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Bare and stretched string tyre models with distributed FrBD dynamics

Luigi Romano

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.

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.
Paper Structure (21 sections, 5 theorems, 43 equations, 8 figures, 3 tables)

This paper contains 21 sections, 5 theorems, 43 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model derivation
  3. Constitutive equations, kinematic constraints, and friction model
  4. Constitutive differential equations and kinematic constraints
  5. Friction model
  6. String tyre model with FrBD dynamics (FrSD)
  7. String dynamics
  8. Tyre forces and moments
  9. Mathematical properties
  10. Well-posedness
  11. Stability and passivity
  12. Stability
  13. Dissipativity and passivity
  14. Numerical and experimental validation
  15. Numerical simulations
  16. ...and 6 more sections

Key Result

Theorem 2.1

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

Figures (8)

  • Figure 1: Steady-state lateral deflection and shear stresses. Model parameters as in Table \ref{['tab:parameters']}: (a) P1; (b) P2.
  • Figure 2: Steady-state tyre characteristics. Model parameters as in Table \ref{['tab:parameters']} (P1).
  • Figure 3: Steady-state tyre characteristics. Model parameters as in Table \ref{['tab:parameters']} (P2).
  • Figure 4: Transient response to different step slip inputs. Model parameters as in Table \ref{['tab:parameters']}.
  • Figure 5: Transient lateral response to a sinusoidal slip input $\sigma_y(s) = \bar{\sigma}_y[1 + 0.5\sin(\omega s)]$ for different values of $\bar{\sigma}_y$ and $\omega = 5$$\textnormal{m}^{-1}$. Model parameters as in Table \ref{['tab:parameters']}.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 2.1: Edwards Edwards
  • Remark 2.1
  • Theorem 3.1: Existence and uniqueness of regular solutions
  • proof
  • Lemma 3.1: Input-to-state stability (ISS)
  • proof
  • Corollary 3.1: Input-to-output stability (IOS)
  • Remark 3.1
  • Lemma 3.2: Passivity
  • proof