Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law

Arne Nordmark, Harry Dankowicz, Alan Champneys

Research output: Contribution to journalArticlepeer-review


This paper concerns the non-smooth dynamics of planar mechanical systems with isolated contact in the presence of Coulomb friction. Following Stronge [Impact Mechanics, Cambridge University Press, Cambridge, 2000], a set of closed-form analytic formulae is derived for a rigid-body impact law based on an energetic coefficient of restitution and a resolution of the impact phase into distinct segments of relative slip and stick. Thus, the impact behavior is consistent both with the assumption of Coulomb friction and with the dissipative nature of impacts. The analysis highlights the presence of boundaries between open regions of initial conditions and parameter values corresponding to distinct forms of the impact law and investigates the smoothness properties of the impact law across these boundaries. It is shown how discontinuities in the impact law are associated with discontinuity-induced bifurcations of periodic trajectories, including non-smooth folds and persistence scenarios. Numerical analysis of an example mechanical model is used to illustrate and validate the conclusions.

Original languageEnglish (US)
Pages (from-to)1011-1023
Number of pages13
JournalInternational Journal of Non-Linear Mechanics
Issue number10
StatePublished - Dec 2009


  • Bifurcation
  • Coulomb friction
  • Discontinuities
  • Impact
  • Non-smooth
  • Rigid body mechanics

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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