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Dugoff Wheel 2DOF

Dugoff Wheel 2DOF wheel with disc, drum, or mapped brake

Since R2023a

  • Dugoff Wheel 2DOF block

Libraries:
Vehicle Dynamics Blockset / Wheels and Tires

Description

The Dugoff Wheel 2DOF block implements a simplified tire with lateral and longitudinal slip capability based on the H. Dugoff 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 analysis, 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.

You can set your own user-defined model parameter values or use a built-in tire model.

If you install the Extended Tire Features for Vehicle Dynamics Blockset support package, you can click the Plot steady state force, moment response button to generate these plots:

  • Lateral force [N] vs Slip angle [rad]

  • Self-aligning moment [Nm] vs Slip angle [rad]

  • Longitudinal force [N] vs Longitudinal slip []

  • Longitudinal force [N] vs Lateral force [N]

With the support package, you can also import tire parameter values defined in the Dugoff Wheel 2DOF block to a tireModel object or export tire parameter values from a tireModel object to the Dugoff Wheel 2DOF block. For more information, see tireModel.get and set.

Use the Tire type parameter to either input tire parameter values or select a fitted tire parameter set.

GoalAction

Input user-defined tire parameter values.

Update the block parameters with user-defined parameter values:

  1. Set Tire type to User defined.

  2. In the Wheel and Tire Parameters section, input user-defined values.

  3. Click Apply.

Select one of the built-in Dugoff tire models to drive the lateral and longitudinal calculations. [Link].

Update the applicable block parameters with values from a built-in tire model:

  1. Set Tire type to the tire that you want to implement. Options include:

    • Light passenger car 205/60R15

    • Light passenger car 245/60R16

    • Mid-size passenger car 235/45R18

    • Performance car 225/40R19

    • SUV 265/50R20

    • Light truck 275/65R18

    • Commercial truck 295/75R22.5

  2. Click Update block with applicable tire values. In the Wheel and Tire Parameters section, the block updates the applicable Longitudinal and Lateral parameters.

  3. Click Apply.

Use the Model slip type parameter to select slip type.

ActionModel Slip Type Setting

Calculate longitudinal and lateral forces under nominal slip conditions

Nominal slip

Calculate longitudinal and lateral forces with additional correction factors for a more accurate response at higher slip values

Extended slip

Use the Brake Type parameter to select the brake.

ActionBrake Type Setting

No braking

None

Implement brake that converts the brake cylinder pressure into a braking force

Disc

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

Drum

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

Mapped

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.

External deflection

The block uses the defined sidewall deflection directly in the effective radius calculation.

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.

Ti=TaTb+Td

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.

Td(s)=1Le|ω|Res+1(FxRe+My)

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:

My=Re{a+b|Vx|+cVx2}{Fzβpiα}tanh(4Vx)

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:

My=Re(FzCr1+Kt(TambTmeas)Fpl)tanh(ω)

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.

EquationLock-Up ConditionFriction ModelDynamic Model

ω0orTS<|Ti+Tfωb|

Unlocked

Tf=Tk,whereTk=FcReffμktanh[4(ωd)]Ts=FcReffμsReff=2(Ro3Ri3)3(Ro2Ri2)

ω˙J=ωb+Ti+To

ω=0andTS|Ti+Tfωb|

Locked

Tf=Ts

ω=0

The equations use these variables.

VariableValue
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

Nominal Slip

Fx=Ckκ1κf(z),wheref(z)={z(2z)                when z<11                           when z1z=μFz(1κ)2(Ckκ)2+(Cαtan(α))2

Extended

Fx=Ckκ1κf(z)gx,wheref(z)={z(2z)                when z<11                           when z1z=μFz(1κ)2(Ckκ)2+(Cαtan(α))2gx=(gx1+gx2μ)κ2(gx3+gx4μ)κ+gx5

Friction coefficient

μ=μ0(1AsVs),whereVs=uκ2+tan2(α)

The equations use these variables.

VariableValue
Fx

Longitudinal force acting on axle along tire-fixed x-axis

Cκ

Longitudinal stiffness

Cα

Lateral stiffness per slip angle

k

Longitudinal slip ratio of tires

Fz

Vertical contact patch normal force along tire-fixed z-axis

u

Velocity component in the wheel plane

μ

Maximum friction coefficient

μ0

Maximum friction scaling coefficient

As

Friction reduction factor

Vs

Friction reduction magnitude

α

Side slip angle of tires

gx

Longitudinal correction factor

gx1

Longitudinal squared slip correction factor

gx2

Longitudinal squared slip friction correction factor

gx3

Longitudinal linear slip correction factor

gx4

Longitudinal linear slip friction correction factor

gx5

Longitudinal offset correction factor

Lateral Force

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

CalculationEquation

Nominal Slip

Fy=Cαtan(α)1κf(z)+γCγ,wheref(z)={z(2z)                 when z<11                            when z1z=μFz(1κ)2(Ckκ)2+(Cαtan(α))2

Extended Slip

Fy=Cαtan(α)1κf(z)gy+γCγ,wheref(z)={z(2z)                when z<11                           when z1z=μFz(1κ)2(Ckκ)2+(Cαtan(α))2gy=(μ+gy1)tan(α)+gy2

Friction Coefficient

μ=μ0(1AsVs),whereVs=uκ2+tan2(α)

The equations use these variables.

