WIAS Preprint No. 2405, (2017)

Three examples concerning the interaction of dry friction and oscillations


  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 34C55 47J20 49J40 74N30


  • Rate-independent friction, averaging of friction, play operator with time-dependent thresholds, locomotion, rate-and-state friction, Hopf bifurcation




We discuss recent work concerning the interaction of dry friction, which is a rate independent effect, and temporal oscillations. First, we consider the temporal averaging of highly oscillatory friction coefficients. Here the effective dry friction is obtained as an infimal convolution. Second, we show that simple models with state-dependent friction may induce a Hopf bifurcation, where constant shear rates give rise to periodic behavior where sticking phases alternate with sliding motion. The essential feature here is the dependence of the friction coefficient on the internal state, which has an internal relaxation time. Finally, we present a simple model for rocking toy animal where walking is made possible by a periodic motion of the body that unloads the legs to be moved.

Appeared in

  • Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., vol. 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 159--177, DOI 10.1007/978-3-319-75940-1_8 .

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