One-dimensional chaos in a system with dry friction: analytical approach

Abstract

We introduce a new analytical method, which allows to find chaotic regimes in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered. The corresponding mathematical model is being studied. We show that the considered dynamical system is a skew product over a piecewise smooth mapping of a segment (the so-called base map). For this base map we demonstrate existence of a domain of parameters where a chaotic dynamics can be observed. We prove existence of an infinite set of periodic points of arbitrarily big period. Moreover, a reduction of the considered map to a compact subset of the segment is semi-conjugated to a shift on the set of one-sided infinite boolean sequences. We find conditions, sufficient for existence of a superstable periodic point of the base map. The obtained result partially solves a general problem: theoretical confirmation of chaotic and periodic regimes numerically and experimentally observed for models of percussion drilling.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    di Bernardo M, Budd ChJ, Champneys AR, Kowalczyk P, Nordmark AB, Tost GO, Piiroinen PT (2008) Bifurcations in nonsmooth dynamical systems. SIAM Rev 50:629–701

    MathSciNet  ADS  Article  MATH  Google Scholar 

  2. 2.

    di Bernardo M, Kowalczyk P, Nordmark AB (2003) Sliding bifurcations: a novel mechanism for a sudden onset of chaos in dry friction oscillators. Int J Bifurc Chaos 13:2935–2948

    Article  MATH  Google Scholar 

  3. 3.

    Blazejczyk-Okolewska B, Kapitanak T (1996) Dynamics of impact oscillator with dry friction. Chaos Solitons Fractals 7:1455–1459

    ADS  Article  Google Scholar 

  4. 4.

    Casapulla C, Portioli F, Maione A, Landolfo R (2013) A macro-block model for in-plane loaded masonry walls with non-associative Coulomb friction. Meccanica 48:2107–2126

    Article  MATH  Google Scholar 

  5. 5.

    Csernák G, Stépán G, Shaw SW (2007) Sub-harmonic resonant solutions of a harmonically excited dry friction oscillator. Nonlinear Dyn 50:93–109

    Article  MATH  Google Scholar 

  6. 6.

    Feeny B, Moon FC (1994) Chaos in a forced dry-friction oscillator: experiments and numerical modelling. J Sound Vib 170:303–323

    ADS  Article  MATH  Google Scholar 

  7. 7.

    Kiseleva M (2013) Oscillations of dynamical systems applied in drilling: analytical and numerical methods. PhD Thesis, Jyväskylä University Printing House

  8. 8.

    Krivtsov AM, Wiercigroch M (1999) Dry friction model of percussive drilling. Meccanica 34:425–434

    Article  MATH  Google Scholar 

  9. 9.

    Krivtsov AM, Wiercigroch M (2000) Penetration rate prediction for percussive drilling via dry friction model. Chaos Solitons Fractals 11:2479–2485

    ADS  Article  MATH  Google Scholar 

  10. 10.

    Kowalczyk P, Piiroinen PT (2008) Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator. Phys D Nonlinear Phenom 237:1053–1073

    MathSciNet  ADS  Article  MATH  Google Scholar 

  11. 11.

    Makarenkov O, Lamb JSW (2012) Dynamics and bifurcations of nonsmooth systems: a survey. Phys D Nonlinear Phenom 241:1826–1844

    MathSciNet  ADS  Article  Google Scholar 

  12. 12.

    Pugno NM, Qifang Yin, Xinghua Shi, Capozza R (2013) A generalization of the Coulombs friction law: from graphene to macroscale. Meccanica 48:1845–1851

    Article  MATH  Google Scholar 

  13. 13.

    Stefański A, Wojewoda J, Wiercigroch M, Kapitaniak T (2003) Chaos caused by non-reversible dry friction. Chaos Solitons Fractals 16:661–664

    ADS  Article  MATH  Google Scholar 

  14. 14.

    Wiercigroch M, de Kraker A (eds) (2000) Applied nonlinear dynamics and chaos of mechanical systems with discontinuities. World Scientific, Singapore, New Jersey, London, Hong Kong

  15. 15.

    Wojewoda J, Kapitanak T, Barron R, Brindley J (1993) Complex behaviour of a quasiperiodically forced experimental system with dry friction. Chaos Solitons Fractals 3:35–46

    ADS  Article  MATH  Google Scholar 

  16. 16.

    Filippov AF (1998) Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  17. 17.

    Hogan SJ, Higham L, Griffin TCL (2007) Dynamics of a piecewise linear map with a gap Proc. R Soc A 463:49–65

    MathSciNet  ADS  Article  MATH  Google Scholar 

  18. 18.

    Simpson DJW, Meiss JD (2010) Aspects of bifurcation theory for piecewise-smooth, continuous systems. arXiv:1006.4123v1

  19. 19.

    Awrejcevicz J (1988) Chaotic motion in a non-linear oscillator with friction. KSME J 2:104–109

    Google Scholar 

  20. 20.

    Galvanetto U (2005) Unusual chaotic attractors in nonsmooth dynamic systems. Int J Bifurc Chaos 15:4081–4086

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Pavlovskaia EM, Wiercigroch M (2007) Low-dimensional maps for piecewise smooth oscillators. J Sound Vib 305:750–771. doi:10.1063/1.2904774

    MathSciNet  ADS  Article  Google Scholar 

  22. 22.

    Szalai R, Osinga HM (2008) Invariant polygons in systems with grazing–sliding. Chaos 28:023121

    MathSciNet  ADS  Article  Google Scholar 

  23. 23.

    Szalai R, Osinga HM (2009) Arnold tongues arising from a grazing–sliding bifurcation. SIAM J Appl Dyn Syst 8:1434–1461

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Block LS, Coppel, WA (1992) Dynamics in one dimension. Lecture notes in mathematics, 1513. Springer-Verlag, Berlin, 1992. viii+249 pp. ISBN 3-540-55309-6

  25. 25.

    Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82:49–68

    MathSciNet  Article  Google Scholar 

  26. 26.

    Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge

    Google Scholar 

  27. 27.

    Sharkovskii OM (1964) Co-existence of cycles of a continuous mapping of a line onto itself. ukranian Math Z 16:61–71

    Google Scholar 

  28. 28.

    Wiercigroch M, Wojevoda AJ, Krivtsov AM (2005) Dynamics of ultrasonic percussive drilling of hard rocks. J Sound Vib 280:739–757

    ADS  Article  Google Scholar 

  29. 29.

    Banerjee S, Grebogi C (1999) Border collision bifurcations in two-dimensional piecewise smooth maps. Physical Rev E 59:4052–4061

    ADS  Article  Google Scholar 

  30. 30.

    Chin W, Ott E, Nusse HN, Grebogi C (1999) Grazing bifurcations in impact oscillators. Physical Rev E 50:4427–4444

    MathSciNet  ADS  Article  Google Scholar 

  31. 31.

    Devaney RL (1987) An introduction to chaotic dynamical systems. Addison-Wesley, Redwood City

    Google Scholar 

  32. 32.

    Mayergoyz ID (2003) Mathematical models of hysteresis and their applications: second edition (Electromagnetism). Academic Press. ISBN 978-0-12-480873-7

  33. 33.

    Krasnosel’skii M, Pokrovskii A (1989) Systems with hysteresis. Springer-Verlag, New York. ISBN 978-0-387-15543-2

  34. 34.

    Mease KD, Bharadwaj S, Iravanchy S (2003) Timescale analysis for nonlinear dynamical systems. J Guid Control Dyn 26:318–330

    ADS  Article  Google Scholar 

  35. 35.

    Litak G, Arkadiusz S, Rusinek R, Sen Asok K (2013) Intermittency and multiscale dynamics in milling of fiber reinforced composites. Meccanica 48:783–789

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work has been partially supported by Russian Foundation for Basic Researches, Grants 12-01-00275-a and 14-01-00202, by Saint-Petersburg State University under Thematic Plans 6.0.112.2010 and 6.38.223.2014, by the UK Royal Society (joint research project of University of Aberdeen and Saint-Petersburg State University), by Centre for Research and by FEDER funds through COMPETE Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (FCT \(-\) Fundação para a Ciência e a Tecnologia), within projects PTDC/MAT/113470/2009 FCT and PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. Authors are grateful to Prof. Ron Chen, Prof. Marian Wiercigroch, Prof. Viktor Avrutin, Dr. James Ing and anonymous Referees for their precious remarks.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sergey Kryzhevich.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Begun, N., Kryzhevich, S. One-dimensional chaos in a system with dry friction: analytical approach. Meccanica 50, 1935–1948 (2015). https://doi.org/10.1007/s11012-014-0071-2

Download citation

Keywords

  • Chaos
  • Mappings of segments
  • Dry friction
  • Filippov systems
  • Turbulence