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Transitions and singularities during slip motion of rigid bodies

Published online by Cambridge University Press:  23 February 2018

P. L. VÁRKONYI
P. L. VÁRKONYI*
Affiliation:
Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Budapest, Hungary email: [email protected]
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Abstract

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The dynamics of moving solids with unilateral contacts are often modelled by assuming rigidity, point contacts, and Coulomb friction. The canonical example of a rigid rod with one endpoint slipping in two dimensions along a fixed surface (sometimes referred to as Painlevé rod) has been investigated thoroughly by many authors. The generic transitions of that system include three classical transitions (slip-stick, slip reversal, and liftoff) as well as a singularity called dynamic jamming, i.e., convergence to a codimension 2 manifold in state space, where rigid body theory breaks down. The goal of this paper is to identify similar singularities arising in systems with multiple point contacts, and in a broader setting to make initial steps towards a comprehensive list of generic transitions from slip motion to other types of dynamics. We show that – in addition to the classical transitions – dynamic jamming remains a generic phenomenon. We also find new forms of singularity and solution indeterminacy, as well as generic routes from sliding to self-excited microscopic or macroscopic oscillations.

Keywords

Contact dynamicsfrictionrigid body dynamicssingularitypiecewise smooth systems37J5570F4070E1870K7070F35
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 778 - 804
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was supported by Grant 124002 of the National Research, Development, and Innovation Office of Hungary.

References

Anh, L. X. (2000) Dynamics of Mechanical Systems with Coulomb Friction (Foundations of Engineering Mechanics), Springer, New York.Google Scholar
Brogliato, B. (1990) Nonsmooth Mechanics., Springer, London, 2nd edition.Google Scholar
Butlin, T. & Woodhouse, J. (2013) Friction-induced vibration: Model development and comparison with large-scale experimental tests. J. Sound Vib. 332 (21), 5302–5321.Google Scholar
Champneys, A. R. & Várkonyi, P. L. (2016) The Painlevé paradox in contact mechanics. IMA J. Appl. Math., 81 (3), 538–588.Google Scholar
di Bernardo, M., Budd, C. J., Champneys, A. R. & Kowalczyk, P. (2008) Piecewise-smooth Dynamical Systems; Theory and Applications, Springer, New York.Google Scholar
Génot, F. & Brogliato, B. (1999) New results on Painlevé paradoxes. Eur. J. Mech. A/Solids, 18, 653–677.Google Scholar
Hogan, S. J. & Kristiansen, K. U. (2017) On the regularization of impact without collision: The painlevé paradox and compliance. In: Proc. R. Soc. A, 473, p. 20160773.Google Scholar
Ibrahim, R. A. (1994) Friction-induced vibration, chatter, squeal, and chaos part II: Dynamics and modeling. Appl. Mech. Rev. 47 (7), 227–253.Google Scholar
Jellet, J. H. (1872) Treatise on the Theory of Friction, Hodges, Foster and Co., Dublin.Google Scholar
Kinkaid, N. M., O'reilly, O. M. & Papadopoulos, P. (2003) Automotive disc brake squeal. J. Sound Vib. 267 (1), 105–166.Google Scholar
Kristiansen, K. U. & Hogan, S. J. (2017) Le canard de Painlevé. arXiv preprint arXiv:1703.07665.Google Scholar
Kruse, S., Tiedemann, M., Zeumer, B, Reuss, P. L., Hetzler, H. & Hoffmann, N. (2015) The influence of joints on friction induced vibration in brake squeal. J. Sound Vib. 340, 239–252.Google Scholar
Le Rouzic, J., Le Bot, A., Perret-Liaudet, J., Guibert, M., Rusanov, A., Douminge, L., Bretagnol, F. & Mazuyer, D. (2013) Friction-induced vibration by Stribecks law: Application to wiper blade squeal noise. Tribol. Lett. 49 (3), 563–572.Google Scholar
Leine, R. & Nijmeijer, H. (2004) Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer, New York.Google Scholar
Leine, R. I., Brogliato, B. & Nijmeijer, H. (2002) Periodic motion and bifurcations induced by the Painlevé paradox. Eur. J. Mech. A/Solids 21, 869–896.Google Scholar
Mo, J. L., Wang, Z. G., Chen, G. X., Shao, T. M., Zhu, M. H. & Zhou, Z. R. (2013) The effect of groove-textured surface on friction and wear and friction-induced vibration and noise. Wear 301 (1), 671–681.Google Scholar
Nordmark, A., Dankowicz, H. & Champneys, A. (2011) Friction-induced reverse chatter in rigid-body mechanisms with impacts. IMA J. Appl. Math. 76, 85–119.Google Scholar
Nordmark, A, Várkonyi, P. L. & Champneys, A. R. (2017) Dynamics beyond dynamic jam; unfolding the Painlevé paradox singularity. SIAM J. Appl. Dyn. Sys., accepted for publication.Google Scholar
Painlevé, P. (1895) Sur les loi du frottement de glissement. Comptes Rendu des Séances de l'Academie des Sci. 121, 112–115.Google Scholar
Sinou, J. & Jezequel, L. (2007) Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping. Eur. J. Mech.-A/Solids 26 (1), 106–122.Google Scholar
Szalai, R. & Jeffrey, M. R. (2014) Nondeterministic dynamics of a mechanical system. Phys. Rev. E 90 (2), 022914.Google Scholar
Várkonyi, P. L. (2017) Dynamics of mechanical systems with two sliding contacts: New facets of Painlevés paradox. Archive Appl. Mech. 87 (5), 785–799.Google Scholar
Zhao, Z., Liu, C., Chen, B. & Brogliato, B. (2015) Asymptotic analysis of Painlevés paradox. Multibody Syst. Dyn. 35 (3), 299–319.Google Scholar