Formula: Created page with "This article specifies a race-engineering tyre model for Formula One: combined-slip force generation, transient build-up, thermal coupling, wear kinetics, circuit energy characterisation, and stint optimisation. Symbols follow motorsport literature; parameters are given as calibrated ranges suitable for lap-time simulation and strategy work. == Force generation (nonlinear, combined slip) == Baseline lateral force (Magic Formula representation): $F_y = D \sin\!\bi..."$

2025-08-06T07:31:23Z

Created page with "This article specifies a race-engineering tyre model for Formula One: combined-slip force generation, transient build-up, thermal coupling, wear kinetics, circuit energy characterisation, and stint optimisation. Symbols follow motorsport literature; parameters are given as calibrated ranges suitable for lap-time simulation and strategy work. == Force generation (nonlinear, combined slip) == Baseline lateral force (Magic Formula representation): <math> F_y = D \sin\!\bi..."

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This article specifies a race-engineering tyre model for Formula One: combined-slip force generation, transient build-up, thermal coupling, wear kinetics, circuit energy characterisation, and stint optimisation. Symbols follow motorsport literature; parameters are given as calibrated ranges suitable for lap-time simulation and strategy work.

== Force generation (nonlinear, combined slip) ==

Baseline lateral force (Magic Formula representation):
<math>
F_y = D \sin\!\big( C \arctan\!\big( B \alpha - E ( B \alpha - \arctan(B \alpha) ) \big) \big),
</math>
with <math>D=\mu F_z</math>; <math>B,C,E</math> shape parameters; slip angle <math>\alpha</math>; vertical load <math>F_z</math>; peak friction <math>\mu</math>.

Combined-slip admissible set (anisotropic ellipse):
<math>
\left(\frac{F_x}{\mu_x F_z}\right)^{n} + \left(\frac{F_y}{\mu_y F_z}\right)^{n} \le 1,\quad n \in [1.6,2.2].
</math>

Load-sensitivity (decreasing peak friction with load):
<math>
\mu(F_z)=\mu_0 \left(\frac{F_z}{F_{z0}}\right)^{a_\mu},\quad a_\mu \in [-0.05,-0.02].
</math>

Camber thrust (additive near-linear term for moderate camber <math>\gamma</math>):
<math>
F_{y,\gamma}=C_\gamma\, \gamma\, F_z,\qquad F_y(\alpha,\gamma)\approx F_y(\alpha,0)+F_{y,\gamma}.
</math>

Indicative calibrated ranges (per front/rear tyre, race trim; to be refined per event):
{| class="wikitable"
! Parameter !! Typical range !! Notes
|-
| Peak lateral friction <math>\mu</math> || 1.60–1.95 || Effective (includes compound & road); varies with temperature and wear
|-
| Cornering stiffness <math>C_\alpha</math> || 80–140 kN/rad || Increases with pressure & load to a limit; front < rear asymmetry common
|-
| Camber thrust coeff. <math>C_\gamma</math> || 2–6 kN/rad || Increases with vertical load; risk of shoulder over-temp if too high
|-
| Stiffness factor <math>B</math> || 8–15 1/rad || Magic-Formula fit; car- and compound-dependent
|-
| Shape factor <math>C</math> || 1.2–1.6 || Controls curvature to peak
|-
| Curvature <math>E</math> || 0.0–0.3 || Lower E → sharper peak; sensitive to wear state
|}

== Transient build-up and rolling losses ==

Relaxation-length dynamics (distance form):
<math>
\frac{dF_y}{ds}+\frac{1}{\lambda}F_y=\frac{C_\alpha}{\lambda}\,\alpha,\qquad \lambda \in [0.3,0.8]\ \mathrm{m}.
</math>

Rolling resistance (per wheel) for lap-sim:
<math>
F_{\mathrm{rr}} = C_{\mathrm{rr}} F_z,\qquad C_{\mathrm{rr}} \in [0.012,0.018],
</math>
with temperature/wear sensitivity modelled as
<math>
C_{\mathrm{rr}} = C_{\mathrm{rr0}} + k_T (T_c - T^{\ast}) + k_w w.
</math>

