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[Iacopo]

"superimposed" does come in degrees.

Still this confuses me.  These "partial superimpositions" are about whirling, or can happen in all kind of motions ?
Could it exist a partial vectorial sum of two different motions ?
I am not understanding how it works.

[Russpin]

Thinking it a little better, I am now not so sure that going bigger would solve the gimbal problem. 

Gimbal gyros are not used much anymore. Your phone probably has mems gyros in it.
One system I worked on had three ring laser gyros.

[ta0]

For our purpose of seeing free inertial behavior, I would guess that if the moment of inertia of the flywheel is more than ten times that of the gimbals, it would suffice. Do you recall what is a typical ratio?

PS: I worked on fiber gyros for my dissertation, over a quarter of a century ago . . . (how can it be that long ago?  )

[Russpin]

For our purpose of seeing free inertial behavior, I would guess that if the moment of inertia of the flywheel is more than ten times that of the gimbals, it would suffice. Do you recall what is a typical ratio?

No I don't. The moments of inertia for the flywheel, inner and outer gimbals could of course be estimated. The dynamics analysis will probably require a simulation.

PS: I worked on fiber gyros for my dissertation, over a quarter of a century ago . . . (how can it be that long ago?  )

Dr. Ta0 ?

[Jeremy McCreary]

"superimposed" does come in degrees.

Still this confuses me.  These "partial superimpositions" are about whirling, or can happen in all kind of motions ?
Could it exist a partial vectorial sum of two different motions ?
I am not understanding how it works.

I'm doing a lousy job of explaining this -- partly because I should have used "superposed" instead of "superimposed". Go to the Wikipedia page on the "superposition principle" and see if that helps. It has some very useful figures.

When I said "'superimposed' does come in degrees", I was referring to approximate superpositions. So let me try again...

I view the total observed motion U of an unbalanced top as an approximate superposition of W, pure whirl, and B, the gyroscopically controlled motion of the top's balanced equivalent. Hence, distinctive features of both W and B are observed in U and remain recognizable as such.

However, U isn't just a simple sum of W and B, vector or otherwise, because W and B interact via the unbalanced top itself. Not enough for either one to totally mask the contributions of the other, but W modifies B in some significant way and/or vice versa.

Hence, the observed unbalanced motion U ends up being somewhat different from the sum of its parts.

Now let me build on the definitions of U, W, and B above to make this idea more concrete. Let B include all of the usual balanced top motions due to (i) gravity acting on the CM and (ii) any sliding friction or rolling resistance acting on the tip. These common tip forces affect B and ultimately the total unbalanced motion U via torques about the CM. Ignore air resistance.

Now suppose that the tip scrubbing I observe in my unbalanced tops really is due to whirl alone -- i.e., that it's part of W. Because the scrubbing modifies the tip forces in real time, it feeds back into the motions lumped under B. Not enough to make B unrecognizable in U, but enough to alter U both directly and via B.

Granted, dividing the unbalanced top system like this is artificial, but it's still useful. And we've been doing it all along in this thread -- mostly without explicitly saying so.

And yes, approximate superposition can happen in any system subject to fundamentally different processes weakly coupled (i.e., weakly interacting) via the system itself. I have one more example involving the trajectory of a projectile in air vs. vacuum if this doesn't work for you.

[Iacopo]

dividing the unbalanced top system like this is artificial, but it's still useful.

Thank you, Jeremy, now I understand.  Weekly interacting motions could be complicated and difficult to unravel and fully understand.
We can just try to understand the basic principles, but l am not asking for more than this.

[Russpin]

A very simplified model of the gimbals would be to add a moment of inertia (Ig) representing both gimbals to the transverse moment of inertia of the flywheel (I₁). The moment of inertia about the symmetry axis stays the same (I₃). Assuming a small angle between the angular velocity vector an the axis of symmetry of the flywheel the precession simpifies to p = wI₃/(I₁ + Ig).
Where w is the spin.
For a thin disk I₃ = 2I₁
if Ig = 0.1I₃
then p = w/(.6) = 1.67w

[Jeremy McCreary]

dividing the unbalanced top system like this is artificial, but it's still useful.

Thank you, Jeremy, now I understand.  Weekly interacting motions could be complicated and difficult to unravel and fully understand.
We can just try to understand the basic principles, but l am not asking for more than this.

You're welcome, Iacopo. Just to be clear, unraveling weakly interacting motions just by watching them and recognizing the underlying components can be quite doable, at least qualitatively, but strongly coupled components may be impossible to recognize for what they are.

