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[Jeremy McCreary]

Nerd alert!

Ran into some interesting spin time trade-offs between aerodynamics and mass distribution with this new LEGO top...



The top consists of the black chassis below with or without fairings. The chassis has a central hub (not well seen) surrounded by a polyhedral space frame. The hub and frame were designed to cover the other's structural weaknesses. Together they're strong enough to hold the top's size and shape against torsion and centrifugal expansion to at least 2,000 RPM.



The frame's idealized geometry is that of a truncated octahedron with 24 identical vertices, 36 identical edges, and 14 faces of 2 different kinds (6 square and 8 hexagonal). The small black clamps mark the vertices. The square faces are embedded within the octagonal parts seen above. The hexagonal faces are empty.



Case 1. The very dirty bare chassis (above) turns in predictably miserable spin times: 7 sec by hand, and 9 sec with the starter in the Case 3 photo.

Reducing total aerodynamic braking torque (ABT) by fairing the chassis improves spin times significantly. However, the accompanying changes in mass distribution (especially AMI, AMI/TMI ratio, and CM height) also impact the results -- mainly via critical speed and the top's ability to resist any ABT present.

Case 2. With the addition of 48 mm lateral fairings (white below), spin times more than doubled to 16 and 24 sec, resp. Relative to Case 1, these fairings reduced total ABT and increased both AMI and AMI/TMI with little change in CM height. The happy result: Slower spin decay and lower critical speed.





So, any further spin-time gains to be had by adding axial fairings to the lateral fairings above? Well, that depends on the fairing size used!

Case 3: With 48 mm axial fairings, spin times actually dropped by ~20% (to 13 and 20 sec) relative to Case 2. Though AMI increased relative to Case 2, I suspect a net increase in critical speed due to a 10-15% increase in CM height and a reduction in AMI/TMI. How much these 48 mm axial fairings actually reduced total ABT relative to Case 2 is hard to say. (I can imagine a lot of interference drag at the triangular openings between fairings here.)



Case 4. The best spin times by far (30 and 51 sec!) came with the orange 64 mm axial fairings below (not in video). Relative to Case 2, adding these larger axial fairings reduced total ABT and increased AMI. Relative to Case 3, the decrease in AMI/TMI was less, and CM height was the same. Such a large boost in spin time probably required improvements in both ABT and critical speed.



Of course, release speeds (1,100 to 1,800 RPM) varied with spin-up method and from case to case. But my tests indicate that their contributions to the spin-time variations observed with a particular method weren't decisive.

[ta0]

Just for its symmetry, I like your truncated octahedron top. 

My gut feeling is that the improvement with the side fairings is mostly due to the larger moment of inertial. It's not clear to me that aerodynamics would improve substantially.

My guess is that the main difference between 2, 3 and 4 have to do to the details of the air flow and therefore drag.

Perhaps you could compare the spinning drags by attaching them to a drill using a torsion spring and comparing the torque they produce at high speed (with flash photography) 

[Iacopo]

Case 4. The best spin times by far (30 and 51 sec!)

It's a much longer spin time, even if the top apparently isn't very much different from that of case 3.
Have you tried to repeat the spin, to check if the difference is real ?
At times tops spin for a longer or a shorter time without any apparent reason.  I think this is related to the contact points, with friction changing a bit because of wear of the contact points, or the top spinning on a different spot of the base, or maybe even some difference in lubrication.
When I make spin time comparisons with my tops, I have made a habit to calculate an average of more spin times, because the single spin time can be misleading. 

[Jeremy McCreary]

Iacopo:  Yes, the spin time trends I reported are quite reproducible.

ta0: I have to think that aerodynamics figured significantly in the Case 4 result. In my experience with nearly 1,000 LEGO tops now, drag is usually the main  limit on spin times.

That's not to say that mass properties don't count. They do -- especially CM height and AMI/TMI ratio. It's just hard to make the kind of progress seen here without reducing drag in some substantial way.

[Iacopo]

Perhaps you could compare the spinning drags by attaching them to a drill using a torsion spring and comparing the torque they produce at high speed (with flash photography) 

You inspired me an experiment for air drag.  Not very accurate but decent enough.
I made a video about it:




Jeremy, after the experiment I see more clearly how spin time in your tops is lowered mainly by air drag, by far, and not tip friction.

[Jeremy McCreary]

I like where you're going with this video, Iacopo! Funny, when ta0 came to visit last week, we discussed this very method of comparing total braking torques (TBTs) over breakfast. Great minds think alike.

Looks like you used a DC motor in your experiments, and that's good, because DC motor characteristics allow the following analysis...

Nerd alert!

