adams_tire_mdr3_help

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Welcome to Adams/Tire
Adams/Tire12
 
Introducing Adams/Tire
Adams/Tire software is a module you use with Adams/Car, Adams/Chassis, Adams/Solver, or 
Adams/View to add tires to your mechanical model and to simulate maneuvers such as braking, steering, 
acceleration, free-rolling, or skidding. Adams/Tire lets you model the forces and torques that act on a tire 
as it moves over roadways or irregular terrain.
Adams/Tire is a set of shared object libraries that Adams/Solver calls through the Adams DIFSUB, 
GFOSUB, GSESUB subroutines. These subroutines calculate the forces and moments that tires exert on 
a vehicle as a result of the interaction between the tires and road surface.
You can use Adams/Tire to model tires for either vehicle-handling, ride and comfort, and vehicle-
durability analyses.
• Handling analyses are useful for studying vehicle dynamic responses to steering, braking, and 
throttle inputs. For example, you can analyze the lateral accelerations produced for a given 
steering input at a given vehicle speed.
• Ride and comfort analyses are useful for assessing the vehicle's vibrations due to uneven roads 
with short wavelength obstacles (shorter than tire circumference), such as level crossings, 
grooves, or brick roads.
• 3D contact analyses are useful for generating road load histories and stress and fatigue studies 
that require component force and acceleration calculation. These studies can help you calculate 
the effects of road profiles, such as pothole, curb, or Belgian block.
Adams/Tire Modules
Adams/Tire has a line of tire modules that you can use with Adams/View, Adams/Solver, Adams/Car, 
and Adams/Chassis. The modules let you model the rubber tires found on many kinds of vehicles, such 
as cars, trucks, and planes. More specifically, the modules let you model the force and torque that tires 
produce to accelerate, brake, and steer vehicles. The five modules available in Adams/Tire are:
• Adams/Tire Handling Module 
• Adams/Tire 3D Spline Road Module 
• Adams/Tire 3D Shell Road Module
• Specific Tire Models 
• Features in Adams/Tire Modules 
Adams/Tire Handling Module
Adams/Tire Handling incorporates the following tire models for use in vehicle dynamic studies:
• Using the PAC2002Tire Model*
• Using the PAC-TIME Tire Model
• Using Pacejka '89 and '94 Models*
• Using the Fiala Handling Force Model
• Using the UA-Tire Model
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Welcome to Adams/Tire
• 521-Tire Model
Adams/Tire Handling uses a point-follower method to calculate tire normal force. Standard Adams/Tire 
is limited to two-dimensional roads, but can be extended with the three-dimensional capabilities of 
Adams/Tire 3D Spline Road.
*The formulae used in the Pacejka tire models are derived from publications, and are commonly referred 
to as the Pacejka method in the automotive industry. Dr. Pacejka himself is not personally associated with 
the development of these tire models, nor does he endorse them in any way.
Adams/Tire 3D Spline Road Module
Adams/3D Spline Road lets you define an arbitrary three-dimensional smooth road surface. In addition, 
you can place three-dimensional road obstacles, such as a curb, pothole, ramp, or road crown, on top of 
the underlying smooth road surface. You can use the 3D Spline Road Module with any of the tire models 
in Adams/Tire. Use the smooth road part in combination with any of the handling tire models, or use the 
more advanced FTire to deal with road obstacles for ride and comfort and durability analysis.
Adams/Tire 3D Shell Road Module
Adams/Tire 3D Shell Road uses a three-dimensional equivalent-volume method to calculate tire normal 
force on three-dimensional roads for use in predicting vehicle loads for durability studies. You can use 
the Pacejka 2002, Pacejka TIME, Pacejka '89, Pacejka '94, or Fiala models to calculate the tire handling 
forces and moments (lateral force, longitudinal force, aligning torque, and so on).
Specific Tire Models
In addition to the tire models in the Adams/Tire Handling Module, Adams/Tire supplies specific tire 
models:
• Pacejka Motorcycle Tire Model 
A Pacejka tire model suitable for motorcycle handling analysis. It describes the tire-road 
interaction forces with tire-road inclination angles up to 60 degrees.
• Adams/Tire FTire Module 
FTire can describe the 3D tire dynamic response up to 120 Hz and beyond, due to its flexible 
ring approach for the tire belt. It can handle any road obstacle.
All tire models support the Adams/Linear functionality.
Adams/Tire14
 
Features in Adams/Tire Modules
The table below lists the features available in Adams/Tire modules.
Which Tire Model Should You Use?
Each tire model is valid in a specific area. Using a tire model outside this area can result in non-realistic 
analysis results. The next table indicates the tire model(s) that are the best to use for a number of 
applications.
In general, the Adams/Tire Handling models are valid on rather smooth roads only: the wavelength of 
road obstacles should not be smaller than the tire circumference. If the wavelengths are shorter, you 
should use the FTire model to cope with the non-linear tire enveloping effects.
Some of the Handling Tire models can describe the first-order response of a tire, but do not take the 
eigenfrequencies of the tire itself into account. Therefore, the Handling Tire models are valid up to 
approximately 8 Hz. The PAC2002 uses a contact mass method that enables it to describe tire behavior 
up to 15 Hz. Beyond that, a tire model should be used, including modeling the tire belt, as FTire does.
Typical Applications for Each Tire Model
Adams/Tire modules: Features: Requirements:
Adams/Tire Handling Fiala Tire Model
Pacejka '89* Tire Model
Pacejka '94* Tire Model
Pacejka 2002 Tire Model
Pacejka TIME Tire Model
UA-Tire Tire Model
2D Road Models
-----------------------------------
5.2.1 Tire 
5.2.1 Tire Methods:
- Equation Method
- Interpolation Method
5.2.1 Road Methods:
- Point Follower
- Equivalent Plane
Full Simulation Package
Adams/Tire 3D Spline Road Fiala Tire Model
3D Smooth Road and Road 
pertubations
Full Simulation Package
Adams/Tire FTire 2D FTire Model
3D FTire Model
2D Road Models
Full Simulation Package
Adams/Tire Motorcycle Tire Pacejka Motorcycle Tire Model
2D Road Models
Full Simulation Package
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Welcome to Adams/Tire
Adams/Tire16
 
Learning Adams/Tire Basics
Adams/Tire12
 
Use and Understanding of Adams/Tire
How to Use Adams/Tire
The Tire Basic help section provides overview material for using Adams/Tire to add tires to a mechanical 
system model. It assumes that you know how to run Adams/Car, Adams/Solver, Adams/View, or 
Adams/Chassis. It also assumes that you have a moderate level of tire-modeling proficiency
You use Adams/Tire to simulate tires according to your analysis requirements. You can create your own 
tire models or you can use the tire models that come with Adams/Tire.
Adams/Tire Steps
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Learning Adams/Tire Basics
 
To use Adams/Tire:
1. Define tires. How you define tires depends on the product you are using (Adams/Chassis, 
Adams/Car, or Adams/Solver). For more information on defining tires, see Defining Tires. 
Regardless of the product you use, the product creates an Adams dataset (.adm), which contains 
the necessary statements that represent the tires in your Adams model, as well as other elements 
of the vehicle, such as the wheel, suspension, and landing gear strut. The primary statement for 
each tire is a GFORCE that applies the tire force to the wheel in your suspension.
2. Reference an existing tire property file from:
Adams/Tire14
 
• Adams/Tire (/install_dir/solver/atire)
• Tire manufacturers or testing companies.
• Files that you create. For example, you can create your own tire property filefor simple kinds 
of tire models, such as the Fiala model.
You can find examples of tire property files for all tire models in the Adams installation directory 
at:
install_dir/solver/atire
where install_dir/ is the path to the installation directory for Adams/Tire.
3. Reference an existing road property file. 
You can find an example road property file for a flat road in the Adams installation directory:
install_dir/solver/atire/mdi_2d_flat.rdf
where install_dir/ is the path to the installation directory for Adams/Tire.
4. Run a simulation of your model. 
You can run a simulation using Adams/Car’s version of Adams/Solver (you do not need an 
Adams/Car license) or you can create an Adams/Solver user library and then run your simulation 
using this library and Adams/Solver. For more information, see Performing Simulations.
5. View the results of the simulation in a postprocessor, such as Adams/PostProcessor. 
Understanding Adams/Tire Processes
When you add tires to your Adams model, three processes occur:
• Adams/Solver invokes Adams/Tire.
• Adams/Tire determines the tire and road model to use.
• Adams/Tire performs any calculations the tire model requires.
Flow of Information in Adams/Tire
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Learning Adams/Tire Basics
Invoking Adams/Tire
• When you perform an analysis, Adams/Solver investigates your .adm file to find elements that 
represent a tire. For example, it looks for a GFORCE with the necessary parameters to define the 
force to the wheel in your suspension. When it finds these parameters, it invokes Adams/Tire.
• Adams/Solver obtains the names of the tire property file (.tir) and road property file (.rdf) from 
the STRING statements in the .adm file.
Determining Tire and Road Model to Use
Inside Adams/Tire, the Tire Object Manager examines the tire property file to determine the tire model 
(for example, Fiala or Pacejka ‘89) to use and examines the road property file to determine the road 
model (for example, 2D or 3D) to use.
Performing Calculations
• The Tire Object Manager calls the selected tire model to calculate the tire forces and moments.
• The tire model reads the tire property file to obtain data for calculating the tire forces and 
moments. It then calls the road model to evaluate where the road is in relation to the tire.
• The road model reads the road property file to obtain data about the road.
Adams/Tire16
 
• The tire model returns the forces and moments to Adams/Solver.
• Adams/Solver applies the forces and moments to the wheel part. 
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Learning Adams/Tire Basics
Defining Tires 
If you use Adams/Car or Adams/Chassis, typically the models you work with will already include tires 
(for example, the statements necessary to invoke Adams/Tire). Therefore, you do not need to add tires 
to your model. If you work with Adams/View, however, you will need to define the tires, and for 
Adams/Solver, add statements to your Adams model using Adams/View or a text editor. Learn how to 
work with:
• Adams/View
• Adams/Car
• Adams/Chassis
• Adams/Solver
• Defining Wheel Inertia
Defining Tires in Adams/Car
Adams/Car includes a wheel-tire subsystem and template that you can use in any full-vehicle assembly. 
The wheel-tire subsystem includes all the elements necessary to start Adams/Tire. You can modify the 
wheel inertia and change the property files.
To modify tires in a subsystem:
1. Select the wheel/tire on the screen, right-click, and then select Modify. 
Adams/Tire18
 
The Modify Wheel dialog box appears with options that allow you to modify the tire property file 
and wheel inertia..
2. Change the values as desired, and then select OK. Learn about entering values in Create/Modify 
Wheel.
Defining Tires in Adams/Chassis
Adams/Chassis includes wheels and tires in all body-tire subsystems.
To modify tires in Adams/Chassis:
1. In Build mode, in the treeview, select the wheel subsystem.
2. In the property editor, select the Tires tab. 
The property editor displays options for changing the wheels and tires as shown below.
Tires Tab in Adams/Chassis
Note: You can also use the Display Tire Property File tool to display the tire property file in 
an Information window. You cannot, however, specify or display the road property file 
from this dialog box. In Adams/Car, you specify the road property file when you submit a 
full-vehicle analysis
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Learning Adams/Tire Basics
 
3. Edit the wheels and tires, such as edit the tire property files and change the scaling coefficients. 
Learn about tire subsystems in ADAMS/Chassis.
Defining Tires in Adams/Solver
If you use Adams/Solver, you must add a set of statements to your Adams model for each tire as 
described in the table, Statements Needed for Adding Tires to Your Model. Once you have added these 
statements to your model, you change the tire or road property file by entering new file names in the 
STRING statements holding the file names. You can do this in your Adams dataset (.adm) or from an 
Adams/Solver command file (.acf) using a STRING command. In an .acf file, the STRING command 
must appear before any simulation commands. For example:
test_rig.adm
mytest
STRING/99, STRING=/usr/mdi/solver/atire/mdi_fiala01.tir
SIMULATE/STATIC
SIMULATE/DYNAMIC, DURATION=1.0,STEPS=50
STOP
• Statements Needed for Defining Tires
• Example Dataset
Statements Needed for Defining Tires
For each tire you want to add to your model, you must create a set of statements in your model. This can 
be done using the dialog box in Adams/View (see Defining Tires in Adams/View) or manually using a text 
editor. For a car with four tires, you need four sets of statements. The table below describes the sets of 
statements. The table, MARKER Locations and Orientations, describes how to locate and orient the three 
MARKERs.
Adams/Tire20
 
Statements Needed for Adding Tires to Your Model
Statement types: Purpose in dataset:
MARKER (3) • Wheel center marker - Identify the wheel part, the wheel center 
location and orientation, and the location for applying tire force 
movements. Use as the GFORCE I marker.
• Road floating marker - Identify the road part to the GFORCE for 
applying reaction forces. Use as the GFORCE JFLOAT marker.
• Road reference marker - Identify the origin and orientation of the 
road. Use as the GFORCE RM marker.
You must locate and orient the MARKER statements as described in MARKER 
Locations and Orientations.
GFORCE (1) Apply the tire force and moments to the wheel part.
DIFF (2) Integrate internal tire states for lag effects.
REQUEST (Up to 11) Output tire kinematics and forces (longitudinal slip, slip angle, camber angle, 
contact patch forces, and moments). For more information, see Performing 
Simulations and Viewing Results.
STRING (5) Identify the tire property file, road property file, and other miscellaneous 
information.
Note: The STRING for "contact type" is required for Adams to correctly create 
the STI tire, but it does not change the contact method, which is based entirely 
on the road model.
ARRAY (1) Holds the IDs of the GFORCE, DIFF, and STRING statements.
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Learning Adams/Tire Basics
MARKER Locations and Orientations
Example Dataset
This section gives you an example dataset fragment that includes a complete set of statements for a single 
tire. The example is based on the following assumptions:
• PART/21 is the wheel and PART/99 is ground.
• The orientations assume that the ground part's x-axis points towards the rear of the vehicle, the 
y-axis points towards the right side of the vehicle, and the z-axis points upward.
! adams_view_name='wheel_center_marker'
MARKER/1
, PART=21
, QP = 0,0,0
, REU = 180D, 0D, 0D 
! 
adams_view_name='road_floating_marker'
Marker statements 
required in dataset: Location and orientation:Wheel center marker Because the tire applies forces to the wheel center marker, you must define 
the wheel center so that it belongs to the wheel part and is located at the 
wheel center. You orient the wheel center as follows:
• x-axis lies in the wheel plane and points in the forward direction of 
the wheel.
• y-axis lies along the wheel's spin axis and points towards the left 
side of the vehicle.
• z-axis lies in the wheel plane and points upward.
Road floating marker The tire applies the reaction forces to the road floating marker. The road 
floating marker must belong to the road part, usually ground, and must be 
defined as FLOATING. Because the marker floats, you do not enter a 
location or orientation.
Road reference marker The road reference marker gives the location and orientation of the road. 
You define the road reference marker so that it belongs to the road part, 
usually ground. In addition, the road reference marker’s z-axis must be 
directed upward, meaning the z-axis is parallel to, but points in the opposite 
direction of, the gravity vector.
Locations of the points on the road contained in the road property file are 
given relative to this marker. Generally, the road reference marker should 
be located on the road surface and below the wheel center by approximately 
the static loaded radius of the tire.
MARKER/2
, PART = 99
Adams/Tire22
 
, FLOATING 
! 
adams_view_name='road_reference_marker'
MARKER/3
, PART = 99 
! adams_view_name='tire_forces'
GFORCE/1
, I = 1
, JFLOAT = 2
, RM = 3
, FUNCTION = USER(900,1,100)/
, ROUTINE=abgTire::gfo900 
!
adams_view_name='tire_force_dif1'
DIFF/2
, IC = 0
, FUNCTION = USER(900,1,100)/
, ROUTINE=abgTire::dif900 
!
adams_view_name='tire_force_dif2'
DIFF/3
, IC = 0
, FUNCTION = USER(900,1,100)/
, ROUTINE=abgTire::dif900 
!Map for GFORCE/DIFF USER Functions:
!-----------------------------------
!par(1): dispatcher branch for tire request (always 900).
!par(2): tire GFORCE statement id.
!par(3): tire ARRAY statement id. 
!
adams_view_name='tire_input_array'
ARRAY/100
,IC
,SIZE=9
,NUM= 2, 3, 1, 99, 100, 101, 102, 103, 0
!array[ 1] : 1st DIFF statement id
!array[ 2] : 2nd DIFF statement id
!array[ 3] : side flag (0 left, 1 right)
!array[ 4] : tire_minor_role STRING id
!array[ 5] : tire_property_file STRING id
!array[ 6] : simulation_type STRING id
!array[ 7] : road_property_file STRING id
!array[ 8] : road_contact_type STRING id
!array[ 9] : RIGID_WHEEL Radius (SUSPENSION analysis tire only) 
!
adams_view_name='tire_rolling_states'
REQUEST/1,
, FUNCTION = USER(902,1,1) 
!
adams_view_name='tire_kinematic_states_ISO'
REQUEST/2,
, FUNCTION = USER(902,2,1)
!
adams_view_name='tire_forces_contact_patch_ISO'
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Learning Adams/Tire Basics
REQUEST/3,
, FUNCTION = USER(902,3,1) 
!Map for REQUEST USER Functions:
!-------------------------------
!par(1) = branch for tire request (always 902).
!par(2) = reqtyp = {1,2,3,4,5,6,7.8.9.10,11}
!par(3) = tire GFORCE statement id.
!String Statements Description of use:
!------------------
!
! adams_view_name='tire_minor_role'
! Used by Adams/Car to determine minor role (for example, FRONT or 
REAR).
STRING/99
,S=front
! adams_view_name='tire_property_file'
! Used by TYRxxx routines. Name of tire property file including 
full path that
! contains tire data or 'RIGID_WHEEL' for use in a suspension 
analysis.
STRING/100
,S=mdi_tire01.tir
!
adams_view_name='simulation_type'
! Used by Adams/Car to determine analysis to be performed one of
'VEHICLE_HANDLING_DYNAMIC'
or 'SUSPENSION'
STRING/101
,S=VEHICLE_HANDLING_DYNAMIC
!
adams_view_name='road_property_file'
! Used by ARCxxx routines. Name of road property file including 
full path that
! contains road data or 'BEDPLATE' for a flat, rigid road used
! with suspension analysis.
STRING/102
,S=example_2d_flat.rdf 
! adams_view_name='road_contact_type'
! handling/durability 
!
STRING/103
, STRING =handling 
Defining Tires in Adams/View
Adams/View provides a dialog box that introduces a tire-wheel assembly in your model. You can also 
use the dialog box to create a road.
• Creating a Tire-Wheel Assembly 
Adams/Tire24
 
• Creating a Road 
Creating a Tire-Wheel Assembly
To create a tire-wheel assembly in Adams/View:
1. Do one of the following: 
• From the Forces tool stack or palette, select the Tire tool.
• From the Build menu, point to Forces, and then select Special Force: Tire. 
The Create Wheel and Tire dialog box appears with options that allow you to introduce the 
wheel inertia, tire property file, and side of the vehicle.
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Learning Adams/Tire Basics
 
2. Enter the values as desired to define the tire, and then select OK. 
Creating a Road
If your model includes tires, you must specify a road because each tire must reference a road. The road 
determines the surface friction, bumps, and other inputs to tires.
To create the road:
1. Display the Create Wheel and Tire dialog box as explained in step 1 above.
Adams/Tire26
 
2. Right-click the Road text box, point to vpg_road, and then select Create.
The Create Road dialog box appears. 
3. Enter the values as desired, and then select OK..
Defining Wheel Inertia
The input values for the wheel part inertia are different depending on the tire model you are using. There 
are differences among the tire models in the Adams/Tire Handling Module, including Adams/Tire 
Motorcycle Tire, and FTire, as explained in the next sections:
• Adams/Tire Handling and Motorcycle Modules
• Adams/Tire FTire 
Adams/Tire Handling and Motorcycle Modules
For tire models in the Adams/Tire Handling Module and the Pacejka Motorcycle Tire model, the inertia 
given for the wheel part must be equal to the total inertia of the tire and the rim.
Note: This dialog box generates a tire interface based on the general-state equation subroutine. A 
more simple interface is shown in Defining Tires Using Adams/Solver.
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Learning Adams/Tire Basics
Adams/Tire FTire 
In FTire, a part of the tire can move with respect to the rim. Therefore, the tire mass and moments of 
inertia have to be split into two parts: a part that is moving with the rim (wheel part) and a part that is 
moving with the tire itself. This subdivision is performed during preprocessing of the tire property (.tir) 
file. When a simulation begins with FTire, the following lines appear in the .msg file:
CTI: add the following mass properties to the rim in your MBS 
model
CTI: (the 'rim-fixed' tire parts which are not accounted for in 
FTire):
The inertia data printed after this message has to be added to the rim inertia and used to defined the wheel 
part inertia. Modification of the wheel part is not done automatically.
Adams/Tire28
 
Simulations and Results
Performing Simulations
Once you have incorporated the required statements for modeling a tire into your dataset, you can submit 
the dataset for simulation. If you have Adams/Car installed, you can submit your dataset to the 
Adams/Car version of Adams/Solver, or you can create an Adams/Solver user library and then run your 
simulation using this library and standard Adams/Solver. 
To submit your dataset to the Adams/Car version of Adams/Solver, do one of the 
following:
In a command window, submit your dataset for simulation using the following commands:
• For UNIX, enter: 
mdi -c acar ru-solver
• For Windows, enter: 
mdi acar ru-solver
• On Windows, from the Start menu, point to Programs, point to MSC.Software, point to MD 
R2 Adams, point to ACar, and then select Adams - Car (solver).
• On UNIX, from the Adams Toolbar, right-click the Adams/Car tool, and then select 
Adams/Car - Solver. 
To create an Adams/Solver user library: 
1. Copy the file install_dir/solver/atire/atire.fto your local directory. 
2. Using atire.f, create a user Adams/Solver library:
a. In a command window, enter the command, where mysol.dll is the name of the library: 
• For UNIX, enter: 
mdi -c cr-user n atire.f -n mysol.dll exit
• For Windows, enter: 
mdi cr-user n atire.f -n mysol.dll exit
b. On Windows, from the Start button, point to Programs, point to MSC.Software, point to 
MD R2 Adams, point to ASolver, and then select Create Custom Solver. Follow the menu 
selections to create a private or site library. 
Note: You can also set the Adams/Car tool on the Adams Toolbar to automatically run 
Adams/Car with Adams/Solver.
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Learning Adams/Tire Basics
c. On UNIX, from the Adams Toolbar, right-click the Adams/Solver tool, point to New, 
and then select Adams/Solver User Library. Enter the parameters to define how to create 
the library. .
To submit your dataset to Adams/Solver using your Adams/Solver user library:
• In a command window, submit your dataset for simulation using the following command 
(assuming your library was mysol): 
a. For UNIX, enter: 
mdi -c ru-user i mysol .
b. For Windows, enter: 
mdi -c ru-user i mysol .
• On Windows, from the Start menu, point to Programs, point to MSC.Software, point to MD 
R2 Adams, point to ASolver, and then select Run Custom Solver. Enter the name of the library 
you want to run. 
• On UNIX, from the Adams Toolbar, right-click the Adams/Solver tool, point to Select 
Library, and then select a library, such as mysol. 
Outputting Results
If you combine requests with a USER function, you can output tire results to the request (.req) and results 
(.res) files. The form of the request statement is:
REQUEST/id
, FUNCTION = USER(902, REQTYP, TIR_ID)/
, ROUTINE = abgTire::req902
where:
• 902 - Branch flag for tire request subroutine.
• REQTYP - Integer code fixing the information output to the request file. Valid values are 
{1,2,3,4,5,6,7,8,9,10,11}. The output for each value of REQTYP is described in the table, Tire 
Outputs.
• TIR_ID - Tire GFORCE statement ID.
For information on the axis systems and sign conventions for these outputs, see About Axis Systems and 
Sign Conventions.
Note: On Windows, you can now enter the FORTRAN file directly without first compiling it.
Note: You can also set the AdamsAdams/Solver tool on the Adams Toolbar to automatically run 
with your user library. 
Example of a request in a dataset.
Adams/Tire30
 
Tire Outputs
Output:
REQTYP 
Request: Component definitions:
Tire rolling states 1 x = rolling radius
y = (rad/sec)
z = free (rad/sec)
 is the actual angular velocity about the wheel's axis, while 
 free is the velocity of the wheel's axle center divided by 
the radius to the instantaneous center of rotation. The 
difference between the two is, therefore, a measure of the slip 
when the vehicle is accelerating or decelerating.
Tire kinematic properties 
in TYDEX-W axis (ISO) 
system.
2 x = longitudinal slip (%)
y = lateral slip angle (degrees)
z = inclination angle (degrees)
Tire contact patch forces 
in TYDEX-W axis (ISO) 
system
3 x = longitudinal force (model units)
y = lateral force (model units)
z = vertical force (model units)
r1 = residual overturning moment (model units)
r2 = rolling resistance moment (model units)
r3 = aligning moment (model units)
Tire contact patch forces 
in SAE axis system
4 x = longitudinal force (model units)
y = lateral force (model units)
z = vertical force (model units)
r1 = residual overturning moment (model units)
r2 = rolling resistance moment (model units)
r3 = aligning moment (model units)
Tire kinematic properties 
in SAE axis system
5 x = longitudinal slip (%)
y = lateral slip angle (radians)
ω
ω
ωω
z = inclination angle (radians)
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Learning Adams/Tire Basics
Forces at hub, in 
TYDEX-C axis system
6 x = longitudinal force (model units)
y = lateral force (model units)
z = vertical force (model units)
r1 = overturning moment (model units)
r2 = rolling resistance moment (model units)
r3 = aligning moment (model units)
Miscellaneous tire states 
#1
7 x = longitudinal lag (du/dt)*
y = lateral lag (du/dt)*
z = longitudinal coefficient of friction
r1 = lateral coefficient of friction
r2 = FXMAX = DX + SVX (peak from Pacejka + vertical 
shift)
r3 = FYMAX = DY + SVY (peak from Pacejka + vertical 
shift)
Miscellaneous tire states 
#2
8 x = pneumatic trail *
y = residual aligning moment at contact patch in ISO
z = FX moment arm*
r1= longitudinal relaxation length*
r2 = lateral relaxation length*
r3 = gyroscopic moment*
Miscellaneous tire states 
#3
9 x = inclination angle induced side force*
y = surface friction
Output:
REQTYP 
Request: Component definitions:
μ
μ
Adams/Tire32
 
Miscellaneous tire states 
#4
14 x = distance traveled*
y = effective plane height*
z = effective plane angle*
r1= effective plane curvature*
r2 = contact length*
Contact patch locations 
(the contact patch 
location along the plane 
of the tire in the 
GFORCE reference 
marker’s coordinate 
system.)
10 x = road contact point X location
y = road contact point Y location
z = road contact point Z location
r1 = tire radial penetration into the road surface
r2 = tire radial penetration velocity into the road surface
Hub and wheel velocities 11 x = hub longitudinal velocity in wheel carrier (TYDEX-C) 
axis system
y = tire longitudinal velocity at the contact patch in the 
contact patch axis system
z = tire lateral velocity at the contact patch in the contact 
patch axis system
Output:
REQTYP 
Request: Component definitions:
33
Learning Adams/Tire Basics
About Axis Systems and Sign Conventions
The following sections describe the tire axis systems and the sign conventions for tire kinematic and 
force outputs.
• Tire Axis Systems
• About Tire Kinematic and Force Outputs
• Sign Conventions for Tire Outputs
Tire Axis Systems
The following sections describe the ISO coordinate systems to which Adams/Tire conforms. The ISO 
coordinates are shown as follows:
• ISO-C (TYDEX C) Axis System
• ISO-W (TYDEX W) Contact-Patch Axis System
• Road Reference Marker Axis System
ISO-C (TYDEX C) Axis System
The TYDEX STI specifies the use of the ISO-C axis system for calculating translational and rotational 
velocities, and for outputting the force and torque at the tire hub.
The properties of the ISO-C axis system are:
• The origin of the ISO-C axis system lies at the wheel center.
• The + x-axis is parallel to the road and lies in the wheel plane.
• The + y-axis is normal to the wheel plane and, therefore, parallel to the wheel’s spin axis.
• The + z-axis lies in the wheel plane and is perpendicular to x and y (such as z = x x y).
TYDEX-C Axis System Used in Adams/Tire
Adams/Tire34
 
 
ISO-W (TYDEX W) Contact-Patch Axis System
The properties of the ISO-W (TYDEX W) axis system are:
• The origin of the ISO-W contact-patch system lies in the local road plane at the tire contact 
point.
• The + x-axis lies in the local road plane along the intersection of the wheel plane and the local 
road plane.
• The + z-axis is perpendicular (normal) to the local road plane and points upward.
• The + y-axis lies in the local road plane and is perpendicular to the + x-axis and + z-axis (such as 
y = z x x).
TYDEX W-Axis System Used in Adams/Tire
35
Learning Adams/Tire Basics
 
Road Reference Marker Axis System
The road reference marker axis system is the underlying coordinate system that Adams/Tire uses 
internally. For example, the tire translational displacement and local road normal for a three-dimensional 
road are expressed in the axis system of the road reference marker.The properties of the reference marker axis system are:
• The GFORCE reference marker defines the axis system.
• The + z-axis points upward.
About Tire Kinematic and Force Outputs
Adams/Tire calculates the kinematic quantities of slip angle, inclination angle, and longitudinal slip. 
These are based on the location, orientation, and velocity of the tire relative to the road. In turn, 
Adams/Tire calculates the forces and moments of the tire using the tire kinematics as inputs to the tire 
mode you select. 
Sign Conventions for Tire Outputs
The table below, Conventions for Naming Variables, and the figure, ISO Coordinate System, show the 
sign conventions for tire kinematic and force outputs.
Adams/Tire36
 
Conventions for Naming Variables
Variable name and 
abbreviation: Description:
Slip angle The angle formed between the direction of travel (velocity) of 
the center of the tire contact patch and the ISO-W: x-axis. If 
the wheel-travel direction has a component in the ISO-W: +y 
direction, a is positive. This produces a negative lateral force 
(Fy). Note that the steer angle, or the vehicle attitude angle, 
plays no part in defining the slip angle.
Inclination 
angle
The angle formed between the ISO-W: x-z plane and the 
wheel plane. If the wheel plane has a component lying in the 
direction of ISO-W, the inclination angle is positive. 
Longitudinal 
slip (Wactual-
Wfree)/Wfree
The ratio of the longitudinal-slip velocity of the contact patch 
to the longitudinal velocity of the wheel. The longitudinal slip 
is positive during acceleration of a moving tire and negative 
during braking. Longitudinal slip is limited to the range -1 to 
+1.
Longitudinal 
force at 
contact patch
Fx The x-component of the force exerted by the road or tire.
Lateral force 
at contact 
patch
Fy The y-component of the force exerted by the road or tire. 
Lateral force may be produced by one or any combination of 
the following: slip angle, inclination angle, conicity, or 
plysteer.
Normal force 
at contact 
patch
Fz The z-component of the force exerted by the road or tire. The 
direction of this force is up. 
Overturning 
moment at 
contact patch
Mx The moment of the forces at the contact patch acting on the 
tire by the road with respect to the ISO-W: x-axis.
Rolling 
resistance 
moment
My The moment of the forces at the contact patch acting on the 
tire by the road with respect to the y-axis.
Aligning 
moment
Mz The moment of the forces at the contact patch acting on the 
tire by the road with respect to the z-axis. 
Spin axis Spin Axis The axis about which the wheel rotates. Perpendicular to the 
wheel plane, not necessarily about the ISO-C: y-axis (only if 
inclination angle is zero).
The central 
plane of the 
Wheel plane The wheel plane is normal to the wheel spin axis.
α
γ
κ
tire and wheel
37
Learning Adams/Tire Basics
ISO Coordinate System
 
Wheel 
heading along 
road
ISO W:X This is not the same as the direction in which the wheel is 
traveling. If the tire reverses its direction, the axis system flips 
180 degrees about the z'-axis.
Direction to 
the left along 
the road
ISO W:Y The direction to the left along ground as viewed from behind 
a forward rolling tire. Expressed as right-hand orthogonal to 
the definitions of x' and z' (such as y = Z x X). 
Z-coordinate ISO W:Z Perpendicular to the road in the neighborhood of the origin of 
the tire axis system in a positive (downward) direction. (If the 
road is flat and in the x-y plane, this is negative global z.)
Variable name and 
abbreviation: Description:
Adams/Tire38
 
Units Supported in Tire Property Files
The following tables list the valid choices for the parameters in the UNITS section of a tire property file. 
Note the following:
• You must enter the choices in single quotes, such as 'METER' for meter. 
• The choices are case-insensitive. Therefore, 'MeTeR,' 'meter,' and 'METER' are all equivalent. 
• The strings are limited to 12 characters and the minimum abbreviation is shown in the tables. So, 
for example, 'millisecond' is valid and is interpreted as 'MILLI.’ 
Length Units
Time Units
Angle Units
Note: For some tire models, the [UNITS] section is not applied consistently to all tire parameters. 
The exceptions are the Magic Formula coefficients for the Pacejka ‘89 and ‘94 model and 
spline data for the 521 model, where the unit conversion factors have to be defined 
explicitly.
The unit: Can be abbreviated:
Kilometers 'KM' 
Meters 'METER' 
Centimeters 'CM' 
Millimeters 'MM'
Miles 'MILE' 
Feet 'FOOT' 
Inches 'IN'
The unit: Can be abbreviated:
Milliseconds 'MILLI'
Seconds 'SEC'
Minutes 'MIN'
Hours 'HOUR'
The units: Can be abbreviated:
Degrees 'DEG' 
Radians 'RAD'
39
Learning Adams/Tire Basics
Mass Units
Force Units
The unit: Can be abbreviated:
Kilograms 'KG'
Grams 'GRAM' 
Pound-Mass 'POUND_MASS' 
Kilo-Pound-Mass 'KPOUND_MASS' 
Slugs 'SLUG' 
Ounce-mass OUNCE_MASS' 
The unit: Can be abbreviated:
Kilograms-Force ‘KG_FORCE' 
Newtons 'NEWTON' or 'N'
Kilo-Newtons 'KNEWTON' or 'KN' 
Pounds-force 'POUND_FORCE'
Kilo-Pound-Force 'KPOUND_FORCE' 
Dynes (gram-cm/sec2) ‘DYNE' 
Ounce-force 'OUNCE_FORCE'
Adams/Tire40
 
Tire Models
Adams/Tire12
 
Using the Fiala Handling Force Model
This section of the help provides detailed technical reference material for defining tires on a mechanical 
system model using Adams/Tire. It assumes that you know how to run Adams/Car, Adams/Solver, or 
Adams/Chassis. It also assumes that you have a moderate level of tire-modeling proficiency. 
The Fiala tire model is the standard tire model that comes with all Adams/Tire modules. This chapter 
contains information for using the Fiala tire model:
• Assumptions
• Inputs
• Tire Slip Quantities and Transient Tire Behavior
• Force Evaluation
• Tire Carcass Shape
• Property File Format Example
• Contact Methods 
Fiala Tire Assumptions
The background of the Fiala tire model is a physical tire model, where the carcass is modeled as a beam 
on an elastic foundation in the lateral direction. Elastic brush elements provide the contact between 
carcass and road. Under these assumptions, analytical expressions for the steady-state slip characteristics 
can be derived, which are the basis for the calculation of the longitudinal and lateral forces in Adams.
• Rectangular contact patch or footprint.
• Pressure distribution uniform across contact patch.
• No tire relaxation effects are considered.
• Camber angle has no effect on tire forces.
Fiala Tire Inputs
The inputs to the Fiala tire model come from two sources:
• Input parameters from the tire property file (.tir), such as tire undeflected radius, that the tire 
references.
• Tire kinematic states, such as slip angle ( ), which Adams/Tire calculates.
The following table summarizes the input that the Fiala tire model uses to calculate force.
α
13
Tire Models
Input for Calculating Tire Forces
Quantity: Description: Use by Fiala: Source:
Mt Mass of tire • Damping 
• Vertical force (Fz) 
Alpha Slip angle Lateral force (Fy) Tire kinematic state 
from Adams/Solver
Ss Longitudinal slip ratio Longitudinal force (Fx) Tire kinematic state 
from Adams/Solver
pen Penetration (tire deflection) Vertical force (Fz) Tire kinematic state 
from Adams/Solver
Vpen d/dt (penetration) Vertical force (Fz) Tire kinematic state 
from Adams/Solver
Vertical_dampin
g
Vertical damping 
coefficient
• Damping
• Vertical force (Fz)
Tire property file (.tir)
Vertical_stiffness Vertical tire stiffness Vertical force (Fz) Tire property file (.tir)
CSLIP Partial derivative of 
longitudinal force (Fx) with 
respect to longitudinal slip 
ratio (S) at zero longitudinalslip
Longitudinal force (Fx) Tire property file (.tir)
CALPHA Partial derivative of lateral 
force (Fy) with respect to 
slip angle ( ) at zero slip 
angle
Lateral force (Fy) Tire property file (.tir)
UMIN Coefficient of friction with 
full slip (slip ratio = 1)
Fx, Fy, Tz Tire property file (.tir)
UMAX Coefficient of friction at 
zero slip
Fx, Fy, Tz Tire property file (.tir)
Rolling_resistanc
e
Rolling resistance 
coefficient
Rolling resistance moment 
(Ty)
Tire property file (.tir)
α
Adams/Tire14
 
Tire Slip Quantities and Transient Tire Behavior
Definition of Tire Slip Quantities
Slip Quantities at combined cornering and braking/traction
The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the 
wheel rotational velocity , and the loaded rolling radius Rl:
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road 
plane:
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point:
 and 
Note that for realistic tire forces the slip angle is limited to 900 and the longitudinal slip in 
between -1 (locked wheel) and 1.
Lagged longitudinal and lateral slip quantities (transient tire behavior)
In general, the tire rotational speed and lateral slip will change continuously because of the changing 
interaction forces in between the tire and the road. Often the tire dynamic response will have an important 
role on the overall vehicle response. For modeling this so-called transient tire behavior, a first-order 
system is used both for the longitudinal slip ? as the side slip angle, ?. Considering the tire belt as a 
stretched string, which is supported to the rim with lateral springs, the lateral deflection of the belt can 
be estimated (see also reference [1]). The figure below shows a top-view of the string model.
Ω Vsx Vx ΩRl–=
Vsy Vy=
κ α
κ VsxVx
--------–= αtan VsyVx
---------=
α κ
15
Tire Models
Stretched String Model for Transient Tire behavior
When rolling, the first point having contact with the road adheres to the road (no sliding assumed). 
Therefore, a lateral deflection of the string will arise that depends on the slip angle size and the history 
of the lateral deflection of previous points having contact with the road.
For calculating the lateral deflection v1 of the string in the first point of contact with the road, the 
following differential equation is valid during braking slip:
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger than 
10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the equation 
can be transformed to:
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill. 
A similar approach yields the following for the deflection of the string in longitudinal 
1
Vx
------
dv1
dt
--------
v1
σα
------+ αtan aφ+=
σα φ
σα
dv1
dt
-------- Vx v1+ σαVsy=
du1
direction:σκ dt-------- Vx u1+ σ– κVsx=
Adams/Tire16
 
Now the practical slip quantities, and , are defined based on the tire 
deformation:
These practical slip quantities and are used instead of the usual and definitions for steady-
state tire behavior.
The longitudinal and lateral relaxation length are read from the tire property file, see Fiala Tire Property 
File Format Example
Fiala Tire Force Evaluation
Types of force evaluation:
• Normal Force of Road on Tire
• Longitudinal Force
• Lateral Force
• Rolling Resistance Moment
• Aligning Moment
• Smoothing
Normal Force of Road on Tire
The normal force of a road on a tire at the contact patch in the SAE coordinates (+Z downward) is always 
negative (directed upward). The normal force is:
Fz = min (0.0, {Fzk + Fzc})
where:
• Fzk is the normal force due to tire vertical stiffness
• Fzc is the normal force due to tire vertical damping
• Fzk = - vertical_stiffness × pen
• Fzc = - vertical_damping × Vpen
Instead of the linear vertical tire stiffness, also an arbitrary tire deflection - load curve can be defined in 
the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Property File Format 
Example). If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection datap oints 
with a cubic spline for inter- and extrapolation are used for the calculation of the vertical force of the tire. 
Note that you must specify VERTICAL_STIFFNESS in the tire property file, but it does not play any 
κ' α'
κ' u1σκ
------ Vx( )sgn=
α' v1σα
------⎝ ⎠⎛ ⎞atan=
κ' α' κ α
role.
17
Tire Models
Longitudinal Force
The longitudinal force depends on the vertical force (Fz), the current coefficient of friction (U), the 
longitudinal slip ratio (Ss), and the slip angle (Alpha). The current coefficient of friction depends on the 
static (UMAX) and dynamic (UMIN) friction coefficients and the comprehensive slip ratio (SsAlpha).
UMAX specifies the tire/road coefficient of friction at zero slip and represents the static friction 
coefficient. This is the y-intercept on the friction coefficient versus slip graph. Note that this value is an 
unobtainable maximum friction value, because there is always slip within a footprint. This value is used 
in conjunction with UMIN to define a linear friction versus slip relation. UMAX will normally be larger 
than UMIN.
UMIN specifies the tire/road coefficient of friction for the full slip case and represents the sliding friction 
coefficient. This is the friction coefficient at 100% slip, or pure sliding. This value is used in conjunction 
with UMAZ to define a linear friction versus slip relationship.
The comprehensive slip ( ):
The current value coefficient of friction (U):
Fiala defines a critical longitudinal slip ( ):
This is the value of longitudinal slip beyond which the tire is sliding.
Case 1. Elastic Deformation State: |Ss| S_critical
Fx = -CSLIP × Ss
Case 2. Complete Sliding State: |Ss| S_critical
Fx = -sign(Ss)(Fx1- Fx2)
where:
 
 
Ssα
Ssα Ss2 tan2 α( )+=
U Umax Umax Umin–( ) Ssα⋅( )–=
Scritical
Scritical
U Fz⋅
2 CSLIP⋅-------------------------=
<
>
Fx1 U Fz⋅=
Fx2
U Fz⋅( )2
4 Ss CSLIP⋅ ⋅
--------------------------------------=
Adams/Tire18
 
Lateral Force
Like the longitudinal force, the lateral force depends on the vertical force (Fz) and the current coefficient 
of friction (U). And similar to the longitudinal force calculation, Fiala defines a critical lateral slip 
( ):
The lateral force peaks at a value equal to U × |Fz| when the slip angle (Alpha) equals the critical slip 
angle ( ).
Case 1. Elastic Deformation State: |Alpha| 
Fy = - U × |Fz|× (1-H3) × sign(Alpha)
where:
Case 2. Sliding State: |Alpha| Alpha_critical
Fy = -U|Fz|sign(Alpha)
Rolling Resistance Moment
When the tire is rolling forward: Ty = -rolling_resistance * Fz
When the tire is rolling backward: Ty = rolling_resistance * Fz
Aligning Moment
Case 1. Elastic Deformation State: |Alpha| 
Mz = U × |Fz|× WIDTH × (1-H) × H3 × sign(Alpha)
where:
Case 2. Complete Sliding State: |Alpha| 
Mz= 0.0
αcritical
αcritical
3 U Fz⋅ ⋅
CALPHA
-------------------------⎝ ⎠⎛ ⎞atan=
αcritical
αcritical≤
H 1 CALPHA α( )tan⋅3 U Fz⋅ ⋅
--------------------------------------------------–=
αcritical≤
H 1 CALPHA α( )tan⋅3 U Fz⋅ ⋅
--------------------------------------------------–=
αcritical>
19
Tire ModelsSmoothing
Adams/Tire can smooth initial transients in the tire force over the first 0.1 seconds of simulation. The 
longitudinal force, lateral force, and aligning torque are multiplied by a cubic step function of time. (See 
STEP in the Adams/Solver online help).
Longitudinal Force FLon = S*FLon
Lateral Force FLat = S*FLat
Aligning Torque Mz = S*Mz
The USE_MODE parameter in the tire property file allows you to switch smoothing on or off:
• USE_MODE = 1, smoothing is off
• USE_MODE = 2, smoothing is on
Fiala Tire Carcass Shape
Using Fiala tire, you can optionally supply a tire carcass cross-sectional shape in the tire property file in 
the [SHAPE] block. The 3D-durability, tire-to-road contact algorithm uses this information when 
calculating the tire-to-road volume of interference. To learn more about this topic, see Applying the Tire 
Carcass Shape. If you omit the [SHAPE] block from a tire property file, the tire carcass cross-section 
defaults to the rectangle that the tire radius and width define.
You specify the tire carcass shape by entering points in fractions of the tire radius and width. Because 
Adams/Tire assumes that the tire cross-section is symmetrical about the wheel plane, you only specify 
points for half the width of the tire. The following apply:
• For width, a value of zero (0) lies in the wheel center plane.
• For width, a value of one (1) lies in the plane of the side wall.
• For radius, a value of one (1) lies on the tread.
For example, suppose your tire has a radius of 300 mm and a width of 185 mm and that the tread is joined 
to the side wall with a fillet of 12.5 mm radius. The tread then begins to curve to meet the side wall at 
+/- 80 mm from the wheel center plane. If you define the shape table using six points with four points 
along the fillet, the resulting table might look like the shape block that is at the end of the following 
property format example.
Fiala Tire Property File Format Example
$---------------------------------------------------------MDI_HEADER
[MDI_HEADER]
 FILE_TYPE = 'tir'
 FILE_VERSION = 2.0
 FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'Tire - XXXXXX'
Adams/Tire20
 
'Pressure - XXXXXX'
'Test Date - XXXXXX'
'Test tire'
'New File Format v2.1'
$---------------------------------------------------------------units
[UNITS]
 LENGTH = 'mm'
 FORCE = 'newton'
 ANGLE = 'degree'
 MASS = 'kg'
 TIME = 'sec'
$--------------------------------------------------------------model
[MODEL]
! use mode 12 11 12 
! -------------------------------------------- 
! smoothingX X
! transient X X
!
 PROPERTY_FILE_FORMAT = 'FIALA'
 USE_MODE = 2.0
$----------------------------------------------------------dimension
[DIMENSION]
 UNLOADED_RADIUS = 309.9
 WIDTH = 235.0
 ASPECT_RATIO = 0.45
$----------------------------------------------------------parameter
[PARAMETER]
 VERTICAL_STIFFNESS = 310.0
 VERTICAL_DAMPING = 3.1
 ROLLING_RESISTANCE = 0.0
 CSLIP = 1000.0
 CALPHA = 800.0
 UMIN = 0.9
 UMAX = 1.0
 RELAX_LENGTH_X = 0.05
 RELAX_LENGTH_Y = 0.15
$---------------------------------------------carcass shape
[SHAPE]
{radius width}
1.0000 0.0000
1.0000 0.5000
1.0000 0.8649
0.9944 0.9235
0.9792 0.9819
0.9583 1.0000
$------------------------------------------------load_curve
$ Maximum of 100 points (optional)
[DEFLECTION_LOAD_CURVE]
{pen fz}
0 0.0
1 212.0
2 428.0
3 648.0
5 1100.0
21
Tire Models
10 2300.0
20 5000.0
30 8100.0
Fiala Tire Contact Methods
The Fiala tire model supports the following roads:
• 2D Roads, see Using the 2D Road Model.
• 3D Spline Roads, see Adams/3D Spline Road Model
• 3D Shell Roads, see Adams/Tire 3D Shell Road Model
Adams/Tire22
 