VariableValue
α

Side slip angle of tires

Fy

Lateral force acting on axle along tire-fixed y-axis

Fz

Vertical contact patch normal force along tire-fixed z-axis

ɣ

Camber angle

Cɣ

Camber stiffness

Cα

Lateral stiffness per angle slip

Ck

Longitudinal stiffness

k

Longitudinal slip ratio of tires

u

Velocity component in the wheel plane

μ

Maximum friction coefficient

μ0

Maximum friction scaling coefficient

Vs

Friction reduction magnitude

As

Friction reduction factor

gy

Lateral correction factor

gy1

Lateral maximum friction correction factor

gy2

Lateral offset correction factor

Vertical Dynamics

The block implements these equations for the vertical dynamics.

CalculationEquation

Vertical response

z¨m=Fztire+mgFz

Tire normal force

Fztire=ρzkbz˙

Vertical sidewall deflection

ρz=zgndz,z0

The equations use these variables.

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

Mx=FyRecos(γ)

Aligning moment

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

Mz={ψ˙bMz                                               when |α'|>α'Criticaltanh(4α')wμ|Fz|(1ξ)ξ3+ψ˙bMz    when |α'|α'Criticalξ=1Ca|tan(α')|3μ|Fz|

Friction scaling

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

The equations use these variables.

VariableValue
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

ψ˙

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 Orientation1

Z-Up tire and wheel coordinate systems showing wheel plane and road plane

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.

Front and side view of disc brake, showing pad, disc, and caliper

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.

T={μPπBa2RmNpads4                when N0μstaticPπBa2RmNpads4         when N=0

Rm=Ro+Ri2

The equations use these variables.

VariableValue
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 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.

Trshoe=(πμcr(cosθ2cosθ1)Ba22μ(2r(cosθ2cosθ1)+a(cos2θ2cos2θ1))+a(2θ12θ2+sin2θ2sin2θ1))PTlshoe=(πμcr(cosθ2cosθ1)Ba22μ(2r(cosθ2cosθ1)+a(cos2θ2cos2θ1))+a(2θ12θ2+sin2θ2sin2θ1))P

T={Trshoe+Tlshoe                 when N0(Trshoe+Tlshoe)μstaticμ   when N=0

Side view of drum brake

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

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

T={fbrake(P,N)                   when N0(μstaticμ)fbrake(P,N)    when N=0

The equations use these variables.

VariableValue
T

Brake torque

fbrake(P,N)

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, fbrake(P,N), 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.

Plot of brake torque as a function of wheel speed and applied brake pressure

Examples

Ports

Input

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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, set the Brake Type parameter, to 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.

Camber angle, ɣ, or inclination angle, ε, in rad.

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.

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 parameter, set Vertical Motion to None or Mapped stiffness and damping.

Tire radial deflection, RadialDeflct. This value is used directly in the effective radius calculation.

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, set Vertical Motion to External Deflection.

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

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

SignalDescriptionUnits

AxlTrq

Axle torque about wheel-fixed y-axis

N·m

Omega

Wheel angular velocity about wheel-fixed y-axis

rad/s

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

Overturning moment about tire-fixed x-axis

N·m

My

Rolling resistance torque about tire-fixed y-axis

N·m

Mz

Aligning moment about tire-fixed z-axis

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

Loaded effective radius

m

Kappa

Longitudinal slip ratio

NA

Alpha

Side slip angle

rad

a

Contact patch half length

m

b

Contact patch half width

m

Gamma

Camber angle

rad

psidot

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

rad/s

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

WhlTrq

Wheel torque

N·m

RL

Loaded radius

m

RadialDeflct

Tire radial deflection

m

Wheel angular velocity, ω, about wheel-fixed y-axis, 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.

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

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Block Options

Use the Tire type parameter to either input tire parameter values or select a fitted tire parameter set.

GoalAction

Input user-defined tire parameter values.

Update the block parameters with user-defined parameter values:

  1. Set Tire type to User defined.

  2. In the Wheel and Tire Parameters section, input user-defined values.

  3. Click Apply.

Select one of the built-in Dugoff tire models to drive the lateral and longitudinal calculations. [Link].

Update the applicable block parameters with values from a built-in tire model:

  1. Set Tire type to the tire that you want to implement. Options include:

    • Light passenger car 205/60R15

    • Light passenger car 245/60R16

    • Mid-size passenger car 235/45R18

    • Performance car 225/40R19

    • SUV 265/50R20

    • Light truck 275/65R18

    • Commercial truck 295/75R22.5

  2. Click Update block with applicable tire values. In the Wheel and Tire Parameters section, the block updates the applicable Longitudinal and Lateral parameters.

  3. Click Apply.

Use the Model slip type parameter to select slip type.

