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.