Using the PAC2002Tire Model
The PAC2002 Magic-Formula tire model has been developed by MSC.Software according to Tyre and 
Vehicle Dynamics by Pacejka [1]. PAC2002 is latest version of a Magic-Formula model available in 
Adams/Tire.
• Learn about:
• When to Use PAC2002
• Modeling of Tire-Road Interaction Forces
• Axis Systems and Slip Definitions
• Contact Point and Normal Load Calculation
• Basics of Magic Formula
• Steady-State: Magic Formula
• Transient Behavior
• Gyroscopic Couple
• Left and Right Side Tires
• USE_MODES OF PAC2002: from Simple to Complex
• Quality Checks for Tire Model Parameters
• Contact Methods
• Standard Tire Interface (STI)
• Definitions
• References
• Example of PAC2002 Tire Property File
• Contact Methods
When to Use PAC2002
Magic-Formula (MF) tire models are considered the state-of-the-art for modeling tire-road interaction 
forces in vehicle dynamics applications. Since 1987, Pacejka and others have published several versions 
of this type of tire model. The PAC2002 contains the latest developments that have been published in 
Tyre and Vehicle Dynamics by Pacejka [1].
In general, a MF tire model describes the tire behavior for rather smooth roads (road obstacle 
wavelengths longer than the tire radius) up to frequencies of 8 Hz. This makes the tire model applicable 
for all generic vehicle handling and stability simulations, including:
• Steady-state cornering
• Single- or double-lane change
• Braking or power-off in a turn
Adams/Tire12
 
• Split-mu braking tests
• J-turn or other turning maneuvers
• ABS braking, when stopping distance is important (not for tuning ABS control strategies)
• Other common vehicle dynamics maneuvers on rather smooth roads (wavelength of road 
obstacles must be longer than the tire radius)
For modeling roll-over of a vehicle, you must pay special attention to the overturning moment 
characteristics of the tire (Mx) and the loaded radius modeling. The last item may not be sufficiently 
accurate in this model.
The PAC2002 model has proven to be applicable for car, truck, and aircraft tires with camber 
(inclination) angles to the road not exceeding 15 degrees.
PAC2002 and Previous Magic Formula Models
Compared to previous versions, PAC2002 is backward compatible with all previous versions of 
PAC2002, MF-Tyre 5.x tire models, and related tire property files.
New Features
The enhancements for PAC2002 in Adams/Tire 2005 r2 are:
• More advanced tire-transient modeling using a contact mass in the contact point with the road. 
This results in more realistic dynamic tire model response at large slip, low speed, and standstill 
(usemode > 20).
• Parking torque and turn-slip have been introduced: the torque around the vertical axis due to 
turning at standstill or at low speed (no need for extra parameters).
• Extended loaded radius modeling (see Contact Point and Normal Load Calculation) are suitable 
for driving under extreme conditions like roll-over events and racing applications. 
• The option to use a nonlinear spline for the vertical tire load-deflection instead of a linear tire 
stiffness. See Contact Point and Normal Load Calculation.
• Modeling of bottoming of the tire to the road by using another spline for defining the bottoming 
forces. Learn more about wheel bottoming.
• Online scaling of the tire properties during a simulation; the scaling factors of the PAC2002 can 
now be changed as a function of time, position, or any other variable in your model dataset. See 
Online Scaling of Tire Properties. 
Modeling of Tire-Road Interaction Forces
For vehicle dynamics applications, an accurate knowledge of tire-road interaction forces is inevitable 
because the movements of a vehicle primarily depend on the road forces on the tires.These interaction 
forces depend on both road and tire properties, and the motion of the tire with respect to the road.
In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear 
damper with one point of contact with the road surface. The contact point is determined by considering 
the tire and wheel as a rigid disc. In the contact point between the tire and the road, the contact forces in 
13
longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the 
road.
The figure, Input and Output Variables of the Magic Formula Tire Model, presents the input and output 
vectors of the PAC2002 tire model. The tire model subroutine is linked to the Adams/Solver through the 
Standard Tire Interface (STI) [3]. The input through the STI consists of:
• Position and velocities of the wheel center
• Orientation of the wheel
• Tire model (MF) parameters
• Road parameters
The tire model routine calculates the vertical load and slip quantities based on the position and speed of 
the wheel with respect to the road. The input for the Magic Formula consists of the wheel load (Fz), the 
longitudinal and lateral slip ( , ), and inclination angle ( ) with the road. The output is the forces 
(Fx, Fy) and moments (Mx, My, Mz) in the contact point between the tire and the road. For calculating 
these forces, the MF equations use a set of MF parameters, which are derived from tire testing data.
The forces and moments out of the Magic Formula are transferred to the wheel center and returned to 
Adams/Solver through STI.
Input and Output Variables of the Magic Formula Tire Model
 
Axis Systems and Slip Definitions
κ α γ
• Axis Systems
Adams/Tire14
 
• Units
• Definition of Tire Slip Quantities
Axis Systems
The PAC2002 model is linked to Adams/Solver using the TYDEX STI conventions, as described in the 
TYDEX-Format [2] and the STI [3].
The STI interface between the PAC2002 model and Adams/Solver mainly passes information to the tire 
model in the C-axis coordinate system. In the tire model itself, a conversion is made to the W-axis system 
because all the modeling of the tire behavior as described in this help assumes to deal with the slip 
quantities, orientation, forces, and moments in the contact point with the TYDEX W-axis system. Both 
axis systems have the ISO orientation but have different origin as can be seen in the figure below.
TYDEX C- and W-Axis Systems Used in PAC2002, Source [2]
 
The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel to the road and in 
the wheel plane (xc-zc-plane). The origin of the C-axis system is the wheel center.
The origin of the W-axis system is the road contact-point defined by the intersection of the wheel plane, 
the plane through the wheel carrier, and the road tangent plane.
The forces and moments calculated by PAC2002 using the MF equations in this guide are in the W-axis 
system. A transformation is made in the source code to return the forces and moments through the STI 
to Adams/Solver.
The inclination angle is defined as the angle between the wheel plane and the normal to the road tangent 
plane (xw-yw-plane).
Units
The units of information transferred through the STI between Adams/Solver and PAC2002 are according 
to the SI unit system. Also, the equations for PAC2002 described in this guide have been developed for 
15
use with SI units, although you can easily switch to another unit system in your tire property file. Because 
of the non-dimensional parameters, only a few parameters have to be changed.
However, the parameters in the tire property file must always be valid for the TYDEX W-axis system 
(ISO oriented). The basic SI units are listed in the table below.
SI Units Used in PAC2002
Definition of Tire Slip Quantities
The longitudinal slip velocity Vsx in the contact point (W-axis system, see Slip Quantities at Combined 
Cornering and Braking/Traction) is defined using the longitudinal speed Vx, the wheel rotational velocity 
, and the effective rolling radius Re:
(1)
Slip Quantities at Combined Cornering and Braking/Traction
Variable type: Name: Abbreviation: Unit:
Angle Slip angle
Inclination angle
Radians
Force Longitudinal force
Lateral force
Vertical load
Fx
Fy
Fz
Newton
Moment Overturning moment
Rolling resistance moment
Self-aligning moment
Mx
My
Mz
Newton.meter
Speed Longitudinal speed
Lateral speed
Longitudinal slip speed
Lateral slip speed
Vx
Vy
Vsx
Vsy
Meters per second
Rotational speed Tire rolling speed Radians per second
α
γ
ω
ω
Vsx Vx ΩRe–=
Adams/Tire16
 
 
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
(2)
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point with:
(3)
(4)
The rolling speed Vr is determined using the effective rolling radius Re:
(5)
Turn-slip is one of the two components that form the spin of the tire. Turn-slip is calculated using the 
tire yaw velocity :
(6)
The total tire spin is calculated using:
(7)
The total tire spin has contributions of turn-slip and camber. denotes the camber reduction factor for 
Vsy Vy=
κ α
κ VsxVx
--------–=
αtan VsyVx
---------=
Vr ReΩ=
φ
ψ·
Wt
ψ·
Vx
------=
ϕ
ψ1 1Vx
------ ψ· 1 εγ–( )Ω γsin–{ }=
ε
the camber to become comparable with turn-slip.
17
Contact Point and Normal Load Calculation
• Contact Point
• Loaded and Effective Tire Rolling Radius
• Wheel Bottoming
Contact Point
In the vertical direction, the tire is modeled as a parallel linear spring and damper having one point of 
contact (C) with the road. This is valid for road obstacles with a wavelength larger than the tire radius 
(for example, for car tires 1m).
For calculating the kinematics of the tire relative to the road, the road is approximated by its tangent 
plane at the road point right below the wheel center (see the figure below).
Contact Point C: Intersection between Road Tangent Plane, Spin Axis Plane, and Wheel Plane
The contact point is determined by the line of intersection of the wheel center-plane with the road tangent 
(ground) plane and the line of intersection of the wheel center-plane with the plane through the wheel 
spin axis. The normal load Fz of the tire is calculated with the tire deflection as follows:ρ
Adams/Tire18
 
(8)
Using this formula, the vertical tire stiffness increases due to increasing rotational speed and decreases 
by longitudinal and lateral tire forces. If qFz1 is zero, qFz1 will be CzR0/Fz0.
When you do not provide the coefficients qV2, qFcx, qFcy, qFz1, qFz2 and qFc in the tire property file, the 
normal load calculation is compatible with previous versions of PAC2002, because, in that case, the 
normal load is calculated using the linear vertical tire stiffness Cz and tire damping Kz according to:
(9)
Instead of the linear vertical tire stiffness Cz (= qFz1Fz0/R0), you can define an arbitrary tire deflection - 
load curve in the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Example of 
PAC2002 Tire Property File). If a section called [DEFLECTION_LOAD_CURVE] exists, the load 
deflection data points with a cubic spline for inter- and extrapolation are used for the calculation of the 
vertical force of the tire. Note that you must specify Cz in the tire property file, but it does not play any 
role.
Loaded and Effective Tire Rolling Radius
With the loaded tire radius Rl defined as the distance of the wheel center to the contact point of the tire 
with the road, the tire deflection can be calculated using the free tire radiusR0 and a correction for the 
tire radius growth due to the rotational tire speed :
(10)
The effective rolling radius Re (at free rolling of the tire), which is used to calculate the rotational speed 
of the tire, is defined by:
(11)
For radial tires, the effective rolling radius is rather independent of load in its load range of operation 
because of the high stiffness of the tire belt circumference. Only at low loads does the effective tire radius 
decrease with increasing vertical load due to the tire tread thickness. See the figure below.
Fz 1 qV2 Ω
Ro
Vo
------ qFcxl
Fx
Fz0
--------⎝ ⎠⎛ ⎞
2
– qFcyl
Fy
Fz0
--------⎝ ⎠⎛ ⎞
2
– qFcylγ2+ +⎩ ⎭⎨ ⎬
⎧ ⎫
qFzl
ρ
R0
------ qFz2
ρ
R0
------⎝ ⎠⎛ ⎞
2
+ Fz0 Kz ρ·•+
=
ω
γ
Fz CzρλCz Kzρ·+=
ω
ρ R0 R1– qV1R0 Ω
R0
V0
------⎝ ⎠⎛ ⎞
2
+=
Re
Vx
Ω------=
19
Effective Rolling Radius and Longitudinal Slip 
To represent the effective rolling radius Re, a MF-type of equation is used:
(12)
in which Fz0 is the nominal tire deflection:
Rf R0 qV1R0
ΩR0
V0
-----------⎝ ⎠⎛ ⎞
2
RFz0 DPeffarc BReffρd( ) FReffρd+tan[ ]–+=
ρ
Adams/Tire20
 
(13)
and is called the dimensionless radial tire deflection, defined by:
(14)
Example of Loaded and Effective Tire Rolling Radius as Function of Vertical Load
Normal Load and Rolling Radius Parameters
Name:
Name Used in Tire Property 
File: Explanation:
Fz0 FNOMIN Nominal wheel load
Ro UNLOADED_RADIUS Free tire radius
B BREFF Low load stiffness effective rolling radius
D DREFF Peak value of effective rolling radius
F FREFF High load stiffness effective rolling radius
Cz VERTICAL_STIFFNESS Tire vertical stiffness (if qFz1=0)
ρFz0
Fz0
CzλCz
----------------=
ρ
ρd ρρFz0
---------=
Kz VERTICAL_DAMPING Tire vertical damping
21
Wheel Bottoming 
You can optionally supply a wheel-bottoming deflection, that is, a load curve in the tire property file in 
the [BOTTOMING_CURVE] block. If the deflection of the wheel is so large that the rim will be hit 
(defined by the BOTTOMING_RADIUS parameter in the [DIMENSION] section of the tire property 
file), the tire vertical load will be increased according to the load curve defined in this section. 
Note that the rim-to-road contact algorithm is a simple penetration method (such as the 2D contact) based 
on the tire-to-road contact calculation, which is strictly valid for only rather smooth road surfaces (the 
length of obstacles should have a wavelength longer than the tire circumference). The rim-to-road 
contact algorithm is not based on the 3D-volume penetration method, but can be used in combination 
with the 3D Contact, which takes into account the volume penetration of the tire itself. If you omit the 
[BOTTOMING_CURVE] block from a tire property file, no force due to rim road contact is added to the 
tire vertical force.
You can choose a BOTTOMING_RADIUS larger than the rim radius to account for the tire's material 
remaining in between the rim and the road, while you can adjust the bottoming load-deflection curve for 
the change in stiffness.
qFz1 QFZ1 Tire vertical stiffness coefficient (linear)
qFz2 QFZ2 Tire vertical stiffness coefficient 
(quadratic)
qFcx1 QFCX1 Tire stiffness interaction with Fx
qFcy1 QFCY1 Tire stiffness interaction with Fy
qFc 1 QFCG1 Tire stiffness interaction with camber
qV1 QV1 Tire radius growth coefficient
qV2 QV2 Tire stiffness variation coefficient with 
speed
Name:
Name Used in Tire Property 
File: Explanation:
γ
Adams/Tire22
 
If (Pentire - (Rtire - Rbottom) - ½·width ·| tan(g) |) < 0, the left or right side of the rim has contact with the 
road. Then, the rim deflection Penrim can be calculated using: 
 = max(0 , ½·width ·| tan( ) | ) + Pentire- (Rtire - Rbottom) δ γ
Penrim= 2/(2 · width ·| tan( ) |) δ γ
23
Srim= ½·width - max(width , /| tan( ) |)/3 
with Srim, the lateral offset of the force with respect to the wheel plane. 
If the full rim has contact with the road, the rim deflection is: 
Penrim = Pentire - (Rtire - Rbottom)
Srim = width2 · | tan( ) | · /(12 · Penrim) 
Using the load - deflection curve defined in the [BOTTOMING_CURVE] section of the tire property 
file, the additional vertical force due to the bottoming is calculated, while Srim multiplied by the sign of 
the inclination is used to calculate the contribution of the bottoming force to the overturning moment. 
Further, the increase of the total wheel load Fz due to the bottoming (Fzrim) will not be taken into account 
in the calculation for Fx, Fy, My, and Mz. Fzrim will only contribute to the overturning moment Mx using 
the Fzrim·Srim. 
Basics of the Magic Formula in PAC2002
The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics 
for the interaction forces between the tire and the road under several steady-state operating conditions. 
We distinguish:
• Pure cornering slip conditions: cornering with a free rolling tire
• Pure longitudinal slip conditions: braking or driving the tire without cornering
• Combined slip conditions: cornering and longitudinal slip simultaneously
For pure slip conditions, the lateral force Fy as a function of the lateral slip , respectively, and the 
longitudinal force Fx as a function of longitudinal slip , have a similar shape (see the figure, 
Characteristic Curves for Fx and Fy Under Pure Slip Conditions). Because of the sine - arctangent 
combination, the basic Magic Formula equation is capable of describing this shape:
(15)
where Y(x) is either Fx with x the longitudinal slip , or Fy and x the lateral slip .
δ γ
γ
γ
Note: Rtire is equal to the unloaded tire radius R0; Pentire is similar to effpen (= ).ρ
α
κ
Y x( ) D Carc Bx E Bx arc Bx( )tan–( )–{ }tan[ ]cos=
κ α
Adams/Tire24
 
Characteristic Curves for Fx and Fy Under Pure Slip Conditions 
The self-aligning moment Mz is calculated as a product of the lateral force Fy and the pneumatic trail t 
added with the residual moment Mzr. In fact, the aligning moment is due to the offset of lateral force Fy, 
called pneumatic trail t, from the contact point. Because the pneumatic trail t as a function of the lateral 
slip α has a cosine shape, a cosine version the Magic Formula is used:
(16)
in which Y(x) is the pneumatic trail t as function of slip angle .
The figure, The Magic Formula and the Meaning of Its Parameters, illustrates the functionality of the B, 
C, D, and E factor in the Magic Formula:
• D-factor determines the peak of the characteristic, and is called the peak factor.
• C-factor determines the part used of the sine and, therefore, mainly influences the shape of the 
curve (shape factor).
• B-factor stretches the curve and is called the stiffness factor.
• E-factor can modify the characteristic around the peak of the curve (curvature factor).
Y x( ) D Carc Bx E Bx arc Bx( )tan–( )–{ }tan[ ]cos=
α
25
The Magic Formula and the Meaning of Its Parameters 
In combined slip conditions, the lateral force Fy will decrease due to longitudinal slip or the opposite, the 
longitudinal force Fx will decrease due to lateral slip. The forces and moments in combined slip 
conditions are based on the pure slip characteristics multiplied by the so-called weighing functions. 
Again, these weighting functions have a cosine-shaped MF equation.
Adams/Tire26
 
The Magic Formula itself only describes steady-state tire behavior. For transient tire behavior (up to 8 
Hz), the MF output is used in a stretched string model that considers tire belt deflections instead of slip 
velocities to cope with standstill situations (zero speed).
Input Variables
The input variables to the Magic Formula are:
Output Variables
The output variables are defined in the W-axis system of TYDEX.
Basic TireParameters
All tire model parameters of the model are without dimension. The reference parameters for the model 
are:
As a measure for the vertical load, the normalized vertical load increment dfz is used:
(17)
with the possibly adapted nominal load (using the user-scaling factor, ):
Longitudinal slip [-]
Slip angle [rad]
Inclination angle [rad]
Normal wheel load Fz [N]
Longitudinal force Fx [N]
Lateral force Fy [N]
Overturning couple Mx [Nm]
Rolling resistance 
moment
My [Nm]
Aligning moment Mz [Nm]
Nominal (rated) load Fz0 [N]
Unloaded tire radius R0 [m]
Tire belt mass mbelt [kg]
κ
α
γ
fzd
Fz F'z0–
F'z0
--------------------=
γΦz0
27
(18)
Nomenclature of the Tire Model Parameters
In the subsequent sections, formulas are given with non-dimensional parameters aijk with the following 
logic:
Tire Model Parameters
User Scaling Factors
A set of scaling factors is available to easily examine the influence of changing tire properties without 
the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. 
You can change the factors in the tire property file. The peak friction scaling factors, and , are 
also used for the position-dependent friction in 3D Road Contact and 3D Road. An overview of all 
scaling factors is shown in the following tables.
Parameter: Definition:
a = p Force at pure slip
q Moment at pure slip
r Force at combined slip
s Moment at combined slip
i = B Stiffness factor
C Shape factor
D Peak value
E Curvature factor
K Slip stiffness = BCD
H Horizontal shift
V Vertical shift
s Moment at combined slip
t Transient tire behavior
j = x Along the longitudinal axis
y Along the lateral axis
z About the vertical axis
k = 1, 2, ...
F'z0 Fz0 λFz0•=
λμξ λμψ
Adams/Tire28
 
Scaling Factor Coefficients for Pure Slip
Name:
Name used in tire 
property file: Explanation:
Fzo
LFZO Scale factor of nominal (rated) load
Cz
LCZ Scale factor of vertical tire stiffness
Cx
LCX Scale factor of Fx shape factor
x
LMUX Scale factor of Fx peak friction coefficient
Ex
LEX Scale factor of Fx curvature factor
Kx
LKX Scale factor of Fx slip stiffness
Hx
LHX Scale factor of Fx horizontal shift
Vx
LVX Scale factor of Fx vertical shift
x
LGAX Scale factor of inclination for Fx
Cy
LCY Scale factor of Fy shape factor
y
LMUY Scale factor of Fy peak friction coefficient
Ey
LEY Scale factor of Fy curvature factor
Ky
LKY Scale factor of Fy cornering stiffness
Hy
LHY Scale factor of Fy horizontal shift
Vy
LVY Scale factor of Fy vertical shift
gy
LGAY Scale factor of inclination for Fy
t
LTR Scale factor of peak of pneumatic trail
Mr
LRES Scale factor for offset of residual moment
LGAZ Scale factor of inclination for Mz
Mx
LMX Scale factor of overturning couple
VMx
LVMX Scale factor of Mx vertical shift
My
LMY Scale factor of rolling resistance moment
λ
λ
λ
λμ
λ
λ
λ
λ
λγ
λ
λμ
λ
λ
λ
λ
λ
λ
λ
λγz
λ
λ
λ
29
Scaling Factor Coefficients for Combined Slip
Scaling Factor Coefficients for Transient Response
Note that the scaling factors change during the simulation according to any user-introduced function. See 
the next section, Online Scaling of Tire Properties.
Online Scaling of Tire Properties
PAC2002 can provide online scaling of tire properties. For each scaling factor, a variable should be 
introduced in the Adams .adm dataset. For example:
!lfz0 scaling
! adams_view_name='TR_Front_Tires until wheel_lfz0_var'
VARIABLE/53
, IC = 1
, FUNCTION = 1.0
This lets you change the scaling factor during a simulation as a function of time or any other variable in 
your model. Therefore, tire properties can change because of inflation pressure, road friction, road 
temperature, and so on. 
You can also use the scaling factors in co-simulations in MATLAB/Simulink. 
For more detailed information, see Knowledge Base Article 12732.
Name:
Name used in tire 
property file: Explanation:
LXAL Scale factor of alpha influence on Fx
LYKA Scale factor of alpha influence on Fx
LVYKA Scale factor of kappa-induced Fy
LS Scale factor of moment arm of Fx
Name:
Name used in tire 
property file: Explanation:
σκ LSGKP Scale factor of relaxation length of Fx
σα LSGAL Scale factor of relaxation length of Fy
gyr
LGYR Scale factor of gyroscopic moment
λxα
λyκ
λVyκ
λs
λ
λ
λ
Adams/Tire30
 
Steady-State: Magic Formula in PAC2002
• Steady-State Pure Slip
• Steady-State Combined Slip
Steady-State Pure Slip
• Longitudinal Force at Pure Slip
• Lateral Force at Pure Slip
• Aligning Moment at Pure Slip
• Turn-slip and Parking
Formulas for the Longitudinal Force at Pure Slip
For the tire rolling on a straight line with no slip angle, the formulas are:
(19)
(20)
(21)
(22)
with following coefficients:
(23)
(24)
(25)
(26)
the longitudinal slip stiffness:
(27)
(28)
(29)
Fx Fx0 κ Fz γ, ,( )=
Fx0 Dx Cxarc Bxκx Ex Bxκx arc Bxκx( )tan–( )–{ }tan[ ] SVx+( )sin=
κx κ SHx+=
γx γ λγx⋅=
Cx pCx1 λCx⋅=
Dx μx Fz ζ1⋅ ⋅=
μx pDx1 pDx2 fzd+( ) 1 pDx3 γ2⋅–( )λμx⋅=
Ex pEx1 pEx2 fz
2d+( ) 1 pEx4 κx( )sgn–{ } λEx with Ex 1≤⋅ ⋅=
Kx Fz pKx1 pKx2 fzd+( ) pKx3 fzd( ) λKx⋅exp⋅ ⋅=
Kx BxCxDx κx∂
∂Fx0at κ 0= = =
Bx Kx CxDx( )⁄=
S p p df⋅+( )λ=
Hx Hx1 Hx2 z Hx
31
(30)
Longitudinal Force Coefficients at Pure Slip
Formulas for the Lateral Force at Pure Slip
(31)
(32)
(33)
The scaled inclination angle:
(34)
with coefficients:
(35)
(36)
Name:
Name used in tire 
property file: Explanation:
pCx1 PCX1 Shape factor Cfx for longitudinal force
pDx1 PDX1 Longitudinal friction Mux at Fznom
pDx2 PDX2 Variation of friction Mux with load
pDx3 PDX3 Variation of friction Mux with inclination
pEx1 PEX1 Longitudinal curvature Efx at Fznom
pEx2 PEX2 Variation of curvature Efx with load
pEx3 PEX3 Variation of curvature Efx with load squared
pEx4 PEX4 Factor in curvature Efx while driving
pKx1 PKX1 Longitudinal slip stiffness Kfx/Fz at Fznom
pKx2 PKX2 Variation of slip stiffness Kfx/Fz with load
pKx3 PKX3 Exponent in slip stiffness Kfx/Fz with load
pHx1 PHX1 Horizontal shift Shx at Fznom
pHx2 PHX2 Variation of shift Shx with load
pVx1 PVX1 Vertical shift Svx/Fz at Fznom
pVx2 PVX2 Variation of shift Svx/Fz with load
SVx Fz pHx1 pHx2 dfz⋅+( )λVx λHμx ζ1⋅ ⋅⋅=
Fy Fy0 α γ Fz, ,( )=
Fy0 Dy Cyarc Byαy Ey Byαy arc Byαy( )tan–( )–{ }tan[ ] SVy+sin=
αy α SHy+=
γy γ λγy⋅=
Cy pCy1 λCy⋅=
D μ F ζ⋅ ⋅=
y y z 2
Adams/Tire32
 
(37)
(38)
The cornering stiffness:
(39)
(40)
(41)
(42)
(43)
The camber stiffness is given by:
(44)
Lateral Force Coefficients at Pure Slip
Name:
Name used in tire 
property file: Explanation:
pCy1 PCY1 Shape factor Cfy for lateral forces
pDy1 PDY1 Lateral friction Muy
pDy2 PDY2 Variation of friction Muy with load
pDy3 PDY3 Variation of friction Muy with squared inclination
pEy1 PEY1 Lateral curvature Efy at Fznom
pEy2 PEY2 Variation of curvature Efy with load
pEy3 PEY3 Inclination dependency of curvature Efy
pEy4 PEY4 Variation of curvature Efy with inclination
pKy1 PKY1 Maximum value of stiffness Kfy/Fznom
pKy2 PKY2 Load at which Kfy reaches maximum value
pKy3 PKY3 Variation of Kfy/Fznom with inclination
pHy1 PHY1 Horizontal shift Shy at Fznom
pHy2 PHY2 Variation of shift Shy with load
μy pDy1 pDy2dfz+( ) 1 pDy3γy2–( ) λμy⋅ ⋅=
Ey pEy1 pEy2dfz+( ) 1 pEy3 pEy4γy+( ) αy( )sgn–{ } γEy with Ey 1≤⋅ ⋅=
Ky0 PKy1 Fz0 2ac
Fz
PKy2F0λFz0
---------------------------
⎩ ⎭⎨ ⎬
⎧ ⎫
tan λFz0 λKy⋅ ⋅⎝ ⎠⎜ ⎟
⎛ ⎞
sin⋅ ⋅=
Ky Ky0 1 pKy3 γy–( ) ζ3⋅ ⋅=
By Ky CyDy( )⁄=
SHy pHy1 pHy2dfz+( ) λHy pHy3γy ζ0 ζ4 1–+⋅+⋅=
SVy Fz pVy1 pVy2dfz+( ) λVy pVy3 pVy4dfz+( ) γy⋅+⋅{ } λμy ζ4⋅ ⋅ ⋅=
Kyγ0 PHy3Ky0 Fz pνy3 pνy4dfz+( )+=33
Formulas for the Aligning Moment at Pure Slip
(45)
with the pneumatic trail t:
(46)
(47)
and the residual moment Mzr:
(48)
(49)
(50)
The scaled inclination angle:
(51)
with coefficients:
(52)
(53)
(54)
pHy3 PHY3 Variation of shift Shy with inclination
pVy1 PVY1 Vertical shift in Svy/Fz at Fznom
pVy2 PVY2 Variation of shift Svy/Fz with load
pVy3 PVY3 Variation of shift Svy/Fz with inclination
pVy4 PVY4 Variation of shift Svy/Fz with inclination and load
Name:
Name used in tire 
property file: Explanation:
M'z Mz0 α γ Fz, ,( )=
Mz0 t Fy0 Mzr+⋅–=
t αt( ) Dt Ctarc Btαt Et Btαt arc Btαt( )tan–( )–{ }tan[ ] α( )coscos=
αt α SHt+=
Mzr αr( ) Dr Crarc Brαr( )tan[ ] α( )cos⋅cos=
αr α SHf+=
SHf SHy SVy Ky⁄+=
γz γ λγz⋅=
Bt qBz1 qBz2dfz qBz3dfz
2+ +( ) 1 qBz4γz qBz5 γz+ +( ) λKy λμy⁄⋅ ⋅=
Ct qCz1=
Dt Fz qDz1 qDz2dfz+( ) 1 qDz3γz qDz4γz2+ +( )
R0
Fz0
-------- λt ζ5⋅ ⋅ ⋅ ⋅ ⋅=
Adams/Tire34
 
(55)
(56)
(57)
(58)
An approximation for the aligning moment stiffness reads:
(59)
Aligning Moment Coefficients at Pure Slip
Name:
Name used in tire 
property file: Explanation:
qBz1 QBZ1 Trail slope factor for trail Bpt at Fznom
qBz2 QBZ2 Variation of slope Bpt with load
qBz3 QBZ3 Variation of slope Bpt with load squared
qBz4 QBZ4 Variation of slope Bpt with inclination
qBz5 QBZ5 Variation of slope Bpt with absolute inclination
qBz9 QBZ9 Slope factor Br of residual moment Mzr
qBz10 QBZ10 Slope factor Br of residual moment Mzr
qCz1 QCZ1 Shape factor Cpt for pneumatic trail
qDz1 QDZ1 Peak trail Dpt = Dpt*(Fz/Fznom*R0)
qDz2 QDZ2 Variation of peak Dpt with load
qDz3 QDZ3 Variation of peak Dpt with inclination
qDz4 QDZ4 Variation of peak Dpt with inclination squared.
qDz6 QDZ6 Peak residual moment Dmr = Dmr/ (Fz*R0)
qDz7 QDZ7 Variation of peak factor Dmr with load
Et qEz1 qEz2dfz qEz3dfz
2+ +( )=
1 qEz1 qEz2γz+( ) 2π---⎝ ⎠⎛ ⎞ arc Bt Ct αt⋅ ⋅( )tan⋅⎝ ⎠⎛ ⎞+⎩ ⎭⎨ ⎬
⎧ ⎫
w ith Et 1≤( )
SHt qHz1 qHz2dfz qHz3 qHz4 dfz⋅+( )γz+ +=
Br qBz9
λKy
λμy
--------- qBz10 By Cy⋅ ⋅+⋅⎝ ⎠⎛ ⎞ ζ6⋅=
Cr ζ7=
Dr Fz qDz6 qDz7dfz+( ) γr⋅ qDz8 qDz9dfz+( ) γz⋅+[ ] Ro λμγ ζ8 1–+⋅ ⋅ ⋅=
Kz t Ky α∂
∂– Mz at α≈⎝ ⎠⎛ ⎞⋅– 0 )= =
qDz8 QDZ8 Variation of peak factor Dmr with inclination
35
Turn-slip and Parking
For situations where turn-slip may be neglected and camber remains small, the reduction factors that 
appear in the equations for steady-state pure slip, are to be set to 1:
For larger values of spin, the reduction factors are given below.
The weighting function is used to let the longitudinal force diminish with increasing spin, according 
to:
with:
The peak side force reduction factor reads:
 
with:
The cornering stiffness reduction factor is given by:
qDz9 QDZ9 Variation of Dmr with inclination and load
qEz1 QEZ1 Trail curvature Ept at Fznom
qEz2 QEZ2 Variation of curvature Ept with load
qEz3 QEZ3 Variation of curvature Ept with load squared
qEz4 QEZ4 Variation of curvature Ept with sign of Alpha-t
qEz5 QEZ5 Variation of Ept with inclination and sign Alpha-t
qHz1 QHZ1 Trail horizontal shift Sht at Fznom
qHz2 QHZ2 Variation of shift Sht with load
qHz3 QHZ3 Variation of shift Sht with inclination
qHz4 QHZ4 Variation of shift Sht with inclination and load
Name:
Name used in tire 
property file: Explanation:
ζi
ζi 1= i 0.= 1.…8
ζ1
ζi arc BxϕR0ϕ( )tan[ ]cos=
Bxϕ pDxϕ1 1 pDxϕ2dfz+( ) arc pDxϕ3κ( )tan[ ]cos=
ζ2
ζ2 arc Byϕ R0 ϕ pDyϕ4 R0 ϕ+( ){ }tan[ ]cos=
Byϕ pDxϕ1 1 pDxϕ2dfz+( ) arc pDxϕ3 αtan( )tan[ ]cos=
ζ
3
Adams/Tire36
 
The horizontal shift of the lateral force due to spin is given by:
The factors are defined by:
The spin force stiffness KyRϕ0 is related to the camber stiffness Kyy0:
in which the camber reduction factor is given by:
The reduction factors and for the vertical shift of the lateral force are given by:
The reduction factor for the residual moment reads:
The peak spin torque Dr is given by:
ζ3 arc pKyϕ1R02ϕ2( )tan[ ]cos=
SHyϕ DHyϕ CHyϕarc BHyϕRoϕ EHyϕ BHyϕ arc BHyϕR0ϕ( )tan–( )–{ }tan[ ]sin=
CHyϕ pHyϕ1
DHyϕ pHyϕ2 pHyϕ3dfz+( ) Vx( )
EHyϕ
sin⋅
PHyϕ4
BHyϕ
KyRϕ0
CyDyKy0
-----------------------
=
=
=
=
KyRϕ0
Kyγ0
1 εγ–
-------------=
εγ pεγϕ1 1 pεγϕ2dfz+( )=
ζ0 ζ4
ζ0 0
ζ4 1 SHyϕ SVyγ Ky⁄–+
=
=
ζ8 1 Drϕ+=
ϕ
Drϕ DDrϕ e CDrϕarc BDrϕR0ϕ EDrϕ BDrϕR0ϕ arc BDrϕR0ϕ( )tan–( )–{ }tan[ ]sin=
The maximum value is given by:
37
The moment at vanishing wheel speed at constant turning is given by:
The shape factors are given by:
in which:
The reduction factor reads:
 
The spin moment at 90º slip angle is given by:
The spin moment at 90º slip angle is multiplied by the weighing function to account for the action 
of the longitudinal slip (see steady-state combined slip equations).
The reduction factor is given by:
DDrϕ
Mzϕ∞
π
2
---CDrϕ⎝ ⎠⎛ ⎞sin
-----------------------------=
Mzϕ∞ qCrϕ1μyR0Fz Fz Fz0⁄=
CDrϕ qDrϕ1
EDrϕ qDrϕ2
BDrϕ
Kzγr0
CDrϕDDrϕ 1 εy–( )
--------------------------------------------
=
=
=
Kzγr0 FzR0 qDz8 qDz9dfz+( )=
ζ6
ζ6 arc qBrϕ1R0ϕ( )tan[ ]cos=
Mzϕ90 Mzϕ∞
2
π--- arc qCrϕ2R0ϕ( ) Gyx κ( )⋅tan⋅ ⋅=
Gyκ
ζ7
ζ7 2π--- arc Mzϕ90 DDrϕ⁄[ ]cos⋅=
Adams/Tire38
 
Turn-Slip and Parking Parameters
The tire model parameters for turn-slip and parking are estimated automatically. In addition, you can 
specify each parameter individually in the tire property file (see example).
Steady-State Combined Slip
PAC2002 has two methods for calculating the combined slip forces and moments. If the user supplies the 
coefficients for the combined slip cosine 'weighing' functions, the combined slip is calculated according 
to Combined slip with cosine 'weighing' functions (standard method). If no coefficients are supplied, the 
Name:
Name used in 
tire property file: Explanation:
p 1 PECP1 Camber spin reduction factor parameter in camber stiffness
p 2 PECP2 Camber spin reduction factor varying with load parameter in 
camber stiffness
pDx 1 PDXP1 Peak Fx reduction due to spin parameter
pDx 2 PDXP2 Peak Fx reduction due to spin with varying load parameter
pDx 3 PDXP3 Peak Fx reduction due to spin with kappa parameter
pDy 1 PDYP1 Peak Fy reduction due to spin parameter
pDy 2 PDYP2 Peak Fy reduction due to spin with varying load parameter
pDy 3 PDYP3 Peak Fy reduction due to spin with alpha parameter
pDy 4 PDYP4 Peak Fy reduction due to square root of spin parameter
pKy 1 PKYP1 Cornering stiffness reduction due to spin
pHy 1 PHYP1 Fy-alpha curve lateral shift limitation
pHy 2 PHYP2 Fy-alpha curve maximum lateral shift parameter
pHy 3 PHYP3 Fy-alpha curve maximum lateral shift varying with load 
parameter
pHy 4 PHYP4 Fy-alpha curve maximum lateral shift parameter
qDt 1 QDTP1 Pneumatic trail reduction factor due to turn slip parameter
qBr 1 QBRP1 Residual (spin) torque reduction factor parameter due to side 
slip
qCr 1 QCRP1 Turning moment at constant turning and zero forward speed 
parameter
qCr 2 QCRP2 Turn slip moment (at alpha=90deg) parameter for increase 
with spin
qDr 1 QDRP1 Turn slip moment peak magnitude parameter
qDr 2 QDRP2 Turn slip moment peak position parameter
εϕ
εϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
39
so-called friction ellipse is used to estimate the combined slip forces and moments, see section Combined 
Slip with friction ellipse
Combined slip with cosine 'weighing' functions
• Longitudinal Force at Combined Slip
• Lateral Force at Combined Slip
• Aligning Moment at Combined Slip
• Overturning Moment at Pure and Combined Slip
• Rolling Resistance Moment at Pure and Combined Slip
Formulas for the Longitudinal Force at Combined Slip
(60)
with Gx the weighting function of the longitudinal force for pure slip.We write:
(61)
(62)
with coefficients:
(63)
(64)
(65)
(66)
(67)
The weighting function follows as:
(68)
Fx Fx0 Gxα α κ Fz, ,( )⋅=
α
Fx Dxα Cxαarc Bxααs Exα Bxααs arc Bxααs( )tan–( )–{ }tan[ ]cos=
αs α SHxα+=
Bxα rBx1 arc rBx2κ{ }tan[ ] λxα⋅cos=
Cxα
Dxα
Fxo
Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–{ }tan[ ]cos
------------------------------------------------------------------------------------------------------------------------------------------------------------------=
Exα rEx1 rEx2dfz with Exα 1≤+=
SHxα rHx1=
Gxα
Cxαarc Bxααs Exα Bxααs arc Bxααs( )tan–( )–{ }tan[ ]cos
Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–[ ]tan[ ]cos
----------------------------------------------------------------------------------------------------------------------------------------------------------------=
Adams/Tire40
 
Longitudinal Force Coefficients at Combined Slip
Formulas for Lateral Force at Combined Slip
(69)
with Gyk the weighting function for the lateral force at pure slip and SVyk the ' -induced' side force; 
therefore, the lateral force can be written as:
(70)
(71)
with the coefficients:
(72)
(73)
(74)
(75)
(76)
(77)
Name:
Name used in tire 
property file: Explanation:
rBx1 RBX1 Slope factor for combined slip Fx reduction
rBx2 RBX2 Variation of slope Fx reduction with kappa
rCx1 RCX1 Shape factor for combined slip Fx reduction
rEx1 REX1 Curvature factor of combined Fx
rEx2 REX2 Curvature factor of combined Fx with load
rHx1 RHX1 Shift factor for combined slip Fx reduction
Fy Fy0 Gyκ α κ γ Fz, , ,( )⋅=
Fy Dyκ Cyκarc Byκκs Eyκ Byκκs arc Byκκs( )tan–( )–{ }tan[ ] SVyκ+cos=
κs κ SHyκ+=
Byκ rBy1 arc rBy2 α rBy3–( ){ }tan[ ] λyκ⋅cos=
Cyκ rCy1=
Dyκ
Fyo
Cyκarc ByκSHyκ Eyκ ByκSHyκ arc ByκSHyκ( )tan–( )–{ }tan[ ]cos
---------------------------------------------------------------------------------------------------------------------------------------------------------------=
Eyκ rEy1 rEy2dfz with Eyκ 1≤+=
SHyκ rHy1 rHy2dfz+=
SVyκ DVyκ rVy5arc rVy6κ( )tan[ ]sin=
41
(78)
The weighting function appears is defined as:
(79)
Lateral Force Coefficients at Combined Slip
Formulas for Aligning Moment at Combined Slip
(80)
with:
(81)
Name:
Name used in 
tire property file: Explanation:
rBy1 RBY1 Slope factor for combined Fy reduction
rBy2 RBY2 Variation of slope Fy reduction with alpha
rBy3 RBY3 Shift term for alpha in slope Fy reduction
rCy1 RCY1 Shape factor for combined Fy reduction
rEy1 REY1 Curvature factor of combined Fy
rEy2 REY2 Curvature factor of combined Fy with load
rHy1 RHY1 Shift factor for combined Fy reduction
rHy2 RHY2 Shift factor for combined Fy reduction with load
rVy1 RVY1 Kappa induced side force Svyk/Muy*Fz at Fznom
rVy2 RVY2 Variation of Svyk/Muy*Fz with load
rVy3 RVY3 Variation of Svyk/Muy*Fz with inclination
rVy4 RVY4 Variation of Svyk/Muy*Fz with alpha
rVy5 RVY5 Variation of Svyk/Muy*Fz with kappa
rVy6 RVY6 Variation of Svyk/Muy*Fz with atan (kappa)
DVyκ μyFz rVy1 rVy2dfz rVy3γ+ +( ) arc rVy4α( )tan[ ]cos⋅ ⋅=
Gyκ
Cyκarc Byκκs Eyκ Byκκsarc Byκκs( )tan( )–{ }tan[ ]cos
Cyκarc ByκSHyκ Eyκ ByκSHyκarc ByκSHyκ( )tan( )–{ }tan[ ]cos
---------------------------------------------------------------------------------------------------------------------------------------------------------=
M'z t Fy' Mzr s Fx⋅+ +⋅–=
t t αt eq,( )=
Adams/Tire42
 
(82)
(83)
(84)
(85)
with the arguments:
(86)
(87)
Aligning Moment Coefficients at Combined Slip
Formulas for Overturning Moment at Pure and Combined Slip
For the overturning moment, the formula reads both for pure and combined slip situations:
(88)
Name:
Name used in 
tire property file: Explanation:
ssz1 SSZ1 Nominal value of s/R0 effect of Fx on Mz
ssz2 SSZ2 Variation of distance s/R0 with Fy/Fznom
ssz3 SSZ3 Variation of distance s/R0 with inclination
ssz4 SSZ4 Variation of distance s/R0 with load and inclination
Dt Ctarc Btαt eq, Et Btαt eq, arc Btαt eq,( )tan–( )–{ }tan[ ] α( )coscos=
F'y γ, 0= Fy SVyκ–=
Mzr Mzr αr eq,( ) Dr arc Brαr eq,( )tan[ ] α( )coscos= =
t t αt eq,( )=
αt eq, arc α2 t
Kx
Ky
------⎝ ⎠⎛ ⎞
2κ2 αt( )sgn⋅+tantan=
αr eq, arc α2 r
Kx
Ky
------⎝ ⎠⎛ ⎞
2κ2 αr( )sgn⋅+tantan=
Mx Ro Fz qSx1λVMx qSx2 λ qSx3
Fy
Fz0
--------⋅+⋅–
⎩ ⎭⎨ ⎬
⎧ ⎫λMx⋅ ⋅=
43
Overturning Moment Coefficients
Formulas for Rolling Resistance Moment at Pure and Combined Slip
The rolling resistance moment is defined by:
(89)
If qsy1 and qsy2 are both zero and FITTYP is equal to 5 (MF-Tyre 5.0), then the rolling resistance is 
calculated according to an old equation:
(90)
Rolling Resistance Coefficients
Combined Slip with friction ellipse
In case the tire property file does not contain the coefficients for the 'standard' combined slip method 
(cosine 'weighing functions), the friction ellipse method is used, as described in this section. Note that 
the method employed here is not part of one of the Magic Formula publications by Pacejka, but is an in-
house development of MSC.Software. 
Name:
Name used in tire 
property file: Explanation:
qsx1 QSX1 Lateral force induced overturning couple
qsx2 QSX2 Inclination induced overturning couple
qsx3 QSX3 Fy induced overturning couple
Name:
Name used in tire 
property file: Explanation:
qsy1 QSY1 Rolling resistance moment coefficient
qsy2 QSY2 Rolling resistance moment depending on Fx
qsy3 QSY3 Rolling resistance moment depending on speed
qsy4 QSY4 Rolling resistance moment depending on speed^4
Vref LONGVL Measurement speed
My Ro Fz qSy1 qSy3Fx Fz0⁄ qSy3 Vx Vref( )⁄ qSy4Vx Vref( )⁄ 4+ + +{ }⋅ ⋅=
My R0 SVx Kx SHx⋅+( )=
κc κ SHx
SVx
Kx
---------+ +=
Adams/Tire44
 
