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Mechanics of the Wheel with Tire

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The Science of Vehicle Dynamics

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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.

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Notes

  1. 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 almost equivalent to air [16], except for the cost.

  2. 2.

    As pointed out by Jon W. Mooney in his review, in Noise Control Engineering Journal, Vol. 62, 2014, the explanation and the figure provided in the first edition of this book were not correct. A similar (incorrect) explanation has appeared in [7, Fig. 1.19].

  3. 3.

    A rigid wheel is essentially an axisymmetric convex rigid surface. The typical rigid wheel is a toroid.

  4. 4.

    \({\mathsf {S}}_w\) is the system recommended by ISO (see, e.g., [18, Appendix 1]).

  5. 5.

    In the SAE terminology, it is \(\omega _c \,\mathbf {j}_c\) that is called spin velocity [4, 14].

  6. 6.

    The two symbols \(\mathbf {V\!\!\,}_o\) and \(\mathbf {V\!\!\,}_O\) are equivalent. Using \(\mathbf {V\!\!\,}_o\) is just a matter of taste.

  7. 7.

    What is relevant in vehicle dynamics is the moment of \((\mathbf {F\!},\mathbf {M}_O)\) with respect to the steering axis of the wheel. But this is another story (Fig. 3.1).

  8. 8.

    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.

  9. 9.

    We remark that, as discussed in Chap. 11, steady-state kinematics of the rim does not necessarily implies steady-state behavior of the tire.

  10. 10.

    We have basically a steady-state behavior even if the operating conditions do not change “too fast”.

  11. 11.

    As a general rule, a subscript, or a superscript, r means “pure rolling”.

  12. 12.

    However, in the brush model , and precisely on p. 467, the effect on C of the elastic compliance of the carcass is taken into account.

  13. 13.

    All other angles are positive angles if measured counterclockwise, as usually done in mathematical writing.

  14. 14.

    Once again, we called tire slips what should be called rim slips.

  15. 15.

    In a step steer the steering wheel of a car may reach \(\omega _z=20^{\circ }/\mathrm{s}=0.35\,\mathrm{rad}/\mathrm{s}\). At a forward speed of \(20\,\mathrm{m/s}\), the same wheels have about \(\omega _c=80\,\mathrm{rad/s}\). The contribution of \(\omega _z\) to \(\varphi \) is therefore like a camber angle \(\gamma \simeq 0.5^{\circ }\).

  16. 16.

    \(\sin (C\uppi /2)=\sin ((2-C)\uppi /2)\), since \(1<C<2\).

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

    Massimo Guiggiani

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  1. Massimo Guiggiani
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Correspondence to Massimo Guiggiani .

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Guiggiani, M. (2018). Mechanics of the Wheel with Tire. In: The Science of Vehicle Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-73220-6_2

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  • DOI: https://doi.org/10.1007/978-3-319-73220-6_2

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  • Print ISBN: 978-3-319-73219-0

  • Online ISBN: 978-3-319-73220-6

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