Combined Slip Wheel 2DOF

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

• Library:

• Vehicle Dynamics Blockset / Wheels and Tires

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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.

Goal	Action
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: Set Tire type to External file. On the Wheel and Tire Parameters > External tire source pane, select Select file. Select the tire coefficient file. Select Update mask values from file. In the dialog box that prompts you for confirmation, click OK. The block updates the parameters. 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: 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 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. Select Apply.

Use the Brake Type parameter to select the brake.

Goal	Brake 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.

Calculation	Equations
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.

$$T_d(s)=\frac{1}{\frac{\left|\omega \right|R_e}{L_e}s+1}(F_x{R}_e+M_y)$$

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.

If	Lockup Condition	Friction Model	Dynamic Model
$\begin{array}{l}\omega \ne 0\\ \text{or}\\ T_S<\left|T_i+T_f-\omega b\right|\end{array}$	Unlocked	$$\begin{array}{l}{T}_f=T_k\\ \text{where,}\\ T_k=F_c{R}_{eff}{\mu }_k\tanh \left[4\left(-{\omega }_d\right)\right]\\ T_s=F_c{R}_{eff}{\mu }_s\\ R_{eff}=\frac{2(R_o{}^3-R_i{}^3)}{3(R_o{}^2-R_i{}^2)}\end{array}$$	$$\dot{\omega }J=-\omega b+T_i+T_o$$
$\begin{array}{l}\omega =0\\ \text{and}\\ T_S\ge \left|T_i+T_f-\omega b\right|\end{array}$	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 (X_T, Y_T, Z_T) are fixed in a reference frame attached to the tire. The origin is at the tire contact with the ground.

• Wheel coordinate system axes (X_W, Y_W, Z_W) 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.

$T=\{\begin{array}{c}\frac{\mu P\pi B_a{}^2{R}_m{N}_{pads}}{4} \text{ when }N\ne 0\\ \frac{{\mu }_{static}P\pi B_a{}^2{R}_m{N}_{pads}}{4} \text{ when }N=0\end{array}$

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

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.

$\begin{array}{l}{T}_{rshoe}=\left(\frac{\pi \mu cr(\cos {\theta }_2-\cos {\theta }_1)B_a{}^2}{2\mu (2r\left(\cos {\theta }_2-\cos {\theta }_1)+a\left({\cos }^2{\theta }_2-{\cos }^2{\theta }_1\right)\right)+ar\left(2{\theta }_1-2{\theta }_2+\sin 2{\theta }_2-\sin 2{\theta }_1\right)}\right)P\\ \\ T_{lshoe}=\left(\frac{\pi \mu cr(\cos {\theta }_2-\cos {\theta }_1)B_a{}^2}{-2\mu (2r\left(\cos {\theta }_2-\cos {\theta }_1)+a\left({\cos }^2{\theta }_2-{\cos }^2{\theta }_1\right)\right)+ar\left(2{\theta }_1-2{\theta }_2+\sin 2{\theta }_2-\sin 2{\theta }_1\right)}\right)P\end{array}$

$T=\{\begin{array}{c}{T}_{rshoe}+T_{lshoe}\text{ when }N\ne 0\\ (T_{rshoe}+T_{lshoe})\frac{{\mu }_{static}}{\mu }\text{ when }N=0\end{array}$

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.

$T=\{\begin{array}{c}{f}_{brake}(P,N) \text{ when }N\ne 0\\ \left(\frac{{\mu }_{static}}{\mu }\right)f_{brake}(P,N) \text{ when }N=0\end{array}$

The equations use these variables.

T	Brake torque
$f_{brake}^{}(P,N)$	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, $f_{brake}^{}(P,N)$, 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.

Ports

Input

BrkPrs — Brake pressure

scalar | N-by-1 vector

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

AxlTrq — Axle torque

scalar | N-by-1 vector

Axle torque, T_a, 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.

Vx — Longitudinal velocity

scalar | N-by-1 vector

Axle longitudinal velocity, V_x, 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.

Vy — Lateral velocity

scalar | N-by-1 vector

Axle lateral velocity, V_y, 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 — Inclination angle

scalar | N-by-1 vector

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.

YawRate — Tire angular velocity

scalar | N-by-1 vector

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.

Prs — Tire inflation pressure

scalar | N-by-1 vector

Tire inflation pressure, p_i, 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.

