Archive of Applied Mechanics

, Volume 84, Issue 6, pp 851–880 | Cite as

Study of a driven and braked wheel using maximal monotone differential inclusions: applications to the nonlinear dynamics of wheeled vehicles

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Abstract

A wheel subjected to two friction forces, one exerted by the ground and one exerted by the brake pad, is studied. A formalism using multivalued operators allows us to write the constitutive laws of this wheel in the form of a differential inclusion, for which we have the existence and the uniqueness of the solution, as well as the convergence of the associated numerical scheme. This wheel may be connected by itself to a chassis. The chassis may also be connected to two of these wheels. In these two cases, we again obtain a well-posed differential inclusion, in terms of existence, uniqueness, and convergence of the numerical scheme. In a more general manner, numerous applications can be proposed in the domain of the nonlinear dynamics of wheeled vehicles.

Keywords

Wheel Brake Friction law Nonlinear dynamics Vehicle 

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References

  1. 1.
    Andrzejewski, R., Awrejcewicz, J.: Nonlinear Dynamics of a Wheeled Vehicle, Advances in Mechanics and Mathematics, vol. 10. Springer, Berlin (2005)Google Scholar
  2. 2.
    Badji, B.: Caractérisation du comportement non linéaire en dynamique du véhicule. PhD thesis, Université de Technologie de Belfort Montbéliard, URL:http://tel.archives-ouvertes.fr/tel-00606485 (2009)
  3. 3.
    Bakker, E., Nybord, L., Pacejka, H.: Tyre modelling for use in vehicle dynamics studies. In: SAE 870421 (1987)Google Scholar
  4. 4.
    Ballard, P.: Contributions à l’étude de quelques problèmes unilatéraux de la mécanique des solides, report of “Habilitation à Diriger des Recherches” (Habilitation). Université de la Méditerranée Aix-Marseille II. Available on URL:http://tel.archives-ouvertes.fr/docs/00/46/15/25/PDF/HDR-Ballard.pdf (2010a)
  5. 5.
    Ballard P.: Frictional contact problems for thin elastic structures and weak solutions of sweeping processes. Arch. Ration. Mech. Anal. 198(3), 789–833 (2010). doi:10.1007/s00205-010-0373-z CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Ballard P., Basseville S.: Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. M2AN Math. Model. Numer. Anal. 39(1), 59–77 (2005). doi:10.1051/m2an:2005004 CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bastien, J.: ètude thèorique et numèrique d’inclusions diffèrentielles maximales monotones. applications à des modèles èlastoplastiques. PhD thesis, Universitè Lyon I, serial number 96-2000 (2000)Google Scholar
  8. 8.
    Bastien J.: Convergence order of implicit euler numerical scheme for maximal monotone differential inclusions. Z. Angew. Math. Phys. 64, 955–966 (2013). doi:10.1007/s00033-012-0276-y CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bastien J.: Description multivoque d’une roue freinèe et applications à la dynamique de véhicules à roues. (français) [multivalued description of a braked wheel and applications to dynamics of wheeled vehicles]. Comptes Rendus de l’Académie des Sci. (Mécanique) 341(9-10), 653–658 (2013). doi:10.1016/j.crme.2013.09.003 CrossRefGoogle Scholar
  10. 10.
    Bastien J., Lamarque C.H.: Persoz’s gephyroidal model described by a maximal monotone differential inclusion. Arch. Appl. Mech. (Ingenieur Archiv) 78(5), 393–407 (2008). doi:10.1007/s00419-007-0171-8 CrossRefMATHGoogle Scholar
  11. 11.
    Bastien J., Schatzman M.: Numerical precision for differential inclusions with uniqueness. M2AN Math. Model. Numer. Anal. 36(3), 427–460 (2002). doi:10.1051/m2an:2002020 CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Bastien J., Schatzman M., Lamarque C.H.: Study of some rheological models with a finite number of degrees of freedom. Eur. J. Mech. A Solids 19(2), 277–307 (2000). doi:10.1016/S0997-7538(00)00163-7 CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    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). doi:10.1016/j.jsv.2006.12.013 CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Bastien, J., Bernardin, F., Lamarque C.H.: Systèmes dynamiques discrets non réguliers déterministes ou stochastiques. Collection Mécanique des structures, Hermès Science Publications, URL:http://www.lavoisier.fr/livre/h3908.html, book translated into English (see [15]) (2012)
  15. 15.
    Bastien, J., Bernardin, F.: Lamarque CH non smooth deterministic or stochastic discrete dynamical systems. Wiley-ISTE, URL:http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1848215258.html, English translation of [14] (2013)
  16. 16.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50) (1973)Google Scholar
  17. 17.
    Brossard, J.P.: Dynamique du véhicule. Collection des sciences appliquées de l’INSA de Lyon, Presses polytechniques et universitaires romandes (2006)Google Scholar
  18. 18.
    Brossard, J.P. Dynamique du freinage. Collection des sciences appliquées de l’INSA de Lyon, Presses polytechniques et universitaires romandes (2009)Google Scholar
  19. 19.
