iTopSpin

Building challenge: We have some really talented top designers and makers around here. Anyone interested in tackling a working tank top?

No, not the strappy kind. I'm talking about a spinning tank top (TT) designed to carry a liquid in a leak-proof coaxial "tank". Tuning all the parameters for maximum play value or a particular starting method won't be easy, but I'm scheming to give it a try in LEGO.

The Hydro Gyro at 3:16 below is a rather elegant TT carrying colored water in a very cleverly shaped see-through tank. In this case, the tank spins inside a low-inertia coaxial frame partly meant for handling (like the cage around a toy gyro). An easy-to-use ripcord starter lets the user experiment with gradual to abrupt spin-ups and low to high release speeds. Gobs of play value here!



A simpler TT from collectop's collection, first posted here, with important similarities to the Hydro Gyro solution...




In these 2 see-through examples, watching the colored liquid react to the motions of the TT is a big part of the fun. But watching a TT react to the motions of a concealed liquid could also get pretty interesting. Think Mexican jumping bean.

Optional background

Based on a dive into the vast literature on the stability of spinning liquid containers, the key design parameters to be tuned for play value include
1. Tank shape.
2. Choice of liquid, with mass density and viscosity being the important mechanical properties.
3. Liquid "fill fraction" f = (liquid volume) / (tank volume).
4. Mass properties of the empty TT and the centrifugally formed liquid mass at operating speed.
5. Range of operating speeds leading to stable sleep or steady precession -- generally with as little wobble as possible.
6. Starting method capable of getting the TT to operating speed with little fuss.

For stability against gravity, an especially important mass property seems to be the moment ratio Q = AMI / TMI at the chosen f, where the TMI is taken at the pivot, and both moments at operating speed. Also key are the relative locations of the liquid and empty top CMs along the spin axis.

Addendum: And, of course, the axial location of the pivot relative to both CMs.

Through this lens, the Hydro Gyro is a low-f, intermediate-Q solution able to stand smoothly over a wide range of operating speeds. The Hydro Gyro's tank superimposes empty top and liquid CMs at speed, and I gather that that's generally the safest bet for stability.

A useful counterexample comes from a recent post by Daveid...



The empty Avon Bubble Bath dispenser on the right starts easily with the provided wind-up starter. And ta0 can boomerang it. But no luck turning it into a working TT filled with water or liquid soap near f = 100%. Specifically, the loaded Avon dispenser wouldn't stand at all with either liquid and 3 different starting methods: The wind-up starter, throwing it on a string, and an electric drill. Apparently, Avon really meant that red "WHEN EMPTY!" at the top of the box.

From the literature, the abrupt spin-ups delivered by the first 2 methods likely stirred up complex 3D flows in the liquid. Such flows have been known to destabilize spinning spacecraft with onboard fuel tanks with disastrous results. The sloshing free air/liquid interfaces formed at f < 100% tend to make matters much worse, but dangerous instabilities can also occur with no free surface at all (f = 100%).

These instabilities in attitude get really dangerous when liquid motions resonate with the loaded TT's natural nutation frequencies. They're fed by the very spin used to stabilize the TT against gravity in the first place. And they can totally overcome the gyroscopic stabilization you'd get if the liquid were solid instead!

One defense is to spin up the liquid slowly and over a long period of time in order to get a more coaxial steady-state tank flow. ta0's drill start tried that tack, but the loaded Avon TT fell right over anyway. A better bet is to tune operating speeds, f, Q away from nutation resonances.

But the failed Avon TT may have had a much simpler problem -- at least with the wind-up starter and the drill: Failure to release above the critical speed needed at f ~ 100%. The dispenser's very light, but a much smaller fill fraction might have worked -- even with some slosh.

Since the literature's full of guidance on coaxial tanks with circular cross-sections, I plan to start my own Project TT there. Just need the right (grocery store?) container to fit with a LEGO stem and tip assembly.

Based on a dive into the vast literature on the stability of spinning liquid containers . . .

I doubt that they tackle gyroscopic stabilization with respect to the pull of gravity.
I don't think a liquid, unless extremely viscous or very constrained, could really precess. I think it would become a pretty mess of flows.

ta0 wrote:

Based on a dive into the vast literature on the stability of spinning liquid containers . . .

