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Combined Slip Wheel 2DOF

Combined slip 2DOF wheel with disc, drum, or mapped brake

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  • Vehicle Dynamics Blockset / Wheels and Tires

  • Combined Slip Wheel 2DOF block

Description

The Combined Slip Wheel 2DOF block implements the longitudinal and lateral behavior of a wheel characterized by the Magic Formula[1] and [2]. You can import your own tire data or use fitted tire data sets provided by the Global Center for Automotive Performance Simulation (GCAPS). Use the block in driveline and vehicle simulations where low frequency tire-road and braking forces are required to determine vehicle acceleration, braking, and wheel-rolling resistance. The block is suitable for applications that require combined lateral slip, for example, in lateral motion and yaw stability studies.

Based on the driveline torque, brake pressure, road height, wheel camber angle, and inflation pressure, the block determines the wheel rotation rate, vertical motion, forces, and moments in all six degrees of freedom (DOF). Use the vertical DOF to study tire-suspension resonances from road profiles or chassis motion.

Use the Tire type parameter to select the source of the tire data.

GoalAction

Implement the Magic Formula using empirical equations1 and 2. The equations use fitting coefficients that correspond to the block parameters.

Update the block parameters with fitting coefficients from a file:

  1. Set Tire type to External file.

  2. On the Wheel and Tire Parameters > External tire source pane, select Select file.

  3. Select the tire coefficient file.

  4. Select Update mask values from file. In the dialog box that prompts you for confirmation, click OK. The block updates the parameters.

  5. Select Apply.

Implement fitted tire data sets provided by the Global Center for Automotive Performance Simulation (GCAPS).

Update the applicable block parameters with GCAPS fitted tire data:

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

    • Light passenger car 205/60R15

    • Mid-size passenger car 235/45R18

    • Performance car 225/40R19

    • SUV 265/50R20

    • Light truck 275/65R18

    • Commercial truck 295/75R22.5

  2. Select Update applicable Tire Parameters with tire type values. On the Tire Parameters tab, the block updates the applicable parameters, including Wheel width, Rim radius, and Wheel mass.

  3. Select Apply.

Use the Brake Type parameter to select the brake.

GoalBrake 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

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

To implement the Magic Formula, the block uses these equations.

CalculationEquations

Longitudinal force

Tire and Vehicle Dynamics[2] equations 4.E9 through 4.E57

Lateral force - pure sideslip

Tire and Vehicle Dynamics[2] equations 4.E19 through 4.E30

Lateral force - combined slip

Tire and Vehicle Dynamics[2] equations 4.E58 through 4.E67

Vertical dynamics

Tire and Vehicle Dynamics[2] equations 4.E68, 4.E1, 4.E2a, and 4.E2b

Overturning couple

Tire and Vehicle Dynamics[2] equation 4.E69

Rolling resistance

  • An improved Magic Formula/Swift tyre model that can handle inflation pressure changes[1] equation 6.1.2

  • Tire and Vehicle Dynamics[2] equation 4.E70

Aligning moment

Tire and Vehicle Dynamics[2] equation 4.E31 through 4.E49

Aligning torque - combined slip

Tire and Vehicle Dynamics[2] equation 4.E71 through 4.E78

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)=1|ω|ReLes+1(Fx Re+My)

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 lockup condition, the block implements these friction and dynamic models.

IfLockup ConditionFriction ModelDynamic Model

ω0orTS<|Ti+Tfωb|

Unlocked

Tf=Tkwhere,Tk=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.

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 about wheel spin axis

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

ɑ

Tire pressure exponent

β

Normal force exponent

pi

Tire pressure

μs

Coefficient of static friction

μk

Coefficient of kinetic friction

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]

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

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.

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.

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.

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

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

Side view of drum brake

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.

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

The equations use these variables.

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

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, 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.

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.

Magic Formula scale factor array. Array dimensions are 27 by the number of wheels, N.

The Magic Formula equations use scale factors to account for static or simulation run-time variations. Nominally, most are set to 1.

