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Author Topic: The slipping unbalanced top: a riddle and an experiment  (Read 308 times)

Iacopo

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The slipping unbalanced top: a riddle and an experiment
« on: June 27, 2020, 09:16:26 AM »

I made an experiment with the aim to try to shed some light about the behaviour of tops that spin while slipping, for the rotational axis not passing through the contact point, so that the contact point is dragged along the spinning surface making a circle around the rotational axis.

I wanted to collect some data which maybe could be helpful for to understand the tippe top reversal.
But I found something else.  Something which seems to me interesting enough to share with you.
----------------------------------------------------------------------------------------------------------------

Do you remember, long time ago I made a particular experiment about unbalance:

We know that large and low tops spin unbalanced staying leaned towards the heavy side.
And we know that, tall and narrow tops, at the countrary, spin unbalanced staying leaned towards the light side.

What about tops with intermediate proportions ?  Maybe they would not lean, so they would spin without wobbling even if unbalanced.
I made a top with an adjustable axis and I set it to match the intermediate proportions, (photo below), to see how it behaves.

It turned out that even with these intermediate proportions the top wobbled.
It wobbled staying leaned not towards the heavy side, nor towards the light side, but sideways:
the marks of the brush on the stem appeared about 90 degrees after the heavy side, (like in the photo below; the top was spun clockwise. If spun counterclockwise, the marks appear on the other side of the stem, which is in any case 90 degrees after the added weight, following the spin direction).
Why ?  This was a total mistery.
Even more puzzling, why the marks always appear about 90 degrees after the heavy side, and never before the heavy side?

The next experiment I am going to expose will give the answer, (or, at least, a key point for the answer).




 

« Last Edit: June 27, 2020, 10:46:29 AM by Iacopo »
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #1 on: June 27, 2020, 10:13:58 AM »

My tops generally do not spin slipping, in the sense that the axis of rotation passes through the tip in them, so the tip is not dragged along the spinning surface.

But if I add a lot of weight to a side to make them badly unbalanced, and I spin them fast enough, they will wobble intensely enough and in this case the tip will be dragged on the spinning surface.  Still the intense wobble is not a messy wobble, the tip makes a clear circular trajectory about the rotational axis in its new position.

I tried to mark the stem of these tops with the paint and brush, while slipping and wobbling intensely.
Unexpectedly, and interestingly, the marks are 90 degrees after the heavy side, (photo below, the tops were spun clockwise), the same like in tops with intermediate proportions;

unexpectedly, because these two tops have not intermediate proportions:
that at the left is a low center of mass top which normally spins unbalanced staying leaned towards the heavy side.
That at the right is tall and narrow enough to spin unbalanced staying leaned towards the light side instead.

The window of the intermediate proportions is very little, and making the tip longer or shorter by just 1 or 2 mm is enough for to exit from these intermediate proportions, in which case the top will behave normally, spinning unbalanced staying leaned towards the heavy or the light side.

But, when there is slipping and the tip is dragged on the spinning surface, the picture changes, and it seems that in this case whatever unbalanced top will spin staying leaned sideways.

The drag seems essential for the sideways tilt.  These tops, with the added weight, if spinning slowly enough so that they don't slip and stay with the tip in the spin axis, tend to receive the marks from the brush in their normal sides, the heavy and the light ones.
The tip must be dragged for to have sideways marks.
 
Next I will ask a little riddle, then I will expose the experiment.


 
« Last Edit: June 27, 2020, 10:50:48 AM by Iacopo »
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #2 on: June 27, 2020, 10:42:04 AM »

RIDDLE:

When the slipping unbalanced top spins dragging its tip on the spinning surface, it means that the rotational axis is not passing through the tip, as it should.

The question is:

Where is located the rotational axis in the slipping unbalanced top ?

A - The rotational axis becomes shifted towards the light side of the top.  This is because the centrifugal force of the added weight pulls the top in its direction while the top spins.

B - The rotational axis is shifted towards the added weight.  This happens because the top with the added weight is acting like a whole spinning system trying to spin about its shifted center of mass.

C - The rotational axis is shifted sideways.



The circumferences in the drawing are the top seen from above, their centers are the position of the tip, the darkened areas are the added weight.

The next experiment will show the answer.  I used the two tops in the last photo for the experiment, with the added weight as you see it, (static imbalance essentially).
« Last Edit: June 27, 2020, 10:58:17 AM by Iacopo »
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ta0

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #3 on: June 27, 2020, 11:57:31 AM »

Fantastic experiments and insights!  8)
Thank you very much for doing the experiments and sharing what you find.

My intuitive guess to the riddle would have been B but I have the feeling you are going to say I'm wrong.

That picture off the tops with such quantity of unbalancing mass is funny!
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Jeremy McCreary

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #4 on: June 27, 2020, 12:15:09 PM »

As usual, skillful experiments on a very complicated practical issue in top engineering.

Based on my own experiments and my foray into the extensive engineering literature on unbalace in real rotating machines, I think an engineer trying to understand your observations would also want at least rough estimates of the following:

1. Mass and eccentricity of each putty wad -- the latter being the mass of the wad times the distance of its own CM from the actual spin axis.