Gnd — Ground displacement

scalar | N-by-1 vector

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.

Fext — Axle force applied to tire

scalar | N-by-1 vector

Axle force applied to tire, F_ext, 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.

ScaleFctrs — Scale factors

27-by-N array

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 Element	Variable	Scale 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

Info — Block data

bus

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

Signal	Description	Units
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

Omega — Wheel angular velocity

scalar | N-by-1 vector

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.

Fx — Longitudinal axle force

scalar | N-by-1 vector

Longitudinal force acting on axle, F_x, 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.

Fy — Lateral axle force

scalar | N-by-1 vector

Lateral force acting on axle, F_y, 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.

Fz — Vertical axle force

scalar | N-by-1 vector

Vertical force acting on axle, F_z, 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.

Mx — Overturning moment

scalar | N-by-1 vector

Longitudinal moment acting on axle, M_x, 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.

My — Rolling resistive moment

scalar | N-by-1 vector

Lateral moment acting on axle, M_y, 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.

Mz — Aligning moment

scalar | N-by-1 vector

Vertical moment acting on axle, M_z, 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

Block Options

Tire Type — Select type

External file (default) | 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

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

Goal	Action
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: Set Tire type to External file. On the Wheel and Tire Parameters > External tire source pane, select Select file. Select the tire coefficient file. Select Update mask values from file. In the dialog box that prompts you for confirmation, click OK. The block updates the parameters. 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: 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 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. Select Apply.

Brake Type — Select type

None | Disc | Drum | Mapped

Use the Brake Type parameter to select the brake.

Goal	Brake 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, mu_static — Static friction

.3 (default) | scalar

Static friction coefficient, dimensionless.

Dependencies

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

• Disc

• Drum

• Mapped

Kinetic friction coefficient, mu_kinetic — Kinetic friction

.2 (default) | scalar

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, disc_abore — Bore distance

.05 (default) | scalar

Disc brake actuator bore, in m.

Dependencies

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

Brake pad mean radius, Rm — Radius

.177 (default) | scalar

Brake pad mean radius, in m.

Dependencies

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

Number of brake pads, num_pads — Count

2 (default) | scalar

Number of brake pads.

Dependencies

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

Drum

Drum brake actuator bore, disc_abore — Bore distance

0.0508 (default) | scalar

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, drum_a — Distance

0.123 (default) | scalar

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, drum_c — Distance

0.212 (default) | scalar

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, drum_r — Radius

0.15 (default) | scalar

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, drum_theta1 — Angle

0 (default) | scalar

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, drum_theta2 — Angle

126 (default) | scalar

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, brake_p_bpt — Breakpoints

vector

Brake actuator pressure breakpoints, in bar.

Dependencies

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

Wheel speed breakpoints, brake_n_bpt — Breakpoints

vector

Wheel speed breakpoints, in rpm.

Dependencies

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

Brake torque map, f_brake_t — Lookup table

array

The lookup table for the brake torque, $f_{brake}^{}(P,N)$, 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.

Dependencies

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

Tire

Tire file or object, tireParamSet — Tire file

vdynPassCar.mat (default) | .tir | .txt

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 — Pressure

scalar

Maximum pressure, PRESMAX, in Pa.

Minimum pressure, PRESMIN — Pressure

scalar

Minimum pressure, PRESMIN, in Pa.

Maximum normal force, FZMAX — Force

10000 (default) | scalar

Maximum normal force, FZMAX, in N.

Minimum normal force, FZMIN — Force

100 (default) | scalar

Minimum normal force, FZMIN, in N.

Velocity tolerance used to handle low velocity situations, VXLOW — Tolerance

0.1 (default) | scalar

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

Max allowable slip ratio (absolute), KPUMAX — Ratio

scalar

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

Minimum allowable slip ratio (absolute), KPUMIN — Ratio

scalar

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

Max allowable slip angle (absolute), ALPMAX — Angle

scalar

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

Minimum allowable slip angle (absolute), ALPMIN — Angle

scalar

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

Maximum allowable camber angle, CAMMAX — Angle

scalar

Maximum allowable camber angle CAMMAX, in rad.

Minimum allowable camber angle, CAMMIN — Angle

scalar

Minimum allowable camber angle, CAMMIN, in rad.