I doubt that they tackle gyroscopic stabilization with respect to the pull of gravity.

Oh, but they do -- some quite explicitly.

Variations on this problem come up in lots of contexts where gravity counts, not just in spacecraft in microgravity. Rocket stages often separate while gravity's still strong. Other examples of some relevance include liquid filled projectiles and the drums in top-loading washing machines.

One directly applicable article dealing mainly with fully filled tank tops (no free surface)...

https://www.sciencedirect.com/science/a ... 0X84713083

Eventually found a free PDF, but can't find it now.

ta0 wrote:

Based on a dive into the vast literature on the stability of spinning liquid containers . . .

I don't think a liquid, unless extremely viscous or very constrained, could really precess. I think it would become a pretty mess of flows.

I gather that the liquid in a tank top (TT) can sometimes do some of both.

Question is, do the real tops we play with ever see the parameter combos required? Since proportions seem to matter more than the TT's absolute mass or size, I think so.

Thanks Jeremy. I confess I'm surprised this has been studied, more so in so many papers.

I found this pdf: Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity. I didn't try to follow the analysis (probably I can't) but this conclusion was surprising to me:

As an illustration of the results obtained in the previous theorem, consider a “classical” symmetric top, T, spinning at sufficiently fast rate around its axis a in the vertical direction, d, passing through the fixed point O and center of mass G (in its highest position). It is then well known that a small disorientation of a from d will produce a stable precession of T around d with a performing small oscillations (nutation). If, however, T possess an interior cavity filled up with a viscous liquid, Theorem 6.1 tells us that under the same above circumstances, the axis a will eventually reposition itself in the vertical direction d, at an exponential rate. This fact provides a further example of the stabilizing influence of an interior liquid-filled cavity on the motion of a rigid body.

So, it appears that the liquid of a filled cavity, if spinning fast enough, can stabilize the top!

ta0 wrote: I found this pdf: Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity. I didn't try to follow the analysis (probably I can't) but this conclusion was surprising to me:

As an illustration of the results obtained in the previous theorem, consider a “classical” symmetric top, T, spinning at sufficiently fast rate around its axis a in the vertical direction, d, passing through the fixed point O and center of mass G (in its highest position). It is then well known that a small disorientation of a from d will produce a stable precession of T around d with a performing small oscillations (nutation). If, however, T possess an interior cavity filled up with a viscous liquid, Theorem 6.1 tells us that under the same above circumstances, the axis a will eventually reposition itself in the vertical direction d, at an exponential rate. This fact provides a further example of the stabilizing influence of an interior liquid-filled cavity on the motion of a rigid body.

So, it appears that the liquid of a filled cavity, if spinning fast enough, can stabilize the top!

Found the same article, that being one of the few paragraphs I could actually understand. The conclusion's strictly theoretical, and only for 100% fill at non-zero viscosity.

From other papers and my own experiments with tank tops (TTs) carrying, not liquids, but loose granular materials like small LEGO parts or ball bearings, stable sleep's also part of the TT repertoire at partial filling and low viscosity -- given enough speed. That's certainly the case for the Hydro Gyro and the ball-bearing TT below.



The low-fill, low-viscosity Hydro Gyro always balances itself centrifugally over a certain speed -- in part due to tank shape. Interestingly, granular TTs are generally pretty good at self-balancing, too.

I think the TT has the potential to be tuned for many interesting behaviors -- stable sleep at full capacity being just one of them. And speed has to be a huge factor in all of them. As a topmaker, that potential really excites me.

Intuitively, I could see that a viscous fluid inside could dampen a vibration or an oscillation, like nutation. Perhaps, by that mechanism it could help stabilize a top. But I don't see a fluid making a top stand: it seems to me that only the solid mass can do that. My guess is that most of the mass has to be in solid form, although these papers seem to say otherwise.
In the case of the bubble wash top, the shell has very low mass with respect to the filling liquid, so I would be very surprised if it was stable at any speed.

I think the TT has the . . .

