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Semilinear single-track vehicle models with distributed tyre friction dynamics

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

Abstract

This paper introduces a novel family of single-track vehicle models that incorporate a distributed representation of transient tyre dynamics, whilst simultaneously accounting for nonlinear effects induced by friction. The core of the proposed framework is represented by the distributed Friction with Bristle Dynamics (FrBD) model, which unifies and extends classical formulations such as Dahl and LuGre by describing the rolling contact process as a spatially distributed system governed by semilinear partial differential equations (PDEs). This model is systematically integrated into a single-track vehicle framework, where the resulting semilinear ODE-PDE interconnection captures the interaction between lateral vehicle motion and tyre deformation. Two main variants are considered: one with rigid tyre carcass and another with flexible carcass, each admitting a compact state-space representation. Local and global well-posedness properties for the coupled system are established rigorously, highlighting the dissipative and physically consistent properties of the distributed FrBD model. A linearisation procedure is also presented, enabling spectral analysis and transfer function derivation, and potentially facilitating the synthesis of controllers and observers. Numerical simulations demonstrate the model's capability to capture micro-shimmy oscillations and transient lateral responses to advanced steering manoeuvres. The proposed formulation advances the state-of-the-art in vehicle dynamics modelling by providing a physically grounded, mathematically rigorous, and computationally tractable approach to incorporating transient tyre behaviour in lateral vehicle dynamics, when accounting for the effect of limited friction.

Semilinear single-track vehicle models with distributed tyre friction dynamics

Abstract

This paper introduces a novel family of single-track vehicle models that incorporate a distributed representation of transient tyre dynamics, whilst simultaneously accounting for nonlinear effects induced by friction. The core of the proposed framework is represented by the distributed Friction with Bristle Dynamics (FrBD) model, which unifies and extends classical formulations such as Dahl and LuGre by describing the rolling contact process as a spatially distributed system governed by semilinear partial differential equations (PDEs). This model is systematically integrated into a single-track vehicle framework, where the resulting semilinear ODE-PDE interconnection captures the interaction between lateral vehicle motion and tyre deformation. Two main variants are considered: one with rigid tyre carcass and another with flexible carcass, each admitting a compact state-space representation. Local and global well-posedness properties for the coupled system are established rigorously, highlighting the dissipative and physically consistent properties of the distributed FrBD model. A linearisation procedure is also presented, enabling spectral analysis and transfer function derivation, and potentially facilitating the synthesis of controllers and observers. Numerical simulations demonstrate the model's capability to capture micro-shimmy oscillations and transient lateral responses to advanced steering manoeuvres. The proposed formulation advances the state-of-the-art in vehicle dynamics modelling by providing a physically grounded, mathematically rigorous, and computationally tractable approach to incorporating transient tyre behaviour in lateral vehicle dynamics, when accounting for the effect of limited friction.
Paper Structure (33 sections, 8 theorems, 92 equations, 12 figures, 2 tables)

This paper contains 33 sections, 8 theorems, 92 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A generalised distributed friction model
  3. Model equations
  4. Distributed bristle dynamics
  5. Frictional force
  6. Vertical pressure distribution
  7. Model solution
  8. Bristle dynamics
  9. Frictional force
  10. Semilinear single-track models
  11. Model equations
  12. Global equilibrium
  13. Distributed tyre dynamics
  14. Rigid relative velocities and apparent slip angles
  15. State-space representation
  16. ...and 18 more sections

Key Result

Theorem 3.1

Suppose that $\mathbf{\Sigma} \in C^0(\mathbb{R}^{2 };\mathbf{M}_{2 }(\mathbb{R}))$ and $\bm{h}_1, \bm{h}_2\in C^0(\mathbb{R}^{2 };\mathbb{R}^{2 })$ are locally Lipschitz continuous, and $\bm{\delta} \in C^0([0,T];\mathbb{R}^{2})$. Then, for all initial conditions (ICs) $(\bm{x}_0,\bm{z}_0) \triangl

Figures (12)

  • Figure 1: A schematic representation of the distributed FrBD model.
  • Figure 2: Schematic of the distributed FrBD friction model particularised for a tyre-wheel system during acceleration. The wheel travels in longitudinal direction with speed $V_x\in \mathbb{R}_{>0}$, and has angular velocity $\Omega \in \mathbb{R}_{>0}$. A local coordinate frame $(O;\xi,\eta,\zeta)$ is attached to the tyre, with the origin $o$ coincident with the leading-edge point. The coordinates $\xi$ and $\eta$ lie in the road plane, whilst $\zeta$ points downward (into the road). The longitudinal coordinate $\xi$ is aligned with the transport velocity $V = \dfrac{V_\textnormal{r}}{L}$, where $V_\textnormal{r} \in \mathbb{R}_{>0}$ denotes the free-rolling speed of the tyre Guiggiani. The $\eta$-axis is oriented so that the local frame $(O;\xi,\eta,\zeta)$ is left-handed.
  • Figure 3: Bristle deflection and frictional force predicted according to the FrBD model, with $\chi_1 = 1$ and $\chi_2 = 0$. Model parameters as in Table \ref{['tab:param1']}.
  • Figure 4: Left: Schematic of the single track model, with its kinematic (blue), dynamic (red), and geometric (black) variables; right: schematic of a single tyre with corresponding reference frame and lateral bristle deflection (in red).
  • Figure 5: Stability charts for two single-track models with constant pressure distribution and flexible tyre carcass linearised around the zero equilibrium $(\bm{x}^\star, \bm{z}^\star(\xi), \bm{\delta}^\star) = \bm{0}$, for different values of the understeer index $\chi \triangleq C_{1}l_1/(C_{2}l_2)$ and longitudinal speed $v_x$. The unstable regions (in white) correspond to combinations of parameters for which the characteristic function $D(\lambda) = \det \tilde{\mathbf{A}}(\lambda)$ has two roots with positive real part. Model parameters: $m = 1300$ kg, $I_z = 2000$$\text{kg}\,\text{m}^2$, $l_1 =1$ m, $l_2 = 1.6$ m, $L_1 = 0.11$ m, $L_2 = 0.9$ m, $\sigma_{0,1} = 163$$\textnormal{m}^{-1}$, $\sigma_{0,2} = 408$$\textnormal{m}^{-1}$, $F_{z1} = 3924$ N, $F_{z2} = 2453$ N; (a) $\lambda_{1} = 0.195$ m, $\lambda_{2} = 0.225$ m; (b) $\lambda_{1} = 0.390$ m, $\lambda_{2} = 0.450$ m.
  • ...and 7 more figures

Theorems & Definitions (21)

  • Theorem 3.1: Local existence and uniqueness of mild solutions
  • proof
  • Theorem 3.2: Local existence and uniqueness of classical solutions
  • proof
  • Theorem 3.3: Global existence and uniqueness of mild solutions
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1: Global existence and uniqueness of solutions
  • ...and 11 more