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

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.

💡 Research Summary

This paper introduces a physically‑derived class of first‑order dynamic friction models that are built by inverting the friction characteristic of a bristle element. Starting from a simple rheological description of a bristle (modeled as a linear spring‑damper), the authors formulate an implicit relation between the bristle force and the sliding velocity at the tip. By applying Edwards’ implicit function theorem, they obtain an explicit ordinary differential equation for the bristle deflection, which yields a state‑space model of the form  ż = h(z, v). The resulting “Friction with Bristle Dynamics” (FrBD) framework is general; any constitutive law for the bristle can be inserted, and the corresponding dynamic equation follows from the inversion of a chosen friction curve (e.g., a generalized Coulomb law with Stribeck and viscous terms).

A specific instance of the FrBD family is derived that closely mirrors the widely used LuGre model. The bristle force is taken as F_b = k z + σ ż, while the friction coefficient μ(v) includes static, dynamic, Stribeck, and viscous contributions. The inversion leads to the state equation
 h(z, v) = −|v|_ε g(v)