Array ElementVariableScale Factor
ScaleFctrs(1,1)lam_Fzo

Nominal load

ScaleFctrs(2,1)lam_mux

Longitudinal peak friction coefficient

ScaleFctrs(3,1)lam_muy

Lateral peak friction coefficient

ScaleFctrs(4,1)lam_muV

Slip speed, Vs, decaying friction

ScaleFctrs(5,1)lam_Kxkappa

Brake slip stiffness

ScaleFctrs(6,1)lam_Kyalpha

Cornering stiffness

ScaleFctrs(7,1)lam_Cx

Longitudinal shape factor

ScaleFctrs(8,1)lam_Cy

Lateral shape factor

ScaleFctrs(9,1)lam_Ex

Longitudinal curvature factor

ScaleFctrs(10,1)lam_Ey

Lateral curvature factor

ScaleFctrs(11,1)lam_Hx

Longitudinal horizontal shift

ScaleFctrs(12,1)lam_Hy

Lateral horizontal shift

ScaleFctrs(13,1)lam_Vx

Longitudinal vertical shift

ScaleFctrs(14,1)lam_Vy

Lateral vertical shift

ScaleFctrs(15,1)lam_Kygamma

Camber force stiffness

ScaleFctrs(16,1)lam_Kzgamma

Camber torque stiffness

ScaleFctrs(17,1)lam_t

Pneumatic trail (effecting aligning torque stiffness)

ScaleFctrs(18,1)lam_Mr

Residual torque

ScaleFctrs(19,1)lam_xalpha

Alpha influence on Fx (kappa)

ScaleFctrs(20,1)lam_ykappa

Kappa influence on Fy (alpha)

ScaleFctrs(21,1)lam_Vykappa

Induced ply steer Fy

ScaleFctrs(22,1)lam_s

Moment arm of Fx

ScaleFctrs(23,1)lam_Cz

Radial tire stiffness

ScaleFctrs(24,1)lam_Mx

Overturning couple stiffness

ScaleFctrs(25,1)lam_VMx

Overturning couple vertical shift

ScaleFctrs(26,1)lam_My

Rolling resistance moment

ScaleFctrs(27,1)lam_Mphi

Parking torque Mz

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 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

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 select the source of the tire data.

GoalAction

Implement the Magic Formula using empirical equations1 and 2. The equations use fitting coefficients that correspond to the block parameters.

Update the block parameters with fitting coefficients from a file:

  1. Set Tire type to External file.

  2. On the Wheel and Tire Parameters > External tire source pane, select Select file.

  3. Select the tire coefficient file.

  4. Select Update mask values from file. In the dialog box that prompts you for confirmation, click OK. The block updates the parameters.

  5. Select Apply.

Implement fitted tire data sets provided by the Global Center for Automotive Performance Simulation (GCAPS).

Update the applicable block parameters with GCAPS fitted tire data:

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

    • Light passenger car 205/60R15

    • Mid-size passenger car 235/45R18

    • Performance car 225/40R19

    • SUV 265/50R20

    • Light truck 275/65R18

    • Commercial truck 295/75R22.5

  2. Select Update applicable Tire Parameters with tire type values. On the Tire Parameters tab, the block updates the applicable parameters, including Wheel width, Rim radius, and Wheel mass.

  3. Select Apply.

Use the Brake Type parameter to select the brake.

GoalBrake 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

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.

Brake pad mean radius, in m.

Dependencies

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

Number of brake pads.

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.

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

Tire

Tire file .tir or object containing empirical data to model tire longitudinal and lateral behavior with the Magic Formula. If you provide an .txt file, make sure the file contains names that correspond to the block parameters.

Update the block parameters with fitting coefficients from a file:

  1. Set Tire type to External file.

  2. On the Wheel and Tire Parameters > External tire source pane, select Select file.

  3. Select the tire coefficient file.

  4. Select Update mask values from file. In the dialog box that prompts you for confirmation, click OK. The block updates the parameters.

  5. Select Apply.

Simulation

Maximum pressure, PRESMAX, in Pa.

Minimum pressure, PRESMIN, in Pa.

Maximum normal force, FZMAX, in N.

Minimum normal force, FZMIN, in N.

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

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.

Nominal longitudinal speed, LONGVL, in m/s.

Wheel

Initial rotational velocity, omegao, in rad/s.

Rotational damping, br, in N·m·s/rad.

Unloaded radius, UNLOADED_RADIUS, in m.

Nominal pressure, NOMPRES, in Pa.

Nominal normal force, FNOMIN, in N.

Wheel width, WIDTH, in m.

Rim radius, RIM_RADIUS, in m.

Inertial

Wheel mass, MASS, in kg.

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

Gravity, GRAVITY, in m/s^2.

Vertical

Initial tire displacement, zo, in m.

Initial wheel vertical velocity (wheel fixed frame), zdoto, in m/s.

