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298 Multibody Systems Approach to Vehicle Dynamics
It is possible to consider an angular velocity vector { S } 1 that only considers
the spinning motion of the wheel and does not contain the contributions
due to and . This vector for angular velocity that only considers
spin is given by:
{ } ˙{ U }
(5.41)
S 1
S
Using this it is possible to determine V C the ‘circumferential velocity’
component of point P relative to the centre of the wheel W and measured
parallel to {X SAE } 1 :
V C ({ S } 1 {R PW } 1 ) • {X SAE } 1 (5.42)
At this stage it may be worth considering the usual two-dimensional representation
of longitudinal slip for straight-line braking. Referring back to
section 5.4.4 a definition of slip ratio, S, during braking was given by
V R
S B e
(5.43)
V
Based on the velocities which have been determined for the threedimensional
case it is now possible to calculate a longitudinal slip ratio, S,
during braking which is given by:
VXC
S
(5.44)
| VX
|
For this formulation of slip ratio V XC can be considered to be the contact
patch velocity relative to the road surface. This is equivalent to V B R e
in the two-dimensional model, albeit using the loaded radius in the vectorbased
formulation. The circumferential velocity V C of P measured relative
to the wheel centre can be subtracted from V XC to give V X the actual longitudinal
velocity of P ignoring the rotation effect. This can be thought of as
the velocity of an imaginary point in the ground that follows the contact
patch and is also equivalent to V in the two-dimensional model. During
traction, the longitudinal slip ratio is formulated using:
VXC
S (5.45)
| Vc
|
The lateral slip of the contact patch relative to the road is defined by the slip
angle where
arctan{V Y /V X } (5.46)
During braking a lateral slip ratio S is computed as:
S |tan | |V Y /V X | (5.47)
During braking S will have a value of zero when V Y is zero and can have a
maximum value of 1.0, which equates to a slip angle of 45 degrees. Slip
angles in excess of this are not usual for vehicle handling but may occur in
other applications where a tyre model is used, for example, to simulate aircraft
taxiing on a runway.
During traction the formulation becomes:
S (1 S)|tan | |V Y /V C | (5.48)