Nominal longitudinal speed, LONGVL — Speed

16.7 (default) | scalar

Nominal longitudinal speed, LONGVL, in m/s.

Wheel

Initial rotational velocity, omegao — Velocity

scalar

Initial rotational velocity, omegao, in rad/s.

Rotational damping, br — Damping

scalar

Rotational damping, br, in N·m·s/rad.

Unloaded radius, UNLOADED_RADIUS — Radius

0.309 (default) | scalar

Unloaded radius, UNLOADED_RADIUS, in m.

Nominal pressure, NOMPRES — Pressure

224006 (default) | scalar

Nominal pressure, NOMPRES, in Pa.

Nominal normal force, FNOMIN — Force

4025 (default) | scalar

Nominal normal force, FNOMIN, in N.

Wheel width, WIDTH — Width

scalar

Wheel width, WIDTH, in m.

Rim radius, RIM_RADIUS — Radius

.19 (default) | scalar

Rim radius, RIM_RADIUS, in m.

Inertial

Wheel mass, MASS — Mass

scalar

Wheel mass, MASS, in kg.

Rotational inertia (rolling axis), IYY — Inertia

scalar

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

Gravity, GRAVITY — Gravity

scalar

Gravity, GRAVITY, in m/s^2.

Vertical

Initial tire displacement, zo — Displacement

0 (default) | scalar

Initial tire displacement, zo, in m.

Initial wheel vertical velocity (wheel fixed frame), zdoto — Velocity

0 (default) | scalar

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

Effective rolling radius at low load stiffness, BREFF — Stiffness

8.25094594147963 (default) | scalar

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

Effective rolling radius peak value, DREFF — Radius

0.260468730454265 (default) | scalar

Effective rolling radius peak value, DREFF, dimensionless.

Effective rolling radius at high load stiffness, FREFF — Radius

0.0735298544471851 (default) | scalar

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

Unloaded to nominal rolling radius ratio, Q_RE0 — Ratio

1.00866439868088 (default) | scalar

Unloaded to nominal rolling radius ratio, Q_RE0, dimensionless.

Radius rotational speed dependence, Q_V1 — Speed

0.000760413786224011 (default) | scalar

Radius rotational speed dependence, Q_V1, dimensionless.

Stiffness rotational speed dependence, Q_V2 — Speed

0.0463384792019201 (default) | scalar

Stiffness rotational speed dependence, Q_V2, dimensionless.

Linear load change with deflection, Q_FZ1 — Load change

0 (default) | scalar

Linear load change with deflection, Q_FZ1, dimensionless.

Quadratic load change with deflection, Q_FZ2 — Load change

15.6870832810226 (default) | scalar

Quadratic load change with deflection, Q_FZ2, dimensionless.

Linear load change with deflection and quadratic camber, Q_FZ3 — Load change

0 (default) | scalar

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

Load response to longitudinal force, Q_FCX — Force

0.138643970247602 (default) | scalar

Load response to longitudinal force, Q_FCX, dimensionless.

Load response to lateral force, Q_FCY — Force

0.10843499565426 (default) | scalar

Load response to lateral force, Q_FCY, dimensionless.

Vertical stiffness change due to lateral load dependency on lateral stiffness, Q_FCY2 — Stiffness

-0.465763352339538 (default) | scalar

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

Stiffness response to pressure, PFZ1 — Stiffness

0.69958166705601 (default) | scalar

Stiffness response to pressure, PFZ1, dimensionless.

Vertical tire stiffness, VERTICAL_STIFFNESS — Stiffness

207885.061134007 (default) | scalar

Vertical tire stiffness, VERTICAL_STIFFNESS, in N/m.

Vertical tire damping, VERTICAL_DAMPING — Damping

494.649255786991 (default) | scalar

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

Rim bottoming out offset, BOTTOM_OFFST — Offset

.01 (default) | scalar

Rim bottoming out offset, BOTTOM_OFFST, in m.

Bottoming out stiffness, BOTTOM_STIFF — Stiffness

2e6 (default) | scalar

Bottoming out stiffness, BOTTOM_STIFF, in N/m.

Structural

Longitudinal stiffness, LONGITUDINAL_STIFFNESS — Stiffness

scalar

Longitudinal stiffness, LONGITUDINAL_STIFFNESS, in N/m.

Lateral stiffness, LATERAL_STIFFNESS — Stiffness

scalar

Longitudinal stiffness, LATERAL_STIFFNESS, in N/m.

