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[Iacopo]

I have made a brass gyroscope:



Playing with it I saw movements that puzzled me:
what is the movement in the last sequence in the video ? Is it nutation ?  It disappeared spontaneously after a while.

[ta0]

First of all, that gyro looks like a laboratory grade instrument and a work of art at the same time: congratulations!

I was also surprised and puzzled by that oscillating behavior at the end. After thinking a bit, this is my attempt at an explanation.
I assume the gyro was spinning balanced without precession before you perturbed it (so gravity has nothing to do with its behavior). You applied a torque to raise the tip of the gyro. This forced the gyro to precess and gave it a small initial angular momentum with respect to the vertical. So when you removed your hand it is spinning fast around its axis and slowly with respect to the vertical. The resulting angular momentum is around an axis slightly shifted with respect to its principal axis (along the axle) towards the vertical. In your case the angular momentum seems to be pointing to the front, slightly below the horizontal. After you release your finger you have a case of inertial precession like described on this post (based on Butikov's paper): link.

I very much want a gyro of that quality!

Edit: I believe some may call this nutation (e.g. for satellites) but it is different to the nutation of tops.

[Jeremy McCreary]

Beautifully designed and executed, as usual, with some interesting innovations. I've never seen balancing screws on the gimbals of a free (aka universally mounted or 2-DOF) gyro before, but they make good sense here.

[Jeremy McCreary]

Ginsberg's Engineering Dynamics textbook (2008) has become my go-to source on all things gyroscopic, including tops. It even has a section on free gyros like yours, and it's free online.

Per Ginsberg, rotation of the outer gimbal relative to the base is precession, and rotation of the inner gimbal relative to the outer is nutation. The outer and inner gimbals are then the precession and nutation gimbals, respectively. These conventions are consistent with the way precession and nutation are applied to tops.

I think we can call the inner gimbal motions you induced  "nutation"  with confidence. The oscillations reflect the "rigidity" (gyroscopic stability) of the rotor's orientation WRT small disturbances.

[Jeremy McCreary]

Forgot to ask: What kind of initial speeds and spin-down times are you getting?

[ta0]

I think we can call the inner gimbal motions you induced  "nutation"  with confidence. The oscillations reflect the "rigidity" (gyroscopic stability) of the rotor's orientation WRT small disturbances.

In the case of the precessing and nutating tops, you can decompose the total movement into those two separate movements along the two gimbals. The periods of precession and nutation are generally different.
But in Iacopo's case, I believe that the axis of the flywheel is tracing a circle (or cone) around a fixed direction. This is very much the same simple movement that we call precession when it happens around a vertical axis.

In the aerospace industry they seem to call nutation what everybody else calls precession. Satellites have nutation dampers. This confused me for a long time.

[Jeremy McCreary]

I think we can call the inner gimbal motions you induced  "nutation"  with confidence. The oscillations reflect the "rigidity" (gyroscopic stability) of the rotor's orientation WRT small disturbances.

In the case of the precessing and nutating tops, you can decompose the total movement into those two separate movements along the two gimbals. The periods of precession and nutation are generally different.
But in Iacopo's case, I believe that the axis of the flywheel is tracing a circle (or cone) around a fixed direction. This is very much the same simple movement that we call precession when it happens around a vertical axis.

Just to be clear, I'm proposing that we use "nutation" for the isolated motion of the inner gimbal, not the combined motion of the rotor's spin axis.

Granted, after Iacopo pokes the outer gimbal at 1:38, the end of the spin axis closer to the camera traces out a closed figure centered on a fixed direction, which appears to be the equilibrium spin axis direction prior to the disturbance. I see the resemblance to a precession of sorts, but I think the resemblance is misleading.

For one thing, this figure looks more like an ellipse than a circle to me. (Could just be the camera angle, but I doubt it.) If so, then the motion of the spin axis after 1:38 isn't a pure rotation about the fixed direction. Hence, it can't be described as either a precession or a nutation, as each term refers to a pure rotation of a spin axis about some other axis. (In a top, the pertinent axes are respectively the vertical and the line of nodes.)

