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A Quadratic Nonlinear Oscillator

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Galloping Instability to Chaos of Cables

Part of the book series: Nonlinear Physical Science ((NPS))

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Abstract

As previously discussed nonlinear vibration and periodic motions of galloping cables, analytical solutions for period-m motions in a periodically forced, and quadratic nonlinear oscillator are presented first through the Fourier series solutions with finite harmonic terms, and the stability and bifurcation analyses of the corresponding period-m motions are carried out. The bifurcation trees of period-1 motions to chaos are presented for a better understanding of complex motions in such a quadratic nonlinear oscillator . Trajectories and amplitude spectrums are illustrated numerically.

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References

  • Luo, A. C. J. (2012). Continuous dynamical systems. Beijing/Glen Carbon: Higher Education Press/L&H Scientific.

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  • Luo, A. C. J., & Yu, B. (2013a). Analytical solutions for stable and unstable period-1 motion in a periodically forced oscillator with quadratic nonlinearity. ASME Journal of Vibration and Acoustics, 135(Article no: 034503), 5 pages.

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  • Luo, A. C. J., & Yu, B. (2013b). Period-m motions and bifurcation trees in a periodically excited, quadratic nonlinear oscillator. Discontinuity, Nonlinearity, and Complexity, 3, 263–288.

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  • Luo, A. C. J. (2014). Toward analytical chaos in nonlinear systems. New York: Wiley.

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  • Luo, A. C. J., & Yu, B. (2015). Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. Journal of Vibration and Control, 21(5), 896–906.

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Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA

    Albert C. J. Luo

  2. Department of Mechanical and Industrial Engineering, University of Wisconsin–Platteville, Platteville, WI, USA

    Bo Yu

Authors
  1. Albert C. J. Luo
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  2. Bo Yu
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Corresponding author

Correspondence to Albert C. J. Luo .

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© 2017 Springer Nature Singapore Pte Ltd. and Higher Education Press, Beijing

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Luo, A.C.J., Yu, B. (2017). A Quadratic Nonlinear Oscillator. In: Galloping Instability to Chaos of Cables. Nonlinear Physical Science. Springer, Singapore. https://doi.org/10.1007/978-981-10-5242-2_4

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