Game Development Reference

In-Depth Information

Notice how the calculations for the
I
cg
of the driver and the fuel are dominated by their

md
2
terms. In this example, the local inertia of the driver and fuel is only 2.7% and 2.1%,

respectively, of their corresponding
md
2
terms.

Finally, we can obtain the total moment of inertia of the body about its own neutral axis

by summing the
I
cg
contributions of each component as follows:

I
cg total
= I
cg car
+ I
cg driver
+ I
cg fuel

I
cg total
= 3783.34 N − s
2
− m + 153.27 N − s
2
− m + 602.07 N −

s
2
− m = 4538.68 N − s
2
− m

The mass properties of the body—that is, the combination of the car, driver, and full

tank of fuel—are shown in
Table 1-3
.

Table 1-3. Example summary of mass properties

Property

Computed value

Total mass (weight)

1972 kg (19,343 N)

Combined center of mass location

(
x
,
y
) = (30.42 m, 30.53 m)

Mass moment of inertia

4538.68 N - s
2
- m

It is important that you understand the concepts illustrated in this example well because

as we move on to more complicated systems and especially to general motion in 3D,

these calculations are only going to get more complicated. Moreover, the motion of the

bodies to be simulated are functions of these mass properties, where mass will determine

how these bodies are affected by forces, center of mass will be used to track position,

and mass moment of inertia will determine how these bodies rotate under the action of

noncentroidal forces.

So far, we have looked at moments of inertia about the three coordinate axes in 3D space.

However, in general 3D rigid-body dynamics, the body may rotate about any axis—not

necessarily one of the coordinate axes, even if the local coordinate axes pass through

the body center of mass. This complication implies that we must add a few more terms

to our set of
I
's for a body to handle this generalized rotation. We will address this topic

further later in this chapter, but before we do that we need to go over Newton's second

law of motion in detail.