# Fiala Wheel 2DOF

Fiala wheel 2DOF wheel with disc, drum, or mapped brake

• Library:
• Vehicle Dynamics Blockset / Wheels and Tires

## Description

The Fiala Wheel 2DOF block implements a simplified tire with lateral and longitudinal slip capability based on the E. Fiala model[1]. The block uses a translational friction model to calculate the forces and moments during combined longitudinal and lateral slip, requiring fewer parameters than the Combined Slip Wheel 2DOF block. If you do not have the tire coefficients needed by the Magic Formula, consider using this block for studies that do not involve extensive nonlinear combined lateral slip or lateral dynamics. If your study does require nonlinear combined slip or lateral dynamics, consider using the Combined Slip Wheel 2DOF block.

The block determines the wheel rotation rate, vertical motion, and forces and moments in all six degrees-of-freedom (DOFs) based on the driveline torque, brake pressure, road height, wheel camber angle, and inflation pressure. You can use this block for these types of analyses:

• Driveline and vehicle simulations that require low frequency tire-road and braking forces for vehicle acceleration, braking, and wheel rolling resistance calculations with minimal tire parameters.

• Wheel interaction with an idealized road surface.

• Ride and handling maneuvers for vehicles undergoing mild combined slip. For this analysis, you can connect the block to driveline and chassis components such as differentials, suspension, and vehicle body systems.

• Yaw stability. For this analyses, you can connect this block to more detailed braking system models.

• Tire stiffness and unsprung mass interactions with ground variations, load transfer, or chassis motion using the block vertical DOF.

The block integrates rotational wheel, vertical mass, and braking dynamics models. For the slip-dependent tire forces and moments, the block implements the Fiala tire model.

Use the Brake Type parameter to select the brake.

Brake Type SettingBrake Implementation

`None`

None

`Disc`

Brake that converts the brake cylinder pressure into a braking force

`Drum`

Simplex drum brake that converts the applied force and brake geometry into a net braking torque

`Mapped`

Lookup table that is a function of the wheel speed and applied brake pressure

To calculate the rolling resistance torque, specify one of these Rolling Resistance parameters.

SettingBlock Implementation

`None`

None

`Pressure and velocity`

Method in Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity.

`ISO 28580`

Method specified in ISO 28580:2018, Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results.

`Magic Formula`

Magic formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients.

`Mapped torque`

Lookup table that is a function of the normal force and spin axis longitudinal velocity.

To calculate vertical motion, specify one of these Vertical Motion parameters.

