Distributed Rolling Contact FrBD Framework

• The framework unifies classical friction models such as brush, Dahl, and LuGre into a scalable, multidimensional system using spatially distributed, dynamic bristle models.
• It employs a coupled ODE–PDE cascade to capture transient force responses, state estimation, and stability properties through rigorous well-posedness and passivity analyses.
• An adaptive observer design with exponential convergence is developed to estimate both distributed bristle states and friction parameters in real-time applications.

The Distributed Rolling Contact FrBD Framework generalizes and rigorously formalizes frictional dynamics in rolling contact interfaces through the concept of spatially distributed, dynamic bristle models, governed by hyperbolic partial differential equations (PDEs). This paradigm mathematically unifies formerly separate lumped brush, Dahl, and LuGre-type friction representations into a scalable, multidimensional, physically consistent framework. By modeling the contact patch as a continuum populated by massless bristle elements with viscoelastic rheology, coupled to a rigid body’s slip, spin, and motion, the framework addresses transient force responses, action surfaces, and internal state estimation in rolling phenomena. Key results establish well-posedness, passivity, and exponential observer convergence for both linear and semilinear, ODE–PDE interconnected rolling contact systems.

1. Mathematical Structure of the Distributed FrBD Rolling Contact Model

The fundamental FrBD (Friction with Bristle Dynamics) framework describes friction via distributed bristle deformation, coupled to rigid-body kinematics and slip (Romano et al., 11 Jan 2026, Romano, 11 Jan 2026, Romano, 20 Jan 2026). In the single-track vehicle setting, the system is modeled as an ODE–PDE cascade:

• ODE (rigid body):

$$\dot X(t) = A_1 X(t) + G_1 (\mathscr{K}_1 z)(t) + G_1 \Theta \Sigma(v(X(t), U(t))) (\mathscr{K}_2 z)(t) + G_1 h_1(v(X(t), U(t)))$$

where $X$ denotes lumped rigid-body states, $U$ control input, and $z$ distributed bristle deflections.

• Homodirectional hyperbolic PDE (distributed bristle dynamics):

$$\partial_t z(\xi, t) + \Lambda \partial_\xi z(\xi, t) = \Theta \Sigma(v(X(t), U(t))) z(\xi, t) + h_2(v(X(t), U(t)))$$

on spatial domain $\xi \in [0,1]$, with boundary condition $z(0, t) = 0$. $\Lambda$ encodes transport speeds, $\Theta$ is a diagonal matrix of friction coefficients, and $\Sigma$, $h_2$ are nonlinear friction terms.

Coupling operators $\mathscr{K}_1$, $\mathscr{K}_2$ combine distributed states to produce lumped force contributions. The underlying uncertainty in friction is parameterized by the matrix $\Theta$, entering the PDE nonlinearly (Romano et al., 14 Jan 2026).

2. Bristle Rheology: Generalized Maxwell and Kelvin–Voigt Extensions

Recent advances extend FrBD models beyond basic elastic-damped bristle laws to arbitrary linear viscoelasticity, using Generalized Maxwell (GM) and Kelvin–Voigt (GKV) networks (Romano, 20 Jan 2026):

• GM model: Bristle is an elastic spring in series with $n$ Maxwell branches (spring + dashpot).
• GKV model: Bristle is an elastic spring in parallel with $n$ Kelvin–Voigt branches.

This leads to systems of $2(n+1)$ hyperbolic PDEs for bristle deflections and internal force/deformation states, which rigorously capture relaxation and rate-dependent phenomena. Analytical stress–strain relations reduce these rheologies to explicit ODE/PDE form. This extension enables direct utilization of Prony series data and systematic refinement toward fractional-order models.

3. Two-Dimensional Distributed Formulations and Contact Geometry

The framework generalizes to full two-dimensional contact patches $\mathscr{C} \subset \mathbb{R}^2$ (Romano, 11 Jan 2026, Romano, 20 Jan 2026). Each material point $x \in \mathscr{C}$ carries a local bristle state evolving under transport velocity $V(x, s)$ and rigid-body slip/spin velocity $v_r(x, s)$. Distributed PDEs capture anisotropic, direction-dependent slip and spin excitation (inclusion of $A_\varphi$ matrices). Boundary conditions are imposed on the inflow (leading edge) of the contact, $z|_{\text{inflow}} = 0$. Variations include:

• Standard linear model: Constant transport direction, simple spin.
• Linear model with large spin slips: Transport and slip velocities allow coupling between bristle deflection and effective spin terms.
• Semilinear model: Fully nonlinear coupling between distributed bristle states and state-dependent rigid-body slip.

