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[Iacopo]

At first I thought to post only a few pictures about this issue, but when I began taking the pictures you see in the video I have been amazed at the variety of the possible movements a spinning top can have.  It is worth a video, I thought.
I analyzed the shapes of the trajectories and I have tried to offer some basic explanations about these shapes.

As I say from time to time, I have not studied physics, nor mathematics, so what I try to explain in the video is only by my personal intuition, and it is written in the poor language of an uneducated person.  I hope there are not big errors in my explanation.

 

[Jeremy McCreary]

From a visual standpoint, absolutely stunning, Iacopo! From a mathematical standpoint, the simpler tracings (e.g., just precession + nutation) look a lot like trochoid curves (the kind made by spirographs) when viewed from above -- and probably not by accident. From a physical standpoint, these images deserve careful study.

[ta0]

Truly beautiful work!   

It is going to take me a while to digest your results.

I would like to see images for an unbalanced top like the ones you used for synchronicity, but with no nutation, in which the axle in the direction of the extra weight is colored one color. I possible, If would like to see if for the top with variable center of mass height.

[Iacopo]

the simpler tracings (e.g., just precession + nutation) look a lot like trochoid curves (the kind made by spirographs) when viewed from above -- and probably not by accident.

I didn't know this curve has a name..  but I thought the same thing, it seems the result of the interlapsing of the two basic movements.
The physics of it I don't understandy with clarity, but the geometry seems evident.  There is a name also for the kind of trochoid, it can be curtate or prolate or common...  I described this as a less or more stretched spring..
The complex movements with all three basic movements seem random but they aren't and an order is recognizable if observed carefully; in their case the spring is stretched not along a circumference but along the spirals of another spring, which makes the resulting trajectory complex.

[Iacopo]

I would like to see images for an unbalanced top like the ones you used for synchronicity, but with no nutation, in which the axle in the direction of the extra weight is colored one color. I possible, If would like to see if for the top with variable center of mass height.

I haven't thought to this but it is interesting..  I will make some more pictures.

[Iacopo]

Nutation can last for a minute or more time, in very oblate tops, but in prolate tops it disappears rapidly after the stem has been kicked with a finger: it can last for maybe one, two seconds or something so.  So prolate tops are not suitable for observing nutation movements and their relationships with the other movements of the top.

If the contact point of a top is large enough, for example if it has a weared ball tip, it can start wobbling spontaneously at a certain speed.
Then this top can spin again perfectly vertical at slower speed. 
This wobbling is not unbalance because the marks of a brush left on its stem are randomly distributed along the stem.
The trajectory of this kind of wobbling is a circle.
If I kick the stem of this top while it is wobbling, the trajectory continues to ba a circle, maybe of a different width, but still a circle;
so I think that the two movements have the same nature, and I called both of them "nutation" in the video.

Nutation mixed with unbalance can origin what in the video I called a "pulsar":
here you can see a top moving in that way:


 

[Russpin]

From the theory of a heavy symmetric top with the contact point fixed, this web page shows the effect of top parameters on top motion. Look about half way down the page.

http://www.entropy.energy/scholar/node/heavy-symmetrical-top

[Iacopo]

From the theory of a heavy symmetric top with the contact point fixed, this web page shows the effect of top parameters on top motion. Look about half way down the page.

http://www.entropy.energy/scholar/node/heavy-symmetrical-top

Thank you, Russpin; I like the ball with the trajectories, but I don't understand the meaning of the letters:  "I" should be the two moments of inertia, but omega and theta I don't know.  The math parts are over my capabilities.

[Russpin]

Thank you, Russpin; I like the ball with the trajectories, but I don't understand the meaning of the letters:  "I" should be the two moments of inertia, but omega and theta I don't know.  The math parts are over my capabilities.

Theta is the angle the axis of symmetry of the top makes with vertical. Theta0 is the initial theta (at time zero).
Omega is the rate the axis of symmetry of the top rotates about the vertical. Omega0 is the initial Omega (at time zero)
w3 is the spin rate about the axis of symmetry. This is a constant. (no fricition)
I1 is the transverse moment of inertia. Note this should be I1 prime ( the transverse moment of inertia about the pivot point)
I3 is the moment of inertia about the symmetry axis

[Iacopo]

Some time ago I have made this top:

It has a movable axis by which I can adjust the height of center of mass.

Unbalanced prolate tops spin staying leaned towards the light side, while unbalanced oblate tops instead spin staying leaned towards the heavy side.
 
I used this top to observe what it happens with a in between distribution of weight;
the result is that an unbalanced top with a transition shape spins staying leaned towards the sector of the top 90 degrees after the heavy side, following the direction of the spin.  The transition extension is of a few millimeters along the vertical axis, during which the direction of leaning of the heavy side of the top changes gradually from position 0 degrees to 180 degrees.
Rotational speed has some effect in the direction of leaning, higher speed favouring the heavy side of the top staying leaned a bit more towards the heavy side.

The following pictures document the effect of different heights of the center of mass on direction of leaning:
the upper part of the stem of the top has been painted half white and half yellow;
a strong light is directed at its side, so that from the color of the luminous trail of the stem it is possible to know the position of the top.

The yellow side is the heavy side of the top.

The light direction in the pictures is from above.

The position of the central part of the yellow sector of the trail indicates the direction of leaning of the top when the heavy side of the top is directed towards the light, the upper part of the picture.
 
In each picture I have indicated the height of center of mass, (mm).

The top has been spun in clockwise direction.

In pictures mm 12, 13 and 14 the yellow sector is above, towards the direction of light: this means that the top is leaned towards its heavy side.
In the following pictures it can be seen the gradual change in the direction of leaning.
 
The effect of speed is not very strong, but you can see some difference in the position of the yellow side in the two pictures at mm 15 and in the two at mm 16: this is due to different rotational speed of the top.
   

[Russpin]

Very nice experiment Iacopo. I'm trying to model an unbalanced top. When a mass is added to a static and dynamic balanced symmetric top the top becomes both static and dynamically unbalanced. The cg shift causes the cg to be off the axis of symmetry an the two transverse moments of inertia become different. 

[Iacopo]

There are two things I find interesting and apparently bizarre in this experiment:

- I expected the top to spin temporarily in vertical position in the central transition shape/speed phase.
This doesn't happen.  The top always wobbles also during the transition phase, and the direction of leaning changes gradually but always in the same direction, relatively to the direction of spinning.
The shift of direction of leaning happens always in counterclockwise direction if the top has been spun in clockwise direction, while the top from oblate becomes prolate.
If the top has been spun in counterclockwise direction, the shift of direction of leaning happens in clockwise direction, while the top from oblate becomes prolate. 
I have tried in various different ways and I always had this behaviour, but the reasons of this I can't understand them.

- In the transition phase, at higher speed the top leans a bit more towards the heavy side, and at slower speed a bit more towards the heavy side.
Alan too had this same result in his top.
But also in this case I find difficult to have a clue about the reasons of this, and also I find this a bit counterintuitive.

[Iacopo]

This is a schema about the transition phase in the same top:
in my last test the transition happened at a bit lower height of center of mass, this I think is due to the fact I have added the modeling clay to unbalance the top above the flywheel, while instead in the test the schema refers to, the clay was added to the side of the flywheel.

[Turegano]

exelente  iformation    good  job