Linear vertical deflection influence on longitudinal stiffness, PCFX1 — Deflection influence

scalar

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

Quadratic vertical deflection influence on longitudinal stiffness, PCFX2 — Deflection influence

scalar

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

Pressure dependency on longitudinal stiffness, PCFX3 — Pressure dependency

scalar

Pressure dependency on longitudinal stiffness, PCFX3, dimensionless.

Linear vertical deflection influence on lateral stiffness, PCFY1 — Deflection influence

scalar

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

Quadratic vertical deflection influence on lateral stiffness, PCFY2 — Deflection influence

scalar

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

Pressure dependency on longitudinal stiffness, PCFY3 — Pressure dependency

scalar

Pressure dependency on longitudinal stiffness, PCFY3, dimensionless.

Contact Patch

Contact length square root term, Q_RA1 — Length term

scalar

Contact length square root term, Q_RA1, dimensionless.

Contact length linear term, Q_RA2 — Length term

scalar

Contact length linear term, Q_RA2, dimensionless.

Contact width root term, Q_RB1 — Width term

scalar

Contact width root term, Q_RB1, dimensionless.

Contact width linear term, Q_RB2 — Width term

scalar

Contact width linear term, Q_RB2, dimensionless.

Longitudinal

Cfx shape factor, PCX1 — Shape factor

scalar

Shape factor, C_fx, PCX1, dimensionless.

Longitudinal friction at nominal normal load, PDX1 — Friction

scalar

Longitudinal friction at nominal normal load, PDX1, dimensionless.

Frictional variation with load, PDX2 — Friction variation

scalar

Frictional variation with load, PDX2, dimensionless.

Frictional variation with camber, PDX3 — Friction variation

scalar

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

Longitudinal curvature at nominal normal load, PEX1 — Curvature

scalar

Longitudinal curvature at nominal normal load, PEX1, dimensionless.

Variation of curvature factor with load, PEX2 — Curvature variation

scalar

Variation of curvature factor with load, PEX2, dimensionless.

Variation of curvature factor with square of load, PEX3 — Curvature variation

scalar

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

Longitudinal curvature factor with slip, PEX4 — Curvature

scalar

Longitudinal curvature factor with slip, PEX4, dimensionless.

Longitudinal slip stiffness at nominal normal load, PKX1 — Stiffness

scalar

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

Variation of slip stiffness with load, PKX2 — Stiffness variation

scalar

Variation of slip stiffness with load, PKX2, dimensionless.

Slip stiffness exponent factor, PKX3 — Slip stiffness

scalar

Slip stiffness exponent factor, PKX3, dimensionless.

Horizontal shift in slip ratio at nominal normal load, PHX1 — Slip ratio shift

scalar

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

Variation of horizontal slip ratio with load, PHX2 — Slip variation

scalar

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

Vertical shift in load at nominal normal load, PVX1 — Load shift

scalar

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

Variation of vertical shift with load, PVX2 — Load variation

scalar

Variation of vertical shift with load, PVX2, dimensionless.

Linear variation of longitudinal slip stiffness with tire pressure, PPX1 — Stiffness variation

scalar

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

Quadratic variation of longitudinal slip stiffness with tire pressure, PPX2 — Stiffness variation

scalar

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

Linear variation of peak longitudinal friction with tire pressure, PPX3 — Friction variation

scalar

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

Quadratic variation of peak longitudinal friction with tire pressure, PPX4 — Friction variation

scalar

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

Combined slip Fx slope factor reduction, RBX1 — Combined slip longitudinal force slope factor reduction

scalar

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

Slip ratio Fx slope reduction variation, RBX2 — Slip ratio longitudinal force slope reduction variation

scalar

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

Camber influence on combined slip Fx stiffness, RBX3 — Camber influence on combined slip longitudinal force stiffness

scalar

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

Shape factor for combined slip Fx reduction, RCX1 — Shape factor for combined slip longitudinal force reduction

scalar

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

Combined Fx curvature factor, REX1 — Combined longitudinal force curvature factor

scalar

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

Combined Fx curvature factor with load, REX2 — Combined longitudinal force curvature factor

scalar

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

Combined slip Fx shift factor reduction, RHX1 — Combined slip longitudinal force slip factor

scalar

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

Overturning

Vertical shift of overturning moment, QSX1 — Overturning moment

scalar

Vertical shift of overturning moment, QSX1, dimensionless.

