Conservation of momentum problems are usually resolved by going back to
Newton's 3rd & identifying which two objects are involved. In this case,
it is the flywheel appartus and the Earth. In the tidal case, is is the
generating station and the ocean. The moon never enters into the system
(except that the moon provides the restoring effect to keep the tides
going)
Does this sound right:
The "Power out" will drive a generator (attached to the Earth) because the
gimbal axis is rotating (with a period of 1 day) relative to the ground.
In order to drive a generator, a torque must be provided by the axle, and
a counter-torque (reaction) applied to the axle in response. Since,
relative to the ground, the axle is rotating with a South-angular-
velocity, the generator will be driven to the South and the gimbal axle
therefore driven to the North.
This torque will be transfered to the flywheel, which will start to move
in that direction. However, it cannot. The gimbal axle will not allow
rotation in that direction.
Let's assume for a moment, wlog, that this takes place on the equator and
the flywheel starts pointing East (a cosmological East, not an Earth-based
rotating East). To prevent rotation to the North-East, the gimbal axle
bearings will apply a downward torque. This will also be transfered to
the flywheel, which will start to change its angular velocity "downwards",
causing the gimbal to spin to the North - the same way it was originally
driven by the generator.
The "ultimate" torque will always be perpendicular to the momentum of the
flywheel (try this in other orientations), so the flywheel axis will
continue to spin with a North-velocity. The original "East" momentum will
be transfered back to the Earth through the gimbal bearings, cyclically -
the flywheel will point East, then down, then West, then up, then East.
The end result will be a flywheel that is spinning on a rotating axis, the
direction of that axis will be rotating with a North-angular velocity.
The angular momentum that has been taken out of the Earth is now tied up
in the flywheel-gimbal system, as expected. Unfortunately, that rotation
will lock the generator and gimbal together and there will be no more
source of energy.
Once again, it is a case of a differential velocity and two objects being
brought to an intermediate state. The trick is, to figure out how to stop
the spinning, extract the kinetic energy in the process and then restart
the system with flywheel spinning to the East again. If that can be done,
then I think Fred has a winner here. Unfortunately, I fear that doing so
would completely reverse the process, and require just as much energy as
was obtained at the start (ignoring friction).
Back to the tides, this is where they differ - the moon restores the tidal
effect for free, we would have to restore the flywheel, and that won't be
free!
On a secondary note, I have no problem at all with the reification of
concepts such as energy and momentum. It is common to refer to energy
being "liberated", "obtained", or "transfered" from one form to another,
for example when wood is burned in a fireplace (ok, "obtained" is pushing
it a little bit!).
The extraction of energy from storage into useful form is similar across a
vast number of processes, the difference lies in the labeling of the
storage "container" - is it labeled bulk kinetic, gravitational potential,
chemical potential, etc.
Similarly, momentum can be "transfered" from one object to another, with
one object "extracting" momentum from, or "yielding" momentum to, the
other.
Reification is the basis of those most valuable conservation principles.
If energy and momentum are not "quantities", then what exactly is being
conserved?