Moment of inertia
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
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 a 3D rigid body
its three axis.
given by Ixy, Ixz and Iyz where
The moment of momentum
for an explanation of how this is obtained)
is the Inertia Matrix
where the moment of momentum vector,
is parallel to
are easier to
solve, so the moment of momentum can be expressed as
this expression for is substituted into equation
hen the following expression is obtained.
seen to be an eigenvalue problem, the three
define the axis about which the body can spin maintaining h
The three eigenvalues are the
moments of inertia
and are known
as A B
The three eigenvectors are the principle axis of inertia
and are orthogonal.
When the axis are aligned with the principle axis Ip can be
axis aligned with principle are useful in solving
Moments of Inertia of a gyroscope
A gyroscope is an axisymmetric body
axisymmetry of a gyroscope all axis in the i-j
principle. A gyroscope can be thought of as an AAC body.
principle moments of inertia A A