2. Spin rate, total mass, CM height, AMI, and central TMI at each of your data points. Or at least the TMI/AMI ratio.

3. The relative amplitudes of the wobbles when you get paint marks at phase angles of 0°, 90°, and 180° ahead of the heavy side. This could be eye-balled from the radius of the bounding circle around the tip or stem trace.

4. Are the observed phase angles and amplitudes in any way speed-dependent?

Let me explain (4). Suppose you give a test top a reasonably pure static unbalance with a putty wad of a particular mass and eccentricity, as you've done.

Now suppose you get a phase of 0° at speed w0. Is there also a speed w90 or w180 where you get a phase of 90° or 180° instead -- with no other change in your setup? And if so, what were the relative wobble amplitudes?

The answer to you riddle could vary with the parameter combo actually involved.
« Last Edit: June 27, 2020, 10:59:50 PM by Jeremy McCreary »
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #5 on: June 27, 2020, 02:49:20 PM »

My intuitive guess to the riddle would have been B but I have the feeling you are going to say I'm wrong.

That picture off the tops with such quantity of unbalancing mass is funny!

No, you are not wrong, but there is another possible answer..

I tried also with less unbalancing mass, there is less intense wobble, but the direction of the shift of the rotation axis is similar.
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #6 on: June 27, 2020, 03:12:10 PM »

The answer to you riddle could vary with the parameter combo actually involved.

Your questions made me remember that you see the unbalance wobble as related to whirling, which makes the picture complicated.
I don't think there is whirling here.  The set of rules seems different. 

Basically, the tops receive the marks sideways, about 90 degrees after the putty wad, everytime the tip is dragged and slips.
The two tops in the photo were used.
The speed and the amount of putty wad must be enough to make the tip slip, this is the condition for to have the marks at the side of the stem and not in the heavy or light side; that's all.

As for the riddle, A is always wrong, B and C can both be true.  But we will see this better, later.


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the Earl of Whirl

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #7 on: June 27, 2020, 04:11:51 PM »

Amazing!!!
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Jeremy McCreary

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #8 on: June 27, 2020, 11:05:22 PM »

I don't think there is whirling here.  The set of rules seems different.

In what way?
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #9 on: June 28, 2020, 04:38:11 AM »

I don't think there is whirling here.  The set of rules seems different.

In what way?

We already discussed this in the past, we have a different interpretation of the unbalance wobble.

Imagine a seesaw with two persons sitting on it;
the heavier person will make that side of the seesaw to tilt down:



Imagine that the seesaw could spin about a vertical axis at its center;
this would not change things a lot, and the seesaw would spin remaining tilted down where it is heavier.

In the same way, an unbalanced spinning top spins with its heavier side remaining tilted down, simply because it is heavier, nothing else.  There is no need for complicated explanations here.
This is valid for tops which are large and low.

Tops tall and narrow behave differently: they spin about the center of mass, so the light side in them protrude towards the outside.

Why the tall and narrow tops behave differently from the large and low ones ?

Imagine an unbalanced, (static unbalance), very low top with the center of mass at the contact point or very near to it, even a spindulum;
because of the unbalance, the tip and the center of mass do not stay on the same vertical axis.
If the tip doesn't slip and is in the rotational axis, the center of mass is necessarily pushed out from the rotational axis, and the heavier side of the top tilts down.
In tall and narrow tops instead the CM is far from the tip so it is free to stay where it wants, that is in the rotation axis.
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #10 on: June 28, 2020, 05:34:44 AM »

I spun the unbalanced tops on a glass pane and recorded a video from below the glass pane itself, so to see exactly how the tip moves and where is the rotational axis.

This is the first top, the one more tall/narrow.
Slow motion, 1/20 x:

https://www.youtube.com/watch?v=Frvy6_Anycw

As Ta0 predicted, the rotational axis results shifted towards the added putty wad.

The tip of this top is made of teflon and is relatively slippery.
« Last Edit: June 28, 2020, 07:48:18 AM by Iacopo »
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #11 on: June 28, 2020, 06:51:47 AM »


This leads to a possible explanation of the marks at 90° from the added weight.

In the drawing below, the blue vertical dotted line is the rotational axis.
The directions A,B,C and D are at straight angle one to each other.

The top is spinning unbalanced, (unbalance wobble), about the vertical rotational axis.

The added weight is towards A.
As we have seen in the video, the tip,(the orange ball), becomes pushed out of the rotational axis, towards the opposite direction, which is B.

The top spins clockwise, (blue arrows).
The tip is dragged along the spinning surface, tracing a circle, (red dotted circle), about the rotational axis.

The drag, with its dynamic sliding friction, causes a force on the top, (red arrow), slowing down the rotation of the top:
this braking force, (the red arrow), is directed towards D.

It seems like the braking force, pivoting on the CM, makes the stem to tilt towards the opposite side, C, (yellow arrow).

The marks of the brush appear on the side C, which is 90° after the heavy side, A.