The following friction coefficients are defined:
The forces corrected for the combined slip conditions are: 
For aligning moment Mx, rolling resistance My and aligning moment Mz the formulae (76) until and 
αc α SHy
SVy
Ky
---------+ +=
α∗ αc( )sin=
β κc
κc2 α∗2+
-------------------------⎝ ⎠⎜ ⎟
⎛ ⎞
acos=
μx act,
Fx 0, SVx–
Fz
-------------------------= μy act,
Fy 0, SVy–
Fz
-------------------------=
μx max,
Dx
Fz
------= μy max,
Dy
Fz
------=
μx 1
1
μx act,
-------------⎝ ⎠⎛ ⎞
2 βtan
μy max,
----------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------------=
μy βtan
1
μx max,
----------------⎝ ⎠⎛ ⎞
2 βtan
μy act,
-------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------------=
Fx
μx
μx act,
-------------Fx 0,= Fy
μy
μy act,
-------------Fy 0,=
including (85) are used with =0.SVyκ
45
Transient Behavior in PAC2002
The previous Magic Formula equations are valid for steady-state tire behavior. When driving, however, 
the tire requires some response time on changes of the inputs. In tire modeling terminology, the low-
frequency behavior (up to 15 Hz) is called transient behavior. PAC2002 provides two methods to model 
transient tire behavior:
• Stretched String 
• Contact Mass 
Stretched String Model
For accurate transient tire behavior, you can use the stretched string tire model (see reference [1]). The 
tire belt is modeled as stretched string, which is supported to the rim with lateral (and longitudinal) 
springs. Stretched String Model for Transient Tire Behavior shows a top-view of the string model. When 
rolling, the first point having contact with the road adheres to the road (no sliding assumed). Therefore, 
a lateral deflection of the string arises that depends on the slip angle size and the history of the lateral 
deflection of previous points having contact with the road.
Stretched String Model for Transient Tire Behavior
For calculating the lateral deflection v1 of the string in the first point of contact with theroad, the 
following differential equation is valid:
Adams/Tire46
 
(91)
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger 
than 10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the 
equation can be transformed to:
(92)
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill.
A similar approach yields the following for the deflection of the string in longitudinal direction:
(93)
Both the longitudinal and lateral relaxation length are defined as of the vertical load:
(94)
(95)
Now the practical slip quantities, and , are defined based on the tire deformation:
(96)
(97)
Using these practical slip quantities, and , the Magic Formula equations can be used to calculate the 
tire-road interaction forces and moments:
(98)
(99)
1
Vx
------
td
dv1 v1
σα
------+ α( ) aφ+tan=
σα φ
σα td
dv1 Vx v1+ σαVsy=
σx td
du1 Vx u1+ σxVsx=
σx Fz pTx1 pTx2dfz+( ) pTx3dfz( ) R0 Fz0⁄( )λσx⋅exp⋅ ⋅=
σα pTy1Fz0 2arc
Fz
pTy2Fz0λFz0( )
---------------------------------
⎩ ⎭⎨ ⎬
⎧ ⎫
tan 1 pKy3 γy–( ) R0λFz0 λσα⋅⋅ ⋅sin=
κ α
κ' u1σx
------ Vx( )sin⋅=
α' v1σα
------⎝ ⎠⎛ ⎞atan=
κ α
Fx Fx α' κ' Fz, ,( )=
Fy Fy α' κ' γ Fz, , ,( )=
47
(100)
Coefficients and Transient Response
Contact Mass Model
The contact mass model is based on the separation of the contact patch slip properties and the tire carcass 
compliance (see reference [1]). Instead of using relaxation lengths to describe compliance effects, the 
carcass springs are explicitly incorporated in the model. The contact patch is given some inertia to ensure 
computational causality. This modeling approach automatically accounts for the lagged response to slip 
and load changes that diminish at higher levels of slip. The contact patch itself uses relaxation lengths to 
handle simulations at low speed.
The contact patch can deflect in longitudinal, lateral, and yaw directions with respect to the lower part 
of the wheel rim. A mass is attached to the contact patch to enable straightforward computations.
The differential equations that govern the dynamics of the contact patch body are:
The contact patch body with mass mc and inertia Jc is connected to the wheel through springs cx, cy, and 
c and dampers kx, ky, and k in longitudinal, lateral, and yaw direction, respectively.
The additional equations for the longitudinal u, lateral v, and yaw deflections are:
Name:
Name used in tire 
property file: Explanation:
pTx1 PTX1 Longitudinal relaxation length at Fznom
pTx2 PTX2 Variation of longitudinal relaxation length with load
pTx3 PTX3 Variation of longitudinal relaxation length with exponent 
of load
pTy1 PTY1 Peak value of relaxation length for lateral direction
pTy2 PTY2 Shape factor for lateral relaxation length
qTz1 QTZ1 Gyroscopic moment constant
Mbelt MBELT Belt mass of the wheel
M'z M'z α' κ' γ Fz, , ,( )=
mc V
·
cx Vcyψ· c–( ) kxu· cxu+ + F=
mc V
·
cy Vcxψ· c–( ) kyu· cyu+ + F=
Jcψ·· c kψβ· cψβ+ + Mz=
ψ ψ
β
u· Vcx Vsx–=
v· Vcy Vsy–=
Adams/Tire48
 
in which Vcx, Vcy and are the sliding velocity of the contact body in longitudinal, lateral, and yaw 
directions, respectively. Vsx, Vsy, and are the corresponding velocities of the lower part of the wheel.
The transient slip equations for side slip, turn-slip, and camber are:
where the calculated deflection angle has been used:
The tire total spin velocity is:
With the transient slip equations, the composite transient turn-slip quantities are calculated:
The tire forces are calculated with and the tire moments with .
β· ψ· c ψ–=
ψ· c
ψ·
σc td
d α' Vx α'+ Vcy Vxβ– Vx βst+=
σc td
dα't Vx α't+ Vx α'=
σc td
dϕ'c Vx ϕ'c+ ψ· γ=
σF2 td
dϕ'F2 Vx ϕ'cF2+ ψ· γ=
σϕ1 td
dϕ'1 Vx ϕ'1+ ψ· γ=
σϕ2 td
dϕ'2 Vx ϕ'2+ ψ· γ=
βst
Mz
cφ
-------=
ψ· γ ψc 1 εγ–( )Ω γsin–=
ϕ'F 2ϕ'c ϕ'F2–=
ϕ'M εϕϕ'c εϕ12 ϕ'1 ϕ'2–( )+=
ϕ' ϕ'
F M
49
The relaxation lengths are reduced with slip:
Here a is half the contact length according to:
The composite tire parameter reads:
 
and the equivalent slip:
Coefficients and Transient Response
Name:
Name used in 
tire property file: Explanation:
mc MC Contact body mass
Ic IC Contact body moment of inertia
kx KX Longitudinal damping
ky KY Lateral damping
k KP Yaw damping
cx CX Longitudinal stiffness
cy CY Lateral stiffness
σc a 1 θζ–( )⋅=
σ2
t0
a
----σc=
σF2 bF2σc=
σϕ1 bϕ1σc=
σϕ2 bϕ2σc=
a pA1R0 ρ zR0
------ pA2
ρz
R0
------+⎝ ⎠⎜ ⎟
⎛ ⎞
=
θ Ky02μyFx
---------------=
ζ 11 κ'+------------- α' aεϕ12 ϕ'1 ϕ'2–+{ }
2 Kx0
Ky0
---------⎝ ⎠⎛ ⎞
2 κ' 23---b ϕ'c+⎩ ⎭⎨ ⎬
⎧ ⎫2
+=
ϕ
c CP Yaw stiffnessϕ
Adams/Tire50
 
The remaining contact mass model parameters are estimated automatically based on longitudinal and 
lateral stiffness specified in the tire property file.
Gyroscopic Couple in PAC2002
When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead 
to gyroscopic effects. To cope with this additional moment, the following contribution is added to the 
total aligning moment:
(101)
with the parameter (in addition to the basic tire parameter mbelt):
(102)
and:
(103)
The total aligning moment now becomes:
(104)
pA1 PA1 Half contact length with vertical tire deflection
pA2 PA2 Half contact length with square root of vertical tire 
deflection
EP Composite turn-slip (moment)
EP12 Composite turn-slip (moment) increment
bF2 BF2 Second relaxation length factor
b 1 BP1 First moment relaxation length factor
b 2 BP2 Second moment relaxation length factor
Name:
Name used in 
tire property file: Explanation:
εϕ
εϕ12
ϕ
ϕ
Mz gyr, cgyrmbeltVrl td
dv arc Brαr eq,( )tan[ ]cos=
cgyr qTz1 λgyr⋅=
arc Brαr eq,( )tan[ ]cos 1=
Mz M'z Mz gyr,+=
51
Coefficients and Transient Response
Left and Right Side Tires
In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire 
construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for 
positive and negative slip angles.
A tire property file with the parameters for the model results from testing with a tire that is mounted in 
a tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used 
for both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering 
wheel angle.
The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that 
indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or 
TIRESIDE = 'RIGHT').
If this keyword is available, Adams/Car corrects for the conicity and plysteer and asymmetry when using 
a tire property file on the opposite side of the vehicle. In fact, the tire characteristics are mirrored with 
respect to slip angle zero. In Adams/View, this option can only be used when the tire is generated by the 
graphical user interface: select Build -> Forces -> Special Force: Tire.
Next to the LEFT and RIGHT side option of TYRESIDE, you can also set SYMMETRIC: then the tire 
characteristics are modified during initialization to show symmetric performance for left and right side 
corners and zero conicity and plysteer (no offsets).Also, when you set the tire property file to 
SYMMETRIC, the tire characteristics are changed to symmetric behavior.
Create Wheel andTire Dialog Box in Adams/View
Name:
Name used in 
tire property file: Explanation:
pTx1 PTX1 Longitudinal relaxation length at Fznom
pTx2 PTX2 Variation of longitudinal relaxation length with load
pTx3 PTX3 Variation of longitudinal relaxation length with exponent of 
load
pTy1 PTY1 Peak value of relaxation length for lateral direction
pTy2 PTY2 Shape factor for lateral relaxation length
qTz1 QTZ1 Gyroscopic moment constant
Mbelt MBELT Belt mass of the wheel
Adams/Tire52
 
USE_MODES of PAC2002: from Simple to Complex
The parameter USE_MODE in the tire property file allows you to switch the output of the PAC2002 tire 
model from very simple (that is, steady-state cornering) to complex (transient combined cornering and 
braking).
The options for the USE_MODE and the output of the model have been listed in the table below.
53
USE_MODE Values of PAC2002 and Related Tire Model Output
Quality Checks for the Tire Model Parameters
Because PAC2002 uses an empirical approach to describe tire - road interaction forces, incorrect 
parameters can easily result in non-realistic tire behavior. Below is a list of the most important items to 
ensure the quality of the parameters in a tire property file:
• Rolling Resistance
• Camber (Inclination) Effects
• Validity Range of the Tire Model Input
USE_MODE: State: Slip conditions:
PAC2002 output
(forces and 
moments):
0 Steady state Acts as a vertical spring & 
damper
0, 0, Fz, 0, 0, 0
1 Steady state Pure longitudinal slip Fx, 0, Fz, 0, My, 0
2 Steady state Pure lateral (cornering) slip 0, Fy, Fz, Mx, 0, Mz
3 Steady state Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
4 Steady state Combined slip Fx, Fy, Fz, Mx, My, Mz
11 Transient Pure longitudinal slip Fx, 0, Fz, 0, My, 0
12 Transient Pure lateral (cornering) slip 0, Fy, Fz, Mx, 0, Mz
13 Transient Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
14 Transient Combined slip Fx, Fy, Fz, Mx, My, Mz
15 Transient Combined slip and turn-slip Fx, Fy, Fz, Mx, My, Mz
21 Advanced transient Pure longitudinal slip Fx, 0, Fz, My, 0
22 Advanced transient Pure lateral (cornering slip) 0, Fy, Fz, Mx, 0, Mz
23 Advanced transient Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
24 Advanced transient Combined slip Fx, Fy, Fz, Mx, My, Mz
25 Advanced transient Combined slip and turn-
slip/parking
Fx, Fy, Fz, Mx, My, Mz
Note: Do not change Fz0 (FNOMIN) and R0 (UNLOADED_RADIUS) in your tire property file. 
It will change the complete tire characteristics because these two parameters are used to 
make all parameters without dimension.
Adams/Tire54
 
Rolling Resistance
For a realistic rolling resistance, the parameter qsy1 must be positive. For car tires, it can be in the order 
of 0.006 - 0.01 (0.6% - 1.0%); for heavy commercial truck tires, it can be around 0.006 (0.6%).
Tire property files with the keyword FITTYP=5 determine the rolling resistance in a different way (see 
equation (85)). To avoid the ‘old’ rolling resistance calculation, remove the keyword FITTYP and add a 
section like the following:
$---------------------------------------------------rolling 
resistance[ROLLING_COEFFICIENTS]
QSY1 = 0.01
QSY2 = 0
QSY3 = 0
QSY4 = 0
 
Camber (Inclination) Effects
Camber stiffness has not been explicitly defined in PAC2002; however, for car tires, positive inclination 
should result in a negative lateral force at zero slip angle. If positive inclination results in an increase of 
the lateral force, the coefficient may not be valid for the ISO but for the SAE coordinate system. Note 
that PAC2002 only uses coefficients for the TYDEX W-axis (ISO) system.
Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System
The table below lists further checks on the PAC2002 parameters.
55
Checklist for PAC2002 Parameters and Properties
Validity Range of the Tire Model Input
In the tire property file, a range of the input variables has been given in which the tire properties are 
supposed to be valid. These validity range parameters are (the listed values can be different):
$--------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip 
KPUMAX = 1.5 $Maximum valid wheel slip 
$-------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip angle 
ALPMAX = 1.5708 $Maximum valid slip angle 
$--------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
CAMMIN = -0.26181 $Minimum valid camber angle 
CAMMAX = 0.26181 $Maximum valid camber angle 
$----------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 225 $Minimum allowed wheel load 
FZMAX = 10125 $Maximum allowed wheel load 
If one of the input parameters exceeds a minimum or maximum validity value, the calculation in the tire 
model is performed with the minimum or maximum value of this range to avoid non-realistic tire 
behavior. In that case, a message appears warning you that one of the inputs exceeds a validity value.
Standard Tire Interface (STI) for PAC2002
Parameter/property: Requirement: Explanation:
LONGVL 1 m/s Reference velocity at which parameters are 
measured
VXLOW Approximately 1 m/s Threshold for scaling down forces and moments
Dx > 0 Peak friction (see equation (24))
pDx1/pDx2 < 0 Peak friction Fx must decrease with increasing load
Kx > 0 Long slip stiffness (see equation (27))
Dy > 0 Peak friction (see equation (36))
pDy1/pDy2 < 0 Peak friction Fx must decrease with increasing load
Ky < 0 Cornering stiffness (see equation (39))
qsy1 > 0 Rolling resistance, in the range of 0.005 - 0.015
Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to 
Adams/Solver, below is a brief background of the STI history (see also reference [4]).
Adams/Tire56
 
At the First International Colloquium on Tire Models for Vehicle Dynamics Analysis on October 21-22, 
1991, the International Tire Workshop working group was established (TYDEX).
The working group concentrated on tire measurements and tire models used for vehicle simulation 
purposes. For most vehicle dynamics studies, people used to develop their own tire models. Because all 
car manufacturers and their tire suppliers have the same goal (that is, development of tires to improve 
dynamic safety of the vehicle) it aimed for standardization in tire behavior description.
In TYDEX, two expert groups, consisting of participants of vehicle industry (passenger cars and trucks), 
tire manufacturers, other suppliers and research laboratories, had been defined with following goals:
• The first expert group's (Tire Measurements - Tire Modeling) main goal was to specify an 
interface between tire measurements and tire models. The result was the TYDEX-Format [2] to 
describe tire measurement data.
• The second expert group's (Tire Modeling - Vehicle Modeling) main goal was to specify an 
interface between tire models and simulation tools, which resulted in the Standard Tire Interface 
(STI) [3]. The use of this interface should ensure that a wide range of simulation software can be 
linked to a wide range of tire modeling software.
Definitions
• General
• Tire Kinematics
• Slip Quantities
• Force and Moments
General
General Definitions
Term: Definition:
Road tangent plane Plane with the normal unit vector (tangent to the road) in the tire-road contact 
point C.
C-axis system Coordinate system mounted on the wheel carrier at the wheel center 
according to TYDEX, ISO orientation.
Wheel plane The plane in the wheel center that is formed by the wheel when considered a 
rigid disc with zero width.
Contact point C Contact point betweentire and road, defined as the intersection of the wheel 
plane and the projection of the wheel axis onto the road plane.
W-axis system Coordinate system at the tire contact point C, according to TYDEX, ISO 
orientation.
57
Tire Kinematics
Tire Kinematics Definitions
Slip Quantities
Slip Quantities Definitions
Forces and Moments
Force and Moment Definitions
Parameter: Definition: Units:
R0 Unloaded tire radius [m]
R Loaded tire radius [m]
Re Effective tire radius [m]
Radial tire deflection [m]
d Dimensionless radial tire deflection [-]
Fz0 Radial tire deflection at nominal load [m]
mbelt Tire belt mass [kg]
Rotational velocity of the wheel [rads-1]
Parameter: Definition: Units:
V Vehicle speed [ms-1]
Vsx Slip speed in x direction [ms-1]
Vsy Slip speed in y direction [ms-1]
Vs Resulting slip speed [ms-1]
Vx Rolling speed in x direction [ms-1]
Vy Lateral speed of tire contact center [ms-1]
Vr Linear speed of rolling [ms-1]
Longitudinal slip [-]
Slip angle [rad]
Inclination angle [rad]
Abbreviation: Definition: Units:
Fz Vertical wheel load [N]
Fz0 Nominal load [N]
ρ
ρ
ρ
ω
κ
α
γ
dfz Dimensionless vertical load [-]
Adams/Tire58
 
References
1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5.
2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, 
Initiated by the International Tire Working Group, July 1995.
3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995.
4. J.J.M. van Oosten, H.-J. Unrau, G. Riedel, E. Bakker, TYDEX Workshop: Standardisation of 
Data Exchange in Tyre Testing and Tyre Modelling, Proceedings of the 2nd International 
Colloquium on Tyre Models for Vehicle Dynamics Analysis, Vehicle System Dynamics, Volume 
27, Swets & Zeitlinger, Amsterdam/Lisse, 1996.
Example of PAC2002 Tire Property File
[MDI_HEADER]
FILE_TYPE ='tir'
FILE_VERSION =3.0
FILE_FORMAT ='ASCII'
! : TIRE_VERSION : PAC2002
! : COMMENT : Tire 235/60R16
! : COMMENT : Manufacturer 
! : COMMENT : Nom. section with (m) 0.235 
! : COMMENT : Nom. aspect ratio (-) 60
! : COMMENT : Infl. pressure (Pa) 200000
! : COMMENT : Rim radius (m) 0.19 
! : COMMENT : Measurement ID 
! : COMMENT : Test speed (m/s) 16.6 
! : COMMENT : Road surface 
! : COMMENT : Road condition Dry
! : FILE_FORMAT : ASCII
! : Copyright MSC.Software, Fri Jan 23 14:30:06 2004
!
! USE_MODE specifies the type of calculation performed:
! 0: Fz only, no Magic Formula evaluation
! 1: Fx,My only
! 2: Fy,Mx,Mz only
! 3: Fx,Fy,Mx,My,Mz uncombined force/moment calculation
! 4: Fx,Fy,Mx,My,Mz combined force/moment calculation
! +10: including relaxation behaviour
Fx Longitudinal force [N]
Fy Lateral force [N]
Mx Overturning moment [Nm]
My Braking/driving moment [Nm]
Mz Aligning moment [Nm]
Abbreviation: Definition: Units:
! *-1: mirroring of tyre characteristics
59
!
! example: USE_MODE = -12 implies:
! -calculation of Fy,Mx,Mz only
! -including relaxation effects
! -mirrored tyre characteristics
!
$--------------------------------------------------------------units
[UNITS]
LENGTH ='meter'
FORCE ='newton'
ANGLE ='radians'
MASS ='kg'
TIME ='second'
$--------------------------------------------------------------model
[MODEL]
PROPERTY_FILE_FORMAT ='PAC2002'
USE_MODE = 14 $Tyre use switch (IUSED)
VXLOW = 1 
LONGVL = 16.6 $Measurement speed 
TYRESIDE = 'LEFT' $Mounted side of tyre 
at vehicle/test bench
$---------------------------------------------------------dimensions
[DIMENSION]
UNLOADED_RADIUS = 0.344 $Free tyre radius 
WIDTH = 0.235 $Nominal section width 
of the tyre 
ASPECT_RATIO = 0.6 $Nominal aspect ratio
RIM_RADIUS = 0.19 $Nominal rim radius 
RIM_WIDTH = 0.16 $Rim width 
$----------------------------------------------------------parameter
[VERTICAL]
VERTICAL_STIFFNESS = 2.1e+005 $Tyre vertical 
stiffness 
VERTICAL_DAMPING = 50 $Tyre vertical damping 
BREFF = 8.4 $Low load stiffness 
e.r.r. 
DREFF = 0.27 $Peak value of e.r.r. 
FREFF = 0.07 $High load stiffness 
e.r.r. 
FNOMIN = 4850 $Nominal wheel load
$----------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel 
slip 
KPUMAX = 1.5 $Maximum valid wheel 
slip 
$---------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip 
angle 
ALPMAX = 1.5708 $Maximum valid slip 
angle 
$---------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
Adams/Tire60
 
CAMMIN = -0.26181 $Minimum valid camber 
angle 
CAMMAX = 0.26181 $Maximum valid camber 
angle 
$------------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 225 $Minimum allowed wheel 
load 
FZMAX = 10125 $Maximum allowed wheel 
load 
$-------------------------------------------------------------scaling
[SCALING_COEFFICIENTS]
LFZO = 1 $Scale factor of nominal 
(rated) load 
LCX = 1 $Scale factor of Fx 
shape factor 
LMUX = 1 $Scale factor of Fx 
peak friction coefficient 
LEX = 1 $Scale factor of Fx 
curvature factor 
LKX = 1 $Scale factor of Fx 
slip stiffness 
LHX = 1 $Scale factor of Fx 
horizontal shift 
LVX = 1 $Scale factor of Fx 
vertical shift 
LGAX = 1 $Scale factor of camber 
for Fx 
LCY = 1 $Scale factor of Fy 
shape factor 
LMUY = 1 $Scale factor of Fy 
peak friction coefficient 
LEY = 1 $Scale factor of Fy 
curvature factor 
LKY = 1 $Scale factor of Fy 
cornering stiffness 
LHY = 1 $Scale factor of Fy 
horizontal shift 
LVY = 1 $Scale factor of Fy 
vertical shift 
LGAY = 1 $Scale factor of camber 
for Fy 
LTR = 1 $Scale factor of Peak 
of pneumatic trail 
LRES = 1 $Scale factor for offset 
of residual torque 
LGAZ = 1$Scale factor of camber 
for Mz 
LXAL = 1 $Scale factor of alpha 
influence on Fx 
LYKA = 1 $Scale factor of alpha 
influence on Fx 
LVYKA = 1 $Scale factor of kappa 
induced Fy 
61
LS = 1 $Scale factor of Moment 
arm of Fx 
LSGKP = 1 $Scale factor of 
Relaxation length of Fx 
LSGAL = 1 $Scale factor of 
Relaxation length of Fy 
LGYR = 1 $Scale factor of 
gyroscopic torque 
LMX = 1 $Scale factor of 
overturning couple 
LVMX = 1 $Scale factor of Mx 
vertical shift 
LMY = 1 $Scale factor of rolling 
resistance torque 
$-------------------------------------------------------longitudinal
[LONGITUDINAL_COEFFICIENTS]
PCX1 = 1.6411 $Shape factor Cfx for 
longitudinal force 
PDX1 = 1.1739 $Longitudinal friction 
Mux at Fznom 
PDX2 = -0.16395 $Variation of friction 
Mux with load 
PDX3 = 0 $Variation of friction 
Mux with camber 
PEX1 = 0.46403 $Longitudinal curvature 
Efx at Fznom 
PEX2 = 0.25022 $Variation of curvature 
Efx with load 
PEX3 = 0.067842 $Variation of curvature 
Efx with load squared 
PEX4 = -3.7604e-005 $Factor in curvature 
Efx while driving 
PKX1 = 22.303 $Longitudinal slip 
stiffness Kfx/Fz at Fznom 
PKX2 = 0.48896 $Variation of slip 
stiffness Kfx/Fz with load 
PKX3 = 0.21253 $Exponent in slip 
stiffness Kfx/Fz with load 
PHX1 = 0.0012297 $Horizontal shift Shx 
at Fznom 
PHX2 = 0.0004318 $Variation of shift 
Shx with load 
PVX1 = -8.8098e-006 $Vertical shift Svx/Fz 
at Fznom 
PVX2 = 1.862e-005 $Variation of shift 
Svx/Fz with load 
RBX1 = 13.276 $Slope factor for 
combined slip Fx reduction 
RBX2 = -13.778 $Variation of slope Fx 
reduction with kappa 
RCX1 = 1.2568 $Shape factor for 
combined slip Fx reduction 
REX1 = 0.65225 $Curvature factor of 
combined Fx 
Adams/Tire62
 
REX2 = -0.24948 $Curvature factor of 
combined Fx with load 
RHX1 = 0.0050722 $Shift factor for 
combined slip Fx reduction 
PTX1 = 2.3657 $Relaxation length 
SigKap0/Fz at Fznom 
PTX2 = 1.4112 $Variation of SigKap0/Fz 
with load 
PTX3 = 0.56626 $Variation of SigKap0/Fz 
with exponent of load 
$--------------------------------------------------------overturning
[OVERTURNING_COEFFICIENTS]
QSX1 = 0 $Lateral force induced 
overturning moment 
QSX2 = 0 $Camber induced 
overturning couple 
QSX3 = 0 $Fy induced overturning 
couple 
$------------------------------------------------------------lateral
[LATERAL_COEFFICIENTS]
PCY1 = 1.3507 $Shape factor Cfy for 
lateral forces 
PDY1 = 1.0489 $Lateral friction Muy 
PDY2 = -0.18033 $Variation of friction 
Muy with load 
PDY3 = -2.8821 $Variation of friction 
Muy with squared camber 
PEY1 = -0.0074722 $Lateral curvature Efy 
at Fznom 
PEY2 = -0.0063208 $Variation of curvature 
Efy with load 
PEY3 = -9.9935 $Zero order camber 
dependency of curvature Efy 
PEY4 = -760.14 $Variation of curvature 
Efy with camber 
PKY1 = -21.92 $Maximum value of 
stiffness Kfy/Fznom 
PKY2 = 2.0012 $Load at which Kfy 
reaches maximum value 
PKY3 = -0.024778 $Variation of Kfy/Fznom 
with camber 
PHY1 = 0.0026747 $Horizontal shift Shy 
at Fznom 
PHY2 = 8.9094e-005 $Variation of shift Shy 
with load 
PHY3 = 0.031415 $Variation of shift Shy 
with camber 
PVY1 = 0.037318 $Vertical shift in 
Svy/Fz at Fznom 
PVY2 = -0.010049 $Variation of shift 
Svy/Fz with load 
PVY3 = -0.32931 $Variation of shift 
Svy/Fz with camber 
63
PVY4 = -0.69553 $Variation of shift 
Svy/Fz with camber and load 
RBY1 = 7.1433 $Slope factor for 
combined Fy reduction 
RBY2 = 9.1916 $Variation of slope Fy 
reduction with alpha 
RBY3 = -0.027856 $Shift term for alpha 
in slope Fy reduction 
RCY1 = 1.0719 $Shape factor for 
combined Fy reduction 
REY1 = -0.27572 $Curvature factor of 
combined Fy 
REY2 = 0.32802 $Curvature factor of 
combined Fy with load 
RHY1 = 5.7448e-006 $Shift factor for 
combined Fy reduction 
RHY2 = -3.1368e-005 $Shift factor for 
combined Fy reduction with load 
RVY1 = -0.027825 $Kappa induced side 
force Svyk/Muy*Fz at Fznom 
RVY2 = 0.053604 $Variation of 
Svyk/Muy*Fz with load 
RVY3 = -0.27568 $Variation of 
Svyk/Muy*Fz with camber 
RVY4 = 12.12 $Variation of 
Svyk/Muy*Fz with alpha 
RVY5 = 1.9 $Variation of Svyk/Muy*Fz 
with kappa 
RVY6 = -10.704 $Variation of 
Svyk/Muy*Fz with atan(kappa) 
PTY1 = 2.1439 $Peak value of 
relaxation length SigAlp0/R0 
PTY2 = 1.9829 $Value of Fz/Fznom 
where SigAlp0 is extreme 
$-------------------------------------------------rolling resistance
[ROLLING_COEFFICIENTS]
QSY1 = 0.01 $Rolling resistance 
torque coefficient 
QSY2 = 0 $Rolling resistance 
torque depending on Fx 
QSY3 = 0 $Rolling resistance 
torque depending on speed 
QSY4 = 0 $Rolling resistance 
torque depending on speed ^4 
$-----------------------------------------------------------aligning
[ALIGNING_COEFFICIENTS]
QBZ1= 10.904 $Trail slope factor 
for trail Bpt at Fznom 
QBZ2 = -1.8412 $Variation of slope 
Bpt with load 
QBZ3 = -0.52041 $Variation of slope 
Bpt with load squared 
QBZ4 = 0.039211 $Variation of slope 
Bpt with camber 
Adams/Tire64
 
QBZ5 = 0.41511 $Variation of slope Bpt 
with absolute camber 
QBZ9 = 8.9846 $Slope factor Br of 
residual torque Mzr 
QBZ10 = 0 $Slope factor Br of 
residual torque Mzr 
QCZ1 = 1.2136 $Shape factor Cpt for 
pneumatic trail 
QDZ1 = 0.093509 $Peak trail Dpt" = 
Dpt*(Fz/Fznom*R0) 
QDZ2 = -0.0092183 $Variation of peak Dpt" 
with load 
QDZ3 = -0.057061 $Variation of peak Dpt" 
with camber 
QDZ4 = 0.73954 $Variation of peak Dpt" 
with camber squared 
QDZ6 = -0.0067783 $Peak residual torque 
Dmr" = Dmr/(Fz*R0) 
QDZ7 = 0.0052254 $Variation of peak 
factor Dmr" with load 
QDZ8 = -0.18175 $Variation of peak 
factor Dmr" with camber 
QDZ9 = 0.029952 $Variation of peak 
factor Dmr" with camber and load 
QEZ1 = -1.5697 $Trail curvature Ept 
at Fznom 
QEZ2 = 0.33394 $Variation of curvature 
Ept with load 
QEZ3 = 0 $Variation of curvature 
Ept with load squared 
QEZ4 = 0.26711 $Variation of curvature 
Ept with sign of Alpha-t 
QEZ5 = -3.594 $Variation of Ept with 
camber and sign Alpha-t 
QHZ1 = 0.0047326 $Trail horizontal shift 
Sht at Fznom 
QHZ2 = 0.0026687 $Variation of shift Sht 
with load 
QHZ3 = 0.11998 $Variation of shift Sht 
with camber 
QHZ4 = 0.059083 $Variation of shift Sht 
with camber and load 
SSZ1 = 0.033372 $Nominal value of s/R0: 
effect of Fx on Mz 
SSZ2 = 0.0043624 $Variation of distance 
s/R0 with Fy/Fznom 
SSZ3 = 0.56742 $Variation of distance 
s/R0 with camber 
SSZ4 = -0.24116 $Variation of distance 
s/R0 with load and camber 
QTZ1 = 0.2 $Gyration torque constant 
MBELT = 5.4 $Belt mass of the wheel 
$-----------------------------------------------turn-slip parameters
[TURNSLIP_COEFFICIENTS]
65
PECP1 = 0.7 $Camber stiffness reduction factor
PECP2 = 0.0 $Camber stiffness reduction factor with load
PDXP1 = 0.4 $Peak Fx reduction due to spin
PDXP2 = 0.0 $Peak Fx reduction due to spin with load
PDXP3 = 0.0 $Peak Fx reduction due to spin with longitudinal 
slip
PDYP1 = 0.4 $Peak Fy reduction due to spin
PDYP2 = 0.0 $Peak Fy reduction due to spin with load
PDYP3 = 0.0 $Peak Fy reduction due to spin with lateral slip
PDYP4 = 0.0 $Peak Fy reduction with square root of spin
PKYP1 = 1.0 $Cornering stiffness reduction due to spin
PHYP1 = 1.0 $Fy lateral shift shape factor
PHYP2 = 0.15 $Maximum Fy lateral shift
PHYP3 = 0.0 $Maximum Fy lateral shift with load
PHYP4 = -4.0 $Fy lateral shift curvature factor
QDTP1 = 10.0 $Pneumatic trail reduction factor
QBRP1 = 0.1 $Residual torque reduction factor with lateral 
slip
QCRP1 = 0.2 $Turning moment at constant turning with zero 
speed
QCRP2 = 0.1 $Turning moment at 90 deg lateral slip
QDRP1 = 1.0 $Maximum turning moment
QDRP2 = -1.5 $Location of maximum turning moment
$--------------------------------------------contact patch parameters
[CONTACT_COEFFICIENTS]
PA1 = 0.4147 $Half contact length dependency on Fz)
PA2 = 1.9129 $Half contact length dependency on sqrt(Fz/R0)
$--------------------------------------------contact patch slip model
[DYNAMIC_COEFFICIENTS]
MC = 1.0 $Contact mass
IC = 0.05 $Contact moment of inertia
KX = 409.0 $Contact longitudinal damping
KY = 320.8 $Contact lateral damping
KP = 11.9 $Contact yaw damping
CX = 4.350e+005 $Contact longitudinal stiffness
CY = 1.665e+005 $Contact lateral stiffness
CP = 20319 $Contact yaw stiffness
EP = 1.0
EP12 = 4.0
BF2 = 0.5
BP1 = 0.5
BP2 = 0.67
$------------------------------------------------------loaded radius
[LOADED_RADIUS_COEFFICIENTS]
QV1 = 0.000071 $Tire radius growth coefficient
QV2 = 2.489 $Tire stiffness variation coefficient with speed
QFCX1 = 0.1 $Tire stiffness interaction with Fx
QFCY1 = 0.3 $Tire stiffness interaction with Fy
QFCG1 = 0.0 $Tire stiffness interaction with camber
QFZ1 = 0.0 $Linear stiffness coefficient, if zero, 
VERTICAL_STIFFNESS is taken
QFZ2 = 14.35 $Tire vertical stiffness coefficient (quadratic)
Adams/Tire66
 
Contact Methods
The PAC2002 model supports the following roads:
• 2D Roads, see Using the 2D Road Model
• 3D Spline Roads, see Adams/3D Spline Road Model
Note that the PAC2002 model has only one point of contact with the road; therefore, the 
wavelength of road obstacles must be longer than the tire radius for realistic output of the model. 
In addition, the contact force computed by this tire model is normal to the road plane. Therefore, 
the contact point does not generate a longitudinal force when rolling over a short obstacle, such 
as a cleat or pothole.
• 3D Shell Roads, see Adams/Tire 3D Shell Road Model
For ride and comfort analyses, we recommend more sophisticated tire models, such as Ftire.
Using the PAC-TIME Tire Model
The PAC-TIME Magic-Formula tire model has been developed by MSC.Software according to a 
publication, A New Tyre Model for TIME Measurement Data, by J.J.M. van Oosten e.a. [5]. PAC-TIME 
has improved equations for side force and aligning moment under pure slip conditions. For longitudinal 
pure slip and combined slip, the tire model is similar to PAC-TIME.
Learn about:
• When to Use PAC-TIME
• Modeling of Tire-Road Interaction Forces
• Axis Systems and Slip Definitions
• Contact Point and Normal Load Calculation
• Basics of Magic Formula
• Steady-State: Magic Formula
• Transient Behavior
• Gyroscopic Couple
• Left and Right Side Tires
• USE_MODES OF PAC-TIME: from Simple to Complex
• Quality Checks for Tire Model Parameters
• Standard Tire Interface (STI)
• Definitions
• References
• Example of PAC-TIME Tire Property File
• Contact Methods
When to Use PAC-TIME
Magic-Formula (MF) tire models are considered the state-of-the-art for modeling tire-road interaction 
forces in vehicle dynamics applications. Since 1987, Pacejka and others have published several versions 
of this type of tire model. The PAC-TIME model is similar to PAC2002, but has improved equations for 
side force (Fy) and aligning moment (Mz) under pure side slip conditions.
The following is background information about the PAC-TIME tire model, as stated in the paper, A New 
Tyre Model for TIME MeasurementData, J.J.M. van Oosten, E. Kuiper, G. Leister, D. Bode, H. 
Schindler, J. Tischleder, S. Köhne [5]:
In 1999 a new method for tyre Force and Moment (F&M) testing has been developed by a consortium of 
European tyre and vehicle manufacturers: the TIME procedure. For Vehicle Dynamics studies often a 
Magic Formula (MF) tyre model is used based upon such F&M data. However when calculating MF 
parameters for a standard MF model out of the TIME F&M data, several difficulties are observed. These 
Adams/Tire12
 
are mainly due to the non-uniform distribution of the data points over the slip angle, camber and load 
area and the mutual dependency in between the slip angle, camber and load. A new MF model for pure 
cornering slip conditions has been developed that allows the calculation of the MF parameters despite 
of the dependency of the three input variables in the F&M data and shows better agreement with the 
measured F&M data points. From mathematical point of view the optimisation process for deriving MF 
parameters is better conditioned with the new MF-TIME, resulting in less sensitivity to starting values 
and better convergence to a global minimum. In addition the MF-TIME has improved extrapolation 
performance compared to the standard MF models for areas where no F&M data points are available. 
Next to the use for TIME F&M data, the new model is expected to have interesting prospects for 
converting ‘on-vehicle’ measured tyre data into a robust set of MF parameters.
In general, an MF tire model describes the tire behavior for rather smooth roads (road obstacle 
wavelengths longer than the tire radius) up to frequencies of 8 Hz. This makes the tire model applicable 
for all generic vehicle handling and stability simulations, including:
• Steady-state cornering
• Single- or double-lane change
• Braking or power-off in a turn
• Split-mu braking tests
• J-turn or other turning maneuvers
• ABS braking, when stopping distance is important (not for tuning ABS control strategies)
• Other common vehicle dynamics maneuvers on rather smooth roads (wavelength of road 
obstacles must be longer than the tire radius)
For modeling roll-over of a vehicle, you must pay special attention to the overturning moment 
characteristics of the tire (Mx), and the loaded radius modeling. The last item may not be sufficiently 
addressed in this model.
The PAC-TIME model has been developed for car tires with camber (inclination) angles to the road not 
exceeding 15 degrees.
Modeling of Tire-Road Interaction Forces
For vehicle dynamics applications, an accurate knowledge of tire-road interaction forces is inevitable 
because the movements of a vehicle primarily depend on the road forces on the tires. These interaction 
forces depend on both road and tire properties, and the motion of the tire with respect to the road.
In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear 
damper with one point of contact with the road surface. The contact point is determined by considering 
the tire and wheel as a rigid disc. In the contact point between the tire and the road, the contact forces in 
longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the 
road.
The figure, Input and Output Variables of the Magic Formula Tire Model, presents the input and output 
vectors of the PAC2002 tire model. The tire model subroutine is linked to the Adams/Solver through the 
Standard Tire Interface (STI) [3]. The input through the STI consists of:
13
• Position and velocities of the wheel center
• Orientation of the wheel
• Tire model (MF) parameters
• Road parameters
The tire model routine calculates the vertical load and slip quantities based on the position and speed of 
the wheel with respect to the road. The input for the Magic Formula consists of the wheel load (Fz), the 
longitudinal and lateral slip ( , ), and inclination angle ( ) with the road. The output is the forces 
(Fx, Fy) and moments (Mx, My, Mz) in the contact point between the tire and the road. For calculating 
these forces, the MF equations use a set of MF parameters, which are derived from tire testing data.
The forces and moments out of the Magic Formula are transferred to the wheel center and returned to 
Adams/Solver through STI.
Input and Output Variables of the Magic Formula Tire Model
 
Axis Systems and Slip Definitions
• Axis Systems
• Units
• Definition of Tire Slip Quantities
κ α γ
Adams/Tire14
 
Axis Systems
The PAC-TIME model is linked to Adams/Solver using the TYDEX STI conventions, as described in the 
TYDEX-Format [2] and the STI [3].
The STI interface between the MF-TIME model and Adams/Solver mainly passes information to the tire 
model in the C-axis coordinate system. In the tire model itself, a conversion is made to the W-axis system 
because all the modeling of the tire behavior, as described in this help, assumes to deal with the slip 
quantities, orientation, forces, and moments in the contact point with the TYDEX W-axis system. Both 
axis systems have the ISO orientation but have different origin as can be seen in the figure below.
TYDEX C- and W-Axis Systems Used in PAC-TIME , Source [2]
 
The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel to the road and in 
the wheel plane (xc-zc-plane). The origin of the C-axis system is the wheel center.
The origin of the W-axis system is the road contact-point defined by the intersection of the wheel plane, 
the plane through the wheel carrier, and the road tangent plane.
The forces and moments calculated by PAC-TIME using the MF equations in this guide are in the W-axis 
system. A transformation is made in the source code to return the forces and moments through the STI 
to Adams/Solver.
The inclination angle is defined as the angle between the wheel plane and the normal to the road tangent 
plane (xw-yw-plane).
Units
The units of information transferred through the STI between Adams/Solver and PAC-TIME are 
according to the SI unit system. Also, the equations for PAC-TIME described in this guide have been 
developed for use with SI units, although you can easily switch to another unit system in your tire 
property file. Because of the non-dimensional parameters, only a few parameters have to be changed.
15
However, the parameters in the tire property file must always be valid for the TYDEX W-axis system 
(ISO oriented). The basic SI units are listed in the table below.
SI Units Used in PAC-TIME
Definition of Tire Slip Quantities
Slip Quantities at Combined Cornering and Braking/Traction 
 
Variable type: Name: Abbreviation: Unit:
Angle Slip angle
Inclination angle
Radians
Force Longitudinal force
Lateral force
Vertical load
Fx
Fy
Fz
Newton
Moment Overturning moment
Rolling resistance moment
Self-aligning moment
Mx
My
Mz
Newton.meter
Speed Longitudinal speed
Lateral speed
Longitudinal slip speed
Lateral slip speed
Vx
Vy
Vsx
Vsy
Meters per second
Rotational speed Tire rolling speed Radians per second
α
γ
ω
Adams/Tire16
 
The longitudinal slip velocity Vsx in the contact point (W-axis system, see Slip Quantities at Combined 
Cornering and Braking/Traction) is defined using the longitudinal speed Vx, the wheel rotational velocity 
, and the effective rolling radius Re:
(1)
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
(2)
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point with:
(3)
(4)
The rolling speed Vr is determined using the effective rolling radius Re:
(5)
Contact Point and Normal Load Calculation
• Contact Point• Loaded and Effective Tire Rolling Radius
Contact Point
In the vertical direction, the tire is modeled as a parallel linear spring and damper having one point of 
contact (C) with the road. This is valid for road obstacles with a wavelength larger than the tire radius 
(for example, for car tires 1m).
For calculating the kinematics of the tire relative to the road, the road is approximated by its tangent plane 
at the road point right below the wheel center (see the figure below).
Contact Point C: Intersection between Road Tangent Plane, Spin Axis Plane, and Wheel Plane
ω
Vsx Vx ΩRe–=
Vsy Vy=
κ α
κ VsxVx
--------–=
αtan VsyVx
---------=
Vr ReΩ=
17
The contact point is determined by the line of intersection of the wheel center-plane with the road tangent 
(ground) plane and the line of intersection of the wheel center-plane with the plane though the wheel spin 
axis.
The normal load Fz of the tire is calculated with:
(6)
where is the tire deflection and is the deflection rate of the tire.
Instead of the linear vertical tire stiffness Cz, you can also define an arbitrary tire deflection - load curve 
in the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Example of PAC-TIME 
Tire Property File). If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection data 
points with a cubic spline for inter- and extrapolation are used for the calculation of the vertical force of 
the tire. Note that you must specify Cz in the tire property file, but it does not play any role.
Loaded and Effective Tire Rolling Radius
With the loaded rolling tire radius R defined as the distance of the wheel center to the contact point of 
the tire with the road, where is the deflection of the tire, and R0 is the free (unloaded) tire radius, then 
the loaded tire radius Rl is:
(7)
In this tire model, a constant (linear) vertical tire stiffness Cz is assumed; therefore, the tire deflection 
Fz Czρ Kz ρ·⋅+=
ρ ρ
ρ
R1 R0 ρ–=
ρ
can be calculated using:
Adams/Tire18
 
(8)
The effective rolling radius Re (at free rolling of the tire), which is used to calculate the rotational speed 
of the tire, is defined by:
(9)
For radial tires, the effective rolling radius is rather independent of load in its load range of operation 
because of the high stiffness of the tire belt circumference. Only at low loads does the effective tire radius 
decrease with increasing vertical load due to the tire tread thickness. See the figure below.
Effective Rolling Radius and Longitudinal Slip
To represent the effective rolling radius Re, an MF type of equation is used:
ρ FzCz
------=
Re
Vx
Ω------=
(10)Re R0 ρFz0 Darc Bρ
d( ) Fρd+tan( )–=
19
in which Fz0 is the nominal tire deflection:
(11)
and d is called the dimensionless radial tire deflection, defined by:
(12)
Effective Rolling Radius and Longitudinal Slip
Normal Load and Rolling Radius Parameters
Name:
Name Used in Tire 
Property File: Explanation:
Fz0 FNOMIN Nominal wheel load
Ro UNLOADED_RADIUS Free tire radius
B BREFF Low load stiffness effective rolling radius
D DREFF Peak value of effective rolling radius
ρ
ρFz0
Fz0
Cz
--------=
ρ
ρd ρρFz0
---------=
Adams/Tire20
 
Basics of the Magic Formula in PAC-TIME
The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics 
for the interaction forces between the tire and the road under several steady-state operating conditions. 
We distinguish:
• Pure cornering slip conditions: cornering with a free rolling tire
• Pure longitudinal slip conditions: braking or driving the tire without cornering
• Combined slip conditions: cornering and longitudinal slip simultaneously
For pure slip conditions, the lateral force Fy as a function of the lateral slip , respectively, and the 
longitudinal force Fx as a function of longitudinal slip , have a similar shape. Because of the sine - 
arctangent combination, the basic Magic Formula example is capable of describing this shape:
(13)
where Y(x) is either Fx with x the longitudinal slip , or Fy and x the lateral slip .
Characteristic Curves for Fx and Fy Under Pure Slip Conditions
F FREFF High load stiffness effective rolling radius
Cz VERTICAL_STIFFNESS Tire vertical stiffness
Kz VERTICAL_DAMPING Tire vertical damping
Name:
Name Used in Tire 
Property File: Explanation:
α
κ
Y x( ) D Carc Bx E Bx arc Bx( )tan–( )–{ }tan[ ]cos=
κ α
21
The self-aligning moment Mz is calculated as a product of the lateral force Fy and the pneumatic trail t 
added with the residual moment Mzr. In fact, the aligning moment is due to the offset of lateral force Fy, 
called pneumatic trail t, from the contact point. Because the pneumatic trail t as a function of the lateral 
slip has a cosine shape, a cosine version the Magic Formula is used:
(14)
in which Y(x) is the pneumatic trail t as function of slip angle .
The figure, The Magic Formula and the Meaning of Its Parameters, illustrates the functionality of the B, 
C, D, and E factor in the Magic Formula:
• D-factor determines the peak of the characteristic, and is called the peak factor.
• C-factor determines the part used of the sine and, therefore, mainly influences the shape of the 
curve (shape factor).
• B-factor stretches the curve and is called the stiffness factor.
• E-factor can modify the characteristic around the peak of the curve (curvature factor). 
The Magic Formula and the Meaning of Its Parameters
α
Y x( ) D Carc Bx E Bx arc Bx( )tan–( )–{ }tan[ ]cos=
α
Adams/Tire22
 