== Thermal model (two-node surface/carcass) ==

Surface <math>T_s</math> and carcass <math>T_c</math> temperatures:
<math>
C_s \frac{dT_s}{dt} = \eta_s\, \mu\, F_z\, v\, \gamma_s - h_s A_s (T_s - T_\infty) - k_{sc}(T_s - T_c),
</math>
<math>
C_c \frac{dT_c}{dt} = k_{sc}(T_s - T_c) - h_c A_c (T_c - T_\infty).
</math>

Temperature modifiers (Gaussian around target carcass temperature <math>T^{\ast}</math>):
<math>
\phi(T_c)=\exp\!\big(-k_\phi (T_c - T^{\ast})^2\big),\qquad
\psi(T_c)=\exp\!\big(-k_\psi (T_c - T^{\ast})^2\big),
</math>
used to scale <math>\mu</math> and <math>C_\alpha</math> respectively.

== Wear and degradation kinetics ==

Wear state <math>w \in [0,1]</math> (0 fresh, 1 end-of-life). Sliding-power law:
<math>
\frac{dw}{dt}=k_w\!\left(\left|F_x v_{s,x}\right|+\left|F_y v_{s,y}\right|\right).
</math>

Performance decay:
<math>
\mu(w,T_c)=\mu_0 \left(1-a_w w\right)\phi(T_c),\qquad
C_\alpha(w,T_c)=C_{\alpha0}\left(1-b_w w\right)\psi(T_c),
</math>
where <math>a_w,b_w \in [0.15,0.45]</math> give typical end-of-stint losses.

Failure/artefact modes: thermal fade (high <math>T_c</math>), graining (low <math>T_s</math> and high shear; reversible), blistering (subsurface; irreversible), flat-spot (lock-up; step change in <math>w</math>), pick-up (reversible after cleaning laps).

== Compounds, operating windows, prescriptions ==

Pirelli supplies five slick compounds (C1–C5). Three are nominated per event (H/M/S labels). Indicative (modelling) windows—replace with event notes when available:

{| class="wikitable"
! Compound !! Nominal grip (rel.) !! Wear rate (rel.) !! Indicative carcass window (°C) !! Typical use
|-
| C1 (hard) || Low || Low || 95–115 || High-energy/abrasive; long stints; hot ambient
|-
| C2 || Low–Med || Low–Med || 95–115 || Long stints; high-load corners
|-
| C3 || Med || Med || 90–110 || Baseline race compound on many tracks
|-
| C4 || Med–High || Med–High || 85–105 || Qualifying bias; cooler ambient; short stints
|-
| C5 (soft) || High || High || 85–100 || Street circuits; low energy; peak grip runs
|}

Prescriptions (event bulletins): minimum starting pressures (front/rear), maximum static camber (per axle), blanket rules (where allowed). These override generic models for compliance and must be enforced in setup.

== Circuit energy characterisation and degradation rates ==

Use a lateral/longitudinal energy index and texture/abrasion rating to forecast wear and window risk. Pirelli’s track severity scale (1–5) maps well to model priors:

{| class="wikitable"
! Circuit !! Track energy (1–5) !! Dominant load !! Surface & notes !! Expected stint shape (S/M/H)
|-
| Monaco || 1 || Low lateral; traction-limited || Smooth street; low energy || Very shallow degradation; thermal management critical at low speeds
|-
| Monza || 2 || Braking/traction; low lateral || Smooth; long straights || Low deg; front-tyre warm-up critical; flat-spot risk
|-
| Bahrain || 3–4 || Traction + braking; heat || Abrasive; hot ambient || Medium–high deg; rear thermal fade risk
|-
| Barcelona (Catalunya) || 4 || Sustained lateral || Medium-abrasive; long corners || Medium–high deg; front-left graining if cool
|-
| Silverstone || 5 || Very high lateral || Fast, high-energy || High deg; carcass temperature control is limiting
|-
| Suzuka || 5 || Mixed; long-radius lateral || Smooth-medium; esse complex || High deg; front-limited early, rear-limited late
|-
| Zandvoort || 4 || Banked lateral loading || Fresh abrasive after resurfacing || Medium–high deg; camber window tight
|}

