Combined Slip Wheel CPI
Combined slip wheel compliant with CPI Tydex standard
 Library:
Vehicle Dynamics Blockset / Wheels and Tires
Description
The Combined Slip Wheel CPI block implements the longitudinal and lateral behavior of a wheel characterized by the Magic Formula^{1, 2} that complies with the contact point interface (CPI) 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 lowfrequency tireroad interactions are required to determine vehicle acceleration, braking, and wheelrolling 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 tiresuspension 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 equations^{1, 2}. The equations use fitting coefficients that correspond to the block parameters.  Update the block parameters with fitting coefficients from a file:

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:

Rotational Wheel Dynamics
The block calculates the inertial response of the wheel subject to:
Axle losses
Tire rolling resistance
Ground contact through the tireroad 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 

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 If you clear Include turn slip, the block sets some of these equations to 1. 
CPI Tire Coordinate System
The block uses tire coordinate system axes (X_{T}, Y_{T}, Z_{T}) that are fixed in a reference frame attached to the tire. The origin is at the tire contact with the ground.
The CPI tire coordinate system is shown in red.
Note
The CPI tire coordinate system (red) is equivalent to the TYDEX wheelaxis coordinate system.
^{1}
Axis  Description 

X_{T}  X_{T} and Y_{T} are parallel to the road plane. The intersection of the wheel plane and the road plane define the orientation of the X_{T} axis. 
Y_{T}  Y_{T} is the projection of the wheel spin axis on the ground. 
Z_{T}  Z_{T} points upward. 
Ports
Input
Omega
— Rotational velocity
scalar
 N
by1
vector
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.
Vx
— Longitudinal velocity
scalar
 N
by1
vector
Axle longitudinal velocity, V_{x}, along tirefixed xaxis, 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
by1
vector
Axle lateral velocity, V_{y}, along tirefixed yaxis, 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
by1
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
by1
vector
Tire angular velocity, r, about the tirefixed zaxis (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.
Fext
— Axle force applied to tire
scalar
 N
by1
vector
Axle force applied to tire, F_{ext}, along vehiclefixed zaxis (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
— Road friction scale factors
2
byN
array
Magic formula road friction scale factor array. Array dimensions are 2
by the number of wheels, N.
The Magic Formula equations use scale factors to account for static or simulation runtime variations. Nominally, most are set to 1
.
Array Element  Variable  Scale Factor 

ScaleFctrs(1,1)  lam_mux  Longitudinal peak friction coefficient 
ScaleFctrs(2,1)  lam_muy  Lateral peak friction coefficient 
Prs
— Tire inflation pressure
scalar
 N
by1
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.
Dependencies
To create this port, select Input tire pressure.
Output
Info
— Block data
bus
Block data, returned as a bus signal containing these block values.
Signal  Description  Units 

 Wheel angular velocity about wheelfixed yaxis  rad/s 
 Longitudinal vehicle force along tirefixed xaxis  N 
 Lateral vehicle force along tirefixed yaxis  N 
 Vertical vehicle force along tirefixed zaxis  N 
 Overturning moment about tirefixed xaxis  N·m 
 Rolling resistance torque about tirefixed yaxis  N·m 
Mz  Aligning moment about tirefixed zaxis  N·m 
 Vehicle longitudinal velocity along tirefixed xaxis  m/s 
 Vehicle lateral velocity along tirefixed yaxis  m/s 
 Loaded effective radius  m 
 Longitudinal slip ratio  NA 
 Side slip angle  rad 
 Contact patch half length  m 
 Contact patch half width  m 
 Camber angle  rad 
 Tire angular velocity about the tirefixed zaxis (yaw rate)  rad/s 
 Axle vertical displacement along tirefixed zaxis  m 
 Vertical sidewall force on ground along tirefixed zaxis  N 
 Tire inflation pressure  Pa 
Fx
— Longitudinal axle force
scalar
 N
by1
vector
Longitudinal force acting on axle, F_{x}, along tirefixed xaxis, 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
by1
vector
Lateral force acting on axle, F_{y}, along tirefixed yaxis, 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
by1
vector
Vertical force acting on axle, F_{z}, along tirefixed zaxis, 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
by1
vector
Longitudinal moment acting on axle, M_{x}, about tirefixed xaxis, 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
by1
vector
Lateral moment acting on axle, M_{y}, about tirefixed yaxis, 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
by1
vector
Vertical moment acting on axle, M_{z}, about tirefixed zaxis, 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
— Type
External file
(default)  Light passenger car 205/60R15
 Midsize 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 equations^{1, 2}. The equations use fitting coefficients that correspond to the block parameters.  Update the block parameters with fitting coefficients from a file:

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:

