed Belhaq
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilham Kirrou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kirrou, I., Belhaq, M. On the quasi-periodic response in the delayed forced Duffing oscillator. Nonlinear Dyn 84, 2069–2078 (2016). https://doi.org/10.1007/s11071-016-2629-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2629-0

Keywords

Search

Navigation