Two-dimensional FrBD friction models for rolling contact: extension to linear viscoelasticity

TL;DR

This paper extends the FrBD friction model to include linear viscoelasticity, capturing complex relaxation phenomena through PDEs and analyzing properties like well-posedness and passivity.

Contribution

It introduces a systematic extension of the FrBD framework to linear viscoelasticity, including new formulations and rigorous analysis of their properties.

Findings

Models can capture complex relaxation phenomena.

Well-posedness and passivity are proven for the linear variants.

Numerical experiments demonstrate steady-state and transient behaviors.

Abstract

This paper extends the distributed rolling contact FrBD framework to linear viscoelasticity by considering classic derivative Generalised Maxwell and Kelvin-Voigt rheological representations of the bristle element. With this modelling approach, the dynamics of the bristle, generated friction forces, and internal deformation states are described by a system of 2(n+1) hyperbolic partial differential equations (PDEs), which can capture complex relaxation phenomena originating from viscoelastic behaviours. By appropriately specifying the analytical expressions for the transport and rigid relative velocity, three distributed formulations of increasing complexity are introduced, which account for different levels of spin excitation. For the linear variants, well-posedness and passivity are analysed rigorously, showing that these properties hold for any physically meaningful parametrisation. Numerical experiments complement the theoretical results by illustrating steady-state characteristics and transient relaxation effects. The findings of this paper substantially advance the FrBD paradigm by enabling a unified and systematic treatment of linear viscoelasticity.