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Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam

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  • Zhang, Wei

Abstract

This paper presents an analysis of the chaotic motion and its control for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. A new method of controlling chaotic motion for the nonlinear nonplanar oscillations of the cantilever beam, refereed as to the force control approach, is proposed for the first time. The governing nonlinear equations of nonplanar motion under combined parametric and external excitations are obtained. The Galerkin procedure is applied to the governing equation to obtain a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations for the in-plane and out-of-plane modes. The work is focused on the case of 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance-primary resonance for the out-of-plane mode. The method of multiple scales is used to transform the parametrically and externally excited system to the averaged equations which have a constant perturbation force. Based on the averaged equations obtained here, numerical simulation is utilized to discover the periodic and chaotic motions for the nonlinear nonplanar oscillations of the cantilever beam. The numerical results indicate that the transverse excitation in the z direction at the free end can control the chaotic motion to a period n motion or a static state for the nonlinear nonplanar oscillations of the cantilever beam. The methodology of controlling chaotic motion by using the transverse excitation is proposed. The transverse excitation in the z direction at the free end may be thought about to be an open-loop control. For the problem investigated in this paper, this approach is an effective methodology of controlling chaotic motion to a period n motion or a static state for the nonlinear nonplanar oscillations of the cantilever beam.

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  • Zhang, Wei, 2005. "Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 731-745.
    • Handle: RePEc:eee:chsofr:v:26:y:2005:i:3:p:731-745
    DOI: 10.1016/j.chaos.2005.01.042
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    References listed on IDEAS

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    1. Emans, Joseph & Wiercigroch, Marian & Krivtsov, Anton M., 2005. "Cumulative effect of structural nonlinearities: chaotic dynamics of cantilever beam system with impacts," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1661-1670.
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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. F. B. PELAP & Y. L. Makenne & R. Kengne, 2015. "Effects of a Non-Sinusoidal Wind on Plants," Asian Engineering Review, Asian Online Journal Publishing Group, vol. 2(1), pages 1-14.
    3. 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.
    4. Zhou, Liangqiang & Ji, Peng & Chen, Fangqi, 2021. "Chaos and subharmonic bifurcation of a composite laminated buckled beam with a lumped mass," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).

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