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Single-mode control and chaos of cantilever beam under primary and principal parametric excitations

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  • El-Bassiouny, A.F.

Abstract

A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.

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  • El-Bassiouny, A.F., 2006. "Single-mode control and chaos of cantilever beam under primary and principal parametric excitations," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1098-1121.
    • Handle: RePEc:eee:chsofr:v:30:y:2006:i:5:p:1098-1121
    DOI: 10.1016/j.chaos.2005.09.015
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    References listed on IDEAS

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    1. Musielak, D.E. & Musielak, Z.E. & Benner, J.W., 2005. "Chaos and routes to chaos in coupled Duffing oscillators with multiple degrees of freedom," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 907-922.
    2. El-Bassiouny, A.F. & Abdelhafez, H.M., 2001. "Prediction of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic and quartic non-linearities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 57(1), pages 61-80.
    3. Cao, Hongjun, 2005. "Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1387-1398.
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    Cited by:

    1. El-Bassiouny, A.F., 2009. "On methods for continuous systems with quadratic, cubic and quantic nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1308-1316.

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