Suppose you have a DC motor with no-load shaft speed N₀ measured in RPM. Now let this motor spin up 2 different tops we'll call "1" and "2" until their speeds plateau at N₁ and N₂, respectively. Then the ratio of the TBTs Q₁ and Q₂ responsible for these plateaus is just

Q₁ / Q₂ = (N₀ - N₁) / (N₀ - N₂)

With a suitable test rig and a laser tachometer, it shouldn't be hard to measure speeds N₀, N₁, and N₂ fairly accurately.

For example, if N₀ = 1,000 RPM, N₁ = 600 RPM, and N₂ = 800 RPM, then Q₁ / Q₂ = 2. In other words, Top 1 generates twice the total braking torque at speed N₁ as Top 2 does at speed N₂.

Now, what we'd really like to know is how the TBTs compare when both tops are spinning at the same speed. Unfortunately, we won't be able to quantify that, but we can say at least this much: In the example above, Top 1's TBT would have to be greater than Top 2's if both were spinning at the faster slower plateau speed -- here, N₂ N₁. Otherwise, Top 1 2 couldn't have continued to slow the motor accelerate from there.

To get absolute values for Q₁ and Q₂ at the respective plateau speeds N₁ and N₂, we'd also need to know the motor's stalled torque Q_s. Then for Top i,

Q_i = Q_s (N₀ - N_i) / N₀ = s (N₀ - N_i),

where s == Q_s / N₀ is known as the motor's "steepness". When you play with lots of different DC motors as I do, steepness becomes a very handy thing to know.

[Jeremy McCreary]

Watched your video again, Iacopo -- really well done. I would have expected the tape to cut the metal top's spin time, but not by that much!

Translating my last post to your experiments and terminology...

The motor speed with just the tape is not quite N₀ but close, as the tape creates its own small aerodynamic load on the motor shaft at speed.

The RPM loss due to Flywheel 1 (say, squared rotor edge, no tape) is L₁ == N₀ - N₁

The RPM loss due to Flywheel 2 (say, rounded edge, no tape) is L₂ == N₀ - N₂

The fact that L₁ exceeds L₂ by an amount hard to attribute to experimental error alone allows you to say with confidence that at the lower plateau speed N₁, Flywheel 2 generates less total braking torque than Flywheel 1.

And since you performed that experiment with the same 2 flywheels at 3 different voltages, you now say that about 3 different N₁ values -- namely, 2116, 3310, and 3893 RPM.

[Iacopo]

For example, if N₀ = 1,000 RPM, N₁ = 600 RPM, and N₂ = 800 RPM, then Q₁ / Q₂ = 2. In other words, Top 1 generates twice the total braking torque at speed N₁ as Top 2 does at speed N₂.

This relationship is interesting.
Unfortunately I don't know the motor's stalled torque, so we are stuck with calculations.

The metal top's spin time was cut by the tape especially at high speed;
with the tape, it slowed down from 1750 rpm to 850 rpm in only one minute, so it lost more than 50 % of its speed in that lapse.  At that speed I can hear the noise of the air on the tape.

I believe your tops would spin much longer in a vacuum chamber.

[ta0]

Great experiment Iacopo! Somebody had to do it and you took the initiative! 
Great idea Jeremy on using those relationships. Another possibility might be to measure the current of the motor required to keep it at a certain fixed speed, but I don't have time now to look into the details (there will be some change in the losses of the motor).

[Iacopo]

Thank you, Ta0.  If calculations become simpler, it could be interesting.  Even if I am not sure how to change gradually the voltage, I have a transformer with only three outputs, 3, 5 and 6 volts.

[Jeremy McCreary]

Just finishing up a LEGO version of Iacopo's method to rank my tops in terms of total braking torque (TBT) at speed. I'm eager to see what trends emerge from the many different designs and sizes I plan to test. Already there have been surprises.

So far, getting enough signal-to-noise ratio (SNR) to detect small differences in TBT has been the big challenge. The signal here is the loss of motor speed due to TBT. The noise comes mainly from unsteady tachometer readings.

Iacopo made all the right choices in his setup. He tested at speeds high enough (over 2,000 RPM) to generate a lot of TBT and used an ungeared DC motor of low steepness (as defined at the end of my last post) to increase the associated speed loss. These measures work to improve SNR, but some of my LEGO tops fly apart at 2,000 RPM, and only one LEGO motor has a steepness low enough to get a decent SNR after the necessary external gearing. The rig's final no-load speed (NLS) of 1,435 RPM at the chuck (for the top stems) is safe for ~90% of my tops with a speed resolution of ~5 RPM.

Preliminary results confirm that my TBTs become overwhelmingly aerodynamic at speeds well below NLS. So far, the cleanest tops have equilibrated at over 99% NLS and the dirtiest at ~50% NLS. For any given pair of tops, I can usually predict the one with more loss, but I've been fooled. In my rig, the tops rest on their tips during testing. Swapping among my 4 most common tip designs (radius of curvature 0.5 to 5.0 mm) makes no clear difference in speed loss at current SNR.

