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[ta0]

Practically, the names in the simulator tell the direction of the motions, and not the nature of the motions.
So, "initial precession angular velocity" in the last simulation is "torque free precession".

That is very true. From a physical point of view, what your top mostly does when hit hard is inertial, in that sense more similar to nutation than torque-induced precession. I guess that's the reason satellite engineers call it nutation even though it looks like precession.

And its "nutation" is not real nutation, (the inertial motion), but it is simply the vertical oscillation, in this case due to residual precession, (the circle is not perfectly horizontal).

I would say that that small nutation is real nutation. It's an inertial oscillation (due to initial conditions) around the residual gravity-induced precession.

[Iacopo]

From a physical point of view, what your top mostly does when hit hard is inertial, in that sense more similar to nutation than torque-induced precession. I guess that's the reason satellite engineers call it nutation even though it looks like precession.

Maybe those engineers call nutation the torque free wobbles and precession the torque induced wobbles, making a difference based on the nature of the motion and not of its kinematics ?

And its "nutation" is not real nutation, (the inertial motion), but it is simply the vertical oscillation, in this case due to residual precession, (the circle is not perfectly horizontal).

I would say that that small nutation is real nutation. It's an inertial oscillation (due to initial conditions) around the residual gravity-induced precession.

Eh.. yes, maybe..  what I thought is that, since the circle is not perfectly horizontal, and it's nearly steady, (because the gravity induced precession of this top is very slow), the stem, practically, goes up and down following this circle, once every about one turn of the torque free precession. 

[ta0]

Eh.. yes, maybe..  what I thought is that, since the circle is not perfectly horizontal, and it's nearly steady, (because the gravity induced precession of this top is very slow), the stem, practically, goes up and down following this circle, once every about one turn of the torque free precession.

Good thinking. The period of this "nutation" is about equal to the "precession" period, so you would be right: this is not real nutation but just the vertical variation due to the slightly slanted circle of the free torque orbit. On the other hand, the top is gaining and losing speed at the expense of the precession and 90 degrees out of phase of the nutation, what is what you would expect with real nutation. I could be that this also work in your almost free torque case. 

[Jeremy McCreary]

On Earth, gravity is alive and well in all of the behaviors seen in any top supported anywhere other than exactly at its CM. So why seek explanations in torque-free motions in this context? There are none -- especially in tops as heavy as Iacopo's. Really.

[Jeremy McCreary]

<soapbox> Please forgive this rant...

When discussing technical things, it's essential to use the same terms for the same things, and to do so precisely and consistently. Otherwise, we end up going in circles, as we've done here and in several previous discussions of precession and nutation.

Don't get me wrong. I find the observations and photos and videos in this thread quite valuable. But I had to give up on the discussion of possible explanations for intermittent wobble, and for only one reason: Couldn't tell for sure how key technical terms were being used from one post to the next.

Maybe those engineers call nutation the torque free wobbles and precession the torque induced wobbles, making a difference based on the nature of the motion and not of its kinematics ?

Not sure what possessed engineers to define precession and nutation as they do. But I for one would love to leave their definitions far behind. For good.

After all, the engineering definitions are generally applied to systems only tangentially related to this forum's primary concern: Tops with a single point of support in a strong gravitational field.

So I have a heartfelt plea: Let's agree to stick with the definitions used throughout the literature dealing directly with our primary concern. (See above.) Then we can take full advantage of that vast body of work with no translation involved. And we can make our own headway without the circles.

</soapbox>

[ta0]

On Earth, gravity is alive and well in all of the behaviors seen in any top supported anywhere other than exactly at its CM. So why seek explanations in torque-free motions in this context? There are none -- especially in tops as heavy as Iacopo's. Really.

I disagree in the case of Iacopo's tops. They may be heavy, but the CM is so close to the support that for fast initial conditions the movement is better explained by a torque-free movement.
In fact, I changed the CM from 1 mm to 0.2 mm (and the angle to 12 degrees), as Iacopo said was the case for #15, and got this:







An almost perfect torque free movement!

I think we have a philosophical difference, Jeremy. Equations don't speak to me as powerful as intuitive models of reality. I love these discussions with Iacopo on intuitive physics. It's very unusual that somebody with no formal training or background is willing and capable of discussing in detail these technical things. I believe he provides new fresh perspectives that force me to think. For example, I now find it more reasonable that in some engineering fields, what top people would call precession they call nutation.

[Jeremy McCreary]

Iacopo and ta0: I love and learn from our 3-way physics discussions, too. And far more raw data on real top behavior has come out of Iacopo's ingenious experiments than from any other source. I really value that, too.

I'm not advocating for equations. Just hoping to get even more out of these exchanges by settling on a few key definitions that come up all the time. If that's not possible, we should state the definitions we're using up front when the chance of confusion is high.

ta0: Will have to think about your case for torque-free behavior here.

