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Tire with longitudinal behavior given by Magic Formula coefficients

**Library:**Simscape / Driveline / Tires & Vehicles

The Tire (Magic Formula) block models a tire with longitudinal behavior given by the Magic Formula [1], an empirical equation based on four fitting coefficients. The block can model tire dynamics under constant or variable pavement conditions.

The longitudinal direction of the tire is the same as its direction of motion as it rolls on pavement. This block is a structural component based on the Tire-Road Interaction (Magic Formula) block.

To increase the fidelity of the tire model, you can specify properties such as tire compliance, inertia, and rolling resistance. However, these properties increase the complexity of the tire model and can slow down simulation. Consider ignoring tire compliance and inertia if simulating the model in real time or if preparing the model for hardware-in-the-loop (HIL) simulation.

The Tire (Magic Formula) block models the tire as a rigid wheel-tire combination in contact with the road and subject to slip. When torque is applied to the wheel axle, the tire pushes on the ground (while subject to contact friction) and transfers the resulting reaction as a force back on the wheel. This action pushes the wheel forward or backward. If you include the optional tire compliance, the tire also flexibly deforms under load.

The figure shows the forces acting on the tire. The table defines the tire model variables.

**Tire Model Variables**

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

r_{w} | Wheel radius |

V_{x} | Wheel hub longitudinal velocity |

u | Tire longitudinal deformation |

Ω | Wheel angular velocity |

Ω′ | Contact point angular velocity. If there is no tire longitudinal deformation, that is, if $$u=0$$, $${\Omega}^{\prime}=\Omega $$. |

$${r}_{w}{\Omega}^{\prime}$$ | Tire tread longitudinal velocity |

$${V}_{sx}={r}_{w}\Omega -{V}_{x}$$ | Wheel slip velocity |

$${{V}^{\prime}}_{sx}={r}_{w}{\Omega}^{\prime}-{V}_{x}$$ | Contact slip velocity. If there is no tire longitudinal deformation, that is, if $$u=0$$, $${{V}^{\prime}}_{sx}={V}_{sx}$$. |

$$k=\frac{{V}_{sx}}{\left|{V}_{x}\right|}$$ | Wheel slip |

$${k}^{\prime}=\frac{{{V}^{\prime}}_{sx}}{\left|{V}_{x}\right|}$$ | Contact slip. If there is no tire longitudinal deformation, that is, if $$u=0$$, $${k}^{\prime}=k$$. |

V_{th} | Wheel hub threshold velocity |

F_{z} | Vertical load on tire |

F_{x} | Longitudinal force exerted on the tire at the contact point |

$${C}_{{F}_{x}}={\left(\frac{\partial {F}_{x}}{\partial u}\right)}_{0}$$ | Tire longitudinal stiffness under deformation |

$${b}_{{F}_{x}}={\left(\frac{\partial {F}_{x}}{\partial \dot{u}}\right)}_{0}$$ | Tire longitudinal damping under deformation |

I_{w} | Wheel-tire inertia, such that the effective mass is equal to $$\frac{{I}_{w}}{{r}_{w}^{2}}$$ |

τ_{drive} | Torque applied by the axle to the wheel |

A nonslipping tire would roll and translate as $${V}_{x}={r}_{w}\Omega $$. However, as tires do slip, they respond by developing a
longitudinal force, *F _{x}*.

The wheel slip velocity is $${V}_{sx}={r}_{w}\Omega -{V}_{x}$$. The *wheel slip* is $$k=\frac{{V}_{sx}}{\left|{V}_{x}\right|}$$. For a locked, sliding wheel, $$k=-1$$. For perfect rolling, $$k=0$$.

For low speeds, as defined by $$\left|{V}_{x}\right|\le \left|{V}_{th}\right|$$, the wheel slip becomes:

$$k=\frac{2{V}_{sx}}{\left({V}_{th}+\frac{{V}_{x}^{2}}{{V}_{th}}\right)}$$

This modification allows for a nonsingular, nonzero slip at zero wheel velocity. For example, for perfect slipping, that is, in the case of a nontranslating spinning tire, $${V}_{x}=0$$ while $$k=\frac{2{r}_{w}\Omega}{{V}_{th}}$$ is finite.

If the tire is modeled with compliance, it is also flexible. In this case,
because the tire deforms, the tire-road contact point turns at a slightly
different angular velocity, *Ω′*, from the wheel,
*Ω*, and requires, instead of the wheel slip, the
*contact point* or *contact patch
slip*
*κ'*. The block models the deforming tire as a translational
spring-damper of stiffness,
*C _{Fx}*,
and damping,

If you model a tire without compliance, that is, if $$u=0$$, then there is no tire longitudinal deformation at any time in the simulation and:

$${k}^{\prime}=k$$

$${{V}^{\prime}}_{sx}={V}_{sx}$$

$${\Omega}^{\prime}=\Omega $$

The full tire model is equivalent to this Simscape™/Simscape
Driveline™ component diagram. It simulates both transient and steady-state
behavior and correctly represents starting from, and coming to, a stop. The
Translational Spring and
Translational Damper are equivalent to the tire
stiffness *C _{Fx}* and
damping

The Wheel and Axle radius is the wheel radius *r _{w}*.
The Mass value is the effective mass, $$\frac{{I}_{w}}{{r}_{w}^{2}}$$. The tire characteristic function

Without tire compliance, the Translational Spring and Translational Damper are
omitted, and contact variables revert to wheel variables. In this case, the tire
effectively has infinite stiffness, and port **P** of Wheel and
Axle connects directly to port **T** of Tire-Road Interaction
(Magic Formula).

Without tire inertia, the Mass is omitted.

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

Tire compliance implies a time lag in the tire response to the forces on it. Time lag simulation increases model fidelity but reduces simulation performance. See Adjust Model Fidelity.

[1] Pacejka, H. B. *Tire and
Vehicle Dynamics.* Elsevier Science, 2005.