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<p>The nonlinear vibrations of a cantilever beam model and the dynamic analysis of several resonances (primary and secondary) controlled by the proportional-derivative (PD) control are studied and discussed. The multiple time-scales method is applied to solve the dynamical system of the cantilever beam where the response curves before and after control are obtained. Through these curves, the stability of the beam’s oscillatory behavior is tested under the effect of different physical parameters. The&#xa0;beam suffers&#xa0;from&#xa0;high&#xa0;vibration amplitudes in each resonance case, especially in the superharmonic and subharmonic resonance cases. Besides the nontrivial amplitude of oscillation, a trivial one appears in some secondary resonance cases. Tuning the control gains can lead to acceptable vibration mitigation levels besides approaching the trivial amplitude in some resonance cases. Accordingly, the analytical and numerical results are compared before and after control to validate the control law.</p>

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Primary, superharmonic and subharmonic resonances control of an oscillatory cantilever beam excited transversely at its free end

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月18日

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  • Eman Desoky,
  • Magdy Kamel,
  • Ali Kandil

摘要

The nonlinear vibrations of a cantilever beam model and the dynamic analysis of several resonances (primary and secondary) controlled by the proportional-derivative (PD) control are studied and discussed. The multiple time-scales method is applied to solve the dynamical system of the cantilever beam where the response curves before and after control are obtained. Through these curves, the stability of the beam’s oscillatory behavior is tested under the effect of different physical parameters. The beam suffers from high vibration amplitudes in each resonance case, especially in the superharmonic and subharmonic resonance cases. Besides the nontrivial amplitude of oscillation, a trivial one appears in some secondary resonance cases. Tuning the control gains can lead to acceptable vibration mitigation levels besides approaching the trivial amplitude in some resonance cases. Accordingly, the analytical and numerical results are compared before and after control to validate the control law.