Intermittent wobbling in a spinning top

[Iacopo]

Spinning tops can wobble because of unbalance or nutation.
Often tops are not very free to nutate so they do not nutate or they nutate only for a short time.
So, usually, the wobble we see in spinning tops is due to unbalance.

Tops with the tip near the center of gravity are particular in this sense because they nutate easily and they can maintain the nutation even for minutes.  This allows to these tops to perform a movement of intermittent wobble.

This particular movement is due to the presence at the same time in the top spinning of both nutation and unbalance wobble.
nutation alone and  unbalance wobble alone make the stem of the top to trace a conical trajectory.
Nutation is always a bit faster than unbalance wobble, (at least in the kind of tops I make); when these two wobbles are both present at the same time, and have the same intensity,  in counterphase the two wobbles cancel each other out and for a short while the top spins apparently balanced.
By the time, being nutation faster, the two wobbles go in phase, and they become synergistic in making the top to wobble.
The alternation of the phases and counterphases make the top to wobble intermittently.

My tops with a recessed tip can perform this kind of motion.
The screws in the flywheel which are used for to fine tune the balance of the top, can also be used for to purposely unbalance the top, and kicking the stem of the spinning unbalanced top with a finger will make it to wobble intermittently.

Probably I never showed it before in my videos.
You can see it in my new video, below.

You will see a sequence with five different movements in succession:

The first is precession mixed with unbalance wobble.
The second is pure precession.
The third and the fourth are the ones about intermittent wobbling;
in the fourth the motion is slower and wider.
The fifth is the top spinning in sleeping position.

https://www.youtube.com/watch?v=LuAC2IEQ2z0

[Iacopo]

Long time ago, since I was impressed by the variety of the movements of spinning tops, I wanted to understand the reason of this variety.
I made many photos of my tops moving in different ways, from above, keeping the top in the darkness and directing a strong light to the upper extremity of the knurl.  The traces left in the photos are the trajectories of the upper extremity of the stem while the top spins. The tiny and bright reflection on the shining metallic hemispheric upper part of knurl of some of my tops produced the most clean traces.

So I discovered that precession, nutation and unbalance have all exactly the same trace shape, (a perfect circle).

The superposition of more than one of the three basic movements causes more complex shapes.
I show a few photos:


This is a photo of the top with the intermittent wobbling, (nutation and unbalance wobble).
The brighter parts of the trace are when the top slows down, in nearly vertical position, and for a short while it seems not to wobble anymore.  There is a bit of precession too in the photo, otherwise the bright cusps should overlap all in the same point, at the center.



This is again a case of intermittent wobbling.
While wobbling intermittently, here the top is also precessing. 



This is precession with unbalance wobble, (the same kind of movement as the first one showed in the video).



Precession with unbalance wobble again; here the top is spinning faster, and precessing more slowly, so the figure of the spring is less stretched. The hard, sharp spiked tip and the hard, smooth spinning surface contributed to the tidiness of the movements of the used tops. 

[ta0]

First of all, congratulations Iacopo on two pieces of art: the top and the video!      
The top, of course, is also an engineering masterpiece! 

Thanks a lot for showing all these movements. Some of the traces you get are so clean that look like idealized drawings from a book instead of experimental results! 

The fact that you can control the unbalance and therefore know for sure when unbalance is a factor, is very useful.

nutation alone and  unbalance wobble alone make the stem of the top to trace a conical trajectory.

I'm not exactly sure what you mean here. Precession will always be present, unless the top is sleeping or supported at the center of mass. Nutation is an up and down oscillation around the angle of stable precession and will make loops when combined with precession.

I don't know how to differentiate between the traces left by only nutation or only unbalance (always in the presence of precession). It's something to think about.

A behavior reminiscent to the one you show with the combined unbalance and nutation, I sometimes see on dying tops and it has always puzzled me. The top starts to fall but then momentarily straightens, to start to fall again, and can repeat this several times. But this behavior is much more complicated as it also depends on the rolling to the tip on the floor.

[Iacopo]

First of all, congratulations Iacopo on two pieces of art: the top and the video!      
The top, of course, is also an engineering masterpiece! 

Thank you very much.  Someone In China will be happy to receive this top. 
I am having some fun making the videos a bit differently now, with Chroma Key, masks and layers. 

nutation alone and  unbalance wobble alone make the stem of the top to trace a conical trajectory.

I'm not exactly sure what you mean here. Precession will always be present, unless the top is sleeping or supported at the center of mass. Nutation is an up and down oscillation around the angle of stable precession and will make loops when combined with precession.

In the video below there is a sample of what I call "pure nutation".
I kick the stem of a balanced top while it is spinning;
if I kick it too hard, or too weakly, precession usually appears together with nutation.
But if I kick it well, only nutation appears.
In the video I also measured the speed of the spin and that of the wobble;
the speed of the wobble is very fast, about 1.7 times that of the spin, which is compatible with the wobble being nutation.   

https://www.youtube.com/watch?v=JqoZkuvzqAE

I don't know how to differentiate between the traces left by only nutation or only unbalance (always in the presence of precession). It's something to think about.

I would not be able to distinguish them with ease, at least in my tops. Unbalance wobble becomes wider by the time, while nutation at the countrary becomes weaker by the time, so with some accurate measurements and knowing the direction of the precession, maybe it can be known, but I never tried.

A behavior reminiscent to the one you show with the combined unbalance and nutation, I sometimes see on dying tops and it has always puzzled me. The top starts to fall but then momentarily straightens, to start to fall again, and can repeat this several times. But this behavior is much more complicated as it also depends on the rolling to the tip on the floor.

I saw it in some videos but I don't know the reason exactly. I am not familiar with higher CM tops.
I suppose that, in a similar way like in my tops, there is an interaction between two wobbles, but I am not sure which wobbles;
in higher CM tops, precession becomes fast at slow spin speed, so the precession itself could be one of the two wobbles.

[Jeremy McCreary]

Agree with ta0 about both top and video. I think the top is one of your best visually.

I also observe intermittent wobble in LEGO tops -- more often with large relative CM height, but also with small. (Mostly at speeds near critical in the latter case.) As ta0 implied, a larger tip radius of curvature strengthens self-righting tendency. And that appears to be an important factor as well.

In this context, "relative CM height" = h = H / R = (CM-contact distance) / (max radius). In your tops, h is typically <<1 by design. In a typical peg top, h ~ 1. And I've seen intermittent wobble in tops with h > 1.

In an ideal symmetric top without tip travel, precession rate, nutation rate, TMI at the tip, and critical speed all depend strongly on this key proportion --  especially the last. Very handy concept when trying to understand behaviors like this one spanning a wide range of sizes and shapes.

The exact souces of wobble may not be the most important governing factors. The main souces of wobble in my somewhat flexible LEGO tops are...
1. Slight misalignment of rotor symmetry axis and CM-contact axis -- e.g., due to a damaged tip or a central axle imperceptibly out of true.
2. Strucural oscillations excited during spin-up or on release.

In contrast, your tops are effectively rigid, and the experiments you described perturb only static balance. LEGO precision and dimensional stability are such that static unbalance is insignificant when all parts are fully seated in an arrangement with rotational or mirror symmetry.

[ta0]

On the last video, the stem of the top seems to be doing a circular motion after you hit it. Is the axis of the circle vertical or at an angle to the vertical?
Before you kick it with the fingers, it appears to be precessing very slow clockwise. I am guessing that you not only kicked it in the vertical plane but horizontally, initially starting it a faster than normal precession.
If the period of nutation equals the period of precession, it can perform a slanted circle.

By the way, are you sure that you are not measuring with the tachometer the stem moving on the front and on the back of the circle? In other words, could you be measuring 2x the "wobbling" rate?

[Iacopo]

On the last video, the stem of the top seems to be doing a circular motion after you hit it. Is the axis of the circle vertical or at an angle to the vertical?
Before you kick it with the fingers, it appears to be precessing very slow clockwise. I am guessing that you not only kicked it in the vertical plane but horizontally, initially starting it a faster than normal precession.
If the period of nutation equals the period of precession, it can perform a slanted circle.

By the way, are you sure that you are not measuring with the tachometer the stem moving on the front and on the back of the circle? In other words, could you be measuring 2x the "wobbling" rate?

In the video, the nutating top had the axis of the circle vertical, (or very near to the vertical).
The trace is a perfect circle when seen from above.

I am sure that I was not measuring 2x the wobbling rate.  I pointed the tachometer towards a side of the trajectory of the stem, (the red light in the video), where the stem passes only once every one turn of the wobbling.
Also, when the top spins more slowly, at 100-200 RPM or less, moving in this way, it is easy to see that the wobble is faster than the spin, even simply looking at it.

Yes, the top was precessing when I kicked its stem.
For to hope to have the pure nutation I have to kick the stem, (red arrow in the drawing), in the direction of the precession, (blue arrow), like trying to accelerate the precession.
With the kick the stem starts a new trajectory, circular, towards the inner side of the previous precession trajectory; this is the fast wobble, (the nutation).  At the same time, the center of this new circular trajectory continues to precess, (green arrow), around the center, (X), but the diameter of the new precession trajectory is littler now.

The harder I kick the stem with the finger, the larger the diameter of the nutation circle, and, at the same time, the littler that of the precession.  If I kick the stem hard enough, the diameter of the nutation circle will be the same of that of the previous precession, while that of the precession will be reduced to zero.  This produces the pure nutation. 
If I kick the stem too hard, the center of the nutation circle will overstep the x point and there will be some precession again, together with a wide nutation circle. 

 

[Iacopo]

LEGO precision and dimensional stability are such that static unbalance is insignificant when all parts are fully seated in an arrangement with rotational or mirror symmetry.

If so, LEGO pieces are very precise, since a few hundreds mm are sufficient to cause some unbalance, as for what I see in my tops.

[Iacopo]

I also observe intermittent wobble in LEGO tops -- more often with large relative CM height, but also with small. (Mostly at speeds near critical in the latter case.)

I too saw intermittent wobble in small diameter/CM height tops, in videos.
But I am not sure how it happens exactly, I don't have such tops and I haven't observed them firsthand.

I see that tops with an external tip tend to refuse to nutate; so maybe in those cases the intermittent wobble could be due to the superposition of two other wobbles, unbalance, and precession, (which is fast near critical speed), and not nutation.

[James]

I have also noticed this movement in my recessed spinning tops - usually at the very end of the spin. Fascinating!

[Jeremy McCreary]

LEGO precision and dimensional stability are such that static unbalance is insignificant when all parts are fully seated in an arrangement with rotational or mirror symmetry.

If so, LEGO pieces are very precise, since a few hundreds mm are sufficient to cause some unbalance, as for what I see in my tops.

Tolerances are said to be about 1 part in 10,000.

Solid ABS plastic is about 5% denser than water. Since most parts include holes or hollows, their effective density is significantly less. Your rotors, on the other hand, are of dense solid metal more than 5 times denser than water.

For a given dimensional error at a given radius, this density difference would result in a larger amplitude of whirl (static unbalance wobble) in your case than mine.

[Iacopo]

I have also noticed this movement in my recessed spinning tops - usually at the very end of the spin. Fascinating!

I agree, fascinating, beautiful and hypnotic. 
One of the movements I like more in my tops.  I never showed it before maybe because I was afraid that this intermittent wobble may make the top to seem defective, but it's not the case.

For to have the most spectacular and long lasting intermittent wobble use a top with a deeply recessed tip, unbalance it, (I removed one of the three screws from the flywheel for the aim, you can add some modeling clay at one side of it), and kick the stem of the top with a finger while the top is spinning. 

[Iacopo]

For a given dimensional error at a given radius, this density difference would result in a larger amplitude of whirl (static unbalance wobble) in your case than mine.

In fact I noticed that littler tops require more precision of the tip centering.

Tolerances to be about 1 part in 10,000 is much better than I thought.  Recently I saw calipers made of plastic, (I was surprised, can a plastic caliper be reliable ?), maybe they are made of the same plastic as Legos.

[ta0]

We discussed previously the simulator at Wolfram Alpha. I tried it for a top with low center of mass that it pushed in the direction of precession at 1.7 times the spin value:



and got:



It does a circle but very slanted. If I decrease the initial precession push to about 1/3 of the spin rate (30 rad/sec), it does an almost horizontal circle (also increased simulation time to 2 seconds to see more orbits):



But if I also increase the center of mass height from 1 mm to 4 cm it does this:



You should enter the values of your top and see if the experimental results correspond to the simulation.

[ta0]

Nutation is created when the top precesses at a different angle than the stable angle for a given spin, so that it oscillates around that stable angle. But if the center of mass is very low, I am guessing that the nutation force will be small as the change of torque with angle with the vertical will be small. That could explain that you see a circle with no vertical oscillations (e.g. nutations).