Nonlinear Dynamics

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

On the mathematical basis of solid friction

Original Paper

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. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)Google Scholar
  2. 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. 3.
    Akay, A.: Acoustics of friction. J. Acoust. Soc. Am. 111(4), 1525–1548 (2002)CrossRefGoogle Scholar
  4. 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. 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. 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. 7.
    Berry, M.V.: Stokes’ phenomenon; smoothing a Victorian discontinuity. Publ. Math. Inst. Hautes Études Sci. 68, 211–221 (1989)CrossRefGoogle Scholar
  8. 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. 9.
    Bowden, F.P., Tabor, D.: The friction and lubrication of solids. Oxford University Press (1964)Google Scholar
  10. 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. 11.
    Brogliato, B.: Nonsmooth Mechanics—Models, Dynamics and Control. Springer, New York (1999)MATHGoogle Scholar
  12. 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. 13.
    Studer, C.: Numerics of Unilateral Contacts and Friction. Lecture Notes in Applied and Computational Mechanics, vol. 47. Springer, Berlin (2009)MATHGoogle Scholar
  14. 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. 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. 16.
    Dahl, P.R.: A solid friction model. TOR-158(3107–18), The Aerospace Corporation, El Segundo, CA (1968)Google Scholar
  17. 17.
    Derjaguin, B.: Molekulartheorie der äusseren Reibung. Z. Phys. 88(9–10), 661–675 (1934)CrossRefGoogle Scholar
  18. 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. 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. 20.
    Dingle, R.B.: Asymptotic Expansions: Their derivation and interpretation. Academic Press, London (1973)MATHGoogle Scholar
  21. 21.
    Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. Lect. Notes Math. 985, 449–494 (1983)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fall, C.P., Marland, E.S., Wagner, J.M., Tyson, J.J.: Computational Cell Biology. Springer, New York (2002)Google Scholar
  23. 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. 24.
    Fenichel, N.: Geometric singular perturbation theory. J. Differ. Equ. 31, 53–98 (1979)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publisher, Dortrecht (1988)CrossRefGoogle Scholar
  26. 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. 27.
    Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)MathSciNetCrossRefGoogle Scholar
  28. 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. 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. 30.
    Hinrichs, N., Oestreich, M., Popp, K.: On the modelling of friction oscillators. J. Sound Vib. 216(3), 435–459 (1998)CrossRefGoogle Scholar
  31. 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. 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. 33.
    Jeffrey, M.R.: Non-determinism in the limit of nonsmooth dynamics. Phys. Rev. Lett. 106(25), 254103 (2011)CrossRefGoogle Scholar
  34. 34.
    Jeffrey, M.R.: Hidden dynamics in models of discontinuity and switching. Phys. D 273–274, 34–45 (2014)MathSciNetCrossRefGoogle Scholar
  35. 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. 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. 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. 38.
    Krim, J.: Friction at macroscopic and microscopic length scales. Am. J. Phys. 70, 890–897 (2002)CrossRefGoogle Scholar
  39. 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. 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. 41.
    Novaes, D.D., Jeffrey, M.R.: Hidden nonlinearities in nonsmooth flows, and their fate under smoothing (submitted) (2015)Google Scholar
  42. 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. 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. 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. 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. 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. 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. 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. 49.
    Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs (1991)MATHGoogle Scholar
  50. 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. 51.
    Tabor, D.: Triobology—the last 25 years. A personal view. Tribol. Int. 28(1), 7–10 (1995)CrossRefGoogle Scholar
  52. 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. 53.
    Tomlinson, G.A.: A molecular theory of friction. Philos. Mag. 7(7), 905–939 (1929)CrossRefGoogle Scholar
  54. 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. 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

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Engineering MathematicsUniversity of BristolBristolUK

Personalised recommendations