# On the mathematical basis of solid friction

## Abstract

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.

## Keywords

Dynamics Sticking Discontinuous Friction Filippov Perturbation Piecewise-smooth Nonsmooth## References

- 1.Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)Google Scholar
- 2.Aizerman, M.A., Pyatnitskii, E.S.: Fundamentals of the theory of discontinuous systems I. II. Autom. Remote Control
**35**, 1066–1079, 1242–1292 (1974)Google Scholar - 3.Akay, A.: Acoustics of friction. J. Acoust. Soc. Am.
**111**(4), 1525–1548 (2002)CrossRefGoogle Scholar - 4.Al-Bender, F., Lampaert, V., Swevers, J.: A novel generic model at asperity level for dry friction force dynamics. Tribol. Lett.
**16**(1), 81–93 (2004)CrossRefGoogle Scholar - 5.Bachar, G., Segev, E., Shtempluck, O., Buks, E., Shaw, S.W.: Noise induced intermittency in a superconducting microwave resonator. EPL
**89**(1), 17003 (2010)CrossRefMATHGoogle Scholar - 6.Bastien, J., Michon, G., Manin, L., Dufour, R.: An analysis of the modified Dahl and Masing models: application to a belt tensioner. J. Sound Vib.
**302**(4–5), 841–864 (2007)MathSciNetCrossRefGoogle Scholar - 7.Berry, M.V.: Stokes’ phenomenon; smoothing a Victorian discontinuity. Publ. Math. Inst. Hautes Études Sci.
**68**, 211–221 (1989)CrossRefGoogle Scholar - 8.Bliman, P.-A., Sorine, M.: Easy-to-use realistic dry friction models for automatic control. In: Proceedings of 3rd European Control Conference, pp. 3788–3794 (1995)Google Scholar
- 9.Bowden, F.P., Tabor, D.: The friction and lubrication of solids. Oxford University Press (1964)Google Scholar
- 10.Braun, O.M., Dauxois, T., Peyrard, M.: Friction in a thin commensurate contact. Phys. Rev. B
**56**(8), 4987–4995 (1997)CrossRefGoogle Scholar - 11.Brogliato, B.: Nonsmooth Mechanics—Models, Dynamics and Control. Springer, New York (1999)MATHGoogle Scholar
- 12.Brogliato, B., Acary, V.: Numerical Methods for Nonsmooth Dynamical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008)MATHGoogle Scholar
- 13.Studer, C.: Numerics of Unilateral Contacts and Friction. Lecture Notes in Applied and Computational Mechanics, vol. 47. Springer, Berlin (2009)MATHGoogle Scholar
- 14.Cieplak, M., Smith, E.D., Robbins, M.O.: Molecular origins of friction: the force on adsorbed layers. Science
**265**(5176), 1209–1212 (1994)CrossRefMATHGoogle Scholar - 15.Csernák, G., Stépán, G.: On the periodic response of a harmonically excited dry friction oscillator. J. Sound Vib.
**295**, 649–658 (2006)CrossRefMATHGoogle Scholar - 16.Dahl, P.R.: A solid friction model. TOR-158(3107–18), The Aerospace Corporation, El Segundo, CA (1968)Google Scholar
- 17.Derjaguin, B.: Molekulartheorie der äusseren Reibung. Z. Phys.
**88**(9–10), 661–675 (1934)CrossRefGoogle Scholar - 18.di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, Berlin (2008)Google Scholar
- 19.Dieci, L., Lopez, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side. J. Comput. Appl. Math.
**236**(16), 3967–3991 (2012)MathSciNetCrossRefMATHGoogle Scholar - 20.Dingle, R.B.: Asymptotic Expansions: Their derivation and interpretation. Academic Press, London (1973)MATHGoogle Scholar
- 21.Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. Lect. Notes Math.
**985**, 449–494 (1983)MathSciNetCrossRefGoogle Scholar - 22.Fall, C.P., Marland, E.S., Wagner, J.M., Tyson, J.J.: Computational Cell Biology. Springer, New York (2002)Google Scholar
- 23.Feeny, B., Moon, F.C.: Chaos in a forced dry-friction oscillator: experiments and numerical modelling. J. Sound Vib.
**170**(3), 303–323 (1994)CrossRefMATHGoogle Scholar - 24.Fenichel, N.: Geometric singular perturbation theory. J. Differ. Equ.
**31**, 53–98 (1979)MathSciNetCrossRefGoogle Scholar - 25.Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publisher, Dortrecht (1988)CrossRefGoogle Scholar
- 26.Gnecco, E., Bennewitz, R., Gyalog, T., Loppacher, C., Bammerlin, M., Meyer, E., Güntherodt, H.-J.: Velocity dependence of atomic friction. PRL