SettingBlock Implementation

`None`

Block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations.

```Mapped stiffness and damping```

Vertical motion depends on wheel stiffness and damping. Stiffness is a function of tire sidewall displacement and pressure. Damping is a function of tire sidewall velocity and pressure.

### Rotational Wheel Dynamics

The block calculates the inertial response of the wheel subject to:

• Axle losses

• Brake and drive torque

• Tire rolling resistance

• Ground contact through the tire-road interface

The input torque is the summation of the applied axle torque, braking torque, and moment arising from the combined tire torque.

`${T}_{i}={T}_{a}-{T}_{b}+{T}_{d}$`

For the moment arising from the combined tire torque, the block implements tractive wheel forces and rolling resistance with first-order dynamics. The rolling resistance has a time constant parameterized in terms of a relaxation length.

To calculate the rolling resistance torque, you can specify one of these Rolling Resistance parameters.

SettingBlock Implementation

`None`

Block sets rolling resistance, `My`, to zero.

`Pressure and velocity`

Block uses the method in SAE Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity. Specifically,

`${M}_{y}={R}_{e}\left\{a+b|{V}_{x}|+c{V}_{x}{}^{2}\right\}\left\{{F}_{z}{}^{\beta }{p}_{i}{}^{\alpha }\right\}\mathrm{tanh}\left(4{V}_{x}\right)$`

`ISO 28580`

Block uses the method specified in ISO 28580:2018, Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. The method accounts for normal load, parasitic loss, and thermal corrections from test conditions. Specifically,

`${M}_{y}={R}_{e}\left(\frac{{F}_{z}{C}_{r}}{1+{K}_{t}\left({T}_{amb}-{T}_{meas}\right)}-{F}_{pl}\right)\mathrm{tanh}\left(\omega \right)$`

`Magic Formula`

Block calculates the rolling resistance, `My`, using the Magic Formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients.

`Mapped torque`

For the rolling resistance, `My`, the block uses a lookup table that is a function of the normal force and spin axis longitudinal velocity.

If the brakes are enabled, the block determines the braking locked or unlocked condition based on an idealized dry clutch friction model. Based on the lock-up condition, the block implements these friction and dynamic models.

IfLock-Up ConditionFriction ModelDynamic Model

$\begin{array}{l}\omega \ne 0\\ \text{or}\\ {T}_{S}<|{T}_{i}+{T}_{f}-\omega b|\end{array}$

Unlocked

$\begin{array}{l}{T}_{f}={T}_{k}\\ \text{where,}\\ {T}_{k}={F}_{c}{R}_{eff}{\mu }_{k}\mathrm{tanh}\left[4\left(-{\omega }_{d}\right)\right]\\ {T}_{s}={F}_{c}{R}_{eff}{\mu }_{s}\\ {R}_{eff}=\frac{2\left({R}_{o}{}^{3}-{R}_{i}{}^{3}\right)}{3\left({R}_{o}{}^{2}-{R}_{i}{}^{2}\right)}\end{array}$

$\stackrel{˙}{\omega }J=-\omega b+{T}_{i}+{T}_{o}$

$\begin{array}{l}\omega =0\\ \text{and}\\ {T}_{S}\ge |{T}_{i}+{T}_{f}-\omega b|\end{array}$

Locked

${T}_{f}={T}_{s}$

$\omega =0$

The equations use these variables.

 ω Wheel angular velocity a Velocity-independent force component b Linear velocity force component c Quadratic velocity force component Le Tire relaxation length J Moment of inertia My Rolling resistance torque Ta Applied axle torque Tb Braking torque Td Combined tire torque Tf Frictional torque Ti Net input torque Tk Kinetic frictional torque To Net output torque Ts Static frictional torque Fc Applied clutch force Fx Longitudinal force developed by the tire road interface due to slip Reff Effective clutch radius Ro Annular disk outer radius Ri Annular disk inner radius Re Effective tire radius while under load and for a given pressure Vx Longitudinal axle velocity Fz Vehicle normal force Cr Rolling resistance constant Tamb Ambient temperature Tmeas Measured temperature for rolling resistance constant Fpl Parasitic force loss Kt Thermal correction factor ɑ Tire pressure exponent β Normal force exponent pi Tire pressure μs Coefficient of static friction μk Coefficient of kinetic friction

### Longitudinal Force

The block implements the longitudinal force as a function of wheel slip relative to the road surface using these equations.

CalculationEquation

Critical slip

$\kappa {\text{'}}_{Critical}=|\frac{\mu {F}_{z}}{2{C}_{\kappa }}|$

Longitudinal force

Friction coefficient

Slip coefficient

${\kappa }_{k\alpha }=\sqrt{\kappa {\text{'}}^{2}+{\text{tan}}^{2}\left(\alpha \text{'}\right)}$

The equations use these variables.

 κ' Slip state Fx Longitudinal force acting on axle along tire-fixed x-axis, Cκ Longitudinal stiffness Fz Vertical contact patch normal force along tire-fixed z-axis, μ Friction coefficient μs Coefficient of static friction μk Coefficient of kinetic friction κka Comprehensive slip coefficient α' Slip angle state λμ Friction scaling

### Lateral Force

The block implements the lateral force as a function of wheel slip angle state using these equations.

CalculationEquation

Critical slip angle

$\alpha {\text{'}}_{Critical}=\text{atan}\left(\frac{3\mu |{F}_{z}|}{{C}_{a}}\right)$

Lateral force

The equations use these variables.