[Iacopo]

Just finishing up a LEGO version of Iacopo's method to rank my tops in terms of total braking torque (TBT) at speed.

It is not very easy to obtain accurate readings with that method;
the weight of the top, acting on the bearings of the electric engine, increases their friction, so, if you suspend the tops to the engine like I did, the tested tops should have the same weight, or an error would be introduced.
The tops must be accurately balanced, otherwise they vibrate, less or more, and the vibration, transmitted to the axis of the engine, makes it run slower; the more intense the vibration, the more the engine is slowed down.

But you say "total breaking torque", while that my test was about air drag alone;
are you testing in a different way from me ?  It could be interesting if you could make a video about it.

Now I am making a little vacuum chamber for my tops.  Many months ago I ordered one from China but I never received it, so I decided to make one by myself.
When it will be ready I will start some tests, comparing spin decay of different tops with and without air. 

[Jeremy McCreary]

It is not very easy to obtain accurate readings with that method

Totally agree. I've taken many steps to improve accuracy, but those pesky fluctuations in laser tachometer readings remain. I see this as noise having nothing to do with any real speed fluctuations, and it's probably the main source of error in my test rig. We'll see if the new tach on the way makes any difference.

...the weight of the top, acting on the bearings of the electric engine, increases their friction, so, if you suspend the tops to the engine like I did, the tested tops should have the same weight, or an error would be introduced.
... But you say "total breaking torque", while that my test was about air drag alone; are you testing in a different way from me ?  It could be interesting if you could make a video about it.

Hope to have a video up in a week or so. Yes, side-loading the motor shaft is a potentially significant source of error. Since I'm ranking tops in the 20-140 g range, I'm spinning them vertically on their tips on a standard glass surface. In principle, the braking torques I'm assessing are then due to both drag and friction -- hence "total braking torque" (TBT).

Now, if the frictional (tip) part of TBT were actually measurable under test conditions, it would muddy the drag signal. But I have good evidence to the contrary -- at least at my top masses, equilibrium speeds (roughly 700-1,400 RPM), and current speed resolution (maybe 5-10 RPM). Since substantial changes in supporting surfaces and tip profiles make no detectable difference in my readings, even at high mass, I'm comfortable in assuming that, to an acceptable degree of accuracy, an observed difference in equilibrium speed reflects a real difference in drag.

TBT shapes a free top's spin decay curve and limits its spin time. The drag part of TBT clearly decreases as speed falls, but the friction component probably varies little in a vertical top spinning in place. Drag definitely dominates TBT at high speed, but tip friction could become important when the decay curve starts to flatten out at low speed. I think the equilibrium speeds you and I are measuring are well into the drag-dominated range.

The tops must be accurately balanced, otherwise they vibrate, less or more, and the vibration, transmitted to the axis of the engine, makes it run slower; the more intense the vibration, the more the engine is slowed down.

Absolutely. Fortunately, LEGO precision guarantees that any symmetrical arrangement of fully seated parts will balance well enough to give a vibration-free spin -- provided the structure is also too stiff to flex, and the central axle is true. In my tops, wobble of any kind -- whether due to unbalance, flexure, or misalignment -- is also bad for spin time.

Now I am making a little vacuum chamber for my tops. Many months ago I ordered one from China but I never received it, so I decided to make one by myself.
When it will be ready I will start some tests, comparing spin decay of different tops with and without air.

Eager to see your results! The only thing better than theorizing about tops is getting real data. Unfortunately, you'll have to get down to a small fraction of a millibar to see any significant decrease in air viscosity with pressure. (Doesn't sound right, I know, but that's the reality.) The other factors affecting drag are all about the top with no dependence on air properties.

[Iacopo]

fluctuations in laser tachometer readings remain.

One thing I found that helps the accuracy of the reading is to hold the tachometer as steady as possible relatively to the top. When I can do so I see the readings are more stable.

Since substantial changes in supporting surfaces and tip profiles make no detectable difference in my readings, even at high mass, I'm comfortable in assuming that, to an acceptable degree of accuracy, an observed difference in equilibrium speed reflects a real difference in drag.

I think you are correct.  In my case the flywheels were relatively little, and smooth, and with low air drag;  it hasn't been very easy to achieve the necessary accuracy for to see differences of their air drag. In your case air drag should be higher and easier to detect.  I think that generally in your tops air drag is quite more important than tip friction, so I agree with your reasoning.

you'll have to get down to a small fraction of a millibar to see any significant decrease in air viscosity with pressure.

I can't do this but the residual pressure will be low enough for having interesting results anyway, I think.
Some calculations could correct this error.

[Jeremy McCreary]

One thing I found that helps the accuracy of the reading is to hold the tachometer as steady as possible relatively to the top. When I can do so I see the readings are more stable.

Yes, I think that's at least part of it. Perhaps the pistol-style tach coming in the mail will help here. It will surely be more ergonomic than the one I have.