Game Development Reference
In-Depth Information
Using the simplified torque curve, the torque for the Boxster S can be modeled by three
equations. The units for engine torque in all three equations are in
N-m
.
T
=
220
Ω≤
1000
(8.17a)
e
T
=
0.025
Ω+
195
1000
<Ω <
4600
(8.17b)
e
e
e
T
=−
0.032
Ω +
457.2
Ω≥
4600
(8.17c)
e
e
e
All three of the lines described by Equations (8.17a) through (8.17c) are specific cases of
the general equation for a straight line.
Tb
=Ω+
d
(8.18)
e
e
The
b
parameter in Equation (8.18) is the slope of the line. Of course, straight lines aren't
the only way to mathematically model a torque curve. Depending on the shape of the curve, a
parabolic or exponential function could also be used to approximate a torque curve.
Having a mathematical expression for the torque curve is all well and good, but to solve for
the acceleration of the car what we really need is an equation for the wheel torque as a function
of the current velocity of the car. An equation that relates wheel torque to car velocity can be
derived if the assumption is made that the tires roll without slipping. Under this condition, the
velocity of the car,
v
, is equal to the wheel radius,
r
w
, multiplied by the angular velocity of the
wheel,
w
w
. As seen in Equation (8.12), the angular velocity is a function of the engine turnover
rate and the gear and final drive ratios.
2
60
p
r
Ω
vr
=
w
=
we
(8.19)
ww
gG
k
As a reminder, the “60” term in Equation (8.19) converts the engine turnover rate from
rpm
to
rev/s
. Plugging Equations (8.18) and (8.19) into Equation (8.11) results in an expression
for the acceleration of the car as a function of the current velocity of the car.
22
2
60
gGbv
gGd
1
CvA
r
a
=
k
+
k
−
D
−
m
g
cos
q
−
g
sin
q
(8.20)
r
mr
2
mr
m
2
p
2
w
w
Equation (8.20) looks really messy, but it's really just an algebraic equation. The constants
can be grouped together to form a simpler equation in which the acceleration of the car is a
function of the current velocity of the car.
dv
a
d
== ++
c v
2
c v
c
(8.21)
1
2
3
The constants,
c
1
,
c
2
, and
c
3
, in Equation (8.21) are the following:
1
2
D
CA
r
c
=−
(8.22a)
1
m
22
60
2
gGb
c
=
k
(8.22b)
2
2
p
mr
w
gGd
k
c
=
−
m
g
cos
q
−
g
sin
q
(8.22c)
3
r
mr
w