For now, having a hard time getting past 2 key observations...

1. When Iacopo's tops with recessed tips are released at an angle to the vertical, they always precess about the same axis -- the vertical.

From a non-magnetic top's perspective, the only thing magic about this direction in space is that it follows gravity's local line of action.

2. When the CM is above the contact, the usual case in Iacopo's tops, the precession is always prograde, as expected in a top subject to gravitational torque.

I've experimented with tops with sliding bell-shaped rotors allowing easy placement of the CM on either side of the contact. Only when the CM and contact exactly coincide does precession disappear in keeping with the absence of a net gravitational torque. In this case, the symmetry axis holds its direction in space even after the spin comes to rest.

But with even a fraction of a millimeter of offset in either direction, the precession reappears in the appropriate direction. And aside from precession rate, the motion is kinematically identical to that with a large offset and wholly consistent with forced precession.

[Iacopo]

On Earth, gravity is alive and well in all of the behaviors seen in any top supported anywhere other than exactly at its CM. So why seek explanations in torque-free motions in this context? There are none -- especially in tops as heavy as Iacopo's. Really.
...
But with even a fraction of a millimeter of offset in either direction, the precession reappears in the appropriate direction. And aside from precession rate, the motion is kinematically identical to that with a large offset and wholly consistent with forced precession.

Jeremy, it seems like you believe that there cannot be torque free wobbles in our tops.

This is not correct, there are indeed some wobbles that are torque free in them.

The most common one is nutation itself;
these vertical oscillations in spinning tops are inertial and torque free.
The fast wobble in my top Nr. 15 I showed here is too torque free.

Gravity is taken care by the "gravity induced precession" and does not prevent the existence of a superposed torque free wobble.

If you are skeptical, consider this:

You say "the motion is kinematically identical to.. forced precession".
Well, if I use my top Nr. 12, which has the tip slightly above the CM, (spindulum), if I spin it clockwise, it precesses counterclockwise.
Now, if I kick the stem of this top, to cause the fast wobble, the fast wobble happens clockwise.
This does not depend on the way I kick the stem, because, whatever the force/direction of the kick, the fast wobble happens always clockwise.
So, the fast wobble has the opposite direction it should have if it was "gravity induced precession".
This tells that the fast wobble is not a "gravity induced precession" but something else.

Consider the deceleration of the wobble:
The "gravity induced precession" becomes faster and faster as the spin speed slows down.
The fast wobble in my top instead becomes slower and slower" as the spin speed slows down.
Again, the fast wobble does not behave like "gravity induced precession".

Consider the speed of the wobble:
The speed of the fast wobble resulted always consistent, at different tested speeds, with the formula:
wobble speed : spin speed = AMI : TMI
This is the formula of the "torque free precession".
I consider this a strong evidence that the fast wobble is indeed "torque free precession".
 

[ta0]

I recommend re-reading the second half of Butikov's paper, Precession and Nutation of a Gyroscope: http://butikov.faculty.ifmo.ru/Gyroscope.pdf
He explicitly addresses the gyroscope with and without gravity and the "superposition of torque-induced regular precession and torque-free nutation". 
You may find very interesting what happens if a gyroscope is originally in steady precession and you turn off gravity (not as difficult to do as you might think: just let if free fall!  ).

[Jeremy McCreary]

I recommend re-reading the second half of Butikov's paper, Precession and Nutation of a Gyroscope: http://butikov.faculty.ifmo.ru/Gyroscope.pdf
He explicitly addresses the gyroscope with and without gravity and the "superposition of torque-induced regular precession and torque-free nutation". 
You may find very interesting what happens if a gyroscope is originally in steady precession and you turn off gravity (not as difficult to do as you might think: just let if free fall!  ).

We have a switch in our house for that. Will review the article.

[Iacopo]

I made a quick animation of the intermittent wobble.
This is for tops with a recessed and spiked tip.

The intermittent wobble happens when there are two wobbles of different nature at the same time, having similar amplitude and a not very different speed.  In this case the two wobbles are unbalance wobble and the torque free precession.

The video represents the motion of the upper extremity of the stem of the top, (big white dot), seen from above.
The yellow dot is the center, when the stem of the top passes here, it is in vertical position.

The blue dot has the motion that the stem would have with only unbalance wobble;
the stem would make a circular trajectory around the center.

Adding the torque free precession, the stem revolves around the torque free precession axis, (blue dot), which, in turn, revolves around the vertical axis, (yellow dot), which maybe could be called "unbalance wobble axis".

