How does a gyrocompass in a moving craft sense the Earth's rotation? Various descriptions of gyrocompasses that I can find (such as the Wikipedia article) claim that the compass will seek to align itself with the earth's rotational axis.
I think I can understand how that works if we place the compass on the ground, at rest with respect to the surface. Then the entire compass housing changes direction in space at a steady 15 degrees per hour, and it makes sense that one can use gyroscopic torques to sense the axis of that rotation and align a readout with it.
However, gyrocompasses are not usually employed at rest on the ground -- they're used on ships and aircraft. That's a moving, rolling, pitching, platform. Ships roll on the waves. Aircraft roll to maneuver, and pitch up and down to modulate lift -- all much more violently than the rate of the earth's rotation. How does a gyrocompass manage to pick out the slow overlaid rotation of the earth out from all that noise, accurately enough to be of any use for navigation?
In a (relatively slow-moving) ship I suppose you can put the compass close to the center of the rolling motion and use an outer set of weighted gimbals to keep one axis pointing straight down. But that won't work in an aircraft which moves so fast that the lateral accelerations while turning displace the direction of "down" (as measured by an onboard plumb line or accelerometer) by tens of degrees -- which again seems to drown out any hope of detecting the rotation of the earth.
The aircraft's orientation in space isn't even tied to the earth's rotation on average; you can fly it to the other side of the planet in about one planetary revolution's time.
Is there some additional effect at work here? Newtonian gravity and mechanics don't appear to offer anything the the moving compass could hang on to.
 A: Here's what I get from the sources that David (+1) pointed to:
Averaging
One of David's references describes a true gyrocompass for use aboard ships. It deals with short-period "noise" from rolling and pitching by averaging them out. It uses (essentially) two nested sets of gimbals with different damping constants to filter the motion of the compass mount to remove the high-frequency components from rolling and pitching (which start at several mHz), leaving the 11.6 μHz component from the earth's rotation.
Since the frequency difference between noise and signal is about three orders of magnitude it really shouldn't have surprised me that it is simple to filter out.
Apart from that: It doesn't!
The ship's gyrocompass cannot automatically compensate for changes in the ship's absolute orientation caused by sailing to a different place on the earth's surface. It needs to be explicitly told the latitude and its time derivative so it can correct for the influence of Coriolis forces on the apparent direction of gravity.
(Actually, the gyrocompass in the reference just gets told the speed of the ship, presumed to travel in the direction the compass itself shows, but that's enough to derive the N/S velocity component).
In more modern systems these inputs can probably come automatically from a GPS unit.
Aircraft gyrocompasses?
The other of David's reference is a 1960 instructional film on aircraft instruments. What it describes are not gyrocompasses, but mere directional gyros. They're used (from a high-level point of view) as short term directional standards for filtering out short-period noise from an on-board magnetic compass.
A gyrocompass designed for ships would probably not work in an airplane. In contrast to a ship, a plane routinely undergoes significant horizontal accelerations that won't average out when integrated relative to the airframe -- e.g. in a racetrack holding pattern.
On the other hand, modern solid state or fiber-optic directional gyros are apparently precise enough that they won't drift significantly before the plane needs to land for refueling anyway. Once the plane is known to be stationary on the ground, the true north direction can be calibrated using the gyrocompass principle. (Wikipedia also mentions that some inertial navigation systems can bootstrap themselves in flight based on GPS position fixes).
A: I had the same question and found the following 2 resources very useful:


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*https://www.youtube.com/watch?v=JnKloSdUJLo.


This video talks about:
a) how a free mounted gyro can be used for attitude indication (roll, pitch) by using it's rigidity in space property.
b) It also discusses how the second property of a gyro (precession) can be used for rate of turn (yaw) indication.
c) Finally (and more related to your question) it discusses 2 gyrocompass systems:
The C-1 system uses a free mounted gyro but has important limitations, including having no drift compensation and so has to be periodically adjusted using a magnetic compass as reference.
The second system (MA-1) overcomes this problem by providing an electronic system that continuously tracks the error between the compass and the gyro, adjusting it's heading accordingly. Also, the short term small oscillations of the magnetic needle are effectively filtered out due to the gyro rigidity in space which tends to align to the average of these oscillations.


*http://ed-thelen.org/SperryManual-05.pdf
This is the manual of the Sperry Gyrocompass (mentioned in the Wikipedia article). It goes into great detail on the gyro properties and how the North seeking function is achieved, with wonderful illustrations.