 
In combined slip conditions, the lateral force Fy will decrease due to longitudinal slip or the opposite, the 
longitudinal force Fx will decrease due to lateral slip. The forces and moments in combined slip 
conditions are based on the pure slip characteristics multiplied by the so-called weighting functions. 
Again, these weighting functions have a cosine-shaped MF examples.
The Magic Formula itself only describes steady-state tire behavior. For transient tire behavior (up to 8 
Hz), the MF output is used in a stretched string model that considers tire belt deflections instead of slip 
velocities to cope with standstill situations (zero speed).
Inclination Effects in the Lateral Force
From a historical point of view, the camber stiffness always has been modeled implicit in the Magic 
Formulas. For deriving coefficients of a Pacejka tire model usually so-called tire tests with slip angle 
sweeps at various values of constant load and inclination are performed. In the resulting Force & Moment 
measurement data, the effects of camber on the side force Fy are relatively small compared to side force 
23
effects by slip angle, which can easily result in non-realistic camber stiffness properties. Because there 
is no explicit definition of the camber stiffness, the effects on camber stiffness cannot be controlled in 
the coefficient optimization process.
The TIME measurement procedure guarantees more realistic tire test data, because they are performed 
under realistic tire operating conditions and specific parts of the test program concentrate on getting 
accurate cornering and camber stiffness. Because the inputs to the test program (side and longitudinal 
slip, inclination, and load) are not independent, for the parameter optimization process, a Pacejka tire 
model was required that has a better definition of cornering and camber stiffness from mathematical 
point of view (for a more detailed explanation, see [5]).
Therefore, the PAC-TIME tire model has an explicit definition of camber effects, similar to the tire 
model for motorcycle tires (PAC_MC). The basic Magic Formula sine function for the lateral force Fy 
has been extended with an argument for the inclination as follows:
(15)
In the PAC-TIMEtire model, C has been set to ½, and E is not used (zero value). This approach 
results in an explicit definition of the camber stiffness, because:
(16)
Input Variables
The input variables to the Magic Formula are:
Input Variables
Output Variables
Its output variables are:
Output Variables.
Longitudinal slip [-]
Slip angle [rad]
Inclination angle [rad]
Normal wheel load Fz [N]
Longitudinal force Fx [N]
Lateral force Fy [N]
γ
Fy0 Dy Cyarc Byαy Ey Byαy arc Byαy( )tan–( )–{ }
Cγarc Byαγ Eγ Bγαγ arc Bγαγ( )tan–( )–{ }tan+
tan[
]
cos=
γ γ
Kγ BγCγDγ
Fyo
γδ-------- at αγ∂ 0= = =
κ
α
γ
Adams/Tire24
 
The output variables are defined in the W-axis system of TYDEX.
Basic Tire Parameters
All tire model parameters of the model are without dimension. The reference parameters for the model 
are:
Basic Tire Parameters
As a measure for the vertical load, the normalized vertical load increment dfz is used:
(17)
with the possibly adapted nominal load (using the user-scaling factor, Fz0):
(18)
Nomenclature of the Tire Model Parameters
In the subsequent sections, formulas are given with non-dimensional parameters aijk with the following 
logic:
Tire Model Parameters
Overturning couple Mx [Nm]
Rolling resistance moment My [Nm]
Aligning moment Mz [Nm]
Nominal (rated) load Fz0 [N]
Unloaded tire radius R0 [m]
Tire belt mass mbelt [kg]
Parameter: Definition:
a = p Force at pure slip
q Moment at pure slip
r Force at combined slip
s Moment at combined slip
dfz
Fz F'z0–
F'z0
--------------------=
λ
F'z0 Fz0 λFz0⋅=
25
User Scaling Factors
A set of scaling factors is available to easily examine the influence of changing tire properties without 
the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. 
You can change the factors in the tire property file. The peak friction scaling factors, factors, and 
', are also used for the position-dependent friction in 3D Road Contact and Adams/3D Road. An 
overview of all scaling factors is shown in the next tables.
Scaling Factor Coefficients for Pure Slip
i = B Stiffness factor
C Shape factor
D Peak value
E Curvature factor
K Slip stiffness = BCD
H Horizontal shift
V Vertical shift
s Moment at combined slip
t Transient tire behavior
j = x Along the longitudinal axis
y Along the lateral axis
z About the vertical axis
k = 1, 2, ...
Name:
Name used in 
tire property file: Explanation:
Fzo LFZO Scale factor of nominal (rated) load
Cx LCX Scale factor of Fx shape factor
LMUX Scale factor of Fx peak friction coefficient
Ex LEX Scale factor of Fx curvature factor
Kx LKX Scale factor of Fx slip stiffness
Vx LVX Scale factor of Fx vertical shift
Hx LHX Scale factor of Fx horizontal shift
x LGAX Scale factor of camber for Fx
Cy LCY Scale factor of Fy shape factor for side slip
Parameter: Definition:
λμξ
λγψ
λ
λ
λμξ
λ
λ
λ
λ
λγ
λ
y LMUY Scale factor of Fy peak friction coefficientλμ
Adams/Tire26
 
Scaling Factor Coefficients for Combined Slip
Scaling Factor Coefficients for Transient Response
Steady-State: Magic Formula in PAC-TIME
• Steady-State Pure Slip
• Steady-State Combined Slip
Ey LEY Scale factor of Fy curvature factor
Ky LKY Scale factor of Fy cornering stiffness
Vy LVY Scale factor of Fy vertical shift
Hy LHY Scale factor of Fy horizontal shift
K LKC Scale factor of camber stiffness (K-factor)
LGAY Scale factor of camber force stiffness
t LTR Scale factor of peak of pneumatic trail
Mr LRES Scale factor for offset of residual torque
z LGAZ Scale factor of camber torque stiffness
Mx LMX Scale factor of overturning couple
VMx LVMX Scale factor of Mx vertical shift
My LMY Scale factor of rolling resistance torque
Name:
Name used in tire 
property file: Explanation:
x LXAL Scale factor of alpha influence on Fx
y LYKA Scale factor of alpha influence on Fx
Vy LVYKA Scale factor of kappa induced Fy
s LS Scale factor of moment arm of Fx
Name:
Name used in tire 
property file: Explanation:
LSGKP Scale factor of relaxation length of Fx
LSGAL Scale factor of relaxation length of Fy
gyr LGYR Scale factor of gyroscopic moment
Name:
Name used in 
tire property file: Explanation:
λ
λ
λ
λ
λ γ
λγψ
λ
λ
λγ
λ
λ
λ
λ α
λ κ
λ κ
λ
λσκ
λσα
λ
27
Steady-State Pure Slip
• Longitudinal Force at Pure Slip
• Lateral Force at Pure Slip
• Aligning Moment at Pure Slip
Formulas for the Longitudinal Force at Pure Slip
For the tire rolling on a straight line with no slip angle, the formulas are:
(19)
(20)
(21)
(22)
with following coefficients:
(23)
(24)
(25)
(26)
the longitudinal slip stiffness:
(27)
(28)
(29)
Fx Fx0 κ Fz γ, ,( )=
Fx0 Dx Cxarc Bxκx Ex Bxκx arc Bxκs( )tan–( )–{ }tan[ ] SVx+cos=
κx κ SHx+=
γx γ λγx⋅=
Cx pCx1 λcx⋅=
Dx μx Fz⋅=
μx pDx1 pDx2dfz+( ) 1 pDx3 γx2⋅( )γμx–⋅=
Ex pEx1 pEx2dfz pEx2dfz2+ +( ) 1 pEx4 κx( )sgn–{ } λEx with Ex 1≤⋅ ⋅=
Kx Fz pKx1 pKx2dfz+( ) pKx3dfz( )λK
Kx
exp⋅
BxCxDx κx∂
∂Fx0 at κx 0
=
= = =
Bx Kx CxDx( )⁄=
SHx pHx1 pHx2dfz+( )λHx=
SVx Fz pVx1 pVx2dfz+( ) λVx λμx⋅ ⋅ ⋅=
Adams/Tire28
 
Longitudinal Force Coefficients at Pure Slip
Formulas for the Lateral Force at Pure Slip
(30)
(31)
(32)
The scaled inclination angle:
(33)
with coefficients:
(34)
(35)
(36)
 (37)
The cornering stiffness:
Name:
Name used in tire 
property file: Explanation:
pCx1 PCX1 Shape factor Cfx for longitudinal force
pDx1 PDX1 Longitudinal friction Mux at Fznom
pDx2 PDX2 Variation of friction Mux with load
pDx3 PDX3 Variation of friction Mux with inclination
pEx1 PEX1 Longitudinal curvature Efx at Fznom
pEx2 PEX2 Variation of curvature Efx with load
pEx3 PEX3 Variation of curvature Efx with load squared
pEx4 PEX4 Factor in curvature Efx while driving
pKx1 PKX1 Longitudinal slip stiffness Kfx/Fz at Fznom
pKx2 PKX2 Variation of slip stiffness Kfx/Fz with load
Fy Fy0 α γ Fz, ,( )=
Fy0 Dy Cyarc Byαy Ey Byαy arc Byαy( )tan–( )–{ } 12---arc Bγγy( )tan+tan SVy+sin=
αy α SHy+=
γy γ λγy⋅=
Cy pCy1 λCy⋅=
Dy μy Fz⋅=
μy pDy1 pDy2dfz+( ) 1 pDy3γy2–( ) λμy⋅ ⋅=
Ey pEy1 pEy2dfz pEy3 pEy4γy+( ) αy( )sgn+ +{ } λEy⋅=
29
() (38)
(39)
(40)
(41)
 (42)
(43)
Lateral Force Coefficients at Pure Slip
Name:
Name used in tire 
property file: Explanation:
pCy1 PCY1 Shape factor Cfy for lateral forces
pDy1 PDY1 Lateral friction Muy
pDy2 PDY2 Variation of friction Muy with load
pDy3 PDY3 Variation of friction Muy with squared inclination
pEy1 PEY1 Lateral curvature Efy at Fznom
pEy2 PEY2 Variation of curvature Efy with load
pEy3 PEY3 Inclination dependency of curvature Efy
pEy4 PEY4 Variation of curvature Efy with inclination
pKy1 PKY1 Maximum value of stiffness Kfy/Fznom
pKy2 PKY2 Load at which Kfy reaches maximum value
pKy3 PKY3 Variation of Kfy/Fznom with inclination
pKy4 PKY4 Shape factor of Kfy
pKy5 PKY5 Linear variation of Kγ with load
Ky α∂
∂Fy pKy1Fz0 pKy4arc
Fz
pKy2Fz0λFz0
-------------------------------
⎩ ⎭⎨ ⎬
⎧ ⎫
tan 1 pKy3γy2–( ) λKyλFz0⋅ ⋅sin= =
with pKy4 2≤
Kγ γ∂
∂Fy Fz pKy5 pKy5dfz+( ) λKy⋅= =
By
Ky
CyDy
-------------=
Bγ
2Kγ
Dy
---------=
SHy pHy1 pHy2dfz+( ) λHy⋅=
SHy Fz pVy1 pVy2dfz+( ) λVyλμy⋅=
pKy6 PKY6 Quadratic variation of Kγ with load
Adams/Tire30
 
Formulas for the Aligning Moment at Pure Slip
(44)
with the pneumatic trail t:
(45)
and the residual moment Mzr:
(46)
(47)
(48)
The scaled inclination angle:
(49)
with coefficients:
(50)
(51)
(52)
pHy1 PHY1 Horizontal shift Shy at Fznom
pHy2 PHY2 Variation of shift Shy with load
pVy1 PVY1 Vertical shift in Svy/Fz at Fznom
pVy2 PVY2 Variation of shift Svy/Fz with load
Name:
Name used in tire 
propertyfile: Explanation:
M'z Mz0 α γ Fz, ,( )=
Mz0 t Fy0 SVy–( )γ 0= Mzr+⋅–=
t αt( ) Dt Ctarc Btαt Et Btαt arc Btαt( )tan–( )–{ }tan[ ] α( )coscos=
αt α SHt+=
Mzr αr( ) Dr arc Brαr( )tan[ ] α( )coscos=
αr α SHt+=
SHr 0=
γz γ λγz⋅=
Bt qBz1 qBz2dfz+( ) 1 qBz4γz2 qBz5 γz+ +( ) λk λμy⁄⋅( )=
Ct qCz1=
Dt Fz qDz1 qDz2dfz+( ) R0 Fz0⁄( ) λt⋅=
31
(53)
(54)
(55)
(56)
An approximation for the aligning moment stiffness reads:
(57)
and the aligning stiffness for inclination is:
(58)
Aligning Moment Coefficients at Pure Slip
Name:
Name used in tire 
property file: Explanation:
qBz1 QBZ1 Trail slope factor for trail Bpt at Fznom
qBz2 QBZ2 Variation of slope Bpt with load
qBz4 QBZ4 Variation of slope Bpt with inclination
qBz5 QBZ5 Variation of slope Bpt with absolute inclination
qCz1 QCZ1 Shape factor Cpt for pneumatic trail
qDz1 QDZ1 Peak trail Dpt = Dpt*(Fz/Fznom*R0)
qDz2 QDZ2 Variation of peak Dpt with load
qDz6 QDZ6 Peak residual moment Dmr = Dmr/ (Fz*R0)
qDz7 QDZ7 Variation of peak factor Dmr with load
qDz8 QDZ8 Variation of peak factor Dmr with inclination
qDz9 QDZ9 Variation of Dmr with inclination and load
Et qEz1 qEz2dfz+( ) 1 qEz42π---arc BtCtαt( )tan+⎩ ⎭⎨ ⎬
⎧ ⎫
with Et 1≤
=
SHt qHz1 qHz2dfz qHz3 qHz4dfz+( )γz+ +=
Br λKz λμy⁄=
Dr Fz qDz6 qDz7dfz+( )λr qDz8 qDz9dfz+( )γz+{ }R0λμy=
α∂
∂Mz tKy– Fz qDz1 qDz2dfz+( ) R0 Fz0⁄( ) pKy5 pKy6dfz+( )Fz λtλKy⋅= =
γd
dMz qDz8 qDz9dfz+( )R0Fzλμy=
qEz1 QEZ1 Trail curvature Ept at Fznom
Adams/Tire32
 
Steady-State Combined Slip
PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies 
the coefficients for the combined slip cosine 'weighing' functions, the combined slip is calculated 
according to Combined slip with cosine 'weighing' functions (standard method). If no coefficients are 
supplied, the so-called friction ellipse is used to estimate the combined slip forces and moments, see 
section Combined Slip with friction ellipse
Combined slip with cosine 'weighing' functions
• Longitudinal Force at Combined Slip
• Lateral Force at Combined Slip
• Aligning Moment at Combined Slip
• Overturning Moment at Pure and Combined Slip
• Rolling Resistance Moment at Pure and Combined Slip
Formulas for the Longitudinal Force at Combined Slip
(59)
with Gx the weighting function of the longitudinal force for pure slip.
We write:
(60)
(61)
with coefficients:
(62)
(63)
qEz2 QEZ2 Variation of curvature Ept with load
qEz4 QEZ4 Variation of curvature Ept with sign of Alpha-t
qHz1 QHZ1 Trail horizontal shift Sht at Fznom
qHz2 QHZ2 Variation of shift Sht with load
qHz3 QHZ3 Variation of shift Sht with inclination
qHz4 QHZ4 Variation of shift Sht with inclination and load
Name:
Name used in tire 
property file: Explanation:
Fx Fx0 Gxα α κ Fz, ,( )⋅=
α
Fx Dxα Cxαarc Bxααs Exα Bxααs arc Bxααs( )tan–( )–{ }tan[ ]cos=
αs α SHxα+=
Bxα rBx1 arc rBx2κ{ }tan[ ] λxα⋅cos=
Cxα rCx1=
33
(64)
(65)
(66)
The weighting function follows as:
(67)
Longitudinal Force Coefficients at Combined Slip
Formulas for Lateral Force at Combined Slip
(68)
with Gyk the weighting function for the lateral force at pure slip and SVyk the ' -induced' side force; 
therefore, the lateral force can be written as:
(69)
(70)
with the coefficients:
(71)
(72)
Name:
Name used in tire 
property file: Explanation:
rBx1 RBX1 Slope factor for combined slip Fx reduction
rBx2 RBX2 Variation of slope Fx reduction with kappa
rCx1 RCX1 Shape factor for combined slip Fx reduction
rEx1 REX1 Curvature factor of combined Fx
rEx2 REX2 Curvature factor of combined Fx with load
rHx1 RHX1 Shift factor for combined slip Fx reduction
Dxα
Fxo
Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–{ }tan[ ]cos
------------------------------------------------------------------------------------------------------------------------------------------------------------------=
Exα rEx1 rEx2dfz with Exα 1≤+=
SHxα rHx1=
Gxα
Cxαarc Bxααs Exα Bxααs arc Bxααs( )tan–( )–{ }tan[ ]cos
Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–{ }tan[ ]cos
------------------------------------------------------------------------------------------------------------------------------------------------------------------=
Fy Fy0 Gyκ α κ γ Fz, , ,( ) SVyκ+⋅=
κ
Fy Dyκ Cyκarc Byκκs Eyκ Bxακs arc Byκκs( )tan–( )–{ }tan[ ] SVyκ+cos=
κs κ SHyκ+=
Byκ rBy1 arc rBy2 α rBy3–( ){ }tan[ ] λyκ⋅cos=
Cyκ rCy1=
Adams/Tire34
 
(73)
(74)
(75)
(76)
The weighting function appears is defined as:
(77)
Lateral Force Coefficients at Combined Slip
Formulas for Aligning Moment at Combined Slip
(78)
with:
Name:
Name used in tire 
property file: Explanation:
rBy1 RBY1 Slope factor for combined Fy reduction
rBy2 RBY2 Variation of slope Fy reduction with alpha
rBy3 RBY3 Shift term for alpha in slope Fy reduction
rCy1 RCY1 Shape factor for combined Fy reduction
rEy1 REY1 Curvature factor of combined Fy
rEy2 REY2 Curvature factor of combined Fy with load
rHy1 RHY1 Shift factor for combined Fy reduction
rHy2 RHY2 Shift factor for combined Fy reduction with load
rVy1 RVY1 Kappa induced side force SVyk/μy·Fz at Fznom
rVy2 RVY2 Variation of SVyk/μy·Fz with load
rVy3 RVY3 Variation of SVyk/μy·Fz with inclination
rVy4 RVY4 Variation of SVyk/μy·Fz with α
rVy5 RVY5 Variation of SVyk/μy·Fz with κ
rVy6 RVY6 Variation of SVyk/μy·Fz with atan(κ)
Dyκ
Fyo
Cyκarc ByκSHyκ Eyκ BxαSHyκ arc ByκSHyκ( )tan–( )–{ }tan[ ]cos
---------------------------------------------------------------------------------------------------------------------------------------------------------------=
Eyκ rEy1 rEy2dfz with Eyκ 1≤+=
SHyκ rHy1 rHy2dfz+=
DVyκ μyFz rVy1 rVy2dfz rVy3γ+ +( ) arc rVy4α( )tan[ ]cos⋅ ⋅=
Gyκ
Cyκarc Byκκs Eyκ Bxακs arc Byκκs( )tan–( )–{ }tan[ ]cos
Cyκarc ByκSHyκ Eyκ BxαSHyκ arc ByκSHyκ( )tan–( )–{ }tan[ ]cos
---------------------------------------------------------------------------------------------------------------------------------------------------------------=
M'z t F'y Mzr s Fx⋅+ +⋅–=
(79)t t αt eq,( )=
35
(80)
(81)
(82)
with the arguments:
(83)
(84)
Aligning Moment Coefficients at Combined Slip
Formulas for Overturning Moment at Pure and Combined Slip
For the overturning moment, the formula reads both for pure and combined slip situations:
(85)
Overturning Moment Coefficients
Name:
Name used in tire 
property file: Explanation:
ssz1 SSZ1 Nominal value of s/R0 effect of Fx on Mz
ssz2 SSZ2 Variation of distance s/R0 with Fy/Fznom
ssz3 SSZ3 Variation of distance s/R0 with inclination
ssz4 SSZ4 Variation of distance s/R0 with load and inclination
Name:
Name used in tire 
property file: Explanation:
qsx1 QSX1 Lateral force induced overturning couple
qsx2 QSX2 Inclination induced overturning couple
qsx3 QSX3 Fy induced overturning couple
Dt Ctarc Btαt eq, Et Btαt eq, arc Btαt eq,( )tan–( )–{ }tan[ ] α( )coscos=
F'y γ, 0= Fy SVyκ–=
Mzr Mzr αr eq,( ) Dr arc Brαr eq,( )tan[ ] α( )coscos= =
t t αt eq,( )=
αt eq, arc α2 t
Kx
Ky
------⎝ ⎠⎛ ⎞
2κ2 αt( )sgn⋅+tantan=
αr eq, arc α2 r
Kx
Ky
------⎝ ⎠⎛ ⎞
2κ2 αr( )sgn⋅+tantan=
Mx Ro Fz qsx1λVMx qsx2 qsx3
Fy
Fz0
--------⋅+–
⎩ ⎭⎨ ⎬
⎧ ⎫λMx⋅ ⋅=
Adams/Tire36
 
Formulas for Rolling Resistance Moment at Pure and Combined Slip
The rolling resistance moment is defined by:
(86)
Rolling Resistance Coefficients
Combined Slip with friction ellipse
In case the tire property file does not contain the coefficients for the 'standard' combined slip method 
(cosine 'weighing functions), the friction ellipse method is used, as described in this section. Note that 
the method employed here is not part of one of the Magic Formula publications by Pacejka, but is an in-
house development of MSC.Software.The following friction coefficients are defined:
Name:
Name used in tire 
property file: Explanation:
qsy1 QSY1 Rolling resistance moment coefficient
qsy2 QSY2 Rolling resistance moment depending on Fx
qsy3 QSY3 Rolling resistance moment depending on speed
qsy4 QSY4 Rolling resistance moment depending on speed^4
Vref LONGVL Measurement speed
My Ro Fz qSy1 qSy2Fx Fz0⁄ qSy3 Vx Vref( )⁄ qSy4 Vx Vref⁄( )4+ +{ }⋅ ⋅=
κc κ SHx
SVx
Kx
---------+ +=
αc α SHy
SVy
Ky
---------+ +=
α∗ αc( )sin=
β κc
κc2 α∗2+
-------------------------⎝ ⎠⎜ ⎟
⎛ ⎞
acos=
μx act,
Fx 0, SVx–
Fz
-------------------------= μy act,
Fy 0, SVy–
Fz
-------------------------=
μx max,
Dx------= μy max,
Dy------=
Fz Fz
37
The forces corrected for the combined slip conditions are: 
For aligning moment Mx, rolling resistance My and aligning moment Mz the formulae (76) until and 
including (84) are used with =0.
Transient Behavior in PAC-TIME
The previous Magic Formula equations are valid for steady-state tire behavior. When driving, however, 
the tire requires some response time on changes of the inputs. In tire modeling terminology, the low-
frequency behavior (up to 8 Hz) is called transient behavior.
For accurate transient tire behavior, you can use the stretched string tire model (see reference [1]). The 
tire belt is modeled as stretched string, which is supported to the rim with lateral (and longitudinal) 
springs. Stretched String Model for Transient Tire Behavior shows a top-view of the string model. When 
rolling, the first point having contact with the road adheres to the road (no sliding assumed). Therefore, 
a lateral deflection of the string arises that depends on the slip angle size and the history of the lateral 
deflection of previous points having contact with the road.
Stretched String Model for Transient Tire Behavior
μx 1
1
μx act,
-------------⎝ ⎠⎛ ⎞
2 βtan
μy max,
----------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------------=
μy βtan
1
μx max,
----------------⎝ ⎠⎛ ⎞
2 βtan
μy act,
-------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------------=
Fx
μx
μx act,
-------------Fx 0,= Fy
μy
μy act,
-------------Fy 0,=
SVyκ
Adams/Tire38
 
 
For calculating the lateral deflection v1 of the string in the first point of contact with the road, the 
following differential equation is valid:
(87)
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger 
than 10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the 
equation can be transformed to:
(88)
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill.
A similar approach yields the following for the deflection of the string in longitudinal direction:
1
Vx
------
td
dv1 v1
σα
------+ α( ) aφ+tan=
σα φ
σα td
dv1 Vx v1+ σαVsy=
39
(89)
Both the longitudinal and lateral relaxation length are defined as of the vertical load:
(90)
(91)
Now the practical slip quantities, and , are defined based on the tire deformation:
(92)
(93)
Using these practical slip quantities, and , the Magic Formula equations can be used to calculate 
the tire-road interaction forces and moments:
(94)
(95)
(96)
Gyroscopic Couple in PAC-TIME
When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead 
to gyroscopic effects. To cope with this additional moment, the following contribution is added to the 
total aligning moment:
(97)
with the parameters (in addition to the basic tire parameter mbelt):
(98)
σx td
du1 Vx u1+ σxVsx–=
σx Fz pTx1 pTx2dfz+( ) pTx3dfz( ) R0 Fz0⁄( )λσκ⋅exp⋅ ⋅=
σα pTy1Fz0 pKy4arc
Fz
pTy2Fz0λFz0
-----------------------------
⎩ ⎭⎨ ⎬
⎧ ⎫
tan 1 pKy3γ2–( ) R0λFz0λσα⋅ ⋅sin=
κ' α'
κ' u1σx
------ Vx( )sin=
α' v1σα
------⎝ ⎠⎛ ⎞atan=
κ' α'
Fx Fx α' κ' Fz, ,( )=
Fy Fy α' κ' γ Fz, , ,( )=
Mz Mz α' κ' γ Fz, , ,( )=
Mz gyr, cgyrmbeltVr1 td
dv arc Brαr eq,( )tan[ ]cos=
cgyr qTz1 λgyr⋅=
and:
Adams/Tire40
 
(99)
The total aligning moment now becomes:
Coefficients and Transient Response
Left and Right Side Tires
In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire 
construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for 
positive and negative slip angles.
A tire property file with the parameters for the model results from testing with a tire that is mounted in a 
tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used for 
both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering 
wheel angle.
The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that 
indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or 
TIRESIDE = 'RIGHT').
If this keyword is available, Adams/Car corrects for the conicity, plysteer, and asymmetry when using a 
tire property file on the opposite side of the vehicle. In fact, the tire characteristics are mirrored with 
respect to slip angle zero. In Adams/View, this option can only be used when the tire is generated by the 
graphical user interface: select Build -> Forces -> Special Force: Tire, as shown in the figure below. 
Create Wheel and Tire Dialog Box in Adams/View
Name:
Name used in tire 
property file: Explanation:
pTx1 PTX1 Longitudinal relaxation length at Fznom
pTx2 PTX2 Variation of longitudinal relaxation length with load
pTx3 PTX3 Variation of longitudinal relaxation length with exponent 
of load
pTy1 PTY1 Peak value of relaxation length for lateral direction
pTy2 PTY2 Shape factor for lateral relaxation length
qTz1 QTZ1 Gyroscopic moment constant
Mbelt MBELT Belt mass of the wheel
arc Brαr eq,( )tan[ ]cos 1=
Mz Mz' Mz gyr,+=
41
Next to the LEFT and RIGHT side option of TYRESIDE, you can also set SYMMETRIC: then the tire 
characteristics are modified during initialization to show symmetric performance for left and right side 
corners and zero conicity and plysteer (no offsets). Also, when you set the tire property file to 
SYMMETRIC, the tire characteristics are changed to symmetric behavior.
USE_MODES of PAC-TIME: from Simple to Complex
The parameter USE_MODE in the tire property file allows you to switch the output of the PAC-TIME 
tire model from very simple (that is, steady-state cornering) to complex (transient combined cornering 
and braking).
Adams/Tire42
 
The options for the USE_MODE and the output of the model have been listed in the table below.
USE_MODE Values of PAC-TIME and Related Tire Model Output
Quality Checks for the Tire Model Parameters
Because PAC-TIME uses an empirical approach to describe tire - road interaction forces, incorrect 
parameters can easily result in non-realistic tire behavior. Below is a list of the most important items to 
ensure the quality of the parameters in a tire property file:
• Rolling Resistance
• Camber (Inclination) Effects
• Validity Range of the Tire Model Input
Rolling Resistance
For a realistic rolling resistance, the parameter qsy1 must be positive. For car tires, it can be in the order 
of 0.006 - 0.01 (0.6% - 1.0%).
$---------------------------------------------------rolling 
resistance
[ROLLING_COEFFICIENTS]
QSY1 = 0.01
USE_MODE: State: Slip conditions:PAC-TIME output
(forces and moments):
0 Steady state Acts as a vertical spring and 
damper
0, 0, Fz, 0, 0, 0
1 Steady state Pure longitudinal slip Fx, 0, Fz, 0, My, 0
2 Steady state Pure lateral (cornering) slip 0, Fy, Fz, Mx, 0, Mz
3 Steady state Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
4 Steady state Combined slip Fx, Fy, Fz, Mx, My, Mz
11 Transient Pure longitudinal slip Fx, 0, Fz, 0, My, 0
12 Transient Pure lateral (cornering) slip 0, Fy, Fz, Mx, 0, Mz
13 Transient Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
14 Transient Combined slip Fx, Fy, Fz, Mx, My, Mz
Note: Do not change Fz0 (FNOMIN) and R0 (UNLOADED_RADIUS) in your tire property 
file. It will change the complete tire characteristics because these two parameters are used 
to make all parameters without dimension.
QSY2 = 0
QSY3 = 0
43
QSY4 = 0
Camber (Inclination) Effects
Camber stiffness has been explicitly defined in PAC-TIME, so camber stiffness can be easily checked 
by the tire model parameters itself, see the table, Checklist for PAC-TIME Parameters and Properties, 
below. For car tires, positive inclination should result in a negative lateral force at zero slip angle (see 
Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System below). If positive 
inclination results in an increase of the lateral force, the coefficient may not be valid for the ISO, but for 
the SAE coordinate system. Note that PAC-TIME only uses coefficients for the TYDEX W-axis (ISO) 
system.
Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System
 
The following table lists further checks on the PAC-TIME parameters.
Checklist for PAC-TIME Parameters and Properties
Parameter/property: Requirement: Explanation:
LONGVL 1 m/s Reference velocity at which parameters are measured
VXLOW Approximately 1 m/s Threshold for scaling down forces and moments
Dx > 0 Peak friction (see equation (24))
Adams/Tire44
 
Validity Range of the Tire Model Input
In the tire property file, a range of the input variables has been given in which the tire properties are 
supposed to be valid. These validity range parameters are (the listed values can be different):
$-----------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip 
KPUMAX = 1.5 $Maximum valid wheel slip 
$----------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip angle 
ALPMAX = 1.5708 $Maximum valid slip angle 
$----------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
CAMMIN = -0.26181 $Minimum valid camber angle 
CAMMAX = 0.26181 $Maximum valid camber angle 
$------------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 225 $Minimum allowed wheel load 
FZMAX = 10125 $Maximum allowed wheel load 
If one of the input parameters exceeds a minimum or maximum validity value, the calculation in the tire 
model is performed with the minimum or maximum value of this range to avoid non-realistic tire 
behavior. In that case, a message appears warning you that one of the inputs exceeds a validity value.
Standard Tire Interface (STI) for PAC-TIME
Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to 
Adams/Solver, below is a brief background of the STI history (see reference [4]).
At the First International Colloquium on Tire Models for Vehicle Dynamics Analysis on October 21-22, 
1991, the International Tire Workshop working group was established (TYDEX).
The working group concentrated on tire measurements and tire models used for vehicle simulation 
purposes. For most vehicle dynamics studies, people previously developed their own tire models. 
pDx1/pDx2 < 0 Peak friction Fx must decrease with increasing load
Kx > 0 Long slip stiffness (see equation (27))
Dy > 0 Peak friction (see equation (35))
pDy1/pDy2 < 0 Peak friction Fx must decrease with increasing load
Ky < 0 Cornering stiffness (see equation (38))
Kg < 0 Camber stiffness (see equation (39))
qsy1 > 0 Rolling resistance, in the range of 0.005 - 0.015
Parameter/property: Requirement: Explanation:
45
Because all car manufacturers and their tire suppliers have the same goal (that is, development of tires 
to improve dynamic safety of the vehicle), it aimed for standardization in the tire behavior description.
In TYDEX, two expert groups, consisting of participants of vehicle industry (passenger cars and trucks), 
tire manufacturers, other suppliers and research laboratories, had been defined with following goals:
• The first expert group's (Tire Measurements - Tire Modeling) main goal was to specify an 
interface between tire measurements and tire models. The result was the TYDEX-Format [2] to 
describe tire measurement data.
• The second expert group's (Tire Modeling - Vehicle Modeling) main goal was to specify an 
interface between tire models and simulation tools, which resulted in the Standard Tire Interface 
(STI) [3]. The use of this interface should ensure that a wide range of simulation software can be 
linked to a wide range of tire modeling software.
Definitions
• General
• Tire Kinematics
• Slip Quantities
• Force and Moments
General
General Definitions
Term: Definition:
Road tangent plane Plane with the normal unit vector (tangent to the road) in the tire-road contact 
point C.
C-axis system Coordinate system mounted on the wheel carrier at the wheel center according to 
TYDEX, ISO orientation.
Wheel plane The plane in the wheel center that is formed by the wheel when considered a rigid 
disc with zero width.
Contact point C Contact point between tire and road, defined as the intersection of the wheel 
plane and the projection of the wheel axis onto the road plane.
W-axis system Coordinate system at the tire contact point C, according to TYDEX, ISO 
orientation.
Adams/Tire46
 
Tire Kinematics
Tire Kinematics Definitions
Slip Quantities
Slip Quantities Definitions
Forces and Moments
Force and Moment Definitions
Parameter: Definition: Units:
R0 Unloaded tire radius [m]
R Loaded tire radius [m]
Re Effective tire radius [m]
Radial tire deflection [m]
d Dimensionless radial tire deflection [-]
Fz0 Radial tire deflection at nominal load [m]
mbelt Tire belt mass [kg]
Rotational velocity of the wheel [rads-1]
Parameter: Definition: Units:
V Vehicle speed [ms-1]
Vsx Slip speed in x direction [ms-1]
Vsy Slip speed in y direction [ms-1]
Vs Resulting slip speed [ms-1]
Vx Rolling speed in x direction [ms-1]
Vy Lateral speed of tire contact center [ms-1]
Vr Linear speed of rolling [ms-1]
Longitudinal slip [-]
Slip angle [rad]
Inclination angle [rad]
Abbreviation: Definition: Units:
Fz Vertical wheel load [N]
Fz0 Nominal load [N]
dfz Dimensionless vertical load [-]
ρ
ρ
ρ
ω
κ
α
γ
Fx Longitudinal force [N]
47
References
1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5.
2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, 
Initiated by the International Tire Working Group, July 1995.
3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995.
4. J.J.M. van Oosten, H.-J. Unrau, G. Riedel, E. Bakker, TYDEX Workshop: Standardisation of 
Data Exchange in Tyre Testing and Tyre Modelling, Proceedings of the 2nd International 
Colloquium on Tyre Models for Vehicle Dynamics Analysis, Vehicle System Dynamics, Volume 
27, Swets & Zeitlinger, Amsterdam/Lisse, 1996.
5. J.J.M. van Oosten, E. Kuiper, G. Leister, D. Bode, H. Schindler, J. Tischleder, S. Köhne,A new 
tyre model for TIME measurement data,TireTechnology Expo 2003, Hannover.
Example of PAC-TIME Tire Property File
[MDI_HEADER]
FILE_TYPE ='tir'
FILE_VERSION =3.0
FILE_FORMAT ='ASCII'
! : TIRE_VERSION : MF-TIME
! : COMMENT : Tire 205/55 R16 90H 
! : COMMENT : Manufacturer Continental
! : COMMENT : Nom. section with (m) 0.205 
! : COMMENT : Nom. aspect ratio (-) 55
! : COMMENT : Infl. pressure (Pa) 250000
! : COMMENT : Rim radius (m) 0.2032 
! : COMMENT : Measurement ID 
! : COMMENT : Test speed (m/s) 11.11 
! : COMMENT : Road surface 
! : COMMENT : Road condition 
! : FILE_FORMAT : ASCII
! : Copyright MSC.Software, Thu Oct 14 13:52:26 2004
!
! USE_MODE specifies the type of calculation performed:
! 0: Fz only, no Magic Formula evaluation
! 1: Fx,My only
! 2: Fy,Mx,Mz only
! 3: Fx,Fy,Mx,My,Mz uncombined force/moment calculation
Fy Lateral force [N]
Mx Overturning moment [Nm]
My Braking/driving moment [Nm]
Mz Aligning moment [Nm]
Abbreviation: Definition: Units:
! 4: Fx,Fy,Mx,My,Mz combined force/moment calculation
! +10: including relaxation behaviour
Adams/Tire48
 
! *-1: mirroring of tyre characteristics
!
! example: USE_MODE = -12 implies:
! -calculation of Fy,Mx,Mz only
! -including relaxation effects
! -mirrored tyre characteristics
!
$---------------------------------------------------------------units
[UNITS]
LENGTH ='meter'
FORCE ='newton'
ANGLE ='radians'
MASS ='kg'
TIME ='second'
$---------------------------------------------------------------model
[MODEL]
PROPERTY_FILE_FORMAT ='PAC-TIME'
USE_MODE = 14 $Tyre use switch (IUSED)
VXLOW = 2 
LONGVL = 30 $Measurement speed 
TYRESIDE = 'LEFT' $Mounted side of tyre 
at vehicle/test bench
$----------------------------------------------------------dimensions
[DIMENSION]
UNLOADED_RADIUS = 0.317 $Free tyre radius 
WIDTH = 0.205 $Nominal section width 
of the tyre 
ASPECT_RATIO = 0.55 $Nominal aspect ratio
RIM_RADIUS = 0.203 $Nominal rim radius 
RIM_WIDTH = 0.165 $Rim width 
$-----------------------------------------------------------parameter
[VERTICAL]
VERTICAL_STIFFNESS = 2.648e+005 $Tyre vertical 
stiffness 
VERTICAL_DAMPING = 500 $Tyre vertical damping 
BREFF = 4.90 $Low load stiffness 
e.r.r. 
DREFF = 0.41 $Peak value of e.r.r. 
FREFF = 0.09 $High load stiffness 
e.r.r. 
FNOMIN = 4704 $Nominal wheel load
$-----------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip 
KPUMAX = 1.5 $Maximum valid wheel slip 
$----------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip 
angle 
ALPMAX = 1.5708 $Maximum valid slip 
angle 
$----------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
49
CAMMIN = -0.26181 $Minimum valid camber 
angle 
CAMMAX = 0.26181 $Maximum valid camber 
angle 
$-----------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 140 $Minimum allowed wheel 
load 
FZMAX = 10800 $Maximum allowed wheel 
load 
$------------------------------------------------------------scaling
[SCALING_COEFFICIENTS]
LFZO = 1 $Scale factor of nominal 
load 
LCX = 1 $Scale factor of Fx 
shape factor 
LMUX = 1 $Scale factor of Fx 
peak friction coefficient 
LEX = 1 $Scale factor of Fx 
curvature factor 
LKX = 1 $Scale factor of Fx 
slip stiffness 
LHX = 1 $Scale factor of Fx 
horizontal shift 
LVX = 1 $Scale factor of Fx 
vertical shift 
LGAX = 1 $Scale factor of camber 
for Fx 
LCY = 1 $Scale factor of Fy 
shape factor 
LMUY = 1 $Scale factor of Fy 
peak friction coefficient 
LEY = 1 $Scale factor of Fy 
curvature factor 
LKY = 1 $Scale factor of Fy 
cornering stiffness 
LHY = 1 $Scale factor of Fy 
horizontal shift 
LVY = 1 $Scale factor of Fy 
vertical shift 
LKC = 1 $Scale factor of camber 
stiffness
LGAY = 1 $Scale factor of camber 
for Fy 
LTR = 1 $Scale factor of Peak 
of pneumatic trail 
LRES = 1 $Scale factor of Peak 
of residual torque 
LGAZ = 1 $Scale factor of camber 
torque stiffness 
LXAL = 1 $Scale factor of alpha 
influence on Fx 
LYKA = 1 $Scale factor of kappa 
influence on Fy 
Adams/Tire50
 
LVYKA = 1 $Scale factor of kappa 
induced Fy 
LS = 1 $Scale factor of Moment 
arm of Fx 
LSGKP = 1 $Scale factor of 
Relaxation length of Fx 
LSGAL = 1 $Scale factor of 
Relaxation length of Fy 
LGYR = 1 $Scale factor of 
gyroscopic torque 
LMX = 1 $Scale factor of 
overturning couple 
LVMX = 1 $Scale factor of Mx 
vertical shift 
LMY = 1 $Scale factor of rolling 
resistance torque 
$--------------------------------------------------------longitudinal
[LONGITUDINAL_COEFFICIENTS]
PCX1 = 1.3178 $Shape factor Cfx for 
longitudinal force 
PDX1 = 1.0455 $Longitudinal friction 
Mux at Fznom 
PDX2 = 0.063954 $Variation of friction 
Mux with load 
PDX3 = 0 $Variation of friction 
Mux with camber 
PEX1 = 0.15798 $Longitudinal curvature 
Efx at Fznom 
PEX2 = 0.41141 $Variation of curvature 
Efx with load 
PEX3 = 0.1487 $Variation ofcurvature 
Efx with load squared 
PEX4 = 3.0004 $Factor in curvature 
Efx while driving 
PKX1 = 23.181 $Longitudinal slip 
stiffness Kfx/Fz at Fznom 
PKX2 = -0.037391 $Variation of slip 
stiffness Kfx/Fz with load 
PKX3 = 0.80348 $Exponent in slip 
stiffness Kfx/Fz with load 
PHX1 = -0.00058264 $Horizontal shift Shx 
at Fznom 
PHX2 = -0.0037992 $Variation of shift Shx 
with load 
PVX1 = 0.045118 $Vertical shift Svx/Fz 
at Fznom 
PVX2 = 0.058244 $Variation of shift 
Svx/Fz with load 
RBX1 = 13.276 $Slope factor for 
combined slip Fx reduction 
RBX2 = -13.778 $Variation of slope Fx 
reduction with kappa 
RCX1 = 1.0 $Shape factor for 
combined slip Fx reduction 
51
REX1 = 0 $Curvature factor of 
combined Fx 
REX2 = 0 $Curvature factor of 
combined Fx with load 
RHX1 = 0 $Shift factor for 
combined slip Fx reduction 
PTX1 = 0.85683 $Relaxation length 
SigKap0/Fz at Fznom 
PTX2 = 0.00011176 $Variation of SigKap0/Fz 
with load 
PTX3 = -1.3131 $Variation of SigKap0/Fz 
with exponent of load 
$--------------------------------------------------------overturning
[OVERTURNING_COEFFICIENTS]
QSX1 = 0 $Lateral force induced 
overturning moment 
QSX2 = 0 $Camber induced 
overturning moment 
QSX3 = 0 $Fy induced overturning 
moment 
$------------------------------------------------------------lateral
[LATERAL_COEFFICIENTS]
PCY1 = 1.18 $Shape factor Cfy for 
lateral forces 
PDY1 = 0.90312 $Lateral friction Muy 
PDY2 = -0.17023 $Exponent lateral 
friction Muy 
PDY3 = -0.76512 $Variation of friction 
Muy with squared camber 
PEY1 = -0.57264 $Lateral curvature Efy 
at Fznom 
PEY2 = -0.13945 $Variation of curvature 
Efy with load 
PEY3 = 0.030873 $Zero order camber 
dependency of curvature Efy 
PEY4 = 0 $Variation of curvature 
Efy with camber 
PKY1 = -25.128 $Maximum value of 
stiffness Kfy/Fznom 
PKY2 = 3.2018 $Load with peak of 
cornering stiffness 
PKY3 = 0 $Variation with camber 
squared of cornering stiffness 
PKY4 = 1.9998 $Shape factor for 
cornering stiffness with load 
PKY5 = -0.50726 $Camber stiffness/Fznom 
PKY6 = 0 $Camber stiffness 
depending on Fz squared 
PHY1 = 0.0031414 $Horizontal shift Shy 
at Fznom 
PHY2 = 0 $Variation of shift 
Shy with load 
PVY1 = 0.0068801 $Vertical shift in 
Svy/Fz at Fznom 
Adams/Tire52
 
PVY2 = -0.0051 $Variation of shift Shv 
with load 
RBY1 = 7.1433 $Slope factor for 
combined Fy reduction 
RBY2 = 9.1916 $Variation of slope Fy 
reduction with alpha 
RBY3 = -0.027856 $Shift term for alpha 
in slope Fy reduction 
RCY1 = 1.0 $Shape factor for 
combined Fy reduction 
REY1 = 0 $Curvature factor of 
combined Fy 
REY2 = 0 $Curvature factor of 
combined Fy with load 
RHY1 = 0 $Shift factor for 
combined Fy reduction 
RHY2 = 0 $Shift factor for 
combined Fy reduction with load 
RVY1 = 0 $Kappa induced side 
force Svyk/Muy*Fz at Fznom 
RVY2 = 0 $Variation of Svyk/Muy*Fz 
with load 
RVY3 = 0 $Variation of Svyk/Muy*Fz 
with camber 
RVY4 = 0 $Variation of Svyk/Muy*Fz 
with alpha 
RVY5 = 0 $Variation of Svyk/Muy*Fz 
with kappa 
RVY6 = 0 $Variation of Svyk/Muy*Fz 
with atan(kappa) 
PTY1 = 4.1114 $Peak value of relaxation 
length SigAlp0/R0 
PTY2 = 6.1149 $Value of Fz/Fznom where 
SigAlp0 is extreme 
$--------------------------------------------------rolling resistance
[ROLLING_COEFFICIENTS]
QSY1 = 0.01 $Rolling resistance 
torque coefficient 
QSY2 = 0 $Rolling resistance 
torque depending on Fx 
QSY3 = 0 $Rolling resistance 
torque depending on speed 
QSY4 = 0 $Rolling resistance 
torque depending on speed^4 
$-----------------------------------------------------------aligning
[ALIGNING_COEFFICIENTS]
QBZ1 = 5.6241 $Trail slope factor for 
trail Bpt at Fznom 
QBZ2 = -2.2687 $Variation of slope Bpt 
with load 
QBZ4 = 6.891 $Variation of slope Bpt 
with camber 
QBZ5 = -0.35587 $Variation of slope Bpt 
with absolute camber 
53
QCZ1 = 1.0904 $Shape factor Cpt for 
pneumatic trail 
QDZ1 = 0.082871 $Peak trail Dpt = 
Dpt*(Fz/Fznom*R0) 
QDZ2 = -0.012677 $Variation of peak Dpt 
with load 
QDZ6 = 0.00038069 $Peak residual torque 
Dmr = Dmr/(Fz*R0) 
QDZ7 = 0.00075331 $Variation of peak 
factor Dmr with load 
QDZ8 = -0.083385 $Variation of peak 
factor Dmr with camber 
QDZ9 = 0 $Variation of peak 
factor Dmr with camber and load 
QEZ1 = -34.759 $Trail curvature Ept 
at Fznom 
QEZ2 = -37.828 $Variation of curvature 
Ept with load 
QEZ4 = 0.59942 $Variation of curvature 
Ept with sign of Alpha-t 
QHZ1 = 0.0025743 $Trail horizontal shift 
Sht at Fznom 
QHZ2 = -0.0012175 $Variation of shift 
Sht with load 
QHZ3 = 0.038299 $Variation of shift 
Sht with camber 
QHZ4 = 0.044776 $Variation of shift 
Sht with camber and load 
SSZ1= 0.0097546 $Nominal value of s/R0: 
effect of Fx on Mz 
SSZ2 = 0.0043624 $Variation of distance 
s/R0 with Fy/Fznom 
SSZ3 = 0 $Variation of distance 
s/R0 with camber 
SSZ4 = 0 $Variation of distance 
s/R0 with load and camber 
QTZ1 = 0 $Gyroscopic torque 
constant 
MBELT = 0 $Belt mass of the wheel 
-kg- 
Contact Methods
The PAC-TIME model supports the following roads:
• 2D Roads, see Using the 2D Road Model
• 3D Spline Roads, see Adams/3D Spline Road Model
Adams/Tire54
 