 α' Slip angle state Fy Lateral force acting on axle along tire-fixed y-axis, Fz Vertical contact patch normal force along tire-fixed z-axis Cɣ Camber stiffness Cα Lateral stiffness per slip angle μ Friction coefficient

### Vertical Dynamics

For the vertical dynamics, the block implements these equations.

CalculationEquation

Vertical response

$\stackrel{¨}{z}m={F}_{ztire}+mg-Fz$

Tire normal force

${F}_{ztire}={\rho }_{z}k-b\stackrel{˙}{z}$

Vertical sidewall deflection

${\rho }_{z}={z}_{gnd}-z,z\ge 0$

The equations use these variables.

 z Tire deflection along tire-fixed z-axis zgnd Ground displacement along tire-fixed z-axis Fztire Tire normal force along tire-fixed z-axis Fz Vertical force acting on axle along tire-fixed z-axis ρz Vertical sidewall deflection along tire-fixed z-axis k Vertical sidewall stiffness b Vertical sidewall damping

### Overturning, Aligning, and Scaling

This table summarizes the overturning, aligning, and scaling implementation.

CalculationImplementation

Overturning moment

The Fiala model does not define an overturning moment. The block implements this equation, requiring minimal parameters.

${M}_{x}={F}_{y}{R}_{e}\text{cos}\left(\gamma \right)$

Aligning moment

The block implements the aligning moment as a combination of yaw rate damping and slip angle state.

Friction scaling

To vary the coefficient of friction, use the `ScaleFctr` input port.

The equations use these variables.

 Mx Overturning moment acting on axle about tire-fixed x-axis Mz Aligning moment acting on axle about tire-fixed z-axis Re Effective contact patch to wheel carrier radial distance ɣ Camber angle k Vertical sidewall stiffness b Vertical sidewall damping $\stackrel{˙}{\psi }$ Tire angular velocity about the tire-fixed z-axis (yaw rate) w Tire width α' Slip angle state bMz Linear yaw rate resistance Fy Lateral force acting on axle along tire-fixed y-axis Cɣ Camber stiffness Cα Lateral stiffness per slip angle μ Friction coefficient Fz Vertical contact patch normal force along tire-fixed z-axis

### Tire and Wheel Coordinate Systems

To resolve the forces and moments, the block uses the Z-Up orientation of the tire and wheel coordinate systems.

• Tire coordinate system axes (XT, YT, ZT) are fixed in a reference frame attached to the tire. The origin is at the tire contact with the ground.

• Wheel coordinate system axes (XW, YW, ZW) are fixed in a reference frame attached to the wheel. The origin is at the wheel center.

Z-Up Orientation[1]

### Brakes

Disc

If you specify the Brake Type parameter `Disc`, the block implements a disc brake. This figure shows the side and front views of a disc brake.

A disc brake converts brake cylinder pressure from the brake cylinder into force. The disc brake applies the force at the brake pad mean radius.

The block uses these equations to calculate brake torque for the disc brake.

`$Rm=\frac{Ro+Ri}{2}$`

The equations use these variables.

 T Brake torque P Applied brake pressure N Wheel speed Npads Number of brake pads in disc brake assembly μstatic Disc pad-rotor coefficient of static friction μ Disc pad-rotor coefficient of kinetic friction Ba Brake actuator bore diameter Rm Mean radius of brake pad force application on brake rotor Ro Outer radius of brake pad Ri Inner radius of brake pad

Drum

If you specify the Brake Type parameter `Drum`, the block implements a static (steady-state) simplex drum brake. A simplex drum brake consists of a single two-sided hydraulic actuator and two brake shoes. The brake shoes do not share a common hinge pin.

The simplex drum brake model uses the applied force and brake geometry to calculate a net torque for each brake shoe. The drum model assumes that the actuators and shoe geometry are symmetrical for both sides, allowing a single set of geometry and friction parameters to be used for both shoes.

The block implements equations that are derived from these equations in Fundamentals of Machine Elements.

`$\begin{array}{l}{T}_{rshoe}=\left(\frac{\pi \mu cr\left(\mathrm{cos}{\theta }_{2}-\mathrm{cos}{\theta }_{1}\right){B}_{a}{}^{2}}{2\mu \left(2r\left(\mathrm{cos}{\theta }_{2}-\mathrm{cos}{\theta }_{1}\right)+a\left({\mathrm{cos}}^{2}{\theta }_{2}-{\mathrm{cos}}^{2}{\theta }_{1}\right)\right)+ar\left(2{\theta }_{1}-2{\theta }_{2}+\mathrm{sin}2{\theta }_{2}-\mathrm{sin}2{\theta }_{1}\right)}\right)P\\ \\ {T}_{lshoe}=\left(\frac{\pi \mu cr\left(\mathrm{cos}{\theta }_{2}-\mathrm{cos}{\theta }_{1}\right){B}_{a}{}^{2}}{-2\mu \left(2r\left(\mathrm{cos}{\theta }_{2}-\mathrm{cos}{\theta }_{1}\right)+a\left({\mathrm{cos}}^{2}{\theta }_{2}-{\mathrm{cos}}^{2}{\theta }_{1}\right)\right)+ar\left(2{\theta }_{1}-2{\theta }_{2}+\mathrm{sin}2{\theta }_{2}-\mathrm{sin}2{\theta }_{1}\right)}\right)P\end{array}$`

The equations use these variables.

 T Brake torque P Applied brake pressure N Wheel speed μstatic Disc pad-rotor coefficient of static friction μ Disc pad-rotor coefficient of kinetic friction Trshoe Right shoe brake torque Tlshoe Left shoe brake torque a Distance from drum center to shoe hinge pin center c Distance from shoe hinge pin center to brake actuator connection on brake shoe r Drum internal radius Ba Brake actuator bore diameter Θ1 Angle from shoe hinge pin center to start of brake pad material on shoe Θ2 Angle from shoe hinge pin center to end of brake pad material on shoe

