# Utilities to Calculate Tire Kinematics, Forces, and Moments

The following utilities to calculate tire kinematics, forces, and moments are available:

## ACTCLC

C Calling Sequence:
SUBROUTINE ACTCLC(TRAMAT, VEL, OMEGA, OMEGAR, RADIUS, RNORM,VLON, VCPLON, VCPLAT, VCPVRT,ALPHA, GAMMA, KAPPA,URAD, CPMTX)
Arguments
Argument Type   Description Symbol
TRAMAT 3,3 array Input Transformation Matrix from ISO to Road Axis system
VEL 3 array Input Velocity of Wheel Carrier in ISO
OMEGA 3 array Input Angular Velocity of Wheel Carrier in ISO
OMEGAR Double Precision Input Angular velocity of the Wheel
RNORM 3 array Input Road Normal Unit Vector in road axis system
VLON Double Precision Output Longitudinal velocity of the tire in ISO
VCPLON Double Precision Output Contact-patch longitudinal velocity in SAE
VCPLAT Double Precision Output Contact-patch lateral velocity in SAE
VCPRVT Double Precision Output Contact-patch vertical velocity in SAE
ALPHA Double Precision Output Slip angle is SAE
GAMMA Double Precision Output Inclination angle
KAPPA Double Precision Output Slip Ratio in SAE
URAD 3 array Output Unit vector from wheel centre to CP in road
CPMTX 3,3 array Output Transformation Matrix from SAE to Road Axis system
Calculations
Velocity of the Wheel Center in the Road reference frame is:

The Spin Axis (S) in the road reference frame is:

The coordinate system at the contact patch is defined by the transformation Matrix:

Here D represents unit vectors the directions corresponding to the Contact Patch:

The Longitudinal velocity of the Wheel center is given by:

Direction of the Unit Vector in the Direction of the CP from the wheel center:

The radius of the tire along the plane of the tire is:

Rigid body laws are applied to the tire to find the velocities at a different point on a rigid body in the road Frame of reference:

The Velocities in the Contact Patch Frame of reference is:

The SAE axis system is defined such that the Y and Z axis of the ISO axis system are flipped. This results in the following transformation matrix for moving any Vector from the ISO system to the SAE system:

The different components of the Contact patch velocities in the SAE axis system can be calculated from the following equation:

Since the Longitudinal velocity component is same in the SAE and the ISO system, they can be used interchangeably, which leads to the following SAE definition of Slip Angle and Slip Ratio:

The inclination angle is the angle between the road normal and the Wheel center to Contact Patch Vector:

## XCP2HB

C Calling Sequence:
SUBROUTINE XCP2HB(FCP, TCP, RAD, TRNMTX, FORCES, TORQUE)
Arguments
Argument Type   Description Symbol
FCP 3 array Input Contact Patch Forces in SAE
TCP 3 array Input Contact Patch Torques in SAE
RAD 3 array Input Radius Vector from Wheel Centre to Contact Patch
CPMTX 3,3 array Input Transformation Matrix from SAE to Road Axis system
FORCE 3 array Output Hub Forces in road reference frame
TORQUE 3 array Output Hub Torques in road reference frame
Calculations
Force Translation to wheel center and move to road frame of reference:

Move to road reference frame followed by Torque translation to wheel center:

## XCP2HB_ISO

C Calling Sequence:
SUBROUTINE XCP2HB (FCP, TCP, RAD, TRAMAT, FORCE, TORQUE)
Arguments
Argument Type   Description Symbol
FCP 3 array Input Contact Patch Forces in ISO
TCP 3 array Input Contact Patch Torques in ISO
RAD 3 array Input Radius Vector from Wheel Centre to Contact Patch
TRAMAT 3,3 array Input Transformation Matrix from ISO to Road Axis system
FORCE 3 array Output Hub Forces in road ISO
TORQUE 3 array Output Hub Torques in road ISO
Calculations
Force Translation to wheel center and move to road frame of reference:

Move to road reference frame followed by Torque translation to wheel center:

## ACTFZ

C Calling Sequence:
SUBROUTINE ACTFZ(VCPVRT, RADIUS, TIREK, TIREC, UNLRAD,FRCRAD, ERRMSG, IERR)
Arguments
Argument Type   Description Symbol
VCPVRT Double Precision Input Vertical contact patch velocity (SAE co-ordinates)