# Tire (Magic Formula)

Tire with longitudinal behavior given by magic formula coefficients

## Library

Simscape / Driveline / Tires & Vehicles

## Description

This block models a tire with longitudinal behavior given by the magic formula, an empirical equation based on four fitting coefficients. 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.

Port A represents the axle on which the tire sits. Port H represents the wheel hub that transmits the thrust generated by the tire to the remainder of the vehicle. Port N accepts a physical signal input of the normal force acting on the tire. The normal force is positive if it acts downward on the tire, pressing it against the pavement. Port S outputs a physical signal with the tire slip measured during simulation.

The block provides two friction variants. The default variant, ```Fixed friction coefficients```, accepts the magic formula coefficients as block parameters. This variant treats the coefficients as constants or load-dependent parameters. Use this variant to model tire dynamics under constant pavement conditions.

The alternative variant, `Variable friction coefficients`, accepts the magic formula coefficients as a physical signal input. Use this variant to model tire dynamics under variable pavement conditions. Selecting this variant exposes physical signal port M. Use this port to provide the magic formula coefficients as a four-element vector, specified in the order [b, c, d, e].

To change block variants:

1. Right-click the tire block.

2. In the context-sensitive menu, select Simscape > Block choice.

3. Select the desired block variant.

To increase the fidelity of the tire model, the block enables you to 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-Loop (HIL) simulation.

## Tire Model

The Tire 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

SymbolDescription and Unit
VxWheel hub longitudinal velocity
uTire longitudinal deformation
ΩWheel angular velocity
Ω′Contact point angular velocity = Ω if u = 0
Vsx = rWΩ – VxWheel slip velocity
V'sx = rWΩ' – VxContact slip velocity = Vsx if u = 0
κ = Vsx/|Vx|Wheel slip
κ'= Vsx/|Vx|Contact slip = κ if u = 0
VthWheel hub threshold velocity
FxLongitudinal force exerted on the tire at the contact point.
CFx = (∂Fx/∂u)0Tire longitudinal stiffness under deformation
bFx = (∂Fx/∂ů)0Tire longitudinal damping under deformation
IwWheel-tire inertia; effective mass = Iw/rw2
τdriveTorque applied by the axle to the wheel

### Tire Kinematics and Response

#### Roll and Slip

If the tire did not slip, it would roll and translate as Vx = rwΩ. But the tire actually does slip and develops a longitudinal force Fx only in response to slip.

The wheel slip velocity is Vsx = rWΩ – Vx. The wheel slip is κ = Vsx/|Vx|. For a locked, sliding wheel, κ = –1. For perfect rolling, κ = 0.

#### Slip at Low Speed

For low speeds, |Vx| ≤ |Vth|, the wheel slip is modified to:

κ = 2Vsx/(Vth + Vx2/Vth) .

This modification allows for a nonsingular, nonzero slip at zero wheel velocity. For example, for perfect slipping (nontranslating spinning tire), Vx = 0 while κ = 2rwΩ/Vth is finite.

#### Deformation

If the tire is modeled with compliance, it is also flexible. Because in this case, 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 CFx and damping bFx.

If the tire is not modeled with compliance, then Ω′ = Ω, V'sx = Vsx, and κ' = κ. In this case, the tire starts simulation undeformed and remains undeformed.

### Tire and Wheel Dynamics

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 CFx and damping bFx. Tire-Road Interaction (Magic Formula) models the longitudinal force Fx on the tire as a function of Fz and κ′ using the Magic Formula, with κ′ as the independent slip variable [1].

The Wheel and Axle radius is the wheel radius rw. The Mass value is the effective mass, Iw/rw2. The tire characteristic function f(κ′, Fz) determines the longitudinal force Fx. Together with the driveshaft torque applied to the wheel axis, Fx determines the wheel angular motion and longitudinal motion.

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.

## Limitations

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

### Tire Compliance

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.

## Ports

PortDescription

A

Conserving mechanical rotational port associated with the axle of the tire

H

Conserving mechanical translational port associated with the hub of the tire

N

Physical signal input port associated with the normal force on the tire

M

Physical signal input associated with the static and kinetic friction coefficients of the tire

S

Physical signal output port associated with the relative slip between the tire and road

## Parameters

### Tire Force

Parameterize by

Select how to use the Magic Formula to model the tire-road interaction. The default is ```Peak longitudinal force and corresponding slip```. This tab appears only when the block variant is set to ```Fixed friction coefficients```. For more information about the block variants, see the block description.

• ```Peak longitudinal force and corresponding slip``` — Parametrize the Magic Formula with physical characteristics of the tire.

• `Constant Magic Formula coefficients` — Parameterize the Magic Formula directly with its 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.

### Dimensions

Unloaded tire-wheel radius rw. The default is `0.3`.

From the drop-down list, choose units. The default is meters (`m`).

### Dynamics

Compliance

Select how to model the dynamical compliance of the tire. The default is ```No compliance — Suitable for HIL simulation```.

• ```No compliance — Suitable for HIL simulation``` — Tire is modeled with no dynamical compliance.

• `Specify stiffness and damping`—Tire is modeled as a stiff, dampened spring and deforms under load. If you select this option, the panel changes from its default.

Inertia

Select how to model the rotational inertia of the tire. The default is `No inertia`.

• `No inertia`—Tire is modeled with no inertia.

• ```Specify inertia and initial velocity```—Tire is modeled with rotational inertia. If you select this option, the panel changes from its default.

### Rolling Resistance

Method used to specify the rolling resistance acting on a rotating wheel hub. The default value is ```No rolling resistance```. Options include:

• `No rolling resistance`

Select this option to ignore the effect of rolling resistance on a model.

• ```Specify rolling resistance```

Select between two rolling resistance models: `Constant coefficient` and ```Pressure and velocity dependent```.

The default value is ```Constant coefficient```.

### Slip Calculation

Velocity threshold

Wheel hub velocity Vth 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`).

## Extended Capabilities

### Real-Time and Hardware-in-the-Loop Simulation

For optimal simulation performance, set the Dynamics > Compliance parameter to ```No compliance - Suitable for HIL simulation```.

## References

[1] Pacejka, H. B. Tire and Vehicle Dynamics, Society of Automotive Engineers and Butterworth-Heinemann, Oxford, 2002, chapters 1,4,7, and 8