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Effect of fast harmonic excitation on a self-excited motion in Van der Pol oscillator

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  • Bourkha, Rachid
  • Belhaq, Mohamed

Abstract

This paper studies the effect of horizontal and vertical fast harmonic parametric excitation on periodic self-excited motion in Van der Pol pendulum. The method of direct partition of motion is first applied to derive the equation governing the slow motion of the oscillator which is the main equation of vibrational mechanics. Then, a perturbation method is used on the slow motion equation to obtain analytical approximation for the periodic solution and its frequency in the slow dynamic. A relationship between the frequency of the limit cycle and the frequency of the fast harmonic excitation is derived. It is shown that in the case where the point of suspension of the pendulum is subjected to horizontal fast harmonic excitation, the periodic self-excitation can disappear. In contrast, this periodic solution persists in the case of the vertical fast harmonic excitation. Numerical simulations on the slow flow and on the original equation confirms the analytical prediction.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:2:p:621-627
    DOI: 10.1016/j.chaos.2006.03.099
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    Cited by:

    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. Fahsi, Abdelhak & Belhaq, Mohamed, 2009. "Hysteresis suppression and synchronization near 3:1 subharmonic resonance," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1031-1036.
    3. 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.

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