    Capatina A., Cocou M., Raous M.: A class of implicit variational inequalities and applications to frictional contact. Math. Methods Appl. Sci. 32(14), 1804–1827 (2009). doi:10.1002/mma.1112 CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Coudeyras, N.: Analyse non-linéaire des instabilités multiples aux interfaces frottantes: application au crissement de frein. PhD thesis, École Centrale de lyon, URL:http://bibli.ec-lyon.fr/exl-doc/ncoudeyras.pdf, supervised by L. Jézéquel and J-J. Sinou. Serial number 2009-35 (2009)
  21. 21.
    Deimling, K.: Multivalued differential equations, de Gruyter series in nonlinear analysis and applications, vol. 1. Walter de Gruyter & Co., Berlin. URL:http://dx.doi.org/10.1515/9783110874228 (1992)
  22. 22.
    Dontchev A., Lempio F.: Difference methods for differential inclusions: a survey. SIAM Rev. 34(2), 263–294 (1992). doi:10.1137/1034050 CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Dontchev A.L., Farkhi E.M.: Error estimates for discretized differential inclusion. Computing 41(4), 349–358 (1989). doi:10.1007/BF02241223 CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Hoffmann, M.: Dynamics of european two-axle freight wagons. PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby, URL:http://www2.imm.dtu.dk/pubdb/p.php?4853, supervised by Ph.D. Hans True, Docent Per Grove Thomsen, IMM, DTU, and Associate Professor Mads Peter Sørensen, MAT, DTU (2006)
  25. 25.
    Hoffmann M.: The dynamics of european two-axle railway freight wagons with UIC standard suspension. Veh. Syst. Dyn. 46(1), 225–236 (2008). doi:10.1080/00423110801935848 CrossRefGoogle Scholar
  26. 26.
    Lempio F., Veliov V.: Discrete approximations of differential inclusions. Bayreuth Math. Schr. 54, 149–232 (1998)MATHMathSciNetGoogle Scholar
  27. 27.
    Martins, J.A.C., Barbarin, S., Raous, M., Pinto da Costa, A.: Dynamic stability of finite-dimensional linearly elastic systems with unilateral contact and Coulomb friction. Comput. Methods Appl. Mech. Eng. 177(3-4), 289–328, URL:http://dx.doi.org/10.1016/S0045-7825(98)00386-7, computational modeling of contact and friction (1999)
  28. 28.
    Pacejka, H.: Modelling complex vehicle systems using bond graphs. J. Frankl. Inst. 319(1-2), 67–81. URL:http://www.sciencedirect.com/science/article/pii/0016003285900651(1985)Google Scholar
  29. 29.
    Pacejka, H.: The tyre as a vehicle component. In: XXVI FISITA Congres, Prague, République Tchèque (1996)Google Scholar
  30. 30.
    Pacejka, H.: Tyre and Vehicle Dynamics. SAE International, Butterworth-Heinemann, Oxford (2006)Google Scholar
  31. 31.
    Pacejka, H., Bakker, E., Lidner, L.: A new tire model with an application in vehicle dynamics studies (1989)Google Scholar
  32. 32.
    Piednoir, F.: Pédaler intelligent, La biomécanique du cycliste. Fédération Française de Cyclotourisme, URL:http://www.piednoir.com/index.html (2007)
  33. 33.
    Piotrowski, J.: Model of the UIC link suspension for freight wagons. Arch. Appl. Mech. 73, URL:http://dx.doi.org/10.1007/s00419-003-0305-6 (2003)
  34. 34.
    Rey, G., Clair, D., Fogli, M., Bernardin, F.: Reliability analysis of roadway departure risk using stochastic processes. Mech. Syst. Signal Process. 25(4), 1377–1392. URL:http://www.sciencedirect.com/science/article/pii/S0888327010004188 (2011)
  35. 35.
    Schiehlen, W. (ed.): Dynamical Analysis of Vehicle Systems, vol. 497, Springer, chap Dynamics of Railway Vehicles and Rail/Wheel Contact, pp. 75–128. URL:http://www2.imm.dtu.dk/pubdb/p.php?5617, cISM International Centre for Mechanical Sciences (2009)
  36. 36.
    True, H., Asmund, R.: The dynamics of a railway freight wagon wheelset with dry friction damping. Veh. Syst. Dyna. 38(2), 149–163. URL:http://www2.imm.dtu.dk/pubdb/p.php?1642 (2002)
  37. 37.
    True, H., Thomsen, G.P. (eds.): Non-smooth problems in vehicle systems dynamics, Euromech 500 Colloquium, Springer, URL:http://www2.imm.dtu.dk/pubdb/p.php?5851 (2010)
  38. 38.
    True H., Engsig-Karup A., Bigoni D.: On the numerical and computational aspects of non-smoothnesses that occur in railway vehicle dynamics. Math. Comput. Simul. 95, 78–97 (2014). doi:10.1016/j.matcom.2012.09.016 CrossRefMathSciNetGoogle Scholar
  39. 39.
    Veliov V.: Second-order discrete approximation to linear differential inclusions. SIAM J. Numer. Anal. 29(2), 439–451 (1992). doi:10.1137/0729026 CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Xia, F.: The dynamics of the three-piece-freight truck. PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby, URL:http://orbit.dtu.dk/en/publications/the-dynamics-of-the-threepiecefreight-truck%288c457427-4d4c-4c89-87a3-f2e9fee11f68%29.html,supervisedbyPh.D.HansTrue (2002)
  41. 41.
    Xia, F., True, H.: The dynamics of the three-piece-freight truck. In: Abe, M. (ed.) Proceedings of the 18th IAVSD Symposium, pp. 212–221. URL:http://www2.imm.dtu.dk/pubdb/p.php?3540 (2004)

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Centre de Recherche et d’Innovation sur le Sport, U.F.R.S.T.A.P.S.Université Claude Bernard - Lyon 1Villeurbanne CedexFrance

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