Overturning moment due to camber, QSX2 — Overturning moment due to camber

scalar

Overturning moment due to camber, QSX2, dimensionless.

Overturning moment due to Fy, QSX3 — Overturning moment due to lateral force

scalar

Overturning moment due to lateral force, QSX3, dimensionless.

Mx combined lateral force load and camber, QSX4 — Overturning moment

scalar

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

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

scalar

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

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

scalar

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

Mx due to camber and load, QSX7 — Overturning moment

scalar

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

Mx due to lateral force and load, QSX8 — Overturning moment

scalar

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

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

scalar

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

Mx due to vertical force and camber, QSX10 — Overturning moment

scalar

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

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

scalar

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

Mx due to squared camber, QSX12 — Overturning moment

scalar

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

Mx due to lateral force, QSX13 — Overturning moment

scalar

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

Mx due to lateral force with camber, QSX14 — Overturning moment

scalar

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

Mx due to inflation pressure, PPMX1 — Overturning moment due to pressure

scalar

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

Lateral

Cfy shape factor for lateral force, PCY1 — Lateral force shape factor

scalar

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

Lateral friction muy, PDY1 — Lateral friction

scalar

Lateral friction, μ_y, PDY1, dimensionless.

Lateral friction variation of muy with load, PDY2 — Lateral friction variation

scalar

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

Lateral friction variation of muy with squared camber, PDY3 — Lateral friction variation

scalar

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

Efy lateral curvature at nominal force FZNOM, PEY1 — Lateral curvature at nominal force

scalar

Lateral curvature, Ef_y, at nominal force, F_ZNOM, PEY1, dimensionless.

Efy curvature variation with load, PEY2 — Lateral curvature variation

scalar

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

Efy curvature constant camber dependency, PEY3 — Lateral curvature constant

scalar

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

Efy curvature variation with camber, PEY4 — Lateral curvature variation

scalar

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

Efy curvature variation with camber squared, PEY5 — Lateral curvature variation

scalar

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

Maximum KFy/FZNOM stiffness, PKY1 — Maximum stiffness

scalar

Maximum lateral force stiffness, KF_y, to nominal force, F_ZNOM, ratio, PKY1, dimensionless.

Load at maximum KFy/FZNOM stiffness, PKY2 — Load

scalar

Load at maximum lateral force stiffness, KF_y, to nominal force, F_ZNOM, ratio, PKY2, dimensionless.

KFy/FZNOM stiffness variation with camber, PKY3 — Stiffness variation

scalar

Lateral force stiffness, KF_y, to nominal force, F_ZNOM, stiffness variation with camber, PKY3, dimensionless.

KFy curvature, PKY4 — Lateral force stiffness curvature

scalar

Lateral force stiffness, KF_y curvature, PKY4, dimensionless.

Variation of peak stiffness with squared camber, PKY5 — Stiffness variation

scalar

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

Fy camber stiffness factor, PKY6 — Lateral force camber stiffness factor

scalar

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

Camber stiffness vertical load dependency, PKY7 — Stiffness

scalar

Camber stiffness vertical load dependency, PKY7, dimensionless.

SHY horizontal shift at FZNOM, PHY1 — Horizontal shift at nominal force

scalar

Horizontal shift, S_HY, at nominal force, F_ZNOM, PHY1, dimensionless.

SHY variation with load, PHY2 — Horizontal shift variation

scalar

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

Svy/Fz vertical shift at FZNOM, PVY1 — Vertical shift at nominal force

scalar

Vertical shift, S_vy, at nominal force, F_ZNOM, PVY1, dimensionless.

Svy/Fz variation with load, PVY2 — Vertical shift variation with load

scalar

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

Svy/Fz variation with camber, PVY3 — Vertical shift variation with camber

scalar

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

Svy/Fz variation with load and camber, PVY4 — Vertical shift variation with load and camber

scalar

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

Cornering stiffness variation with inflation pressure, PPY1 — Stiffness variation with pressure

scalar

Cornering stiffness variation with inflation pressure, PPY1, dimensionless.

