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2018 | OriginalPaper | Chapter

11. Tire Models

Author : Massimo Guiggiani

Published in: The Science of Vehicle Dynamics

Publisher: Springer International Publishing

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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.

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Footnotes
1
Actually, the formulation presented here of the brush model is quite general, and hence it is a bit involved.
 
2
For the very first time we look at the kinematics of points in the contact patch.
 
3
Not to be confused with the global friction coefficients (2.​90) and (2.​92).
 
4
The use of the practical slip \(\varvec{\kappa }\) would not have provided an equally neat formula.
 
5
The total time derivative is evaluated within \(\hat{{\mathsf {S}}}\), that is as if \(\,\mathbf {i}\) and \(\,\mathbf {j}\) were fixed.
 
6
As reported in [11, p. 4], this approach is actually due to d’Alembert.
 
7
In the brush model, \(\hat{y}\) is more a parameter than a variable.
 
8
More convenient governing equations for the sliding state are given in (11.59) and (11.60).
 
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
Since the tangential force is constant in time, it is possible to exploit its dependence on the given slips.
 
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
More generally, in tilting vehicles, which may have three wheels, like MP3 by Piaggio, or even four.
 
13
Of course, the effect cannot be to “add” the camber force, that is to translate the curve vertically.
 
14
The crucial aspects are: \(\mathbf {e}_s\) not depending on time, \(\mathbf {e}_a(\hat{x}_s,t)=\mathbf {e}_s(\hat{x}_s)\).
 
Literature
1.
Clark SK (ed) (1971) Mechanics of pneumatic tires. National Bureau of Standards, Washington
2.
Deur J, Asgari J, Hrovat D (2004) A 3D brush-type dynamic tire friction model. Veh Syst Dyn 42:133–173CrossRef
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):508–524CrossRef
4.
5.
Kreyszig E (1999) Advanced Engineering Mathematics, 8th edn. John Wiley & Sons, New YorkMATH
6.
Lugner P, Pacejka H, Plöchl M (2005) Recent advances in tyre models and testing procedures. Veh Syst Dyn 43:413–436CrossRef
7.
Milliken WF, Milliken DL (1995) Race Car Veh Dyn. SAE International, Warrendale
8.
Pacejka HB (2002) Tyre and vehicle dynamics. Butterworth-Heinemann, Oxford
9.
Pacejka HB, Sharp RS (1991) Shear force development by pneumatic tyres in steady state conditions: a review of modelling aspects. Veh Syst Dyn 20:121–176CrossRef
10.
Savaresi SM, Tanelli M (2010) Active braking control systems design for vehicles. Springer, LondonCrossRef
11.
Truesdell C, Rajagopal KR (2000) An introduction to the mechanics of fluids. Birkhäuser, BostonCrossRef
12.
Zwillinger D (ed) (1996) CRC standard mathematical tables and formulae, 30th edn. CRC Press, Boca RatonMATH
Metadata
Title
Tire Models
Author
Massimo Guiggiani
Copyright Year
2018
Book
The Science of Vehicle Dynamics

Print ISBN: 978-3-319-73219-0

Electronic ISBN: 978-3-319-73220-6

Copyright Year: 2018

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

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