Skip to main content
SpringerLink
Log in

A maximal monotone impact law for the 3-ball Newton’s cradle

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

The 3-ball Newton’s cradle is used as a stepping stone to divulge the structure of impact laws. A continuous conewise linear impact law that maps the preimpact contact velocities to the postimpact contact velocities is proposed for the 3-ball Newton’s cradle. The proposed impact law is kinematically, kinetically, and energetically consistent. It reproduces the outcomes of experimental observation. Moreover, it is in accordance with the outcome of the collision of three identical linear-elastic thin rods for which the impact process is governed by the one-dimensional wave equation. The proposed impact law is shown to be nonexpansive. Therefore, the relationship between the mean contact velocity and its dual, the impulsive force, is maximal monotone. A counterexample to maximal cyclical monotonicity of this relationship allows us to conclude that no dissipation function exists for the proposed impact law.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Acary, V., Brogliato, B.: Concurrent multiple impacts modelling: case-study of a 3-ball chain. In: Computational Fluid and Solid Mechanics, Second MIT Conference. Elsevier, Amsterdam (2003)

    Google Scholar 

  2. Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008)

    MATH  Google Scholar 

  3. Baumann, M., Leine, R.I.: Convergence based synchronisation of unilaterally constrained multibody systems. In: Proceedings of the ENOC 2014 Conference, Vienna (2014)

    Google Scholar 

  4. Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control, 3rd edn. Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  5. Darboux, G.: Étude géométrique sur les percussions et les chocs des corps. Bul. Sci. Math. Astron., Deux. Sér. 4(1), 126–160 (1880)

    MATH  Google Scholar 

  6. Glocker, Ch.: An introduction to impacts. In: Nonsmooth Mechanics of Solids. CISM Courses and Lectures, vol. 485, pp. 45–101. Springer, Vienna (2006)

    Chapter  Google Scholar 

  7. Glocker, Ch.: Energetic consistency conditions for standard impacts, Part I: Newton-type inequality impact laws and Kane’s example. Multibody Syst. Dyn. 29(1), 77–117 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Glocker, Ch.: Energetic consistency conditions for standard impacts, Part II: Poisson-type inequality impact laws. Multibody Syst. Dyn. 32(4), 445–509 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Glocker, Ch., Aeberhard, U.: The geometry of Newton’s cradle. In: Alart, P., Maisonneuve, O., Rockafellar, R.T. (eds.) Nonsmooth Mechanics and Analysis. Theoretical and Numerical Advances, AMMA, vol. 12, pp. 185–194. Springer, Berlin (2006)

    Chapter  Google Scholar 

  10. Graff, K.F.: Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975)

    MATH  Google Scholar 

  11. Hertz, H.: Über die Berührung fester elastischer Körper. J. Reine Angew. Math. 92, 156–171 (1882)

    MathSciNet  Google Scholar 

  12. Keller, J.B.: Impact with friction. J. Appl. Mech. 53, 1–4 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Leine, R.I., Baumann, M.: Variational analysis of inequality impact laws. In: Proceedings of the ENOC 2014 Conference, Vienna (2014)

    Google Scholar 

  14. Leine, R.I., van de Wouw, N.: Stability and Convergence of Mechanical Systems with Unilateral Constraints. Lecture Notes in Applied and Computational Mechanics, vol. 36. Springer, Berlin (2008)

    MATH  Google Scholar 

  15. Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. I. Theoretical framework. Proc. R. Soc. A 464(2100), 3193–3211 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nguyen, N.S., Brogliato, B.: Multiple Impacts in Dissipative Granular Chains. Lecture Notes in Applied and Computational Mechanics, vol. 72. Springer, Berlin (2014)

    Google Scholar 

  17. Nguyen, N.S., Zhang, H., Brogliato, B.: Multiple impacts with friction in the rocking block and tapered chains. In: Proceedings of the ENOC 2011 Conference, Rome (2011)

    Google Scholar 

  18. Payr, M.: An experimental and theoretical study of perfect multiple contact collisions in linear chains of bodies. Dissertation, ETH No. 17808, Zurich (2008)

  19. Rockafellar, R.T., Wets, R.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  20. Seifried, R., Schiehlen, W., Eberhard, P.: The role of the coefficient of restitution on impact problems in multi-body dynamics. J. Multi-Body Dyn. 224(3), 279–306 (2010)

    Google Scholar 

  21. Stronge, W.J.: Rigid body collisions with friction. Proc. R. Soc. Lond. A 431(1881), 169–181 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Winandy, T., Leine, R.I.: Towards a maximal monotone impact law for Newton’s cradle. In: Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, Barcelona, Spain (2015)

    Google Scholar 

  23. Wittenburg, J.: Schwingungslehre. Springer, Berlin (1996)

    Book  Google Scholar 

Download references

Acknowledgements

This research is supported by the Fonds National de la Recherche, Luxembourg (Proj. Ref. 8864427).

Author information

Authors and Affiliations

  1. Institute for Nonlinear Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569, Stuttgart, Germany

    Tom Winandy & Remco I. Leine

Authors
  1. Tom Winandy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Remco I. Leine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tom Winandy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Winandy, T., Leine, R.I. A maximal monotone impact law for the 3-ball Newton’s cradle. Multibody Syst Dyn 39, 79–94 (2017). https://doi.org/10.1007/s11044-016-9533-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-016-9533-8

Keywords

Search

Navigation