Euler's Equations
The
reference frame
i, j and k is
aligned with the principle axis of the body. The axes are body fixed
and so the i, j, k
reference frame
moves with the body.
The moment
of momentum can be defined as
Reference
frame
Using a body fixed reference frame, aligned with the
principle axis ensures that the elements of (inertia matrix) don't
change
The I, J, K axes are space fixed
In the body fixed reference frame (i, j, k)
the instantaneous angular velocity,
can be defined as
Using the inertia matrix
and the fact that along with
the above definition for
gives the following expression for
moment of momentum.
(A)
Since i,
j and k
move with the body this equation always holds.
Differentiating the above expression gives
(B)
From the application of
Newton's Laws of Motion to circular motion we know
the following is true
If P is stationary
or at the center of
mass, G then
(C)
can be expressed
as
From
(A)
From this
(D)
Using the expanded definitions of and hp the following
expression for
is obtained
Where
x
represents the
vector cross product. Expanding the above equation gives
(E)
Substituting (E),
(C) and (D)
into equation (B)
gives:
These are EULER'S EQUATIONS
in a body fixed axis frame. They determine the angular motion of a body
subject to an external couple Q about a point P which MUST either be a
FIXED
POINT or at G.
Axisymmetric
bodies: The Gyroscope
equations
For an
axisymmetric body (e.g.: rotor of a gyroscope) the moment of
inertia is the same for all i and j within the i-j plane. This is
true for all AAC bodies. The k axis is aligned with C and so
i, j
and
k
are always aligned with principle axis.
The reference frame moves with angular velocity
where
The body itself has angular velocity
where
In addition to being axisymmetric about the k axis
the a gyroscope
spins within the reference frame about the k axis. Therefore
For
AAC bodies (such as gyroscopes)
EULER'S
EQUATIONS
become
These are
the
GYROSCOPE
EQUATIONS. Euler's angles
are used to solve them.