Effective rolling radius at low load stiffness, BREFF, dimensionless.

Effective rolling radius peak value, DREFF, dimensionless.

Effective rolling radius at high load stiffness, FREFF, dimensionless.

Unloaded to nominal rolling radius ratio, Q_RE0, dimensionless.

Radius rotational speed dependence, Q_V1, dimensionless.

Stiffness rotational speed dependence, Q_V2, dimensionless.

Linear load change with deflection, Q_FZ1, dimensionless.

Quadratic load change with deflection, Q_FZ2, dimensionless.

Linear load change with deflection and quadratic camber, Q_FZ3, dimensionless.

Load response to longitudinal force, Q_FCX, dimensionless.

Load response to lateral force, Q_FCY, dimensionless.

Vertical stiffness change due to lateral load dependency on lateral stiffness, Q_FCY2, dimensionless.

Stiffness response to pressure, PFZ1, dimensionless.

Vertical tire stiffness, VERTICAL_STIFFNESS, in N/m.

Vertical tire damping, VERTICAL_DAMPING, in N·s/m.

Rim bottoming out offset, BOTTOM_OFFST, in m.

Bottoming out stiffness, BOTTOM_STIFF, in N/m.

Structural

Longitudinal stiffness, LONGITUDINAL_STIFFNESS, in N/m.

Longitudinal stiffness, LATERAL_STIFFNESS, in N/m.

Linear vertical deflection influence on longitudinal stiffness, PCFX1, dimensionless.

Quadratic vertical deflection influence on longitudinal stiffness, PCFX2, dimensionless.

Pressure dependency on longitudinal stiffness, PCFX3, dimensionless.

Linear vertical deflection influence on lateral stiffness, PCFY1, dimensionless.

Quadratic vertical deflection influence on lateral stiffness, PCFY2, dimensionless.

Pressure dependency on longitudinal stiffness, PCFY3, dimensionless.

Contact Patch

Contact length square root term, Q_RA1, dimensionless.

Contact length linear term, Q_RA2, dimensionless.

Contact width root term, Q_RB1, dimensionless.

Contact width linear term, Q_RB2, dimensionless.

Longitudinal

Shape factor, Cfx, PCX1, dimensionless.

Longitudinal friction at nominal normal load, PDX1, dimensionless.

Frictional variation with load, PDX2, dimensionless.

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

Longitudinal curvature at nominal normal load, PEX1, dimensionless.

Variation of curvature factor with load, PEX2, dimensionless.

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

Longitudinal curvature factor with slip, PEX4, dimensionless.

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

Variation of slip stiffness with load, PKX2, dimensionless.

Slip stiffness exponent factor, PKX3, dimensionless.

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

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

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

Variation of vertical shift with load, PVX2, dimensionless.

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

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

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

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

Combined slip longitudinal force, Fx, slope factor reduction, RBX1, dimensionless.

Slip ratio longitudinal force, Fx, slope reduction variation, RBX2, dimensionless.

Camber influence on combined slip longitudinal force, Fx, stiffness, RBX3, dimensionless.

Shape factor for combined slip longitudinal force, Fx, reduction, RCX1, dimensionless.

Combined longitudinal force, Fx, curvature factor, REX1, dimensionless.

Combined longitudinal force, Fx, curvature factor with load, REX2, dimensionless.

Combined slip longitudinal force, Fx, shift factor reduction, RHX1, dimensionless.

Overturning

Vertical shift of overturning moment, QSX1, dimensionless.

Overturning moment due to camber, QSX2, dimensionless.

Overturning moment due to lateral force, QSX3, dimensionless.

Overturning moment, Mx, combined lateral force load and camber, QSX4, dimensionless.

Overturning moment, Mx, load effect due to lateral force and camber, QSX5, dimensionless.

Overturning moment, Mx, load effect due to B-factor, QSX6, dimensionless.

Overturning moment, Mx, due to camber and load, QSX7, dimensionless.

Overturning moment, Mx, due to lateral force and load, QSX8, dimensionless.

Overturning moment, Mx, due to B-factor of lateral force and load, QSX9, dimensionless.

Overturning moment, Mx, due to vertical force and camber, QSX10, dimensionless.

Overturning moment, Mx, due to B-factor of vertical force and camber, QSX11, dimensionless.

Overturning moment, Mx, due to squared camber, QSX12, dimensionless.

Overturning moment, Mx, due to lateral force, QSX13, dimensionless.

