Tire tread in contact with road surface approximated by Magic Formula

Tires & Vehicles/Tire Subcomponents

The Tire-Road Interaction (Magic Formula) block models the longitudinal tire-road contact force using the Magic Formula. Use this block as a component in tire models.

The convention for the vertical load is positive downward. If the vertical load is zero or negative, the horizontal tire force vanishes. In that case, the tire is just touching the ground or has left the ground. The longitudinal direction lies along the forward-backward axis of the tire. For model details, see Tire-Road Interaction Model.

You specify the downward vertical load *F*_{z} (in
newtons) through a physical signal input at port N. The block reports
the developed tire slip *κ*, as a decimal fraction,
through a physical signal output at port S.

The tire tread in contact with the road and the transfer of horizontal thrust reaction to the vehicle at the wheel hub are represented by the translational conserving ports T and H, respectively.

**Parametrize by**Select how to use the Magic Formula to model the tire-road interaction. The default is

`Peak longitudinal force and corresponding slip`

.`Peak longitudinal force and corresponding slip`

— Parametrize the Magic Formula with physical characteristics of the tire.`Constant Magic Formula coefficients`

— Parametrize the Magic Formula directly with constant coefficients. If you select this option, the panel changes from its default.`Load-dependent Magic Formula coefficients`

— Parametrize the Magic Formula directly with load-dependent coefficients. If you select this option, the panel changes from its default.

**Velocity threshold**The wheel hub velocity

*V*_{th}below which the slip calculation is modified to avoid singular evolution at zero velocity. Must be positive. The default is`0.1`

.From the drop-down list, choose units. The default is meters per second (

`m/s`

).

The Tire-Road Interaction (Magic Formula) block models the longitudinal forces at the tire-road contact patch using the Magic Formula of Pacejka [7].

The figure displays the forces on the tire. The table defines the tire model variables.

**Tire-Road Contact Variables**

Symbol | Description and Unit |
---|---|

Ω | Wheel angular velocity |

r_{w} | Wheel radius |

V_{x} | Wheel hub longitudinal velocity |

r_{w}Ω | Tire tread longitudinal velocity |

V_{sx} =
r_{w}Ω – V_{x} | Wheel slip velocity = tread velocity T – hub velocity H |

κ = V_{sx}/|V_{x}| | Wheel slip |

F_{z}, F_{z0} | Vertical load and nominal vertical load on tire |

F_{x} = f(κ, F_{z}) | Longitudinal force exerted on the tire at the contact point. Also a characteristic function f of the
tire. |

A tire model provides a steady-state *tire characteristic
function* *F*_{x} = *f*(*κ*, *F*_{z}),
the longitudinal force *F*_{x} on
the tire, based on:

Vertical load

*F*_{z}Wheel slip

*κ*

The Magic Formula is a specific form for the tire characteristic
function, characterized by four dimensionless coefficients (*B*, *C*, *D*, *E*),
or stiffness, shape, peak, and curvature:

*F*_{x} = *f*(*κ*, *F*_{z})
= *F*_{z}·*D*·sin( *C*·arctan[
{ *B**κ* – *E*·[ *B**κ* –
arctan(*B**κ*) ] } ] ) .

The slope of *f* at *κ* =
0 is *B**C**D*·*F*_{z}.

A more general Magic Formula uses dimensionless coefficients that are functions of the tire load. A more complex set of parameters p_i, entered in the dialog box, specifies these functions:

*F*_{x0} = *D*_{x}·sin( *C*_{x}·arctan[
{ *B*_{x}*κ*_{x} – *E*_{x}·[ *B*_{x}*κ*_{x} –
arctan(*B*_{x}*κ*_{x})
] } ] ) + *S*_{Vx} ,

where

*df*_{z} = (*F*_{z} – *F*_{z0})/*F*_{z} ,

*κ*_{x} = *κ* +
S_{Hx} ,

*C*_{x} = p_Cx1
,

*D*_{x} = *μ*_{x}·*F*_{z} ,

*μ*_{x} =
p_Dx1 + p_Dx2·*df*_{z} ,

*E*_{x} = (p_Ex1
+ p_Ex2·*df*_{z} + p_Ex3·*df*_{z}^{2})[1
– p_Ex4·sgn(*κ*_{x})]
,

*K*_{xκ} = *F*_{z}·(p_Kx1
+ p_Kx2·*df*_{z})·exp(p_Kx3·*df*_{z})
,

*B*_{x} = *K*_{xκ}/(*C*_{x}*D*_{x} + *ε*_{x})
,

*S*_{Hx} = p_Hx1
+ p_Hx2·*df*_{z} ,

*S*_{Vx} = *F*_{z}·(p_Vx1
+ p_Vx2·*df*_{z}) .

*S*_{Hx} and *S*_{Vx} represent
offsets to the slip and longitudinal force in the force-slip function,
or horizontal and vertical offsets if the function is plotted as a
curve. *μ*_{x} is the longitudinal
load-dependent friction coefficient. *ε*_{x} is
a small number inserted to prevent division by zero as *F*_{z} approaches
zero.

The block uses a representative set of Magic Formula coefficients
The block scales the coefficients to yield the peak longitudinal force *F*_{x0} at
the corresponding slip *κ*_{0} that
you specify, for rated vertical load *F*_{z0}.

Numerical values are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Surface | B | C | D | E |
---|---|---|---|---|

Dry tarmac | 10 | 1.9 | 1 | 0.97 |

Wet tarmac | 12 | 2.3 | 0.82 | 1 |

Snow | 5 | 2 | 0.3 | 1 |

Ice | 4 | 2 | 0.1 | 1 |

The Tire-Road Interaction (Magic Formula) block assumes longitudinal motion only and includes no camber, turning, or lateral motion.

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