# Tire (Magic Formula)

Tire defined by Magic Formula coefficients

• Libraries:
Simscape / Driveline / Tires & Vehicles

## Description

The Tire (Magic Formula) block represents a tire with longitudinal behavior given by the Magic Formula , an empirical equation based on four fitting coefficients. You 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 compliance, inertia, rolling resistance, and varying effective rolling radius. However, these properties increase the complexity of the tire model and can slow down simulation. Consider neglecting tire compliance and inertia if simulating the model in real time or if preparing the model for hardware-in-the-loop (HIL) simulation.

### Tire Model

The block treats the tire as a rigid wheel-tire combination that is in contact with the road and subject to slip. When torque drives the wheel axle, the tire transmits longitudinal force, Fx, to the road. The tire transfers the resulting reaction as a force back on the wheel. This action rotates the wheel to generate longitudinal motion. If you model tire compliance, the tire also flexibly deforms under load. If you set the Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```, the tire radius also changes depending on the load and rotational velocity.

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

SymbolDescription
VxWheel hub longitudinal velocity.
uTire longitudinal deformation.
ΩWheel angular velocity.
${\Omega }^{\prime }=\Omega +\frac{\stackrel{˙}{u}}{{r}_{W}}$Contact point angular velocity. If there is no tire longitudinal deformation, $u=0$.
${V}_{T}={r}_{w}{\Omega }^{\prime }={r}_{w}\Omega +\stackrel{˙}{u}$Tire tread longitudinal velocity. Typically, the tire tread longitudinal velocity has a component due to tire rotation, rwΩ, and an optional component due to tire deformation, $\stackrel{˙}{u}$.
${V}_{sx}={V}_{x}-{V}_{T}$Contact patch slip velocity. If there is no tire longitudinal compliance, $u=0$.
$k=-\frac{{V}_{sx}}{{|{V}_{x}|}_{smooth}}$Wheel slip for a tire without compliance.
VthWheel hub threshold velocity.
VXLOWLower boundary of slip denominator.
FxLongitudinal 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.
${b}_{{F}_{x}}={\left(\frac{\partial {F}_{x}}{\partial \stackrel{˙}{u}}\right)}_{0}$Tire longitudinal damping under deformation.
IwWheel-tire inertia, such that the effective mass is equal to $\frac{{I}_{w}}{{r}_{w}^{2}}$
τdriveTorque applied by the axle to the wheel.

### Tire Kinematics and Response

You can model roll, slip, and deformation.

Roll and Slip

The equation for translational motion of a non-slipping, non-compliant tire is ${V}_{x}={r}_{w}\Omega$. When tires experience slip, they develop a longitudinal force, Fx.

The contact patch slip velocity is ${V}_{sx}={V}_{x}-{r}_{w}\Omega -\stackrel{˙}{u}$. For a tire without compliance, u = 0. The unsmoothed contact patch slip is

`$k=-\frac{{V}_{sx}}{|{V}_{x}|},$`

and the block saturates the slip denominator as

where VXLOW is the Lower boundary of slip denominator, VXLOW parameter. The block smoothly transitions |Vx| to VXLOW over the transition regions -VXLOW - Vth/2 < Vx < -VXLOW + Vth/2 and VXLOW - Vth/2 < Vx < VXLOW + Vth/2 . The block saturates slip according to

where kpumin is the Minimum valid wheel slip, KPUMIN parameter and kpumax is the Maximum valid wheel slip, KPUMAX parameter. The block transitions k over the regions kpumin - kth/2 < k < kpumin + kth/2 and kpumax - kth/2 < k < kpumax + kth/2 . The block defines the slip smoothing threshold as

`${k}_{th}=\frac{{V}_{th}}{1m/s}.$`

For this equation, a locked, sliding wheel has k = -1. For perfect rolling, k = 0.

Deformation

When you set Compliance to ```Specify stiffness and damping```, the block treats the tire as flexible. When the tire deforms, the tire-road contact point turns at a slightly different angular velocity, Ω′, from the wheel velocity, Ω, which generates contact patch slip. The block defines the deforming tire as a translational spring-damper of stiffness, CFx, and damping, bFx.

When you set Compliance to ```No compliance - Suitable for HIL simulation```, $u=0$, and there is no tire longitudinal deformation at any time in the simulation, such that ${\Omega }^{\prime }=\Omega$.

### Tire and Wheel Dynamics

The block is equivalent to this component diagram. The block simulates both transient and steady-state behavior and represents starting and stopping. The Translational Spring and Translational Damper blocks are equivalent to the tire stiffness CFx and damping bFx, respectively. The Tire-Road Interaction (Magic Formula) block represents the longitudinal force Fx on the tire as a function of Fz, and k using the Magic Formula, where k is the independent slip variable and Fz is the input signal at port N. The block labeled `Rolling Radius` is the tire rolling radius rw. The inertia value is the effective inertia, $\frac{{I}_{w}}{{r}_{w}^{2}}$. The tire characteristic function f(k′, 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.

