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2022 | OriginalPaper | Chapter

3. Steady-State Brush Theory

Author : Luigi Romano

Published in: Advanced Brush Tyre Modelling

Publisher: Springer International Publishing

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Abstract

In steady-state conditions, explicit expressions for tyre characteristics may be derived using the theoretical framework provided by the brush theory. This chapter is, thus, dedicated to addressing the stationary problem from both the local and global perspectives. The fundamental concepts of critical slip and spin are introduced with respect to an isotropic tyre, and the deformation of the bristles inside the contact patch is investigated for different operating conditions of the tyre. Analytical functions describing the tyre forces and moment acting inside the contact patch are obtained for the particular case of a rectangular contact patch. The analysis is qualitative in nature.

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previous chapter Tyre-Road Contact Mechanics Equations

next chapter Unsteady-State Brush Theory

A D-convex set is a set convex along a specified direction. Analogously, a D-convex function is a function which is convex along a given direction. The reader may refer to [19] for additional details.

In turn, the existence of a continuous leading edge ensures that the steady-state displacement of the bristle is at least \(C^0(\mathscr {P})\).

From a global perspective, full and partial sliding conditions have not been defined rigorously in the literature. An intuitive way of stating mathematically full sliding conditions would be to consider the Lebesgue measures \(\lambda ^*(\cdot )\) of the contact patch \(\mathscr {P}\) and its sliding region \(\mathscr {P}^\text {(s)}\). Then full sliding would occur if \(\lambda ^*(\mathscr {P}^\text {(s)})\equiv \lambda ^*(\mathscr {P})\), and partial sliding if \(\lambda ^*(\mathscr {P}^\text {(s)})< \lambda ^*(\mathscr {P})\). With the same rationale, vanishing sliding conditions would correspond to \(\lambda ^*(\mathscr {P}^\text {(a)})\equiv \lambda ^*(\mathscr {P})\).

When the tyre is anistoropic, a constant sliding direction cannot be found.

It should be noticed that, since it is assumed that the contact patch has no lateral dimension, the analytical expressions for the stiffnesses in Eq. (3.15) should be reduced by a factor of 2b. This consideration also holds for Sect. 3.5.

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Title: Steady-State Brush Theory
Author: Luigi Romano
Publisher: Springer International Publishing
Book: Advanced Brush Tyre Modelling
Print ISBN: 978-3-030-98434-2

Electronic ISBN: 978-3-030-98435-9

Copyright Year: 2022
DOI: https://doi.org/10.1007/978-3-030-98435-9_3

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