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Theoretical analysis of co-dimension-two grazing bifurcations in \(\varvec{n}\)-degree-of-freedom impact oscillator with symmetrical constrains

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Abstract

The co-dimension-two grazing bifurcation in n-degree-of-freedom impact oscillator with bilateral constraints is investigated. Using a classical approach of discontinuity mappings, the existence conditions for co-dimension-two grazing bifurcation are obtained and simplified skillfully. For the impact oscillator has two discontinuity surfaces, the existence conditions become complex compared with the unilateral constraints and are discussed in four different cases. Furthermore, The deduced theoretical results are employed to identify and explore the distribution of co-dimension-two grazing bifurcation points for a two-degree-of-freedom impact oscillator with bilateral constraints as an example. The co-dimension-two grazing bifurcation points are presented, and the complex dynamic behaviors in the vicinity of the co-dimension-two grazing bifurcation point are displayed numerically by unfolding chart and phase diagrams.

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References

  1. Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Grazing bifurcations in impact oscillators. Phys. Rev. E 50(6), 4427–4444 (1994)

    Article  MathSciNet  Google Scholar 

  2. Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145(2), 279–297 (1991)

    Article  Google Scholar 

  3. Nordmark, A.B.: Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators. Nonlinearity 14(6), 1517–1542 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Nordmark, A.B., Dankowica, H., Champneys, A.R.: Discontinuity-induced bifurcations in systems with impacts and friction: discontinuities in the impact law. Int. J. Nonlinear Mech. 44(10), 1011–1023 (2009)

    Article  MATH  Google Scholar 

  5. di Bernardo, M., Budd, C.J., Champneys, A.R.: Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, New York (2008)

    Google Scholar 

  6. Dankowicz, H., Katzenbath, M.: Discontinuity-induced bifurcations in models of mechanical contact, capillary, adhesion and cell division: a common framework. Phys. D 241(22), 1869–1881 (2012)

    Article  MathSciNet  Google Scholar 

  7. Humphries, N., Piiroinen, P.T.: A discontinuity-geometry view of the relationship between saddle-node and grazing bifurcations. Phys. D 241(22), 1911–1918 (2012)

    Article  MathSciNet  Google Scholar 

  8. Kryzhevich, S., Wiercigroch, M.: Topology of vibro-impact systems in the neighborhood of grazing. Phys. D 241(22), 1919–1931 (2012)

    Article  MathSciNet  Google Scholar 

  9. Masona, J.F., Humphriesa, N., Piiroinen, P.T.: Numerical analysis of codimension-one, -two and -three bifurcations in a periodically-forced impact oscillator with two discontinuity surfaces. Math. Comput. Simul. 95, 98C110 (2014)

    Google Scholar 

  10. di Bernardo, M., Budd, C.J., Champneys, A.R.: Normal form map for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Phys. D 160(3), 222–254 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Zhao, X., Reddy, C.K., Nayfeh, A.H.: Nonlinear dynamics of an electrically driven impact microactuator. Nonlinear Dyn. 40(3), 227–239 (2005)

    Article  MATH  Google Scholar 

  12. Thota, P., Zhao, X., Dankowicz, H.: Co-dimension-two grazing bifurcations in single-degree-of-freedom impact oscillators. J. Comput. Nonlinear Dyn. 1(4), 328–335 (2006)

    Article  Google Scholar 

  13. Csaba, H., Champneys, A.R.: Grazing bifurcations and chatter in a pressure relief valve model. Phys. D 241(22), 2068–2076 (2012)

    Article  MathSciNet  Google Scholar 

  14. Elmegard, M., Krauskopf, B., Osinga, H.M., Starke, J., Thomsen, J.: Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment. Nonlinear Dyn. 77(3), 951–966 (2014)

  15. Thota, P., Dankowica, H.: TC-HAT(\(\hat{TC}\)): a novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems. SIAM J. Appl. Dyn. Syst. 7(4), 1283–1322 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kang, W., Thota, P., Wilcox, B., Dankowicz, H.: Bifurcation analysis of a microactuator using a new toolbox for continuation of hybrid system trajectories. J. Comput. Nonlinear Dyn. 4(1), 1–8 (2009)

    Article  Google Scholar 

  17. Dankowicz, H., Schilder, F.: A extended continuation problem for bifurcation analysis in the presence of constraints. J. Comput. Nonlinear Dyn. 6(3), 1–8 (2011)

    Article  Google Scholar 

  18. Chvez, J.P., Pavlovskaia, E., Wiercigroch, M.: Bifurcation analysis of a piecewise-linear impact oscillator with drift. Nonlinear Dyn. 77(1–2), 213–227 (2014)

    Article  Google Scholar 

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Acknowledgments

This work is supported by the Project Sponsored by the National Natural Science Foundation of China (Nos. 11372077 and 10972059) , the Guangxi Natural Science Foundation (Nos. 2013GXNSFAA- 019017, 2010GXNSFA013110 and 2014GXNSFBA118024), and the Scientific Research Foundation of Guangxi University (No. XBZ120251).

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  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Jieqiong Xu, Pu Chen & Qunhong Li

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  1. Jieqiong Xu
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  2. Pu Chen
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Correspondence to Qunhong Li.

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Xu, J., Chen, P. & Li, Q. Theoretical analysis of co-dimension-two grazing bifurcations in \(\varvec{n}\)-degree-of-freedom impact oscillator with symmetrical constrains. Nonlinear Dyn 82, 1641–1657 (2015). https://doi.org/10.1007/s11071-015-2266-z

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  • DOI: https://doi.org/10.1007/s11071-015-2266-z

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