Tank Tops is an new category name for tops that you just introduced, and you already converted it into an acronym! Elon Musk forbade the use of acronyms at SpaceX unless he personally approved them. I completely agree with his rant: Acronyms Seriously Suck

Better Understand The Terminology

ta0 wrote: Tank Tops is an new category name for tops that you just introduced, and you already converted it into an acronym! Elon Musk forbade the use of acronyms at SpaceX unless he personally approved them. I completely agree with his rant: Acronyms Seriously Suck

Acronyms have their pros and cons and also their place. Agree, no great reason to abbreviate "tank top" as "TT" here. Just did it out of habit. But at least I redefined TT in each post before using it.

Also agree that too many different acronyms in one post can be a problem. But acronyms can also make for easier reading by giving certain concepts or long phrases their own short, easy-to-spot symbols. in my 2 main career fields, you'd be shot for spelling everything out every time, Mr. Musk not withstanding.

ta0 wrote: Intuitively, I could see that a viscous fluid inside could dampen a vibration or an oscillation, like nutation. Perhaps, by that mechanism it could help stabilize a top. But I don't see a fluid making a top stand: it seems to me that only the solid mass can do that. My guess is that most of the mass has to be in solid form, although these papers seem to say otherwise.

Now that you mention it, that part puzzles me as well.

Also surprising how little the article actually said about the empty top, the cavity, and the liquid in its model. The filled top had total mass M, body frame moments of inertia A, B, and C, and an inertial fixed point at distance l from its overall CM. The only liquid variables were density and kinematic viscosity. Nowhere could I find symbols for top or cavity dimensions -- not even for proportions.

As for solid vs. liquid behavior, folks in the spinning liquid biz talk a lot about "solid body rotation", wherein all relative flow within the liquid has ceased. Spin a real liquid in a tank long enough at constant speed and orientation, and viscous dissipation will eventually bring the liquid to this lowest possible energy state. To reintroduce internal flow at that point, you'd have to add energy.

From a dynamic standpoint, once the liquid in a tank top reaches steady-state solid body rotation, the tank top will behave as if it were completely solid -- at least until disturbed. Guessing that was the goal of your Avon drill experiment.

Question is, did the liquid actually reach that steady state in the spin-up time allowed? Probably so. A formula I found suggests that it would take at most a few seconds for every part of the soap in your Avon top to reach at least 99% of its solid body rotation speed after an impulsive start from rest. And that leaves me betting on your original suspicion -- namely, that the drill just wasn't fast enough to reach the Avon top's filled critical speed.

ta0 wrote: Intuitively, I could see that a viscous fluid inside could dampen a vibration or an oscillation, like nutation. Perhaps, by that mechanism it could help stabilize a top. But I don't see a fluid making a top stand: it seems to me that only the solid mass can do that. My guess is that most of the mass has to be in solid form, although these papers seem to say otherwise.
In the case of the bubble wash top, the shell has very low mass with respect to the filling liquid, so I would be very surprised if it was stable at any speed.

I think the same. I too was surprised about these papers statement.

Maybe there is a different mechanism which could make the liquid in a tank top to help it to rise.
Not the gyroscopic effect because this effect is reduced by the freedom of movement of the liquid.
I think to something else, (but this is just a vague idea, I am not sure whether it works in this way):

Let's imagine a top with a spherical cavity completely filled with liquid, (blue in the drawing).
At the start, (A), the liquid spins together with the top with its same speed and same spin axis orientation.
While precessing, the spin axis of the top changes orientation, but the spin axis of the liquid will tend to maintain its orientation, because of inertia, (B).
In position B the liquid spins in a direction so to push the walls of the tank top in a way to make it to tilt backwards.
This push is a rising torque, it makes the top to rise.

Well, the spin axis of the liquid should move too, driven by the movements of the top, which will tend to align the spin axis of the liquid to the spin axis of the top. But since it takes some time for the top to transfer the movement to the bulk of the liquid, there is a delay, so, practically, the two axes are not superposed.
The misalignment between the two axes would produce a feedback torque from the liquid to the top.

With the correct delay, (which supposedly depends on various parameters among which the dimensions and design of the top, viscosity of the liquid, spin and precession speeds), the torque from the liquid to the top could make it rise.