ActionModel Slip Type Setting

Calculate longitudinal and lateral forces under nominal slip conditions

Nominal slip

Calculate longitudinal and lateral forces with additional correction factors for a more accurate response at higher slip values

Extended slip

Dependencies

Setting Model slip type to Extended slip enables these parameters:

SettingParameters Enabled

Extended slip

  • Longitudinal squared slip correction factor, gx1

  • Longitudinal squared slip friction correction factor, gx2

  • Longitudinal linear slip correction factor, gx3

  • Longitudinal linear slip friction correction factor, gx4

  • Longitudinal offset correction factor, gx5

  • Lateral maximum friction correction factor, gy1

  • Lateral offset correction factor, gy2

Use the Brake Type parameter to select the brake.

ActionBrake Type Setting

No braking

None

Implement brake that converts the brake cylinder pressure into a braking force

Disc

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

Drum

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

Mapped

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

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.

External deflection

The block uses the defined sidewall deflection directly in the effective radius calculation.

Dependencies

Setting Vertical Motion to Mapped stiffness and damping enables these parameters:

SettingParameters Enabled

Mapped stiffness and damping

  • Wheel mass, MASS

  • Initial tire displacement, 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, specified as a scalar or N-by-1 vector, in N. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Lateral stiffness per slip angle, Cα, specified as a scalar or N-by-1 vector, in N/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Camber stiffness, Cɣ, specified as a scalar or N-by-1 vector, in N/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Maximum friction scaling coefficient, μ0, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Friction reduction factor, As, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Longitudinal relaxation length, Lrelx, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Lateral relaxation length, Lrely, specified as a scalar or N-by-1 vector, in m/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Extended slip

Lateral offset correction factor, gy2, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Lateral maximum friction correction factor, gy1, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal offset correction factor, gx5, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal linear slip friction correction factor, gx4, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal linear slip correction factor, gx3, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal squared slip friction correction factor, gx2, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal squared slip correction factor, gx1, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Rolling Resistance

Pressure and Velocity

Velocity-independent force coefficient, a, dimensionless.

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.

Aligning

Tire nominal section width, WIDTH, in m.

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

Brake

Static friction coefficient, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc, Drum, or Mapped.

Kinematic friction coefficient, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc, Drum, or Mapped.

Disc

Disc brake actuator bore, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc.

Brake pad mean radius, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc.

Number of brake pads, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc.

Drum

Drum brake actuator bore, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin to drum center distance, in m.

Dependencies

To enable this parameter, set Brake Type to Drum.

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

Dependencies

To enable this parameter, set Brake Type to Drum.

Drum internal radius, in m.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin to pad start angle, in deg.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin to pad end angle, in deg.

Dependencies

To enable this parameter, set Brake Type to Drum.

Mapped

Brake actuator pressure breakpoints, in bar.

Dependencies

To enable this parameter, set Brake Type to Mapped.

Wheel speed breakpoints, in rpm.

Dependencies

To enable this parameter, set Brake Type to Mapped.

The lookup table for the brake torque, fbrake(P,N), 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.

Plot showing brake torque as a function of wheel speed and applied brake pressure

Dependencies

To enable this parameter, set Brake Type to Mapped.

Wheel

Rotational damping, specified as a scalar or N-by-1 vector, in N·m·s/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other rotational parameters.

N is the number of wheels and must match the input signal dimensions.

Tire rotational inertia (rolling axis), specified as a scalar or N-by-1 vector, in kg·m2. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other rotational parameters.

N is the number of wheels and must match the input signal dimensions.

Initial wheel rotational velocity, specified as a scalar or N-by-1 vector, in rad/s. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other rotational parameters.

N is the number of wheels and must match the input signal dimensions.

Tire unloaded radius, in m.

Vertical

Tire mass, specified as a scalar or N-by-1 vector, in kg. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other vertical parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

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

Initial tire displacement, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other vertical parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

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

Initial wheel vertical velocity, specified as a scalar or N-by-1 vector, in m/s. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other vertical parameters.

N is the number of wheels and must match the input signal dimensions.

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 Setup

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 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.

Plotting

Click Install Extended Tire Features to install the Extended Tire Features for Vehicle Dynamics Blockset support package. With the support package, you can plot steady-state force and moment tire responses from the Dugoff Wheel 2DOF Block Parameters dialog box.

Click Plot steady state force, moment response to generate these plots:

  • Lateral force [N] vs Slip angle [rad]

  • Self-aligning moment [Nm] vs Slip angle [rad]

  • Longitudinal force [N] vs Longitudinal slip []

  • Longitudinal force [N] vs Lateral force [N]

Dependencies

To enable this parameter, click Install Extended Tire Features.

References

[1] Bhoraskar, A. and P. Sakthivel. "A Review and a Comparison of Dugoff and Modified Dugoff Formula with Magic Formula." 2017 International Conference on Nascent Technologies in Engineering (ICNTE)(2017): 1–4. https://doi.org/10.1109/ICNTE.2017.7947898.

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

[3] International Organization for Standardization. Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. ISO 28580: 2018. https://www.iso.org/standard/67531.html.

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2023a

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1 Reprinted with permission Copyright © 2008 SAE International. Further distribution of this material is not permitted without prior permission from SAE.