Cornering stiffness variation with inflation pressure induced nominal load dependency, PPY2 — Stiffness variation with pressure

scalar

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

Linear inflation pressure on peak lateral friction, PPY3 — Pressure

scalar

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

Quadratic inflation pressure on peak lateral friction, PPY4 — Pressure

scalar

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

Inflation pressure effect on camber stiffness, PPY5 — Pressure

scalar

Inflation pressure effect on camber stiffness, PPY5, dimensionless.

Combined Fy reduction slope factor, RBY1 — Combined lateral force reduction slope factor

scalar

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

Fy slope reduction with slip angle, RBY2 — Lateral force slope reduction with slip angle

scalar

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

Fy shift reduction with slip angle, RBY3 — Lateral force shift reduction with slip angle

scalar

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

Fy combined stiffness variation from camber, RBY4 — Lateral force combined stiffness variation from camber

scalar

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

Fy combined reduction shape factor, RCY1 — Lateral force combined reduction shape factor

scalar

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

Fy combined curvature factor, REY1 — Lateral force combined curvature factor

scalar

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

Fy combined curvature factor with load, REY2 — Lateral force combined curvature factor with load

scalar

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

Fy combined reduction shift factor, RHY1 — Lateral force combined reduction shift factor

scalar

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

Fy combined reduction shift factor with load, RHY2 — Lateral force combined reduction shift factor with load

scalar

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

Slip ratio side force Svyk/Muy*Fz at FZNOM, RVY1 — Slip ratio slide force at nominal force

scalar

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

Side force Svyk/Muy*Fz variation with load, RVY2 — Side force variation with load

scalar

Side force variation with load, RVY2, dimensionless.

Side force Svyk/Muy*Fz variation with camber, RVY3 — Side force variation with camber

scalar

Side force variation with camber, RVY3, dimensionless.

Side force Svyk/Muy*Fz variation with slip angle, RVY4 — Side force variation with slip angle

scalar

Side force variation with slip angle, RVY4, dimensionless.

Side force Svyk/Muy*Fz variation with slip ratio, RVY5 — Side force variation with slip ratio

scalar

Side force variation with slip ratio, RVY5, dimensionless.

Side force Svyk/Muy*Fz variation with slip ratio arctangent, RVY6 — Side force variation with slip ratio arctangent

scalar

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

Rolling

Torque resistance coefficient, QSY1 — Torque resistance

scalar

Torque resistance coefficient, QSY1, dimensionless.

Torque resistance due to Fx, QSY2 — Torque resistance due to longitudinal force

scalar

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

Torque resistance due to speed, QSY3 — Torque resistance due to speed

scalar

Torque resistance due to speed, QSY3, dimensionless.

Torque resistance due to speed^4, QSY4 — Torque resistance due to speed

scalar

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

Torque resistance due to square of camber, QSY5 — Torque resistance due to camber

scalar

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

Torque resistance due to square of camber and load, QSY6 — Torque resistance due to camber and load

scalar

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

Torque resistance due to load, QSY7 — Torque resistance due to load

scalar

Torque resistance due to load, QSY7, dimensionless.

Torque resistance due to pressure, QSY8 — Torque resistance due to pressure

scalar

Torque resistance due to pressure, QSY8, dimensionless.

Aligning

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

scalar

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

Bpt slope variation with load, QBZ2 — Slope variation with load

scalar

Slope variation with load, QBZ2, dimensionless.

Bpt slope variation with square of load, QBZ3 — Slope variation with load

scalar

Slope variation with square of load, QBZ3, dimensionless.

Bpt slope variation with camber, QBZ4 — Slope variation with camber

scalar

Slope variation with camber, QBZ4, dimensionless.

Bpt slope variation with absolute value of camber, QBZ5 — Slope variation with camber

scalar

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

Bpt slope variation with square of camber, QBZ6 — Slope variation with camber

scalar

Slope variation with square of camber, QBZ6, dimensionless.

Br of Mzr slope scaling factor, QBZ9 — Slope scaling factor

scalar

Slope scaling factor, QBZ9, dimensionless.

Br of Mzr cornering stiffness factor, QBZ10 — Cornering stiffness factor

0 (default) | scalar

Br of Mzr cornering stiffness factor, QBZ10, dimensionless.

Cpt pneumatic trail shape factor, QCZ1 — Pneumatic trail shape factor

scalar

Pneumatic trail shape factor, C_pt, QCZ1, dimensionless.

