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2022 | OriginalPaper | Buchkapitel

# 4. Unsteady-State Brush Theory

verfasst von: Luigi Romano

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## Abstract

Many transient phenomena concerning the tyre-road interaction are effectively explained within the theoretical framework of the brush theory. The analysis in vanishing sliding conditions is relatively simple and may be conducted with respect to any time-varying slip input. The case of limited friction available inside the contact patch is rather involving. In this context, the investigations proposed in this chapter are limited to small spin slips under the assumption of a thin tyre. The situation further complicates when considering a flexible tyre carcass, but it may be still approached using some intuition from Chap. 3. A rather general formulation of the transient problem is proposed, which allows to gain some preliminary insights about the relaxation behaviour of the tyre.
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Fußnoten
1
It should be observed that if the initial conditions are described by a function $$\boldsymbol{u}_{\boldsymbol{t}0}(\boldsymbol{x})$$ using the original coordinates $$\boldsymbol{x}$$, then in the coordinate system $$\boldsymbol{\xi }$$ they should be represented by a new function $$\boldsymbol{u}_{\boldsymbol{t}0}^\prime (\boldsymbol{\xi }) \triangleq \boldsymbol{u}_{\boldsymbol{t}0}(\boldsymbol{x}(\boldsymbol{\xi }))$$. Similar considerations also hold for other functions. However, the same notation is used in the remaining of the chapter for the sake of simplicity.

2
Therefore, the transient brush theory may be seen as a weak one, in the sense that the solutions are always $$C^0(\mathscr {P}\times \mathbb {R}_{\ge 0};\mathbb {R}^2)$$, but higher regularity cannot be required.

3
The coordinates for which and correspond to the points where the slope of the lateral shear stress equals that of the friction bound, and for which the micro-sliding velocity vanishes, that is $$\bar{\boldsymbol{v}}_\text {s}(\boldsymbol{\xi },s) = \boldsymbol{0}$$.

4
It should be noticed that in Eqs. (4.19) the adhesion solution $$u_y^\text {(a)}(\boldsymbol{\xi },s)$$ has been extended analytically over the whole contact patch $$\mathscr {P}$$. This makes it possible to restate $$\mathscr {P}^\text {(a)}$$ and $$\mathscr {P}^\text {(s)}$$ as in Eqs. (4.20).

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