Two-dimensional FrBD friction models for rolling contact

Luigi Romano

arxiv: 2601.06811 · v10 · pith:U4IBX7QFnew · submitted 2026-01-11 · ⚛️ physics.app-ph

Two-dimensional FrBD friction models for rolling contact

Luigi Romano This is my paper

Pith reviewed 2026-05-16 15:38 UTC · model grok-4.3

classification ⚛️ physics.app-ph

keywords rolling contactfriction modelbristle dynamicspassivitytwo-dimensional modelimplicit function theoremdistributed contactspin slip

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The pith

A two-dimensional FrBD model for rolling contact eliminates sliding velocity via the Implicit Function Theorem and preserves passivity in its linear formulations for nearly all practical parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes the one-dimensional Friction with Bristle Dynamics framework to two dimensions so that rolling contact can include simultaneous longitudinal and lateral slips together with spin and arbitrary transport motion across a finite patch. It starts from a rheological bristle representation plus an analytical local sliding-friction law, then applies the Implicit Function Theorem to remove sliding velocity entirely, leaving a dynamic model driven only by the rigid relative velocity. Three distributed versions of increasing complexity are constructed for standard linear rolling, linear rolling with large spin, and semilinear rolling. Analysis of the linear cases establishes well-posedness, stability, and passivity under standard assumptions and almost any practical parametrization. Simulations display steady-state surfaces, transient relaxation, and the influence of changing normal loads, giving a unified description usable across many rolling systems.

Core claim

By representing bristle elements rheologically and pairing them with an analytical local sliding-friction law, the sliding velocity is eliminated through the Implicit Function Theorem, producing a fully dynamic friction model driven solely by rigid relative velocity. Distributed formulations of increasing complexity are then obtained for linear rolling contact, linear rolling with large spin slips, and semilinear cases; the linear versions are shown to be well-posed, stable, and passive for almost any practical choice of parameters.

What carries the argument

Implicit Function Theorem applied to the local sliding-friction law inside a bristle-dynamics representation, used to remove sliding velocity and obtain velocity-driven distributed models over the contact region.

If this is right

Linear rolling-contact formulations remain passive for almost any practical parametrization.
Well-posedness and stability hold for the linear models under standard assumptions.
Steady-state action surfaces and transient relaxation phenomena can be computed numerically.
The model accommodates arbitrary combinations of longitudinal, lateral, and spin slips over finite contact regions.
Semilinear extensions continue to handle large spin slips while retaining the core dynamic structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Passivity preservation would let the model be inserted directly into energy-based Lyapunov arguments for vehicle or manipulator control.
The same elimination step might be tried on non-rolling contact problems if their local laws also meet the invertibility requirement.
Quantitative comparison of simulated relaxation times against tire or wheel test data would check whether the transients match observed behavior.
When the Implicit Function Theorem does not apply, a hybrid numerical solution of the local law could still be substituted without losing the overall distributed structure.

Load-bearing premise

The local sliding-friction law must satisfy invertibility conditions that allow the Implicit Function Theorem to eliminate sliding velocity from the governing equations.

What would settle it

An explicit local friction law and parameter set for which the implicit function relating friction force to rigid velocity fails to exist locally, or a linear distributed model that produces negative energy dissipation for some practical parameter values.

Figures

Figures reproduced from arXiv: 2601.06811 by Luigi Romano.

**Figure 2.** Figure 2: Free-body diagram of the bristle element in the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Rolling contact problem between: (a) two spheres with angular velocities [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Schematic of a tyre rolling over a rigid road. For a rigid wheel, the spin components would [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: Force-slip surfaces in the absence of spin slips (parabolic pressure distribution): (a) rectangular [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Action surfaces for different combinations of spin slips [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗

**Figure 7.** Figure 7: Friction circles (parabolic pressure distribution): (a) rectangular contact patch; (b) elliptical [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

**Figure 8.** Figure 8: Steady-state characteristics in the absence of spin slips (rectangular contact area with parabolic [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗

**Figure 9.** Figure 9: Steady-state characteristics in the absence of spin slips (elliptical contact area with parabolic [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗

**Figure 10.** Figure 10: Steady-state characteristics in the presence of large spin slips (rectangular contact area with [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗

**Figure 11.** Figure 11: Steady-state characteristics in the presence of large spin slips (elliptical contact area with [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗

**Figure 12.** Figure 12: Transient forces predicted by Model 3 for step slip inputs in the absence of spin (parabolic pressure distribution). Line styles: rectangular contact area (solid thick line), elliptical contact area (dashed lines). Model parameters as in [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗

**Figure 13.** Figure 13: Transient forces predicted by Model 3 for step slip inputs in the presence of large spins (parabolic pressure distribution). Line styles: rectangular contact area (solid thick line), elliptical contact area (dashed lines). Model parameters as in [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗

**Figure 14.** Figure 14: Transient longitudinal force in the presence of an oscillating normal load and a time-varying [PITH_FULL_IMAGE:figures/full_fig_p033_14.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends FrBD to two dimensions with spin and slips by using the Implicit Function Theorem to drop sliding velocity, yielding passivity for linear cases, but the required invertibility of the local law is not explicitly checked for combined 2D slips.

