Skip to main content
SpringerLink
Log in

Nonlinear vibration and of the Duffing oscillator to parametric excitation with time delay feedback

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

Abstract

The aim of this paper was to study the nonlinear vibrations of a parametric excited Duffing oscillator with time delay feedback. At some values of the time delay can be used to suppress the vibration of the nonlinear system. The method of multiple scales perturbation is applied to obtain the analytical solution of the system and the frequency response equation near sub-harmonic resonance case. The stability of the obtained nonlinear solution is studied and solved numerically. The effects of the different parameters of the system behavior are investigated. Analytical solution in this paper is in good agreement with the numerical simulation.

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

Similar content being viewed by others

References

  1. El-Bassiouny, A.F., El-kholy, S.: Resonances of a nonlinear SDOF system with time-delay in linear feedback control. Phys. Scr. 81, 015007 (12pp) (2010)

  2. El-Ganaini, W.A.A., Elgohary, H.A.: Vibration suppression via time-delay absorber described by non-linear differential equations. Adv. Theor. Appl. Mech. 4(2), 49–67 (2011)

    MATH  Google Scholar 

  3. El-Gohary, H.A., El-Ganaini, W.A.A.: Vibration suppression of a dynamical system to multi-parametric excitations via time-delay absorber. Appl. Math. Model. 36, 35–45 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Saeed, N.A., El-Ganini, W.A., Eissa, M.: Nonlinear time delay saturation-based controller for suppression of nonlinear beam vibrations. Appl. Math. Model. 37, 8846–8864 (2013)

    Article  MathSciNet  Google Scholar 

  5. El-Sayed, A.T., Bauomy, H.S.: Vibration control of helicopter blade flapping via time-delay absorber. Meccanica 49, 587–600 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nbendjo, B.R.N., Tchoukuegno, R., Woafo, P.: Active control with delay of vibration and chaos in a double-well Duffing oscillator. Chaos Solitons Fractals 18, 345–353 (2003)

    Article  MATH  Google Scholar 

  7. Zhao, Y.Y., Xu, J.: Effects of delayed feedback control on nonlinear vibration absorber system. J Sound Vib. 308, 212–230 (2007)

    Article  Google Scholar 

  8. Lu, W., Liu, Y.: Vibration control for the primary resonance of the Duffing oscillator by a time delay state feedback. Int. J. Nonlinear Sci. 8(3), 324–328 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Naik, R.D., Singru, P.M.: Resonance stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback. Commun. Nonlinear Sci. Numer. Simul. 16, 3397–3410 (2011)

    Article  MATH  Google Scholar 

  10. Sun, Z., Xu, W., Yang, X., Fang, T.: Effects of time delays on bifurcation and chaos in anon-autonomous system with multiple time delays. Chaos Solitons Fractals 31, 39–53 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. El-Bassiouny, A.F.: Fundamental and subharmonic resonances of harmonically oscillation with time delay state feedback. Shock Vib. 13, 65–83 (2006)

    Article  Google Scholar 

  12. El-Bassiouny, A.F.: Vibration control of a cantilever beam with time delay statefeedback. Verlag der ZeitschriftfürNaturforschung61a, 1–1220 (2006)

  13. El-Bassiouny, A.F.: Stability and oscillation of two coupled Duffing equations with time delay state feedback. Phys. Scr. 75, 726–735 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nayfeh, A.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

    Y. A. Amer

  2. Department of Basic Sciences, Modern Academy for Engineering and Technology, Maadi, Egypt

    A. T. EL-Sayed

  3. Department of Mathematics, Faculty of Science, Damietta University, Damietta, Egypt

    A. A. Kotb

Authors
  1. Y. A. Amer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. T. EL-Sayed
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Kotb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Kotb.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amer, Y.A., EL-Sayed, A.T. & Kotb, A.A. Nonlinear vibration and of the Duffing oscillator to parametric excitation with time delay feedback. Nonlinear Dyn 85, 2497–2505 (2016). https://doi.org/10.1007/s11071-016-2840-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2840-z

Keywords

Search

Navigation