When you do not model tire compliance, the block omits the ```Longitudinal Stiffness``` and `Longitudinal Damper` blocks in the diagram, and contact variables revert to wheel variables. In this case, the tire effectively has infinite stiffness, and port P of Rolling Radius connects directly to port C of Ideal Force Sensor block.

When you set Rolling Resistance to a setting other than `Pressure and velocity dependent (Magic Formula)`, the block omits the Ideal Force Sensor block and connection T of the Rolling Resistance block. When you set Compliance to ```No compliance - Suitable for HIL simulation```, Port P of the Rolling Radius connects directly to the Ideal force sensor port C or port T of the Tire-Road Interaction (Magic Formula) block.

The block determines an effective rolling radius to account for centrifugal growth and reduction due to loading such that

`$\begin{array}{l}{r}_{w}={R}_{\Omega }-\frac{{F}_{z0}}{{c}_{z}}\left({D}_{reff}\mathrm{arctan}\left({B}_{reff}\frac{{F}_{z}}{{F}_{z0}}\right)+{F}_{reff}\frac{{F}_{z}}{{F}_{z0}}\right)\\ {R}_{\Omega }={R}_{0}\left({q}_{re0}+{q}_{v1}{\left(\frac{\Omega {R}_{0}}{{V}_{0}}\right)}^{2}\right)\end{array}$`

where:

• rw is the effective rolling radius.

• RΩ is the centrifugal growth of the free tire radius.

• Fz is the vertical load on the tire.

• Fz0 is the Tire nominal vertical load, FNOMIN parameter.

• cz is the Vertical stiffness parameter.

• Ω is the wheel rotational velocity.

The quantities Fz and Ω vary with time.

These quantities correspond to TIR file values and block parameters:

VariableTIR File IdentifierParameter Name
Breff`BREFF`Low load stiffness effective rolling radius, BREFF
Dreff`DREFF`Peak value of effective rolling radius, DREFF
Freff`FREFF`High load stiffness effective rolling radius, FREFF
qre0`Q_RE0`Ratio of nominal tire radius with non-rolling free tire radius, Q_RE0
qv1`Q_V1`Tire radius increase with speed, Q_V1

Note

If your TIR file does not provide `Q_RE0` and `Q_V1`, set these values to 1 and 0, respectively. The `sdlUtility.tirread` function handles this automatically.

### Assumptions and Limitations

• The block assumes only longitudinal motion and includes no camber, turning, or lateral motion.

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

## Ports

### Input

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Physical signal input port associated with the normal force acting on the tire, in N. The normal force is positive if it acts downward on the tire, pressing it against the pavement.

Physical signal input port associated with the Magic Formula coefficients. Provide the Magic Formula coefficients as a four-element vector, specified in the order [B, C, D, E].

#### Dependencies

To enable this port, set Parameterize by to ```Physical signal Magic Formula coefficients```.

### Output

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

### Conserving

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Mechanical rotational port associated with the axle that the tire sits on.

Mechanical translational port associated with the wheel hub that transmits the thrust generated by the tire to the remainder of the vehicle.

## Parameters

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### Main

Magic Formula tire simulation option. Select how the block parameterizes the tire using the Magic Formula:

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

• ```Constant Magic Formula coefficients``` — Specify the parameters that define the constant B, C, D, and E coefficients as scalars.

• ```Load-dependent Magic Formula coefficients``` — Specify the parameters that define the load-dependent C, D, E, K, H, and V coefficients as vectors, one for each coefficient.

• ```Physical signal Magic Formula coefficients``` — Specify the Magic Formula coefficients using port M as a four-element vector in the order [B, C, D,E].

#### Dependencies

To enable this parameter, set Parameterize by to ```Peak longitudinal force and corresponding slip```.

Maximum longitudinal force Fx0 that the tire exerts on the wheel when the vertical load equals its rated value Fz0.

#### Dependencies

To enable this parameter, set Parameterize by to ```Peak longitudinal force and corresponding slip```.

Contact slip, k0, expressed as a percentage (%), when the longitudinal force equals its maximum value Fx0 and the vertical load equals its rated value Fz0.

#### Dependencies

To enable this parameter, set Parameterize by to ```Peak longitudinal force and corresponding slip```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Constant Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Constant Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Constant Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Constant Magic Formula coefficients```.

Nominal normal force Fz0 on the tire.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients```.

### Geometry

Whether to treat the rolling radius as constant or dependent on load and velocity.

#### Dependencies

To enable this parameter, set Parameterize to ```Load-dependent Magic Formula coefficients```.

#### Dependencies

To enable this parameter, either set:

• Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to `Constant radius`, or

• Parameterize by to ```Peak longitudinal force and corresponding slip```, ```Constant Magic Formula coefficients```, or ```Physical signal Magic Formula coefficients```.

Non-rolling free tire radius value associated with the Magic Formula. `R0` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Effective rolling radius at low load stiffness. `BREFF` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Peak value of effective rolling radius. `DREFF` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Effective rolling radius at high load stiffness. `FREFF` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Ratio of the nominal tire radius to the non-rolling free tire radius. `Q_RE0` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Tire radius increase with respect to speed. `Q_V1` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Nominal hub longitudinal speed. `LONGVL` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

Vertical stiffness of the tire.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and Effective rolling radius model to ```Load and velocity dependent (Magic Formula)```.