Tire file or object, tireParamSet
— Tire file
.mat
 .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:
Set Tire type to
External file
.On the 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.
Tire side
— Select tire side
Right
(default)  Left
 Symmetric
Specify the tire side.
Tire pressure
— Pressure
220000
(default)  scalar
Tire inflation pressure, p, in Pa.
Dependencies
To enable this parameter, clear Input tire pressure.
Ply steer
— Include ply steer
on
(default)  off
Select to include ply steer in the Magic Formula equations.
By default, the blocks include ply steer and turn slip in the Magic Formula equations. The equations are fit to flatbelt test data and predict a number of tire effects, including ply steer and turn slip. Consider removing the effects if your:
Test data does not include ply steer or turn slip data.
Analysis does not require ply steer or turn slip effects.
If you clear Ply steer, the block internally sets these parameters to 0:
Vertical shift of overturning moment, QSX1
Combined slip Fx shift factor reduction, RHX1
Efy curvature constant camber dependency, PEY3
SHY horizontal shift at FZNOM, PHY1
SHY variation with load, PHY2
Svy/Fz vertical shift at FZNOM, PVY1
Svy/Fz variation with load, PVY2
Fy shift reduction with slip angle, RBY3
Slip ratio side force Svyk/Muy*Fz at FZNOM, RVY1
Side force Svyk/Muy*Fz variation with load, RVY2
Bpt slope variation with camber, QBZ4
Dpt peak trail variation with camber, QDZ3
Dmr peak residual torque, QDZ6
Dmr peak residual torque variation with load, QDZ7
Ept variation with sign of alphat, QEZ4
Sht horizontal trail shift at FZNOM, QHZ1
Sht variation with load, QHZ2
Nominal value of s/R0: effect of Fx on Mz, SSZ1
Turn slip
— Include turn slip
on
(default)  off
Select to include ply steer in Magic Formula equations.
By default, the blocks include ply steer and turn slip in the Magic Formula equations. The equations are fit to flatbelt test data and predict a number of tire effects, including ply steer and turn slip. Consider removing the effects if your:
Test data does not include ply steer or turn slip data.
Analysis does not require ply steer or turn slip effects.
If you clear Turn slip, the block internally:
Sets the Magic Formula turn slip equations to 1. Specifically, equations 4.E77, 4.E79, 4.E81, 4.E83, 4.E84, 4.E92, 4.E102, 4.E101, and 4.E105.^{2}.
Uses Magic Formula terms that effect horizontal shift.
Uses Magic Formula small turn slip values in 4.E27^{2}.
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
scalar
Maximum normal force, FZMAX, in N.
Minimum normal force, FZMIN
— Force
scalar
Minimum normal force, FZMIN, in N.
Velocity tolerance used to handle low velocity situations, VXLOW
— Tolerance
scalar
Velocity tolerance used to handle lowvelocity 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
scalar
Nominal longitudinal speed, LONGVL, in m/s.
Default tyre side, tyreside
— Side
'Right'
(default)  char
Default tyre side, tyreside, dimensionless.
Wheel
Initial rotational velocity, omegao
— Velocity
scalar
Initial rotational velocity, specified as a scalar, in rad/s.
Rotational damping, br
— Damping
scalar
Rotational damping, specified as a scalar, in N·m·s/rad.
Unloaded radius, UNLOADED_RADIUS
— Radius
scalar
Unloaded radius, UNLOADED_RADIUS, in m.
Nominal pressure, NOMPRES
— Pressure
scalar
Nominal pressure, NOMPRES, in Pa.
Nominal normal force, FNOMIN
— Force
scalar
Nominal normal force, FNOMIN, in N.
Wheel width, WIDTH
— Width
scalar
Wheel width, WIDTH, in m.
Rim radius, RIM_RADIUS
— Radius
scalar
Rim radius, RIM_RADIUS, in m.
Nominal aspect ratio, ASPECT_RATIO
— Ratio
scalar
Nominal aspect ratio, ASPECT_RATIO, dimensionless.
Inertial
Wheel mass, MASS
— Mass
scalar
Wheel mass, specified as a scalar, in kg.
Rotational inertia (rolling axis), IYY
— Inertia
scalar
Rotational inertia (rolling axis), specified as a scalar, in kg·m^{2}.
Gravity, GRAVITY
— Gravity
scalar
Gravity, GRAVITY, in m/s^2.
Vertical
Initial tire displacement, zo
— Displacement
scalar
Initial tire displacement, zo, in m.
Initial wheel vertical velocity (wheel fixed frame), zdoto
— Velocity
scalar
Initial wheel vertical velocity (wheel fixed frame), zdoto, in m/s.
Effective rolling radius at low load stiffness, BREFF
— Stiffness
scalar
Effective rolling radius at low load stiffness, BREFF, dimensionless.
Effective rolling radius peak value, DREFF
— Radius
scalar
Effective rolling radius peak value, DREFF, dimensionless.
Effective rolling radius at high load stiffness, FREFF
— Radius
scalar
Effective rolling radius at high load stiffness, FREFF, dimensionless.
Unloaded to nominal rolling radius ratio, Q_RE0
— Ratio
scalar
Unloaded to nominal rolling radius ratio, Q_RE0, dimensionless.
Radius rotational speed dependence, Q_V1
— Speed
scalar
Radius rotational speed dependence, Q_V1, dimensionless.
Stiffness rotational speed dependence, Q_V2
— Speed
scalar
Stiffness rotational speed dependence, Q_V2, dimensionless.
Linear load change with deflection, Q_FZ1
— Load change
scalar
Linear load change with deflection, Q_FZ1, dimensionless.
Quadratic load change with deflection, Q_FZ2
