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[Russpin]

The fact that the spin rate and the precession rate are equivalent cancel out whatever gyro effect.  Each single mass point of a spinning unbalanced top, is traveling at a uniform and constant speed, (let's assume an ideal top without any frictions), being their trajectories simple circles around the vertical axis.
There are not the accelerations and decelerations of the single mass points, typical of the gyro effect, without which it is impossible to trigger a gyro effect.

I don't understand what you are saying. The motion of an unbalanced top is very complex and hard to describe in just words. It's much better to show it with quantitative plots. This is a plot from my unbalanced top simulation. An unbalancing point mass of 0.1 percent of the tops mass is placed on the rim.
As can be seen the motion of points on the top are not simple circles around the vertical and they are undergoing acceleration and deceleration and they are subject to gyroscopic effects.

[Jeremy McCreary]

Ah, but that's where we disagree. For starters, the most common whirl modes don't involve deformation, either by bending or twisting.

Can you give an example of this?

Sure! In a rotating machine with an unbalanced rotor on a shaft much stiffer than its bearings, the first 2 whirl modes (in order of ascending critical speed) are called "rigid body" modes.

Basically, the critical speeds in these modes are low enough that it's easier for the very stiff shaft to orbit inside the less-than-perfect bearings than to bend between them.

Whirl amplitudes max out at the resonances, which occur whenever shaft and critical speeds match. However, rigid-body whirl can still be observed between critical speeds. IMO, that's ultimately the basis for the paintbrush method..

A truly rigid spinning top in some ways resembles an infinitely stiff overhung (cantilevered) rotor with much softer bearings. In machines like this, gyroscopic effects arising from the rotor modulate the critical speeds to some extent, but the whirling forced by unbalance still dominates the dynamics. The critical speed shifts depend strongly on the  AMI/TMI (I₃ / I₁) ratio.

You'll see what I mean about "rigid body modes" in the figures in any of these free online PDFs...

Swanson, E., et al., 2005, A Practical Review of Rotating Machinery Critical Speeds and Modes

Nelson, F., 2007, Rotor Dynamics without equations

Freese, T.D., and Grazier, P.E., 2004, Balance this!

If you look at only one, I recommend Swanson. For something more mathematical, and with discussions of machines closer to spinning tops, try

Nelson, H.D., and Talbert, P.B., 2003, Rotordynamic considerations (chapter 10 of unknown book)

IMO, the top's lack of a 2nd bearing at the stem doesn't eliminate the possibility of whirl and may even encourage it. The single-bearing case just allows gyroscopic effects to play a much more important -- though not necessarily dominant -- role in the motion of the stem when unbalance is present.

[Russpin]

Sure! In a rotating machine with an unbalanced rotor on a shaft much stiffer than its bearings, the first 2 whirl modes (in order of ascending critical speed) are called "rigid body" modes.

Basically, the critical speeds in these modes are low enough that it's easier for the very stiff shaft to orbit inside the less-than-perfect bearings than to bend between them. Whirl amplitudes max out at the resonances, which occur whenever shaft and critical speeds match. However, rigid-body whirl can still be observed between critical speed. That's ultimately the basis for the paintbrush method..

In these so called "rigid body modes" there is a vibration mode with the bearings. this does not happen with tops. My unbalanced top simulation displays the paintbrush behavior without bearing vibration modes.

[Jeremy McCreary]

In these so called "rigid body modes" there is a vibration mode with the bearings. this does not happen with tops.

Sure it does -- at least in real tops. A top has one friction bearing at its tip. It's not the usual revolute bearing found in rotating machinery, but it still constrains lateral motion of the tip relative to the floor. The fact that the constraint has its limits is OK, because machine bearings can be soft, too.

Unbalanced rotor + stiff shaft + soft bearings is the recipe for rigid body whirl. This also applies to the more top-like case of an "overhung" rotor with both shaft bearings on the same side of the rotor.

When I purposely unbalance one of my tops and twirl it on a surface that amplifies sound, I can often hear the tip scrubbing (not rolling) back and forth across the surface as the top whirls. I can also see this action in unbalanced tops with very low critical speeds. The scrubbing goes away when the unbalance is removed.

My unbalanced top simulation displays the paintbrush behavior without bearing vibration modes.

Perhaps your simulation is generating whirl automatically. The governing Newtonian equations are given in the last reference I listed.

Capturing paintbrush behavior in a simulation is quite a feat! But tell me: If gyroscopic precession and nutation are the only motions involved, how do you think the requisite synchronization of these motions comes about? Ditto for the 0°, 90°, and 180° phase angles observed between the paint marks and the azimuth of the unbalance on the rotor?

Bear in mind that the pure precession rate is much slower than the pure spin rate in most real tops most of the time -- even unbalanced ones. I know of no gyroscopic effect capable of forcing one of these rates to be an integral multiple of the other. Ditto for synchronization of the precession and nutation rates.

All of these observations are easily explained with rigid-body whirl.

[Iacopo]

As can be seen the motion of points on the top are not simple circles around the vertical and they are undergoing acceleration and deceleration and they are subject to gyroscopic effects.

I don't know the reason.  My sensation is that in your plot there are at least two movements mixed together that make the picture complex.

What I think could be uncorrect, but what I see with my eyes in real tops shouldn't be wrong:
I find nearly impossible to spin a top without having at least a bit of precession and nutation at the start.
But by the time they both disappear.  Apparently (in my very low CM tops) they seem to disappear completely.
If the top is well balanced at that point it will spin in perfectly vertical position, in sleeping position.
If then there is unbalance, instead of the sleeping top I see that the trajectory of the stem is a circle:
this is a picture of it; I obtained it in a dark room, taking the picture from above the top, directing a strong light at the side of the upper part of the stem, which is made of polished metal; the reflection on it left the circular trail on the picture below:   



It can be seen that it is a clean circle.  Then, the top moves staying leaned always towards the same side of itself, so it is clear that all its mass points are travelling in simple circles around the rotation axis.