Dpt peak trail, QDZ1 — Peak trail

scalar

Peak trail, D_pt, QDZ1, dimensionless.

Dpt peak trail variation with load, QDZ2 — Peak trail variation with load

scalar

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

Dpt peak trail variation with camber, QDZ3 — Peak trail variation with camber

scalar

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

Dpt peak trail variation with square of camber, QDZ4 — Peak trail variation with camber

scalar

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

Dmr peak residual torque, QDZ6 — Peak residual torque

scalar

Peak residual torque, D_mr, QDZ6, dimensionless.

Dmr peak residual torque variation with load, QDZ7 — Peak residual torque variation with load

scalar

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

Dmr peak residual torque variation with camber, QDZ8 — Peak residual torque variation with camber

scalar

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

Dmr peak residual torque variation with camber and load, QDZ9 — Peak residual torque variation with camber and load

scalar

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

Dmr peak residual torque variation with square of camber, QDZ10 — Peak residual torque variation with camber

scalar

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

Dmr peak residual torque variation with square of load, QDZ11 — Peak residual torque variation with load

scalar

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

Ept trail curvature at FZNOM, QEZ1 — Trail curvature at nominal force

scalar

Trail curvature, E_pt, at nominal force, F_ZNOM, QEZ1, dimensionless.

Ept variation with load, QEZ2 — Trail curvature variation with load

scalar

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

Ept variation with square of load, QEZ3 — Trail curvature variation with load

scalar

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

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

scalar

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

Ept variation with sign of alpha-t and camber, QEZ5 — Variation

scalar

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

Sht horizontal trail shift at FZNOM, QHZ1 — Horizontal trail shift at nominal load

scalar

Horizontal trail shift, Sh_t, at nominal load, F_ZNOM, QHZ1, dimensionless.

Sht variation with load, QHZ2 — Horizontal trail shift variation with load

scalar

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

Sht variation with camber, QHZ3 — Horizontal trail shift variation with camber

scalar

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

Sht variation with load and camber, QHZ4 — Horizontal trail shift variation with load and camber

scalar

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

Inflation pressure influence on trail length, PPZ1 — Pressure influence on trail length

scalar

Inflation pressure influence on trail length, PPZ1, dimensionless.

Inflation pressure influence on residual aligning torque, PPZ2 — Pressure influence on aligning torque

scalar

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

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

scalar

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

s/R0 variation with lateral to nominal force ratio, SSZ2 — Variation with lateral to nominal force ratio

scalar

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

s/R0 variation with camber, SSZ3 — Variation with camber

scalar

Variation with camber, SSZ3, dimensionless.

s/R0 variation with camber and load, SSZ4 — Variation with camber and load

scalar

Variation with camber and load, SSZ4, dimensionless.

Turnslip

Fx peak reduction due to spin, PDXP1 — Longitudinal force peak reduction due to spin

scalar

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

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

scalar

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

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

scalar

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

Cornering stiffness reduction due to spin, PKYP1 — Stiffness reduction due to spin

scalar

Cornering stiffness reduction due to spin, PKYP1, dimensionless.

Fy peak reduction due to spin, PDYP1 — Lateral force peak reduction due to spin

scalar

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

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

scalar

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

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

scalar

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

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

scalar

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

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

scalar

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

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

scalar

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

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

scalar

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

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

scalar

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

Camber stiffness reduction due to spin, PECP1 — Camber stiffness reduction

scalar

Camber stiffness reduction due to spin, PECP1, dimensionless.

Camber stiffness reduction due to spin with load, PECP2 — Camber stiffness reduction

scalar

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

Turn slip pneumatic trail reduction factor, QDTP1 — Turn slip pneumatic trail reduction factor

scalar

Turn slip pneumatic trail reduction factor, QDTP1, dimensionless.

Turn moment for constant turning and zero longitudinal speed, QCRP1 — Turn moment for constant turning

scalar

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

Turn slip moment increase with spin at 90deg slip angle, QCRP2 — Turn slip moment

scalar

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

Residual spin torque reduction from side slip, QBRP1 — Residual spin torque reduction

scalar

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

Turn slip moment peak magnitude, QDRP1 — Turn slip moment peak magnitude

scalar

Turn slip moment peak magnitude, QDRP1, dimensionless.

Turn slip moment curvature, QDRP2 — Turn slip moment curvature

scalar

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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Version History

Introduced in R2018a