Two-dimensional FrBD friction models for rolling contact

TL;DR

This paper develops a comprehensive two-dimensional generalisation of the FrBD friction framework for rolling contact, unifying bristle-based rheology with a local friction law and eliminating sliding velocity through an implicit-inversion approach. It introduces three distributed models—standard linear, semilinear for large spin slips, and a linear large-spin approximation—built on a PDE formulation that captures transport, boundary inflow, and state-dependent bristle deflection across a contact patch. The authors establish well-posedness (for fixed-domain linear models), ISS and IOS stability results, and dissipativity/passivity properties, showing that the friction-based system dissipates energy under broad conditions. Numerical simulations reveal steady-state force-slip and action surfaces, as well as transient relaxation and responses to time-varying normal loads, demonstrating the framework’s ability to reproduce key tyre/rail dynamics while remaining amenable to control-oriented analysis and potential model-order reductions.

Abstract

This paper develops a comprehensive two-dimensional generalisation of the recently introduced Friction with Bristle Dynamics (FrBD) framework for rolling contact problems. The proposed formulation extends the one-dimensional FrBD model to accommodate simultaneous longitudinal and lateral slips, spin, and arbitrary transport kinematics over a finite contact region. The derivation combines a rheological representation of the bristle element with an analytical local sliding-friction law. By relying on an application of the Implicit Function Theorem, the notion of sliding velocity is then eliminated, and a fully dynamic friction model, driven solely by the rigid relative velocity, is obtained. Building upon this local model, three distributed formulations of increasing complexity are introduced, covering standard linear rolling contact, as well as linear and semilinear rolling in the presence of large spin slips. For the linear formulations, well-posedness, stability, and passivity properties are investigated under standard assumptions. In particular, the analysis reveals that the model preserves passivity under almost any parametrisation of practical interest. Numerical simulations illustrate steady-state action surfaces, transient relaxation phenomena, and the effect of time-varying normal loads. The results provide a unified and mathematically tractable friction model applicable to a broad class of rolling contact systems.

Figures (14)

• 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 reference frame $(O;x,y,z)$ with unit vectors $(\hat{\bm{e}}_x, \hat{\bm{e}}_y, \hat{\bm{e}}_z)$.
• Figure 2: Free-body diagram of the bristle element in the $x$ and $z$-directions, along with its reaction on the upper and lower bodies in the absence of inertial effects.
• 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: Schematic of a tyre rolling over a rigid road. For a rigid wheel, the spin components would be defined as $\varphi_1 \triangleq \frac{\sin \gamma}{R_\delta}$ and $\varphi_2 \triangleq -\frac{\dot{\psi}}{\Omega R_\delta}$, where $\gamma \in \mathbb{R}$ denotes the camber angle, $\dot{\psi} \in \mathbb{R}$ the turning speed, $\Omega \in \mathbb{R}_{>0}$ the angular velocity, and $R_\delta \in \mathbb{R}_{>0}$ the deformed radius. From simple trigonometrical considerations, it follows that $C_{\varphi_1}$ always lies outside $\mathscr{C}$. In reality, for a deformable tyre, the spin slips are often defined as $\varphi_1 = \varphi_\gamma \triangleq \frac{(1-\varepsilon_\gamma)\sin\gamma}{R_\textnormal{r}}$ and $\varphi_2 = \varphi_\psi \triangleq -\frac{\dot{\psi}}{V_\textnormal{r}}$, where $\varepsilon_\gamma \in [0,1)$ denotes the camber reduction factor, $R_\textnormal{r} \geq R_\delta$ the effective rolling radius, and $V_\textnormal{r} \in \mathbb{R}_{>0}$ is the rolling speed. Also in this case, $C_{\varphi_1}$ must necessarily lie outside $\mathscr{C}$.
• Figure 5: Force-slip surfaces in the absence of spin slips (parabolic pressure distribution): (a) rectangular contact area; (b) elliptical contact area. Model parameters as in Table \ref{['tab:parameters']}.

Theorems & Definitions (24)

• Theorem 2.1
• proof
• Remark 2.1
• Theorem 2.2: Edwards Edwards
• Theorem 3.1: Existence and uniqueness of solutions
• proof : Proof
• Theorem 3.2: Existence and uniqueness of solutions
• proof
• Definition 4.1: Input-to-state-stability (ISS)
• Definition 4.2: Input-to-output stability (IOS)