Environmental Engineering Reference
In-Depth Information
would be expected give rise to a periodic change in the apparent latitude (the
latitude deduced from observations of the stars) of any given point on the Earth's
surface. Detailed observations of the apparent latitude at many locations around
the globe over the last century or so have produced data which suggest that the
situation is more complicated than the above analysis suggests. The data show many
irregularities, but have an underlying periodicity of one year which is thought to be
due to seasonal atmospheric effects. However, in addition to the annual periodicity,
there is a component with a period of 420 days known as the 'Chandler wobble'.
It is this component that is thought to be the effect of the precession of
around
the polar axis. That the observed period is longer than the predicted value may
be a result of the Earth not being a perfectly rigid body. In particular, the Earth's
mantle is thought to be a viscous fluid, the flow of which effectively reduces the
moment of inertia difference I 3
ω
I and extends the period of the wobble. You
can demonstrate an effect that a fluid interior has on the rotational properties of
a body with an experiment with two eggs; simply compare the effort it takes to
spin a raw egg on a flat surface as opposed to that required for a hardboiled egg.
You will observe that it is more difficult to get the raw egg to spin at a given
rate. This is because the fluid interior drags on the shell and dissipates energy
through non-conservative viscous forces. A raw egg can not be usefully described
by a moment of inertia tensor since different parts of the egg generally rotate
with different angular velocities. A proper representation of the rotational motion
therefore requires the use of a function of space to define the local velocity of an
element of matter at any point within the egg, as well as the forces acting on it.
In this chapter we shall not delve any deeper into the physics of rotating fluids,
which is really the domain of fluid dynamics, but shall instead continue to explore
the rich physics of rigid bodies as governed by Euler's equations.
10.7 THE STABILITY OF FREE ROTATION
We determined in Section 10.4 and Section 10.5 that it is theoretically possible to
obtain fixed-axis rotation about any of the three principal axes of a free rigid body.
However, we did not ask a somewhat more advanced question as to whether such
rotation could be maintained for a finite time in a realistic system. Surprisingly,
as we shall discover in this section, the answer is that sustained fixed-axis rotation
about only two of the principal axes is achievable in practice.
For rotation about a principal axis to be stable we require that small deviations
in the alignment of the angular velocity with the principal axis do not produce large
effects with time. Such deviations will inevitably occur no matter how carefully
we try to set a body spinning about a principal axis, but if the effects remain small
we can consider them as perturbations to the motion and we will be able to ignore
them at some level. However, if the perturbations come to dominate the motion
we consider it to be unstable.
To see the effect that perturbations have on a body rotating about a principal
axis, we look directly at the solutions to Euler's equations. Suppose that the body
is rotating with angular velocity
ω
, which is nearly parallel to e 3 . We can then
make the approximation
ω 1 2
ω 3 ,
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