# Longitudinal Wheel

Longitudinal wheel with disc, drum, or mapped brake

Libraries:
Powertrain Blockset / Drivetrain / Wheels
Vehicle Dynamics Blockset / Wheels and Tires

## Description

The Longitudinal Wheel block implements the longitudinal behavior of an ideal wheel. You can specify the longitudinal force and rolling resistance calculation method, and brake type. Use the block in driveline and longitudinal vehicle simulations where low frequency tire-road and braking forces are required to determine vehicle acceleration, braking, and wheel-rolling resistance. For example, you can use the block to determine the torque and power requirements for a specified drive cycle or braking event. The block is not suitable for applications that require combined lateral slip.

There are four types of Longitudinal Wheel blocks. Each block implements a different brake type.

Block NameBrake Type SettingBrake Implementation
Longitudinal Wheel - No Brake

`None`

None

Longitudinal Wheel - Disc Brake

`Disc`

Brake that converts the brake cylinder pressure into a braking force.

Longitudinal Wheel - Drum Brake

`Drum`

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

Longitudinal Wheel - Mapped Brake

`Mapped`

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

The block models longitudinal force as a function of wheel slip relative to the road surface. To calculate the longitudinal force, specify one of these Longitudinal Force parameters.

SettingBlock Implementation

`Magic Formula constant value`

Magic Formula with constant coefficient for stiffness, shape, peak, and curvature.

`Magic Formula pure longitudinal slip`

Magic Formula with load-dependent coefficients that implement equations 4.E9 through 4.E18 in Tire and Vehicle Dynamics.

`Mapped force`

Lookup table that is a function of the normal force and wheel slip ratio.

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.

`${T}_{d}\left(s\right)=\frac{1}{\frac{{L}_{e}}{|\omega |{R}_{e}}s+1}+\left({F}_{x}{R}_{e}+{M}_{y}\right)$`

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

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

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

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

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

### Brakes

Disc

If you specify the Brake Type parameter as `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.

VariableValue
T

Brake torque

P

Applied brake pressure

N

Wheel speed

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

Ri

Drum

If you specify the Brake Type parameter as `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.

VariableValue
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

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 as `Mapped`, the block uses a lookup table to determine the brake torque.

The equations use these variables.

VariableValue
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.

### Longitudinal Force

To model the Longitudinal Wheel block longitudinal forces, you can use the Magic Formula. The model provides a steady-state tire characteristic function Fx = f(κ, Fz), the longitudinal force Fx on the tire, based on:

• Vertical load Fz

• Wheel slip κ

The Magic Formula model uses these variables.

 Ω Wheel angular velocity rw Wheel radius Vx Wheel hub longitudinal velocity rwΩ Tire tread longitudinal velocity Vsx = rwΩ – Vx Wheel slip velocity κ = Vsx/|Vx| Wheel slip Fz, Fz0 Vertical load and nominal vertical load on tire Fx = f(κ, Fz) Longitudinal force exerted on the tire at the contact point. Also a characteristic function f of the tire.

Magic Formula Constant Value

If you set Longitudinal Force to ```Magic Formula constant value```, the block implements the Magic Formula as a specific form of the tire characteristic function, characterized by four dimensionless coefficients (B, C, D, E), or stiffness, shape, peak, and curvature:

The slope of f at κ = 0 is BCD·Fz.

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

SurfaceBCDE
Dry tarmac101.910.97
Wet tarmac122.30.821
Snow520.31
Ice420.11

Magic Formula Pure Longitudinal Slip

If you set Longitudinal Force to ```Magic Formula pure longitudinal slip```, the block implements a more general Magic Formula using dimensionless coefficients that are functions of the tire load. The block implements the longitudinal force equations in Chapter 4 of Tire and Vehicle Dynamics, including 4.E9 through 4.E18:

SHx and SVx 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. μx is the longitudinal load-dependent friction coefficient. εx is a small number inserted to prevent division by zero as Fz approaches zero.

### Vertical Dynamics

If you select no vertical degrees-of-freedom by setting to `None`, the block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations.

