Bare and stretched string tyre models with distributed FrBD dynamics

Abstract: This paper presents a novel class of string tyre models with FrBD friction dynamics. By modelling the distributed carcass and tread deformations with string-like equations, the resulting formulation leads to a system of semilinear parabolic partial differential equations (PDEs) that describe the evolution of the tyre states without explicitly distinguishing between stick and slip regimes. Rigorous stability and passivity analyses are also conducted using a Lyapunov-based approach, establishing boundedness of the distributed states and energy dissipation during rolling contact. The proposed Lyapunov function admits a clear physical interpretation as the total elastic energy stored in the tyre, enabling a direct link between mechanical energy storage and frictional dissipation due to slip losses. The steady-state and transient behaviours of the model are investigated both numerically and experimentally, revealing that the new formulation can satisfactorily reproduce nonlinear relaxation phenomena excited by step slip inputs. The resulting model provides a physically interpretable, mathematically well-posed, and computationally efficient basis for advanced vehicle dynamics simulations and control-oriented applications.

• The paper introduces FrSD tyre models that couple nonlocal carcass and treadband string dynamics with regularised distributed FrBD friction, producing semilinear parabolic PDEs rather than classical hyperbolic rolling-contact equations.
• The paper proves well-posedness, uniform input-to-state stability, input-to-output stability, and passivity using an elastic-energy Lyapunov function, while showing that diffusion-driven relaxation accelerates with slip and filters high-frequency inputs.
• The paper validates the model through simulations and TU/e flat-plank experiments, reproducing continuous stress profiles, force saturation, aligning-moment reversal, and steady-state deflections within roughly ±10° near the contact centre, while identifying slower high-slip transients as a limitation.

The paper "Bare and stretched string tyre models with distributed FrBD dynamics" introduces a new class of tyre models, termed FrSD (friction with string dynamics), which couples a nonlocal, string-like constitutive description of carcass and treadband deformation with a distributed dynamic friction law of the FrBD (friction with bristle dynamics) type. The resulting governing equations form a system of semilinear parabolic PDEs over the contact patch, a structural property that distinguishes the formulation from classical transport-dominated rolling contact models and underpins its stability, passivity, and numerical behaviour (2603.02869).

Modelling framework

The model is built on three ingredients. First, the constitutive equation relates the distributed deformation $\bm{u}(x,t)$ to the tangential force per unit length $\bm{q}(x,t)$ through a string-type operator,

$$\mathbf{S}\,\partial_x^2 \bm{u} - \mathbf{K}\bm{u} = -\bm{q},$$

where $\mathbf{S} = \mathrm{diag}(EA, S)$ collects bending stiffness and effective tension, and $\mathbf{K} = \mathrm{diag}(k_x, k_y)$ contains longitudinal and lateral stiffnesses per unit length. This nonlocal description—stresses depend on spatial derivatives of the deformation field—replaces the local elastic foundation assumed by brush models. Second, kinematic constraints link sliding velocity to rigid relative velocity plus both material and convective derivatives of the deflection. Third, friction is described by a regularised FrBD law in which the nondimensional friction force takes the direction of $\mathbf{M}^2(\bm{v}_s)\bm{v}_s$, with $\mathbf{M}$ a slip-velocity-dependent matrix of friction coefficients incorporating Stribeck and viscous effects.

Combining these via an application of the Implicit Function Theorem (with one Newton step and a standard Jacobian approximation inherited from prior FrBD work), and switching to travelled distance $s$ as the time-like variable, yields the governing PDE:

$$\frac{\partial \bm{u}}{\partial s} - \mathbf{\Gamma}\,\partial_x^2 \bm{u} = \partial_x \bm{u} + \mathbf{\Sigma}\,\bm{u} - \bar{\bm{v}},$$

with Robin boundary conditions at both contact edges derived from exponentially decaying deflections outside the patch. Crucially, the diffusion matrix $\mathbf{\Gamma}$ depends explicitly on the slip and spin inputs through the friction law.

Parabolic structure and its consequences

The central structural claim of the paper is that the FrSD equations are parabolic for all nonzero regularisation parameters $\bm{q}(x,t)$0, whereas classical brush and string models with Coulomb or FrBD friction are hyperbolic. The author argues this appears to be a new finding: injecting higher-order spatial derivatives into the constitutive relation fundamentally alters the mathematical character of rolling contact dynamics. Three consequences follow directly.

Slip-modulated relaxation: because the diffusive term scales with slip magnitude, force build-up transients shorten as slip increases, matching experimental observations that relaxation length effectively decreases at large slip. Boundary condition flexibility: unlike hyperbolic formulations, which can enforce only one boundary condition (typically the no-kink condition at the leading edge), the parabolic problem admits Robin conditions at both edges simultaneously, producing smooth stress profiles without artificial discontinuities. Degeneracy handling: at exactly zero slip with $\bm{q}(x,t)$1 the diffusion vanishes and the system reverts to hyperbolic; the strictly positive regularisation parameter restores uniform parabolicity and enables well-posedness results.

