Advertisement

The Rolling

  • Genady P. CherepanovEmail author
Chapter

Abstract

During two hundred years, the Coulomb’s coefficient of rolling friction and the force of resistance to rolling were being determined experimentally in expensive natural tests. In this Chapter, using the invariant integral of rolling and new exact solutions of the contact problems of the theory of elasticity, Coulomb’s rolling friction coefficient and the resistance force were calculated in all basic cases, including the rolling of: (i) an elastic cylinder on an elastic half-space of another material, or on an elastic plate, or on a sticky membrane; (ii) a ball on an elastic half-space, or an elastic plate, or on a membrane; and (iii) a torus on an elastic half-space, or on an elastic plate, or on a membrane. The theoretical results comply very well with known test probes. This Chapter is a “must-to-know” for automotive engineers.

Literature

  1. 1.
    A. Erman, in Life in Ancient Egypt (Macmillan Co., London, 1894; Dover, New York, 1971) 570 pp. ISBN 0-486-22632-8Google Scholar
  2. 2.
    Special Report 216 (National Academy of Sciences, Transportation Research Board, Washington, 2006)Google Scholar
  3. 3.
    A.A. Coulomb, in Theorie des machines simples (Academie des Sciences, Paris, 1781)Google Scholar
  4. 4.
    R. Cross, Coulomb’s Law for rolling friction. Am. J. Phys. 84(3), 221–230 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Wikipedia: Rolling ResistanceGoogle Scholar
  6. 6.
    B.M. Javorsky, A.A. Detlaf, The Guide on Physics, 7th edn. (Nauka, Moscow, 1979) 942 pp (in Russian)Google Scholar
  7. 7.
    H.R. Hertz, Uber die Beruhrung Fester Elastischer Korper. Zeitschrift fur Reine Angew. Math. 92, 156 (1882)zbMATHGoogle Scholar
  8. 8.
    J.A. Williams, Engineering Tribology (Oxford University Press, London, 1994)Google Scholar
  9. 9.
    R.C. Hibbeler, Engineering Mechanics: Statics and Dynamics (Prentice Hall, Pearson, 2007)zbMATHGoogle Scholar
  10. 10.
    V.L. Popov, Contact Mechanics and Friction: Physical Principles and Applications (Springer-Verlag, Berlin, 2010)CrossRefGoogle Scholar
  11. 11.
    N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of the Elasticity (Noordhoff, Groningen, 1963), p. 718Google Scholar
  12. 12.
    F.D. Gakhov, Boundary Value Problems (Pergamon Press, London, 1966), p. 584Google Scholar
  13. 13.
    G.P. Cherepanov, A non-linear problem in the theory of analytical functions. Doklady USSR Acad. Sci. (Math.) 147(3), 566–568 (1962)Google Scholar
  14. 14.
    G.P. Cherepanov, Stresses in an inhomogeneous plate with cracks. Not. USSR Acad. Sci. (Mech.) 1, 131–138 (1962). (in Russian)Google Scholar
  15. 15.
    G.P. Cherepanov, On a method of the solution of elastic-plastic problems. J. Appl. Math. Mech. (JAMM) 27(3), 428–435 (1963)Google Scholar
  16. 16.
    G.P. Cherepanov, The Riemann-Hilbert problem for cuts along a straight line or circumference. Doklady USSR Acad. Sci. (Math.) 156(2), 275–277 (1964)Google Scholar
  17. 17.
    G.P. Cherepanov, Boundary value problems with analytical coefficients. Doklady of the USSR Academy of Sciences (Mathematics) 161(2), 312–314 (1965)Google Scholar
  18. 18.
    G.P. Cherepanov, Some problems concerning the unknown body boundaries in the theory of elasticity and plasticity, in Applications of the Theory of Functions in Continuum Mechanics, vol. 1. (Nauka, Moscow, 1965), pp. 135–150 (in Russian)Google Scholar
  19. 19.
    G.P. Cherepanov, On modeling in linear reology, in Problems of Hydrodynamics and Continuum Mechanics, L. I. Sedov Anniversary Volume, (Nauka, Moscow, 1969), pp. 553–560Google Scholar
  20. 20.
    L.A. Galin, G.P. Cherepanov, Contact elastic-plastic problems for plates. Doklady USSR Acad. Sci. (Mech.) 177(1), 56–58 (1967)Google Scholar
  21. 21.
    G.P. Cherepanov, On the non-uniqueness problem in the theory of plasticity. Doklady USSR Acad. Sci. (Mech.) 218(4), 1124–1126 (1974)Google Scholar
  22. 22.
    G.P. Cherepanov, Mechanics of Brittle Fracture (McGraw Hill, New York, 1978), p. 950Google Scholar
  23. 23.
    G.P. Cherepanov, Methods of Fracture Mechanics: Solid Matter Physics (Kluwer, Dordrecht, 1997), p. 300CrossRefGoogle Scholar
  24. 24.
    G.P. Cherepanov, Fracture Mechanics (ICR, IzhevsK-Moscow, 2012), p. 872Google Scholar
  25. 25.
    G.P. Cherepanov, Some new applications of the invariant integrals in mechanics. J. Appl. Math. Mech. (JAMM) 76(5), 519–536 (2012)CrossRefGoogle Scholar
  26. 26.
    G.P. Cherepanov, Theory of rolling: solution of the Coulomb problem. J. Appl. Mech. Tech. Phy. (JAMT) 55(1), 182–189 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    G.P. Cherepanov, The contact problem of the mathematical theory of elasticity with stick-and-slip areas: the theory of rolling and tribology. J. Appl. Math. Mech. (JAMM) 79(1), 81–101 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    G.P. Cherepanov, The laws of rolling. J. Phys. Mesomech. 23(5), 25–48 (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.MiamiUSA

Personalised recommendations