If you set to ```Mapped stiffness and damping```, the 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.

`$Fztire\left(z,\stackrel{˙}{z},{P}_{tire}\right)={F}_{zk}\left(z,{P}_{tire}\right)+{F}_{zb}\left(\stackrel{˙}{z},{P}_{tire}\right)$`

The block determines the vertical response using this differential equation.

`$\stackrel{¨}{z}m=Fztire-{F}_{z}-mg$`

When you disable the vertical degree-of-freedom, the input normal force from the vehicle passes directly to the longitudinal and rolling force calculations.

`$\begin{array}{l}\stackrel{¨}{z}=\stackrel{˙}{z}=m=0\\ Fztire=mg\end{array}$`

The block uses the wheel-fixed frame to resolve the vertical forces.

The equations use these variables.

 Fztire Tire normal force along the wheel-fixed z-axis m Axle mass Fzk Tire normal force due to wheel stiffness along the wheel-fixed z-axis Fzb Tire normal force due to wheel damping along the wheel-fixed z-axis Fz Suspension or vehicle normal force along the wheel-fixed z-axis PTire Tire pressure $z,\stackrel{˙}{z},\stackrel{¨}{z}$ Tire displacement, velocity, and acceleration, respectively, along the wheel-fixed z-axis

### Power Accounting

For the power accounting, the block implements these equations.

Bus Signal DescriptionEquations

`PwrInfo`

`PwrTrnsfrd` — Power transferred between blocks

• Positive signals indicate flow into block

• Negative signals indicate flow out of block

`PwrRoad`

Tractive power applied from the axle

${P}_{road}={F}_{x}{V}_{x}$

`PwrAxlTrq`

External torque applied by the axle to the wheel

${P}_{T}=T\omega$

`PwrFz`

Vertical force applied to the wheel by the vehicle or suspension

${P}_{Fz}={F}_{z}\stackrel{˙}{z}$

`PwrNotTrnsfrd` — Power crossing the block boundary, but not transferred

• Positive signals indicate an input

• Negative signals indicate a loss

`PwrSlip`

Tractive power loss

${P}_{\kappa }={F}_{x}{V}_{x}+\left(-{F}_{cp}{R}_{e}+{M}_{y}\right)\omega$

`PwrMyRoll`

Rolling resistance power

${P}_{My}={M}_{y}\omega$

`PwrMyBrk`

Braking power${P}_{brk}={M}_{brk}\omega$

`PwrMyb`

Rolling viscous damping loss

${P}_{b}=-b{\omega }^{2}$

`PwrFzDamp`

Vertical damping power

`PwrStored` — Stored energy rate of change

• Positive signals indicate an increase

• Negative signals indicate a decrease

`PwrStoredzdot`

Rate of change of vertical kinetic energy

${P}_{\stackrel{˙}{z}}=m\stackrel{¨}{z}\stackrel{˙}{z}$

`PwrStoredq`

Rate of change of rotational kinetic energy

${P}_{\omega }={I}_{yy}\stackrel{˙}{\omega }\omega$

`PwrStoredFsFzSprng`

Rate of change of stored sidewall potential energy

${P}_{Fzk}={F}_{zk}{\stackrel{˙}{z}}_{x}$

`PwrStoredGrvty`

Rate of change of gravitational potential energy

${P}_{g}=-mg\stackrel{˙}{Z}$

The equations use these variables.

 ω Wheel angular velocity b Linear velocity force component Fx Longitudinal force developed by the tire road interface due to slip Fcp Tire slip force at contact patch Fz Vehicle normal force Fzb Tire normal force due to wheel damping Fzk Tire normal force due to wheel stiffness Iyy Wheel rotational inertia Mbrk Braking moment My Rolling resistance torque Re Effective tire radius while under load and for a given pressure T Axle torque applied on wheel Vx Longitudinal axle velocity $z,\stackrel{˙}{z},\stackrel{¨}{z}$ Tire displacement, velocity, and acceleration, respectively ω Wheel angular velocity $\stackrel{˙}{Z}$ Vehicle vertical velocity along the vehicle-fixed z-axis

## Ports

### Input

expand all

Brake pressure, in Pa.

