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Using Fourier differential quadrature method to analyze transverse nonlinear vibrations of an axially accelerating viscoelastic beam

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Abstract

In this paper, a Fourier expansion-based differential quadrature (FDQ) method is developed to analyze numerically the transverse nonlinear vibrations of an axially accelerating viscoelastic beam. The partial differential nonlinear governing equation is discretized in space region and in time domain using FDQ and Runge–Kutta–Fehlberg methods, respectively. The accuracy of the proposed method is represented by two numerical examples. The nonlinear dynamical behaviors, such as the bifurcations and chaotic motions of the axially accelerating viscoelastic beam, are investigated using the bifurcation diagrams, Lyapunov exponents, Poincare maps, and three-dimensional phase portraits. The bifurcation diagrams for the in-plane responses to the mean axial velocity, the amplitude of velocity fluctuation, and the frequency of velocity fluctuation are, respectively, presented when other parameters are fixed. The Lyapunov exponents are calculated to further identify the existence of the periodic and chaotic motions in the transverse nonlinear vibrations of the axially accelerating viscoelastic beam. The conclusion is drawn from numerical simulation results that the FDQ method is a simple and efficient method for the analysis of the nonlinear dynamics of the axially accelerating viscoelastic beam.

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Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NSFC) through grant Nos. 11290152, 11072008 and 11172009, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

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Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, Beijing , 100124, People’s Republic of China

    W. Zhang, D. M. Wang & M. H. Yao

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  1. W. Zhang
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  2. D. M. Wang
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  3. M. H. Yao
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Correspondence to W. Zhang.

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Zhang, W., Wang, D.M. & Yao, M.H. Using Fourier differential quadrature method to analyze transverse nonlinear vibrations of an axially accelerating viscoelastic beam. Nonlinear Dyn 78, 839–856 (2014). https://doi.org/10.1007/s11071-014-1481-3

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  • DOI: https://doi.org/10.1007/s11071-014-1481-3

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