This theory would explain why the sideways marks appear only when the tip is dragged on the spinning surface, (because this is the essential condition for to have a sideways force at the tip, the red arrow), and why the marks always appear about 90 degrees after the heavy side.



But, what about the unbalanced tops having intermediate proportions, which too receive the marks of the brush sideways, but apparently do not slip on the spinning surface ?

I suppose that, in imbalanced tops, since the CM and the tip are not on the same vertical axis, there is always a contention between them, the CM and the tip, for to stay in the rotational axis, each one pushing the other out of it.  I suppose that often there is not a clear winner and the rotational axis remains somewhere between them.
Maybe in the imbalanced tops with intermediate proportions, the imbalance is sufficient to push the tip very slightly out of the rotation axis:
apparently the top spins about the tip but reality could be that the tip is a tiny bit shifted from the rotational axis, and makes a tiny circle, (one tenth of mm, for example ?), so there could be a tiny braking sideways drag at the tip. 
The forces which normally make an unbalanced top to tilt towards the heavy or light side are much stronger, but, when they are balanced together, in the tops with intermediate proportions, they cancel each other out, so, at that point, the little sideways force becomes perceptible and the top receives the marks at the side of the stem, 90 degrees after the heavy side.
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ta0

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #12 on: June 28, 2020, 10:18:13 AM »

I like your explanation. It mostly makes sense to me. I need to think more about why proximity of the center of mass to the spinning point would stop it from spinning around B and make the heavy side stick out. But everything on your explanation seems plausible. Thanks for sharing your insights and making the effort to explain them. I'm extremely happy that you are on the forum.
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Iacopo

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #13 on: June 28, 2020, 04:41:22 PM »

I need to think more about why proximity of the center of mass to the spinning point would stop it from spinning around B and make the heavy side stick out. But everything on your explanation seems plausible. Thanks for sharing your insights and making the effort to explain them. I'm extremely happy that you are on the forum.

Thanks, Ta0, I too am happy here, if nothing bad happens I think I will continue to participate for a long time.

I thought to know it, but, thinking some more, I realize that I have not very clear ideas about it.
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Jeremy McCreary

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Re: The slipping unbalanced top: a riddle and an experiment
« Reply #14 on: June 28, 2020, 04:51:33 PM »

I'm extremely happy that you [Iacopo] are on the forum.

Same here. Iacopo: Your tops are fabulous, your experiments are always well-designed and carefully executed, and you're far and away our main purveyor of high-quality empirical data. And I love your diagrams and videos. Wouldn't want to see any of that end.

Please know that I don't mean to come across as antagonistic when I balk at certain interpretations of the empirical data. Maybe I'm overly careful, but I've seen too many of my own pet theories go down in flames with the arrival of new data or a new journal article. The one lesson I keep learning over and over again: Don't get too attached to any explanation. Tops are way more complicated -- and humbling -- than they look.

The questions I asked above were meant as a shopping list for the empirical data points that any dynamic theory of the behaviors observed would ultimately have to explain. My inclination to see some kinds of wobble as the top version of whirl is as much at risk here as any other theory.



So how does the paintbrush method work exactly? No question that it does work in cases of static unbalance, and quite well: The top's heavy side will almost always be at a phase angle of 0°, 90°, or 180° from the paint mark. You just have to add some test masses to figure out which phase angle applies to the particular top under test. Same with all of the equivalent methods I've adapted for LEGO tops.

But this unquestioned success raises some fundamental kinematic and dynamic questions. I think all of them deserve careful answers before results coming out of the paintbrush method can really be understood in dynamic terms. Again, just trying to be careful.

Q1. Why are the marks always on the same side of the stem in the first place? Why don't they just encircle the stem?

Q2. Why just these 3 phase angles in the vast majority of cases? Why are intermediate phases so rare in practice?

Q3. When you take the full range of real top designs into account, the phase angle actually observed seems to depend in a complicated way on CM height, central TMI/AMI ratio, and the spin and/or precession rate at the moment of testing. What's the general relationship? What other factors might be involved?

Regarding Q1, I don't think it's enough to say that the stem "leans" this way or that relative to the heavy side. After all, the gravitational torque acting on any real top generally produces precession, not tilt. And the marks often end up on the light side (180° phase angle), not the heavy.

Speaking strictly kinematically, we can probably all agree that when the paintbrush method is used, the spin rate about the symmetry axis is usually higher than the precession rate about the vertical. And their ratio is generally not a whole number. Yet, we don't see the paint encircle the stem. We see a short arc of paint accumulate on just one side instead. Consistently. And for a given top with static unbalance, that arc's nearly always centered at 0°, 90°, or 180° from what turns out to be the heavy side. Consistently.

You've gotta admit, that's pretty strange.

Regarding Q3, our tops together cover a pretty big chunk of the parameter space in CM height, central TMI/AMI ratio, and spin and precession rates. And perhaps for that very reason, I'm still far from answering the Q3 subquestions to my own satisfaction.
« Last Edit: June 28, 2020, 07:51:21 PM by Jeremy McCreary »
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