#### 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.

Axle longitudinal velocity along vehicle(body)-fixed x-axis, in m/s.

Absolute value of suspension or vehicle normal force along body-fixed z-axis, in N.

Ground displacement, `Grndz`, along negative wheel-fixed z-axis, in m.

#### Dependencies

To create `Gnd`:

• Set Vertical Motion to ```Mapped stiffness and damping```.

• On the Vertical pane, select Input ground displacement.

Longitudinal friction scaling factor, dimensionless.

#### Dependencies

To enable this port, select Input friction scale factor.

Tire pressure, in Pa.

#### Dependencies

To enable this port:

• Set one of these parameters:

• Longitudinal Force to ```Magic Formula pure longitudinal slip```.

• Rolling Resistance to `Pressure and velocity` or ```Magic Formula```.

• Vertical Motion to ```Mapped stiffness and damping```.

• On the Wheel Dynamics pane, select Input tire pressure.

Ambient temperature, Tamb, in K.

The ambient temperature, Tamb, is the temperature near tire in application environment, in K. For example, the measured ambient temperature is the ambient temperature near the tire when the vehicle is on the road.

Select to create input port `Tamb` to input the measured ambient temperature.

#### Dependencies

To enable this port:

1. Set Rolling Resistance to `ISO 28580`.

2. On the Rolling Resistance pane, select to Input ambient temperature.

### Output

expand all

Bus signal containing these block calculations.

SignalDescriptionUnits

`AxlTrq`

Axle torque about body-fixed y-axis

N·m

`Omega`

Wheel angular velocity about body-fixed y-axis

`Omegadot`

Wheel angular acceleration about body-fixed y-axis

`Fx`

Longitudinal vehicle force along body-fixed x-axis

N

`Fz`

Vertical vehicle force along body-fixed z-axis

N

`Fzb`

Tire normal force due to wheel damping along the wheel-fixed z-axis

N

`Fzk`

Tire normal force due to wheel stiffness along the wheel-fixed z-axis

N

`My`

Rolling resistance torque about body-fixed y-axis

N·m

`Myb`

Rolling resistance torque due to damping about body-fixed y-axis

N·m

`Kappa`

Slip ratio

NA

`Vx`

Vehicle longitudinal velocity along body-fixed x-axis

m/s

`Re`

Wheel effective radius along wheel-fixed z-axis

m

`BrkTrq`

Brake torque about body-fixed y-axis

N·m

`BrkPrs`

Brake pressure

Pa

`z`

Wheel vertical deflection along wheel-fixed z-axis

m

`zdot`

Wheel vertical velocity along wheel-fixed z-axis

m/s

`zddot`

Wheel vertical acceleration along wheel-fixed z-axis

m/s^2

`Gndz`

Ground displacement along negative of wheel-fixed z-axis (positive input produces wheel lift)

m

`GndFz`

Vertical wheel force on ground along negative of wheel-fixed z-axis

N

`TirePrs`

Tire pressure

Pa

`Fpatch`

Tractive force (Fx considering relaxation effects). Use for vehicle transient maneuvers.

N

`PwrInfo`

`PwrTrnsfrd`

`PwrRoad`

External torque applied by the axle to the wheel

W

`PwrAxlTrq`

Vertical force applied to the wheel by the vehicle or suspension

W

`PwrFz`

Tractive power loss

W

`PwrNotTrnsfrd`

`PwrSlip`

Rolling resistance power

W

`PwrMyRoll`

Braking powerW

`PwrMyBrk`

Rolling viscous damping loss

W

`PwrMyb`

Vertical damping power

W

`PwrFzDamp`

Rate of change of vertical kinetic energy

W

`PwrStored`

`PwrStoredzdot`

Rate of change of rotational kinetic energy

W

`PwrStoredq`

Rate of change of stored sidewall potential energy

W

`PwrStoredFsFzSprng`

Rate of change of gravitational potential energy

W

`PwrStoredGrvty`

Tractive power applied from the axle

W

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

Wheel angular velocity, about body-fixed y-axis, in rad/s.