Example calibrated degradation slopes (race runs, dry; representative order-of-magnitude for modelling, per compound on “medium” energy tracks):
{| class="wikitable"
! Compound !! Linear deg <math>d</math> (s/lap) !! Nonlinear term at end-of-stint (extra s over last 5 laps)
|-
| C1 || 0.010–0.020 || 0.2–0.6
|-
| C2 || 0.015–0.030 || 0.3–0.8
|-
| C3 || 0.020–0.040 || 0.5–1.2
|-
| C4 || 0.030–0.060 || 0.8–1.8
|-
| C5 || 0.040–0.080 || 1.0–2.5
|}

== Strategy modelling and optimal stints ==

Per-lap loss including wear, thermal offset from target <math>T^{\ast}</math>, and traffic factor <math>Q_k</math>:
<math>
\Delta t_k = \alpha_1 w_k + \alpha_2 (T_{c,k}-T^{\ast})^2 + \alpha_3 Q_k.
</math>

Cumulative stint time to lap <math>N</math>:
<math>
T_{\mathrm{stint}}(N)=\sum_{k=1}^{N}\!\big(t_0+\Delta t_k\big).
</math>

Under linear degradation <math>\Delta t_k \approx d k</math> and pit loss <math>P</math>, the continuous optimum stint length is
<math>
N^{\ast} \approx \sqrt{\frac{2P}{d}},
</math>
useful for first-order stop count decisions. The undercut condition at lap <math>n</math>:
<math>
\big(t(n)-t(n+1)\big)_{\mathrm{old}} \;>\; P - \big(t(n)-t(n+1)\big)_{\mathrm{new}}.
</math>

Fuel burn-off and DRS availability couple into <math>d</math> and <math>\alpha_2</math>; safety-car or VSC resets change optimal <math>N^{\ast}</math> by reducing the opportunity cost of a stop.

== Data acquisition and online identification ==

Typical online state vector and measurements for estimation:
<math>
x=\{C_\alpha,\mu,\lambda,T_s,T_c,w\},\qquad
y=\{F_x,F_y,\omega,\tfrac{d\psi}{dt},a_y,T_{\mathrm{IR}}\}.
</math>
Teams use EKF/UKF observers to update <math>x</math> in real time from wheel speed, brake temps, steering, IMU, and IR cameras; stint-end mass and vibration signatures cross-check wear <math>w</math>.

== Validation workflow (practical) ==

# Identify <math>B,C,D,E,C_\alpha,\lambda</math> on the tyre rig and from clean track segments.
# Fit <math>C_s,C_c,h_sA_s,h_cA_c,k_{sc}</math> from controlled long-runs.
# Calibrate <math>k_w</math> against abrasion metrics and compare to stint mass loss.
# Close the loop in lap-sim; verify predicted vs observed stint shape, undercut thresholds, and pit δ.
# Replace generic windows with event-specific Pirelli prescriptions upon bulletin release.

== See also ==
* [[Race strategy modelling]]
* [[Lap time and delta analysis]]
* [[Chassis and suspension design]]
* [[Aerodynamics in Formula One]]

== References ==
<references />
* FIA Formula One Technical Regulations (Tyres & Wheels; current season).
* Pirelli Motorsport, Event Technical Notes (pressures, camber, nominations, blanket rules).
* Pacejka, H., ''Tire and Vehicle Dynamics'', Elsevier.
* Milliken, W. & Milliken, D., ''Race Car Vehicle Dynamics'', SAE.
* Dixon, J., ''Tires, Suspension, and Handling'', SAE.
* Selected SAE papers on tyre thermal/wear modelling and combined slip in racing applications.
* Bosch, ''Automotive Handbook'' (combined slip, relaxation length, rolling resistance).

Tyres and degradation models - Revision history