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# Combined Slip Wheel 2DOF

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

• Library:
• Vehicle Dynamics Blockset / Wheels and Tires

## 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 > pane, select .

3. Select the tire coefficient file.

4. Select . In the dialog box that prompts you for confirmation, click . The block updates the parameters.

5. Select .

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 . On the Tire Parameters tab, the block updates the applicable parameters, including Wheel width, Rim radius, and Wheel mass.

3. Select .

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.

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

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.

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

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

Unlocked

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

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

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

Locked

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

$\omega =0$

The equations use these variables.

 ω Wheel angular velocity a Velocity independent force component b Linear velocity force component c Quadratic velocity force component Le Tire relaxation length J Moment of inertia My Rolling resistance torque Ta Applied axle torque 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 Orientation1

### Brakes

Disc

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

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

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

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

The equations use these variables.

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

Drum

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

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

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

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

The equations use these variables.

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

Mapped

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

The equations use these variables.

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

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

• T is brake torque, in N·m.

• P is applied brake pressure, in bar.

• N is wheel speed, in rpm.

## Ports

### Input

expand all

Brake pressure, in Pa.

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

#### Dependencies

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

• `Disc`

• `Drum`

• `Mapped`

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

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

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

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

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

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

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

expand all

### 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 > pane, select .

3. Select the tire coefficient file.

4. Select . In the dialog box that prompts you for confirmation, click . The block updates the parameters.

5. Select .

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 . On the Tire Parameters tab, the block updates the applicable parameters, including Wheel width, Rim radius, and Wheel mass.

3. Select .

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, ${f}_{brake}^{}\left(P,N\right)$, 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.

### 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 > pane, select .

3. Select the tire coefficient file.

4. Select . In the dialog box that prompts you for confirmation, click . The block updates the parameters.

5. Select .

### 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.

## Version History

Introduced in R2018a

## See Also

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