Jeremy McCreary wrote: once the liquid in a tank top reaches steady-state solid body rotation, the tank top will behave as if it were completely solid

This is not possible in a precessing spinning top, because the orientation of the top changes continuously while precessing, so the liquid in the tank cannot reach a steady state condition.

Worth keeping in mind...

1. The very opaque Galdi article claiming viscous self-righting in a completely filled tank top dealt only with sleeping tops never far from the vertical.

The more readable Or article I linked is more general in terms of tilt but, with some justification, ignored viscous effects under the assumption that inertial pressures in the liquid pushing on cavity walls would be much more effective in destabilizing the top.

Otherwise, near as I can tell, these 2 articles took different shortcuts to reach similar conclusions about stability against gravity near zero tilt.

2. My reading and experiments suggest that it's safest to think in terms of a spectrum in the ratio of gyroscopic to flow effects. Where these influences coexist, as in a tank top, you'll likely see some of both in the system's behavior, though one can dominate in extreme cases like the Hydro Gyro at high speed.

One of the tricks of the trade is to develop indicators (in the form of scale-independent nondimensional numbers) that tell you how far you are from a simplifying extreme case.

If you don't know where you stand in terms of these indicators, it's hard to figure what to expect. If you're interested in the most important indicators for a tank top, look up the Reynolds, Ekman, and Rossby numbers on Wikipedia.

A solid top that is not vertical precesses due to conservation of angular momentum. Gravity provides no horizontal torque so the momentum around a vertical axis needs to remain constant. When a top falls the vertical component of its spin decreases but this is compensated by precession. And the precession inertial forces are what keep the top from falling. But the bulk of a (low viscosity) liquid is free to remain spinning with a vertical axis when the top falls, so it does not contribute to precession (but it does to the gravitational pull) and therefore should not in my opinion help the top stand up.

ta0 wrote: A solid top that is not vertical precesses due to conservation of angular momentum. Gravity provides no horizontal torque so the momentum around a vertical axis needs to remain constant. When a top falls the vertical component of its spin decreases but this is compensated by precession. And the precession inertial forces are what keep the top from falling. But the bulk of a (low viscosity) liquid is free to remain spinning with a vertical axis when the top falls, so it does not contribute to precession (but it does to the gravitational pull) and therefore should not in my opinion help the top stand up.

Agree with nearly all of that, but I don't think it's as black and white as the last sentence. The liquid, being at least partially confined and having inertia of its own, is not entirely free to defy gyroscopic effects like precession.

The liquid in the Hydro Gyro seems quite happy to precess at high speed, and shortly after spin-up is probably in a steady state closely approximating solid body rotation.

A few more data points: Turns out that I already have a (mostly water-tight) tank top, here partly filled with its original load of LEGO volleyballs...





The empty top's well-balanced, and the balls self-balance quite nicely with any starting method. Through the window, you can see them jam against the lateral walls in solid body rotation. And they enter this state almost immediately after a sufficiently hard crank of the stem.

Not saying that a liquid has to behave in exactly the same way. But the many important similarities between granular and liquid flows shouldn't be dismissed out of hand in a tank top context. Many of these similarities are borne out in both the literature on spinning drums with granular loads and in my own experiments with granular loads in this and many other LEGO tops.

On swapping out the balls for water at ~70% fill by volume, I observed the following:

1. The top fell over immediately with any hand start, and in an unusually jerky way. No exceptions. Through the window, I could see the water sloshing violently the whole time.

2. Very different story with a motorized starter delivering slow accelerations ending in a sustained steady-state speed with tilt held constant throughout. Here I watched the water form a stable ring against the lateral wall and could easily imagine it being in solid body rotation by the time I pulled the starter away. Even at ~15° tilt.

On release at 0° tilt, the high-drag, top-heavy top slept for ~10 s with some normal-looking wobble before falling in a normal-looking way. And at ~15° tilt, it precessed -- not as long as it slept, but with no greater wobble.

3. No self-righting tendency observed with the rather fine tip installed.

Many more experiments planned now, but these initial tests confirm that a liquid-bearing tank top can still behave like a fully rigid top in many respects -- even with a free liquid surface inside the tank. You just need (1) the right combo of parameters like speed, fill fraction, and mass properties, and (2) the right liquid prep before you let go.