Mapped

If you specify the Brake Type parameter `Mapped`, the block uses a lookup table to determine the brake torque.

The equations use these variables.

 T Brake torque ${f}_{brake}^{}\left(P,N\right)$ Brake torque lookup table P Applied brake pressure N Wheel speed μstatic Friction coefficient of drum pad-face interface under static conditions μ Friction coefficient of disc pad-rotor interface

The lookup table for the brake torque, ${f}_{brake}^{}\left(P,N\right)$, is a function of applied brake pressure and wheel speed, where:

• T is brake torque, in N·m.

• P is applied brake pressure, in bar.

• N is wheel speed, in rpm.

## Ports

### Input

expand all

Brake pressure, in Pa.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

#### Dependencies

To enable this port, for the Brake Type parameter, specify one of these types:

• `Disc`

• `Drum`

• `Mapped`

Axle torque, Ta, about wheel spin axis, in N·m.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Axle longitudinal velocity, Vx, along tire-fixed x-axis, in m/s.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Axle lateral velocity, Vy, along tire-fixed y-axis, in m/s.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Tire angular velocity, r, about the tire-fixed z-axis (yaw rate), in rad/s.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Tire inflation pressure, pi, in Pa.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Ground displacement along tire-fixed z-axis, in m. Positive input produces wheel lift.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Axle force applied to tire, Fext, along vehicle-fixed z-axis (positive input compresses the tire), in N·m.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Scale factor to account for variations in the coefficient of friction.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

### Output

expand all

Block data, returned as a bus signal containing these block values.

SignalDescriptionUnits

`AxlTrq`

N·m

`Omega`

Wheel angular velocity about wheel-fixed y-axis

`Fx`

Longitudinal vehicle force along tire-fixed x-axis

N

`Fy`

Lateral vehicle force along tire-fixed y-axis

N

`Fz`

Vertical vehicle force along tire-fixed z-axis

N

`Mx`

N·m

`My`

Rolling resistance torque about tire-fixed y-axis

N·m

`Mz`

N·m

`Vx`

Vehicle longitudinal velocity along tire-fixed x-axis

m/s

`Vy`

Vehicle lateral velocity along tire-fixed y-axis

m/s

`Re`

m

`Kappa`

Longitudinal slip ratio

NA

`Alpha`

Side slip angle

`a`

Contact patch half length

m

`b`

Contact patch half width

m

`Gamma`

Camber angle

`psidot`

Tire angular velocity about the tire-fixed z-axis (yaw rate)

`BrkTrq`

Brake torque about the vehicle-fixed y-axis

N·m

`BrkPrs`

Brake pressure

Pa

`z`

Axle vertical displacement along tire-fixed z-axis

m

`zdot`

Axle vertical velocity along tire-fixed z-axis

m/s

`Gnd`

Ground displacement along tire-fixed z-axis (positive input produces wheel lift)m

`GndFz`

Vertical sidewall force on ground along tire-fixed z-axis

N

`Prs`

Tire inflation pressure

Pa

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Longitudinal force acting on axle, Fx, along tire-fixed x-axis, in N. Positive force acts to move the vehicle forward.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Lateral force acting on axle, Fy, along tire-fixed y-axis, in N.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Vertical force acting on axle, Fz, along tire-fixed z-axis, in N.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Longitudinal moment acting on axle, Mx, about tire-fixed x-axis, in N·m.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Lateral moment acting on axle, My, about tire-fixed y-axis, in N·m.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

Vertical moment acting on axle, Mz, about tire-fixed z-axis, in N·m.

Vector is the number of wheels, N, by `1`. If you provide a scalar value, the block assumes that number of wheels is one.

## Parameters

expand all

### Block Options

Use the Brake Type parameter to select the brake.

Brake Type SettingBrake Implementation

`None`

None

`Disc`

Brake that converts the brake cylinder pressure into a braking force

`Drum`

Simplex drum brake that converts the applied force and brake geometry into a net braking torque

`Mapped`

Lookup table that is a function of the wheel speed and applied brake pressure

To calculate the rolling resistance torque, specify one of these Rolling Resistance parameters.

SettingBlock Implementation

`None`

None

`Pressure and velocity`

Method in Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity.

`ISO 28580`

Method specified in ISO 28580:2018, Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results.

`Magic Formula`

Magic formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients.

`Mapped torque`

Lookup table that is a function of the normal force and spin axis longitudinal velocity.

#### Dependencies

SelectingParameters