Note that the PAC-TIME model has only one point of contact with the road; therefore, the 
wavelength of road obstacles must be longer than the tire radius for realistic output of the model. 
In addition, the contact force computed by this tire model is normal to the road plane. Therefore, 
the contact point does not generate a longitudinal force when rolling over a short obstacle, such 
as a cleat or pothole.
• 3D Shell Roads, see Adams/Tire 3D Shell Road Model
For ride and comfort analyses, we recommend more sophisticated tire models, such as Ftire.
Using Pacejka '89 and '94 Models
Adams/Tire provides you with the handling force models, Pacejka '89 and Pacejka '94. 
• About Pacejka '89 and '94
• Using Pacejka '89 Handling Force Model
• Using Pacejka '94 Handling Force Model
• Combined Slip
• Left and Right Side Tires
• Contact Methods
About Pacejka '89 and '94
The Pacejka '89 and '94 handling models are special versions of the Magic-Formula Tyre model as cited 
in the following publications:
• Pacejka '89 - H.B Pacejka, E. Bakker, and L. Lidner. A New Tire Model with an Application in 
Vehicle Dynamics Studies, SAE paper 890087, 1989.
• Pacejka '94 - H.B Pacejka and E. Bakker. The Magic Formula Tyre Model. Proceedings of the 1st 
International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Swets & Zeitlinger 
B.V., Amsterdam/Lisse, 1993.
PAC2002 is technically superior, continuously kept up to date with latest Magic Formula developments, 
and MSC’s recommended handling model. However, because many Adams/Tire users have pre-existing 
tire data or new data from tire suppliers and testing organizations in a format that is compatible with the 
Pacejka '89 and '94 models, the Adams/Tire Handling module includes these models in addition to the 
PAC2002.
The material in this help is intended to illustrate only the formulas used in the Pacejka '89 and '94 tire 
models. For general information on the PAC2002 and the Magic Formula method, see the papers cited 
above or the PAC2002 help.
• History of the Pacejka Name in Adams/Tire
• About Coordinate Systems
• Normal Force
History of the Pacejka Name in Adams/Tire
The formulas used in the Pacejka '89 and '94 tire models are derived from publications by Dr. H.B. 
Pacejka, and are commonly referred to as the Pacejka method in the automotive industry. Dr. Pacejka 
himself is not personally associated with the development of these tire models, nor does he endorse them 
in any way.
About Coordinate Systems
The coordinate systems used in tire modeling and measurement are sometimes confusing. The coordinate 
systems employed in the Pacejka ’89 and ’94 tire models are no exception. They are derived from the 
Adams/Tire12
 
tire-measurement systems that the majority of Adams/Tire customers were using at the time when the 
models were originally developed.
The Pacejka '89 and '94 tire models were developed before the implementation of the TYDEX STI. As a 
result, Pacejka ’89 conforms to a modified SAE-based tire coordinate system and sign conventions, and 
Pacejka ’94 conforms to the standard SAE tire coordinate system and sign conventions. MSC maintains 
these conventions to ensure file compatibility for Adams/Tire customers.
Future tire models will adhere to one single coordinate system standard, the TYDEX C-axis and W-axis 
system. For more information on the TYDEX standard, see Standard Tire Interface (STI).
Normal Force
The normal force Fz is calculated assuming a linear spring (stiffness: kz) and damper (damping constant 
cz), so the next equation holds: 
If the tire loses contact with the road, the tire deflection and deflection velocity become zero, so the 
resulting normal force Fz will also be zero. For very small positive tire deflections, the value of the 
damping constant is reduced and care is taken to ensure that the normal force Fz will not become 
negative.
In stead of the linear vertical tire stiffness cz , also an arbitrary tire deflection - load curve can be defined 
in the tire property file in the section [DEFLECTION_LOAD_CURVE], see also the example tire 
property files, Example of Pacejka ’89 Property File and Example of Pacejka ’94 Property File. If a section 
called [DEFLECTION_LOAD_CURVE] exists, the load deflection datapoints with a cubic spline for 
inter- and extrapolation are used for the calculation of the vertical force of the tire. Note that you must 
specify VERTICAL_STIFFNESS in the tire property, but it does not play any role.
Definition of Tire Slip Quantities
Slip Quantities at combined cornering and braking/traction
Fz kzρ czρ·+=
ρ ρ·
13
The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the 
wheel rotational velocity , and the loaded rolling radius Rl:
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point:
 and 
Note that for realistic tire forces the slip angle is limited to 900 and the longitudinal slip in 
between -1 (locked wheel) and 1.
Lagged longitudinal and lateral slip quantities (transient tire behavior)
In general, the tire rotational speed and lateral slip will change continuously because of the changing 
interaction forces in between the tire and the road. Often the tire dynamic response will have an 
important role on the overall vehicle response. For modeling this so-called transient tire behavior, a first-
order system is used both for the longitudinal slip ? as the side slip angle, ?. Considering the tire belt as 
a stretched string, which is supported to the rim with lateral springs, the lateral deflection of the belt can 
be estimated (see also reference [1]). The figure below shows a top-view of the string model.
Stretched String Model for Transient Tire behavior
Ω
Vsx Vx ΩRl–=
Vsy Vy=
κ α
κ VsxVx
--------–= αtan VsyVx
---------=
α κ
Adams/Tire14
 
When rolling, the first point having contact with the road adheres to the road (no sliding assumed). 
Therefore, a lateral deflection of the string will arise that depends on the slip angle size and the history 
of the lateral deflection of previous points having contact with the road.
For calculating the lateral deflection v1 of the string in the first point of contact with the road, the 
following differential equation is valid during braking slip:
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger than 
10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the equation 
can be transformed to:
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill. 
A similar approach yields the following for the deflectionof the string in longitudinal direction:
Now the practical slip quantities, and , are defined based on the tire deformation:
These practical slip quantities and are used instead of the usual and definitions for steady-
state tire behavior.
The longitudinal and lateral relaxation length are estimated with the longitudinal and lateral stiffness of 
the non-rolling tire:
1
Vx
------
dv1
dt
--------
v1
σα
------+ αtan aφ+=
σα φ
σα
dv1
dt
-------- Vx v1+ σαVsy=
σκ
du1
dt
-------- Vx u1+ σ– κVsx=
κ' α'
κ' u1σκ
------ Vx( )sgn=
α' v1σα
------⎝ ⎠⎛ ⎞atan=
κ' α' κ α
σκ
BCDx
longitudinal_stiffness
--------------------------------------------------------= and σα
BCDy
lateral_stiffness
-------------------------------------------=
For BCDx and BCDy see section Force and Moment Formulation for Pacejka '89 or '94.
15
In case the longitudinal stiffness is not available in the tire property file the longitudinal stiffness is 
estimated with:
Using Pacejka '89 Handling Force Model
Learn about the Pacejka '89 handling force model:
• Using Correct Coordinate System and Units
• Force and Moment Formulation for Pacejka ’89
• Example of Pacejka ’89 Property File
Using Correct Coordinate System and Units in Pacejka '89
The test data and resulting coefficients that come from the Pacejka '89 tire model conform to a modified 
SAE tire coordinate system. The standard SAE tire coordinate system is shown next and the modified 
sign conventions for Pacejka '89 are described in the table below.
SAE Tire Coordinate System
longitudinal_stiffness 4 lateral_stiffness×=
Note: The section [UNITS] in the tire property file does not apply to the Magic Formula 
coefficients.
Adams/Tire16
 
Conventions for Naming Variables
Force and Moment Formulation for Pacejka '89
• Longitudinal Force for Pacejka '89
• Lateral Force
• Self-Aligning Torque
• Overturning Moment
• Rolling Resistance
• Smoothing
Longitudinal Force for Pacejka '89
C - Shape Factor
C=B0
D - Peak Factor
D=(B1*FZ2+B2*FZ)
BCD
BCD=(B3*FZ2+B4*FZ)*EXP(-B5*FZ)
B - Stiffness Factor
Variable name and abbreviation: Description:
Normal load Fz (kN) Positive when the tire is penetrating the 
road.*
Lateral force Fy (N) Positive in a right turn.
Negative in a left turn.
Longitudinal force Fx (N) Positive during traction.
Negative during braking.
Self-aligning torque Mz (Nm) Positive in a left turn.
Negative in a right turn.
Inclination angle (deg) Positive when the top of the tire tilts to the 
right (when viewing the tire from the 
rear).*
Sideslip angle (deg) Positive in a right turn.*
Longitudinal slip (%) Negative in braking (-100%: wheel lock).
Positive in traction.
* Opposite convention to standard SAE coordinate system shown in SAE Tire Coordinate System.
γ
α
κ
B=BCD/(C*D)
17
Horizontal Shift
Sh=B9*FZ+B10
Vertical Shift
Sv=0.0
Composite
X1=(κ+Sh)
E Curvature Factor
E=(B6*FZ2+B7*FZ+B8)
FX Equation
FX=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Longitudinal Force
Example Longitudinal Force Plot for Pacejka ’89
Parameters: Description:
B0 Shape factor
B1, B2 Peak factor
B3, B4, B5 BCD calculation
B6, B7, B8 Curvature factor
B9, B10 Horizontal shift
Adams/Tire18
 
Lateral Force for Pacejka '89
C - Shape Factor
C=A0
D - Peak Factor
D=(A1*FZ2+A2*FZ)
BCD
BCD=A3*SIN(ATAN(FZ/A4)*2.0)*(1.0-A5*ABS(γ))
B - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=A9*FZ+A10+A8*γ
Vertical Shift
Sv=A11*FZ*γ+A12*FZ+A13
Composite
X1=(α+Sh)
E - Curvature Factor
19
E=(A6*FZ+A7)
FY Equation
FY=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Parameters for Lateral Force
Example Lateral Force Plot for Pacejka ’89
 
Self-Aligning Torque
C - Shape Factor
Parameters: Description:
A0 Shape factor
A1, A2 Peak factor
A3, A4, A5 BCD calculation
A6, A7 Curvature factor
A8, A9, A10 Horizontal shift
A11, A12, A13 Vertical shift
C=C0
Adams/Tire20
 
D - Peak Factor
D=(C1*FZ2+C2*FZ)
BCD
BCD=(C3*FZ2+C4*FZ)*(1-C6*ABS(γ))*EXP(-C5*FZ)
B - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=C11*γ+C12*FZ+C13
Vertical Shift
Sv= (C14*FZ2+C15*FZ)*γ+C16*FZ+C17
Composite
X1=(α+Sh)
E - Curvature Factor
E=(C7*FZ2+C8*FZ+C9)*(1.0-C10*ABS(γ))
MZ Equation
MZ=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Parameters for Self-Aligning Torque
Example Self-Aligning Torque Plot for Pacejka ’89
Parameters: Description:
C0 Shape factor
C1, C2 Peak factor
C3, C4, C5, C6 BCD calculation
C7, C8, C9, C10 Curvature factor
C11, C12, C13 Horizontal shift
C14, C15, C16, C17 Vertical shift
21
 
Overturning Moment
The lateral stiffness is used to calculate an approximate lateral deflection of the contact patch when there 
is a lateral force present:
deflection = Fy / lateral_stiffness
This deflection, in turn, is used to calculate an overturning moment due to the vertical force:
Mx (overturning moment) = -Fz * deflection
And an incremental aligning torque due to longtiudinal force (Fx)
Mz = Mz,Magic Formula + Fx * deflection
Here Mz,Magic Formula is the magic formula for aligning torque and Fx * deflection is the contribution due 
to the longitudinal force.
Rolling Resistance
The rolling resistance moment My is opposite to the wheel angular velocity. The magnitude is given by:
My = Fz * Lrad * rolling_resistance
Where Fz equals the vertical force and Lrad is the tyre loaded radius. The rolling resistance coefficient 
can be entered in the tire property file:
[PARAMETER]
ROLLING_RESISTANCE = 0.01
Adams/Tire22
 
A value of 0.01 introduces a rolling resistance force that is 1% of the vertical load.
Smoothing
When you indicate smoothing by setting the value of use mode in the tire property file, Adams/Tire 
smooths initial transients in the tire force over the first 0.1 seconds of simulation. The longitudinal force, 
lateral force, and aligning torque are multiplied by a cubic step function of time. (See STEP in the 
Adams/Solver online help.)
Longitudinal Force
FLon = S*FLon
Lateral Force
FLat = S*FLat
Overturning Moment
Mx = S*Mx
Rolling resistance moment
My = S*My
Aligning Torque
Mz = S*Mz
The USE_MODE parameter in the tire property file allows you to switch smoothing on or off:
• USE_MODE = 1 or 2, smoothing is off
• USE_MODE = 3 or 4, smoothing is on
Example of Pacejka '89 Property File
$---------------------------------------------------------MDI_HEADER
[MDI_HEADER]
FILE_TYPE = 'tir'
FILE_VERSION = 2.0
FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'Tire - XXXXXX'
'Pressure - XXXXXX'
'Test Date - XXXXXX'
'Test tire'
$-------------------------------------------------------------UNITS
[UNITS]
LENGTH = 'mm'
FORCE = 'newton'
ANGLE = 'radians'
MASS = 'kg'
TIME = 'sec'
23
$-------------------------------------------------------------MODEL
[MODEL]
! use mode 123411121314
! -----------------------------------------------------------------
! smoothingXXXX
! combinedXXXX
! transient X X X X
!
PROPERTY_FILE_FORMAT = 'PAC89'
USE_MODE = 12.0
TYRESIDE = 'LEFT'
$----------------------------------------------------------DIMENSION
[DIMENSION]
UNLOADED_RADIUS = 326.0
WIDTH = 245.0
ASPECT_RATIO = 0.35
$----------------------------------------------------------PARAMETER
[PARAMETER]
VERTICAL_STIFFNESS = 310.0
VERTICAL_DAMPING = 3.1
LATERAL_STIFFNESS = 190.0
ROLLING_RESISTANCE = 0.0
$---------------------------------------------------------LOAD_CURVE
$ For a non-linear tire vertical stiffness (optional)
$ Maximum of 100 points
[DEFLECTION_LOAD_CURVE]
{pen fz}
 0 0.0
 1 212.0
 2 428.0
 3 648.05 1100.0
10 2300.0
20 5000.0
30 8100.0
$-----------------------------------------------LATERAL_COEFFICIENTS
[LATERAL_COEFFICIENTS]
a0 = 1.65000
a1 = -34.0
a2 = 1250.00
a3 = 3036.00
a4 = 12.80
a5 = 0.00501
a6 = -0.02103
a7 = 0.77394
a8 = 0.0022890
a9 = 0.013442
a10 = 0.003709
a11 = 19.1656
a12 = 1.21356
a13 = 6.26206
$--------------------------------------------------------
longitudinal
[LONGITUDINAL_COEFFICIENTS]
Adams/Tire24
 
b0 = 1.67272
b1 = -9.46000
b2 = 1490.00
b3 = 30.000
b4 = 176.000
b5 = 0.08860
b6 = 0.00402
b7 = -0.06150
b8 = 0.20000
b9 = 0.02990
b10 = -0.17600
$----------------------------------------------------------aligning
[ALIGNING_COEFFICIENTS]
c0 = 2.34000
c1 = 1.4950
c2 = 6.416654
c3 = -3.57403
c4 = -0.087737
c5 = 0.098410
c6 = 0.0027699
c7 = -0.0001151
c8 = 0.1000
c9 = -1.33329
c10 = 0.025501
c11 = -0.02357
c12 = 0.03027
c13 = -0.0647
c14 = 0.0211329
c15 = 0.89469
c16 = -0.099443
c17 = -3.336941
$--------------------------------------------------------------shape
[SHAPE]
{radial width}
 1.0 0.0
 1.0 0.2
 1.0 0.4
 1.0 0.5
 1.0 0.6
 1.0 0.7
 1.0 0.8
 1.0 0.85
 1.0 0.9
 0.9 1.0
Using Pacejka '94 Handling Force Model
Learn about the Pacejka '94 handling force model:
• Using Correct Coordinate System and Units
• Force and Moment Formulation for Pacejka ’94
• Example of Pacejka ’94 Property File
25
Using Correct Coordinate System and Units in Pacejka '94
The test data and resulting coefficients that come from the Pacejka '94 tire model conform to the standard 
SAE tire coordinate system. The standard SAE coordinates are shown in SAE Tire Coordinate System. 
(See also About Coordinate Systems.) The corresponding sign conventions for Pacejka '94 are described 
next
Conventions for Naming Variables
Force and Moment Formulation for Pacejka '94
• Longitudinal Force for Pacejka '94
• Lateral Force for Pacejka '94
• Self-Aligning Torque
• Overturning Moment
• Rolling Resistance
• Smoothing
Longitudinal Force for Pacejka '94
C - Shape Factor
C=B0
Note: The section [UNITS] in the tire property file does not apply to the Magic Formula 
coefficients.
Variable name and abbreviation: Description:
Normal load Fz (kN) Positive when the tire is penetrating the road.
Lateral force Fy (N) Positive in a right turn.
Negative in a left turn.
Longitudinal force Fx (N) Positive during traction.
Negative during braking.
Self-aligning torque Mz (Nm) Positive in a left turn.
Negative in a right turn.
Inclination angle (deg) Positive when the top of the tire tilts to the right 
(when viewing the tire from the rear).
Sideslip angle (deg) Positive in a left turn.
Longitudinal slip (%) Negative in braking (-100%: wheel lock).
Positive in traction.
γ
α
κ
D - Peak Factor
Adams/Tire26
 
D=(B1*FZ2+B2*FZ) * DLON
BCD
BCD=((B3*FZ2+B4*FZ)*EXP(-B5*FZ)) * BCDLON
B - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=B9*Fz+B10 
Vertical Shift
Sv=B11*FZ+B12
Composite
X1=(κ+Sh)
E Curvature Factor
E=((B6*FZ+B7)*FZ+B8)*(1-(B13*SIGN(1,X1))))
FX Equation
FX=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Parameters for Longitudinal Force
Lateral Force for Pacejka '94
C - Shape Factor
C=A0
D - Peak Factor
D=((A *F +A ) *(1-A * 2)*F ) * DLAT
Parameters: Description:
B0 Shape factor
B1, B2 Peak factor
B3, B4, B5 BCD calculation
B6, B7, B8, B13 Curvature factor
B9, B10 Horizontal shift
B11, B12 Vertical shift
DLON, BCDLON Scale factor
γ
1 Z 2 15 Z
27
BCD
BCD=(A3*SIN(ATAN(FZ/A4)*2.0)*(1-A5*ABS( )))* BCDLAT
B - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=A8*FZ+A9+A10*
Vertical Shift
Sv=A11*FZ+A12+(A13*FZ2+A14*FZ)*
Composite
X1=(α+Sh)
E - Curvature Factor
E=(A6*FZ+A7)*(1-(((A16*γ)+A17)*SIGN(1,X1))))
FY Equation
FY=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Parameters for Lateral Force
Self-Aligning Torque for Pacejka '94
C - Shape Factor
C=C0
D - Peak Factor
D=(C1*FZ2+C2*FZ)*(1-C18* 2)
Parameters: Description:
A0 Shape factor
A1, A2, A15 Peak factor
A3, A4, A5 BCD calculation
A6, A7, A16, A17 Curvature factor
A8, A9, A10 Horizontal shift
A11, A12, A13, A14 Vertical shift
DLAT, BCDLAT Scale factor
γ
γ
γ
γ
BCD
Adams/Tire28
 
BCD=(C3*FZ2+C4*FZ)*(1-(C6*ABS(γ)))*EXP(-C5*FZ)
B - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=C11*FZ+C12+C13*
Vertical Shift
Sv=C14*FZ+C15+(C16*FZ2+C17*FZ)*
Composite
X1=( +Sh)
E - Curvature Factor
E=(((C7*FZ2)+(C8*FZ)+C9)*(1-(((C19* )+C20)*SIGN(1,X1))))/(1-(C10*ABS( )))
MZ Equation
MZ=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Parameters for Self-Aligning Torque
Overturning Moment
The lateral stiffness is used to calculate an approximate lateral deflection of the contact patch when there 
is a lateral force present:
deflection = Fy / lateral_stiffness
This deflection, in turn, is used to calculate an overturning moment due to the vertical force:
Mx (overturning moment) = -Fz * deflection
And an incremental aligning torque due to longtiudinal force (Fx):
Parameters: Description:
C0 Shape factor
C1, C2, C18 Peak factor
C3, C4, C5, C6 BCD calculation
C7, C8, C9, C19, C20 Curvature factor
C11, C12, C13 Horizontal shift
C14, C15, C16, C17 Vertical shift
γ
γ
α
γ γ
Mz = Mz,Magic Formula + Fx * deflection
29
Here Mz,Magic Formula is the magic formula for aligning torque and Fx * deflection is the contribution due 
to the longitudinal force.
Rolling Resistance
The rolling resistance moment My is opposite to the wheel angular velocity. The magnitude is given by:
My = Fz * Lrad * rolling_resistance
Where Fz equals the vertical force and Lrad is the tyre loaded radius. The rolling resistance coefficient 
can be entered in the tire property file:
[PARAMETER]
ROLLING_RESISTANCE = 0.01
A value of 0.01 will introduce a rolling resistance force, which is 1% of the vertical load.
Smoothing
Adams/Tire smooths initial transients in the tire force over the first 0.1 seconds of simulation. The 
longitudinal force, lateral force, and aligning torque are multiplied by a cubic step function of time. (See 
STEP in the Adams/Solver online help.)
Longitudinal Force
FLon = S*FLon
Lateral Force
FLat = S*FLat
Overturning Moment
Mx = S*Mx
Rolling resistance moment
My = S*My
Aligning Torque
Mz = S*Mz
The USE_MODE parameter in the tire property file allows you to switch smoothing on or off:
• USE_MODE = 1 or 2, smoothing is off
• USE_MODE = 3 or 4, smoothing is on
Example of Pacejka '94 Property File
!:FILE_TYPE: tir
!:FILE_VERSION: 2
!:TIRE_VERSION: PAC94
Adams/Tire30
 
!:COMMENT: New File Format v2.1
!:FILE_FORMAT: ASCII
!:TIMESTAMP: 1996/02/15,13:22:12
!:USER: ncos
$--------------------------------------------------------------units
[UNITS]
 LENGTH = 'inch'
 FORCE = 'pound_force'
 ANGLE = 'radians'
 MASS = 'pound_mass'
 TIME = 'second'
$--------------------------------------------------------------model
[MODEL]
! use mode 12341234
! ---------------------------------------------------------------
! smoothingXXXX
! combinedXXXX
! transient X X X X
!
 PROPERTY_FILE_FORMAT = 'PAC94'
 USE_MODE = 12.0
 TYRESIDE = 'LEFT'
$---------------------------------------------------------dimensions
[DIMENSION]
 UNLOADED_RADIUS = 12.95
 WIDTH = 10.0
 ASPECT_RATIO = 0.30
$---------------------------------------------------------parameter
[PARAMETER]
 VERTICAL_STIFFNESS = 2500
 VERTICAL_DAMPING = 250.0
 LATERAL_STIFFNESS =1210.0
 ROLLING_RESISTANCE = 0.01
$---------------------------------------------------------load_curve
$ Maximum of 100 points (optional)
[DEFLECTION_LOAD_CURVE]
{pen fz}
0.000 0
0.039 943
0.079 1904
0.118 2882
0.197 4893
0.394 10231
0.787 22241
1.181 36031
$-----------------------------------------------------------scaling
[SCALING_COEFFICIENTS]
 DLAT = 0.10000E+01 
 DLON = 0.10000E+01 
 BCDLAT = 0.10000E+01 
 BCDLON = 0.10000E+01 
$-----------------------------------------------------------lateral 
[LATERAL_COEFFICIENTS]
 A0 = 1.5535430E+00
31
 A1 = -1.2854474E+01
 A2 = -1.1133711E+03
 A3 = -4.4104698E+03
 A4 = -1.2518279E+01
 A5 = -2.4000120E-03
 A6 = 6.5642332E-02
 A7 = 2.0865589E-01
 A8 = -1.5717978E-02
 A9 = 5.8287762E-02
 A10 = -9.2761963E-02
 A11 = 1.8649096E+01
 A12 = -1.8642199E+02
 A13 = 1.3462023E+00
 A14 = -2.0845180E-01
 A15 = 2.3183540E-03
 A16 = 6.6483573E-01
 A17 = 3.5017404E-01
$------------------------------------------------------longitudinal
[LONGITUDINAL_COEFFICIENTS]
 B0 = 1.4900000E+00
 B1 = -2.8808998E+01
 B2 = -1.4016957E+03
 B3 = 1.0133759E+02
 B4 = -1.7259867E+02
 B5 = -6.1757933E-02
 B6 = 1.5667623E-02
 B7 = 1.8554619E-01
 B8 = 1.0000000E+00
 B9 = 0.0000000E+00
 B10 = 0.0000000E+00
 B11 = 0.0000000E+00
 B12 = 0.0000000E+00
 B13 = 0.0000000E+00
$----------------------------------------------------------aligning
[ALIGNING_COEFFICIENTS]
 C0 = 2.2300000E+00
 C1 = 3.1552342E+00
 C2 = -7.1338826E-01
 C3 = 8.7134880E+00
 C4 = 1.3411892E+01
 C5 = -1.0375348E-01
 C6 = -5.0880786E-03
 C7 = -1.3726071E-02
 C8 = -1.0000000E-01
 C9 = -6.1144302E-01
 C10 = 3.6187314E-02
 C11 = -2.3679781E-03
 C12 = 1.7324400E-01
 C13 = -1.7680388E-02
 C14 = -3.4007351E-01
 C15 = -1.6418691E+00
 C16 = 4.1322424E-01
 C17 = -2.3573702E-01
 C18 = 6.0754417E-03
Adams/Tire32
 
 C19 = -4.2525059E-01
 C20 = -2.1503067E-01
$--------------------------------------------------------------shape
[SHAPE]
{radial width}
 1.0 0.0
 1.0 0.2
 1.0 0.4
 1.0 0.5
 1.0 0.6
 1.0 0.7
 1.0 0.8
 1.0 0.85
 1.0 0.9
 0.9 1.0
Combined Slip of Pacejka '89 and '94
The combined slip calculation of the Pacejka '89 and '94 tire models is identical. Note that the method 
employed here is not part of the Magic Formula as developed by Professor Pacejka, but is an in-house 
development of MSC.
Inputs:
• Dimensionless longitudinal slip κ (range –1 to 1) and side slip angle α in radians
• Longitudinal force Fx and lateral force Fy calculated using the Magic Formula
• Horizontal/vertical shifts and peak values of the Magic Formula (Sh, Sv, D)
Output:
• Adjusted longitudinal force Fx and lateral force Fy to incorporate the reduction due to combined 
slip:
Friction coefficients:
κ* κ Shx+=
α* α Shy+=
SAG α*( )sin=
β arc κ*
κ*( )2 SAG2+
---------------------------------------⎝ ⎠⎜ ⎟
⎛ ⎞
cos=
33
Forces corrected for combined slip conditions:
Left and Right Side Tires
In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire 
construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for 
positive and negative slip angles.
A tire property file with the parameters for the model results from testing with a tire that is mounted in 
a tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used 
for both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering 
wheel angle.
The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that 
indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or 
TIRESIDE = 'RIGHT').
If this keyword is available, Adams/Car corrects for the conicity and plysteer and asymmetry when using 
a tire property file on the opposite side of the vehicle. In fact, the tire characteristics are mirrored with 
respect to slip angle zero.
In AdamsS/View this option can only be used when the tire is generated by the graphical user interface: 
select Build -> Forces -> Special Force: Tire.
Next to the LEFT and RIGHT side option of TYRESIDE, you can also select SYMMETRIC: then the 
tire characteristics are modified during initialization to show symmetric performance for left and right 
side corners and zero conicity and plysteer (no offsets). Also, when you set the tire property file to 
SYMMETRIC, the tire characteristics are changed to symmetric behavior.
μx act,
Fx Svx–
Fz
-------------------- μy act,
Fy Svy–
Fz
--------------------
μx max,
Dx
Fz
------ μy max,
Dy
Fz
------
= =
= =
μx 11
μx act,
-------------⎝ ⎠⎛ ⎞
2 βtan
μy max,
----------------⎝ ⎠⎛ ⎞
2
+
--------------------------------------------------------- μy βtan1
μx max,
----------------⎝ ⎠⎛ ⎞
2 βtan
μy max,
----------------⎝ ⎠⎛ ⎞
2
+
------------------------------------------------------------= =
Fx comb,
μx
μx act,
------------- Fx Svx+( ) Fy comb,
μy
μy act,
------------- Fy Svy+( )= =
Adams/Tire34
 
Figure 1 Create Wheel and Tire Dialog Box in Adams/View
Contact Methods
The Pacejka '89 and '94 models support the following roads:
• 2D roads, see Using the 2D Road Model.
• 3D Spline roads, see Adams/3D Spline Road Model
These tire models use a one point of contact method; therefore, the wavelength of road obstacles 
must be longer than the tire radius for realistic output of the model.
• 3D Shell roads, see Adams/Tire 3D Shell Road Model
35
Adams/Tire36
 
PAC MC
Learn about using the University of Arizona (UA) tire model:
When to Use PAC Motorcycle
Magic-Formula (MF) tire models are considered the state-of-the-art for modeling of the tire-road 
interaction forces in Vehicle Dynamics applications. First versions of the mode that were published by 
Pacejka considered tire models for car and truck tires. In his book, Tyre and Vehicle Dynamics [1], he 
also described a model for motorcycle tires that is backwards compatible with the MF-MCTyre, 
previously resold by MSC.Software, and contains the latest developments in this field.
In general, a MF tire model describes the tire behavior for rather smooth roads (road obstacle 
wavelengths longer than the tire radius) up to frequencies of 8 Hz. This makes the tire model applicable 
for all generic vehicle handling and stability simulations, including:
• Steady-state cornering
• Lane-change maneuvers
• Braking or power-off in a turn
• Split-mu braking tests
• J-turn or other turning maneuvers
• ABS braking, when stopping distance is important (not for tuning ABS control strategies)
• Shimmy and weave phenomena, which can be analyzed when the tire model is used in transient 
mode (see USE_MODES of PAC MC: from Simple to Complex)
• All other common vehicle dynamics maneuvers on rather smooth road (wavelength of road 
obstacles must be longer than the tire radius)
The PAC MC model has proven to be applicable to motorcycle tires with inclination angles to the road 
up to 60 degrees. In some cases, it can be used for car tires when exposed to large camber.
PAC MC and Previous Magic Formula Models
Compared to previous versions, PAC MC is backward compatible with all MF-MCTyre 1.x tire models, 
generates the same output, and deals with all previous versions of MF-MCTyre property files.
In addition to PAC MC in Adams, the PAC MC in v2 contins a more advanced tire-road contact modeling 
method that takes the tire's cross-section shape intoaccount, which plays an important role at large 
inclination angles of the wheel with the road. Learn more about the tire cross-section profile contact 
method.
Adams/Tire12
 
Modeling Tire-Road Interaction Forces
For vehicle dynamics applications, accurate knowledge of tire-road interaction forces is inevitable 
because the movements of a vehicle primarily depend on the road forces on the tires. These interaction 
forces depend on both road and tire properties and the motion of the tire with respect to the road.
In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear 
damper with one point of contact with the road surface. The contact point is determined by considering 
the tire and wheel as a rigid disc. In the contact point between the tire and the road the contact forces in 
longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the 
road.
The figure, Input and Output Variables of the Magic Formula Tire Model, presents the input and output 
vectors of the PAC MC tire model. The tire model subroutine is linked to the Adams/Solver through the 
Standard Tire Interface (STI) ([3]). The input through the STI consists of the:
• Position and velocities of the wheel center
• Orientation of the wheel
• Tire model (MF) parameters
• Road parameters
The tire model routine calculates the vertical load and slip quantities based on the position and speed of 
the wheel with respect to the road. The input for the Magic Formula consists of the wheel load ( ), the 
longitudinal and lateral slip ( , ), and inclination angle ( ) with the road. The output is the forces 
( , ) and moments ( , , ) in the contact point between the tire and the road. For 
calculating these forces, the MF equations use a set of MF parameters, which are derived from tire testing 
data.
The forces and moments out of the Magic Formula are transferred to the wheel center and returned to 
Adams/Solver through the STI.
Input and Output Variables of the Magic Formula Tire Model
Fz
κ α γ
Fx Fy Mx My Mz
13
 
Axis Systems and Slip Definitions
• Axis System
• Units
• Definition of Tire Slip Quantities
Axis System
The PAC MC model is linked to Adams/Solver using the TYDEX STI conventions as described in the 
TYDEX-Format [2] and the STI [3].
The STI interface between the PAC MC model and Adams/Solver mainly passes information to the tire 
model in the C-axis coordinate system. In the tire model itself, a conversion is made to the W-axis system 
because all the modeling of the tire behavior, as described in this help, assumes to deal with the slip 
quantities, orientation, forces, and moments in the contact point with the TYDEX W-axis system. Both 
axis systems have the ISO orientation but have a different origin as can be seen in the figure below.
TYDEX C- and W-Axis Systems Used in PAC MC, Source[2]
Adams/Tire14
 
 
The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel to the road and in 
the wheel plane (xc-zc-plane). The origin of the C-axis system is the wheel center.
The origin of the W-axis system is the road contact-point defined by the intersection of the wheel plane, 
the plane through the wheel carrier, and the road tangent plane.
The forces and moments calculated by PAC MC using the MF equations in this guide are in the W-axis 
system. A transformation is made in the source code to return the forces and moments through the STI 
to Adams/Solver.
The inclination angle is defined as the angle between the wheel plane and the normal to the road tangent 
plane (xw-yw-plane).
Units
The units of information transferred through the STI between Adams/Solver and PAC MC are according 
to the SI unit system. Also, the equations for PAC MC described in this guide have been developed for 
use with SI units, although you can easily switch to another unit system in your tire property file. Because 
of the non-dimensional parameters, only a few parameters have units to be changed.
However, the parameters in the tire property file must always be valid for the TYDEX W-axis system 
(ISO oriented). The basic SI units are listed in the table below.
15
SI Units Used in PAC MC
Definition of Tire Slip Quantities
Slip Quantities at Combined Cornering and Braking/Traction
 
The longitudinal slip velocity in the contact point (W-axis system, see the figure, Slip Quantities at 
Combined Cornering) is defined using the longitudinal speed , the wheel rotational velocity , and 
the effective rolling radius :
Variable Type: Name: Abbreviation: Unit:
Angle Slip angle
Inclination angle
Radians
Force Longitudinal force
Lateral force
Vertical load
Newton
Moment Overturning moment
Rolling resistance 
moment
Self-aligning moment
Newton.meter
Speed Longitudinal speed
Lateral speed
Longitudinal slip speed
Lateral slip speed
Meters per second
Rotational speed Tire rolling speed Radians per second
α
γ
Fx
Fy
Fz
Mx
My
Mz
Vx
Vy
Vsx
Vsy
ω
Vsx
Vx ω
Re
Adams/Tire16
 
(1)
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
(2)
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point with:
(3)
(4)
The rolling speed Vr is determined using the effective rolling radius Re:
(5)
Contact-Point and Normal Load Calculation
• Contact Point
• Loaded and Effective Tire Rolling Radius
Contact Point
In the vertical direction, the tire is modeled as a parallel linear spring and damper having one point of 
contact (C) with the road. This is valid for road obstacles with a wavelength larger than the tire radius 
(for example, for car tires 1m).
For calculating the kinematics of the tire relative to the road, the road is approximated by its tangent plane 
at the road point right below the wheel center (see figure below).
Contact Point C: Intersection between Road Tangent Plane, Spin Axis Plane, and Wheel Plane
Vsx Vx ΩRe–=
Vsy Vy=
κ α
κ VsxVx
--------–=
α( )tan VsyVx
---------=
Vr ReΩ=
17
 
The contact point is determined by the line of intersection of the wheel center-plane with the road tangent 
(ground) plane and the line of intersection of the wheel center-plane with the plane though the wheel spin 
axis.
The normal load of the tire is calculated with:
(6)
where is the tire deflection and is the deflection rate of the tire.
To take into account the effect of the tire cross-section profile, you can choose a more advanced method 
(see the Tire Cross Section Profile Contact Method).
Instead of the linear vertical tire stiffness Cz, also an arbitrary tire deflection - load curve can be defined 
in the tire property file in the section [DEFLECTION_LOAD_CURVE]. If a section called 
[DEFLECTION_LOAD_CURVE] exists, the load deflection datapoints with a cubic spline for inter- and 
extrapolation are used for the calculation of the vertical force of the tire. Note that you must specify 
in the tire property file, but it does not play any role.
Fz
Fz Czρ Kz ρ·⋅+=
ρ ρ·
Cz
Adams/Tire18
 
Loaded and Effective Tire Rolling Radius
With the loaded rolling tire radius R defined as the distance of the wheel center to the contact point of the 
tire with the road (see Effective Rolling Radius and Longitudinal Slip), where ρ is the deflection of the tire, 
and R0 is the free (unloaded) tire radius, then the loaded tire radius Rl reads:
(7)
In this tire model, a constant (linear) vertical tire stiffness is assumed; therefore, the tire deflection 
 can be calculated using:
(8)
The effective rolling radius Re (at free rolling of the tire), which is used to calculate therotational speed 
of the tire, is defined by:
(9)
For radial tires, the effective rolling radius is rather independent of load in its load range of operation due 
to the high stiffness of the tire belt circumference. Only at low loads does the effective tire radius decrease 
with increasing vertical load due to the tire tread thickness. See the figure below.
Effective Rolling Radius and Longitudinal Slip
R1 R0 ρ–=
Cz
ρ
ρ FzCz
------=
Re
Vx
Ω------=
19
 
To represent the effective rolling radius Re, a MF-type of equation is used:
(10)
in which is the nominal tire deflection:
(11)
and is called the dimensionless radial tire deflection, defined by:
Re R0 ρFz0 D Bρd( ) Fρd+( )atan⋅( )–=
ρFz0
ρFz0
Fz0
Cz
--------=
ρd
Adams/Tire20
 
(12)
Example of the Loaded and Effective Tire Rolling Radius as a Function of the Vertical Load
 
Normal Load and Rolling Radius Parameters
Name:
Name Used in Tire 
Property File: Explanation:
FNOMIN Nominal wheel load
UNLOADED_RADIUS Free tire radius
BREFF Low load stiffness effective rolling radius
DREFF Peak value of effective rolling radius
FREFF High load stiffness effective rolling radius
VERTICAL_STIFFNESS Tire vertical stiffness
VERTICAL_DAMPING Tire vertical damping
ρd ρρFz0
---------=
Fz0
R0
B
D
F
Cz
KZ
21
Basics of the Magic Formula in PAC MC
The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics 
for the interaction forces between the tire and the road under several steady-state operating conditions. 
We distinguish:
• Pure cornering slip conditions: cornering with a free rolling tire
• Pure longitudinal slip conditions: braking or driving the tire without cornering
• Combined slip conditions: cornering and longitudinal slip simultaneously
For pure slip conditions, the lateral force as a function of the lateral slip , respectively, and the 
longitudinal force as a function of longitudinal slip , have a similar shape (see the figure, 
Characteristic Curves for Fx and Fy Under Pure Slip Conditions). Because of the sine - arctangent 
combination, the basic Magic Formula example is capable of describing this shape:
(13)
where Y(x) is either with x the longitudinal slip , or and x the lateral slip .
Characteristic Curves for Fx and Fy Under Pure Slip Conditions
 
The self-aligning moment is calculated as a product of the lateral force and the pneumatic trail t 
added with the residual moment . In fact, the aligning moment is due to the offset of lateral force , 
called pneumatic trail t, from the contact point. Because the pneumatic trail t as a function of the lateral 
slip has a cosine shape, a cosine version the Magic Formula is used:
Fy α
Fx κ
Y x( ) D c bx E bx bx( )atan–( )–( )atan⋅[ ]cos⋅=
Fx κ Fy α
Mz Fy
Mzr Fy
α
Adams/Tire22
 
(14)
in which Y(x) is the pneumatic trail t as function of slip angle .
The figure, The Magic Formula and the Meaning of Its Parameters, illustrates the functionality of the B, 
C, D, and E factor in the Magic Formula:
• D-factor determines the peak of the characteristic, and is called the peak factor.
• C-factor determines the part used of the sine and, therefore, mainly influences the shape of the 
curve (shape factor).
• B-factor stretches the curve and is called the stiffness factor.
• E-factor can modify the characteristic around the peak of the curve (curvature factor).
The Magic Formula and the Meaning of Its Parameters
Y x( ) D C Bx E Bx Bx( )atan–( )–( )atan⋅[ ]cos⋅=
α
23
 
In combined slip conditions, the lateral force decreases due to longitudinal slip or the opposite, the Fy
longitudinal force decreases due to lateral slip. The forces and moments in combined slip conditions Fx
Adams/Tire24
 
are based on the pure slip characteristics multiplied by the so-called weighting functions. Again, these 
weighting functions have a cosine-shaped MF examples.
The Magic Formula itself only describes steady-state tire behavior. For transient tire behavior (up to 8 
Hz), the MF output is used in a stretched string model that considers tire belt deflections instead of slip 
velocities to cope with standstill situations (zero speed).
Inclination Effects in the Lateral Force
From a historical point of view, the basic Magic Formulas have always been developed for car and truck 
tires, which cope with inclinations angles of not more than 10 degrees. To be able to describe the effects 
at large inclinations, an extension of the basic Magic Formula for the lateral force Fy has been developed. 
A contribution of the inclination has also been added within the MF sine function:
(15)
This elegant formulation has the advantage of an explicit definition of the camber stiffness, because this 
results now in:
(16)
Input Variables
The input variables to the Magic Formula are:
Input Variables
Output Variables
Its output variables are:
Output Variables
Longitudinal slip [-]
Slip angle [rad]
Inclination angle [rad]
Normal wheel load Fz [N]
Longitudinal force Fx [N]
Lateral force Fy [N]
γ
Fy0 Dy Cyarc Byαy Ey Byαy arc Byαy( )tan–( )–{ }
Cγarc Bγγy Eγ Bγγy arc Bγγy( )tan–( )–{ }tan+
tan[
]
sin=
Kγ BγCγDγ γ∂
∂Fyo= = at αy 0=
κ
α
γ
25
The output variables are defined in the W-axis system of TYDEX.
Basic Tire Parameters
All tire model parameters of the model are without dimension. The reference parameters for the model 
are:
Basic Tire Parameters
As a measure for the vertical load, the normalized vertical load increment dfz is used:
(17)
with the possibly adapted nominal load (using the user-scaling factor, ):
(18)
Nomenclature of the Tire Model Parameters
In the subsequent sections, formulas are given with non-dimensional parameters aijk with the following 
logic:
Tire Model Parameters
Overturning couple Mx [Nm]
Rolling resistance moment My [Nm]
Aligning moment Mz [Nm]
Nominal (rated) load Fz0 [N]
Unloaded tire radius R0 [m]
Tire belt mass mbelt [kg]
Parameter: Definition:
a = p Force at pure slip
q Moment at pure slip
r Force at combined slip
s Moment at combined slip
dfz
Fz F'z0–
F'z0
--------------------=
λFz0
F'z0 Fz0 λFz0⋅=
Adams/Tire26
 
User Scaling Factors
A set of scaling factors is available to easily examine the influence of changing tire properties without 
the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. 
You can change the factors in the tire property file. The peak friction scaling factors, and , 
are also used for the position-dependent friction in 3D Road Contact and Adams/3D Road. An overview 
of all scaling factors is shown in the next tables.
Scaling Factor Coefficients for Pure Slip
i = B Stiffness factor
C Shape factor
D Peak value
E Curvature factor
K Slip stiffness = BCD
H Horizontal shift
V Vertical shift
s Moment at combined slip
t Transient tire behavior
j = x Along the longitudinal axis
y Along the lateral axis
z About the vertical axis
k = 1, 2, ...
Name:
Name used in 
tire property file: Explanation:
LFZO Scale factor of nominal (rated) load
Cx LCX Scale factor of Fx shape factor
LMUX Scale factor of Fx peak friction coefficient
Ex LEX Scale factor of Fx curvature factor
Kx LKX Scale factor of Fx slip stiffness
Vx LVX Scale factor of Fx vertical shift
x LGAX Scale factor of camber for Fx
Cy LCY Scale factor of Fy shape factor
y LMUY Scale factor of Fy peak friction coefficient
Ey LEY Scale factor of Fy curvature factor
Parameter: Definition:
λμξ λγψ
λFz0λ
λμξ
λ
λ
λ
λγ
λ
λμ
λ
27
Scaling Factor Coefficients for Combined Slip
Scaling Factor Coefficients for Combined Slip
Ky LKY Scale factor of Fy cornering stiffness
C LCC Scale factor of camber shape factorK LKC Scale factor of camber stiffness (K-factor)
E LEC Scale factor of camber curvature factor
Hyy LHY Scale factor of Fy horizontal shift
LGAY Scale factor of camber force stiffness
t LTR Scale factor of peak of pneumatic trail
Mr LRES Scale factor for offset of residual torque
gz LGAZ Scale factor of camber torque stiffness
Mx LMX Scale factor of overturning couple
VMx LVMX Scale factor of Mx vertical shift
My LMY Scale factor of rolling resistance torque
Name:
Name used in tire 
property file: Explanation:
x LXAL Scale factor of alpha influence on Fx
y LYKA Scale factor of alpha influence on Fx
Vy LVYKA Scale factor of kappa induced Fy
s LS Scale factor of Moment arm of Fx
Name:
Name used in tire 
property file: Explanation:
x LXAL Scale factor of alpha influence on Fx
y LYKA Scale factor of alpha influence on Fx
Vy LVYKA Scale factor of kappa induced Fy
s LS Scale factor of Moment arm of Fx
Name:
Name used in 
tire property file: Explanation:
λ
λ γ
λ γ
λ γ
λ
λγψ
λ
λ
λ
λ
λ
λ
λ α
λ κ
λ κ
λ
λ α
λ κ
λ κ
λ
Adams/Tire28
 
