4569846498

Tyre characteristics and modelling 313
an overall effective contact point for the tyre at the given moment of interaction
with the terrain it is necessary to determine an effective normal vector
to the road surface acting through the contact point. Once again
Vesimaki (1997) uses a weighted average approach. The road normal can
therefore be defined by using again the x component X ern as an example:
Amwn
Xern
∑ ∑ Xnm
(5.71)
i1
j1
Veff
where
Xn m is the x component of the road normal for the mth component of area
within the cross-sectional tyre element.
Having found a volume, contact point and road normal vector for the
deformed tyre at the given integration point in time it is necessary to compute
a normal force acting on the tyre from the road. This involves an intermediate
step where the effective volume of penetration, V eff , is related to
effective radial penetration of the tyre. This involves interpolation of a
look-up table held within the tyre model that relates, for the defined tyre
profile, the tyre penetration to penetrated volume when the tyre is compressed
onto a flat surface.
The final computation required by the tyre model is to determine the effective
coefficient of friction, eff , due to contact with the terrain:
n
m
n m
eff ∑ ∑
i=1
j1
Amw
V
eff
n
m
(5.72)
where
m is the coefficient of friction associated with the mth component of area
within the cross-sectional tyre element.
For off-road simulation a useful aspect of this approach is that the coefficient
of friction can be factored to vary for each road element as described
in section 5.6.3.
Another tyre model, FTire, specifically developed for ride and durability
simulations has been developed by COSIN Software in Germany (Gipser,
1999) and made available through an interface in MSC.ADAMS.
The model comprises a rigid rim surrounded by elements with elastic interconnections
that form a surrounding flexible belt or ring and has been
developed to deal with frequencies up to 120 Hz and to encompass obstacles
in the longitudinal direction of rolling with wavelengths half the length
of the tyre contact patch. In the transverse direction the model can handle
inclination of the road surface but not obstacles that vary across the tyre
lateral direction, hence the model is referred to as a ‘2 1 ⁄2-dimensional’ nonlinear
vibration model. The model can also accommodate the effects of
stiffening and radial growth associated with high angular spin velocities.
The model input parameters comprise tyre geometry and measured physical
characteristics, with the optional input of natural frequencies and
damping factors associated with the lower vibration modes of an unloaded
tyre on a rigid rim. The belt or flexible ring is modelled as 50 to 100 lumped
mass elements elastically interconnected and mounted to the rigid rim.