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The Moon is tidally locked to Earth - a long time ago it was spinning, but after a time, the spin stopped - two factors stabilizing it: unequal mass distribution creating an equilibrium position where there's one "preferred" axis by which it faces Earth - an attractor, and the dynamic tidal forces acting as stress upon its structure, dissipating the energy - a friction, reducing the rotation speed.

So, at one point in time, it would no longer complete a full spin, but instead reverse its turn direction, start spinning in opposite direction, with maximum spin speed around the time where the "preferred" axis faces Earth, then slowing down again, until its spin stops, and reverses direction again - each time the angle smaller as the dynamic forces act against the direction of rotation - but also, with the rotation angle and angular velocity dropping, the value of the dynamic forces dropping.

There is a significant apparent wobble caused by the orbit's eccentricity, and possibly some actual continuous wobble, as the result - with the attractor axis missing Earth as result. But is there any long-term oscillation left over from the times when the Moon was spinning - a harmonic motion around its original spin axis, that would likely not be in tune with the orbital period? (I imagine the period of these oscillations would be very long; after all it's about reversing Moon's spin back and forth, by a force that isn't all that strong.)

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This is called "physical libration". The Moon’s physical librations and determination of its free modes (2011) estimates it using the Apollo mission retroreflectors. Their result is very small. They note that the "damping times for these two modes are estimated as $2$x$10^4$ and $2$x$10^6$ years", so it is a mystery why it is even as large as their small result. They say:

Some possible excitation mechanisms have been explored in the past without satisfactory explanation. It has been shown that a recent meteoroid impact is an unlikely source of such excitation Peale (1975). Eckhardt (1993) proposed an excitation process related to a resonance crossing of the longitude normal mode (of 2.9 years) and a close forced frequency. During the evolution of the lunar orbit, the free and forced frequencies change slowly and can cross. However, the mechanism excites only the libration in longitude mode. Yoder (1981) proposed an alternative mechanism, based on turbulent fluid core interaction, to excite the wobble mode. The new determination of the amplitudes of the free librations invites new investigation of their excitation mechanisms.

I did not search to see whether anyone has taken up their invitation.

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You may have gotten your reference frame wrong. The moon never stopped revolving around its own axis, it's just that one revolution takes as long as one orbit around the earth.

So, at one point in time, it would no longer complete a full spin, but instead reverse its turn direction

No, it just slowed down contiually until one side faced earth permanently (discounting libration, which, as you say, is due to the ellipticity of the orbit.).

Tidal forces get weaker as tidal locking progresses, so the change in lunar spin would be greater at earlier times and then asymptotically approach zero as the moon becomes tidally locked.

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    $\begingroup$ Are you saying the spin slowed down to a crawl first, and only then the mass imbalance creating the attractor direction (and finally stopping the Moon for good) appeared? e.g. the moon never turned in the opposite direction? Also, frame of reference is never wrong or right, it's just always what we choose ;) I choose the frame with axis always aimed at Earth; in that frame the Moon is not spinning. :) $\endgroup$
    – SF.
    Aug 17, 2016 at 10:25
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    $\begingroup$ Bad frame of reference can lead to a situation looking more complicated than it is, and lead to drawing the wrong conclusions. Tidal forces will work on any extended object, you don't need an internal asymmetry. If you look "from the top" at the isolated earth-moon system, you'll see that the moon slows down its spin until it is tidally locked. In your scenario, the moon would have to slow down more than is necessary for tidal locking, then spin up again. Why, and how could that work? If you take "stopping" out of the picture, it all gets much simpler. $\endgroup$
    – Alex
    Aug 17, 2016 at 11:08
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    $\begingroup$ That's the second factor I wrote about: dynamic tidal forces acting as friction. The first factor is mass asymmetry, where tidal forces cause oscillation. Place a very long bar in orbit, pointing at Earth (not precisely; at a small angle). it will start turning as the bottom end exceeds its local orbital velocity and is being "ejected outwards" while the top end goes below the local orbital velocity and is pulled down. Its maximum rotary velocity is in horizontal position, then the rotation will reverse once the ends are reversed. $\endgroup$
    – SF.
    Aug 17, 2016 at 12:21
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    $\begingroup$ That effect is much more pronounced than the "frictional" tidal forces (losses of energy through heat due to material stress caused by tidal forces), and it takes a very long time for the bar to stabilize horizontally. Similarly, Moon's asymmetries cause similar effect, though through its age the "frictional" forces extinguished most of the effect - or all of it; the question asks precisely how much. $\endgroup$
    – SF.
    Aug 17, 2016 at 12:25
  • $\begingroup$ uh. I just realized I got the example wrong: lower altitude will result in too low a velocity and a downwards pull, and vice versa. The stable position will be vertical. Horizontal is unstable equilibrium. $\endgroup$
    – SF.
    Aug 17, 2016 at 13:05
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Logically moon has partially completed tidal locking and is on its way to complete its tidal locking in future..ie will eventually have zero angular spin with respect to earth. In past, after moon just stopped spinning, it must have started to go back in reverse directiion and continued back and forth ever since, just like a pendulam having harmonic motion and continued till today. The amplitude of the harmonic motion must be reducing with time due to kinetic losses. When the amplitude finally becomes zero in distant future, at that time it will be in absolute tidal locking with earth.

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  • $\begingroup$ Why? There is nothing what could decrease the amplitude of the pendulum. $\endgroup$
    – peterh
    Jul 20, 2019 at 8:59
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Yes, the Moon it is actually oscillating , and actually it is not really tidally locked, infact it is slowly going far away from Earth (by the time the Moon will leave the orbit the sun would already have expire its life however).

Let's see this photo from Wikipedia:

moon oscillation

In this image the moon phases where accelerated, the full complete oscillation takes about 1 month to complete.

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  • $\begingroup$ This motion is what we see from Earth, in reality the moon is just revolving around its axis in 1 direction. $\endgroup$
    – CoffeDeveloper
    Aug 17, 2016 at 9:57
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    $\begingroup$ The Moon's appearance does oscillate (called libration) but this doesn't mean it isn't tidally locked with Earth. $\endgroup$
    – Dean
    Aug 17, 2016 at 13:13
  • $\begingroup$ From what understand tidal locking result in a stable orbit (I may be wrong), moon orbit is not stable, it is slowing going away. If I'm wrong my apologize :), in that case then the moon orbit is unstable and tidally locked at same time :). I think a detail wrong in an answer does not deserve a downvote anyway $\endgroup$
    – CoffeDeveloper
    Aug 17, 2016 at 13:27
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    $\begingroup$ The Moon's orbit is stable in so much that it doesn't degrade or exhibit complex behavior. Just because the Moon is slowly moving away doesn't mean it isn't stable, all orbits evolve over time, and in this case the drag of the Earths oceans has caused the Moon to recede. In any case, the Moon will stay tidally locked to the Earth as it moves away and the Earth's rotation slows down. $\endgroup$
    – Dean
    Aug 17, 2016 at 14:02

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