### Rolling Resistance

Whether to include rolling resistance in the tire simulation.

Whether to include resistance using a constant coefficient or pressure and velocity dependent resistance:

• `Constant coefficient` — Neglect rolling resistance.

• ```Pressure and velocity dependent SAE J2452``` — Parameterize rolling resistance in accordance with SAE J2452.

• ```Pressure and velocity dependent (Magic Formula)``` — Parameterize rolling resistance in accordance with the Magic Formula.

#### Dependencies

To enable this parameter, select Model rolling resistance.

Coefficient that sets the proportionality between the normal force and the rolling resistance force. The parameter must be greater than zero.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Constant coefficient```.

Inflation pressure of the tire. The parameter must be greater than zero.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Pressure and velocity dependent SAE J2452``` or ```Pressure and velocity dependent (Magic Formula)```.

Exponent of the tire pressure in the model equation.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Pressure and velocity dependent SAE J2452```.

Exponent of the normal force model equation.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Pressure and velocity dependent SAE J2452```.

Velocity-independent force component in the model equation. The parameter must be greater than zero.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Pressure and velocity dependent SAE J2452```.

Velocity-dependent force component in the model equation. The parameter must be greater than zero.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Pressure and velocity dependent SAE J2452```.

Force component that depends on the square of the velocity term in the model equation. The parameter must be greater than zero.

#### Dependencies

To enable this parameter, select Model rolling resistance and set Rolling resistance to ```Pressure and velocity dependent SAE J2452```.

Nominal tire pressure. `NOMPRES` it the TIR file identifier.

#### Dependencies

To enable this parameter, set Resistance model to ```Pressure and velocity dependent (Magic Formula)```.

Nominal normal force, Fz0, on tire. `FNOMIN` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Resistance model to ```Pressure and velocity dependent (Magic Formula)```.

Nominal hub longitudinal speed. `LONGVL` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Resistance model to ```Pressure and velocity dependent (Magic Formula)```.

Magic Formula Q coefficients. `qsy1`,…, `qsy8` correspond to the TIR file identifiers.

#### Dependencies

To enable this parameter, set Resistance model to ```Pressure and velocity dependent (Magic Formula)```.

Velocity at which the full rolling resistance force is transmitted to the rolling hub. The parameter ensures that the force remains continuous during velocity direction changes, which increases the numerical stability of the simulation. The parameter must be greater than zero.

#### Dependencies

To enable this parameter, select Model rolling resistance.

### Scaling

Whether to include scaling coefficients in the Magic Formula parameterization.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` or set Resistance model to ```Pressure and velocity dependent (Magic Formula)```.

Scale factor of the Fx nominal vertical load, λFZO. `LFZO` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the Fx shape factor, λLCX. `LCX` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the Fx peak friction coefficient, λμX. `LMUX` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the Fx curvature factor, λEX. `LEX` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the Fx slip stiffness, λKX. `LKX` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the Fx horizontal shift, λhX. `LHX` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the Fx vertical shift, λVX. `LVX` is the TIR file identifier.

#### Dependencies

To enable this parameter, set Parameterize by to ```Load-dependent Magic Formula coefficients``` and select Enable scaling coefficients.

Scale factor of the rolling resistance, λMY. `LMY` is the TIR file identifier.

#### Dependencies

To enable this parameter, either set

• Parameterize by to ```Load and velocity dependent (Magic Formula)```, or

• Resistance model to ```Pressure and velocity dependent (Magic Formula)```

and select Enable scaling coefficients.

### Dynamics

Whether to the include dynamical compliance of the tire:

• ```No compliance - Suitable for HIL simulation``` — The block neglects dynamical compliance.

• `Specify stiffness and damping` — The block treats the tire as a spring-damper system under load.

Tire longitudinal stiffness CFx.

#### Dependencies

To enable this parameter, set Compliance to ```Specify stiffness and damping```.

Tire longitudinal damping bFx.

#### Dependencies

To enable this parameter, set Compliance to ```Specify stiffness and damping```.

Model for the rotational inertia of the tire.

• `No inertia` — The block neglects dynamical compliance.

• ```Specify inertia and initial velocity``` — The block treats the tire as a stiff, damped spring and deforms under load.

Rotational inertia Iw of the wheel-tire assembly.

#### Dependencies

To enable this parameter, set Inertia to ```Specify inertia and initial velocity```.

Initial angular velocity, Ω(0), of the tire.

#### Dependencies

To enable this parameter, set Inertia to ```Specify inertia and initial velocity```.

Lower boundary of slip denominator, VXLOW. `VXLOW` is the TIR file identifier.

Threshold velocity the block uses to transition between slip regimes, Vth. To learn more, see Roll and Slip.

Minimum valid wheel slip. A negative value represents wheel slip in the reverse rotational direction.

Maximum valid wheel slip.