**84**(6), 1–4 (2000)CrossRefGoogle Scholar - 27.Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev.
**39**(4), 644–668 (1997)MathSciNetCrossRefGoogle Scholar - 28.Guardia, M., Hogan, S.J., Seara, T.M.: An analytical approach to codimension-2 sliding bifurcations in the dry friction oscillator. SIAM J. Dyn. Syst.
**9**, 769–798 (2010)MathSciNetCrossRefMATHGoogle Scholar - 29.He, G., Muser, M.M., Robbins, M.O.: Adsorbed layers and the origin of static friction. Science
**284**(5420), 1650–1652 (1999)CrossRefMATHGoogle Scholar - 30.Hinrichs, N., Oestreich, M., Popp, K.: On the modelling of friction oscillators. J. Sound Vib.
**216**(3), 435–459 (1998)CrossRefGoogle Scholar - 31.Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol.
**117**(4), 500–544 (1952)CrossRefGoogle Scholar - 32.Israelachvili, J.N.: Adhesion, friction and lubrication of molecularly smooth surfaces. In: Singer, I.L., Pollock, H.M. (eds.) Fundamentals of Friction. Kluwer, Dortrecht (1992)Google Scholar
- 33.Jeffrey, M.R.: Non-determinism in the limit of nonsmooth dynamics. Phys. Rev. Lett.
**106**(25), 254103 (2011)CrossRefGoogle Scholar - 34.Jeffrey, M.R.: Hidden dynamics in models of discontinuity and switching. Phys. D
**273–274**, 34–45 (2014)MathSciNetCrossRefGoogle Scholar - 35.Jeffrey, M.R., Simpson, D.J.W.: Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise. Nonlinear Dyn.
**76**(2), 1395–1410 (2014)MathSciNetCrossRefMATHGoogle Scholar - 36.Jones, C.K.R.T.: Geometric singular perturbation theory. Volume 1609 of Lecture Notes in Mathematics, pp. 44–120. Springer, New York (1995)Google Scholar
- 37.Kowalczyk, P., Piiroinen, P.T.: Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator. Phys. D Nonlinear Phenom.
**237**(8), 1053–1073 (2008)MathSciNetCrossRefMATHGoogle Scholar - 38.Krim, J.: Friction at macroscopic and microscopic length scales. Am. J. Phys.
**70**, 890–897 (2002)CrossRefGoogle Scholar - 39.Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bif. Chaos
**13**, 2157–2188 (2003)MathSciNetCrossRefMATHGoogle Scholar - 40.Machina, A., Edwards, R., van den Dreissche, P.: Singular dynamics in gene network models. SIAM J. Dyn. Syst.
**12**(1), 95–125 (2013)CrossRefMATHGoogle Scholar - 41.Novaes, D.D., Jeffrey, M.R.: Hidden nonlinearities in nonsmooth flows, and their fate under smoothing (submitted) (2015)Google Scholar
- 42.Olsson, H., Astrom, K.J., de Wit, C.C., Gafvert, M., Lischinsky, P.: Friction models and friction compensation. Eur. J. Control
**4**, 176–195 (1998)CrossRefGoogle Scholar - 43.Persson, B.N.J., Albohr, U.O., Tartaglino, A.I., Volokitin, E.Tosatti: On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J. Phys. Condens. Matter
**17**, R1–R62 (2005)CrossRefMATHGoogle Scholar - 44.Persson, B.N.J., Zhang, Z.Y.: Theory of friction: Coulomb drag between two closely spaced solids. Phys. Rev. B
**57**(12), 7327–7334 (1998)CrossRefMATHGoogle Scholar - 45.Piiroinen, P.T., Kuznetsov, Y.A.: An event-driven method to simulate Filippov systems with accurate computing of sliding motions. ACM Trans. Math. Softw.
**34**(3), 13:1–13:24 (2008)MathSciNetCrossRefGoogle Scholar - 46.Popov, V.: Phonon contribution to friction stress in an atomically flat contact of crystalline solids at low temperature. Z. Angew. Math. Mech.
**80**(S1), 65–68 (2000)CrossRefMATHGoogle Scholar - 47.Radiguet, M., Kammer, D.S., Gillet, P., Molinari, J.-F.: Survival of heterogeneous stress distributions created by precursory slip at frictional interfaces. PRL
**111**(164302), 1–4 (2013)Google Scholar - 48.Shaw, S.W.: On the dynamics response of a system with dry friction. J. Sound Vib.
**108**(2), 305–325 (1986)CrossRefGoogle Scholar - 49.Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs (1991)MATHGoogle Scholar
- 50.Stokes, G.G.: On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Philos. Soc.
**10**, 106–128 (1864)Google Scholar - 51.Tabor, D.: Triobology—the last 25 years. A personal view. Tribol. Int.
**28**(1), 7–10 (1995)CrossRefGoogle Scholar - 52.Teixeira, M.A., da Silva, P.R.: Regularization and singular perturbation techniques for non-smooth systems. Phys. D
**241**(22), 1948–1955 (2012)MathSciNetCrossRefGoogle Scholar - 53.Tomlinson, G.A.: A molecular theory of friction. Philos. Mag.
**7**(7), 905–939 (1929)CrossRefGoogle Scholar - 54.Weymouth, A.J., Meuer, D., Mutombo, P., Wutscher, T., Ondracek, M., Jelinek, P., Giessibl, F.J.: Atomic structure affects the directional dependence of friction. PRL
**111**(126103), 1–4 (2013)Google Scholar - 55.Wojewoda, J., Andrzej, S., Wiercigroch, M., Kapitaniak, T.: Hysteretic effects of dry friction: modelling and experimental studies. Philos. Trans. R. Soc. A
**366**, 747–765 (2008)CrossRefMATHGoogle Scholar