— Load change
scalar
Quadratic load change with deflection, Q_FZ2, dimensionless.
Linear load change with deflection and quadratic camber, Q_FZ3
— Load change
scalar
Linear load change with deflection and quadratic camber, Q_FZ3, dimensionless.
Load response to longitudinal force, Q_FCX
— Force
scalar
Load response to longitudinal force, Q_FCX, dimensionless.
Load response to lateral force, Q_FCY
— Force
scalar
Load response to lateral force, Q_FCY, dimensionless.
Vertical stiffness change due to lateral load dependency on lateral stiffness, Q_FCY2
— Stiffness
scalar
Vertical stiffness change due to lateral load dependency on lateral stiffness, Q_FCY2, dimensionless.
Stiffness response to pressure, PFZ1
— Stiffness
scalar
Stiffness response to pressure, PFZ1, dimensionless.
Vertical tire stiffness, VERTICAL_STIFFNESS
— Stiffness
scalar
Vertical tire stiffness, VERTICAL_STIFFNESS, in N/m.
Vertical tire damping, VERTICAL_DAMPING
— Damping
scalar
Vertical tire damping, VERTICAL_DAMPING, in N·s/m.
Rim bottoming out offset, BOTTOM_OFFST
— Offset
scalar
Rim bottoming out offset, BOTTOM_OFFST, in m.
Bottoming out stiffness, BOTTOM_STIFF
— Stiffness
scalar
Bottoming out stiffness, BOTTOM_STIFF, in N/m.
Linear load dependent camber angle influence on vertical stiffness, Q_CAM1
— Stiffness
scalar
Linear load dependent camber angle influence on vertical stiffness, Q_CAM1, dimensionless.
Quadratic load dependent camber angle influence on vertical stiffness, Q_CAM2
— Stiffness
scalar
Quadratic load dependent camber angle influence on vertical stiffness, Q_CAM2, dimensionless.
Linear load and camber angle dependent reduction on vertical stiffness, Q_CAM3
— Stiffness
scalar
Linear load and camber angle dependent reduction on vertical stiffness, Q_CAM3, dimensionless.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
Overturning
Vertical shift of overturning moment, QSX1
— Overturning moment
scalar
Vertical shift of overturning moment, QSX1, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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 Bfactor, QSX6
— Overturning moment
scalar
Overturning moment, M_{x}, load effect due to Bfactor, 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 Bfactor of lateral force and load, QSX9
— Overturning moment
scalar
Overturning moment, M_{x}, due to Bfactor 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 Bfactor of vertical force and camber, QSX11
— Overturning moment
scalar
Overturning moment, M_{x}, due to Bfactor 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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
SHY variation with load, PHY2
— Horizontal shift variation
scalar
Horizontal shift, S_{HY}, variation with load, PHY2, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
Svy/Fz vertical shift at FZNOM, PVY1
— Vertical shift at nominal force
scalar
Vertical shift, S_{vy}, at nominal force, F_{ZNOM}, PVY1, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
Svy/Fz variation with load, PVY2
— Vertical shift variation with load
scalar
Vertical shift, S_{vy}, variation with load, PVY2, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
Side force Svyk/Muy*Fz variation with load, RVY2
— Side force variation with load
scalar
Side force variation with load, RVY2, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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 alphat, QEZ4
— Trail curvature variation
scalar
Trail curvature, E_{pt} variation with sign of alphat, QEZ4, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
Ept variation with sign of alphat and camber, QEZ5
— Variation
scalar
Trail curvature, E_{pt} variation with sign of alphat 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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
Sht variation with load, QHZ2
— Horizontal trail shift variation with load
scalar
Horizontal trail shift, Sh_{t}, variation with load, QHZ2, dimensionless.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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.
Dependencies
If you clear Ply steer, the block internally sets this parameter to 0 in the Magic Formula equations.
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 90degree 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.
Model Examples
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 ButterworthHeinemann, 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 2021, 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.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2021bR2022b: New Ply steer and Turn slip Parameters
Starting from R2022b, the Combined Slip Wheel CPI block includes Ply steer and Turn slip parameters. To remove ply steer and turn slip from the Magic Formula implementation of these blocks, clear the Ply steer and Turn slip parameters.
See Also
Combined Slip Wheel 2DOF  Combined Slip Wheel 2DOF STI  Fiala Wheel 2DOF  Longitudinal Wheel
^{1} Reprinted with permission Copyright © 2008 SAE International. Further distribution of this material is not permitted without prior permission from SAE.
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