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296 Multibody Systems Approach to Vehicle Dynamics
{U S }
R 1 R 2
C
{U R }
P
z
Road surface
Fig. 5.53
Inclined tyre geometry
calculations. Consider first the view in Figure 5.53 looking along the wheel
plane at the tyre inclined on a flat road surface.
The vector {U s } is a unit vector acting along the spin axis of the tyre. The
vector {U r } is a unit vector that is normal to the road surface and passes
through the centre of the tyre carcass at C. The contact point P between the
tyre and the surface of the road is determined as the point at which the vector
{U r } intersects the road surface. For the purposes of this document it is
assumed the road is flat and only one point of contact occurs.
The camber angle between the wheel plane and the surface of the road is
calculated using:
/2 (5.31)
where
arccos({U r } • {U s }) (5.32)
The vertical penetration of the tyre z at point P is given by:
z R 2 |CP| (5.33)
In order to calculate the tyre forces and moment it is also necessary to
determine the velocities occurring in the tyre. In Figure 5.54 the SAE coordinate
system is located at the contact point P. This is established by the
three unit vectors {X SAE } 1 , {Y SAE } 1 and {Z SAE } 1 . Note that referring back to
Chapter 2 the subscript 1 indicates that the components of a vector are
resolved parallel to reference frame 1, which in this case is the Ground
Reference Frame (GRF).
Using the triangle law of vector addition it is possible to locate the contact
point P relative to the fixed Ground Reference Frame O 1 :
{R P } 1 {R W } 1 {R PW } 1 (5.34)
At this stage it should be said that the vector {R PW } 1 represents the loaded
radius and not the effective rolling radius of the tyre. Should this be significant
for the work in hand, such as the modelling and simulation of ABS
(Ozdalyan and Blundell, 1998), then further modification of the tyre model
may be necessary.