Moment of inertia
Moment
of inertia is the rotational analogue to mass. The mass moment of
inertia about a fixed axis is the property of a body that measures the
body's resistance to rotational acceleration. The greater its value,
the greater the moment required to provide a given acceleration about a
fixed pivot.
The moment
of inertia must be specified with respect to a
chosen axis of rotation.
The symbols Ixx, Iyy and Izz are frequently used
to express the moments
of inertia
of a 3D rigid body
about
its three axis.
(A)
Products
of
Inertia are
given by Ixy, Ixz and Iyz where
(B)
Inertia
Matrix
The
moment of momentum,
![](images/Imoment.png)
can be
expressed as
(C) (
See PDF
for an explanation of how this is obtained)
Where
![Ip](images/ip.png)
is the Inertia Matrix
![](images/inertiamatrix.png)
Problems
where the moment of momentum vector,
h
is parallel to
![omega](images/omega.png)
are easier to
solve, so the moment of momentum can be expressed as
![](images/Ihp.png)
If
this expression for is substituted into equation
(C)
then the following expression is obtained.
![](images/Ipscaled.png)
This
can be
seen to be an eigenvalue problem, the three
eigenvalues
![](images/lambda.png)
of
![](images/ip.png)
define the axis about which the body can spin maintaining h
parallel to
![omega](images/omega.png)
The three eigenvalues are the
principle
moments of inertia and are known
as
A B and
C
The three eigenvectors are the
principle axis of inertia
and are orthogonal
.
![](images/principleaxis.png)
When the axis are aligned with the principle axis Ip can be
expressed as
![](images/Ippriciple.png)
Therefore
axis aligned with principle are useful in solving
practical
problems
Moments of Inertia of a gyroscope
A gyroscope is an axisymmetric body
![](images/ijkrotor.png)
Due
to the
axisymmetry of a gyroscope all axis in the i-j
plane are
principle. A gyroscope can be thought of as an
AAC body. It
has
principle moments of inertia
A A and
C
![](images/aacrotor.png)