Scaling Factor Coefficients for Transient Response
Steady-State: Magic Formula for PAC MC
• Steady-State Pure Slip
• Steady-State Combined Slip
Steady-State Pure Slip
• Longitudinal Force at Pure Slip
• Lateral Force at Pure Slip
• Aligning Moment at Pure Slip
Formulas for the Longitudinal Force at Pure Slip
For the tire rolling on a straight line with no slip angle, the formulas are:
(19)
(20)
(21)
(22)
with following coefficients:
(23)
(24)
(25)
(26)
the longitudinal slip stiffness:
Name:
Name used in tire 
property file: Explanation:
LSGKP Scale factor of relaxation length of Fx
LSGAL Scale factor of relaxation length of Fy
gyr LGYR Scale factor of gyroscopic moment
σκ
σα
λ
Fx Fx0 κ Fz γ, ,( )=
Fx0 Dx Cxarc Bxκx Ex Bxκx arc Bxκx( )tan–( )–{ }tan[ ] SVx+sin=
κx κ SHx+=
γx γ λγx⋅=
Cx pCx1 λCx⋅=
Dx μx Fz ζ1⋅ ⋅=
μx pDx1 pDx2dfz+( ) 1 pDx3 γx2⋅–( )λμx⋅=
Ex pEx1 pEx2dfz pEx3dfz2+ +( ) 1 pEx4 κx( )sgn–{ } λEx with Ex 1≤⋅ ⋅=
29
(27)
(28)
(29)
(30)
Longitudinal Force Coefficients at Pure Slip
Formulas for the Lateral Force at Pure Slip
(31)
(32)
Name:
Name used in tire 
property file: Explanation:
pCx1 PCX1 Shape factor Cfx for longitudinal force
pDx1 PDX1 Longitudinal friction Mux at Fznom
pDx2 PDX2 Variation of friction Mux with load
pDx3 PDX3 Variation of friction Mux with camber
pEx1 PEX1 Longitudinal curvature Efx at Fznom
pEx2 PEX2 Variation of curvature Efx with load
pEx3 PEX3 Variation of curvature Efx with load squared
pEx4 PEX4 Factor in curvature Efx while driving
pKx1 PKX1 Longitudinal slip stiffness Kfx/Fz at Fznom
pKx2 PKX2 Variation of slip stiffness Kfx/Fz with load
pKx3 PKX3 Exponent in slip stiffness Kfx/Fz with load
pVx1 PVX1 Vertical shift Svx/Fz at Fznom
pVx2 PVX2 Variation of shift Svx/Fz with load
Kx Fz pKx1 pKx2dfz+( ) pKx3dfz( ) λK
(Kx
⋅exp⋅ ⋅
BxCxDx κx∂
∂Fx0 at κx 0 )
=
= = =
Bx Kx CxDx( )⁄=
SHx psy1FzλMy SVx+( ) Kx⁄–=
SVx Fz pVx1 pVx2dfz+( ) λVx λμx ζ1⋅ ⋅ ⋅ ⋅=
Fy Fy0 α γ Fz, ,( )=
Fy0 Dy Cyarc Byαy Ey Byαy arc Byαy( )tan–( )–{ }tan[sin=
Cγarc Bγγy Eγ Bγγy arc Bγγy( )tan–( )–{ }tan+ ]
Adams/Tire30
 
(33)
The scaled inclination angle:
(34)
with coefficients:
(35)
(36)
(37)
(38)
The cornering stiffness:
(39)
 (40)
(41)
(42)
and the explicit camber stiffness:
(43)
(44)
(45)
αy α SHy Cy Cγ 2<+( )+=
γy γ λγy⋅=
Cy pCy1 λCy⋅=
Dy μy Fz ζ2⋅ ⋅=
μy pDy1 pDy2dfz( ) 1 pDy3γy2–( ) λμy⋅ ⋅exp⋅=
Ey pEy1 pEy2γy2 pEy3 pEy4γy+( ) αy( )sin⋅–+{ } λEy with Ey 1≤⋅=
Ky pKy1Fzo pKy2arc
Fz
pKy3 pKy4γy2+( )FzoλFzo
-----------------------------------------------------------
⎩ ⎭⎨ ⎬
⎧ ⎫
tan
1 pKy5γy2–( ) λFzo λKy
(Ky
⋅ ⋅
sin
ByCyDy αy∂
∂Fyo at αy 0 )
=
= = =
By Ky CyDy( )⁄=
SHy pHy1 λHy⋅=
Cγ pCy2 λCγ⋅=
Kγ pKy6 pKy7dfz+( ) Fz λKγ (=BγCγDγ⋅ ⋅ γ∂
∂Fyo at αy 0= = =
Eγ pEy5 λEγ with Eγ 1≤⋅=
Bγ Kγ CγDγ( )⁄=
31
Lateral Force Coefficients at Pure Slip
Formulas for the Aligning Moment at Pure Slip
(46)
with the pneumatic trail t:
(47)
(48)
and the residual moment Mzr:
(49)
Name:
Name used in 
tire property file: Explanation:
pCy1 PCY1 Shape factor Cfy for lateral forces
pCy2 PCY2 Shape factor Cfc for camber forces
pDy1 PDY1 Lateral friction Muy
pDy2 PDY2 Exponent lateral friction Muy
pDy3 PDY3 Variation of friction Muy with squared camber
pEy1 PEY1 Lateral curvature Efy at Fznom
pEy2 PEY2 Variation of curvature Efy with camber squared
pEy3 PEY3 Asymmetric curvature Efy at Fznom
pEy4 PEY4 Asymmetric curvature Efy with camber
pEy5 PEY5 Camber curvature Efc
pKy1 PKY1 Maximum value of stiffness Kfy/Fznom
pKy2 PKY2 Curvature of stiffness Kfy
pKy3 PKY3 Peak stiffness factor
pKy4 PKY4 Peak stiffness variation with camber squared
pKy5 PKY5 Lateral stiffness dependency with camber squared
pKy6 PKY6 Camber stiffness factor Kfc
pKy7 PKY7 Vertical load dependency of camber stiffness Kfc
pHy1 PHY1 Horizontal shift Shy at Fznom
Mz' Mz0 α γ Fz, ,( )=
Mz0 t Fy0 Mzr+⋅–=
t αt( ) Dt Ctarc Btαt Et Btαt arc Btαt( )tan–( )–{ }tan[ ] α( )coscos=
αt α SHt+=
Mzr αr( ) Dr Crarc Brαr( )tan[ ] α( )cos⋅cos=
Adams/Tire32
μy
 
(50)
The scaled inclination angle:
(51)
with coefficients:
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(59)
(60)
An approximation for the aligning moment stiffness reads:
(61)
Aligning Moment Coefficients at Pure Slip
Name:
Name used in tire 
property file: Explanation:
qBz1 QBZ1 Trail slope factor for trail Bpt at Fznom
qBz2 QBZ2 Variation of slope Bpt with load
qBz3 qBz3 Variation of slope Bpt with load squared
qBz4 QBZ4 Variation of slope Bpt with camber
αr α SHr+=
γz γ λγz⋅=
Bt qBx1 qBx2dfz qBx3dfz2+ +( ) 1 qBx4γz qBz5 γz+ +{ } λKy λμy⁄⋅ ⋅=
Ct qCz1=
Dt Fz qDz1 qDz2dfz+( ) 1 qDz3 γz qDz4γz2+ +( ) R0 Fz0⁄( )⋅ ⋅ ⋅=
Et qEx1 qEx2dfz qEx3dfz2+ +( )=
1 qEz4 qEz5γz+( ) 2π---⎝ ⎠
⎛ ⎞ arc Bt Ct αt⋅ ⋅( )tan⋅ ⋅+⎩ ⎭⎨ ⎬
⎧ ⎫
with Et 1≤
SHt 0=
Br qBz9 λKy λμy⁄⋅=
Dr Fz qDz6 qDz7dfz+( )λr qDz8 qDz9dfz+( )γz qDz10 qDz11dfz+( ) γz γz⋅( )+ +[ ]R0λ=
SHr qHz1 qHz2dfz qHz3 qHz4dfz+( )γz+ +=
Kz t Ky αd
dMz at α–≈⎝ ⎠⎛ ⎞⋅– 0 )= =
qBz5 QBZ5 Variation of slope Bpt with absolute camber
33
Steady-State Combined Slip
PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies 
the coefficients for the combined slip cosine 'weighing' functions, the combined slip is calculated 
according to Combined slip with cosine 'weighing' functions (standard method). If no coefficients are 
supplied, the so-called friction ellipse is used to estimate the combined slip forces and moments, see 
section Combined Slip with friction ellipse
Combined slip with cosine 'weighing' functions
• Longitudinal Force at Combined Slip
• Lateral Force at Combined Slip
• Aligning Moment at Combined Slip
qBz9 QBZ9 Slope factor Br of residual torque Mzr
qCz1 QCZ1 Shape factor Cpt for pneumatic trail
qDz1 QDZ1 Peak trail Dpt = Dpt*(Fz/Fznom*R0)
qDz2 QDZ2 Variation of peak Dpt with load
qDz3 QDZ3 Variation of peak Dpt with camber
qDz4 QDZ4 Variation of peak Dpt with camber squared.
qDz6 QDZ6 Peak residual moment Dmr = Dmr/ (Fz*R0)
qDz7 QDZ7 Variation of peak factor Dmr with load
qDz8 QDZ8 Variation of peak factor Dmr with camber
qDz9 QDZ9 Variation of Dmr with camber and load
qDz10 QDZ10 Variation of peak factor Dmr with camber squared
qDz11 QDZ11 Variation of Dmr with camber squared and load
qEz1 QEZ1 Trail curvature Ept at Fznom
qEz2 QEZ2 Variation of curvature Ept with load
qEz3 QEZ3 Variation of curvature Ept with load squared
qEz4 QEZ4 Variation of curvature Ept with sign of Alpha-t
qEz5 QEZ5 Variation of Ept with camber and sign Alpha-t
qHz1 QHZ1 Trail horizontalshift Shr at Fznom
qHz2 QHZ2 Variation of shift Shr with load
qHz3 QHZ3 Variation of shift Shr with camber
qHz4 QHZ4 Variation of shift Sht with camber and load
Name:
Name used in tire 
property file: Explanation:
• Overturning Moment at Pure and Combined Slip
Adams/Tire34
 
• Rolling Resistance Moment at Pure and Combined Slip
Formulas for the Longitudinal Force at Combined Slip
(62)
with Gx o the weighting function of the longitudinal force for pure slip.
We write:
(63)
(64)
with coefficients:
(65)
(66)
(67)
(68)
(69)
The weighting function follows as:
(70)
Longitudinal Force Coefficients at Combined Slip
Name:
Name used in tire 
property file: Explanation:
rBx1 RBX1 Slope factor for combined slip Fx reduction
rBx2 RBX2 Variation of slope Fx reduction with kappa
rBx3 RBX3 Influence of camber on stiffness for Fx reduction
rCx1 RCX1 Shape factor for combined slip Fx reduction
rEx1 REX1 Curvature factor of combined Fx
rEx2 REX2 Curvature factor of combined Fx with load
Fx Fx0 Gxα α κ Fz, ,( )⋅=
α
Fx Dxα Cxαarc Bxααs Exα Bxααs arc Bxααs( )tan–( )–{ }tan[ ]cos=
αs α SHxα+=
Bxα rBx1 rBx3γ2+( ) arc rBx2κ{ }tan[ ] λxα⋅cos=
Cxα rCx1=
Dxα
Fxo
Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–{ }tan[ ]cos
------------------------------------------------------------------------------------------------------------------------------------------------------------------=
Exα rEx1 rEx2dfz with Exα 1≤+=
SHxα rHx1=
Gxα
Cxαarc Bxααs Exα Bxααs arc Bxααs( )tan–( )–{ }tan[ ]cos
Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–{ }tan[ ]cos
------------------------------------------------------------------------------------------------------------------------------------------------------------------=
rHx1 RHX1 Shift factor for combined slip Fx reduction
35
Formulas for Lateral Force at Combined Slip
(71)
with Gyk the weighting function for the lateral force at pure slip and SVyk the ' -induced' side force; 
therefore, the lateral force can be written as:
(72)
(73)
with the coefficients:
(74)
(75)
(76)
(77)
(78)
(79)
(80)
The weighting function appears is defined as:
(81)
Lateral Force Coefficients at Combined Slip
Name:
Name used in tire 
property file: Explanation:
rBy1 RBY1 Slope factor for combined Fy reduction
rBy2 RBY2 Variation of slope Fy reduction with alpha
rBy3 RBY3 Shift term for alpha in slope Fy reduction
rCy1 RCY1 Shape factor for combined Fy reduction
Fy Fy0 Gyκ α κ γ Fz, , ,( ) SVyκ+⋅=
κ
Fy Dyκ Cyκarc Byκκs Eyκ Byκκs arc Byκκs( )tan–( )–{ }tan[ ] SVyκ+cos=
κs κ SHyk+=
Byκ rBy1 rBy4γ2+( ) arc rBy2 α rBy3–( ){ }tan[ ] λyκ⋅cos=
Cyκ rCy1=
Dyκ
Fyo
Cyκarc ByκSHyκ Eyκ ByκSHyκ arc ByκSHyκ( )tan–( )–{ }tan[ ]cos
---------------------------------------------------------------------------------------------------------------------------------------------------------------=
Eyκ rEy1 rEy2dfz with Eyκ 1≤+=
SHyκ rHy1 rHy2dfz+=
SVyk DVyκ rVy5arc rvy6κ( )tan[ ] λVyκ⋅sin=
DVyκ μyFz rVy1 rVy2dfz rVy3γ+ +( ) arc rVy4α( )tan[ ]cos⋅ ⋅=
Gyκ
Cyκarc Byκκs Eyκ Byκκs arc Byκκs( )tan–( )–{ }tan[ ]cos
Cyκarc ByκSHyκ Eyκ ByκSHyκ arc ByκSHyκ( )tan–( )–{ }tan[ ]cos
---------------------------------------------------------------------------------------------------------------------------------------------------------------=
rEy1 REY1 Curvature factor of combined Fy
Adams/Tire36
 
Formulas for Aligning Moment at Combined Slip
(82)
with:
(83)
(84)
(85)
(86)
(87)
with the arguments:
(88)
(89)
rEy2 REY2 Curvature factor of combined Fy with load
rHy1 RHY1 Shift factor for combined Fy reduction
rHy2 RHY2 Shift factor for combined Fy reduction with load
rVy1 RVY1 Kappa-induced side force Svyk/Muy*Fz at Fznom
rVy2 RVY2 Variation of Svyk/Muy*Fz with load
rVy3 RVY3 Variation of Svyk/Muy*Fz with inclination
rVy4 RVY4 Variation of Svyk/Muy*Fz with alpha
rVy5 RVY5 Variation of Svyk/Muy*Fz with kappa
rVy6 RVY6 Variation of Svyk/Muy*Fz with atan (kappa)
Name:
Name used in tire 
property file: Explanation:
Mz' t Fy' Mzr s Fx⋅+ +⋅–=
t t αt eq,( )=
Dt Ctarc Btαt eq, Et Btαt eq, ac Btαt eq,( )tan–( )–{ }tan[ ] α( )coscos=
F'y γ, 0= Fy SVyκ–=
Mzr Mzr αr eq,( ) Dr arc Brαr eq,( )tan[ ] α( )coscos= =
s ssz1 ssz2 Fy Fz0⁄( ) ssz3 ssz4dfz+( )γ+ +{ } R0 λs⋅ ⋅=
αt eq, arc α2 t
Kx
Ky
------⎝ ⎠⎛ ⎞
2κ2+tan αt( )sgn⋅tan=
αr eq, arc α2 r
Kx
Ky
------⎝ ⎠⎛ ⎞
2κ2+tan αr( )sgn⋅tan=
37
Aligning Moment Coefficients at Combined Slip
Formulas for Overturning Moment at Pure and Combined Slip
For the overturning moment, the formula reads both for pure and combined slip situations:
(90)
Overturning Moment Coefficients
Formulas for Rolling Resistance Moment at Pure and Combined Slip
The rolling resistance moment is defined by:
(91)
Rolling Resistance Coefficients
Name:
Name used in tire 
property file: Explanation:
ssz1 SSZ1 Nominal value of s/R0 effect of Fx on Mz
ssz2 SSZ2 Variation of distance s/R0 with Fy/Fznom
ssz3 SSZ3 Variation of distance s/R0 with inclination
ssz4 SSZ4 Variation of distance s/R0 with load and inclination
Name:
Name used in tire 
property file: Explanation:
qsx1 QSX1 Lateral force-induced overturning couple
qsx2 QSX2 Inclination-induced overturning couple
qsx3 QSX3 Fy-induced overturning couple
Name:
Name used in tire 
property file: Explanation:
qsy1 QSY1 Rolling resistance moment coefficient
qsy2 QSY2 Rolling resistance moment depending on Fx
qsy3 QSY3 Rolling resistance moment depending on speed
qsy4 QSY4 Rolling resistance moment depending on speed^4
Vref LONGVL Measurement speed
Mx R0 Fz qsx1λVMx qsx2 γ qsx3
Fy
Fz0
--------⋅+⋅–
⎩ ⎭⎨ ⎬
⎧ ⎫λMx⋅ ⋅=
My R0 Fz qSy1 qSy2Fx Fz0⁄ qSy3 Vx Vref⁄ qSy4 Vx Vref⁄( )4+ + +{ }⋅ ⋅=
Adams/Tire38
 
Combined Slip with friction ellipse
In case the tire property file does not contain the coefficients for the 'standard' combined slip method 
(cosine 'weighing functions), the friction ellipse method is used, as described in this section. Note that 
the method employed here is not part of one of the Magic Formula publications by Pacejka, but is an in-
house development of MSC.Software. 
The following friction coefficients are defined:
The forces corrected for the combined slip conditions are: 
κc κ SHx
SVx
Kx
---------+ +=
αc α SHy
SVy
Ky
---------+ +=
α∗ αc( )sin=
β κc
κc2 α∗2+
-------------------------⎝ ⎠⎜ ⎟
⎛ ⎞
acos=
μx act,
Fx 0, SVx–
Fz
-------------------------= μy act,
Fy 0, SVy–
Fz
-------------------------=
μx max,
Dx
Fz
------= μy max,
Dy
Fz
------=
μx 1
1
μx act,
-------------⎝ ⎠⎛ ⎞
2 βtan
μy max,
----------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------------=
μy βtan
1
μx max,
----------------⎝ ⎠⎛ ⎞
2 βtan
μy act,
-------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------------=
Fx
μx
μx act,
-------------Fx 0,= Fy
μy
μy act,
-------------Fy 0,=
39
For aligning moment Mx, rolling resistance My and aligning moment Mz the formulae (283) until and 
including (291) are used with =0.
Transient Behavior in PAC MC
The previous Magic Formula examples are valid for steady-state tire behavior. When driving, however, 
the tire requires some response time on changes of the inputs. In tire modeling terminology, the low-
frequency behavior (up to 8 Hz) is called transient behavior.
Stretched String Model for Transient Tire Behavior
 
For accurate transient tire behavior, you can use the "stretched string" tire model (see also reference [1]). 
The tire belt is modeled as stretched string, which is supported to the rim with lateral (and longitudinal) 
springs. The figure, Stretched String Model for Transient Tire Behavior, shows a top-view of the string 
model.When rolling, the first point having contact with the road adheres to the road (no sliding 
assumed). Therefore, a lateral deflection of the string arises that depends on the slip angle size and the 
history of the lateral deflection of previous points having contact with the road.
SVyκ
For calculating the lateral deflection v1 of the string in the first point of contact with the road, the 
following differential equation is valid:
Adams/Tire40
 
(92)
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger than 
10 m. This differential example cannot be used at zero speed, but when multiplying with Vx, the example 
can be transformed to:
(93)
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill.
A similar approach yields the following for the deflection of the string in longitudinal direction:
(94)
Both the longitudinal and lateral relaxation length are defined as of the vertical load:
(95)
(96)
Now the practical slip quantities, and , are defined based on the tire deformation:
(97)
(98)
Using these practical slip quantities, and , the Magic Formula examples can be used to calculate 
the tire-road interaction forces and moments:
(99)
(100)
1
Vx
------
td
dv1 v1
σα
------+ α( ) aφ+tan=
σα φ
σα td
dv1 Vx v1+ σαVsy=
σx td
du1 Vx u1+ σxVsx–=
σx Fz pTx1 pTx2dfz+( ) pTx3dfz( ) R0 Fz0⁄( )λσκ⋅exp⋅ ⋅=
σα pTy1 pTy2arc
Fz
pTy3 pKy4γ2+( )Fz0λFz0
---------------------------------------------------------
⎩ ⎭⎨ ⎬
⎧ ⎫
tan 1 pKy5γ2–( ) R0λFz0λσα⋅sin=
κ' α'
κ' u1σx
------ Vx( )sin⋅=
α' v1σα
------⎝ ⎠⎛ ⎞atan=
κ' α'
Fx' Fx α' κ' Fz, ,( )=
Fy' Fy α' κ' γ Fz, , ,( )=
41
(101)
Gyroscopic Couple in PAC MC
When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead 
to gyroscopic effects. To cope with this additional moment, the following contribution is added to the 
total aligning moment:
(102)
with the parameters (in addition to the basic tire parameter mbelt):
(103)
and:
(104)
The total aligning moment now becomes:
(105)
Coefficients and Transient Response
Left and Right Side Tires
In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire 
construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for 
positive and negative slip angles.
Name:
Name used in tire 
property file: Explanation:
pTx1 PTX1 Relaxation length sigKap0/Fz at Fznom
pTx2 PTX2 Variation of sigKap0/Fz with load
pTx3 PTX3 Variation of sigKap0/Fz with exponent of load
pTy1 PTY1 Peak value of relaxation length Sig_alpha
pTy2 PTY2 Shape factor for lateral relaxation length
pTy3 PTY3 Load where lateral relaxation is at maximum
qTz1 QTZ1 Gyroscopic torque constant
Mbelt MBELT Belt mass of the wheel
Mz' Mz' α' κ' γ Fz, , ,( )=
Mz gyr, cgyrmbeltVr1 td
dv arc Brαr eq,( )tan[ ]cos=
cgyr qTz1 λgyr⋅=
arc Brαr eq,( )tan{ }cos 1=
Mz Mz' Mz gyr,+=
Adams/Tire42
 
A tire property file with the parameters for the model results from testing with a tire that is mounted in a 
tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used for 
both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering 
wheel angle.
The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that 
indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or 
TIRESIDE = 'RIGHT').
If this keyword is available, Adams/Car corrects for the conicity and plysteer and asymmetry when using 
a tire property file on the opposite side of the vehicle. In fact, the tire characteristics are mirrored with 
respect to slip angle zero.
In Adams/View this option can only be used when the tire is generated by the graphical user interface: 
select Build -> Forces -> Special Force: Tire (see figure of dialog box below).
Next to the LEFT and RIGHT side option of TYRESIDE, you can also select SYMMETRIC: then the 
tire characteristics are modified during initialization to show symmetric performance for left and right 
side corners and zero conicity and plysteer (no offsets). Also, when you set the tire property file to 
SYMMETRIC, the tire characteristics are changed to symmetric behavior.
Create Wheel and Tire Dialog Box in Adams/View
43
USE_MODES of PAC MC: from Simple to Complex
The parameter USE_MODE in the tire property file allows you to switch the output of the PAC MC tire 
model from very simple (that is, steady-state cornering) to complex (transient combined cornering and 
braking).
The options for USE_MODE and the output of the model are listed in the table below.
Adams/Tire44
 
USE_MODE Values of PAC MC and Related Tire Model Output
Contact Methods
The PAC MC model supports the following roads:
• 2D Roads, see Using the 2D Road Model
• 3D Spline Roads, see Adams/3D Spline Road Model 
By default the PAC-MC uses a one point of contact model similar to all the other Adams/Tire 
Handling models. However the PAC-MC has an option to take the tire cross section shape into 
account:
• 3D Shell Roads, see Adams/Tire 3D Shell Road Model
Tire Cross-Section Profile Contact Method
In combination with the 2D Road Model and the 3D Road Model, you can improve the tire-road contact 
calculation method by providing the tire's cross-section profile, which has an important influence on the 
wheel center height at large inclination angles with the road.
USE MODE: State: Slip conditions:
PAC MC output
(forces and moments)
0 Steady state Acts as a vertical spring and damper 0, 0, Fz, 0, 0, 0
1 Steady state Pure longitudinal slip Fx, 0, Fz, 0, My, 0
2 Steady state Pure lateral (cornering) slip 0, Fy, Fz, Mx, 0, Mz
3 Steady state Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
4 Steady state Combined slip Fx, Fy, Fz, Mx, My, Mz
11 Transient Pure longitudinal slip Fx, 0, Fz, 0, My, 0
12 Transient Pure lateral (cornering) slip 0, Fy, Fz, Mx, 0, Mz
13 Transient Longitudinal and lateral (not 
combined)
Fx, Fy, Fz, Mx, My, Mz
14 Transient Combined slip Fx, Fy, Fz, Mx, My, Mz
45
If the tire model reads a section called [SECTION_PROFILE_TABLE] in the tire property file, the cross 
section profile will be taken into account for the vertical load calculation of the tire. The method assumes 
that the tire deformation will not influence the position of the point with largest penetration (P), which 
is valid for motor cycle tires.
The vertical tire load Fz is calculated using the penetration (effpen = ) of the tire through the tangent 
road plane in the point C, see Figure above, according to:
(106)
Because in this method the tangent to the cross section profile determines the point P, a high accuracy of 
ρ
Fz Czρ Kz ρ·⋅+=
the cross section profile is required. The section height y as function of the tire width x must be a 
continous and monotone increasing function. To avoid singularities and instability, it is highly 
Adams/Tire46
 
recommended to fit measured cross section data with a polynom (for example y = a·x2 + b·x4 + c·x6 + 
..) and provide the y cross section height data (y) from the polynom in the tire property file up to the 
maximum width of the tire. The profile is assumed to be symmetric with respect to the wheel plane.
Note that the PAC MC model has only one point of contact with the road; therefore, the wavelength of 
roadobstacles must be longer than the tire radius for realistic output of the model. In addition, the contact 
force computed by this tire model is normal to the road plane. Therefore, the contact point does not 
generate a longitudinal force when rolling over a short obstacle, such as a cleat or pothole.
For ride and comfort analysis, we recommend more sophisticated tire models, such as Ftire.
Quality Checks for the Tire Model Parameters
Because PAC MC uses an empirical approach to describe tire - road interaction forces, incorrect 
parameters can easily result in non-realistic tire behavior. Below is a list of the most important items to 
ensure the quality of the parameters in a tire property file:
• Camber (Inclination) Effects
• Validity Range of the Tire Model Input
Camber (Inclination) Effects
Camber stiffness has been explicitly defined in PAC MC (see equation (43). For realistic tire behavior, 
the sign of the camber stiffness must be negative (TYDEX W-axis (ISO) system). If the sign is positive, 
the coefficients may not be valid for the ISO but for the SAE coordinate system. Note that PAC MC only 
uses coefficients for the TYDEX W-axis (ISO) system.
Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System
Note: Do not change Fz0 (FNOMIN) and R0 (UNLOADED_RADIUS) in your tire property file. 
It will change the complete tire characteristics because these two parameters are used to 
make all parameters without dimension.
47
 
The table below lists further checks on the PAC MC parameters.
Checklist for PAC MC Parameters and Properties
Validity Range of the Tire Model Input
In the tire property file, a range of the input variables has been given in which the tire properties are 
Parameter/property: Requirement: Explanation:
LONGVL 1 m/s Reference velocity at which parameters are measured
VXLOW Approximately 1m/s Threshold for scaling down forces and moments
Dx 0 Peak friction (see equation (24))
pDx1/pDx2 0 Peak friction Fx must decrease with increasing load
Kx 0 Long slip stiffness (see equation (27))
Dy 0 Peak friction (see equation (36))
pDy1/pDy2 0 Peak friction Fx must decrease with increasing load
Ky 0 Cornering stiffness (see equation (39))
qsy1 0 Rolling resistance, should in range of 0.005 - 0.015
 0 Camber stiffness (see equation (43))
>
>
<
<
>
<
<
<
Kγ <
supposed to be valid. These validity range parameters are (the listed values can be different):
Adams/Tire48
 
$---------------------------------------------------long_slip_range 
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip
KPUMAX = 1.5 $Maximum valid wheel slip
$--------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip angle
ALPMAX = 1.5708 $Maximum valid slip angle
$--------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
CAMMIN = -1.0996 $Minimum valid camber angle
CAMMAX = 1.0996 $Maximum valid camber angle
$----------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 73.75 $Minimum allowed wheel load
FZMAX = 3319.5 $Maximum allowed wheel load
If one of the input parameters exceeds a minimum or maximum validity value, the calculation in the tire 
model will be performed with the minimum or maximum value of this range to avoid non-realistic tire 
behavior. In that case, a message appears warning you that one of the inputs exceeds a validity value.
Standard Tire Interface (STI) for PAC MC
Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to 
Adams/Solver, below is a brief background of the STI history (see reference [4]).
At the First International Colloquium on Tire Models for Vehicle Dynamics Analysis on October 21-22, 
1991, the International Tire Workshop working group was established (TYDEX).
The working group concentrated on tire measurements and tire models used for vehicle simulation 
purposes. For most vehicle dynamics studies, people previously developed their own tire models. 
Because all car manufacturers and their tire suppliers have the same goal (that is, development of tires to 
improve dynamic safety of the vehicle), it aimed for standardization in tire behavior description.
In TYDEX, two expert groups, consisting of participants of vehicle industry (passenger cars and trucks), 
tire manufacturers, other suppliers and research laboratories, had been defined with following goals:
• The first expert group's (Tire Measurements - Tire Modeling) main goal was to specify an 
interface between tire measurements and tire models. The result was the TYDEX-Format [2] to 
describe tire measurement data.
• The second expert group's (Tire Modeling - Vehicle Modeling) main goal was to specify an 
interface between tire models and simulation tools, which resulted in the Standard Tire Interface 
(STI) [3]. The use of this interface should ensure that a wide range of simulation software can be 
linked to a wide range of tire modeling software.
49
Definitions
• General
• Tire Kinematics
• Slip Quantities
• Force and Moments
General
General Definitions
Tire Kinematics
Tire Kinematics Definitions
Term: Definition:
Road tangent plane Plane with the normal unit vector (tangent to the road) in the tire-road 
contact point C.
C-axis system Coordinate system mounted on the wheel carrier at the wheel center 
according to TYDEX, ISO orientation.
Wheel plane The plane in the wheel centre that is formed by the wheel when considered 
a rigid disc with zero width.
Contact point C Contact point between tire and road, defined as the intersection of the wheel 
plane and the projection of the wheel axis onto the road plane.
W-axis system Coordinate system at the tire contact point C, according to TYDEX, ISO 
orientation.
Parameter: Definition: Units:
R0 Unloaded tire radius [m]
R Loaded tire radius [m]
Re Effective tire radius [m]
Radial tire deflection [m]
d Dimensionless radial tire deflection [-]
Fz0 Radial tire deflection at nominal load [m]
mbelt Tire belt mass [kg]
Rotational velocity of the wheel [rads-1]
ρ
ρ
ρ
ω
Adams/Tire50
 
Slip Quantities
Slip Quantities Definitions
Forces and Moments
Force and Moment Definitions
References
1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5.
2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, 
Initiated by the International Tire Working Group, July 1995.
3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995.
Parameter: Definition: Units:
V Vehicle speed [ms-1]
Vsx Slip speed in x direction [ms-1]
Vsy Slip speed in y direction [ms-1]
Vs Resulting slip speed [ms-1]
Vx Rolling speed in x direction [ms-1]
Vy Lateral speed of tire contact center [ms-1]
Vr Linear speed of rolling [ms-1]
Longitudinal slip [-]
Slip angle [rad]
Inclination angle [rad]
Abbreviation: Definition: Units:
Fz Vertical wheel load [N]
Fz0 Nominal load [N]
dfz Dimensionless vertical load [-]
Fx Longitudinal force [N]
Fy Lateral force [N]
Mx Overturning moment [Nm]
My Braking/driving moment [Nm]
Mz Aligning moment [Nm]
κ
α
γ
51
4. J.J.M. van Oosten, H.-J. Unrau, G. Riedel, E. Bakker, TYDEX Workshop: Standardisation of 
Data Exchange in Tyre Testing and Tyre Modelling, Proceedings of the 2nd International 
Colloquium on Tyre Models for Vehicle Dynamics Analysis, Vehicle System Dynamics, Volume 
27, Swets & Zeitlinger, Amsterdam/Lisse, 1996.
Example of PAC MC Tire Property File
[MDI_HEADER]
FILE_TYPE ='tir'
FILE_VERSION =3.0
FILE_FORMAT ='ASCII'
! : TIRE_VERSION : PAC Motorcycle 
! : COMMENT :Tire 180/55R17
! : COMMENT : Manufacturer 
! : COMMENT : Nom. section with (m) 0.18 
! : COMMENT : Nom. aspect ratio (-) 55
! : COMMENT : Infl. pressure (Pa) 200000
! : COMMENT : Rim radius (m) 0.216 
! : COMMENT : Measurement ID 
! : COMMENT : Test speed (m/s) 16.7 
! : COMMENT : Road surface 
! : COMMENT : Road condition Dry
! : FILE_FORMAT : ASCII
! : Copyright MSC.Software, Mon Oct 20 10:46:57 2003
!
! USE_MODE specifies the type of calculation performed:
! 0: Fz only, no Magic Formula evaluation
! 1: Fx,My only
! 2: Fy,Mx,Mz only
! 3: Fx,Fy,Mx,My,Mz uncombined force/moment calculation
! 4: Fx,Fy,Mx,My,Mz combined force/moment calculation
! +10: including relaxation behaviour
! *-1: mirroring of tyre characteristics
!
! example: USE_MODE = -12 implies:
! -calculation of Fy,Mx,Mz only
! -including relaxation effects
! -mirrored tyre characteristics
!
$-------------------------------------------------------------units
[UNITS]
LENGTH ='meter'
FORCE ='newton'
ANGLE ='radians'
MASS ='kg'
TIME ='second'
$-------------------------------------------------------------model
[MODEL]
PROPERTY_FILE_FORMAT ='PAC_MC'
USE_MODE = 14 $Tyre use switch (IUSED)
VXLOW = 1 
Adams/Tire52
 
LONGVL = 16.7 $Longitudinal speed 
during measurements 
TYRESIDE = 'SYMMETRIC' $Mounted side of tyre 
at vehicle/test bench
$---------------------------------------------------------dimensions
[DIMENSION]
UNLOADED_RADIUS = 0.322 $Free tyre radius 
WIDTH = 0.18 $Nominal section width 
of the tyre 
RIM_RADIUS = 0.216 $Nominal rim radius 
RIM_WIDTH = 0.135 $Rim width 
$----------------------------------------------------------parameter
[VERTICAL]
VERTICAL_STIFFNESS = 2e+005 $Tyre vertical 
stiffness 
VERTICAL_DAMPING = 50 $Tyre vertical damping 
BREFF = 8.4 $Low load stiffness 
eff. rolling radius 
DREFF = 0.27 $Peak value of eff. 
rolling radius 
FREFF = 0.07 $High load stiffness 
eff. rolling radius 
FNOMIN = 1475 $Nominal wheel load
$----------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip 
KPUMAX = 1.5 $Maximum valid wheel slip 
$---------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip 
angle 
ALPMAX = 1.5708 $Maximum valid slip 
angle 
$---------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
CAMMIN = -1.0996 $Minimum valid camber 
angle 
CAMMAX = 1.0996 $Maximum valid camber 
angle 
$-----------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 73.75 $Minimum allowed wheel 
load 
FZMAX = 3319.5 $Maximum allowed wheel 
load 
$------------------------------------------------------------scaling
[SCALING_COEFFICIENTS]
LFZO = 1 $Scale factor of nominal 
load 
LCX = 1 $Scale factor of Fx 
shape factor 
LMUX = 1 $Scale factor of Fx 
peak friction coefficient 
53
LEX = 1 $Scale factor of Fx 
curvature factor 
LKX = 1 $Scale factor of Fx 
slip stiffness 
LVX = 1 $Scale factor of Fx 
vertical shift 
LGAX = 1 $Scale factor of camber 
for Fx 
LCY = 1 $Scale factor of Fy 
shape factor 
LMUY = 1 $Scale factor of Fy 
peak friction coefficient 
LEY = 1 $Scale factor of Fy 
curvature factor 
LKY = 1 $Scale factor of Fy 
cornering stiffness 
LCC = 1 $Scale factor of camber 
shape factor 
LKC = 1 $Scale factor of camber 
stiffness (K-factor) 
LEC = 1 $Scale factor of camber 
curvature factor 
LHY = 1 $Scale factor of Fy 
horizontal shift 
LGAY = 1 $Scale factor of camber 
force stiffness 
LTR = 1 $Scale factor of Peak 
of pneumatic trail 
LRES = 1 $Scale factor of Peak 
of residual torque 
LGAZ = 1 $Scale factor of camber 
torque stiffness 
LXAL = 1 $Scale factor of alpha 
influence on Fx 
LYKA = 1 $Scale factor of kappa 
influence on Fy 
LVYKA = 1 $Scale factor of kappa 
induced Fy 
LS = 1 $Scale factor of Moment 
arm of Fx 
LSGKP = 1 $Scale factor of 
Relaxation length of Fx 
LSGAL = 1 $Scale factor of 
Relaxation length of Fy 
LGYR = 1 $Scale factor of 
gyroscopic torque 
LMX = 1 $Scale factor of 
overturning couple 
LVMX = 1 $Scale factor of Mx 
vertical shift 
LMY = 1 $Scale factor of rolling 
resistance torque 
$------------------------------------------------------longitudinal
[LONGITUDINAL_COEFFICIENTS]
Adams/Tire54
 
PCX1 = 1.7655 $Shape factor Cfx for 
longitudinal force 
PDX1 = 1.2839 $Longitudinal friction 
Mux at Fznom 
PDX2 = -0.0078226 $Variation of friction 
Mux with load 
PDX3 = 0 $Variation of friction 
Mux with camber 
PEX1 = 0.4743 $Longitudinal curvature 
Efx at Fznom 
PEX2 = 9.3873e-005 $Variation of curvature 
Efx with load 
PEX3 = 0.066154 $Variation of curvature 
Efx with load squared 
PEX4 = 0.00011999 $Factor in curvature 
Efx while driving 
PKX1 = 25.383 $Longitudinal slip 
stiffness Kfx/Fz at Fznom 
PKX2 = 1.0978 $Variation of slip 
stiffness Kfx/Fz with load 
PKX3 =0.19775 $Exponent in slip 
stiffness Kfx/Fz with load 
PVX1 = 2.1675e-005 $Vertical shift Svx/Fz 
at Fznom 
PVX2 = 4.7461e-005 $Variation of shift 
Svx/Fz with load 
RBX1 = 12.084 $Slope factor for 
combined slip Fx reduction 
RBX2 = -8.3959 $Variation of slope Fx 
reduction with kappa 
RBX3 = 2.1971e-009 $Influence of camber 
on stiffness for Fx combined 
RCX1 = 1.0648 $Shape factor for 
combined slip Fx reduction 
REX1 = 0.0028793 $Curvature factor of 
combined Fx 
REX2 = -0.00037777 $Curvature factor of 
combined Fx with load 
RHX1 = 0 $Shift factor for 
combined slip Fx reduction 
PTX1 = 0.83 $Relaxation length 
SigKap0/Fz at Fznom 
PTX2 = 0.42 $Variation of SigKap0/Fz 
with load 
PTX3 = 0.21 $Variation of SigKap0/Fz 
with exponent of load 
$--------------------------------------------------------overturning
[OVERTURNING_COEFFICIENTS]
QSX1 = 0 $Lateral force induced 
overturning moment 
QSX2 = 0.16056 $Camber induced 
overturning moment 
QSX3 = 0.095298 $Fy induced overturning 
moment 
55
$------------------------------------------------------------lateral
[LATERAL_COEFFICIENTS]
PCY1 = 1.1086 $Shape factor Cfy for 
lateral forces 
PCY2 = 0.66464 $Shape factor Cfc for 
camber forces 
PDY1 = 1.3898 $Lateral friction Muy 
PDY2 = -0.0044718 $Exponent lateral 
friction Muy 
PDY3 = 0.21428 $Variation of friction 
Muy with squared camber 
PEY1 = -0.80276 $Lateral curvature Efy 
at Fznom 
PEY2 = 0.89416 $Variation of curvature 
Efy with camber squared 
PEY3 = 0 $Asymmetric curvature 
Efy at Fznom 
PEY4 = 0 $Asymmetric curvature 
Efy with camber 
PEY5 = -2.8159 $Camber curvature Efc 
PKY1 = -19.747 $Maximum value of 
stiffness Kfy/Fznom 
PKY2 = 1.3756 $Curvature of stiffness 
Kfy 
PKY3 = 1.3528 $Peak stiffness factor 
PKY4 = -1.2481 $Peak stiffness 
variation with camber squared 
PKY5 = 0.3743 $Lateral stiffness 
depedency with camber squared 
PKY6 = -0.91343 $Camber stiffness factor 
Kfc 
PKY7 = 0.2907 $Vertical load 
dependency of camber stiffn. Kfc 
PHY1 = 0 $Horizontal shift Shy 
at Fznom 
RBY1 = 10.694 $Slope factor for 
combined Fy reduction 
RBY2 = 8.9413 $Variation of slope Fy 
reduction with alpha 
RBY3 = 0 $Shift term for alpha 
in slope Fy reduction 
RBY4 = -1.8256e-010 $Influence of camber 
on stiffness of Fy combined 
RCY1 = 1.0521 $Shape factor for 
combined Fy reduction 
REY1 = -0.0027402 $Curvature factor of 
combined Fy 
REY2 = -0.0094269 $Curvature factor of 
combined Fy with load 
RHY1 = -7.864e-005 $Shift factor for 
combined Fy reduction 
RHY2 = -6.9003e-006 $Shift factor for 
combined Fy reduction with load 
Adams/Tire56
 
RVY1 = 0 $Kappa induced side 
force Svyk/Muy*Fz at Fznom 
RVY2 = 0 $Variation of Svyk/Muy*Fz 
with load 
RVY3 = -0.00033208 $Variation of 
Svyk/Muy*Fz with camber 
RVY4 = -4.7907e+015 $Variation of 
Svyk/Muy*Fz with alpha 
RVY5 = 1.9 $Variation of Svyk/Muy*Fz 
with kappa 
RVY6 = -30.082 $Variation of 
Svyk/Muy*Fz with atan(kappa) 
PTY1 = 0.75 $Peak value of relaxation 
length Sig_alpha 
PTY2 = 1 $Shape factor for 
Sig_alpha 
PTY3 = 0.6 $Value of Fz/Fznom where 
Sig_alpha is maximum 
$-------------------------------------------------rolling resistance
[ROLLING_COEFFICIENTS]
QSY1 = 0.01 $Rolling resistance 
torque coefficient 
QSY2 = 0 $Rolling resistance 
torque depending on Fx 
QSY3 = 0 $Rolling resistance 
torque depending on speed 
QSY4 = 0 $Rolling resistance 
torque depending on speed^4 
$-----------------------------------------------------------aligning
[ALIGNING_COEFFICIENTS]
QBZ1 = 9.246 $Trail slope factor for 
trail Bpt at Fznom 
QBZ2 = -1.4442 $Variation of slope Bpt 
with load 
QBZ3 = -1.8323 $Variation of slope Bpt 
with load squared 
QBZ4 = 0 $Variation of slope Bpt 
with camber 
QBZ5 = 0.15703 $Variation of slope Bpt 
with absolute camber 
QBZ9 = 8.3146 $Slope factor Br of 
residual torque Mzr 
QCZ1 = 1.2813 $Shape factor Cpt for 
pneumatic trail 
QDZ1 = 0.063288 $Peak trail Dpt = 
Dpt*(Fz/Fznom*R0) 
QDZ2 = -0.015642 $Variation of peak Dpt 
with load 
QDZ3 = -0.060347 $Variation of peak Dpt 
with camber 
QDZ4 = -0.45022 $Variation of peak Dpt 
with camber squared 
QDZ6 = 0 $Peak residual torque 
Dmr = Dmr/(Fz*R0) 
57
QDZ7 = 0 $Variation of peak 
factor Dmr with load 
QDZ8 = -0.08525 $Variation of peak 
factor Dmr with camber 
QDZ9 = -0.081035 $Variation of peak 
factor Dmr with camber and load 
QDZ10 = 0.030766 $Variation of peak 
factor Dmr with camber squared 
QDZ11 = 0.074309 $Variation of Dmr with 
camber squared and load 
QEZ1 = -3.261 $Trail curvature Ept 
at Fznom 
QEZ2 = 0.63036 $Variation of curvature 
Ept with load 
QEZ3 = 0 $Variation of curvature 
Ept with load squared 
QEZ4 = 0 $Variation of curvatureEpt with sign of Alpha-t 
QEZ5 = 0 $Variation of Ept with 
camber and sign Alpha-t 
QHZ1 = 0 $Trail horizontal shift 
Sht at Fznom 
QHZ2 = 0 $Variation of shift 
Sht with load 
QHZ3 = 0 $Variation of shift 
Sht with camber 
QHZ4 = 0 $Variation of shift 
Sht with camber and load 
SSZ1 = 0 $Nominal value of s/R0: 
effect of Fx on Mz 
SSZ2 = 0.0033657 $Variation of distance 
s/R0 with Fy/Fznom 
SSZ3 = 0.16833 $Variation of distance 
s/R0 with camber 
SSZ4 = 0.017856 $Variation of distance 
s/R0 with load and camber 
QTZ1 = 0 $Gyroscopic torque 
constant 
MBELT = 0 $Belt mass of the wheel 
-kg- $
Adams/Tire58
 
