Environmental Engineering Reference
In-Depth Information
10.2.1 Calculating the moment of inertia tensor
The components of the moment of inertia tensor are calculated using Eq. (10.12)
or Eq. (10.21) and they depend on the co-ordinate system used. In this section we
look at some cases where the rigid body possesses a degree of symmetry that aids
in the calculation. In particular, where there is an axis of rotational symmetry there
is no change in the moment of inertia in the lab as long the body rotates only about
that axis. In which case it does not matter whether we calculate the moment of
inertia in the body-fixed frame or the lab frame, the result will be the same.
x
3
x
2
x
1
Figure 10.3
A cylinder with its axis of symmetry along the
x
3
axis.
For example, consider a solid cylinder with its symmetry axis along the
x
3
axis
as shown in Figure 10.3. If the cylinder rotates about the symmetry axis then the
angular velocity vector is
(
0
,
0
,ω)
and the angular momentum is
L
1
L
2
L
3
I
11
I
12
I
13
I
21
I
22
I
23
I
31
I
32
I
33
0
0
ω
I
13
I
23
I
33
=
=
.
ω
(10.22)
Also, since there is rotational symmetry about the
x
3
axis, the products of inertia
I
13
=
I
32
are identically zero. To see why this is so let us consider
the product of inertia
I
23
. Notice that for each term proportional to
x
2
in the
sum
I
23
in Eq. (10.20), the symmetry of the mass distribution ensures that there
is a term of equal magnitude, but opposite sign, corresponding to position
I
31
and
I
23
=
−
x
2
(see Figure 10.4). These matching terms always cancel giving rise to
I
23
=
0.
Thus Eq. (10.22) becomes
L
1
L
2
L
3
I
13
I
23
I
33
0
0
I
33
=
=
ω
ω
(10.23)
or
L
3
=
0. So in this case the moment of inertia about the
x
3
axis is the only element of the moment of inertia matrix that matters. We could
therefore dispense with the fancy notation and write the equation
L
I
33
ω
and
L
1
=
L
2
=
=
Iω
just as
we did in Chapter 4.
Now let us now consider another special case. Namely, that of a planar object,
i.e. something thin and flat that we can approximate as being two dimensional.
As such it can be described by an area mass density distribution
σ(
r
)
. For such a
