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

TL;DR

This work extends the friction with bristle dynamics ($FrBD$) rolling-contact framework to linear viscoelasticity by employing Generalised Maxwell ($GM$) and Generalised Kelvin-Voigt ($GKV$) bristle representations. It derives distributed, $2(n+1)$-state hyperbolic PDEs that couple bristle deflection, internal stresses, and rolling kinematics, and introduces three model variants to capture different levels of spin. The linear variants are proven well-posed and passive, ensuring stability under typical boundary and input conditions, while numerical simulations reveal steady-state and transient effects of higher-order relaxation on force and moment responses. The results enable direct incorporation of experimentally measured relaxation spectra into rolling-contact simulations and offer a unified framework for viscoelastic friction across diverse rolling elements, with potential extensions to fractional models and broader tribological applications.

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.

Figures (10)

• Figure 1: A schematic representation of the friction model: (a) configuration with a rigid substrate; (b) configuration with a deformable substrate. The problem is studied in a right-handed reference frame $(O;x,y,z)$ with unit vectors $(\hat{\bm{e}}_x, \hat{\bm{e}}_y, \hat{\bm{e}}_z)$.
• Figure 2: A schematic representation of the Generalised Maxwell (GM) and Generalised Kelvin-Voigt (GKV) rheological models. The matrix $\bar{\mathbf{K}}_0$ collects the normalised micro-stiffnesses of the zeroth element, modelled as an elastic spring. The matrices $\bar{\mathbf{K}}_i$ and $\bar{\mathbf{C}}_i$ denote the normalised micro-stiffness and micro-damping matrices of the element $i$, $i \in \{1,\dots, n\}$. For the GM, the corresponding matrix of time constants is given by $\bm{\tau}_i \triangleq \bar{\mathbf{C}}_i\bar{\mathbf{K}}_i^{-1}$, $i \in \{1,\dots,n\}$.
• Figure 3: Rolling contact problem between: (a) two spheres with angular velocities $\bm{\omega}_1, \bm{\omega}_2 \in \mathbb{R}^3$; (b) a sphere translating and rolling over a stationary plane, where $\bm{V}_1\in \mathbb{R}^3$ denotes the translational velocity of its centre, and $\bm{\omega}_1\in \mathbb{R}^3$ its rolling velocity.
• Figure 4: Steady-state characteristics in the absence of spin slips predicted using Models \ref{['semilinmodel']} (rectangular contact area with parabolic pressure distribution). Line styles: FrBD$_1$-KV from FrBDroll (solid thick lines), FrBD$_2$-GM (solid lines), FrBD$_3$-GM (dashed lines). Model parameters as in Table \ref{['tab:parameters']}.
• Figure 5: Steady-state characteristics in the absence of spin slips predicted using Models \ref{['semilinmodel']} (elliptical contact area with parabolic pressure distribution). Line styles: FrBD$_1$-KV from FrBDroll (solid thick lines), FrBD$_2$-GM (solid lines), FrBD$_3$-GM (dashed lines). Model parameters as in Table \ref{['tab:parameters']}.

Theorems & Definitions (13)

• Theorem 2.1: Edwards Edwards
• Theorem 4.1: Existence and uniqueness of solutions of Models \ref{['stdmodel']}
• proof : Proof
• Theorem 4.2: Existence and uniqueness of solutions of Models \ref{['linmodel2']}
• proof
• Definition 4.1: Dissipativity and passivity
• Lemma 4.1: Passivity of the FrBD$_{n+1}$-GM models
• proof
• Lemma 4.2: Passivity of the FrBD$_{n+1}$-GKV models
• proof