Environmental Engineering Reference
In-Depth Information
Notice that
r
α
2
+
r
α
3
is the square of the distance of particle
α
from the
x
1
axis.
Thus
I
11
is the moment of inertia for rotation about the
x
1
axis. Similarly, the other
diagonal elements represent the moments of inertia for rotation about the
x
2
and
x
3
axes, respectively:
m
α
r
α
1
+
r
α
3
,
I
22
=
α
m
α
r
α
1
+
r
α
2
.
I
33
=
(10.19)
α
These are none other than the moments of inertia for rotation about the three
co-ordinate axes, i.e. the objects that we already met in Chapter 4. Notice also that
I
forms a symmetric real matrix, i.e.
I
ij
I
ji
and
I
ij
I
ij
, so there are only three
independent off-diagonal elements. These are known as the products of inertia and
they have the form:
=
=
I
12
=
I
21
=−
m
α
r
α
1
r
α
2
,
α
I
23
=
I
32
=−
m
α
r
α
2
r
α
3
,
(10.20)
α
I
13
=
I
31
=−
m
α
r
α
1
r
α
3
.
α
Thus far we have expressed the elements of the moment of inertia tensor as discrete
sums over all the particles in the body, but as usual the body may be better described
by a continuous density function
ρ(
r
)
, where an element of mass d
m
at position
r
is contained within a volume d
V
such that d
m
ρ(
r
)
d
V
(see Figure 10.2). In
which case the sums in Eq. (10.12) are replaced by integrals over the continuous
mass distribution and Eq. (10.12) should be written
=
d
m
[
r
2
δ
ij
−
d
Vρ(
r
)
[
r
2
δ
ij
−
I
ij
=
r
i
r
j
]
=
r
i
r
j
]
.
(10.21)
V
V
In this way, our picture of a body as being made up of particles is replaced by a
picture in which the body is made up of infinitesimal elements of volume d
V
and
mass d
m
.
We are certainly free to use whatever co-ordinate system we choose for the
evaluation of the matrix elements
I
ij
. However, using a co-ordinate system in the
lab frame of reference will immediately introduce the problem that the matrix
elements
I
ij
will, in general, change as the body rotates. Alternatively we can
choose a co-ordinate system in the body-fixed frame and this has the virtue that
I
ij
are constants in time. This provides an important simplification and we will
therefore tend to calculate the moment of inertia matrix in the body-fixed frame.
The price that we will pay for using the body-fixed frame is that we will have be
careful with the dynamical equations of motion, since this frame of reference is
rotating and is therefore non-inertial.