Well-posedness for sufficiently regular solutions follows from standard analytic-semigroup theory (Lunardi's framework): for $\bm{q}(x,t)$2 coefficient functions and initial data in $\bm{q}(x,t)$3 compatible with the boundary conditions, a unique solution in $\bm{q}(x,t)$4 exists.

Stability and passivity analysis

The Lyapunov function adopted is physically transparent—it equals the total elastic energy stored in the tyre, combining bulk and gradient terms of the deformation with boundary contributions weighted by the inverse relaxation lengths. Differentiating along solutions and integrating by parts shows that the energy derivative decomposes into a negative-definite dissipation term proportional to the inner product of shear stress and nondimensional slip velocity, plus a quadratic term controlled by the regularised diffusion.

Two rigorous results are established. First, the system is uniformly input-to-state stable in the spatial $\bm{q}(x,t)$5 norm: there exist $\bm{q}(x,t)$6 and $\bm{q}(x,t)$7 such that the state norm is bounded by decaying initial energy plus a gain on the sup norm of the slip input. Combined with the output map, this implies input-to-output stability of the forces and aligning moment, with the moment bound containing a quadratic term reflecting large carcass deflections. Second, the system is passive with respect to the slip input and force output, using the elastic energy as storage function; the paper also verifies that physical dissipation—$\bm{q}(x,t)$8 with respect to actual sliding velocity—is preserved despite the Implicit Function Theorem approximation used in deriving the model.

A notable corollary concerns transient slip losses: the difference between dissipation rates computed with sliding versus rigid slip velocities equals exactly the rate of change of stored elastic energy. Consequently, slip losses cannot generally be evaluated as the inner product of global forces with global slip inputs—an equivalence holding only in steady state—a result consistent with earlier brush-theory findings.

Numerical and experimental validation

Steady-state simulations with two parametrisations (relaxation lengths of 0.3/0.5 m versus 0.1/0.2 m, vertical load 3 kN) reproduce expected characteristics: lateral force and aligning moment peaks whose magnitudes decrease under combined slip, asymptotic saturation at large slip, exponential deflection decay outside the contact patch, and continuous shear stresses along the contact length. Under the adopted friction law ($\bm{q}(x,t)$9, $$\mathbf{S}\,\partial_x^2 \bm{u} - \mathbf{K}\bm{u} = -\bm{q},$$0), the total tangential force saturates below 2.5 kN despite the applied normal load, and aligning-moment sign reversal under negative longitudinal force emerges from carcass deflection asymmetries, consistent with Higuchi's measurements.

Transient simulations demonstrate the two signature behaviours enabled by parabolicity: progressively faster relaxation with increasing step slip amplitude—even beyond the peak of the tyre characteristic—and pronounced low-pass filtering of sinusoidal slip excitation ($$\mathbf{S}\,\partial_x^2 \bm{u} - \mathbf{K}\bm{u} = -\bm{q},$$1), with attenuation strengthening at higher frequencies ($$\mathbf{S}\,\partial_x^2 \bm{u} - \mathbf{K}\bm{u} = -\bm{q},$$2, 10, 20 m⁻¹) and larger mean slip levels.

Experimental validation uses flat-plank rig data from TU/e. For steady-state lateral carcass deflection measured optically across sideslip angles, the calibrated model (lateral relaxation length 1.089 m, contact semilength 0.03 m) reproduces the measured deformation profile within roughly ±10° around the contact centre, though deviations appear farther out due to the assumed exponential decay outside the patch. For longitudinal step-slip relaxation tests, the optimised model captures nonlinear relaxation across the tested range and converges to correct steady-state forces, although it exhibits somewhat slower transients at large slip and slight asymmetry between positive and negative inputs.

Limitations and open questions

Several limitations are stated explicitly. The derivation of Eq. (ODEz) via the Implicit Function Theorem involves a first-order truncation and a Jacobian approximation, so the FrSD model is an approximation to the true string dynamics; passivity preservation is verified but full equivalence is not claimed. Compact pressure distributions such as the parabolic profile cannot be used directly because $$\mathbf{S}\,\partial_x^2 \bm{u} - \mathbf{K}\bm{u} = -\bm{q},$$3 is singular at the contact edges, requiring regularisation or a series connection with bristle elements—the latter being deferred to future work. The well-posedness theorem covers only sufficiently regular solutions; weak-solution theory is not developed. Experimental agreement degrades outside the immediate vicinity of the contact patch, and transient accuracy at high slip remains imperfect. Extensions to camber, variable pressure, viscoelastic tread behaviour, and the vehicle-level implications of the parabolic structure remain open.

Conclusion

The paper establishes a mathematically well-posed, physically interpretable tyre model whose defining feature is the parabolic character of its governing equations, induced by coupling a nonlocal string constitutive law with regularised distributed FrBD friction. Rigorous ISS, IOS, and passivity proofs grounded in an energy-based Lyapunov function, together with validation against optical deflection measurements and step-slip relaxation experiments, support the claim that the formulation captures slip-dependent relaxation and low-pass filtering behaviour that transport-dominated models cannot represent without additional mechanisms. The identified limitations—regularised pressure profiles, series bristle extensions, and refined transient fidelity at large slip—define the concrete open problems left by the work (2603.02869).