Wheel vertical deflection along wheel-fixed z-axis, in m.

#### Dependencies

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

Wheel vertical velocity along wheel-fixed z-axis, in m/s.

#### Dependencies

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

## Parameters

expand all

### Block Options

The block models longitudinal force as a function of wheel slip relative to the road surface. To calculate the longitudinal force, specify one of these Longitudinal Force parameters.

SettingBlock Implementation

`Magic Formula constant value`

Magic Formula with constant coefficient for stiffness, shape, peak, and curvature.

`Magic Formula pure longitudinal slip`

Magic Formula with load-dependent coefficients that implement equations 4.E9 through 4.E18 in Tire and Vehicle Dynamics.

`Mapped force`

Lookup table that is a function of the normal force and wheel slip ratio.

#### Dependencies

SelectingEnables These Parameters

```Magic Formula constant value```

Pure longitudinal peak factor, Dx

Pure longitudinal shape factor, Cx

Pure longitudinal stiffness factor, Bx

Pure longitudinal curvature factor, Ex

```Magic Formula pure longitudinal slip```

Cfx shape factor, PCX1

Longitudinal friction at nominal normal load, PDX1

Frictional variation with load, PDX2

Frictional variation with camber, PDX3

Longitudinal curvature at nominal normal load, PEX1

Variation of curvature factor with load, PEX2

Variation of curvature factor with square of load, PEX3

Longitudinal curvature factor with slip, PEX4

Longitudinal slip stiffness at nominal normal load, PKX1

Variation of slip stiffness with load, PKX2

Slip stiffness exponent factor, PKX3

Horizontal shift in slip ratio at nominal normal load, PHX1

Variation of horizontal slip ratio with load, PHX2

Vertical shift in load at nominal normal load, PVX1

Variation of vertical shift with load, PVX2

Linear variation of longitudinal slip stiffness with tire pressure, PPX1

Quadratic variation of longitudinal slip stiffness with tire pressure, PPX2

Linear variation of peak longitudinal friction with tire pressure, PPX3

Quadratic variation of peak longitudinal friction with tire pressure, PPX4

Linear variation of longitudinal slip stiffness with tire pressure, PPX1

Slip speed decay function scaling factor, lam_muV

Brake slip stiffness scaling factor, lam_Kxkappa

Longitudinal shape scaling factor, lam_Cx

Longitudinal curvature scaling factor, lam_Ex

Longitudinal horizontal shift scaling factor, lam_Hx

Longitudinal vertical shift scaling factor, lam_Vx

```Mapped force```

Slip ratio breakpoints, kappaFx

Normal force breakpoints, FzFx

Longitudinal force map, FxMap

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

Each Rolling Resistance setting enables additional parameters.

SettingParameters Enabled

```Pressure and velocity```

• Velocity independent force coefficient, aMy

• Linear velocity force component, bMy

• Quadratic velocity force component, cMy

• 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

There are four types of Longitudinal Wheel blocks. Each block implements a different brake type.

Block NameBrake Type SettingBrake Implementation
Longitudinal Wheel - No Brake

`None`

None

Longitudinal Wheel - Disc Brake

`Disc`

Brake that converts the brake cylinder pressure into a braking force.

