Abstract
The first goal of this chapter is to describe the kinematics of a wheel with tire, mainly under steady-state conditions. This leads to the definitions of slips as a measure of the extent to which the wheel with tire departs from pure rolling conditions. All aspects are discussed in detail and with a critical approach, showing that the use of the slips implies neglecting some phenomena. The slip angle is also defined and discussed. It is shown that a wheel with tire resembles indeed a rigid wheel because slip angles are quite small. The relationships between the kinematics and the forces/couples the tire exchange with the road are investigated by means of experimental tests. The Magic Formula provides a convenient way to represent these functions. Finally, the mechanics of wheels with tire is summarized with the aid of quite a number of plots.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
Only in competitions it is worthwhile to employ special (and secret) gas mixtures instead of air. The use of nitrogen, as often recommended, is in fact completely equivalent to air, except for the cost.
- 2.
A rigid wheel is essentially an axisymmetric convex rigid surface. The typical rigid wheel is a toroid.
- 3.
S is the system recommended by ISO (see, e.g., [14, Appendix 1]).
- 4.
- 5.
What is relevant in vehicle dynamics is the moment of (F,M O ) with respect to the steering axis of the wheel. But this is another story.
- 6.
More precisely, it is necessary to have a mathematical description of the shape of the road surface in the contact patch. The plane just happens to be the simplest.
- 7.
We have basically a steady-state behavior even if the operating conditions do not change “too fast”.
- 8.
However, in the brush model, and precisely at p. 294, the effect of the elastic compliance of the carcass on C is taken into account.
- 9.
In a toroidal rigid wheel with maximum radius r 0 and lateral radius s r we would have r r =r 0−s r (1−cosγ), c r =−tanγs r and ε r =0. It follows that \(\dot{c}_{r} \neq- \dot{\gamma}s_{r}\).
- 10.
Common definitions of the slip angle, like “α being the difference in wheel heading and direction” are not sufficiently precise.
- 11.
All other angles are positive angles if measured counterclockwise, as usually done in mathematical writing.
- 12.
In a step steer the steering wheel of a car may reach ω z =20∘/s=0.35 rad/s. At a forward speed of 20 m/s, the same wheels have about ω c =80 rad/s. The contribution of ω z to φ is therefore like a camber angle γ≈0.5∘.
- 13.
sin(Cπ/2)=sin((2−C)π/2), since 1<C<2.
References
Bastow D, Howard G, Whitehead JP (2004) Car suspension and handling, 4th edn. SAE International, Warrendale
Bergman W (1977) Critical review of the state-of-the-art in the tire and force measurements. SAE Preprint (770331)
Clark SK (ed) (2008) The pneumatic tire. NHTSA–DOT HS 810 561
Dixon JC (1991) Tyres, suspension and handling. Cambridge University Press, Cambridge
Font Mezquita J, Dols Ruiz JF (2006) La Dinámica del Automóvil. Editorial de la UPV, Valencia
Gillespie TD (1992) Fundamentals of vehicle dynamics. SAE International, Warrendale
Meirovitch L (1970) Methods of analytical dynamics. McGraw-Hill, New York
Michelin (2001) The tyre encyclopaedia. Part 1: grip. Société de Technologie Michelin, Clermont–Ferrand [CD-ROM]
Michelin (2002) The tyre encyclopaedia. Part 2: comfort. Société de Technologie Michelin, Clermont–Ferrand [CD-ROM]
Michelin (2003) The tyre encyclopaedia. Part 3: rolling resistance. Société de Technologie Michelin, Clermont–Ferrand [CD-ROM]
Milliken WF, Milliken DL (1995) Race car vehicle dynamics. SAE International, Warrendale
Murray RM, Li Z, Sastry SS (1994) A mathematical introduction to robot manipulation. CRC Press, Boca Raton
Pacejka HB (1996) The tyre as a vehicle component. In: 26th FISITA congress ’96: engineering challenge human friendly vehicles, Prague, June 17–21, pp 1–19
Pacejka HB (2002) Tyre and vehicle dynamics. Butterworth–Heinemann, Oxford
Pacejka HB (2005) Slip: camber and turning. Veh Syst Dyn 43(Supplement):3–17
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
Popov VL (2010) Contact mechanics and friction. Springer, Berlin
Pytel A, Kiusalaas J (1999) Engineering mechanics—statics. Brooks/Cole, Pacific Grove
Wong JY (2001) Theory of ground vehicles. Wiley, New York
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Guiggiani, M. (2014). Mechanics of the Wheel with Tire. In: The Science of Vehicle Dynamics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8533-4_2
Download citation
DOI: https://doi.org/10.1007/978-94-017-8533-4_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-8532-7
Online ISBN: 978-94-017-8533-4
eBook Packages: EngineeringEngineering (R0)