Tire

Linear and Magic Formula tire model component

Note:  This component is used with the Longitudinal Vehicle Dynamics component.

The Tire component models the longitudinal dynamics of a two tires together with the wheels and axle. The tire longitudinal forces are modeled either by a linear formulation (with saturation) or by the Magic formula. Wheel/axle inertia is included but no compliance is considered.

Equations

The tire translation and rotation velocities are given as

$v_x$ = $\frac{\ⅆ}{\ⅆ t}{s}_x$

$\mathrm{\ω}$ = $\frac{\ⅆ}{\ⅆ t}\mathrm{\ϕ}$ ,    $\mathrm{\α}$ = $\frac{\ⅆ}{\ⅆ t}\mathrm{\ω}$

Where  $s_x$  is the tire translation along the road and $\mathrm{\ϕ}$  is the tire rotation.

The equation of motion is

$J · \mathrm{\α} \= \mathrm{\τ} - f · r$

Where  $J$  is the tire inertia, $\mathrm{\tau }$  is the axle torque,  $f$  is the traction/brake force, and $r$  is the tire radius.

The slip ratio is approximated by

$\mathrm{\κ} \= \frac{r · \mathrm{\ω} - v_x}{\max \left(∣r · \mathrm{\ω}∣\, ∣v_x∣\, {\mathrm{\ϵ}}_v\right)}$

Where  ${\mathrm{ϵ}}_v$  is a small positive number added to avoid singularity.

The tire contact force, f, is calculated as

$f \&equals; \&lcub;\begin{array}{cc}0& f_z <0 \\ \&lcub;\begin{array}{cc}{f}_z · \frac{F_{x\mathrm{0}}}{F_{z\mathrm{0}}} · \frac{\mathrm{\&kappa;}}{{\mathrm{\&kappa;}}_0}& ∣\mathrm{\&kappa;}∣< {\mathrm{\&kappa;}}_0 \\ f_z · \frac{F_{x\mathrm{0}}}{F_{z\mathrm{0}}} \&period; \mathrm{sign}\&lpar;\mathrm{\&kappa;}\&rpar;& ∣\mathrm{\&kappa;}∣\ge {\mathrm{\&kappa;}}_0\end{array}& \mathrm{formulation} \&equals; \mathbf{Linear} \\ K·f_z · \mathrm{D} · \sin \left(C · {\tan }^{-1}\left(\left(B · k - E · \&lpar;B · k - {\tan }^{-1}\left(B · k\right)\right)\right)\right)\&rpar;\&plus;S_v& \mathrm{formulation} \&equals; \mathbf{Magic}\end{array}$

where parameters $K$ and $S_v$ are added to the standard Magic formula to adjust the tire force and facilitate fixed step integration, respectively. Also,

$f_x \&plus; f \&equals; 0$

Connections 

Parameters

Initial Conditions

See Also

Driveline Library Overview

MapleSim Library Overview

1-D Mechanical Overview

Tire Component Library

Vehicle and Tire

Longitudinal Vehicle Dynamics

References

Pacejka, Hans B. Tire and Vehicle Dynamics. Second Edition. SAE International and Elsevier, 2005.