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Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator

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  • Sah, Simohamed
  • Belhaq, Mohamed

Abstract

We investigate the interaction effect of fast vertical parametric excitation and time delay on self-oscillation in a van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic and then we apply the averaging technique on this slow dynamic to derive a slow flow. In particular we analyze the slow flow to analytically approximate regions where self-excited vibrations can be eliminated. Numerical integration is performed and compared to the analytical results showing a good agreement for small time delay. It was shown that vertical parametric excitation, in the presence of delay, can suppress self-excited vibrations. These vibrations, however, persist for all values of the excitation frequency in the case of a fast vertical parametric excitation without delay [Bourkha R, Belhaq M. Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos, Solitons & Fractals, 2007;34(2):621–7.].

Suggested Citation

  • Sah, Simohamed & Belhaq, Mohamed, 2008. "Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1489-1496.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:5:p:1489-1496
    DOI: 10.1016/j.chaos.2006.10.040
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    References listed on IDEAS

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    1. Nana Nbendjo, B.R. & Woafo, P., 2007. "Active control with delay of horseshoes chaos using piezoelectric absorber on a buckled beam under parametric excitation," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 73-79.
    2. Ji, J.C. & Hansen, C.H., 2006. "Stability and dynamics of a controlled van der Pol–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 555-570.
    3. Bourkha, Rachid & Belhaq, Mohamed, 2007. "Effect of fast harmonic excitation on a self-excited motion in Van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 621-627.
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    1. Hamdi, Mustapha & Belhaq, Mohamed, 2009. "Self-excited vibration control for axially fast excited beam by a time delay state feedback," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 521-532.
    2. Ji, J.C. & Zhang, N. & Gao, Wei, 2009. "Difference resonances in a controlled van der Pol-Duffing oscillator involving time delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 975-980.
    3. Fahsi, Abdelhak & Belhaq, Mohamed, 2009. "Hysteresis suppression and synchronization near 3:1 subharmonic resonance," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1031-1036.

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