Intermediate Axis Theorem - Intuitive Explanation

[A.T.]

Veritasium posted a video, featuring a visualization of an "intuitive" explanation of the Intermediate Axis Theorem by Terry Tao, based on centrifugal forces in a rotating frame of reference:

Unfortunately, the animation is just as incomplete, as Tao's original explanation from 2011, and suggests the lowest moment of inertia axis would be unstable as well. The video conveniently doesn't show the animation for that case.

Fortunately, Tao has now posted an update, which also accounts for the Coriolis forces, which are crucial to explain why the lowest moment of inertia axis is stable. His explanation makes much more sense now:
https://mathoverflow.net/a/82020
I hope someone will make a similar 3D animation that includes the Coriolis forces.

Obviously, explaining something using a counter intuitive rotating frame of reference might not be really intuitive to many. Ideally you would break it down to linear dynamics in an inertial frame.

 

[Swamp Thing]

For a moment, I was confused about the reference to Coriolis forces. My understanding was that these arise when some element of a system is driven from one diameter orbit to a different diameter orbit -- so how can Coriolis forces arise in a rigid system?

But after a moment, it occurred to me that if a rigid system changes its rotation axis, then certain elements may find themselves nearer or further from the new axis, than they were from the old one. Is this what you meant?

 

[A.T.]

But after a moment, it occurred to me that if a rigid system changes its rotation axis, then certain elements may find themselves nearer or further from the new axis, than they were from the old one. Is this what you meant?

Coriolis forces arise if you are using a rotating reference frame, and something moves in that frame, not parallel to the frame rotation axis. The explanation by Tao uses the rotating frame in which the spinning object is initially at rest. As soon it deviates from the initial axis, parts of it start moving in the rotating initial rest frame, and thus experience Coriolis forces. Read Tao's update.

 

[Swamp Thing]

In Tao's update to his post, he notes that the Coriolis force is $F_{\mathrm{Cor}} = -2 m \Omega \times v$. (The $v$ here has to be a radial velocity, presumably). But if the system is rigid, then $v$ has to be zero.

The third paragraph in the update says:

we have as before that the m-masses experience a little bit of centrifugal force and begin to slide away from the y-axis and into the rest of the yz-plane. When doing so, they will then experience some Coriolis force in a direction parallel to the x-axis (which direction it is depends on the orientation of the rotation, as per the right hand rule formula for the cross product). However, due to the rigidity of the disk, as mediated by tension forces within the disk, it is not possible for the m-masses to actually move in the x-direction without also moving the much heavier M-masses.

It seems to me that Tao is sort of skirting around the fact that $v = 0$ by saying that IF the m's were free to move radially, then they would produce a Coriolis force. He then invokes the rigidity of the disk and says that even if there was a Coriolis force, it would be proportional to the tiny m's, and it would be trying to move the huge M's, so it wouldn't really end up doing much. So the m's "cannot actually experience any significant motion in the direction of the Coriolis force".

It still remains unclear how the introduction of the Coriolis force adds anything to the original explanation. And in any case, the C.F. can only exist if there is a radial velocity v, which there isn't -- is there?

The final paragraph describes the stable mode:

... when it rotates around the x-axis. Working in a rotating frame around this axis, the M-masses are near the x-axis of rotation, while the m-masses lie near the y-axis. As before, the M-masses experience centrifugal force and thus begin drifting slightly away from the x-axis into the xz-plane. But then the Coriolis force on these masses kicks in, which is now proportional to the heavy mass M rather than the light mass m.

But if it's rotating around the X axis, well, the big M's are ON the axis of rotation (see fig below) -- so there can't be any centrifugal force! And even if it were, they wouldn't actually move radially (because rigidity) so there wouldn't be a Coriolis force.

Me real confused!

 

[A.T.]

Coriolis force is $F_{\mathrm{Cor}} = -2 m \Omega \times v$. (The $v$ here has to be a radial velocity, presumably).

No, $v$ here is the total velocity vector in the rotating frame.

But if it's rotating around the X axis, well, the big M's are ON the axis of rotation (see fig below) -- so there can't be any centrifugal force!

Stability is about how the system reacts to a small perturbation. In the real world nothing is ever perfectly on the axis.

 

[Swamp Thing]

No, v here is the total velocity vector in the rotating frame.

Ah, okay.. I hadn't realized that tangential velocity produces a Coriolis force too. So "today I learned..."

But is it correct to say that v=0 for a rigid system? In which case no Coriolis force?

 

[A.T.]

But is it correct to say that v=0 for a rigid system? In which case no Coriolis force?

Did you read the second part of post #5 ?

 

[Swamp Thing]

I did read your remark about stability. What came to mind were examples like, [1] The axis of rotation starting off with a small tilt away from the ideal geometric axis, or [2] the masses in one or both pairs being slightly unequal, or [3] some other tiny geometric asymmetry in the position of the masses.

In all such cases, it's plausible that the system will develop a growing tendency to wobble more and more, and will end up settling down with a flipped rotation sense -- which in turn might later flip back to the initial state, and so on and so forth.

But that is exactly what we set out to demonstrate in an intuitively understandable way. In order to do this, Dr Tao's approach is to allow the masses to be slightly mobile, then invoke Coriolis forces, then go back to the rigid case. Why not just take a case where the initial rotation axis is, say, $0.5^\circ$ away from the geometric axis of symmetry? Would we observe Coriolis forces in this case, or in any other scenario with a small asymmetry but with infinite rigidity?

 

[valba]

3D animation that includes the Coriolis forces:

 

[Swamp Thing]

3D animation that includes the Coriolis forces:

Thanks, that is very helpful. Especially the links below the video on the YouTube page.

One of those links says, "Conservation of angular momentum is a bit questionable in ODE. If your rigid bodies have inertial tensors whose principal axes are all equal, everything is fine. Otherwise, you will get into trouble. A freely rotating body will tend to gain energy rather rapidly, eventually spinning fast enough to crash the simulator".

Now, those discussions are happening in the context of physics simulations for gaming. I'm wondering if this is really a serious headache that afflicts physics and engineering in general. For example, is it all that troublesome to simulate the tumbling of a satellite with a "pathological" asymmetry? And do the "professionals" also have to resort to workarounds like non-physical damping just to tame their simulation problems? What are some state-of-the art solutions to this issue?