521-Tire Model
About 521-Tire
The 521-Tire model is a simple model that requires a small set of parameters or experimental data to 
simulate the behavior of tires. The 521-Tire is the first tire model incorporated in Adams. The name 
“521” (actually “5.2.1”) refers to the version number of Adams/Tire when it was first released.
The slip forces and moments can be calculated in two ways:
• Using the Equation method
• Using the Interpolation method
Two dedicated contact methods exist for the 521-Tire:
• Point Follower, used for Handling analysis models
• Equivalent Plane Method, used for 3D Contact analysis models
Any combination of force and contact method is allowed.
The road data files used for the 521-Tire are unique and cannot be used in combination with any other 
Handling tire model. The 521 road file format is described in Road Data File 521_pnt_follow.rdf.
Note that the capability and generality of the 521-Tire have been superseded by other, newer tire models, 
described throughout this guide. We’ve retained the 521-Tire model primarily for backward 
compatibility. We recommend that you use other tire models for new work. 
Tire Slip Quantities and Transient Tire Behaviour
Definition of Tire Slip Quantities
Slip Quantities at Combined Cornering and Braking/Traction
Adams/Tire12
 
 
The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the 
wheel rotational velocity , and the loaded rolling radius Rl:
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point: 
Note that for realistic tire forces the slip angle is limited to 90 degrees and the longitudinal slip in 
between -1 (locked wheel) and 1.
Lagged longitudinal and lateral slip quantities (transient tire behavior)
In general, the tire rotational speed and lateral slip will change continuously because of the changing 
interaction forces in between the tire and the road. Often the tire dynamic response will have an important 
role on the overall vehicle response. For modeling this so-called transient tire behavior, a first-order 
system is used both for the longitudinal slip as the side slip angle, . Considering the tire belt as a 
stretched string, which is supported to the rim with lateral springs, the lateral deflection of the belt can 
be estimated (see also reference [1]). The figure below shows a top-view of the string model.
Stretched String Model for Transient Tire Behavior
Ω
Vsx Vz ΩR1–=
Vsy Vy=
κ α
κ VsxVx
-------- and αtan– VsyVx
---------= =
α κ
κ α
13
When rolling, the first point having contact with the road adheres to the road (no sliding assumed). 
Therefore, a lateral deflection of the string will arise that depends on the slip angle size and the history 
of the lateral deflection of previous points having contact with the road.
For calculating the lateral deflection v1 of the string in the first point of contact with the road, the 
following differential equation is valid during braking slip:
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger than 
10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the equation 
can be transformed to:
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill. 
A similar approach yields the following for the deflection of the string in longitudinal direction:
1
Vx
------
dv1
dt
--------
v1
σα
------+ α( )tan aφ+=
σα φ
σα
dv1
dt
-------- Vx v1+ σ– κVsx=
σα
du1
dt
-------- Vx u1+ σ– κVsx=
Now the practical slip quantities, and are defined based on the tire deformation:κ′ α′
Adams/Tire14
 
These practical slip quantities and are used instead of the usual and definitions for steady-
state tire behavior.
The longitudinal and lateral relaxation length are read from the tire property file, see Tire Property File 
521_equation.tir and 521_interpol.tir
Force Calculations
You can use the 521-Tire model for handling and durability analyses.
Directional Vectors for the Application of Tire Forces and Torques at the Center of the Tire-Road 
Surface Contact Patch
The forces act along the directional vectors. From the tire spin vector and various information you supply 
in the tire property and the road profile data files, Adams/Tire determines the positions and orientations 
of the tire vertical, lateral, and longitudinal directional vectors. Figure 3 shows these directional vectors.
The tire vertical force acts along the vertical directional vector, the tire aligning torque acts about the 
same vector, the tire lateral force acts along the lateral directional vector, and the tire longitudinal force 
acts along the longitudinal directional vector. At this point, Adams/Tire determines the force directions 
κ' u1σκ
------ Vx( )sin=
α' v1σα
------⎝ ⎠⎛ ⎞atan=
κ′ α′ κ α
15
as if it were going to apply the tire aligning torque and all of the tire forces at the center of the tire-road 
surface contact patch.
The tire-road surface contact patch may deflect laterally. Adams/Tire calculates the lateral deflection in 
the direction (and with the sign) of the lateral force. The magnitude of the deflection is equal to the lateral 
force divided by the tire lateral stiffness you provide in the tire property data file.
The tire vertical, lateral, and longitudinal forces are forces in the tire vertical, lateral, and longitudinal 
directions (as determined at the tire-road surface contact patch). The tire aligning torque is a torque about 
the tire vertical vector. The vehicle durability force has components in both the tire vertical and the tire 
longitudinal directions.
Normal Force
The tire normal force Fz is calculated based on the tire deflection and radial velocity. A progressive 
spring and linear damping constant are employed:
where Fstiff is tire stiffness force and Fdamp is tire damping force. The vertical stiffness force is calculated 
from:
where Kz is the tire vertical stiffness, δ is tire deflection, and is the stiffness exponent. The tire 
damping force is calculated from:
where Cz is the tire damping constant.
The damping constant is reduced for small tire deflections, which are below 5% of the unloaded tire 
radius.
The tirevertical stiffness can also be described using a spline function (force versus deflection) in the 
Adams dataset. The user array is used to switch between tire property file stiffness and spline stiffness. 
If the first value in the user array is equal to '5215', the spline vertical stiffness is used. The second value 
of the user array refers to the ID of the spline. The message, 'Using spline data for the vertical spring', is 
shown in the message file. If the first value in the user array is not equal to '5215', the tire property file 
stiffness is used.
The following is an example of using the spline vertical stiffness:
! adams_view_name='spline_vertical_stiffness'
SPLINE/10
, X = -1,0,10,30
Fz Fstiff Fdamp–=
Fstiff Kzδθ=
θ
Fdamp Cz RadialVelocity×=
, Y = 0,0,2000,6000
!
Adams/Tire16
 
! adams_view_name='wheel_user_array'
ARRAY/102
, NUM=5215,10
Another option for having a non-linear tire stiffness is to introduce a deflection-load table in the tire 
property file in a section called [DEFLECTION_LOAD_CURVE]. See 521-Tire Tire and Road Property 
Files on page 20. If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection 
datapoints with a cubic spline for inter- and extrapolation are used for the calculation of the vertical force 
of the tire. 
Longitudinal Force
The tire longitudinal force Fx can have up to three contributions:
• Traction/braking force
• Rolling resistance force
• Durability force (in case of durability contact)
Traction/Braking Force
Traction force is developed if the vehicle is starting to move and a braking force if the vehicle is 
beginning to stop. In either case, the absolute magnitude of the force is calculated from:
where the friction coefficient μ is a function of the longitudinal slip velocity Vsx in the contact patch. 
Note that this is somewhat unusual, since all the other Handling tire models in Adams/Tire assume that 
the longitudinal force Fx is a function of the slip ratio.
Schematic of Friction Coefficient Versus Local Slip Velocity
Fx μFz=
17
The μ curve as a function of longitudinal slip velocity is created using standard Adams STEP functions 
(see body 4 on page 10). You have to specify two points on the curve to define this characteristic:
• The coordinates of the curve at μstatic: (velocity μstatic, μstatic)
• The coordinates of the curve at μdynamic: (velocity μdynamic, μdynamic)
The friction values may be available to you as function of slip ratio instead of slip velocity. Converting 
Slip Ratio Data to Velocity Data on page 16 explains how the slip ratios can be converted to slip 
velocities.
Rolling Resistance Force
Rolling resistance Moment My is calculated from: 
where coefrr is the rolling resistance coefficient that should be supplied in the tire property data file. 
Durability Force
Durability force, sometimes known as radial planar force, is a special kind of tire vertical force. It is the 
durability force that resists the action of road bumps. This force acts along the instantaneous vertical 
directional vector calculated by Adams/Tire. The Adams/Tire durability tire forces are limited to two-
dimensional forces that lie in the plane of the tire and are directed toward the wheel-center marker. 
Adams/Tire superimposes these forces upon any traction or lateral forces developed in the tire-road 
My coefrr Fz⋅=
surface interaction.
Adams/Tire18
 
You must select the Equivalent Plane Method for generating these durability forces.
Lateral Force and Aligning Torque
Two methods exist for calculating the lateral force Fy and self-aligning moment Mz:
• Interpolation Method
• Equation Method
Interpolation Method
The AKIMA spline is employed to calculate Fy and Mz as a function of the slip angle α, camber angle γ, 
and vertical load Fz. You should provide the data in the SAE axis system.
Note that the slip angle α and vertical load Fz input for the force and moment calculation of Fx, Fy, Mx, 
My, and Mz are limited to minimum and maximum values in the input to avoid unrealistic extrapolated 
values. 
Equation Method
The Equation Method uses the following equation to generate the lateral force Fy:
where Kα denotes the tire cornering stiffness coefficient.
The aligning moment Mz is calculated using the pneumatic trail t according to: 
while the pneumatic trails are calculated with half the contact length a: 
with R0 and Rl are, respectively, the unloaded and loaded tire radius.
Overturning Moment
In both methods, the overturning moment Mx calculation is based on the lateral tire force Fy, the lateral 
tire stiffness Ky, and the vertical load:
Fy μstatFz 1 e Kα α––( )⋅ sign α( )⋅( )–=
Mz t– Fy⋅=
t 13
--- a e
Kα α–⋅⋅=
a R0
2 R1
2–=
Mx
Fy
K
------= Fz
y
19
Tire Lateral Force as a Function of Slip Angle
• The contribution of the camber is disregarded in the Equation Method.
• The cornering stiffness equals .
Combined Slip of 5.2.1
The combined slip calculation of the 5.2.1. is using the friction ellipse and is similar to the combined slip 
calculation of the Pacejka '89 and '94 tire models.
Inputs: 
• Dimensionless longitudinal slip (range -1 to 1) and side slip angle in radians 
• Longitudinal force Fx and lateral force Fy calculated using the equations of 521-Tire
• The vertical shift of Fy,a=0 is Fy calculated at zero slip angle
Output:
• Adjusted longitudinal force Fx and lateral force Fy incorporates the reduction due to combined 
slip:
•
Friction coefficients:
γ
μstatFzKa–
κ α
β k
k2 α2sin+
------------------------------⎝ ⎠⎜ ⎟
⎛ ⎞
acos=
μx act,
Fx
F
-----= μy act,
Fy Fy α 0=,–
F
------------------------------=
z z
Adams/Tire20
 
Forces corrected for combined slip conditions: 
 
Due to the lateral deflection of the tire patch, the aligning moment under combined slip conditions 
increases by the effect of the longitudinal force Fx and the lateral tire stiffness Ky:
and the overturning moment uses the lateral force for combined slip:
Smoothing
When you indicate smoothing by setting the value of USE_MODE in the tire property file, Adams/Tire 
smooths initial transients in the tire force over the first 0.1 seconds of the simulation. The longitudinal 
force, lateral force, and aligning torque are multiplied by a cubic step function of time. (See STEP in the 
Adams/Solver online help.)
• Longitudinal Force Fx = SFx.
• Lateral Force Fy = SFy
• Overturning moment torque Mx = SMz
• Aligning torque Mz = SMz
Changing the Operating Mode: USE_MODE
You can change the behavior of the tire model by changing the value of USE_MODE in the [MODEL] 
section of the tire property file. If USE_MODE equals zero, or when it is absent, the smoothing time 
equals 0.001 seconds and the 
521-Tire model is compatible with the previous Adams/Solver implementation. 
μx 1
1
μx act,
-------------⎝ ⎠⎛ ⎞
2 βtan
μstat
-----------⎝ ⎠⎛ ⎞
2
+
----------------------------------------------------= μy βtan
1
μstat
----------⎝ ⎠⎛ ⎞
2 βtan
μy act,
-------------⎝ ⎠⎛ ⎞
2
+
---------------------------------------------------=
Fx comb,
μx
μx act,
-------------Fx= Fy comb,
μy
μy act,
------------- Fy Fy α 0=,+( )=
Mz comb, Mz pure, Fx comb,+=
Fy comb,
Ky
------------------⋅
Mx comb,
Fy comb,
Ky
------------------Fz=
21
By selecting a value of USE_MODE between 1 and 4, smoothing and combined slip correction can be 
switched on and off, as shown in Table 1. The smoothing time equals 0.1 seconds for these values of 
USE-MODE.
Converting Slip Ratio Data to Velocity Data
Adams/Tire requires that you enter the velocities that correspond to μstatic and μdynamic. You will often 
obtain this information as the coefficient of friction versus slip ratio. You can calculate the velocities 
required by Adams/Tire fromthe coefficient of friction versus slip ratio curve in the following way:
where:
• = Slip ratio
• = Free rolling rotational velocity (no slip)
• = Actual rotational velocity
Kinematic relationships between translational and rotational velocities and the effective rolling radius 
give:
where:
• = Contact patch velocity reletive to road surface
• = Actual longitudinal velocity
USE_MODE: Smoothing: Combined slip correction:
1 off off
2 off on
3 on off
4 on on
κ ωa ωf–ωf
------------------=
κ
ωf
ωa
ωa
Vx Vsx–
Re
---------------------=
ωf
Vx
Re
------=
Vsx
Vx
Adams/Tire22
 
• = Effective rolling radius
Substituting these relationships into the original slip ratio equation with some cancelling of variables 
gives:
Therefore:
During testing for the coefficient of friction as a function of slip ratio, the longitudinal velocity Vx is held 
constant. Therefore, you can obtain Vsx, the relative velocity of the contact patch with respect to the road 
surface, from the test data curves for the static and dynamic values of friction.
Contact Methods
For handling analyses (which use a flat road surface profile), the 521-Tire model uses the point-follower 
contact method. For durability analyses (which use uneven road surface profiles), the Equivalent Plane 
Method yields the instantaneous tire radius directly, while finding the new road surface orientation 
vector.
About the Point-Follower Method
The point-follower contact method assumes a single contact point between the tire and road. The contact 
point is the point nearest to the wheel center that lies on the line formed by the intersection of the tire 
(wheel) plane with the local road plane.
The contact force computed by the point-follower contact method is normal to the road plane. Therefore, 
in a simulation of a tire hitting a pothole, the point-follower contact method does not generate the 
expected longitudinal force.
About the Equivalent Plane Method 
521-Tire uses the Equivalent Plane method to reorient the vertical road surface vector, which gives the 
direction of the vertical force, and to calculate the new tire radius. To do this, a new smooth road surface 
is generated at an angle calculated such that only the shape of the tire is different (see body 6 on page 18).
Equivalent Plane Method
Re
κ VsxVx
--------–=
Vsx Vxκ–=
23
Both the deflected tire area and its centroid remain unchanged. The vector between the deflected area 
centroid and the wheel-center marker then determines the orientation of the. vertical vector 
perpendicular to the road surface.
The Equivalent Plane method is best suited for relatively large obstacles because it assumes the tire 
encompasses the obstacle uniformly. In reality, the pneumatics and the bending stiffness of the tire 
carcass prevent this. The result is an uneven pressure distribution and possibly gaps between the tire and 
the road. If the obstacle is larger than the tire contact patch (such as a pothole or curb), the uniform 
assumption is good. If the obstacle is much smaller than the tire patch, however (such as a tar strip or 
expansion joint), the assumption is poor, and the Equivalent Plane method may greatly underestimate the 
durability force.
Definition of Equivalent Plane Parameters
Adams/Tire24
 
When using the Equivalent Plane method the following parameters need to be specified in the tire 
property file:
Equivalent_plane_angle
Specifies the subtended angle (in degrees) bisected by the z-axis of the wheel-center marker, as shown 
in Figure 7. This angle determines the extent of the road the tire can envelop. The value of the 
equivalent_plane_angle must be between 0 and 180 degrees.
Equivalent_plane_increments
Specifies the number of increments into which the shadow of the tire subtended section is divided, as 
shown in Figure 7.
521-Tire Tire and Road Property Files
This section contains four example input data files. For reference, the files are called:
• 521_equation.tir
• 521_interpol.tir
• 521_pnt_follow.rdf
• 521_equiv_plane.rdf
The first two files are tire property files, and the last two are road files. The file 521_equation.tir 
illustrates the required format and parameters when you use the Equation method. The file 
521_interpol.tir illustrates the Interpolation method. The two *.rdf files show how road data files must 
be specified when either of the contact methods is used.
25
Tire Property File 521_equation.tir and 521_interpol.tir
You can select the method for calculating the normal force by setting the 
VERTICAL_FORCE_METHOD parameter to either POINT_FOLLOWER (for the Point Follower 
method) or EQUIVALENT_PLANE (for the Equivalent Plane method). See Contact Methods on page 
17 for details on these methods. 
You can select the method for calculating the lateral force by setting the LATERAL_FORCE_METHOD 
parameter to either INTERPOLATION or symbol. See Lateral Force and Aligning Torque on page 11 
for details on these calculation methods.
The following table specifies how some of the parameter names used in the tire property file correspond 
to parameters introduced in the equations that were presented in the previous sections. 
521-equation.tir
The 521-equation.tir example tire property file starts here.
$--------------------------------------------------------MDI_HEADER
[MDI_HEADER]
 FILE_TYPE = 'tir'
 FILE_VERSION = 3.0
 FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'Tire - XXXXXX'
'Pressure - XXXXXX'
'Test Date - XXXXXX'
Parameter in file: Used in equation: As parameter:
vertical_stiffness [10] Kz
vertical_damping [11] Cz
lateral_stiffness [18] Ky
cornering_stiffness_coefficient [6] Kα
Mu_Static Figure 4 μstatic
Mu_Dynamic Figure 4 μdynamic
Mu_Static_velocity Figure 4 velocity μstatic
Mu_Dynamic_Velocity Figure 4 velocity μdynamic
rolling_resistance_coefficient [13] coeffrr
vertical_stiffness_exponent [141] Note: If you do not specify 
vertical_stiffness_
exponent in the tire 
property file, 521-Tire 
uses the default value of 
1.1. 
'Test tire'
$-------------------------------------------------------------units
Adams/Tire26
 
[UNITS]
 LENGTH = 'mm'
 FORCE = 'newton'
 ANGLE = 'rad'
 MASS = 'kg'
 TIME = 'second'
$-------------------------------------------------------------model
[MODEL]
! use mode 123411121314
! -----------------------------------------------------------------
! smoothingXXXX
! combinedXXXX
! transient X X X X
!
 PROPERTY_FILE_FORMAT = '5.2.1'
 USE_MODE = 1
$----------------------------------------------------------dimension
[DIMENSION]
 UNLOADED_RADIUS = 310.0
 WIDTH = 195.0
 ASPECT_RATIO = 0.70
 RIM_RADIUS = 195,0
 RIM_WIDTH = 139.7
$---------------------------------------------------------parameters
!
 VERTICAL_FORCE_METHOD = EQUIVALENT_PLANE
 LATERAL_FORCE_METHOD = EQUATION
!
 vertical_stiffness = 206.0
 vertical_stiffness_exponent = 1.1
 vertical_damping = 2.06
!
 lateral_stiffness = 50
 cornering_stiffness_coefficient = 50
!
 Mu_Static = 0.95
 Mu_Dynamic = 0.75
 Mu_Static_Velocity = 3000
 Mu_Dynamic_Velocity = 6000
!
 rolling_resistance_coefficient = 0.01
!
 EQUIVALENT_PLANE_ANGLE= 100
 EQUIVALENT_PLANE_INCREMENTS= 50
!
521_interpol.tir
The 521-interpol.tir example tire property file starts here. In addition to the file for 521_equation.tir, it 
contains data that is used for calculating the lateral force and aligning moment, instead of using formula 
6 to 9. Note that the [DEFLECTION_LOAD_CURVE] can also be used in the tire property file for the 
Equation method.
27
$--------------------------------------------------------MDI_HEADER[MDI_HEADER]
 FILE_TYPE = 'tir'
 FILE_VERSION = 3.0
 FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'Tire - XXXXXX'
'Pressure - XXXXXX'
'Test Date - XXXXXX'
'Test tire'
$-------------------------------------------------------------units
[UNITS]
 LENGTH = 'mm'
 FORCE = 'newton'
 ANGLE = 'rad'
 MASS = 'kg'
 TIME = 'second'
$--------------------------------------------------------------model
[MODEL]
! use mode 123411121314
! ----------------------------------------------------------------
! smoothingXXXX
! combinedXXXX
! transient X X X X
!
 PROPERTY_FILE_FORMAT = '5.2.1'
 USE_MODE = 1
$----------------------------------------------------------dimension
[DIMENSION]
 UNLOADED_RADIUS = 310.0
 WIDTH = 195.0
 ASPECT_RATIO = 0.70
 RIM_RADIUS = 195,0
 RIM_WIDTH = 139.7
$---------------------------------------------------------parameters
!
 VERTICAL_FORCE_METHOD = POINT_FOLLOWER ! or EQUIVALENT_PLANE
 LATERAL_FORCE_METHOD = INTERPOLATION ! or EQUATION
!
 vertical_stiffness = 206.0
 vertical_stiffness_exponent = 1.1
 vertical_damping = 2.06
 lateral_stiffness = 50
 cornering_stiffness_coefficient = 50
!
 Mu_Static = 0.95
 Mu_Dynamic = 0.75
 Mu_Static_Velocity = 3000
 Mu_Dynamic_Velocity = 6000
!
 rolling_resistance_coefficient = 0.01
!
 EQUIVALENT_PLANE_ANGLE= 100
Adams/Tire28
 
 EQUIVALENT_PLANE_INCREMENTS= 50
!
!------------------CAMBER ANGLE VALUES-------------------------------
-----------
! Conversion
! No. of pnts factor(D to R) pnt1 pnt2 pnt3 pnt4 pnt5
!
 CAMBER_ANGLE_DATA_LIST
 5 0.017453292 -3.0 0.0 3.0 6.0 10.0
!
!------------------SLIP ANGLE VALUES---------------------------------
-----------
! Conversion
! No. of pnts factor(D to R) pnt1 ...... pnt9
!
 SLIP_ANGLE_DATA_LIST
 9 0.017453292 -15.0 -10.0 -5.0
 -2.5 0.0 2.5
 5.0 10.0 15.0
!
!-----------------VERTICAL FORCE VALUES------------------------------
-----------
! Conversion 
! No. of pnts factor
! pnt1 pnt2 pnt3 pnt4 pnt5 
!
 VERTICAL_FORCE_DATA_LIST
 5 4.448
 200.0 600.0 1100.0 1500.0 1900.0
!
!-----------------ALLIGNING TORQUE VALUES----------------------------
-----------
! No. of pnts Conversion
! factor
!
! pnt1 .... pnt225
!
 ALIGNING_TORQUE_DATA_LIST
 225 -1355.7504
 5.31 6.52 22.88 26.41 30.58
 0.11 2.84 5.49 -3.92 -14.04
 0.47 -12.44 -37.99 -67.22 -116.07
 0.04 -21.38 -69.04 -111.44 -168.11
 0.80 -3.70 -27.94 -44.25 -53.74
 1.75 17.43 52.20 81.97 145.78
 2.54 11.08 40.53 73.54 95.55
 -1.28 0.02 14.82 2.93 10.35
 1.59 -3.77 -17.17 6.60 -11.91
 0.06 14.23 22.93 11.45 15.74 
 5.95 5.54 13.72 -1.65 -15.64
 -1.29 -9.45 -26.98 -57.25 -107.71
 -5.05 -17.73 -62.62 -109.03 -161.88 
29
 0.46 -2.48 -19.48 -33.54 -49.52
 4.71 26.10 60.80 90.85 119.51
 4.26 16.60 52.46 93.32 141.34
 2.41 4.28 2.21 9.11 30.44
 -0.92 0.22 12.61 2.51 -18.77
 0.43 -4.62 15.36 7.16 11.70
 6.70 15.92 0.14 -4.20 -11.81
 -2.20 -5.53 -13.28 -47.48 -92.88
 -1.39 -17.28 -52.17 -102.80 -161.71 
 2.87 -0.38 -14.27 -29.03 -42.42
 6.99 24.54 66.06 93.27 126.38
 7.10 18.78 58.20 104.51 156.39
 1.63 2.91 8.33 20.32 42.09
 -0.78 10.13 -9.94 -13.02 -11.95
 5.62 4.36 23.16 38.03 8.73
 2.31 6.41 14.10 6.03 -11.66
 7.87 1.33 -16.31 -40.24 -82.58
 1.40 -10.04 -50.94 -93.06 -157.50
 2.10 0.56 -16.15 -27.15 -40.13
 5.60 26.48 62.92 90.16 122.03
 3.56 20.63 60.74 108.26 162.97
 -0.08 1.81 14.39 34.98 59.72
 1.38 -2.13 -2.42 -4.08 -2.72
 3.69 1.71 29.06 10.05 11.38
 3.09 7.15 -7.92 13.53 -5.78
 6.08 0.38 -2.69 -32.10 -62.17
 0.76 -7.65 -37.28 -89.05 -145.09
 0.70 4.37 -7.59 -23.71 -28.49
 5.92 34.39 72.55 92.88 129.34
 4.36 29.81 76.70 118.91 180.59
 -2.03 5.94 26.18 53.59 89.76
 0.39 -5.52 -6.06 10.16 7.81
 !-----------------LATERAL FORCE VALUES------------------------------
---------
 ! No. of pnt Conversion
 ! factor
 ! pnt1 .... pnt225
 !
 LATERAL_FORCE_DATA_LIST
 225 4.448
 
 234.08 585.56 1000.29 1307.77 1603.78
 269.79 628.82 1040.78 1331.72 1624.83
 213.70 565.29 974.49 1198.82 1387.74
 150.79 452.18 752.21 885.23 960.13
 11.52 50.58 199.87 199.50 208.75
 -116.75 -367.42 -618.68 -683.16 -857.81
 -224.15 -588.24 -1001.01 -1235.88 -1488.88
 -242.08 -612.70 -1059.55 -1344.53 -1658.66
 -213.99 -597.29 -988.14 -1343.86 -1689.35
Adams/Tire30
 
 234.40 572.75 981.30 1352.37 1698.90
 239.27 647.77 1007.37 1357.22 1666.30
 252.34 603.75 1033.50 1288.76 1483.64 
 167.55 481.45 826.41 962.64 1028.74
 32.23 78.77 231.31 250.14 254.32
 -122.59 -423.13 -552.58 -613.52 -607.61
 -208.93 -576.28 -948.45 -1149.44 -1314.69
 -261.05 -634.90 -1064.15 -1338.52 -1581.84
 -241.50 -607.16 -1021.87 -1322.30 -1598.25
 210.20 578.56 968.72 1344.05 1730.40
 237.91 600.60 1025.67 1377.57 1733.03
 226.60 629.48 1084.97 1354.12 1575.22
 154.74 496.21 878.72 1028.03 1095.59
 34.37 74.19 240.00 284.42 283.85
 -130.29 -339.00 -509.04 -543.75 -555.05
 -226.48 -557.52 -884.91 -1083.18 -1175.12
 -270.70 -595.22 -1059.76 -1314.74 -1564.43
 -254.64 -602.76 -1032.71 -1313.22 -1609.96
 238.28 531.25 945.70 1305.28 1786.96
 227.13 594.51 1038.87 1365.33 1733.29
 221.76 633.49 1135.31 1375.28 1619.82
 195.50 505.90 899.88 1059.92 1135.28
 28.51 68.59 241.99 311.15 331.84
 -145.10 -319.56 -464.11 -499.27 -500.83
 -230.33 -548.99 -815.88 -991.78 -1108.36
 -230.62 -597.10 -1009.76 -1261.43 -1504.09
 -218.36 -570.13 -1049.72 -1344.94 -1589.60
 228.49 564.69 954.06 1332.84 1687.50
 221.19 595.52 1019.74 1378.35 1749.40
 224.63 590.58 1108.01 1408.87 1707.09
 178.96 474.70 918.87 1125.97 1242.75
 42.58 65.26 230.69 306.58 428.45 
 -144.43 -290.91 -368.02 -398.98 -394.66
 -224.99 -494.65 -761.78 -886.03 -941.20
 -246.51 -563.13 -980.33 -1249.57 -1462.88
 -239.34 -567.10 -1050.56 -1348.66 -1611.11
521-Tire Road Data Files
The road data files used with the 521-Tire are unique and cannot be used with any other tire model. The 
data files are fully described by the following two examples. 
Road Data File 521_pnt_follow.rdf
This example file shows that, if you use the Point Follower method and indicate it in the associated tire 
property file, the road_profile_type parameter must be set to FLAT.
$--------------------------------------------------------MDI_HEADER
[MDI_HEADER]
FILE_TYPE = 'rdf'
FILE_VERSION= 5.00
31
FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'flat 2d contact road for testing purposes'
$-------------------------------------------------------------UNITS
[UNITS]
 LENGTH = 'mm'
 FORCE = 'newton'
 ANGLE = 'radians'
 MASS = 'kg'
 TIME = 'sec'
$-------------------------------------------------------------MODEL
[MODEL]
 METHOD = '5.2.1'
 FUNCTION_NAME = 'ARC913'
$--------------------------------------------------------PARAMETERS
ROAD_PROFILE_TYPE = FLAT
INITIAL_HEIGHT = 0.000
Road Data File 521_equiv_plane.rdf
The following example shows which data the road data file must contain if the Equivalent Plane method 
is used and specified in the associated tire property file. The main difference with the road data file used 
in association with the Point Follower method is that here the ROAD_PROFILE_TYPE parameter is set 
to INPUT and a ROAD_INPUT_DATA_LIST is specified.
$---------------------------------------------------------MDI_HEADER
[MDI_HEADER]
FILE_TYPE = 'rdf'
FILE_VERSION = 5.00
FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'5.2.1 input road for testing purposes'
$--------------------------------------------------------------UNITS
[UNITS]
 LENGTH = 'mm'
 FORCE = 'newton'
 ANGLE = 'radians'
 MASS = 'kg'
 TIME = 'sec'
$--------------------------------------------------------------MODEL
[MODEL]
 METHOD = '5.2.1'
 FUNCTION_NAME = 'ARC913'
$---------------------------------------------------------PARAMETERS
ROAD_PROFILE_TYPE = INPUT
INITIAL_HEIGHT = 0.000
ROAD_INPUT_DATA_LIST
 23, 1
 -10000.00, 00.00
 1740.00, 00.00
 1740.94, 1.92
 1743.73, 3.55
Adams/Tire32
 
 1748.31, 4.59
 1754.55, 4.79
 1762.32, 3.88
 1771.41, 1.65
 1781.61, 7.89
 1792.65, 2.47
 1804.28, 5.26
 1816.20, 6.20
 1828.12, 5.26
 1839.75, 2.47
 1850.79, 7.89
 1860.99, 1.65
 1870.08, 3.88
 1877.85, 4.79
 1884.09, 4.59
 1888.67, 3.55
 1891.46, 1.92
 1892.40, 00.00
 40000.00, 00.00
Using the UA-Tire Model
Learn about using the University of Arizona (UA) tire model:
• Background Information
• Tire Model Parameters
• Force Evaluation
• Operating Mode: USE_MODE
• Tire Carcass Shape
• Property File Format Example
• Contact Methods
Background Information for UA-Tire
The University of Arizona tire model was originally developed by Drs. P.E. Nikravesh and G. Gim. 
Reference documentation: G. Gim, Vehicle Dynamic Simulation with a Comprehensive Model for 
Pneumatic Tires, PhD Thesis, University of Arizona, 1988. The UA-Tire model also includes relaxation 
effects, both in the longitudinal and lateral direction.
The UA-Tire model calculates the forces at the ground contact point as a function of the tire kinematic 
states, see Inputs and Output of the UA-Tire Model. A description of the inputs longitudinal slip κ, side 
slip α and camber angle can be found in About Tire Kinematic and Force Outputs. The tire 
deflection and deflection velocity are determined using either a point follower or durability contact 
model. For more information, see Road Models in Adams/Tire. A description of outputs, longitudinal 
force Fx, lateral force Fy, normal force Fz, rolling resistance moment My and self aligning moment Mz 
is given in About Tire Kinematic and Force Outputs. The required tire model parameters are described in 
Tire Model Parameters.
Inputs and Output of the UA-Tire Model
γ
ρ ρ·
Adams/Tire12
 
Definition of Tire Slip Quantities
Slip Quantities at Combined Cornering and Braking/Traction
13
The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the 
wheel rotational velocity , and the effective rolling radius Re:
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip 
velocities in the contact point:
When the UA Tire is used for the force calculation the slip quantities during positive Vsx (driving) are 
defined as:
The rolling speed Vr is determined using the effective rolling radius Re:
Note that for realistic tire forces the slip angle is limited to 45 degrees and the longitudinal slip Ss 
(= ) in between -1 (locked wheel) and 1.
Lagged longitudinal and lateral slip quantities (transient tire behavior)
In general, the tire rotational speed and lateral slip will change continuously because of the changing 
interaction forces in between the tire and the road. Often the tire dynamic response will have an 
important role on the overall vehicle response. For modeling this so-called transient tire behavior, a first-
order system is used both for the longitudinal slip as the side slip angle, . Considering the tire belt as 
a stretched string, which is supported to the rim with lateral spring, the lateral deflection of the belt can 
be estimated (see also reference [1]). The figure below shows a top-view of the string model.
Stretched String Model for Transient Tire Behavior
ω
Vsx Vx ΩRe–=
Vsy Vy=
κ α
κ VsxVx
-------- and αtan– VsyVx
---------= =
κ VsxVr
-------- and αtan– VsyVr
---------= =
Vr ReΩ=
α
κ
κ
Adams/Tire14
 
When rolling, the first point having contact with the road adheres to the road (no sliding assumed). 
Therefore, a lateral deflection of the string will arise that depends on the slip angle size and the history 
of the lateral deflection of previous points having contact with the road.
For calculating the lateral deflection v1 of the string in the first point of contact with the road, the 
following differential equation is valid during braking slip:
with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger 
than 10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the 
equation can be transformed to:
When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection 
depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with 
this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill. 
When the UA Tire is used for the force calculations, at positive Vsx (traction) the Vx should be replaced 
by Vr in these differential equations.
1
Vx
------
td
dv1 v1
σα
------+ α( ) aφ+tan=
σα φ
σα td
dv1 Vx v1+ σαVsy=
A similar approach yields the following for the deflection of the string in longitudinal direction:
15
Now the practical slip quantities, ’ and ’, are defined based on the tire deformation:
 
 
These practical slip quantities and are used instead of the usual and definitions for steady-
state tire behavior. kVlow_x and kVlow_y are the damping rates at low speed applied below the 
LOW_SPEED_THRESHOLD speed. For the LOW_SPEED_DAMPING parameter in the tire property 
file yields:
kVlow_x= 100 · kVlow_y= LOW_SPEED_DAMPING
Tire Model Parameters
Definition of Tire Parameters
σx td
dv1 Vx v1+ σ– αVsx=
κ α
κ' u1σκ
------ kVlowxVsx–⎝ ⎠⎛ ⎞ Vx( )sin=
α' v1σα
------⎝ ⎠⎛ ⎞ kVlowyvsy–⎝ ⎠⎛ ⎞atan=
κ' α' κ α
Note: If the tire property file's REL_LEN_LON or REL_LEN_LAT = 0, then steady-state tire 
behavior is calculated as tire response on change of the slip and .κ α
Symbol:
Name in tire 
property file: Units*: Description:
r1 UNLOADED_R
ADIUS
L Tire unloaded radius
kz VERTICAL_STI
FFNESS
F/L Vertical stiffness
cz VERTICAL_DA
MPING
FT/L Vertical damping
Cr ROLLING_RES
ISTANCE
L Rolling resistance parameter
CsCSLIP F Longitudinal slip stiffness, 
C CALPHA F/A Cornering stiffness, 
C CGAMMA F/A Camber stiffness,
κ∂
∂Fx
κ 0=α α∂
∂Fy
α 0=γ γ∂
∂Fy
γ 0=
UMIN UMIN - Minimum friction coefficient (Sg=1)
Adams/Tire16
 
* L=length, F=force, A=angle, T=time
Force Evaluation in UA-Tire
• Normal Force
• Slip Ratios
• Friction Coefficient
Normal Force
The normal force F z is calculated assuming a linear spring (stiffness: k z ) and damper (damping constant 
c z ), so the next equation holds:
If the tire loses contact with the road, the tire deflection and deflection velocity become zero so the 
resulting normal force F z will also be zero. For very small positive tire deflections the value of the 
damping constant is reduced and care is taken to ensure that the normal force Fz will not become 
negative.
In stead of the linear vertical tire stiffness cz , also an arbitrary tire deflection - load curve can be defined 
in the tire property file in the section [DEFLECTION_LOAD_CURVE], see also the Property File 
Format Example. If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection 
datapoints with a cubic spline for inter- and extrapolation are used for the calculation of the vertical force 
of the tire. Note that you must specify VERTICAL_STIFFNESS in the tire property file but it does not 
play any role.
Slip Ratios
For the calculation of the slip forces and moments a number of slip ratios will be introduced:
Longitudinal Slip Ratio: Ss
The absolute value of longitudinal slip ratio, Ss, is defined as:
UMAX UMAX - Maximum friction coefficient (Ssg=0)
x REL_LEN_LON L Relaxation length in longitudinal direction
y REL_LEN_LAT L Relaxation length in lateral direction
Symbol:
Name in tire 
property file: Units*: Description:
σ
σ
Fz kzρ czρ·+=
ρ ρ·
Ss κ=
Where κ is limited to be within the range -1 to 1.
17
Lateral Slip Ratios: Sα , Sγ , Sαγ
The lateral slip ratio due to slip angle, , is defined as:
The lateral slip ratio due to inclination angle, S , is defined as:
A combined lateral slip ratio due to slip and inclination angles, S , is defined as:
where is the length of the contact patch.
Comprehensive Slip Ratio: Ssαγ
A comprehensive slip ratio due to longitudinal slip, slip angle, and inclination angle may be defined as:
Sα
Sα*
αtan during braking
1 Ss–( ) αtan during traction
Sα min 1.0 Sα*,( )
=
=
γ
Sγ γsin=
αγ
Sα*
α l γsin2rl
------------–tan during braking
1 Ss–( ) αl γsin2rl
------------tan during traction
=
l 8r1ρ=
Sαγ min 1.0 S*αγ,( )=
S*sαγ Ss2 S2αγ+
Sαγ min 1.0 S*αγ,( )=
=
Adams/Tire18
 
Friction Coefficient
The resultant friction coefficient between the tire tread base and the terrain surface is determined as a 
function of the resultant slip ratio (Ss ) and friction parameters (UMAX and UMIN ). The friction 
parameters are experimentally obtained data representing the kinematic property between the surfaces of 
tire tread and the terrain.
A linear relationship between Ss and , the corresponding road-tire friction coefficient, is assumed. 
The figure below depicts this relationship.
Linear Tire-Terrain Friction Model
 
This can be analytically described as:
μ = UMAX - (UMAX - UMIN) * Ssαγ
The friction circle concept allows for different values of longitudinal and lateral friction coefficients ( x 
and y) but limits the maximum value for both coefficients to . See the figure below.
Friction Circle Concept
αγ
αγ μ
μ
μ μ
19
 
The relationship that defines the friction circle follows:
or and 
where:
Slip Forces and Moments
μx
μ-----⎝ ⎠⎛ ⎞
2 μy
μ-----⎝ ⎠⎛ ⎞
2
+ 1=
μx μ βcos= μy μ βsin=
βcos Ss
Ssαγ
---------- and βsin SαγSsαγ
----------= =
To compute longitudinal force, lateral force, and self-aligning torque in the SAE coordinate system, you 
must perform a test to determine the precise operating conditions. The conditions of interest are:
Adams/Tire20
 
• Case 1: 
• Case 2: and 
• Case 3: and 
• Forces and moments at the contact point
The lateral force Fh can be decomposed into two components: Fha and Fhg. The two components are in 
the same direction if a· g < 0 and in opposite direction if 0.
Case 1. αγ < 0
Before computing the longitudinal force, the lateral force, and the self-aligning torque, some slip 
parameters and a modified lateral friction coefficient should be determined. If a slip ratio due to the 
critical inclination angle is denoted by S c, then it can be evaluated as:
If Ssc represents a slip ratio due to the critical (longitudinal) slip ratio, then it can be evaluated as:
If a slip ratio due to the critical slip angle is denoted by S c, then it can be determined as:
when .
The term critical stands for the maximum value which allows an elastic deformation of a tire during pure 
slip due to pure slip ratio, slip angle, or inclination angle. Whenever any slip ratio becomes greater than 
its corresponding critical value, an elastic deformation no longer exists, but instead complete sliding state 
represents the contact condition between the tire tread base and the terrain surface.
A nondimensional slip ratio Sn is determined as:
where:
αγ 0<
αγ 0≥ CαSα CγSγ≥
αγ 0≥ CαSα CγSγ<
αγ <
γ
Sγc μ
Fz
Cγ
------=
Ssc 3μ
Fz
Cs
-----=
α
Sαc
Cs
Cα
------- Ssc2 Ss2– 3Cγ
Sγ
Cα
-------–=
Ss Ssc≤
Sn
B2 B22 B1B3–+
B1
-------------------------------------------=
21
A nondimensional contact patch length is determined as:
A modified lateral friction coefficient is evaluated as:
where is the available friction as determined by the friction circle.
To determine the longitudinal force, the lateral force, and the self-aligning torque, consider two subcases 
separately. The first case is for the elastic deformation state, while the other is for the complete sliding 
state without any elastic deformation of a tire. These two subcases are distinguished by slip ratios caused 
by the critical values of the slip ratio, the slip angle, and the inclination angle. Specifically, if all of slip 
ratios are smaller than those of their corresponding critical values, then there exists an elastic 
deformation state, otherwise there exists only complete sliding state between the tire tread base and the 
terrain surface.
(i) Elastic Deformation State: , , and 
In the elastic deformation state, the longitudinal force F , the lateral force F , and three components 
of the self-aligning torque are written as functions of the elastic stiffness and the slip ratio as well as the 
normal force and the friction coefficients, such as:
B1 3μFz( )2 3CγSγ( )2
B2
–
2CαSαCγSγ
B3 CsSs( )2 CαSα( )2+[ ]–
=
=
=
ln 1 Sn–=
μym( )
μym( ) μy
CγSγ
Fz
-----------⎝ ⎠⎛ ⎞–=
μy μ βsin=
Sγ Sγc< Ss Ssc< Sα Sαc<
ξ τ
Adams/Tire22
 
where:
• is the offset between the wheel plane center and the tire tread base.
• is set to zero if it is negative.
• the length of the contact patch.
Mz is the portion of the self-aligning torque generated by the slip angle . Mzs and Mzs are other 
components of the self-aligning torque produced by the longitudinal force, which has an offset between 
the wheel center plane and the tire tread base, due to the slip angle and the inclination angle , 
respectively. The self-aligning torque Mz is determined as combinations of Mz , Mzs and Mzs .
(ii) Complete Sliding State: S S c, Ss Ssc, and S S c
In the complete sliding state, the longitudinal force, the lateral force, and three components of the self-
aligning torque are determined as functions of the normal force and the friction coefficients without any 
elastic stiffness and slip ratio as:
Fξ CsSsln2 μxFz 1 3ln2– 2ln3+( )
Fη
+
CαSsln2 μym( )Fz 1 3ln2– 2ln3+( ) CγSγ
Mzα+ +
CαSα
1
2
---– 23
---ln+⎝ ⎠⎛ ⎞
3
2
---μym( )FzSn2+ lln2
Mzsα
2
3
---CsSsSαln3
3μxμyFz2
5Cα
---------------------- 1 10ln3– 15ln4 6ln5–+( )
Mzsγ
+
η Fξ
=
=
=
=
=
η Sγ rl2 l'2 4⁄–=
rl2 l2 4⁄–
l 8rlρ=
α α α γ
α γ
α α γ
γ ≥ γ ≥ α ≥ α
23
Case 2: 0 and C S C S
As in Case 1, a slip ratio due to the critical value of the slip ratio can be obtained as:
A slip ratio due to the critical value of the slip angle can be found as:
when Ss Ssc.
The nondimensional slip ratio Sn, is determined as:
where:
Fξ μxFz
Fη μyFz
Mzα 0
Mzαs
3μxμyFz2l
5Cα
------------------------
Mzsγ η Fξ
=
=
=
=
=
α γ ≥ α α ≥ γ γ
Sγc 3μ
Fz
Cγ
------=
Sαc
Cs
Cα
------- Ssc2 Ss2– 3Cγ
Sγ
Cα
-------+=
≤
Sn
B2 B22 B1B3–+
B1
-------------------------------------------=
Adams/Tire24
 
The nondimensional contact patch length ln is found from the equation ln = 1 - Sn, and the modified 
lateral friction coefficient is expressed as:
For the longitudinal force, the lateral force and the self-aligning torque two subcases should also be 
considered separately. A slip ratio due to the critical value of the inclination angle is not needed here since 
the required condition for Case 2, C S C S , replaces the critical condition of the inclination 
angle.
(i) Elastic Deformation State: Ss Ssc and S Sac
In the elastic deformation state:
(ii) Complete Sliding State: Ss Ssc and S Sαχ
B1 3μFz( )2 3CγSγ( )2
B2
–
2CαSαCγSγ
B3 CsSs( )2 CαSα( )2+[ ]–
=
=
=
μym( )
μym( ) μy
CγSγ
Fz
-----------⎝ ⎠⎛ ⎞+=
α α ≥ γ γ
< α <
Fξ CsSsln2 μxFz 1 3ln2– 2ln3+( )
Fη
+
CαSsln2 μym( )Fz 1 3ln2– 2ln3+( ) CγSγ
Mzα
+ +
CαSα
1
2
---– 23
---ln+⎝ ⎠⎛ ⎞
3
2
---μym( )FzSn2+ lln2
Mzsα
2
3
---CsSsSαln3
3μxμyFz2
5Cα
---------------------- 1 10ln3– 15ln4 6ln5–+( )
Mzsγ
+
η Fξ
=
=
=
=
=
≥ α ≥
25
Case 3: 0 and C S C S
Similar to Cases 1 and 2, slip ratios due to the critical values of the inclination angle and the slip ratio 
are obtained as:
The nondimensional slip ratio Sn, is expressed as:
where:
Fξ μxFz
Fη μyFz
Mzα 0
Mzαs
3μxμyFz2l
5Cα
------------------------
Mzsγ η Fξ
=
=
=
=
=
α γ ≥ α α < γ γ
Sγc
3μFz CαSα+
3Cγ
--------------------------------
Ssc
1
Cs
----- 3μFz( )2 CαSα 3CγSγ–( )–
=
=
Sn
B2 B22 B1B3–+
B1
-------------------------------------------=
B1 3μFz( )2 3CγSγ( )2
B2
–
2CαSαCγSγ
=
=
B3 CsSs( )2 CαSα( )2+[ ]–=
Adams/Tire26
 