Overturning moment, Mx, due to lateral force with camber, QSX14, dimensionless.

Overturning moment, Mx, due to inflation pressure, PPMX1, dimensionless.

Lateral

Shape factor for lateral force, Cfy, PCY1, dimensionless.

Lateral friction, μy, PDY1, dimensionless.

Variation of lateral friction, μy, with load, PDY2, dimensionless.

Variation of lateral friction, μy, with squared camber, PDY3, dimensionless.

Lateral curvature, Efy, at nominal force, FZNOM, PEY1, dimensionless.

Lateral curvature, Efy, variation with load, PEY2, dimensionless.

Lateral curvature, Efy, constant camber dependency, PEY3, dimensionless.

Lateral curvature, Efy, variation with camber, PEY4, dimensionless.

Lateral curvature, Efy, variation with camber squared, PEY5, dimensionless.

Maximum lateral force stiffness, KFy, to nominal force, FZNOM, ratio, PKY1, dimensionless.

Load at maximum lateral force stiffness, KFy, to nominal force, FZNOM, ratio, PKY2, dimensionless.

Lateral force stiffness, KFy, to nominal force, FZNOM, stiffness variation with camber, PKY3, dimensionless.

Lateral force stiffness, KFy curvature, PKY4, dimensionless.

Variation of peak stiffness with squared camber, PKY5, dimensionless.

Lateral force, Fy, camber stiffness factor, PKY6, dimensionless.

Camber stiffness vertical load dependency, PKY7, dimensionless.

Horizontal shift, SHY, at nominal force, FZNOM, PHY1, dimensionless.

Horizontal shift, SHY, variation with load, PHY2, dimensionless.

Vertical shift, Svy, at nominal force, FZNOM, PVY1, dimensionless.

Vertical shift, Svy, variation with load, PVY2, dimensionless.

Vertical shift, Svy, variation with camber, PVY3, dimensionless.

Vertical shift, Svy, variation with load and camber, PVY4, dimensionless.

Cornering stiffness variation with inflation pressure, PPY1, dimensionless.

Cornering stiffness variation with inflation pressure induced nominal load dependency, PPY2, dimensionless.

Linear inflation pressure on peak lateral friction, PPY3, dimensionless.

Quadratic inflation pressure on peak lateral friction, PPY4, dimensionless.

Inflation pressure effect on camber stiffness, PPY5, dimensionless.

Combined lateral force, Fy, reduction slope factor, RBY1, dimensionless.

Lateral force, Fy, slope reduction with slip angle, RBY2, dimensionless.

Lateral force, Fy, shift reduction with slip angle, RBY3, dimensionless.

Lateral force, Fy, combined stiffness variation from camber, RBY4, dimensionless.

Lateral force, Fy, combined reduction shape factor, RCY1, dimensionless.

Lateral force, Fy, combined curvature factor, REY1, dimensionless.

Lateral force, Fy, combined curvature factor with load, REY2, dimensionless.

Lateral force, Fy, combined reduction shift factor, RHY1, dimensionless.

Lateral force, Fy, combined reduction shift factor with load, RHY2, dimensionless.

Slip ratio side force at nominal force, FZNOM, RVY1, dimensionless.

Side force variation with load, RVY2, dimensionless.

Side force variation with camber, RVY3, dimensionless.

Side force variation with slip angle, RVY4, dimensionless.

Side force variation with slip ratio, RVY5, dimensionless.

Side force variation with slip ratio arctangent, RVY6, dimensionless.

Rolling

Torque resistance coefficient, QSY1, dimensionless.

Torque resistance due to longitudinal force, Fx, QSY2, dimensionless.

Torque resistance due to speed, QSY3, dimensionless.

Torque resistance due to speed^4, QSY4, dimensionless.

Torque resistance due to square of camber, QSY5, dimensionless.

Torque resistance due to square of camber and load, QSY6, dimensionless.

Torque resistance due to load, QSY7, dimensionless.

Torque resistance due to pressure, QSY8, dimensionless.

Aligning

Trail slope factor for trail Bpt at nominal force, FZNOM, QBZ1, dimensionless.

Slope variation with load, QBZ2, dimensionless.

Slope variation with square of load, QBZ3, dimensionless.

Slope variation with camber, QBZ4, dimensionless.

Slope variation with absolute value of camber, QBZ5, dimensionless.

Slope variation with square of camber, QBZ6, dimensionless.