But either way, ellipse or circle, the spin axis motion after 1:38 is a linear combination of 2 independent oscillations of equal period: (i) Oscillation of the outer gimbal excited by Iacopo's finger, and (ii) the inner gimbal's oscillatory gyroscopic reaction to (i). Oscillation (ii) then keeps (i) going, and vice-versa, until bearing friction damps them out.

I see at least 2 problems with calling this compound motion "precession" and letting it go at that:
(1) In free gyros like Iacopo's, what should we then call the pure rotation of the outer gimbal relative to the base?
(2) We lose any direct conceptual connection between the precession of a free gyro and that of a top. The latter, as you pointed out, is a pure rotation that can't be decomposed.

In the aerospace industry they seem to call nutation what everybody else calls precession. Satellites have nutation dampers. This confused me for a long time.

Arghh! An equally maddening confusion surrounds tip-speed ratio (TSR) -- arguably, the single most important thing you can know about any kind of propeller. But the TSR used by hydrodynamcists and naval architects (boat engineers) is the reciprocal of the TSR used by aerodynamicists and aeronautical engineers. Never mind that propellers generate thrust from lift in exactly the same way in air and water!

[Iacopo]

Ta0 and Jeremy, thank you for the answers.
I have to go away and I have no time to write now, but you both have writed interesting things, I will write tonight.
I confirm that the gyro was balanced and there was no precession during the last sequence.

[ta0]

The gimbals are there to let the gyro spin freely around its center of mass. If you don't unbalance the gyro, they completely cancel the effect of gravity. So you could take the gyro without gimbals to the space station and have it do the same movement while floating. It would be completely arbitrary to call the component of the movement in one direction nutation and in another precession.

Having said that, I just noticed this curious and perplexing statement in the introduction to Butikov's paper (italics in the original):

For an axially symmetrical body, kinematics of such inertial rotation corresponds to precession. Being applied to a gyroscope, this free precession is called nutation.

 

I really recommend that paper, with (somewhat) physically intuitive explanations instead of math: Inertial Rotation of a Rigid Body

[Jeremy McCreary]

ta0: Sorry, I've lost track of where we agree and disagree at this point. But I do agree that there's a certain degree of arbitrariness in all of this terminology -- both yours and mine.

I have 2 Butikov articles -- the one you recommended on inertial (torque-free) rigid body rotations (RBRs) and his follow-up article on forced (torque-induced) RBRs. Both are valuable, and I've read them several times, including this morning. Since we're all stuck with gravity, I still find the latter easier to apply to the tops we usually talk about on this forum.

Butikov's articles are unique in their attempt to describe 3D RBRs strictly from the observer's reference frame. In contrast, most RBR treatments (including those concerned with tops) refer RBRs to one axis (e.g., the vertical axis in the case of a top) from the observer's frame and 2 more from a frame tied to the rotating body itself.

This difference in perspective creates a lot of confusion by itself, and it underlies your Butikov quote about the use of "nutation". He repeats this quote and does a better job of explaining it in the follow-up paper, but I'm still not entirely clear.

Yet another potential source of confusion is Butikov's use of "gyroscope" to refer, not to a device like Iacopo's 2-DOF gyro, but to  any spinning rigid body with its total angular momentum vector very closely aligned with its spin angular momentum vector. Under that definition, our tops usually qualify as gyroscopes for a time after release but probably not when approaching their toppling speeds. Other authors refer to this special case as the "gyroscopic approximation". The "fast top approximation" implies the same situation.

Another (related) part of the problem is the fact that engineers often use "nutation" for what physicists call "precession" -- hence, the aerospace usage you noted earlier. There's nothing wrong with this as long as potentially confusing terms are clearly defined up front, but that often doesn't happen. Oddly enough, Ginsberg's engineering textbook sides with the physicists all the way, but at least he doesn't leave you wondering about his terms.

