Environmental Engineering Reference
In-Depth Information
minimum spin needed. This is a result of the pencil having
I
I
3
. Gyroscopes
are constructed so that
I
3
>I
in order that the minimum value of
ω
3
is not too
large. To achieve this the flywheel is made to be more massive than the frame that
supports it, and has most of its mass close to its outer radius.
10.8.2 Nutation of a gyroscope
We have seen that if the gyroscope is spinning at the correct frequency for
the angle of tilt then one observes uniform precession. In practice we do not
usually know the correct angle and tend to release at too high or low a value of
θ
.
Moreover, we release the gyroscope from rest, rather than at the correct precession
frequency. The result of these starting conditions is that the gyroscope bounces a
little before settling down into a precession at constant
θ
. To understand this aspect
of a gyroscope's behaviour it is most convenient to look at the motion in a rotating
frame of reference and to use Euler's equations. There is a subtlety here; the rotating
frame that we will use is one that precesses uniformly with angular velocity
ω
p
k
,
where
k
represents the vertical direction in the lab frame. We will call this frame
the precessing frame. It is clearly non-inertial, but it is
not
a body-fixed frame
since the flywheel still spins in it. Essentially, we will use the precessing frame as
a temporary replacement for the lab frame, which we are allowed to do as long as
we introduce the correct fictitious forces. The beauty of using the precessing frame
is that the torque is necessarily zero since the vector
L
stands still in this frame.
It is the cancellation of the real torque and the torque due to the ficticious forces
that ensures that this is just so. Thus, precession at constant angle
θ
to
k
in the
lab is represented by the gyroscope simply spinning about the
e
3
axis with angular
speed
ω
t
in the precessing frame, just like a free body spinning about a principal
axis. However, if we assume that the gyroscope is released with its centre-of-mass
at rest in the lab frame, the initial angular velocity in the precessing frame will be
−
ω
t
e
3
. This has small
8
components in the plane perpendicular to
e
3
and so
the gyroscope does not start off with a simple precession about
k
in the lab frame.
Since in the precessing frame there is no torque, we can use our solution for the
free symmetric top, i.e. Eq. (10.83). This gives us the frequency at which the
e
3
axis revolves about
L
:
ω
p
k
+
I
3
ω
3
I
≈
,
(10.114)
where we have made the approximation that cos
1.
So, in the precessing frame the symmetry axis revolves around
L
. To understand
how things appear in the lab we have to superimpose this motion of the symmetry
axis upon the precession of
L
about the vertical direction in the lab, which occurs
at a frequency given by Eq. (10.99):
≈
τ
I
3
ω
3
sin
θ
ω
p
=
ω
t
.
8
Since
ω
t
is much larger than
ω
p
.