If the unbalance wobble and the torque free wobble had the same angular speed, the resultant trajectory would still be a perfect circle.
But the torque free precession is a bit faster than the unbalance wobble, so the trajectory is more complex, when the two wobbles are in counterphase they cancel each other out and for an instant the stem is vertical, at the center, apparently not wobbling.
When the two wobbles are in phase, the stem is at the greatest distance from the center and moving at the highest speed.
The alternation of phases and counterphases produces the intermittent wobble.   

[ta0]

That's a very nice animation, Iacopo!

Strictly speaking, unbalance does not produce simple circles. It produces loops with cusps (we discussed some here). These are the diagrams from Butikov's other paper, Inertial Rotation of a Rigid Body http://butikov.faculty.ifmo.ru/Precession.pdf:



They show the inertial "precession" of a body (with the two transversal moments of inertia equal) floating in space, that is spun along a direction different to the principal axis. The top precesses around a fixed direction in space (the direction of the angular momentum, which is conserved). At the same time the top rotates around its axis so that a certain cone ("moving axoid") fixed to the top rolls without slipping on a fixed cone ("immovable axoid"). The vector w shows the direction of the axis around which the top is spinning at that instant of time. At time zero it will coincide with the direction in which the top was first spun. The top on the left is elongated (prolate) so the transverse moment of inertia is larger than the axial moment of inertia. The reverse is true for the flat top on the right (oblate).

For a balanced top, the stem will coincide with the principal axis, n, and just make a circle around a fixed direction: simple torque-free precession.
If the top is unbalanced the stem will not be along the direction n. Let's assume the imbalance is not very large, so these diagrams are still valid. If we spun the top using the stem, the stem at time zero will coincide with w. So the stem is attached to the surface of the moving cone. As the top precesses, the stem will make loops with cusps each time it touches again the fixed cone, as it has zero speed at that time (like the cycloid made by a point on the rim of a wheel).

To me, it's much easier to visualize the prolate case on the left than the oblate case on the right, but Iacopo's tops are oblate. In that case the stem makes more than one turn around the fixed direction before touching again the fixed cone. If you imagine that the stem (w) is vertical at the start, you can see that after a period of the time that is the minimum common multiple of the "precession" period and the spin period, the top would become vertical again. This might never happen (if the periods are not commensurable), but if it does, the stem will come to a short stop at that time.

In the more general case where the unbalanced top is randomly hit, the stem will not coincide with w at the start. So even if it happens to come back to vertical, it won't have zero speed and won't give the same visual effect.

[Iacopo]

Strictly speaking, unbalance does not produce simple circles. It produces loops with cusps...
...the stem is attached to the surface of the moving cone. As the top precesses, the stem will make loops with cusps each time it touches again the fixed cone.

You are describing the case of the unbalance wobble mixed with the gravity induced precession another wobble.
It behaves in the way you say.

But the unbalance wobble can also be "pure", and in that case the stem makes a simple circular trajectory, like this:



When detecting the heavy side of the unbalanced top with the paint and brush technique, it works best when the top spins with this simple circular trajectory; 
the marks left by the brush will be consistently always in the same side of the stem.

If there is some "gravity induced precession" and/or "torque free precession" mixed with the unbalance wobble, the loops and irregularities coming from the superposition of these basic circular motions, (like in cycloids), are disturbances that can make the marks to appear in less or more wrong positions.
I never use the brush if I see that there is too much "gravity induced precession" or too much "inertial nutation", I try to spin the top as much vertical as I can, then, in case, I wait for the remnant of these two other wobbles to disappear before to mark the stem.   

[ta0]

You are describing the case of the unbalance wobble mixed with the gravity induced precession.

No, no gravity needed. The rolling cones are purely inertial, no gravity involved. 
If it's unbalanced, the stem will rotate around the principal axis while the principal axis rotates around the fixed direction of the total momentum.
But it will look almost like a circle if the unbalance is small, because the stem will be very close to the principal axis.

I believe the condition to get the apparent periodic stops of the wobble is that the stem is on the surface of the moving axoid or, in other words, that the top is first spun in the direction of the stem, or at least in at direction a the same distance than the stem from the principal axis. I'm assuming that the unbalance is not big enough to make the two transversal axes very different, because in that case Butikov's analysis doesn't work.

[Iacopo]

No, no gravity needed. The rolling cones are purely inertial, no gravity involved. 
If it's unbalanced, the stem will rotate around the principal axis while the principal axis rotates around the fixed direction of the total momentum.
But it will look almost like a circle if the unbalance is small, because the stem will be very close to the principal axis.

Sorry, I didn't read careful enough and I corrected too late.

I should have replied that you described the case of the unbalance wobble mixed with the torque free precession.
It is possible to have pure unbalance wobble too, in which case the stem trajectory is a circle; a bigger unbalance will produce a larger circle, at parity of spin speed.