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Three Examples Concerning the Interaction of Dry Friction and Oscillations

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Trends in Applications of Mathematics to Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 27))

Abstract

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.

Partially supported by DFG via project B01 in SFB 1114.

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Acknowledgements

The authors is grateful to Martin Heida and Elias Pipping for stimulating discussions. The research was partially supported by DFG via the project B01 Fault networks and scaling properties of deformation accumulation within the SFB 1114 Scaling Cascades in Complex Systems.

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Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Alexander Mielke

Authors
  1. Alexander Mielke
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Correspondence to Alexander Mielke .

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Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Augusto Visintin

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Mielke, A. (2018). Three Examples Concerning the Interaction of Dry Friction and Oscillations. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_8

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