Skip to main content
SpringerLink
Log in
Book cover

The Science of Vehicle Dynamics pp 461–538Cite as

Tire Models

  • Chapter
  • First Online:

Abstract

In this chapter a simple, yet significant, tire model is developed. It is basically a brush model, but with some noteworthy additions with respect to more common formulations. For instance, the model takes care of the transient phenomena that occur in the contact patch. A number of Figures show the pattern of the local actions within the contact patch.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Actually, the formulation presented here of the brush model is quite general, and hence it is a bit involved.

  2. 2.

    For the very first time we look at the kinematics of points in the contact patch.

  3. 3.

    Not to be confused with the global friction coefficients (2.90) and (2.92).

  4. 4.

    The use of the practical slip \(\varvec{\kappa }\) would not have provided an equally neat formula.

  5. 5.

    The total time derivative is evaluated within \(\hat{{\mathsf {S}}}\), that is as if \(\,\mathbf {i}\) and \(\,\mathbf {j}\) were fixed.

  6. 6.

    As reported in [11, p. 4], this approach is actually due to d’Alembert.

  7. 7.

    In the brush model, \(\hat{y}\) is more a parameter than a variable.

  8. 8.

    More convenient governing equations for the sliding state are given in (11.59) and (11.60).

  9. 9.

    The solution of \(y' + f(x) y = g(x)\) is

    $$y(x) = \exp \left( -\int ^x f(t) \mathrm{d}t\right) \left[ \int ^x \exp \left( \int ^z f(t) \mathrm{d}t\right) g(z) \mathrm{d}z + C \right] .$$
  10. 10.

    Since the tangential force is constant in time, it is possible to exploit its dependence on the given slips.

  11. 11.

    If, as usual, also \(\hat{x}_0(\hat{x},\hat{y}) = \hat{x}_0(\hat{x},-\hat{y})\) and \(p(\hat{x},\hat{y}) = p(\hat{x},-\hat{y})\).

  12. 12.

    More generally, in tilting vehicles, which may have three wheels, like MP3 by Piaggio, or even four.

  13. 13.

    Of course, the effect cannot be to “add” the camber force, that is to translate the curve vertically.

  14. 14.

    The crucial aspects are: \(\mathbf {e}_s\) not depending on time, \(\mathbf {e}_a(\hat{x}_s,t)=\mathbf {e}_s(\hat{x}_s)\).

References

  1. Clark SK (ed) (1971) Mechanics of pneumatic tires. National Bureau of Standards, Washington

    Google Scholar 

  2. Deur J, Asgari J, Hrovat D (2004) A 3D brush-type dynamic tire friction model. Veh Syst Dyn 42:133–173

    Article  Google Scholar 

  3. Deur J, Ivanovic V, Troulis M, Miano C, Hrovat D, Asgari J (2005) Extension of the LuGre tyre friction model related to variable slip speed along the contact patch length. Veh Syst Dyn 43(Supplement):508–524

    Article  Google Scholar 

  4. Hüsemann T (2007) Tire technology: simulation and testing. Technical report, Institut für Kraftfahrwesen, Aachen, Vehicle: www.ika.rwth-aachen.de/lehre/kfz-labor/2_tires_en.pdf

  5. Kreyszig E (1999) Advanced Engineering Mathematics, 8th edn. John Wiley & Sons, New York

    MATH  Google Scholar 

  6. Lugner P, Pacejka H, Plöchl M (2005) Recent advances in tyre models and testing procedures. Veh Syst Dyn 43:413–436

    Article  Google Scholar 

  7. Milliken WF, Milliken DL (1995) Race Car Veh Dyn. SAE International, Warrendale

    Google Scholar 

  8. Pacejka HB (2002) Tyre and vehicle dynamics. Butterworth-Heinemann, Oxford

    Google Scholar 

  9. Pacejka HB, Sharp RS (1991) Shear force development by pneumatic tyres in steady state conditions: a review of modelling aspects. Veh Syst Dyn 20:121–176

    Article  Google Scholar 

  10. Savaresi SM, Tanelli M (2010) Active braking control systems design for vehicles. Springer, London

    Book  Google Scholar 

  11. Truesdell C, Rajagopal KR (2000) An introduction to the mechanics of fluids. Birkhäuser, Boston

    Book  Google Scholar 

  12. Zwillinger D (ed) (1996) CRC standard mathematical tables and formulae, 30th edn. CRC Press, Boca Raton

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Civile e Industriale, Università di Pisa, Pisa, Italy

    Massimo Guiggiani

Authors
  1. Massimo Guiggiani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Massimo Guiggiani .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Guiggiani, M. (2018). Tire Models. In: The Science of Vehicle Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-73220-6_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-73220-6_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73219-0

  • Online ISBN: 978-3-319-73220-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation