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[Jeremy McCreary]

You didn't mention the TMI but I am not too surprised if it doesn't influence the precession speed.

Didn't know what to make of the TMI's absence from the =slow= precession rate formula above when I first saw it, at least not physically. After looking hard at the faster-than-usual precession of all the high-CM tops we collected here, seems like TMI ought to be involved in precession rate somehow.

Well, that turns out to be case, though only indirectly in the =slow= precession regime we usually observe. At the very least, precession rate covaries with TMI via tip-CM distance.

As you may well know, for a given set of initial conditions (especially but not limited to release speed and tilt), steady precession can proceed at 2 very different rates -- a "slow" one usually a good bit slower than the spin rate, and a "fast" one usually much closer to the spin rate. But the initial conditions leading to the fast rate are very hard to achieve in practice, so we rarely if ever see it in our tops. At least, that's the scientific lore on the subject.

TMI does appear explicitly in the fast rate formula, but curiously, not in the slow one generally applicable to our tops. Per the parallel axis theorem, however, changing tip-CM distance necessarily changes the TMI about the tip. That's the indirect connection between precession rate and TMI: Both increase with CM-tip distance -- especially the TMI.

I haven't seen a good physical explanation of the TMI's absence from the slow rate formula. Perhaps ta0 has one.

[ta0]

If His Orangeness ever decides to dump Milania, this lady would make a perfect 4th wife. 

I haven't seen a good physical explanation of the TMI's absence from the slow rate formula. Perhaps ta0 has one.

Precession is still a mystery to me . . .
I wish I had a good intuitive explanation of why it does not depend of the transverse moment of inertia (for the steady state).

[Iacopo]

As you may well know, for a given set of initial conditions (especially but not limited to release speed and tilt), steady precession can proceed at 2 very different rates

I suppose slow precession is the one we all know.  Fast precession could be what I called "oscillation" in my gyroscope.
There is a movement I can easily obtain in my super low CM tops, kicking their stem while they are spinning: they oscillate very rapidly, and they keep oscillating for some time, even for more than a minute;
I called this movement "nutation" in the past but probably it is just the fast precession you are saying that I observed.
In fact it is easy to induce this movement in tops with a very recessed tip, but in normal tops I can't.
If I kick the stem of a normal top while spinning, it doesn't oscillate rapidly at all, or it does so just for a fraction of second.

TMI does appear explicitly in the fast rate formula, but curiously, not in the slow one generally applicable to our tops. Per the parallel axis theorem, however, changing tip-CM distance necessarily changes the TMI about the tip. That's the indirect connection between precession rate and TMI: Both increase with CM-tip distance -- especially the TMI.

why it does not depend of the transverse moment of inertia (for the steady state).

After some more thinking I changed my idea about the TMI.  It is my understanding that the TMI should influence the precession speed;
a higher TMI should slow down precession.  It should influence both the fast and the slow precession.  I don't see a reason why it shouldn't.  The fact that there is some relationship between CM-tip distance and TMI, is not the answer, because basically CM-tip distance and TMI are independent parameters from one another, even if TMI is measured about the tip.

[ta0]

There is a movement I can easily obtain in my super low CM tops, kicking their stem while they are spinning: they oscillate very rapidly, and they keep oscillating for some time, even for more than a minute;
I called this movement "nutation" in the past but probably it is just the fast precession you are saying that I observed.
In fact it is easy to induce this movement in tops with a very recessed tip, but in normal tops I can't.
If I kick the stem of a normal top while spinning, it doesn't oscillate rapidly at all, or it does so just for a fraction of second.

I believe this is indeed nutation.

[Jeremy McCreary]

I believe this is indeed nutation.

I'm with ta0. The nutation rate is typically of the same order as the spin rate and sometimes exceeds it. Fast precession is said to be exceedingly rare in practice.

[Russpin]

Fast precession is said to be exceedingly rare in practice.

For unbalanced tops, precession rate (phi dot) is the same magnitude as the spin rate (w₃).

[Iacopo]

I believe this is indeed nutation.

I'm with ta0. The nutation rate is typically of the same order as the spin rate and sometimes exceeds it. Fast precession is said to be exceedingly rare in practice.

Ok, it is nutation.  Also I can confirm that at times that oscillation was faster than spinning.
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. 

Fast precession is said to be exceedingly rare in practice.

For unbalanced tops, precession rate (phi dot) is the same magnitude as the spin rate (w₃).

So the correct name for what I called "unbalance wobbling" seems to be "precession". 

[Russpin]

So the correct name for what I called "unbalance wobbling" seems to be "precession". 

I don't want to get into a semantics argument, but in the physics literature I have seen "precession" is related to phi dot and “nutation” is related to theta dot.

Where the Euler angles theta, phi and psi are defined as:




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

[Jeremy McCreary]

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

I think there are at least 2 more distinct motions in the unbalanced case, and engineers lump both of them under "whirl".

[Jeremy McCreary]

Let's face it, guys and gals: Everyone likes to use "precession" and "nutation" in their own way, and we seldom take the time to declare the definitions we've adopted. You'd think that the engineers and physicists who talk about this stuff for a living would have this all sorted out, but they're as guilty as anybody!

Problem is, mixing definitions is good way to get more heat than light out of a conversation on top behavior. We've proven that more than once, most recently here.

Through Reply #33, everyone was using "precession" for the same rather specific motion -- namely, the steady gyroscopic precession of a balanced top. Just a stable rotation of the top's spin axis about the vertical. The only thing driving this particular rotation is gravitational torque about the top's tip.

When we read or write "precession" on the forum, we usually seem to have the motion I just decribed in mind, and that's a good default for us. But competing definitions have their own scenarios and assumptions. In tops, differences at this generally unspoken level often lead to very different behaviors -- or to very different descriptions or interpretations of the same behavior.

Engineers sometimes use "precession" for the distinctive repetitive motion resulting from static unbalance in a spinning rotor. But this motion is qualitatively very different from pure gyroscopic precession in our usual sense. Whatever it's called, engineers consider this particular motion a form of "whirl", and it's usually synchronous with spin. Whirl is driven by revolving inertial forces and not by steady old gravity.

Whirl can also have a gyroscopic component -- especially in the presence of couple unbalance. This component of the total motion can have a stronger resemblance to the purely gyroscopic steady precession of a balanced top, but it's still not quite the same thing. And it's usually not a prominent feature of the observable motion in any event.

[Russpin]

I think there are at least 2 more distinct motions in the unbalanced case, and engineers lump both of them under "whirl".

Whirl is a term used in Rotor Dynamics where vibration modes of mechanical structures are excited by unbalance in a rotor. There is often a critical speed where resonance can occur. At the speeds, sizes and materials of tops I don't think whirl occurs in most tops even if they are unbalanced. My unbalanced top simulation predicts almost all observed unbalanced top behavior and it assumes a perfectly rigid body with no vibration modes.

[Jeremy McCreary]

Whirl is a term used in Rotor Dynamics where vibration modes of mechanical structures are excited by unbalance in a rotor. There is often a critical speed where resonance can occur.

No disagreement there. I'm familiar with the subject.

At the speeds, sizes and materials of tops I don't think whirl occurs in most tops even if they are unbalanced.

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

I've made and tested and played with working finger tops in hundreds of different designs by now. Their behaviors, when unbalanced in one way or another, often look to me more consistent with a form of whirl superimposed on gyroscopic precession and nutation than with precession and nutation alone.

That's especially true of the odd but completely reproducible behavior underlying the "paintbrush method" commonly -- and successfully -- used to locate static unbalances in tops. And that behavior occurs in tops of all stiffnesses.

[Iacopo]

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.     

[Russpin]

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?

I believe there is not gyroscopic effect in the pure motion of an unbalanced top

The motion of an unbalanced top does include gyroscopic effects without them it would just tip over.

[Iacopo]

I believe there is not gyroscopic effect in the pure motion of an unbalanced top

The motion of an unbalanced top does include gyroscopic effects without them it would just tip over.

You are right, I didn't use the correct words. 
I mean that the mechanism of unbalance by itself does not involve a gyroscopic effect.
There is gyroscopic effect in precession and nutation; even when a top is spinning vertical in sleeping position, still there should be a sort of micro gyroscopic effect, (excuse me my poor terminology), or the top would loose balance and it would tip over.
In this sense, a gyroscopic effect is always present in a spinning top, and in an unbalanced top too, of course.

But, (this is my intuition, I have not studied and I may be wrong), unbalance, if present, is superimposed to all this;
in unbalance there is a torque (from gravity) tilting the top down in the direction of leaning of the top.
Then there is an opposed torque to it, which is given (in case of oblate tops) by centrifugal force wanting to align the geometrical axis of the top with its rotation axis.  Then there is the center of mass wanting to stay in the axis of rotation, (for all tops), which too opposes the top to tilt down from the rotation axis.

There is nothing else. Just a balance between these opposed forces, which do not trigger a gyro effect.

When you see an unbalanced top spinning, it is not the axis of rotation that is spinning around that of precession; they are both vertical, and superimposed, (if the top is not precessing).  It is the geometrical axis of the top that spins around the other two ones; this gives the visual sensation of a precession, similar to the other one, with the gyro effect, but this one is different.

If then the top is precessing, (regular precession with gyro effect), the movement due to unbalance would superimpose to it and the resulting trajectory of the top axis would be a composed one, a sort of cycloid; to say that they are two different things.

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 would say, the rotation axis is linked to the precession (vertical) axis through gyro effects, while the geometric axis is linked to the rotation axis through the mechanism I said above and which doesn't involve gyro effects.