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First-order friction models with bristle dynamics: lumped and distributed formulations

Luigi Romano, Ole Morten Aamo, Jan Åslund, Erik Frisk

TL;DR

This work develops Friction with Bristle Dynamics (FrBD), a physically motivated framework that derives first-order rate-dependent friction models by inverting the bristle-friction characteristic. It yields a LuGre-like lumped model and a distributed PDE extension for rolling-contact phenomena, accompanied by rigorous stability and passivity analyses. The lumped model demonstrates robust dissipativity without velocity-dependent damping, and the distributed variant supports ISS/IOS and practical rolling scenarios, with extensive numerical and experimental validation including a diaphragm-valve setup. The results offer a principled approach to friction modeling with clear implications for control, estimation, and observer design in precision mechatronics and rolling-contact applications.

Abstract

Dynamic models, particularly rate-dependent models, have proven effective in capturing the key phenomenological features of frictional processes, whilst also possessing important mathematical properties that facilitate the design of control and estimation algorithms. However, many rate-dependent formulations are built on empirical considerations, whereas physical derivations may offer greater interpretability. In this context, starting from fundamental physical principles, this paper introduces a novel class of first-order dynamic friction models that approximate the dynamics of a bristle element by inverting the friction characteristic. Amongst the developed models, a specific formulation closely resembling the LuGre model is derived using a simple rheological equation for the bristle element. This model is rigorously analyzed in terms of stability and passivity -- important properties that support the synthesis of observers and controllers. Furthermore, a distributed version, formulated as a hyperbolic partial differential equation (PDE), is presented, which enables the modeling of frictional processes commonly encountered in rolling contact phenomena. The tribological behavior of the proposed description is evaluated through classical experiments and validated against the response predicted by the LuGre model, revealing both notable similarities and key differences.

First-order friction models with bristle dynamics: lumped and distributed formulations

TL;DR

This work develops Friction with Bristle Dynamics (FrBD), a physically motivated framework that derives first-order rate-dependent friction models by inverting the bristle-friction characteristic. It yields a LuGre-like lumped model and a distributed PDE extension for rolling-contact phenomena, accompanied by rigorous stability and passivity analyses. The lumped model demonstrates robust dissipativity without velocity-dependent damping, and the distributed variant supports ISS/IOS and practical rolling scenarios, with extensive numerical and experimental validation including a diaphragm-valve setup. The results offer a principled approach to friction modeling with clear implications for control, estimation, and observer design in precision mechatronics and rolling-contact applications.

Abstract

Dynamic models, particularly rate-dependent models, have proven effective in capturing the key phenomenological features of frictional processes, whilst also possessing important mathematical properties that facilitate the design of control and estimation algorithms. However, many rate-dependent formulations are built on empirical considerations, whereas physical derivations may offer greater interpretability. In this context, starting from fundamental physical principles, this paper introduces a novel class of first-order dynamic friction models that approximate the dynamics of a bristle element by inverting the friction characteristic. Amongst the developed models, a specific formulation closely resembling the LuGre model is derived using a simple rheological equation for the bristle element. This model is rigorously analyzed in terms of stability and passivity -- important properties that support the synthesis of observers and controllers. Furthermore, a distributed version, formulated as a hyperbolic partial differential equation (PDE), is presented, which enables the modeling of frictional processes commonly encountered in rolling contact phenomena. The tribological behavior of the proposed description is evaluated through classical experiments and validated against the response predicted by the LuGre model, revealing both notable similarities and key differences.
Paper Structure (27 sections, 9 theorems, 61 equations, 9 figures, 4 tables)

This paper contains 27 sections, 9 theorems, 61 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Basic considerations on rate-dependent friction models
  3. General structure of rate-dependent models
  4. Physical derivation of rate-dependent models
  5. A LuGre-like lumped FrBD model
  6. Model equations and solution
  7. Model equations
  8. Well-posedness and solution
  9. Mathematical properties
  10. Stability
  11. Dissipativity and passivity
  12. Linearized model
  13. A LuGre-like distributed FrBD model
  14. Model equations and solution
  15. Model equations
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose that the mapping $H : \mathbb{R}^{m+n}\mapsto \mathbb{R}^n$ is $C^1$ in a neighborhood of a point $(x^\star,y^\star)$, where $H(x^\star,y^\star) = 0$. If the Jacobian matrix $\nabla_{y}H(x^\star,y^\star)^{\mathrm{T}}$ is nonsingular, there exist a neighborhood $\mathcal{X}$ of $x^\star$ in $ for $x\in \mathcal{X}$.

Figures (9)

  • Figure 1: A schematic representation of the friction model.
  • Figure 2: Two possible friction curves: generalized Coulomb (blue line), and generalized Coulomb with viscous friction (orange line).
  • Figure 3: A schematic representation of the distributed FrBD friction model.
  • Figure 4: Normalized steady-state bristle force $\frac{F_\textnormal{b}}{F_z}$ for the constant pressure distribution (blue line), and exponential one (orange line), for a tire operating in a low-friction environment. Model parameters as in Table \ref{['tab:param1']}.
  • Figure 5: Pre-sliding displacement obtained by simulating the FrBD model, for different values of excitation frequencies $\omega = 1$, 5, and 10 Hz. Model parameters as in Table \ref{['tab:param2']}.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Theorem 1.1: Edwards Edwards
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 3.1: Existence and uniqueness of solutions
  • proof
  • Remark 3.2: Global existence and uniqueness
  • Remark 3.3
  • Lemma 3.1: Stability
  • proof
  • ...and 20 more