If then there are also some micro oscillations or some micro movements superimposed which I can't see, this I don't know.
But, in case, I don't think that this would make it impossible to reason about the main causes of these movements.
Plots are interesting for showing something but also it would be interesting to understand them.

 

[Jeremy McCreary]

The motion of an unbalanced top is in general a combination of both precession and nutation.

I believe there is not gyroscopic effect in the pure motion of an unbalanced top, when spin rate and precession rate are of the same magnitude.  This is a motion of a different nature, I would have liked a word different from "precession" to distinguish it from the common "precession". 

"Whirling" used to indicate wobbling from unbalance, I don't know if it could be the right term: I too don't understand how it can whirl without deformation, or without loose bearings where the axis of the top would vibrate, in some way.   

If you're saying that a prominent component of the motion of a stiff unbalanced top is fundamentally non-gyroscopic in nature, I totally agree. In my experience, this component has all the characteristics of rigid-body whirl. The "loose bearing" part is critical to the extension of whirl to unbalanced tops.

[Russpin]

All of these observations are easily explained with rigid-body whirl.

I would love to see that!

[Jeremy McCreary]

All of these observations are easily explained with rigid-body whirl.

I would love to see that!

Check the references I gave. It's all there.

[Russpin]

Check the references I gave. It's all there.

No it does not I have checked. You say it's easily explained why not do it yourself?

[Jeremy McCreary]

Check the references I gave. It's all there.

No it does not I have checked. You say it's easily explained why not do it yourself?

Honestly, I've been trying, Russpin. Guess we'll have to agree to disagree about the role of whirl -- or at least a whirl-like component of motion -- in unbalanced tops.

[Russpin]

Honestly, I've been trying, Russpin. Guess we'll have to agree to disagree about the role of whirl -- or at least a whirl-like component of motion -- in unbalanced tops.

Fair enough Jeremy.

[Iacopo]

If you're saying that a prominent component of the motion of a stiff unbalanced top is fundamentally non-gyroscopic in nature, I totally agree. In my experience, this component has all the characteristics of rigid-body whirl. The "loose bearing" part is critical to the extension of whirl to unbalanced tops.

I believe that there is not gyroscopic effect related to the component "unbalance wobbling".
There is gyroscopic effect in a spinning unbalanced top, but it is related to some other component of the whole motion, which is the component "precession", and/or the component "nutation".  If precession and nutation are absent, still there should be a sort of "micro-precession", (I don't know how to call it), which keeps the axis of rotation in vertical position, so, at least a very tiny amount of gyroscopic effect is always present in the whole motion.

I think you are into something, when you say "whirling". 
I remember when Aerobie described one of his tops starting to wobble spontaneously at a certain speed, and then ceasing to wobble at a lower speed.  I saw the same in my tops, (when the tip is very weared).
This wobbling is nutation, (at least that I saw in my tops); I can obtain nutation kicking the stem of the top while it is spinning, but in these cases the nutation started by itself, after minutes of the top spinning in sleeping position.
I suspect that whirling could have something to do with all this, triggering nutation through the tip, (in some way a bearing, as you say), and only at a certain speed, because of a resonance effect.

But unbalance seems something different to me.
When you say that the tip scrubs in an unbalanced top, (I see the same in my unbalanced tops), my explanation for this is that the center of mass is not vertically aligned with the tip, (which is what it makes the top unbalanced), so, especially when the top spins fast, the center of mass wanting to stay in the axis of rotation pushes the tip out of the axis of rotation, (since they are not aligned), and for this reason it scrubs on the base.

The component "unbalance wobbling" is the simplest of the components of motion in a spinning top, because it doesn't involve a gyroscopic effect;
think to a perfectly balanced top (with the tip in the axis having the largest, or littlest, moment of inertia, passing through the center of mass, which is what it makes a top balanced) but a distorted shape and a bent stem.
Even in sleeping position, such a top would still seem to precess, but it would not be a real precession, it's only a visual effect.

It's not very different in the unbalance motion: it's not a real precession (involving gyroscopic effect), but just a visual effect.
     

[ta0]

I am behind in reading and processing this topic (I am a slow thinker  ).

I haven't said this yet, but the oscillation in the gyroscope has a spin rate that is twice the precession rate. To me it seems to have the same nature of the nutation of tops. 

That is a very interesting observation. As pointed out by Butikov, the wobble of a (floating) plate is twice as fast as its spin rate (and in the opposite direction).
Or in more technical terms, a very oblate top, spinning around an axle slightly shifted from its principal axis, will precess (inertially) at double the frequency it spins (and in retrograde direction.)
Butikov mentions that Feynman recalled this backwards in his autobiographic book: "Surely You are Joking, Mr. Feynman!" (great read!). According to Feynman, observing the wobble of a plate thrown in a cafeteria put him on the path to the Nobel Prize (in electrodynamics)!

[Russpin]

Or in more technical terms, a very oblate top, spinning around an axle slightly shifted from its principal axis, will precess (inertially) at double the frequency it spins (and in retrograde direction.)

Keep in mind that this top would have to be floating in space for this to be true.

[ta0]

Or in more technical terms, a very oblate top, spinning around an axle slightly shifted from its principal axis, will precess (inertially) at double the frequency it spins (and in retrograde direction.)

Keep in mind that this top would have to be floating in space for this to be true.

That's (virtually) the case for Iacopo's gyro.