Now the inertia matrix will be considered more carefully. It is a
symmetric
matrix, which can be expressed as
|
(13.93) |
For each
, the entry is called a
moment of inertia. The three cases are
|
(13.94) |
|
(13.95) |
and
|
(13.96) |
The remaining entries are defined as follows. For each
such that , the product of inertia is
|
(13.97) |
and
.
One problem with the formulation so far is that the inertia matrix
changes as the body rotates because all entries depend on the
orientation . Recall that it was derived by considering as a
collection of infinitesimal particles in the translating frame. It is
possible, however, to express the inertia matrix in the body frame of
. In this case, the inertia matrix can be denoted as because
it does not depend on the orientation of with respect to the
translational frame. The original inertia matrix is then recovered by
applying a rotation that relates the body frame to the
translational frame:
, in which is a rotation matrix.
It can be shown (see Equation (2.91) and Section 3.2 of [994])
that after performing this substitution, (13.92)
simplifies to
|
(13.98) |
The body frame of must have its origin at the center of
mass ; however, its orientation has not been constrained. For
different orientations, different inertia matrices will be obtained.
Since captures the physical characteristics of , any two
inertia matrices differ only by a rotation. This means for a given
, all inertia matrices that can be defined by different body frame
orientations have the same eigenvalues and eigenvectors. Consider the
positive definite quadratic form
, which represents the
equation of an ellipsoid. A standard technique in linear algebra is
to compute the principle axes of an ellipsoid, which turn out to be
the eigenvectors of . The lengths of the ellipsoid axes are given
by the eigenvalues. An axis-aligned expression of the ellipsoid can
be obtained by defining , in which is the matrix formed
by columns of eigenvectors. Therefore, there exists an orientation of
the body frame in which the inertia matrix simplifies to
|
(13.99) |
and the diagonal elements are the eigenvalues. If the body happens to
be an ellipsoid, the principle axes correspond to the ellipsoid axes.
Moment of inertia tables are given in many texts [690]; in
these cases, the principle axes are usually chosen as the axis of the
body frame because they result in the simplest expression of .
Steven M LaValle
2020-08-14