Longitudinal Wheel - Drum Brake

`Drum`

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

Longitudinal Wheel - Mapped Brake

`Mapped`

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

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 ParametersCreates These Output Ports

```Mapped stiffness and damping```

Wheel and unsprung mass, m

Initial deflection, zo

Initial velocity, zdoto

Gravitational acceleration, g

Vertical deflection breakpoints, zFz

Pressure breakpoints, pFz

Force due to deflection, Fzz

Vertical velocity breakpoints, zdotFz

Force due to velocity, Fzzdot

Ground displacement, Gndz

Input ground displacement

`z`

`zdot`

Longitudinal friction scaling factor, dimensionless.

#### Dependencies

To enable this parameter, clear Input friction scale factor.

Create input port for longitudinal friction scaling factor.

#### Dependencies

Selecting this parameter:

• Creates input port `lam_mux`.

• Disables parameter Longitudinal scaling factor, lam_x.

### Wheel Dynamics

Axle viscous damping coefficient, br, in N·m·s/rad.

Wheel inertia, in kg·m^2.

Initial angular velocity of wheel, along body-fixed y-axis, in rad/s.

Wheel relaxation length, in m.

Loaded wheel radius, `Re`, in m.

#### Dependencies

To create this parameter, set Rolling Resistance to ```Pressure and velocity``` or ```Magic Formula```.

Nominal longitudinal speed along body-fixed x-axis, in m/s.

#### Dependencies

To enable this parameter, set Longitudinal Force to `Magic Formula pure longitudinal slip`.

Nominal camber angle, in rad.

#### Dependencies

To enable this parameter, set either:

• Longitudinal Force to ```Magic Formula pure longitudinal slip```.

• Rolling Resistance to ```Magic Formula```.

Nominal pressure, in Pa.

#### Dependencies

To enable this parameter, set either:

• Longitudinal Force to ```Magic Formula pure longitudinal slip```.

• Rolling Resistance to ```Magic Formula```.

Pressure, in Pa.

#### Dependencies

To enable this parameter:

• Set one of these:

• Longitudinal Force to ```Magic Formula pure longitudinal slip```.

• Rolling Resistance to `Pressure and velocity` or ```Magic Formula```.

• Vertical Motion to ```Mapped stiffness and damping```.

• On the Wheel Dynamics pane, clear Input tire pressure.

### Longitudinal

Magic Formula Constant Value

Pure longitudinal peak factor, dimensionless.

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

SurfaceBCDE
Dry tarmac101.910.97
Wet tarmac122.30.821
Snow520.31
Ice420.11

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Magic Formula constant value`.

Pure longitudinal shape factor, dimensionless.

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

SurfaceBCDE
Dry tarmac101.910.97
Wet tarmac122.30.821
Snow520.31
Ice420.11

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Magic Formula constant value`.

Pure longitudinal stiffness factor, dimensionless.

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

SurfaceBCDE
Dry tarmac101.910.97
Wet tarmac122.30.821
Snow520.31
Ice420.11

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Magic Formula constant value`.

Pure longitudinal curvature factor, dimensionless.

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

SurfaceBCDE
Dry tarmac101.910.97
Wet tarmac122.30.821
Snow520.31
Ice420.11

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Magic Formula constant value`.

Magic Formula Pure Longitudinal Slip

Cfx shape factor, PCX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal friction at nominal normal load, PDX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Frictional variation with load, PDX2, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Frictional variation with camber, PDX3, 1/rad^2.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal curvature at nominal normal load, PEX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Variation of curvature factor with load, PEX2, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Variation of curvature factor with square of load, PEX3, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal curvature factor with slip, PEX4, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal slip stiffness at nominal normal load, PKX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Variation of slip stiffness with load, PKX2, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Slip stiffness exponent factor, PKX3, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Horizontal shift in slip ratio at nominal normal load, PHX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Variation of horizontal slip ratio with load, PHX2, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Vertical shift in load at nominal normal load, PVX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Variation of vertical shift with load, PVX2, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Linear variation of longitudinal slip stiffness with tire pressure, PPX1, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Quadratic variation of longitudinal slip stiffness with tire pressure, PPX2, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Linear variation of peak longitudinal friction with tire pressure, PPX3, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Quadratic variation of peak longitudinal friction with tire pressure, PPX4, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Slip speed decay function scaling factor, lam_muV, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Brake slip stiffness scaling factor, lam_Kxkappa, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal shape scaling factor, lam_Cx, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal curvature scaling factor, lam_Ex, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal horizontal shift scaling factor, lam_Hx, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Longitudinal vertical shift scaling factor, lam_Vx, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter ```Magic Formula pure longitudinal slip```.

Mapped Force

Slip ratio breakpoints, dimensionless.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Mapped force`.

