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Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

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  • Ghayesh, Mergen H.
  • Amabili, Marco
  • Farokhi, Hamed

Abstract

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

Suggested Citation

  • Ghayesh, Mergen H. & Amabili, Marco & Farokhi, Hamed, 2013. "Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 8-29.
    • Handle: RePEc:eee:chsofr:v:52:y:2013:i:c:p:8-29
    DOI: 10.1016/j.chaos.2013.03.005
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    References listed on IDEAS

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    1. Zhang, Neng-Hui & Chen, Li-Qun, 2005. "Nonlinear dynamical analysis of axially moving viscoelastic strings," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1065-1074.
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    Cited by:

    1. Yan, Zhi & Wang, Wei & Liu, Xianbin, 2018. "Analysis of a quintic system with fractional damping in the presence of vibrational resonance," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 780-793.
    2. Farokhi, Hamed & Ghayesh, Mergen H. & Amabili, Marco, 2013. "In-plane and out-of-plane nonlinear dynamics of an axially moving beam," Chaos, Solitons & Fractals, Elsevier, vol. 54(C), pages 101-121.
    3. Li Jiang & Tao Wang & Qing-Xue Huang, 2023. "Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations," Mathematics, MDPI, vol. 11(7), pages 1-15, March.

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