Nonlinear Dynamics

, Volume 81, Issue 4, pp 1699–1716 | Cite as

On the mathematical basis of solid friction

  • Mike R. Jeffrey
Original Paper


A piecewise-smooth ordinary differential equation model of a dry-friction oscillator is studied, as a paradigm for the role of nonlinear and hysteretic terms in discontinuities of dynamical systems. The friction discontinuity is a switch in direction of the contact force in the transition between left- and rightward slipping motion. Nonlinear terms introduce dynamics that is novel in the context of piecewise-smooth dynamical systems theory (in particular they are outside the standard Filippov convention), but are shown to account naturally for static friction, and moreover provide a simple route to including hysteresis. The nonlinear terms are understood in terms of dummy dynamics at the discontinuity, given a formal derivation here. The result is a three-parameter model built on the minimal mathematical features necessary to account for the key characteristics of dry friction. The effect of compliance can be distinguished from the contact model, and numerical simulations reveal that all behaviours persist under smoothing and under small random perturbations, but nonlinear effects can be made to disappear abruptly amid sufficient noise.


Dynamics Sticking Discontinuous Friction Filippov Perturbation Piecewise-smooth Nonsmooth 


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Engineering MathematicsUniversity of BristolBristolUK

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