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


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.


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


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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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