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position of the center of mass, and a dynamic part depending on the acceleration of
the vehicle:
mg
l
f
+
mh
l
f
+
F
zf
=
l
r
(
θ
−
θ)
−
l
r
¨
l
r
cos
h
sin
x
(14.5)
mg
l
f
+
mh
l
f
+
F
zr
=
l
r
(
l
f
cos
θ
+
h
sin
θ)
+
l
r
¨
x
The aerodynamic drag is given by:
1
2
ρ
2
F
a
=
C
d
A
f
(
˙
x
−
v
w
)
(14.6)
ρ
where
is the air density;
A
f
is the equivalent frontal area of the car (the projected
area in the
yz
plane), whose value is about 80% of the area calculated from the
dimensions of the car;
C
d
is the aerodynamic drag coefficient, which can be exper-
imentally calculated (Rajamani
2012
), or obtained from the documentation of the
vehicle; finally
is the wind speed in the direction of the
x
axis.
The rolling resistance depends on the speed as in the expression:
v
w
F
rf
+
F
rr
=
F
r
=
mgC
r
v
(14.7)
where
C
r
is a coefficient experimentally obtained along with the aerodynamic drag
coefficient
C
d
.
14.2.4.2 Tyre
This provides the traction forces according to expression (
14.4
). The friction coeffi-
cient
μ
is given by the experimental expression called
the magic formula
(Pacejka
2006
):
μ
=
D
sin
(
C
arctan
(
B
σ
−
E
(
B
σ
−
arctan
(
B
σ ))))
(14.8)
In this expression
σ
is the
slip factor
, and is defined as:
ω
R
−˙
x
σ
=
(14.9)
max
(ω
R
,
˙
x
)
where
x
is the speed
of the car. The values of the coefficients
B
,
C
,
D
and
E
depend on the type of surface
(dry tarmac, sand, snow, etc.).
ω
and
R
are the angular speed and the radius of the wheel, and
˙
14.2.4.3 Wheel
A wheel spins around under the effect of the torques applied to it:
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