So in the end, it's a "potayto" vs. "potahto" thing. Ginsberg's approach and terminology work better for me -- in part, because I find them easier to relate to what I actually see in our tops and in gyros like Iacopo's as well. I also value Butikov's perspective from the observer's frame but sometimes struggle with his presentation -- in part, because I was already comfortable with Ginsberg-like approaches.

[Jeremy McCreary]

I confirm that the gyro was balanced and there was no precession during the last sequence.

We're used to unidirectional precession in our tops, but that's not the only possibility in a balanced free (2-DOF) gyro like yours.

Whether you think of the inner or outer gimbal as the precession gimbal, there was definitely precession in the final sequence (after 1:38). It was just oscillatory. Real navigational gyros work hard to damp out such oscillations as quickly as possible, though not through bearing friction. At heart, many are still balanced free gyros.

[Iacopo]

I very much want a gyro of that quality!

Thank you for the appreciation.  I am willing to make one for you, if you want, but it is largely handcrafted and it wouldn't be cheap.
When I was looking for pictures of gyros for designing the mine, I found this gyro for sale, which maybe could interest you:
http://www.gyroscope.com/d.asp?product=LECTURE
It costs about $ 750 and is large, it can spin for 20 minutes, the mine would cost more and it is lighter and littler, it spins for 3 minutes (60 mm flywheel) and not much more, (I don't know if this is because of new bearings still in need of running in, but I suppose it will never spin for very much longer than this).

Forgot to ask: What kind of initial speeds and spin-down times are you getting?

It spins for about 3 minutes or little more, starting from 2000 - 2500 RPMs.

I really recommend that paper, with (somewhat) physically intuitive explanations instead of math: Inertial Rotation of a Rigid Body

Thank you. I will try to read it.

this figure looks more like an ellipse than a circle to me.

I think this is due to a difference of the moments of inertia because of the gimbals;  in the vertical oscillations only the internal gimbal moves, instead in the horizontal oscillations both the gimbals have to move, (further, the internal gimbal has higher moment of inertia when moving in the horizontal oscillations), so the vertical oscillations are broader because there is less moment of inertia.
Without the gimbals, in open space, I think the figure would be a circle not an ellipse.

Whether you think of the inner or outer gimbal as the precession gimbal, there was definitely precession in the final sequence (after 1:38).

I agree that an outer gimbal continuous and uniform movement should indicate the presence of torque induced precession.
But the outer gimbal oscillating back and forth seems to me something different:
maybe nutation, or maybe torque free precession. 
I don't know if "nutation" and "torque free precession" are synonyms or there is a difference of meaning between the two terms.
But I am sure that there is not "torque induced precession" in the last sequence.

[Jack]

i have come to the conclusion that you are not in fact human, and that you are actually an alien teasing us with your superior machining
skills 

[Jeremy McCreary]

If we could all agree to define some terms and stick to them, I suspect that most of the apparent disagreements would disappear. Since Iacopo made this beauty, he should have the naming rights. If we start with the pure (single-axis) rotations of the rotor and gimbals -- without regard to how they might arise -- we can build from there.

Iacopo: For the sake of discussion, what would you like to call...

1. Rotation of the rotor relative to the inner gimbal?
2. Rotation of the inner gimbal relative to the outer?
3. Rotation of the outer gimbal relative to the base?

[Iacopo]

For the sake of discussion, what would you like to call...

1. Rotation of the rotor relative to the inner gimbal?
2. Rotation of the inner gimbal relative to the outer?
3. Rotation of the outer gimbal relative to the base?

I am not the most appropriate person for establishing these terms... 
I could arbitrarily abbreviate these expressions as:

1. Rotation A
2. Rotation B
3. Rotation C

But if you prefer to use different abbreviations, do so.