Abstract
This paper establishes new analytical results in the mathematical theory of brush tyre models. In the first part, the exact problem which considers large camber angles is analysed from the perspective of linear dynamical systems. Under the assumption of vanishing sliding, the most salient properties of the model are discussed with some insights on concepts as existence and uniqueness of the solution. A comparison against the classic steady-state theory suggests that the latter represents a very good approximation even in case of large camber angles. Furthermore, in respect to the classic theory, the more general situation of limited friction is explored. It is demonstrated that, in transient conditions, exact sliding solutions can be determined for all the one-dimensional problems. For the case of pure lateral slip, the investigation is conducted under the assumption of a strictly concave pressure distribution in the rolling direction.
Highlights
The mechanics of pneumatic tyres is an ubiquitous topic in vehicle dynamics
The optimisation of the tyre operating conditions is crucial when it comes to enhance the vehicle’s performance and has been object of several studies which led, in the last decades, to the offspread of a large number of ad hoc developed models. Despite their simplistic nature, the so called brush models [1,2,3,4,5,6] represent a solid basis for a basilar understanding of the tyre dynamics, being only grounded on physical assumptions which allow for a straightforward interpretation of the tyre-road interaction
In particular, that the PDEs ruling the dynamics of the brush models may be reinterpreted as a system of simpler ODEs, to which the classic results of existence and uniqueness borrowed from the well-established theory for ordinary differential equations (ODEs) fairly apply
Summary
The mechanics of pneumatic tyres is an ubiquitous topic in vehicle dynamics. tyres almost represent the unique interface which allows ground vehicles to exchange traction forces with the external environment. According to Pacejka [2, 7], the brush models were firstly introduced by Fromm, as reported in [8], and derived starting from the more sophisticated formulation presented in [9] It seems, that their origin may be traced back to the studies pioneered by Kalker few years later on the simplified theory of rolling contact [10,11,12,13,14,15,16,17].2. 3, a broader treatment of the general theory introduced in [45] is given which frames the governing equations of the tyre-rolling contact into the wider context of the linear system theory It is shown, in particular, that the PDEs ruling the dynamics of the brush models may be reinterpreted as a system of simpler ODEs, to which the classic results of existence and uniqueness borrowed from the well-established theory for ordinary differential equations (ODEs) fairly apply.
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