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Combined Slip Wheel STI

Combined slip wheel compliant with STI Tydex standard

  • Library:
  • Vehicle Dynamics Blockset / Wheels and Tires

  • Combined Slip Wheel STI block

Description

The Combined Slip Wheel STI block implements the longitudinal and lateral behavior of a wheel characterized by the Magic Formula1, 2 that complies with the standard tire interface (STI) Tyre Data Exchange Format (TYDEX)3 standard. 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 interactions are required to determine vehicle acceleration 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 wheel rotational velocity, longitudinal and lateral velocity, wheel camber angle, and inflation pressure, the block determines the 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, 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 External tire source pane, Click Select file.

  3. Select the tire coefficient file.

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

  5. Click 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. Click 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. Click Apply.

Rotational Wheel Dynamics

The block calculates the inertial response of the wheel subject to:

  • Axle losses

  • 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 Dynamics2 equations 4.E9 through 4.E57

Lateral force - pure sideslip

Tire and Vehicle Dynamics2 equations 4.E19 through 4.E30

Lateral force - combined slip

Tire and Vehicle Dynamics2 equations 4.E58 through 4.E67

Vertical dynamics

Tire and Vehicle Dynamics2 equations 4.E68, 4.E1, 4.E2a, and 4.E2b

Overturning couple

Tire and Vehicle Dynamics2 equation 4.E69

Rolling resistance

  • An improved Magic Formula/Swift tyre model that can handle inflation pressure changes2 equation 6.1.2

  • Tire and Vehicle Dynamics2 equation 4.E70

Aligning moment

Tire and Vehicle Dynamics2 equation 4.E31 through 4.E49

Aligning torque - combined slip

Tire and Vehicle Dynamics2 equation 4.E71 through 4.E78

STI Wheel Coordinate System

The block uses wheel coordinate system axes (XW, YW, ZW) that are fixed in a reference frame attached to the wheel. The origin is at the wheel center.

The STI wheel coordinate system is shown in blue.

Note

The STI wheel coordinate system (blue) is equivalent to the TYDEX centre-axis coordinate system.

[1]

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

AxisDescription
XW

XW and YW are parallel to the wheel plane:

  • XW is parallel to the local road plane.

  • YW is parallel to the wheel-spin axis.

YW
ZW

ZW points upward.

Ports

Input

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Wheel position along inertial-fixed X-, Y-, Z-axes, respectively, in m.

Vector is the number of wheels, N, by 3.

Transformation matrix from the wheel coordinate system to the Earth-fixed inertial coordinate system.

Rotation angle of rim with respect to the wheel center, in rad.

Wheel velocity along inertial-fixed X-, Y-, and Z-axes, respectively, in m.

Vector is the number of wheels, N, by 3.

Tire rotational velocity, ω, about wheel spin 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.

Rim rotational velocity, ω, about wheel spin axis, in rad/s.

Vector containing wheel position, rotation, and velocity with respect to the Earth-fixed inertial coordinate system.

Vector ElementDescription

Road(1,1)

Road(1,2)

Road(1,3)

Wheel position along inertial-fixed X-, Y-, and Z-axes, respectively, in m.

Road(1,4)

Road(1,5)

Road(1,6)

Road(1,7)

Road(1,8)

Road(1,9)

Road(1,10)

Road(1,11)

Road(1,12)

Transformation matrix from the wheel coordinate system to the Earth-fixed inertial coordinate system.

Road(1,13)

Road(1,14)

Road(1,15)

Wheel velocity along inertial-fixed X-, Y-, and Z-axes, respectively, in m/s.

Road(1,16)

Road(1,17)

Road(1,18)

Wheel angular velocity along inertial-fixed X-, Y-, and Z-axes, respectively, in rad/s.

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

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.

Dependencies

To create this port, select Input tire pressure.

Output

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

SignalDescriptionUnits

CPI_info

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

rhoz

Axle vertical displacement along tire-fixed z-axis

m

FNormal

Vertical sidewall force on ground along tire-fixed z-axis

N

Prs

Tire inflation pressure

Pa

DCM

Transformation matrix from the wheel coordinate system to the Earth-fixed inertial coordinate system

NA
Xe

Wheel position along inertial-fixed X-, Y-, Z-axes, respectively

m
Ang

Rotation angle of the rim with respect to the wheel center

rad
Omega

Tire rotational velocity, ω, about wheel spin axis

rad/s
Ve

Wheel velocity along inertial-fixed X-, Y-, Z-axes, respectively

m/s
OmegaWc

Rim rotational velocity, ω, about wheel spin axis

rad/s
Road

Vector containing wheel position, rotation, and velocity with respect to the Earth-fixed inertial coordinate system

NA

Force applied at wheel center by tire along wheel-fixed x-, y-, z-axes, respectively, in N.

Moment applied at wheel center by tire about wheel-fixed x-, y-, z-axes, respectively, in N·m.

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, 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 External tire source pane, Click Select file.

  3. Select the tire coefficient file.

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

  5. Click 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. Click 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. Click Apply.

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 External tire source pane, click Select file.

  3. Select the tire coefficient file.

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

  5. Click Apply.

Specify the tire side.

Tire inflation pressure, p, in Pa.

Dependencies

To enable this parameter, clear Input tire pressure.

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.

Default tyre side, tyreside, dimensionless.

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.

Nominal aspect ratio, ASPECT_RATIO, dimensionless.

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.

Linear load dependent camber angle influence on vertical stiffness, Q_CAM1, dimensionless.

Quadratic load dependent camber angle influence on vertical stiffness, Q_CAM2, dimensionless.

Linear load and camber angle dependent reduction on vertical stiffness, Q_CAM3, dimensionless.

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, Hans B. Tire and Vehicle Dynamics. 3rd ed. Oxford, United Kingdom: SAE and Butterworth-Heinemann, 2012.

[3] Bohm, F., and H. P. Willumeit, "Tyre Models for Vehicle Dynamic Analysis: Proceedings of the 2nd International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Held at the Technical University of Berlin, Germany, February 20-21, 1997." Vehicle System Dynamics - International Journal of Vehicle Mechanics and Mobility 27, sup. 1, 343–45. https://doi.org/0.1080/00423119708969669.

[4] 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.