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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Notes
- 1.
- 2.
In the brush model there is slip spin velocity \(\varOmega_{s_{z}}\) only within the contact patch, as if it were entirely due to the camber angle.
- 3.
The use of the practical slip κ would not have provided an equally neat formula.
- 4.
The total time derivative is evaluated within \(\hat{\mathsf{S}}\), that is as if i and j were fixed.
- 5.
As reported in [11, p. 4], this approach is actually due to d’Alembert.
- 6.
- 7.
The solution of y′+f(x)y=g(x) is
$$y(x) = \exp \biggl(-\int^x f(t) \mathrm{d}t \biggr) \biggl[\int^x \exp \biggl(\int^z f(t) \mathrm{d}t \biggr)g(z) \mathrm{d}z + C \biggr]. $$ - 8.
Since the tangential force is constant in time, it is possible to exploit its dependence on the given slips.
- 9.
If, as usual, also \(\hat{x}_{0}(\hat{x},\hat{y}) = \hat{x}_{0}(\hat{x},-\hat {y})\) and \(p(\hat{x},\hat{y}) = p(\hat{x},-\hat{y})\).
- 10.
More generally, in tilting vehicles, which may have three wheels, like MP3 by Piaggio, or even four.
- 11.
Of course, the effect cannot be to “add” the camber force, that is to translate the curve vertically.
- 12.
This kind of models are often referred to as single contact point transient tire models [8].
- 13.
The main points are: e s not depending on time, \(\mathbf{e}_{a}(\hat{x}_{s},t)=\mathbf{e}_{s}(\hat {x}_{s})\).
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Guiggiani, M. (2014). Tire Models. In: The Science of Vehicle Dynamics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8533-4_10
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