Slope scaling factor, QBZ9, dimensionless.

Br of Mzr cornering stiffness factor, QBZ10, dimensionless.

Pneumatic trail shape factor, Cpt, QCZ1, dimensionless.

Peak trail, Dpt, QDZ1, dimensionless.

Peak trail, Dpt, variation with load, QDZ2, dimensionless.

Peak trail, Dpt, variation with camber, QDZ3, dimensionless.

Peak trail, Dpt, variation with square of camber, QDZ4, dimensionless.

Peak residual torque, Dmr, QDZ6, dimensionless.

Peak residual torque, Dmr, variation with load, QDZ7, dimensionless.

Peak residual torque, Dmr, variation with camber, QDZ8, dimensionless.

Peak residual torque, Dmr, variation with camber and load, QDZ9, dimensionless.

Peak residual torque, Dmr, variation with square of camber, QDZ10, dimensionless.

Peak residual torque, Dmr, variation with square of load, QDZ11, dimensionless.

Trail curvature, Ept, at nominal force, FZNOM, QEZ1, dimensionless.

Trail curvature, Ept variation with load, QEZ2, dimensionless.

Trail curvature, Ept variation with square of load, QEZ3, dimensionless.

Trail curvature, Ept variation with sign of alpha-t, QEZ4, dimensionless.

Trail curvature, Ept variation with sign of alpha-t and camber, QEZ5, dimensionless.

Horizontal trail shift, Sht, at nominal load, FZNOM, QHZ1, dimensionless.

Horizontal trail shift, Sht, variation with load, QHZ2, dimensionless.

Horizontal trail shift, Sht, variation with camber, QHZ3, dimensionless.

Horizontal trail shift, Sht, variation with load and camber, QHZ4, dimensionless.

Inflation pressure influence on trail length, PPZ1, dimensionless.

Inflation pressure influence on residual aligning torque, PPZ2, dimensionless.

Nominal value of s/R0: effect of longitudinal force, Fx, on aligning torque, Mz, SSZ1, dimensionless.

Variation with lateral to nominal force ratio, SSZ2, dimensionless.

Variation with camber, SSZ3, dimensionless.

Variation with camber and load, SSZ4, dimensionless.

Turnslip

Longitudinal force, Fx, peak reduction due to spin, PDXP1, dimensionless.

Longitudinal force, Fx, peak reduction due to spin with varying load, PDXP2, dimensionless.

Longitudinal force, Fx, peak reduction due to spin with slip ratio, PDXP3, dimensionless.

Cornering stiffness reduction due to spin, PKYP1, dimensionless.

Lateral force, Fy, peak reduction due to spin, PDYP1, dimensionless.

Lateral force, Fy, peak reduction due to spin with varying load, PDYP2, dimensionless.

Lateral force, Fy, peak reduction due to spin with slip angle, PDYP3, dimensionless.

Lateral force, Fy, peak reduction due to square root of spin, PDYP4, dimensionless.

Lateral force, Fy, versus slip angle response lateral shift limit, PHYP1, dimensionless.

Lateral force, Fy, versus slip angle response max lateral shift limit, PHYP2, dimensionless.

Lateral force, Fy, versus slip angle response max lateral shift limit with load, PHYP3, dimensionless.

Lateral force, Fy, versus slip angle response lateral shift curvature factor, PHYP4, dimensionless.

Camber stiffness reduction due to spin, PECP1, dimensionless.

Camber stiffness reduction due to spin with load, PECP2, dimensionless.

Turn slip pneumatic trail reduction factor, QDTP1, dimensionless.

Turn moment for constant turning and zero longitudinal speed, QCRP1, dimensionless.

Turn slip moment increase with spin at 90-degree slip angle, QCRP2, dimensionless.

Residual spin torque reduction from side slip, QBRP1, dimensionless.

Turn slip moment peak magnitude, QDRP1, dimensionless.

Turn slip moment curvature, QDRP2, dimensionless.

References

[1] Besselink, Igo, Antoine J. M. Schmeitz, and Hans B. Pacejka, "An improved Magic Formula/Swift tyre model that can handle inflation pressure changes," Vehicle System Dynamics - International Journal of Vehicle Mechanics and Mobility 48, sup. 1 (2010): 337–52, https://doi.org/10.1080/00423111003748088.

[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. Fundamentals of Machine Elements, SI Version. 3rd ed. Boca Raton: CRC Press, 2014.

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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.