Steady-state forces and moments:

$$F(s) = \iint_{\mathscr{C}} p(x) f(x, s) dx, \quad M_z(s) = \iint_{\mathscr{C}} p(x) [x \times f(x, s)] dx$$

where $p(x)$ is contact pressure.

4. Well-Posedness, Stability, and Passivity Properties

All linear and semilinear distributed FrBD systems possess rigorous well-posedness and stability guarantees under standard regularity, local Lipschitz, and boundedness assumptions (Romano et al., 11 Jan 2026, Romano, 11 Jan 2026, Romano et al., 14 Jan 2026, Romano, 20 Jan 2026).

• Well-posedness: Existence and uniqueness of mild/classical solutions on Hilbert spaces $L^2(\mathscr{C}; \mathbb{R}^d)$, $H^1$, etc. Transport operators generate $C_0$ semigroups; nonlinear terms treated as Lipschitz perturbations (Pazy’s Theorem).
• Input-to-State Stability (ISS) and Input-to-Output Stability (IOS): Energy functionals are shown to decay with appropriate bounds, yielding stability and global existence.
• Passivity: Quadratic Lyapunov/storage functionals demonstrate passivity, i.e., the interface cannot generate net energy under any parametrisation meeting mild conditions ($\dot{W} \leq -\langle pf, \bar v_r \rangle_{L^2}$). This generalizes the fundamental thermodynamic constraint on friction dissipation.

5. Adaptive Observer Design for Distributed FrBD Systems

A central contribution is the synthesis of boundary adaptive observers for ODE–PDE cascades with uncertain friction (Romano et al., 14 Jan 2026). Key elements:

• Finite-dimensional estimator: Uses filtered boundary measurements ($Y_1 = \partial_t z(0, t)$, $Y_2 = \partial_t^2 z(0, t)$) to reconstruct rigid-body velocities and regress friction coefficients $\Theta$ via parameter update laws.
• Infinite-dimensional observer: Recursively estimates distributed bristle states $\hat{z}(\xi, t)$ using the estimated parameters and boundary data.
• Lyapunov analysis: Provides exponential convergence under persistent excitation of the regressor $\varphi(t)$, with simultaneous decay of state and parameter errors.
• Implementation: Vehicle lateral dynamics are modeled with observed PDE/ODE states, with demonstrated rapid convergence in simulation (state errors within $0.3$s, parameter errors within $2$s, and robustness to forward velocity perturbation).

6. Practical Applications and Performance Characterization

Applications span single-track vehicle models, tire/road transients, micro-shimmy oscillations, wheel–rail and robotics rolling contacts (Romano et al., 11 Jan 2026). The distributed FrBD accurately captures:

• Transient force development: Multi-exponential and prolonged relaxation, as observed in Maxwell-type viscoelastic bristles, and micro-shimmy damping.
• Steady-state surfaces and action regions: Generation of force–slip and moment–slip maps, deformation of friction circles under spin.
• High-frequency dynamic response: Phase lag and amplitude attenuation, notably in systems with flexible carcasses or multi-mode bristle laws.
• Robustness to parameter variation: Observer/estimator maintains convergence and accuracy under uncertainty in transport speeds, friction levels, and input excitation.

7. Significance and Implications for Modeling of Rolling Contact Systems

The Distributed Rolling Contact FrBD Framework enables:

• Unified and systematic representation of rolling contact friction for arbitrary viscoelasticity, slip, and spin conditions.
• Integration of experimental (DMA/Prony) data with friction modeling via ODE–PDE parameterization.
• Analytic stability and passivity assessment, guiding control and observer design.
• Real-time simulation and robust parameter estimation in vehicle, railway, and robotics applications, with internal state tracking at each contact point.

This suggests the FrBD paradigm provides a mathematically rigorous and physically transparent foundation for future developments in rolling contact simulation, control, and online identification across diverse engineering platforms (Romano et al., 11 Jan 2026, Romano, 11 Jan 2026, Romano et al., 14 Jan 2026, Romano, 20 Jan 2026).