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[Iacopo]

(I think strictly speaking it's a nutation superposed to a precession, but they cannot be differentiated as they are of similar amplitude and frequency).

That faster wobble in simple circular motion should be pure nutation.
Nutation and precession at the same time in this spindulum look like this:

https://youtu.be/ofaDbADvqVM

By the way, in spite of the precession, I still could measure in the video the speeds of the nutation, (480 RPM), and of the spin, (373 RPM), and if I plot these data in the graphic, the resultant dot is very near to the nutation curve, telling that the presence of the precession doesn't seem to disturb significantly the nutation speed/spin speed ratio.

[Iacopo]

I think that when the spindulum is spinning fast, the regular equations of a fast top apply.

I think I agree.  As I already noticed with nutation, also with precession, the theoretical speeds seem not far from the measured ones, but only at high speed.

If I apply the formula:

Precession speed, (radians/sec) = [mass, (Kg) x gravity acceleration, (Kg m²) x CM-tip distance, (m)] : [axial moment of inertia, (Kg m²) x spin speed, (radians/sec)]

I get:

(0.036 x 9.8 x 0.003) : (0.0000127 x 30.7) = 2.7
30.7 x 9.55 = 293 RPM, spin speed
2.7 x 9.55 = 25.78 RPM, precession speed

which is consistent enough with the measured data in the graphic.

But at slow spin speed I get:

(0.036 x 9.8 x 0.003) : (0.0000127 x 4.2) = 19.8
4.2 x 9.55 = 40.1 RPM, spin speed
19.8 x 9.55 = 189 RPM, precession speed

which is quite far from the graphic data.

My idea is that there is something alterating the speeds of nutation and precession of the spindulums when the spin speed is slow.

This something could be the circular oscillation.
At slow spin speed, the circular oscilllation would still be present and deeply influence the speeds of the nutation and of the precession.
At the increasing of the spin speed, the circular oscillation would gradually fade away.
This would explain why the two curves of the nutation and of the precession in the graphic have the same dot of origin, and why this dot of origin also represents the natural oscillation frequency of the spindulum.


 

[ta0]

Nutation and precession at the same time in this spindulum look like this:

There the nutation is several times faster than the precession, so they can be clearly differentiated.

I was going to post some videos of what I did, but I think I should redo them because there are some difference to what you got. Your experiments were much more thorough, so I trust more your results.

[Iacopo]

... it will nutate strongly (I think strictly speaking it's a nutation superposed to a precession, but they cannot be differentiated as they are of similar amplitude and frequency).

I don't think that what I observed is precession, (in the sense of torque induced precession), superposed to the nutation, (torque free), for the following two reasons:

• Nutation and precession change their own speed in opposite ways, (one accelerates while the other decelerates, as the spindulum slows down), so they can have the same speed only for a short while; their superposition can't show as a simple and stable circular motion of the stem, like I could observe in the tested spindulum, at all the various speeds, from slow to fast. 
• In spindulums nutation and precession have always opposite directions, so their superposition can't produce a circular motion of the stem in any case, even when the two speeds and amplitudes are the same.

I believe instead that it is the circular oscillation, that is superposed and makes the nutation faster, (at slow spin speed).

[Iacopo]

...there are some difference to what you got.

Maybe you spun the Dynamical Top with an elliptical or back and forth oscillation instead of circular... ? This test works with the circular oscillation.

[ta0]

...there are some difference to what you got.

Maybe you spun the Dynamical Top with an elliptical or back and forth oscillation instead of circular... ? This test works with the circular oscillation.

No, I think that was not the problem. But I was using the handheld phone camera and not the best experimental setup.

Here is a video showing the back and forth and the circular oscillation, with (almost) no spin:

https://www.youtube.com/watch?v=1OkbY-hr5qU

[Iacopo]

Here is a video showing the back and forth and the circular oscillation, with (almost) no spin:

It is a nice top, it seems accurate and moves very smoothly. It must be pleasant to experiment with it.

[Iacopo]

I added to the graphic the curve and the line of the theoretical precession and nutation.

I calculated the theoretical nutation using the formula:

Nutation speed = spin speed x (axial moment of inertia : transverse moment of inertia at the tip).



At high spin speed the spindulum has approximately the precession and nutation speeds predicted by the formulas.
At low spin speed, precession and nutation seem to be gradually replaced by circular oscillation, until, at zero spin speed, only circular oscillation remains.
 

[ta0]

Beautiful work, Iacopo. You would have been a fine scientist. I'm always amazed that you had no formal scientific training.

This graph shows the rotational speed (of the axle direction around the vertical axis) vs. spin (of the spindulum body around it's own axis). A negative value of the spin means that it's spinning in the opposite direction as the rotation.



I extended the screws to interfere less with the support, so the oscillating period is larger than on my initial experiment: 0.83 sec, what corresponds to a non-spin rotation frequency of 72.3 RPM. As Iacopo found out, spins in the same direction accelerate the rotation and in the opposite direction slow it down.

[ta0]

https://www.youtube.com/watch?v=53hOeMIfPpQ

[Iacopo]

"Oscillation with nutation" and "oscillation with precession" here seem to be part of the same curve...

[ta0]

"Oscillation with nutation" and "oscillation with precession" here seem to be part of the same curve...

Yes, and same for your curves:

[Iacopo]

https://www.youtube.com/watch?v=53hOeMIfPpQ

The video at the right has an elliptical oscillation, which is interesting because it can be seen that the spin makes the elliptical orbit itself to spin. 

It is mainly the orbit spinning that increases or decreases the speed of the oscillation. The spinning of the orbit is invisible if the orbit is circular, (we only see the resultant change of the oscillation speed in this case), but it becomes visible if the orbit is elliptical, or if the oscillation is back and forth, in which case it is possible, with some patience, to measure the speed of the orbit, (reply #44).

     

[Iacopo]

Yes, and same for your curves:

I see ... and for all the spindulums, probably.
I wonder how would appear the data of a normal top, plotted in a graph in this way.

[ortwin]

https://www.youtube.com/watch?v=53hOeMIfPpQ

Oh, wow! Especially that rotation with backward spin looks sooo cool!
The first few times you look at it, it is hard to believe that one is looking  at the motion of only one single solid body!
How do you initialize that motion? Is that hard? Is there a video about that? Does it look as cool if viewed in person?