Skip to content

21. Tire — belt / carcass transient (first-order relaxation)

Learning objectives

After this chapter you can:

  1. Explain why a steady Magic Formula (Chapter 03) cannot reproduce the lag in measured \(F_y(t)\) after a step steer, and which physical effect supplies it.
  2. Write the first-order belt relaxation ODE and its exact discrete update, and identify the time constant \(\tau = \sigma/|V_x|\).
  3. Distinguish three different transient mechanisms — relaxation length, the LuGre bristle state, and the belt — and say which regime each fixes.
  4. Describe VDSim's shipped adoption: a backend-agnostic slip filter that feeds relaxed \(\kappa,\alpha\) to the MF path and relaxed slip velocity to the LuGre path, opt-in per wheel.
  5. State what is validated and the honest limit (first order only, no higher belt eigenmodes).

Prerequisites

  • Chapter 03 — Pacejka MF96 pure/combined slip; the steady \((\kappa,\alpha)\mapsto F\) map.
  • Chapter 19 — LuGre bristle \(z\) (contact friction state).
  • Chapter 11 — substep integration.
  • Design — design/TIRE_ROADMAP.md §3 (belt phase T2).

Source: core/include/vdsim/belt_tire.hpp (belt_relax), wired in seven_dof_dynamics.cpp and free_3d_dynamics.cpp; params TireParams::belt.

21.1 Motivation — the carcass deflects before the patch slides

A steady Magic Formula maps the instantaneous slip to force. In a step steer or a fast brake release, the measured lateral force lags the geometric slip angle by tens of milliseconds: the carcass / belt must first deflect before the contact patch reaches the steady deflection distribution that the empirical curve fits. An algebraic map produces this lateral force the same instant the slip appears — too early.

The lag length is set by the relaxation length \(\sigma\) [m] (a carcass compliance property): the tire travels roughly \(\sigma\) before the force reaches \(1-1/e\) of its steady value, so the time constant is

\[ \tau = \frac{\sigma}{|V_x|}. \]

It shrinks with speed — the same deflection is covered faster — which is the signature the validation checks (§21.4).

21.2 Three transients, three regimes

Mechanism Physics Fixes VDSim
Relaxation length first-order \(\alpha_{\mathrm{dyn}}\to\alpha_{\mathrm{geom}}\) lateral transient (MF) legacy relaxation_length_lat (lateral, MF only)
LuGre \(z\) bristle stick–slip at the contact rest / presliding / grade hold Chapter 19 (lugre.enabled)
Belt transient carcass compliance filters slip before the law step-steer / brake-in-turn transient shape this chapter (belt.enabled)

LuGre fixes the low-speed / presliding end; the belt fixes the high-speed transient shape. They are complementary, not alternatives — VDSim lets the belt filter the slip that feeds the LuGre bristle (§21.3).

21.3 Model and discrete update

Each wheel carries a relaxed slip component \(s_{\mathrm{eff}}\) (for \(\kappa\) and \(\alpha\), or for the slip velocity on the LuGre path) obeying

\[ \frac{\sigma}{|V_x|}\,\dot s_{\mathrm{eff}} = s_{\mathrm{geom}} - s_{\mathrm{eff}}. \]

Integrated exactly over a substep \(h\) at frozen \(V_x\) (unconditionally stable, and correct at standstill where \(|V_x|\to 0\) freezes the state instead of dividing):

\[ \boxed{\,s_{\mathrm{eff}} \leftarrow s_{\mathrm{geom}} + (s_{\mathrm{eff}} - s_{\mathrm{geom}})\,e^{-|V_x|\,h/\sigma}\,} \]

This is belt_relax(s_eff, s_geom, Vx, sigma, dt). In the dynamics host:

  • MF path (!lugre): the relaxed \(\kappa,\alpha\) feed ITireModel::compute.
  • LuGre path: the relaxed slip velocities \(v_{r,\text{long}}, v_{r,\text{lat}}\) feed the bristle (belt → LuGre, complementary).

The belt state advances once per substep (between integrator steps) and is held frozen inside an RK4 step so all four stages see the same slip linearization — the same pattern as the LuGre \(z\) sub-advance. It is opt-in via TireParams::belt {enabled, sigma_lat, sigma_long}; default off leaves the steady behaviour (and the published ISO baseline) unchanged.

  step steer at t=0, fixed Vx
  s_geom ──┐ (instant)
           │      s_eff(t) = s_geom (1 - e^{-t/tau}),  tau = sigma/|Vx|
  ─────────┴───────────────────────────►  t
           0      tau            5·tau
                  63%            ~100%

21.4 Validation

tests/integration/test_belt_validation.cpp checks the L2 step-steer response against the relaxation analytic (a fixed-time \(|a_y|\) metric — time-to-fraction is corrupted by the underdamped yaw overshoot):

  • Steady state unchanged — the belt is transient-only; \(a_{y,\mathrm{ss}}\) within 8%.
  • Early response suppressed — at \(t=\tau\) the belt-on \(|a_y|\) is below 95% of belt-off (the slip lag), while still building.
  • More lag at lower speed — at a fixed early time the slower car (larger \(\tau\)) is more suppressed, i.e. the belt-on/off ratio is smaller: the \(\sigma/|V_x|\) scaling.

The belt_relax primitive itself is unit-tested against the exact exponential (BeltTire.*): instant when \(\sigma=0\), frozen at standstill, \(63\%\) at \(\tau\) and steady by \(5\tau\), monotone with no overshoot.

21.5 Shipped status and limitations

  • Opt-in, default off — no ISO 7401/4138 drift on the default preset; the belt is a layer on top of the steady law (MF96 / MF2002) or LuGre.
  • Wired on L2 (seven_dof, also the L3 inner) and L5 (free_3d), both the MF and LuGre paths. L1 bicycle wiring is pending (lowest priority).
  • First order only. This is the carcass relaxation (one time constant), not the higher belt eigenmodes / rigid-ring + enveloping behaviour of FTire / MF-Swift (commercial, up to 100–150 Hz). Out of scope; see design/TIRE_ROADMAP.md §2.
  • Longitudinal relaxation uses the same primitive with sigma_long; combined with the MF2002 backend (Chapter 03 / T1) it filters both \(\kappa\) and \(\alpha\).

Cross-references