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Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam

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Abstract

This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.

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Acknowledgments

The authors gratefully acknowledge the support of the State Key Program of National Natural Science Foundation of China (No. 11232009), the National Natural Science Foundation of China (No. 11372171), and Innovation Program of Shanghai Municipal Education Commission (No. 12YZ028).

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, 149 Yan Chang Road, Shanghai, 200072, China

    Qiao-Yun Yan, Hu Ding & Li-Qun Chen

  2. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Li-Qun Chen

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  1. Qiao-Yun Yan
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  2. Hu Ding
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Correspondence to Hu Ding.

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Yan, QY., Ding, H. & Chen, LQ. Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam. Nonlinear Dyn 78, 1577–1591 (2014). https://doi.org/10.1007/s11071-014-1535-6

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  • DOI: https://doi.org/10.1007/s11071-014-1535-6

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