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Extended Abstracts Spring 2016 pp 37–42Cite as

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Single-Impact Orbits Near Grazing Periodic Orbits for an Impact Oscillator

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Part of the book series: Trends in Mathematics ((RPCRMB,volume 8))

Abstract

We describe a geometric approach to understand the mechanism of creation and annihilation of single-impact periodic orbits close to grazing for a general second-order one degree of freedom impact oscillator. Here, non-degenerate grazing (nonzero acceleration) is assumed, with approaches to degenerate grazing also outlined: this is work in progress. The method in principle extends to more degrees of freedom (coupled oscillators, for example) and to a variety of restitution rules such as soft impacts, delays or sticking.

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References

  1. S.R. Bishop, Impact oscillators. Phil. Trans. Roy. Soc. London A 347, 347–351 (1994)

    Article  MATH  Google Scholar 

  2. J.W. Bruce, Seeing – the mathematical viewpoint. Math. Intell. 6, 18–25 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. J.W. Bruce, P.J. Giblin, Curves and Singularities (Cambridge University Press, Cambridge, 1984)

    MATH  Google Scholar 

  4. J.W. Bruce, P.J. Giblin, Outlines and their duals. Proc. London Math. Soc. 50, 552–570 (1984)

    MathSciNet  MATH  Google Scholar 

  5. C.J. Budd, F.J. Dux, Chattering and related behaviour in impact oscillators. Phil. Trans. Roy. Soc. London A 347, 365–389 (1994)

    Article  MATH  Google Scholar 

  6. C.J. Budd, F.J. Dux, A. Cliffe, The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators. J. Sound Vib. 184, 475–502 (1995)

    Article  MATH  Google Scholar 

  7. D.R.J. Chillingworth, Discontinuity geometry for an impact oscillator. Dyn. Syst. 17, 389–420 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. D.R.J. Chillingworth, Dynamics of an impact oscillator near a degenerate graze. Nonlinearity 23, 2723–2748 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. D.R.J. Chillingworth, A.B. Nordmark, Periodic orbits close to grazing for an impact oscillator, in Recent Trends in Dynamical Systems, vol. 35, Springer Proceedings in Mathematics and Statistics, ed. by A. Johann, et al. (2013), pp. 25–37

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. M. Di Bernardo et al., Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50, 629–701 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. J.F. Mason, N. Humphries, P.T. Piiroinen, Numerical analysis of codimension-one, -two and -three bifurcations in a periodically-forced impact oscillator with two discontinuity surfaces. Math. Comput. Simul. 95, 98–110 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. A.B. Nordmark, P.T. Piiroinen, Simulation and stability analysis of impacting systems with complete chattering. Nonlinear Dyn. 58, 85–106 (2009)

    Google Scholar 

  15. J. Sotomayor, M.A. Teixeira, Vector fields near the boundary of a 3-manifold, Dynamical Systems Valparaiso 1986, vol. 1331, Lecture Notes in Math (Springer, Berlin, 1988), pp. 169–195

    Chapter  Google Scholar 

  16. G.W. Whiston, Global dynamics of a vibro-impacting linear oscillator. J. Sound Vib. 118, 395–429 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. G.W. Whiston, Singularities in vibro-impact dynamics. J. Sound Vib. 152, 427–460 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. X. Zhao, H. Dankowicz, Unfolding degenerate grazing dynamics in impact actuators. Nonlinearity 19, 399–418 (2006)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK

    D. R. J. Chillingworth

Authors
  1. D. R. J. Chillingworth
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Correspondence to D. R. J. Chillingworth .

Editor information

Editors and Affiliations

  1. DEIB, Politecnico di Milano , Milano, Italy

    Alessandro Colombo

  2. Department of Engineering Mathematics, University of Bristol , BRISTOL, Avon, United Kingdom

    Mike Jeffrey

  3. Departament de Matemàtiques, Universitat Politècnica de Catalunya , Barcelona, Spain

    J. Tomàs Lázaro

  4. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Josep M. Olm

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Chillingworth, D.R.J. (2017). Single-Impact Orbits Near Grazing Periodic Orbits for an Impact Oscillator. In: Colombo, A., Jeffrey, M., Lázaro, J., Olm, J. (eds) Extended Abstracts Spring 2016. Trends in Mathematics(), vol 8. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55642-0_7

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