read the letter

The core new piece is the two-dimensional FrBD generalization that folds in longitudinal and lateral slip plus spin across a finite patch. The authors apply the Implicit Function Theorem to remove sliding velocity from the equations, leaving a model driven only by rigid-body velocities, then build three distributed versions of increasing complexity and show passivity for the linear ones under nearly any practical parameter set. The numerical examples on steady-state surfaces and transient relaxation under time-varying loads are straightforward and useful for seeing the model's behavior. That part is solid and fills a gap for people who need dynamic rolling-contact friction without excessive computation. The soft spot is the IFT step itself. The derivation assumes the local rheological-plus-friction law has an invertible Jacobian everywhere in the domain, including under simultaneous slips and large spin, but the paper gives no analytic or numeric confirmation that this holds in the two-dimensional setting beyond the one-dimensional case. Without that check, the well-posedness and stability claims for the distributed models rest on an unverified condition. The semilinear large-spin version gets even less formal attention. This is for vehicle-dynamics or robotics researchers who already work with bristle-type models and want a tractable 2D extension. It deserves a serious referee because the extension and passivity results are concrete and the math is laid out clearly, even though the invertibility gap will need addressing in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a two-dimensional generalization of the Friction with Bristle Dynamics (FrBD) framework for rolling contact. It combines a rheological bristle representation with an analytical local sliding-friction law, applies the Implicit Function Theorem to eliminate sliding velocity, and obtains a dynamic model driven solely by rigid-body relative velocity. Three distributed formulations of increasing complexity are introduced for linear rolling contact and for linear/semilinear rolling with large spin. For the linear formulations, well-posedness, stability, and passivity are analyzed under standard assumptions, with the claim that passivity is preserved under almost any practical parameterization. Numerical simulations illustrate steady-state action surfaces, transient relaxation, and effects of time-varying normal loads.

Significance. If the central derivation and passivity results hold, the work supplies a unified, mathematically tractable, and passive friction model for two-dimensional rolling contact that accommodates combined slips and spin. The passivity property under broad parameterization would be a notable strength for stability analysis in control and dynamics applications such as vehicle tires and robotic locomotion.

major comments (1)

[Local model derivation via Implicit Function Theorem] Local-model derivation (IFT step): the claim that sliding velocity can be eliminated via the Implicit Function Theorem rests on the local sliding-friction law (rheological bristle plus analytical friction) having an invertible Jacobian everywhere in the domain, including under simultaneous longitudinal/lateral slips and spin. No explicit analytic verification or numerical check of the required monotonicity/non-singularity conditions is supplied for the two-dimensional case. This step is load-bearing for both the model reduction and the subsequent passivity proofs of the linear distributed formulations.

minor comments (2)

[Abstract] The abstract states the main results but contains no equations or quantitative statements; adding one or two key expressions (e.g., the reduced dynamic friction law or the passivity inequality) would improve readability.
[Introduction / Notation] Notation for the two-dimensional slip and spin vectors should be introduced with an explicit table or diagram early in the manuscript to aid readers unfamiliar with the one-dimensional FrBD precursor.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The primary concern is the lack of explicit verification for the Jacobian invertibility in the Implicit Function Theorem step. We address this below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: Local-model derivation (IFT step): the claim that sliding velocity can be eliminated via the Implicit Function Theorem rests on the local sliding-friction law (rheological bristle plus analytical friction) having an invertible Jacobian everywhere in the domain, including under simultaneous longitudinal/lateral slips and spin. No explicit analytic verification or numerical check of the required monotonicity/non-singularity conditions is supplied for the two-dimensional case. This step is load-bearing for both the model reduction and the subsequent passivity proofs of the linear distributed formulations.

Authors: We agree that an explicit verification strengthens the foundation. The local friction law is constructed as the sum of a linear bristle term (with positive-definite stiffness matrix) and an analytical sliding-friction term that is strictly monotone in the sliding-velocity vector. This composite map is therefore strictly monotone on the entire domain, which directly implies that its Jacobian is positive definite and hence invertible for any combination of longitudinal/lateral slips and spin. The monotonicity holds under the standard assumptions on the analytical friction law (regularized Coulomb or similar) used throughout the paper. To make this transparent, we will add a dedicated appendix containing (i) the analytic proof of monotonicity and Jacobian nonsingularity in the two-dimensional setting and (ii) numerical checks for representative cases with large spin and simultaneous slips. These additions will also clarify the hypotheses underlying the passivity proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central modeling step applies the Implicit Function Theorem to eliminate sliding velocity from the combined rheological bristle and analytical friction law, yielding a dynamic model driven by rigid-body velocity. This relies on stated invertibility conditions under standard assumptions but does not reduce any claimed result to its inputs by construction, nor does it rename a fitted quantity as a prediction. No load-bearing self-citation chain is exhibited that would force the 2D extension or passivity properties; the prior 1D FrBD framework is referenced as a starting point but the 2D generalization, distributed formulations, and passivity analysis introduce independent content. The derivation remains self-contained as a modeling extension without the specific reductions required for circularity flags.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on a rheological bristle representation combined with an analytical local sliding-friction law; no explicit free parameters, axioms, or invented entities are stated.

axioms (1)

domain assumption A rheological representation of the bristle element can be combined with an analytical local sliding-friction law
Invoked to obtain the local model before applying the Implicit Function Theorem

pith-pipeline@v0.9.0 · 5498 in / 1275 out tokens · 37881 ms · 2026-05-16T15:38:24.436674+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By relying on an application of the Implicit Function Theorem, the notion of sliding velocity is then eliminated... G(vr)=Σ1‖M(vr)vr‖2,ε + M²(vr)
IndisputableMonolith/Foundation/ArrowOfTime.lean entropy_monotone unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the model preserves passivity under almost any parametrisation of practical interest

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