Normal force breakpoints, N.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Mapped force`.

Longitudinal force versus slip ratio and normal force, N.

#### Dependencies

To create this parameter, select the Longitudinal Force parameter `Mapped force`.

### Rolling Resistance

Pressure and Velocity

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

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Pressure and velocity```.

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

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Pressure and velocity```.

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

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Pressure and velocity```.

Tire pressure exponent, ɑ, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Pressure and velocity```.

Normal force exponent, β, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Pressure and velocity```.

ISO 28580

Parasitic force loss, Fpl, in N.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```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 enable this parameter, set Rolling Resistance to ```ISO 28580```.

Thermal correction factor, Kt, in 1/degC.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```ISO 28580```.

Measured ambient temperature, Tmeas, near tire during tire testing, in K.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```ISO 28580```.

Measured ambient temperature, Tamb, near tire in application environment, in K. For example, the measured ambient temperature is the ambient temperature near the tire when the vehicle is on the road.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```ISO 28580```.

Select to create input port Tamb to input the measured ambient temperature.

The measured ambient temperature, Tamb, is the temperature near tire in application environment, in K. For example, the measured ambient temperature is the ambient temperature near the tire when the vehicle is on the road.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```ISO 28580```.

Magic Formula

Rolling resistance torque coefficient, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Longitudinal force rolling resistance coefficient, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Linear rotational speed rolling resistance coefficient, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Quartic rotational speed rolling resistance coefficient, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

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

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Load based camber squared rolling resistance torque, in 1/rad^2.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Normal load rolling resistance coefficient, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Pressure load rolling resistance coefficient, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Rolling resistance scaling factor, dimensionless.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Magic Formula```.

Mapped

Spin axis velocity breakpoints, in m/s.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Mapped torque```.

Normal force breakpoints, in N.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Mapped torque```.

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

#### Dependencies

To enable this parameter, set Rolling Resistance to ```Mapped torque```.

### Brake

Static friction coefficient, specified as a scalar, dimensionless.

#### Dependencies

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

• `Disc`

• `Drum`

• `Mapped`

Kinematic friction coefficient, specified as a scalar, dimensionless.

#### Dependencies

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

• `Disc`

• `Drum`

• `Mapped`

Disc

Disc brake actuator bore, specified as a scalar, in m.

#### Dependencies

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

Brake pad mean radius, specified as a scalar, in m.

#### Dependencies

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

Number of brake pads, specified as a scalar, dimensionless.

#### Dependencies

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

Drum

Drum brake actuator bore, specified as a scalar, 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.

Drum internal radius, in m.

#### 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

Nominal rated wheel load along wheel-fixed z-axis, in N.

#### Dependencies

To enable this parameter, set either:

• Longitudinal Force to ```Magic Formula pure longitudinal slip```.

• Rolling Resistance to ```Magic Formula```.

Nominal rated load scaling factor, dimensionless. Used to scale the normal for specific applications and load conditions.

#### Dependencies

To enable this parameter, set Longitudinal Force to `Magic Formula pure longitudinal slip`.

Wheel and unsprung 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```.

Ground displacement, `Grndz`, along negative wheel-fixed z-axis, in m.

#### 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 Setup

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

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

Maximum allowable absolute slip ratio, dimensionless.

Velocity tolerance used to handle low-velocity situations, in m/s.

Minimum ambient temperature, TMIN, in K.

#### Dependencies

To enable this parameter, set Rolling Resistance to ```ISO 28580```.

Maximum ambient temperature, TMAX, in K.

#### Dependencies

To enable this parameter, set Rolling Resistance to `ISO 28580`.

## References

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

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

[3] Schmid, Steven R., Bernard J. Hamrock, and Bo O. Jacobson. "Chapter 18: Brakes and Clutches." Fundamentals of Machine Elements, SI Version. 3rd ed. Boca Raton, FL: CRC Press, 2014.

[4] Shigley, Joseph E., and Larry Mitchel. Mechanical Engineering Design. 4th ed. New York, NY: McGraw Hill, 1983.

[5] 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.

## Version History

Introduced in R2017a