[Paper Review] Two-dimensional FrBD friction models for rolling contact

Luigi Romano|arXiv (Cornell University)|Jan 11, 2026

Dynamics and Control of Mechanical Systems0 citations

TL;DR

This paper presents a comprehensive two-dimensional generalisation of the Friction with Bristle Dynamics (FrBD) framework for rolling contact, deriving a fully dynamic friction law and three distributed models that handle longitudinal and lateral slips, spin, and varying contact kinematics.

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.

Motivation & Objective

• Motivate and extend FrBD to two-dimensional rolling contact scenarios.
• Develop a physically grounded bristle rheology coupled with a local friction law.
• Eliminate sliding velocity via an implicit-function approach to obtain a dynamic friction law driven by rigid relative velocity.
• Derive three distributed rolling contact models with increasing complexity (standard, large-spin, and linear/semi-linear variants).
• Analyze well-posedness, stability, and passivity for linear formulations and illustrate with numerical simulations.

Proposed method

• Adopt a Kelvin-Voigt bristle rheology with deformation z and rate ż to model bristle forces f(z, ż).
• Postulate a local friction force f_r(v_s) that maximises dissipation under a convex admissible set defined by a positive definite matrix M(v_s).
• solve for f_r*(v_s) using an analytic closed-form from the Max-Dissipation principle (including regularisation).
• Use an implicit function theorem framework to relate bristle dynamics to friction, yielding an ODE for ż and a PDE when extended to spatial contact domains.
• Develop three distributed models: a standard FrBD-like linear model, a semilinear model incorporating large spin slips, and a linear/semilinear approximation for spin-dominated regimes.
• Prove well-posedness for linear formulations and examine stability, ISS/IOS, and passivity properties; support findings with numerical simulations showing steady-state surfaces, relaxation, and time-varying loads.

Figur 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 $(\hat{\bm{e}}_{x},\hat{\bm{e}}_{y},\hat{\bm{e}}_{z})$ .

Experimental results

Research questions

• RQ1 How can the FrBD framework be extended from 1D to 2D to capture simultaneous longitudinal and lateral slips and spin?
• RQ2 What is a physically consistent local friction law in 2D that couples with a bristle rheology and remains amenable to analysis?
• RQ3 Under what conditions are the resulting linear 2D FrBD models well-posed and passive, and how do these properties depend on parameter choices?
• RQ4 How do time-varying normal loads and moving contact domains affect the frictional response in the 2D FrBD framework?
• RQ5 What are the differences and trade-offs among the three distributed rolling contact models (standard, semilinear, and a first-order spin-dominated approximation) in terms of accuracy and tractability?

Key findings

• The paper provides a mathematically rigorous two-dimensional FrBD friction model for rolling contact, unifying bristle dynamics with a nonlinear friction law.
• For linear formulations, existence and uniqueness (well-posedness) are established under standard regularity assumptions.
• The models exhibit passivity under almost all practical parameterisations, supporting control-oriented applications.
• Numerical simulations illustrate steady-state action surfaces, transient relaxation phenomena, and the impact of time-varying normal loads.
• Three distributed formulations capture standard rolling contact, large spin slips, and a first-order spin-dominated approximation, enabling a spectrum of modeling fidelity.

Figur 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.