```Pressure and velocity```

Velocity independent force coefficient, aMy

Linear velocity force component, bMy

Tire pressure exponent, alphaMy

Normal force exponent, betaMy

`ISO 28580`

Parasitic losses force, Fpl

Rolling resistance constant, Cr

Thermal correction factor, Kt

Measured temperature, Tmeas

Parasitic losses force, Fpl

Ambient temperature, Tamb

`Magic Formula`

Rolling resistance torque coefficient, QSY

Longitudinal force rolling resistance coefficient, QSY2

Linear rotational speed rolling resistance coefficient, QSY3

Quartic rotational speed rolling resistance coefficient, QSY4

Camber squared rolling resistance torque, QSY5

Load based camber squared rolling resistance torque, QSY6

Normal load rolling resistance coefficient, QSY7

Pressure load rolling resistance coefficient, QSY8

Rolling resistance scaling factor, lam_My

`Mapped torque`

Spin axis velocity breakpoints, VxMy

Normal force breakpoints, FzMy

Rolling resistance torque map, MyMap

To calculate vertical motion, specify one of these Vertical Motion parameters.

SettingBlock Implementation

`None`

Block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations.

```Mapped stiffness and damping```

Vertical motion depends on wheel stiffness and damping. Stiffness is a function of tire sidewall displacement and pressure. Damping is a function of tire sidewall velocity and pressure.

SelectingEnables These Parameters

```Mapped stiffness and damping```

Wheel mass, MASS

Initial tire deflection, zo

Initial velocity, zdoto

Initial wheel vertical velocity (wheel fixed frame), zdoto

Vertical deflection breakpoints, zFz

Pressure breakpoints, pFz

Force due to deflection, Fzz

Vertical velocity breakpoints, zdotFz

Force due to velocity, Fzzdot

### Longitudinal and Lateral

Longitudinal stiffness, Cκ, in N.

Lateral stiffness per slip angle, Cα, in N/rad.

Kinematic friction, μk, dimensionless.

Static friction, μs, dimensionless.

Longitudinal relaxation length, Lrelx, in m.

Lateral relaxation length, Lrely, in m/rad.

### Rolling

Rotational inertia (rolling axis), IYY, in kg·m^2.

Pressure and Velocity

Velocity-independent force coefficient, a, in s/m.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Pressure and velocity`.

Linear velocity force component, b, in s/m.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Pressure and velocity`.

Quadratic velocity force component, c, in s^2/m^2.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Pressure and velocity`.

Tire pressure exponent, ɑ, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Pressure and velocity`.

Normal force exponent, β, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Pressure and velocity`.

ISO 28580

Parasitic force loss, Fpl, in N.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Rolling resistance constant, Cr, in N/kN. ISO 28580 specifies the rolling resistance unit as one newton of tractive resistance for every kilonewtons of normal load.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Thermal correction factor, Kt, in 1/degC.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Measured temperature, Tmeas, in K.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Measured temperature, Tamb, in K.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Select to create input port `Tamb`.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Magic Formula

Rolling resistance torque coefficient, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Longitudinal force rolling resistance coefficient, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Linear rotational speed rolling resistance coefficient, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Quartic rotational speed rolling resistance coefficient, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Camber squared rolling resistance torque, in 1/rad^2.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Normal load rolling resistance coefficient, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Pressure load rolling resistance coefficient, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Rolling resistance scaling factor, dimensionless.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter ```Magic Formula```.

Mapped

Spin axis velocity breakpoints, in m/s.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Mapped torque`.

Normal force breakpoints, in N.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Mapped torque`.

Rolling resistance torque versus axle speed and normal force, in N·m.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `Mapped torque`.

