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

${v}_{x}$ =

$\mathrm{ω}$ =  ,    $\mathrm{α}$ =

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

The equation of motion is

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

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

The tire contact force, f, is calculated as

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,

Connections

 Name Condition Description ID - Drive shaft rotational flange driveshaft ${\mathrm{flange}}_{x}$ - Axle longitudinal flange flange_x ${f}_{z}$ - Real input port for axle vertical force f_z

Parameters

Symbol

Condition

Default

Units

Description

ID

$\mathrm{formulation}$

$-$

$\mathbf{Linear}$

$-$

Defines the tire formulation type:

 • Linear: linear formulation with saturation
 • Magic: Magic formula

tiremode

${\mathrm{ϵ}}_{v}$

$-$

$0.1$

$\frac{\mathrm{m}}{\mathrm{s}}$

A small constant added to the denominator of the slip ratio to prevent singularity

epsilon_v

${F}_{z0}$

$1500$

$\mathrm{N}$

Fz0

${F}_{x0}$

$2000$

$\mathrm{N}$

Traction force at nominal vertical load

Fx0

${k}_{0}$

$0.15$

$-$

Slip ratio at peak traction force

sr0

$J$

$-$

$2$

$\mathrm{kg}·{\mathrm{m}}^{2}$

Inertia of the two wheels and the axle

J

$r$

$-$

$0.35$

$\mathrm{m}$

r

$\left[B,C,\mathrm{D},E\right]$

$-$

Coefficients used in the Magic formula.

BCDE

$K$

$1$

$-$

Extra scaling coefficient for the Magic formula.

K

${S}_{v}$

$0$

N

Residual tire force at zero slip (Use ${S}_{v}\ne 0$ for models with fixed-step integration)

Sv

Initial Conditions

 Symbol Condition Default Units Description ID ${\mathrm{ω}}_{0}$ $-$  $\frac{\mathrm{rad}}{s}$ Initial tire angular velocity w

Longitudinal Vehicle Dynamics

References

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