[ta0]

A very simplified model of the gimbals would be to add a moment of inertia (Ig) representing both gimbals to the transverse moment of inertia of the flywheel (I₁). The moment of inertia about the symmetry axis stays the same (I₃). Assuming a small angle between the angular velocity vector an the axis of symmetry of the flywheel the precession simpifies to p = wI₃/(I₁ + Ig).
Where w is the spin.
For a thin disk I₃ = 2I₁
if Ig = 0.1I₃
then p = w/(.6) = 1.67w

That was easy 
So, the transverse moment of inertia of the flywheel has to be 10 times the moment of inertia of the gimbals for p = 1.82 w or 91% of the ideal value. Perhaps a little challenging but sounds doable. Twenty times would get it to 1.9, what I would prefer.

[Iacopo]

A very simplified model of the gimbals would be to add a moment of inertia (Ig) representing both gimbals to the transverse moment of inertia of the flywheel (I₁). The moment of inertia about the symmetry axis stays the same (I₃). Assuming a small angle between the angular velocity vector an the axis of symmetry of the flywheel the precession simpifies to p = wI₃/(I₁ + Ig).
Where w is the spin.
For a thin disk I₃ = 2I₁
if Ig = 0.1I₃
then p = w/(.6) = 1.67w

Here the transverse moment of inertia matters, if higher it slows down precession. This sounds very logical to me.
This is for a gyroscope, but I don't see why the transverse moment of inertia shouldn't influence the precession speed in spinning tops too, as noted by Jeremy at the beginning of this thread.  I think it should.

[Jeremy McCreary]

...This is for a gyroscope, but I don't see why the transverse moment of inertia shouldn't influence the precession speed in spinning tops too, as noted by Jeremy at the beginning of this thread.  I think it should.

Just for the record, I said that the standard approximate expression for the slow precession rate (the one usually observed) doesn't include TMI explicitly, but the fast precession rate approximation does.

The slow precession rate still grows with TMI, but if the slow rate approximation is to be believed, only because both grow with CM-tip distance per the parallel axis theorem. I also find this rather surprising, but the only assumptions underlying this approximation are as follows:
1. The top is symmetric.
2. The precession is steady -- i.e., no nutation.
3. The total angular speed about the spin axis (a combination of the spin and precession rates) is far above the critical value for steady precession. (Requiring the top's spin kinetic energy to be much greater than its gravitational potential energy pretty much takes care of this, but the two conditions aren't exactly the same.)
4. No dissipation.

No. 4 is clearly a fiction in real-world tops, but I'm not sure it's a deal-killer. Otherwise, nothing patently unreasonable here for the tops we generally use -- at least not early on in their spins.

[Iacopo]

1. The top is symmetric.
2. The precession is steady -- i.e., no nutation.
3. The total angular speed about the spin axis (a combination of the spin and precession rates) is far above the critical value for steady precession. (Requiring the top's spin kinetic energy to be much greater than its gravitational potential energy pretty much takes care of this, but the two conditions aren't exactly the same.)
4. No dissipation.

I have thought to this for some time and still it seems strange to me.  Even with these assumptions.
If I will have some free time for this, I will make a little practical experiment with real tops, to see if this is true.

[Jeremy McCreary]

I have thought to this for some time and still it seems strange to me.  Even with these assumptions.
If I will have some free time for this, I will make a little practical experiment with real tops, to see if this is true.

I'm sure the experiment will be quite elegant, too, Iacopo! I've thought about how one might vary TMI while controlling CM-tip distance in tops. It's not a simple problem.

[Iacopo]

I'm sure the experiment will be quite elegant, too, Iacopo! I've thought about how one might vary TMI while controlling CM-tip distance in tops. It's not a simple problem.

Not too difficult.  I was thinking to make an axis with two wooden discs inserted. 
In the first test I would use this top with the discs at middle height of the axis, adjoined one to the other, and I would see how fast it precesses.
In the second test, in the same top, I would shift the wooden discs along the axis, the upper disc upwards, and the lower disc downwards, by the same distance. 
In this way the top would still have same weight, same diameter, same axial moment of inertia, same center of mass-tip distance:
only the transverse moment of inertia would change.
Then I would see if it precesses slower or not than in the first test.

[Iacopo]

Experiment about influence of transversal moment of inertia on precession speed:

The top in the second picture is the same as the one in the first picture, but the lower wooden disc has been shifted downwards by mm. 12.5 and the upper disc has been shifted upwards by mm 12.5 .  The two discs, after they have been balanced, weigh both 20.3 grams.
The only difference among the two versions of the top should be the transversal moment of inertia.



The first top precesses slightly slower than the second:
At about 650 RPM, and about same angle of tilting, the first precesses in 1.03 seconds, the other precesses in 0.98 seconds.

The angle of tilting seems to have little influence on precession speed; when the top is more tilted, it precesses a bit slower, (I thought the opposite should happen), but the tip scrubs in more tilted position, so this is not clear.

The rotation speed instead influences very much the precession speed.

This experiment seems to demonstrate that an higher transversal moment of inertia does not slow down precession speed.