For the longitudinal force, the lateral force, and the self-aligning torque, two subcases should also be 
considered similar to Cases 1 and 2. A slip ratio due to the critical value of the slip angle is not needed 
here since the required condition for Case 3, C S C S , replaces the critical condition of 
the slip angle. 
(i) Elastic Deformation State: S S c and Ss Ssc
In the elastic deformation state, F and Mz can be written:
(ii) Complete Sliding State: S S c and Ss Ssc
In the complete sliding state, F , F , Mz , Mzs , and Mzs can be determined by using:
α α < γ γ
γ < γ <
η α
Fξ CsSsln2 μxFz 1 3ln2– 2ln3+( )
Fη
+
CαSsln2 μym( )Fz 1 3ln2– 2ln3+( ) CγSγ
Mzα
+ +
CαSα
1
2
---– 23
---ln+⎝ ⎠⎛ ⎞
3
2
---μym( )FzSn2+ lln2
Mzsα
2
3
---CsSsSαln3
3μxμyFz2
5Cα
---------------------- 1 10ln3– 15ln4 6ln5–+( )
Mzsγ
+
η Fξ
=
=
=
=
=
γ ≥ γ ≥
ξ η α α γ
Fξ μxFz
Fη μyFz
Mzα 0
Mzαs
3μxμyFz2l
5Cα
------------------------
Mzsγ η Fξ
=
=
=
=
=
27
respectively. The longitudinal force F , the lateral force F , and three components of the self-aligning 
torques, Mz , Mzs , and Mzs , always have positive values, but they can be transformed to have 
positive or negative values depending on the slip ratio s, the slip angle , and the inclination angle in 
the SAE coordinate system.
Tire Forces and Moments in the SAE Coordinate System
For the general formulations of the longitudinal force Fx, lateral force Fy, and self-aligning torque Mz, 
in the SAE coordinate system, the three possible combinations of the slip ratio, the slip angle, and the 
inclination angle are also considered.
Longitudinal Force:
Fx = sin(κ) F , for all cases
Lateral Force:
Fy = -sin( ) F , for cases 1 and 2
Fy = sin( ) F , for case 3
Self-aligning Torque:
Mz = sin( ) Mz - sin( ) [-sin( ) Mzs + sin( )Mzs ]
Rolling Resistance Moment:
My = -Cr Fz, for a forward rolling tire.
My = Cr Fz, for a backward rolling tire.
Operating Mode: USE_MODE
You can change the behavior of the tire model through the switch USE_MODE in the [MODEL] section 
of the tire property file.
• USE_MODE = 0: Steady-state forces and moments 
• The tire forces and moments react instantaneously to changes in the tire kinematic states.
• USE_MODE = 1: Transient tire behavior
• The tire will have a lagged response because of the so-called relaxation length in both 
longitudinal and lateral direction. See Lagged Longitudinal and Lateral Slip Quantities (transient 
tire behavior). 
• The effect of the relaxation lengths will be most pronounced at low forward velocity and/or high 
excitation frequencies. 
• USE_MODE = 2: Smoothing of forces and moments on startup of the simulation 
ξ η
α α γ
α γ
ξ
α η
γ η
α α κ α α γ γ
Adams/Tire28
 
• When you indicate smoothing by setting the value of use mode in the tire property file, 
Adams/Tire smooths initial transients in the tire force over the first 0.1 seconds of simulation. 
The longitudinal force, lateral force, and aligning torque are multiplied by a cubic step function 
of time. (See STEP in the Adams/Solver online help.)
Longitudinal Force FLon = S*FLon
Lateral Force FLat = S*FLat
Aligning Torque Mz = S*Mz
Tire Carcass Shape
You can optionally supply a tire carcass cross-sectional shape in the tire property file in the [SHAPE] 
block. The 3D-durability, tire-to-road contact algorithm uses this information when calculating the tire-
to-road volume of interference. If you omit the [SHAPE] block from a tire property file, the tire carcass 
cross-section defaults to the rectangle that the tire radius and width define.
You specify the tire carcass shape by entering points in fractions of the tire radius and width. Because 
Adams/Tire assumes that the tire cross-section is symmetrical about the wheel plane, you only specify 
points for half the width of the tire. The following apply:
• For width, a value of zero (0) lies in the wheel center plane.
• For width, a value of one (1) lies in the plane of the side wall.
• For radius, a value of one (1) lies on the tread.
For example, suppose your tire has a radius of 300 mm and a width of 185 mm and that the tread is joined 
to the side wall with a fillet of 12.5 mm radius. The tread then begins to curve to meet the side wall at 
>+/- 80 mm from the wheel center plane. If you define the shape table using six points with four points 
along the fillet, the resulting table might look like the shape block that is at the end of the property format 
example (see SHAPE).
Property File Format Example
$--------------------------------------------------------MDI_HEADER
[MDI_HEADER]
FILE_TYPE = 'tir' FILE_VERSION = 2.0
FILE_FORMAT = 'ASCII'
(COMMENTS) {comment_string}
'Tire - XXXXXX'
'Pressure - XXXXXX'
'TestDate - XXXXXX'
'Test tire'
'New File Format v2.1'
$-------------------------------------------------------------units
[UNITS]
LENGTH = 'meter'
FORCE = 'newton'
ANGLE = 'rad'
29
MASS = 'kg'
TIME = 'sec'
$-------------------------------------------------------------model
[MODEL]
! use mode 12 3 
! ------------------------------------------
! relaxation lengths X 
! smoothing X 
!
PROPERTY_FILE_FORMAT = 'UATIRE'
USE_MODE = 2
$-------------------------------------------------------dimension
[DIMENSION]
UNLOADED_RADIUS = 0.295
WIDTH = 0.195
ASPECT_RATIO = 0.55
$---------------------------------------------------------parameter
[PARAMETER]
VERTICAL_STIFFNESS = 190000
VERTICAL_DAMPING = 50
ROLLING_RESISTANCE = 0.003
CSLIP = 80000
CALPHA = 60000
CGAMMA = 3000
UMIN = 0.8
UMAX = 1.1
REL_LEN_LON = 0.6
REL_LEN_LAT = 0.5
$-------------------------------------------------------------shape
[SHAPE]
{radial width}
1.0 0.0 
1.0 0.2 
1.0 0.4 
1.0 0.6 
1.0 0.8 
0.9 1.0
$---------------------------------------------------------load_curve
$ For a non-linear tire vertical stiffness (optional)
$ Maximum of 100 points
[DEFLECTION_LOAD_CURVE]
{pen fz}
0.000 0.0
0.001 212.0
0.002 428.0
0.003 648.0
0.005 1100.0
0.010 2300.0
0.020 5000.0
0.030 8100.0
Adams/Tire30
 
Contact Methods
The UA-Tire Model supports the following roads
• 2D roads, see Using the 2D Road Model.
• 3D Splie roads, see Adams/3D Spline Road Model
The UA-Tire Model uses a one point of contact method; therefore, the wavelength of road 
obstacles must be longer than the tire radius for realistic output of the model.
• 3D Shell roads, see Adams/Tire 3D Shell Road Model
Using FTire Tire Model
Learn about:
• About FTire
• Modeling Approach
• Using FTire with Road Models
• Using FTire with Adams
• Parameters 
• About FTire Parameters 
• Procedure for Parameterizing FTire
• List of FTire Parameters
• About the Tire Data File
• Choosing Operating Conditions
This help describes the Flexible Ring Tire Model (FTire)™, as it is invoked from Adams.
© Michael Gipser, Cosin Consulting
About FTire
The tire model, FTire (Flexible ring tire model), is a sophisticated tire force element. You can use it in 
MBS-models for vehicle-ride comfort investigations and other vehicle dynamics simulations on even or 
uneven roadways.
Adams/Tire12
 
The main benefits of FTire are:
• Fully nonlinear.
• Valid in frequency domain up to 120 Hz, and beyond.
• Valid for obstacle wave lengths up to half the length of the contact patch, and less.
• Parameters, among others, are the natural frequencies and damping factors of the linearized 
model, and easy-to-obtain global static properties.
• Models both the in-plane and out-of-plane forces and moments.
• Computational effort no more than 5 to 20 times real time, depending on platform and model 
level.
• High accuracy when passing single obstacles, such as cleats and potholes.
• Applicable in extreme situations like many kinds of tire misuse and sudden pressure loss.
• Sufficiently accurate in predicting steady-state tire characteristics.
In contrast to other tire models, FTire does not need any complicated road data preprocessing. Rather, it 
takes and resolves road irregularities, and even extremely high and sharp-edged obstacles, just as they 
are defined.
We recommend that you visit www.ftire.com, to learn more about FTire theory, validation, data supply, 
and application. Also, at the FTire Web site, you will be kept informed about the latest FTire 
improvements, and how to receive them. In the download section, you will find a set of auxiliary 
programs, called FTire/tools for Windows™. These tools help to analyze and parameterize an FTire 
13
Model. FTire/tools is free for FTire licensees. It comprises static, steady-date, and modal analysis, 
linearization, data estimation, identification and validation tools, road data visualization, and more. In 
the site's documentation section, you will find a more detailed and permanently updated FTire 
documentation, together with as some additional literature.
Modeling Approach
FTire uses the following modeling approach:
• The tire belt is described as an extensible and flexible ring carrying bending stiffnesses, 
elastically founded on the rim by distributed, partially dynamic stiffnesses in radial, tangential, 
and lateral directions. The degrees of freedom of the ring are such that rim in-plane as well as 
out-of-plane movements are possible. The ring is numerically approximated by a finite number 
of discrete masses, the belt elements. These belt elements are coupled with their direct neighbors 
by stiff springs and by bending stiffnesses both in-plane and out-of-plane. 
Belt In-Plane and Out-Of-Plane Bending Stiffness outlines in-plane and out-of-plane bending 
stiffness placing. In-plane bending stiffness is realized by means of torsional springs about the 
lateral axis. The torsional deflection of these springs is determined by the angle between three 
consecutive belt elements, projected onto the rim mid-plane. Similarly, the out-of-plane bending 
stiffness is described by means of torsional springs about the radial axis. Here, the torsional 
deflection is determined by the angle between three consecutive belt elements, projected onto 
the belt tangential plane. Note that in the figure, the yellow plates do not represent the belt 
elements themselves, but rather the connecting lines between the elements.
Belt In-Plane (left) and Out-Of-Plane (right) Bending Stiffness
 
• FTire calculates all stiffnesses, bending stiffnesses, and damping factors during preprocessing, 
fitting the prescribed modal properties (see list of data below).
Adams/Tire14
 
• A number of massless tread blocks (5 to 50, for example) are associated with every belt element. 
These blocks carry nonlinear stiffness and damping properties in the radial, tangential, and 
lateral direction. The radial deflections of the blocks depend on the road profile, focus, and 
orientation of the associated belt elements. FTire determines tangential and lateral deflections 
using the sliding velocity on the ground and the local values of the sliding coefficient. The latter 
depends on ground pressure and sliding velocity.
• FTire calculates all six components of tire forces and moments acting on the rim by integrating 
the forces in the elastic foundation of the belt.
Because of this modeling approach, the resulting overall tire model is accurate up to relatively high 
frequencies both in longitudinal and in lateral directions. There are few restrictions in its applicability 
with respect to longitudinal, lateral, and vertical vehicle dynamics situations. FTire deals with large- 
and/or short-wave-length obstacles. It works out of, and up to, a complete standstill, with no additional 
computing effort nor any model switching. Finally, it is applicable with high accuracy in such delicate 
simulations as ABS braking on extremely uneven roadways, and so on.
In a full 3D variant, FTire additionally takes into account belt element rotation and bending about the 
circumferential axis. These new degrees of freedom enable FTire to use contact elements that are 
distributed not only along a single line, but over the whole contact patch. You can choose the arrangement 
of the contact elements to be either randomly distributed, or distributed along several parallel lines.
In the full 3D variant, belt torsion about the circumferential axis is described by:
• Torsional stiffnesses between belt elements and rim, about circumferential axis (represented by 
red torsion springs in the left side of the figure, Belt).
• Torsional stiffnesses between adjacent belt elements, about circumferential axis (represented by 
blue torsion springs in the left side of figure, Belt).
The right sideof the figure, Belt outlines the belt bending stiffness about the circumferential axis. This 
is done in a somewhat simplified manner. Actually, lateral belt bending is taken into account by 
introducing a parabolic shape function for each belt element. The curvature of this shape function is 
treated as a belt elements’ additional degree of freedom.
Belt Torsional and Twisting Stiffness, and Belt Lateral Bending Stiffness
Note: Radial, tangential, and lateral are relative to the orientation of the belt element, 
whereas sliding velocity is the block end-point velocity projected onto the road 
profile tangent plane. By polynomial interpolation, certain precautions have been 
taken not to let the ground pressure distribution mirror the polygonal shape of the 
belt chain.
15
You should chose the full 3D variant, which takes about 30% more computing time, in situations where 
a considerable excitation of tire vibrations in lateral direction is expected. This, for example, will happen 
when the tire runs over cleats that are placed in an oblique direction relative to the tire rolling direction. 
Similarly, such an excitation will happen when the tire is running over obstacles with large camber angle.
Optionally, FTire can take into account tire non-uniformity, that is, a harmonic variation of vertical or 
longitudinal stiffness, as well as static and dynamic imbalance, conicity, ply-steer, and geometrical run-
out.
All stiffness values may depend on the actual inflation pressure. To take full advantage of that option, it 
is necessary to provide basic FTire input data, such as radial stiffness data and natural frequencies at two 
different pressure values. Actual inflation pressure is one of the ‘operating conditions variables,’ which 
can be made time-dependent, and therefore, can be changed even during a simulation.
There are two more operating conditions: tread depth and model level. The latter signal allows you to 
switch between the reduced variant of FTire (all contact elements are arranged in one single line near the 
rim mid-plane), and the full 3D variant (contact elements cover the whole contact patch).
The kernel of the FTire implementation is an implicit integration algorithm (BDF) that calculates the belt 
shape. The integrator runs parallel but synchronized with the Adams main integrator. By using this 
specialized implicit BDF integrator, you can choose the belt extensibility so it is extremely small. This 
also allows the simulation of an inextensible belt without any numerical drawbacks.
Using FTire with Road Models
FTire supports all MSC road definitions, including Motorsports and all 3D roads. It also supports several 
customer-specific and third-party roads. For more information about available road descriptions, please 
contact info@ftire.com.
Adams/Tire16
 
Using FTire with Adams
FTire is a high-resolution tire model, with respect to road irregularities and tire vibration modes. To take 
full advantage of that precision, we recommend that you choose a small step size for the Adams 
integrator. There should be a minimum of 1,000 steps per one second simulation time (that is, an output 
time step of 1 ms or less).
Controlling integrator step size in:
• Adams/Car
• Adams/Chassis
• Adams/View
• Adams/Solver
Controlling Integrator Step Size in Adams/Car
In Adams/Car, you can control the integrator step size by selecting:
Settings → Solver → Dynamics
and entering 1ms in the Hmax text box.
Alternatively, you can edit the driver control file (.dcf) that Adams/Car automatically generates when 
performing a new dynamic maneuver. In that file, override the integrator step size, which is defined in 
[EXPERIMENT] block, by entering the value 0.001 or less. After editing the file, you can launch 
subsequent simulation experiments with the same driver's control (and, of course, the new integrator step 
size) by selecting the following from Adams/Car:
Simulate → Full-Vehicle Analysis → DCF Driven → Driver Control Files → Browse
and selecting the .dcf you just edited.
Controlling Integrator Step Size in Adams/Chassis
In Adams/Chassis, you can control the integrator step size by setting the HMAX value to 0.001 or less. 
HMAX is defined by selecting the following from Adams/Chassis:
System file → Properties → system_parameters → solver → hmax
Controlling Integrator Step Size in Adams/View
In Adams/View, you can control the integrator step size by checking:
Settings → Solver → Dynamics → Customized Settings 
size, Min Step Size, and Max Step Size.
Controlling Integrator Step Size in Adams/Solver
In Adams/Solver, you can control the integrator step size by setting INTEGRATOR/HMAX to the 
desired value in the Adams dataset (adm).
17
FTire Parameters
• About FTire Parameters
• Procedure for Parameterizing FTire
• Listing of FTire Parameters
About FTire Parameters
FTire parameters can be divided into several groups. There are parameters that define:
• Tire size and geometry
• Stiffness, damping, and mass distribution of the belt/sidewall structure
• Tire imperfections (non-uniformity, imbalance, conicity, and so on)
• Stiffness and damping properties of the tread rubber
• Friction characteristics of the tread rubber
• Numerical properties of the model
For convenience, FTire tries to use data that can be measured as easy as possible. As a consequence, the 
number of basic data might be larger than the number of internal parameters defined by these basic 
parameters.
For example, the following four parameters together, after preprocessing, actually result in only two 
values used in FTire: compression and shear stiffness of the idealized blocks that represent tread rubber:
• tread_depth 
• tread_base_height 
• stiffness_tread_rubber 
• tread_positive 
Also, sometimes different combinations of parameters are possible. This is true especially for data of the 
second group, which determine the structural stiffness and damping properties of FTire. Your choice of 
which combination of parameters to supply depends on the types of measurements that are available and 
their accuracy.
Moreover, it is possible to prescribe over-determined subsets of parameters. For example, you may 
define the belt in-plane bending stiffness by prescribing the frequency of the first bending mode, and at 
the same time the radial stiffness on a transversal cleat. Both parameters are strongly influenced by the 
bending stiffness, but might contradict each other.
In such a case, FTire automatically recognizes that the system of equations to be solved is over-
determined, and applies an appropriate solver (Householder QR factorization) to determine the solution 
in the sense of least squares fit. That means, FTire is looking for a compromise to meet both conditions 
as much as possible. Users can control the compromise by optionally defining weights for the 
contradicting conditions.
Adams/Tire18
 
Note that, among others, FTire uses modal data to calculate internal structural stiffness and damping 
coefficients. They are processed in such a way that the mathematical model, for small excitations, shows 
exactly the measured behavior in the frequency domain. FTire is not a modal model, nor is it linear.
First Six Vibration Modes Of An Unloaded Tire With Fixed Rim
 
When parametrizing FTire, the bending mode frequencies rather sensitively influence the respective 
bending stiffness. As an alternative, determining the radial stiffness both on a flat surface and on a short 
obstacle (cleat) is an inexpensive and very accurate way to get both the vertical stiffness between belt 
nodes and rim and the in-plane bending stiffness.
Other ways to determine the bending stiffness (and other data, as well) are to use the software tools 
FTire/fit (time- and frequency-domain parameter identification) and FTire/estim (qualifiedparameter 
estimation by comparison with a reference tire). For more information, see www.ftire.com.
Unfortunately, there is no direct analogy of the ‘radial stiffness on cleat’ measuring procedure to get the 
out-of-plane bending stiffness. But this parameter does not seem to be as relevant as the in-plane bending 
stiffness for ride comfort and durability. An indirect, but also very accurate, way to validate the out-of-
plane bending stiffness is to check resulting side-force and self-aligning characteristic. The cornering 
stiffness, the pneumatic trail, as well as the difference between maximum side force and side force for 
very large side-slip angles, are very sensitively determined both by the tread rubber friction characteristic 
and by the out-of-plane bending stiffness. Similarly, the fourth mode (see figure, First Six Vibration 
Modes Of An Unloaded Tire With Fixed Rim), being itself determined by the stiffness between belt nodes 
and rim in lateral direction, very strongly influences the side-slip angle where maximum side force 
occurs.
Procedure for Parameterizing FTire
A typical procedure to parametrize FTire might be:
1. Either from tire data sheets, by some simple and inexpensive measurements, or directly from the 
tire supplier, obtain: 
• Tire size, load index, and speed symbol 
• Rolling circumference 
• Rim diameter 
• Tread width 
• Tire mass 
19
• Tread depth 
• Rubber height over steel belt 
• Shore-A stiffness or Young's modulus of tread rubber 
• Tread pattern positive
2. Determine the natural frequencies and damping moduli of the first six modes, for an unloaded, 
inflated tire, where the rim is fixed. Normally, you do this by exciting the tire structure with an 
impulse hammer, measuring the time histories of at least four acceleration sensors in all three 
directions, distributed along the tire circumference, and processing these using an FFT signal 
analyzer. Optionally, repeat this step for a second inflation pressure value.
3. Determine the tire radial stiffness on a flat surface and on a short obstacle, for one or two 
inflation pressure value(s).
4. Determine (or estimate) the lateral belt curvature radius from the unloaded tire's cross-section. 
Determine the belt lateral bending stiffness to get a reasonable pressure distribution in the 
lateral direction.
5. Determine (or estimate) tread rubber adhesion and sliding friction coefficients for ground 
pressure values 0.5 bar, 2 bar, and 10 bar.
6. Take natural frequencies and damping moduli of modes 1, 2, and 4, together with the radial 
stiffness on flat surface and on a cleat, for one or two inflation pressure value(s), as well as the 
remaining basic data. These values result in a first, complete FTire input file for the basic variant 
(belt circumferential rotation, twisting, and bending not taken in to account; all contact elements 
are arranged in one line).
7. Let FTire preprocess these data. Compare the resulting additional modal properties of the model 
with the modal data that are not used so far (modes 3, 5, and 6). If necessary, adjust the 
preprocessed data to find a compromise with respect to accuracy.
8. If respective measurements are available, validate the data determined so far by means of side 
force and aligning torque characteristics, and by measurements of vertical and longitudinal 
force variations induced during rolling over cleats both with low and high speed. The validation 
can be extended to a full parameter fitting procedure by using TIRE/fit, as mentioned earlier.
9. Estimate the following additional data that are only relevant for 'out-of-plane' excitation: 
• Belt element torsional stiffness relative to rim (represented by red torsion springs between 
yellow belt elements and gray rim in the figure, Belt)
• Belt twisting stiffness (represented by blue torsion springs between adjacent yellow belt 
elements in the figure, Belt)
• Belt bending stiffness/damping about circumferential direction
• Belt lateral curvature radius
• Coupling coefficient between belt lateral displacement and belt rotation.
Start with the respective values of the sample data file. Then, adjust the values by fitting the model's 
response to obliquely oriented cleats and handling characteristics for large camber angles at the same 
time. This identification procedure can be made easier by using the the additional tool FTire/fit.
Adams/Tire20
 
Clearly, the performance of this procedure is not very easy in practice. On the other hand, every tire 
model that is accurate enough for ride comfort and durability calculations will need as much or even more 
data.
List of FTire Parameters
The following is a comprehensive list of all mandatory and optional FTire parameters. However, many 
items are explained in greater detail in the extended documentation to be downloaded from the restricted 
area in www.ftire.com. You will receive your pass-code from info@ftire.com.
FTIRE_DATA Section Parameters
The parameter: Means:
tire_section_width Tire section width as specified in the tire size designation (using 
length unit as specified in the [UNITS] section).
tire_aspect_ratio Tire aspect ratio as specified in the tire size designation. Unit is %.
rim_diameter Rim diameter as specified in the tire size designation (using length 
unit as specified in the [UNITS] section).
rim_width Inner distance between the two rim flanges.
load_index Load index of tire, as displayed in tire service description.
tread_width Width of tread that comes into contact with the road under normal 
running conditions at LI load, without camber angle.
rolling_circumference Rolling circumference of tire under the following running 
conditions:
• Free rolling at v = 60 km/h and zero camber angle
• Vertically loaded by half of the maximum load
The circumference is the distance traveled with one complete 
wheel revolution.
tire_mass Overall tire mass.
inflation_pressure Inflation pressure, at which tire data measurements have been 
taken.
inflation_pressure_2 Second inflation pressure, at which tire data measurements have 
been taken (optional).
stat_wheel_load_at_10mm_defl Static wheel load of the inflated tire, when it is deflected by 10 
mm, with zero camber angle, on a flat surface, during stand-still, 
at very low friction value.
21
stat_wheel_load_at_20mm_defl Static wheel load of the inflated tire, when it is deflected by 20 
mm, with zero camber angle, on a flat surface, during stand-still, 
at very low friction value.
Note: Instead of using:
stat_wheel_load_at_10mm_defl and 
stat_wheel_load_at_20mm_defl
Note: You can equally define:
stat_wheel_load_at_20mm_defl and 
stat_wheel_load_at_40mm_defl.
Note: This will better fit typical operating conditions of truck 
tires. For extremely heavy vehicles, there are even more 
pairs of deflection values predefined. These can be 
found at the extended documentation at www.ftire.com.
dynamical_stiffening Increase of the overall radial stiffness at high speed as compared 
to radial stiffness during standstill. Unit is %.
speed_at_half_dyn_stiffening Running speed at which dynamic stiffening reaches half of the 
final value.
belt_extension_at_200_kmh Percentage of rolling circumference growth at a running speed of 
200 km/h = 55.55 m/s = 124.3 mph, compared to low speed.
interior_volume Interior tire volume when the tire is mounted on the rim and 
inflated with inflation_pressure.
Note: This parameter is only needed if you specify the next 
parameter (volume_gradient) and it is nonzero.
volume_gradient Relative decrease in volume, of a small tire segment, when that 
segment is deflected vertically.
Note: This parameter is optional and only marginally affects 
the model’s behavior.
rel_long_belt_memb_tension The percentage by which inflation pressure forces in the belt 
region arecompensated with membrane tension in longitudinal 
direction, as compared to the total compensation in lateral and 
longitudinal direction.
Note: This parameter is optional, and can only be calculated 
using a finite-element (FE) model, or estimated by 
parameter identification. A value of 70 to 80% seems to 
be appropriate for many tires. The value will increase 
with increasing belt lateral curvature radius.
The parameter: Means:
Adams/Tire22
 
f1 First natural frequency: in-plane, rigid-body rotation around 
wheel spin axis. Rim is fixed. See the figure, First Six Vibration 
Modes.
f2 Second natural frequency: rigid-body movement in fore-aft 
direction. Rim is fixed. See the figure, First Six Vibration Modes.
f4 Fourth natural frequency: out-of-plane, rigid-body rotation 
around road normal axis. Rim is fixed. See the figure, First Six 
Vibration Modes.
Note: f3 (out-of-plane, rigid-body movement) is not needed 
because it is closely related to f4.
At least one of:
f5 Fifth natural frequency: first in-plane bending mode 
(quadrilateral-shaped). Rim is fixed.
belt_in_plane_bend_stiffn In-plane bending stiffness of the belt ring of deflated and 
unloaded tire.
wheel_load_at_10_mm_defl_clea
t
Static wheel load of the inflated tire, when it is deflected by 10 
mm, with zero camber angle, on a cleat as specified below, during 
stand-still. Cleat must be high enough that the tire does not touch 
the ground apart from the cleat. The cleat is oriented in the lateral 
direction, perpendicular to the tire’s rolling direction.
Note: For truck tires, you can specify 
wheel_load_at_20_mm_defl_ cleat, as well.
weight_f5
weight_in_plane_bend_stiffn
weight_wheel_load_cleat
If you provide at least two of the data on the previous page to 
define the in-plane bending stiffness, they constitute an over-
determined system of equations for the respective FTire's internal 
stiffness values. FTire will try to find a compromise. You can 
control this compromise by setting these weight values. Their 
relative size controls, in a least-squares approach, the contribution 
of the respective parameter. If a weight is set to zero, the related 
parameter is completely ignored.
Note: The weights are optional. Default value is 1.
cleat_width Width of cleat that was used to determine all parameters that 
require a cleat:
wheel_load_at_10_mm_defl_ cleat
wheel_load_at_10_mm_defl_ cl_lo
and so on.
Note: Parameter is optional. Default value is 20 mm.
The parameter: Means:
23
cleat_bevel_edge_width Bevel edge width (measured after projection to x-y-plane) of cleat 
that was used to determine all parameters that require a cleat:
wheel_load_at_10_mm_defl_ cleat
wheel_load_at_10_mm_defl_ cl_lo
and so on.
Note: Parameter is optional. Default value is 0 mm.
At least one of:
f6 Sixth natural frequency: first out-of-plane bending mode (banana-
shaped).
belt_out_of_plane_bend_stiffn Out-of-plane bending stiffness of the belt ring of inflated but 
unloaded tire.
weight_f6
weight_out_of_plane_bend_st
If you provide both data above (f6 and 
belt_out_of_plane_bend_stiff) to define the out-of-plane bending 
stiffness, they constitute an over-determined system of equations 
for the respective FTire's internal stiffness values. FTire will try to 
find a compromise. You can control the compromise by setting 
these weight values. Their relative size controls, in a least-squares 
approach, the contribution of the respective parameter. If a weight 
is set to zero, the related parameter is completely ignored.
Note: The weights are optional. Default value is 1.
D1 Damping of f1, between 0 and 1:
0 = undamped, ..., 1 = aperiodic limit case
D2 Damping of f2.
D4 Damping of f4.
Note: D5 and D6 cannot be prescribed, but result from D1, 
D2, and D4.
belt_twist_stiffn Belt-twisting stiffness: if the mean torsion angle relative to the rim 
is 0, the value is the moment in longitudinal direction per 1 degree 
twist angle for a unit length belt segment. This value is 
independent on the number of belt segments.
Note: Only needed for full 3D variant. Unit is 
force*length2/angle.
The parameter: Means:
Adams/Tire24
 
belt_torsion_stiffn Belt-torsional stiffness: if twist angle is 0, the value is the moment 
in longitudinal direction per 1 degree torsion angle relative to rim, 
for a unit-length belt segment. This value is independent on the 
number of belt segments.
Note: Only needed for full 3D variant.Unit is force/angle.
belt_torsion_lat_displ_coupl If belt twist angle is 0, value is the kinematic belt torsion angle at 
1 mm lateral belt displacement.
Note: Optional, and only needed for full 3D variant. Unit is 
angle/length. Default value is 0.
belt_lat_curvature_radius Curvature radius of belt cross section perpendicular to mid-plane.
Note: Optional, and only needed for full 3D variant. Default 
value is (nearly) infinity.
belt_lat_bend_stiffn Bending stiffness of belt elements about circumferential direction.
Note: Optional, and only needed for full 3D variant. Unit is 
force*length2. Default value is (nearly) infinity.
wheel_load_at_10_mm_defl_lo_c
l
Wheel load at 10 mm deflection on longitudinal cleat. Static 
wheel load of the inflated tire, when it is deflected by 10 mm, with 
zero camber angle, on a cleat as specified above, during stand-
still. Cleat must be high enough that the tire does not touch the 
ground apart from the cleat. The cleat is oriented in longitudinal 
direction, along foot-print centerline.
Note: This parameter is optional and you can specify it instead 
of, or in addition to, belt_lat_bend_stiffn. For truck 
tires, if you specify wheel_load_at_40_mm, FTire looks 
for wheel_load_at_20_mm_defl_ lo_cl instead.
weight_lat_bend_st
weight_wheel_load_lo_cl
If you provide both data on the previous page 
(belt_lat_bend_stiffn and wheel_load_at_10_mm_defl_lo_cl) to 
define the lateral belt bending stiffness, they constitute an over-
determined system of equations for the respective FTire's internal 
stiffness values. FTire will try to find a compromise. You can 
control the compromise by setting these weight values. Their 
relative size controls, in a least-squares approach, the contribution 
of the related parameter. If a weight is set to zero, the respective 
parameter is completely ignored.
Note: The weights are optional. Default value is 1.
belt_lat_bend_damp Quotient of bending damping and bending stiffness of belt 
elements about circumferential direction.
Note: Optional, and only needed for full 3D variant. Unit is 
The parameter: Means:
time. Default value is 1 ms.
25
f1_p2
f2_p2
f4_p2
f5_p2
f6_p2
D1_p2
D2_p2
D4_p2
belt_in_plane_bend_st_p2
wheel_load_at_10_mm_defl_cl_p
2
wheel_load_at_20_mm_defl_cl_p
2
belt_out_of_plane_bend_st_p2
belt_lat_bend_stiffn_p2
belt_twist_st_p2
belt_torsion_st_p2
If measurements for a second inflation pressure 
(inflation_pressure_2) are available, these are the respective 
values of the following taken at that pressure:
• f1 
• f2 
• f4 
• f5 
• f6 
• D1 
• D2 
• D4 
• belt_in_plane_bend_stiffn 
• wheel_load_at_10_mm_defl_cleat 
• wheel_load_at_20_mm_defl_cleat 
• belt_out_of_plane_bend_stiffn 
• belt_lat_bend_stiffn 
• belt_twist_stiffn 
• belt_torsion_stiffn 
Note: These data are optional.
tread_depth Mean groove depth in tread.
tread_base_height Rubber height over steel belt for zero tread depth, which is the 
distance between steel belt and grooves.
stiffness_tread_rubber Stiffness of tread rubber in Shore-A units.
tread_positive Percentage of gross tread contact area with respect to overall 
footprint area (tread pattern positive).
damping_tread_rubber Quotient of tread rubber damping modulus and tread rubber 
elasticity modulus.
Note: Deflection/force phase-lag of elastomers is oftenassumed to be independent of excitation frequency. This 
behavior is not yet implemented in FTire; instead, 
viscous damping is used. The parameter 
damping_tread_rubber is nothing but the quotient of 
damper coefficient and spring stiffness of the coupling 
of blocks and belt. For that reason, the parameter carries 
the unit time.
The parameter: Means:
Adams/Tire26
 
sliding_velocity The sliding velocity of a tread rubber block, when its friction 
coefficient reaches the my_sliding values.
blocking_velocity The sliding velocity of a tread rubber block, when its friction 
coefficient reaches the my_blocking values.
low_ground_pressure The first of three ground-pressure values that defines the pressure 
dependency of the friction coefficient. Default value is 0.1 bar.
med_ground_pressure The second of three ground-pressure values that defines the 
pressure dependency of the friction coefficient. Default value is 2 
bar.
high_ground_pressure The third of three ground-pressure values that defines the pressure 
dependency of the friction coefficient. Default value is 10 bar.
my_adhesion_at_low_p Coefficient of adhesion friction (which is equal to static friction) 
between tread rubber and road, at first ground pressure value.
Note: For this parameter and the parameters in the following 
eight rows, you can still use the parameter names 
my_..._at_..._bar, used in the previous FTire version. To 
avoid confusion with the actual ground pressure values, 
however, we recommend you use the more general 
names.
my_sliding_at_low_p Coefficient of sliding friction, at a sliding velocity defined by 
parameter sliding_velocity, between tread rubber and road, at first 
ground pressure value.
my_blocking_at_low_p Coefficient of sliding friction, at a sliding velocity defined by 
parameter blocking_velocity, between tread rubber and road, at 
first ground pressure value.
my_adhesion_at_med_p Coefficient of adhesion friction (which is equal to static friction) 
between tread rubber and road, at second ground pressure value.
my_sliding_at_med_p Coefficient of sliding friction, at a sliding velocity defined by 
parameter sliding_velocity, between tread rubber and road, at 
second ground pressure value.
static_balance_weight Weight that would have put up on the rim horn for static 
balancing.
Note: Parameter is optional.
static_balance_ang_position The angular position at the rim where the static balance weight 
would have been placed.
Note: Parameter is optional.
The parameter: Means:
27
dynamic_balance_weight One of the two equal weights that would have been placed on the 
rim outer and inner horns for dynamic balancing.
Note: Parameter is optional.
dynamic_balance_ang_position The angular position at the rim where the left dynamic balance 
weight would have been placed.
Note: This parameter is optional.
radial_non_uniformity Amplitude of the harmonic radial stiffness variation as percentage 
of the mean radial stiffness.
Note: Parameter is optional.
radial_non_unif_ang_position Angular position where radial stiffness reaches its maximum.
Note: This parameter is optional.
tang_non_uniformity Amplitude of the harmonic tangential stiffness variation as 
percentage of the mean tangential stiffness.
Note: Parameter is optional.
tang_non_unif_ang_position Angular position where tangential stiffness reaches its maximum.
Note: Parameter is optional.
conicity Small rotation angle of belt elements at zero moment, about 
circumferential axis, resulting in a conical shape of the unloaded 
belt.
Note: Parameter is optional and can only be used with the full 
3D variant. Nonzero conicity will cause a small side-
force without side-slip angle. The sign of that force is 
independent of the tire’s rolling direction.
ply_steer_percentage Percentage of lateral belt displacement relative to radial belt 
displacement, when a radial force is applied.
Note: Ply-steer, besides conicity, is one of the reasons for 
nonzero side forces at zero side-slip angle. In contrast to 
the conicity side-force, this residual side force changes 
sign when the tire rolling direction is reversed.
run_out The maximum deviation of the local tire radius from the mean tire 
radius. Run-out is assumed to be a harmonic function of the 
angular position.
run_out_ang_position The angular belt element position relative to the rim, where 
maximum run-out occurs.
number_belt_segments Number of numerical belt segments. Maximum value is 200, but 
can be changed upon request.
The parameter: Means:
Adams/Tire28
 
About the FTire Tire Data File
As with all TeimOrbit files, entries in the [UNITS] block define the physical units of all parameters.
number_blocks_per_belt_segm Number of numerical blocks (= contact elements) per belt 
segment. Maximum value is 50, but can be changed upon request.
number_tread_strips Number of strips, into which the contact points are arranged in the 
full 3D variant, using an equal spacing.
Note: If value is greater than or equal to 1000, the contact 
points are scattered randomly over the tread. 
Alternatively, it is possible to place tread elements 
according to the actual tread pattern of the tire. This is 
done by specifying a bitmap file of the footprint. For 
more information, see the extended documentation at 
www.ftire.com.
If you specify neither number_tread_strips nor the 
bitmap file, FTire uses the basic FTire variant instead of 
the full 3D variant, regardless of the model-level 
specification in the operating_conditions section.
maximum_time_step Maximum integration time step allowed.
Note: You can call FTire with very large time steps (if this 
makes sense for your model). Internally, FTire uses 
multi-step integration with an internal time step that is 
chosen on basis of maximum_time_step. This internal 
time step is kept constant if the external time step does 
not change.
Changing the external time step can result in 
considerable longer computation time, because certain 
time-consuming preprocessing calculations have to be 
repeated. For that reason, you should avoid changing 
the external time step whenever possible.
BDF_parameter Numerical parameter to control the internal FTire implicit (BDF) 
integration scheme, which is independent of the Adams 
integrator.
0 = Euler explicit 
0.5 = Trapezoidal rule 
1 = Euler implicit 
Theoretically, every value between 0 and 1 are allowed. 0.505 or 
greater is recommended.
The parameter: Means:
29
The basic parameters are preprocessed during initialization, resulting in the preprocessed parameters. 
These parameters are saved in a separate TeimOrbit-style file, which can be used in further simulations 
instead of the basic data file. By this, you can omit the preprocessing calculation phase, which may result 
in a considerable saving of time.
This preprocessed data file is a copy of the original one; the preprocessed data are appended after the 
bottom line, using a hexadecimal, space-saving coding. In contrast to earlier versions of FTire, it is 
possible to use this file for parameter changes instead of the original one.
You should, of course, not change the hexadecimal data but only the readable part of the file. The 
hexadecimal section does not only contain the preprocessed data, but a copy of the original one, as well. 
Moreover, it carries coded information about the FTire version that was used for creation.
This information helps to automatically determine whether or not an update of the preprocessed data is 
required. This means that whenever you change some basic data or you download a new FTire version, 
preprocessing will be repeated automatically, and the preprocessed data file saved in your current 
working directory. You can (and should) replace the FTire data file in your database with this one,without any loss of information.
From www.ftire.com, you can download a tool (being a member of FTire/tools) to carry out preprocessing 
outside of Adams.
The FTire interface routine automatically recognizes whether several wheels of the car share the same 
basic data file. In that case, preprocessing is done only once for all these files. Also, FTire automatically 
recognizes whether the data file contains basic parameters or pre-processed ones.
FTire does not use the data in the section [VERTICAL]. It is only included for compatibility with other 
tire models. It is recommended that you set Vertical_Stiffness to the value of 
stat_wheel_load_at_10_mm_defl, after dividing by 10 mm. For Vertical_Damping, choose 0 (or a small 
nonzero value). The actual vertical damping of FTire is not just one single value, but will depend on 
rolling speed, inflation pressure, load, camber, and so on.
The following is an examples of a basic FTire data file. Note that by far not all possible data are defined. 
For examples, only data for one inflation pressure are provided.
$--------------------------------------------------------MDI_HEADER
[MDI_HEADER]
 FILE_TYPE = 'tir'
 FILE_VERSION = 4.0
 FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
 'Tire Manufacturer - unknown'
 'Tire Type - unknown'
 'Tire Dimension - 195/65 R 15'
 'Pressure - 2.0 bar'
 'File Generation Date - 03/03/11 10:32'
$-------------------------------------------------------------SHAPE
[SHAPE]
{radial width}
 1.0 0.0
 1.0 0.4
Adams/Tire30
 
 1.0 0.9
 0.9 1.0
$-------------------------------------------------------------UNITS
[UNITS]
 FORCE = 'NEWTON'
 MASS = 'GRAM'
 LENGTH = 'MM'
 TIME = 'MILLISECOND'
 ANGLE = 'DEGREE'
$---------------------------------------------------------DIMENSION
[DIMENSION]
 UNLOADED_RADIUS = 326.0 $ [mm]
$----------------------------------------------------------VERTICAL
[VERTICAL]
 VERTICAL_STIFFNESS = 170.0 $ [N/mm]
 VERTICAL_DAMPING = 0.0 $ [Nms/mm]
$-------------------------------------------------------------MODEL
[MODEL]
 PROPERTY_FILE_FORMAT = 'FTIRE' $
 separate_animation = 0 $ [0/1]
 additional_output_file = 0 $ [0/1]
 verbose = 0 $ [0/1]
$----------------------------------------------OPERATING_CONDITIONS
[OPERATING_CONDITIONS]
 inflation_pressure = 2.0 $ [bar] 
 tread_depth = 8.0 $ [m] 
 model_level = 7 $ [-] 
$---------------------------------------------------------PARAMETER
[FTIRE_DATA]
$basic data and geometry *******************************************
 tire_section_width = 195 $ [mm]
 tire_aspect_ratio = 65 $ [%]
 rim_diameter = 381 $ [mm] 
 rim_width = 152.4 $ [mm]
 load_index = 91 $ [-]
 rolling_circumference = 1975 $ [mm]
 tread_lat_curvature_radius = 800 $ [mm]
 tread_width = 160 $ [mm]
 tire_mass = 9000 $ [g]
 interior_volume = 0.03e9 $ [mm^3]
 volume_gradient = 1.0 $ [%/mm]
 belt_torsion_lat_displ_coupl = 0.0 $ [deg/mm]
$
$static and modal data for 1st infl. pressure ***********************
 stat_wheel_load_at_10_mm_defl = 1690 $ [N]
 stat_wheel_load_at_20_mm_defl = 3600 $ [N]
 dynamic_stiffening = 20 $ [%]
 speed_at_half_dyn_stiffening = 5.55 $ [mm/ms]=[m/s]
 radial_hysteretic_stiffening = 0 $ [%]
 radial_hysteresis_force = 0 $ [N]
 tang_hysteretic_stiffening = 0 $ [%]
 tang_hysteresis_force = 0 $ [N]
 belt_extension_at_200_kmh = 1.0 $ [%]
 rel_long_belt_memb_tension = 82.0 $ [%]
31
$
 f1 = 62.1 $ in-plane rotat. [Hz]
 f2 = 81.4 $ in-plane transl. [Hz]
 f4 = 80.0 $ out-of-plane rotat. [Hz]
$
 D1 = 0.05 $ in-plane rotat. [-]
 D2 = 0.08 $ in-plane transl. [-]
 D4 = 0.05 $ out-of-plane rotat. [-]
$
 belt_in_plane_bend_stiffn = 2.0e6 $ [Nmm^2]
 belt_out_of_plane_bend_stiffn = 200.0e6 $ [Nmm^2]
 belt_lat_bend_stiffn = 20.0e6 $ [Nmm^2]
 belt_twist_stiffn = 1.0e6 $ [Nmm^2/deg]
 belt_torsion_stiffn = 100.0 $ [N/deg]
$
 rim_flange_contact_stiffness = 3000.0 $ [N/mm]
 rim_to_flat_tire_distance = 30.0 $ [mm]
$
$tread properties **************************************************
 tread_depth = 8.0 $ [mm]
 tread_base_height = 3.0 $ [mm] 
 stiffness_tread_rubber = 64 $ [Shore A]
 tread_positive = 65 $ [%]
 damping_tread_rubber = 0.025 $ 
 [ms]
$
 sliding_velocity = 0.1 $ [mm/ms]
 blocking_velocity = 50.0 $ [mm/ms]
 low_ground_pressure = 0.01 $ [bar]
 med_ground_pressure = 2.0 $ [bar]
 high_ground_pressure = 10.0 $ [bar]
 mu_adhesion_at_low_p = 1.3 $ [-]
 mu_sliding_at_low_p = 1.1 $ [-]
 mu_blocking_at_low_p = 0.8 $ [-]
 mu_adhesion_at_med_p = 1.3 $ [-]
 mu_sliding_at_med_p = 1.0 $ [-]
 mu_blocking_at_med_p = 0.8 $ [-]
 mu_adhesion_at_high_p = 1.3 $ [-]
 mu_sliding_at_high_p = 1.0 $ [-]
 mu_blocking_at_high_p = 0.8 $ [-]
$
$tire imperfections ************************************************
 static_balance_weight = 0.0 $ [g]
 static_balance_ang_position = 0.0 $ [deg]
 dynamic_balance_weight = 0.0 $ [g]
 dynamic_balance_ang_position = 0.0 $ [deg]
 radial_non_uniformity = 0.0 $ [%]
 radial_non_unif_ang_position = 0.0 $ [deg]
 tang_non_uniformity = 0.0 $ [%]
 tang_non_unif_ang_position = 0.0 $ [deg] 
 ply_steer_percentage = 0.0 $ [%]
 conicity = 0.0 $ [deg]
 run_out = 0.0 $ [mm]
 run_out_angular_position = 0.0 $ [deg]
Adams/Tire32
 
$
$measuring conditions **********************************************
 inflation_pressure = 2.0 $ [bar]
 rim_inertia = 0.25e9 $ [g*mm^2]
$
$numerical data ****************************************************
 number_belt_segments = 80 $
 number_blocks_per_belt_segm = 32 $
 number_tread_strips = 8 $
 maximum_time_step = 0.2 $ [ms]
 BDF_parameter = 0.505 $ 0.5 .. 1.0 [-]
Choosing FTire Operating Conditions
You can control certain tire data during a simulation, without rerunning preprocessing. These parameters, 
listed below, are called operating condition parameters:
• Inflation pressure - The operating condition value of inflation_pressure defines the actual, 
possibly time-dependent inflation pressure, whereas the [FTIRE_DATA] value describes the 
inflation pressure at which the remainder of the data measurements had been taken.
• Tread depth -The operating condition value of tread_depth defines the actual, possibly time-
dependent tread depth, whereas the [FTIRE_DATA] value describes the tread depth at which the 
remainder of the data measurements had been taken.
• Model level - The operating condition value of model_level defines what variant of FTire is to 
be used: the basic version (=6) or the full 3D version (=7). The list of possible variants will be 
extended in the next release.
Also in the next FTire release, ambient temperature, will be added to the list of operating conditions.
To determine the actual operating conditions,