### Aligning

Wheel width, WIDTH, in m.

Linear yaw rate resistance, bMz, in N·m·s/rad.

### Brake

Static friction coefficient, dimensionless.

#### Dependencies

To enable this parameter, for the Brake Type parameter, specify one of these types:

• `Disc`

• `Drum`

• `Mapped`

Kinematic friction coefficient, dimensionless.

#### Dependencies

To enable this parameter, for the Brake Type parameter, specify one of these types:

• `Disc`

• `Drum`

• `Mapped`

Disc

Disc brake actuator bore, in m.

#### Dependencies

To enable the disc brake parameters, select `Disc` for the Brake Type parameter.

#### Dependencies

To enable the disc brake parameters, select `Disc` for the Brake Type parameter.

#### Dependencies

To enable the disc brake parameters, select `Disc` for the Brake Type parameter.

Drum

Drum brake actuator bore, in m.

#### Dependencies

To enable the drum brake parameters, select `Drum` for the Brake Type parameter.

Shoe pin to drum center distance, in m.

#### Dependencies

To enable the drum brake parameters, select `Drum` for the Brake Type parameter.

Shoe pin center to force application point distance, in m.

#### Dependencies

To enable the drum brake parameters, select `Drum` for the Brake Type parameter.

#### Dependencies

To enable the drum brake parameters, select `Drum` for the Brake Type parameter.

Shoe pin to pad start angle, in deg.

#### Dependencies

To enable the drum brake parameters, select `Drum` for the Brake Type parameter.

Shoe pin to pad end angle, in deg.

#### Dependencies

To enable the drum brake parameters, select `Drum` for the Brake Type parameter.

Mapped

Brake actuator pressure breakpoints, in bar.

#### Dependencies

To enable the mapped brake parameters, select `Mapped` for the Brake Type parameter.

Wheel speed breakpoints, in rpm.

#### Dependencies

To enable the mapped brake parameters, select `Mapped` for the Brake Type parameter.

The lookup table for the brake torque, ${f}_{brake}^{}\left(P,N\right)$, is a function of applied brake pressure and wheel speed, where:

• T is brake torque, in N·m.

• P is applied brake pressure, in bar.

• N is wheel speed, in rpm.

#### Dependencies

To enable the mapped brake parameters, select `Mapped` for the Brake Type parameter.

### Vertical

Wheel mass, in kg. Used in the vertical motion calculations.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Initial axle displacement along wheel-fixed z-axis, in m.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Initial axle velocity along wheel-fixed z-axis, in m.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Gravitational acceleration, in m/s^2.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Mapped Stiffness and Damping

Vector of sidewall deflection breakpoints corresponding to the force table, in m.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Vector of pressure data points corresponding to the force table, in Pa.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Force due to sidewall deflection and pressure along wheel-fixed z-axis, in N.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Vector of sidewall velocity breakpoints corresponding to the force due to velocity table, in m.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

Force due to sidewall velocity and pressure along wheel-fixed z-axis, in N.

#### Dependencies

To enable this parameter, set Vertical Motion to ```Mapped stiffness and damping```.

### Simulation

Maximum normal force, in N. Used with all vertical force calculations.

Minimum normal force, in N. Used with all vertical force calculations.

Maximum pressure, PRESMAX, in Pa.

Minimum pressure, PRESMIN, in Pa.

Max allowable slip ratio (absolute), KPUMAX, dimensionless.

Minimum allowable slip ratio (absolute), KPUMIN, dimensionless.

Max allowable slip angle (absolute), ALPMAX, in rad.

Minimum allowable slip angle (absolute), ALPMIN, in rad.

Maximum allowable camber angle CAMMAX, in rad.

Minimum allowable camber angle, CAMMIN, in rad.

Minimum ambient temperature, TMIN, in K.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

Maximum ambient temperature, TMAX, in K.

#### Dependencies

To create this parameter, select the Rolling Resistance parameter `ISO 28580`.

## References

[1] Fiala, E. "Seitenkrafte am Rollenden Luftreifen." VDI Zeitschrift, V.D.I.. Vol 96, 1954.

[2] Highway Tire Committee. Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. Standard J2452_199906. Warrendale, PA: SAE International, June 1999.

[3] ISO 28580:2018. Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. ISO (International Organization for Standardization), 2018.

[4] Pacejka, H. B. Tire and Vehicle Dynamics. 3rd ed. Oxford, UK: SAE and Butterworth-Heinemann, 2012.