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[Iacopo]

So the greatest energy transfer would be to a very heavy top. 

So it seems.
For my fingers, it seems to be maybe a 500-700 grams top. 

But the rate of energy transfer is the mechanical power given by the product of torque and angular speed (in rad/s). Hence, the energy transfer rate must be minimal at the start of the twirl and at the end and must peak somewhere in between.

This is something interesting I never thought before.
So, the highest angular acceleration is not at the start of the twirl, right ?

[Iacopo]

The orange line in this graph represents the maximum energy (joule) I could put in a perfect finger top by a single twirl of the fingers, depending on the moment of inertia of the top.
The red dots are the real, tested spinning tops.
I indicated the weight (grams) of the tops, near the red dots, which is a more intuitive measure than the moment of inertia.

The distance between the dots and the orange line represents the efficiency of the top twirling system;



For example, the top that weighs 329 grams is far from the orange line; it is very inefficient.
Its stem is smooth and relatively slippery. The shape of the stem also is not very good, it is cylindrical, and too large.
So I can spin it only up to little more than 0.6 joule, instead of almost 1 joule, as its size should allow.

The top weighing 165 grams instead is on the orange line, it is very efficient;
it has a tapered stem, it is knurled, there is good grip for the fingers, good diameter, and the stem is very long.
It gets 0.7 joule, which is excellent for a top of this size.

Based on my actual data, I would say:

tops up to 200-300 grams are in the range where, at the increasing of their size, they can receive a higher amount of energy with a single twirl. 
500-600 grams tops instead are in the range where the perfect gear ratio is already reached, so it is not possible anymore to further increase the amount of energy given to them simply increasing their size.
   

[Jeremy McCreary]

So, the highest angular acceleration is not at the start of the twirl, right?

Yes, but the power (energy transfer rate) at a given moment depends as much on the current speed as it does on the angular acceleration at the time.  That means: (i) Of course, no torque, no power. (i) But also, no speed, no power. (iii) High speed can outweigh applied torque in the power game.

My hunch is that power will peak once the top gets up some speed. But angular acceleration is just the applied torque divided by a constant AMI, so what we're really after here is a representative applied torque-speed curve (ATSC).

This hunch comes in part from a mental image of a twirling ATSC that decreases steadily during spin-up. That's how spin-ups feel to me, but I may also be biased by a lot of work with DC electric motors (whose ATSCs decline linearly toward zero at no-load speed). Now, I doubt that twirling ATSCs are linear, but if they do decline roughly with speed, then peak energy transfer will come after start-up.

So, how could we get hold of a representative ATSC? What will it look like? How will it change from top to top and user to user? So many questions. Some "spin-up curves" (speed-time curves during spin-up) would be a good start.

It would be useful to pool the forum's observations on questions like these.

[Jeremy McCreary]

The orange line in this graph represents the maximum energy (joule) I could put in a perfect finger top by a single twirl of the fingers, depending on the moment of inertia of the top.

A very interesting window onto the mechanics of twirling, Iacopo! I'm still giving the graph the careful study it deserves.

[Iacopo]

Yes, but the power (energy transfer rate) at a given moment depends as much on the current speed as it does on the angular acceleration at the time.  That means: (i) Of course, no torque, no power. (i) But also, no speed, no power.

Ok.  Now I understand better.
We could say that energy and speed do not grow with the same rate, because energy is proportional to speed squared.  Energy grows relatively little, at slow speed, at parity of angular acceleration.
So the maximum energy transfer can't be at the start of the twirl, even if the torque is stronger.  I see this clearly now.

I suppose acceleration is different.
Without frictions, a constant torque should produce a constant acceleration;
so, if torque is strongest at the start of the twirl, which seems probable, acceleration too should be strongest at the start.
 

[Jeremy McCreary]

So the maximum energy transfer can't be at the start of the twirl, even if the torque is stronger.  I see this clearly now.

To see how all this plays out in a somewhat analogous case, I highly recommend this well-explained treatment of a disk spun up by a typical brushed permanent magnet DC electric motor. (The author happens to be writing for fellow battlebot builders.) The disk's AMI is the only load on the motor, as drag and friction are ignored. The graph below nicely summarizes the most pertinent results:



The torque output of such a motor declines steadily (and linearly) with speed. My guess is that finger torque also declines steadily as a top stem picks up angular speed -- though probably nonlinearly. Nonetheless, it's instructive to see how a declining torque-speed curve impacts a related spin-up process with a simple mathematical description.

[Jeremy McCreary]

This brings up an interesting design parameter.  What is your maximum angle of lean before scraping the flywheel?  Currently I aim for about 8 degrees.

Totally agree, Alan: Scrape angle, AMI, twirling skill, play value, and spin time are all coupled, and that can make for some tricky audience-dependent design trade-offs. In my experience, the higher the AMI, or the shorter the stem, the larger the scrape angle must be to get a reasonable shot at a clean twirl -- especially in unpracticed hands.

It is about 4 degrees in my latest tops, and I have no problems spinning them hard on a flat surface, with stems about 70 mm long.

Wow, if that 70 mm goes all the way to the tip, you have at most 5 mm of lateral wiggle room at the stem! That's pretty tight at high torque. The high-AMI, high-drag top on the goniometer below has a scrape angle of ~7.5°. LEGO show visitors never get clean twirls out of it, and even I scrape now and then when I really crank it.



There are several ways to play these trade-offs, each with its pro and cons. Usually, a combination works best for me.

• Reducing rotor radius to a comfortable scrape angle at constant ground clearance reduces AMI and CM height -- possibly with an increase in TMI/AMI ratio. If you're after a long-spinning sleeper, only the CM change is good.
• Jacking up the rotor to a comfortable scrape angle at constant rotor radius increases CM height and TMI with (i) a potentially strong adverse effect on spin time, and (ii) some loss of willingness to sleep.
• Elongating the stem with no other change significantly improves tilt control during spin-up at little cost in CM height and TMI. The shorter stem in the foreground works for me, but unpracticed hands would need the longer one.
• Or you could just forget about twirling the top directly and go to a detachable starter. The 2 examples below don't add much launch speed, but they greatly improve tilt control throughout spin-up.

One final point: If I limited myself to tops easily twirled by hand, I'd miss out on a lot of fun designs. Take my "planet tops" Revlon VI and Zargon IV. They're very popular at LEGO shows, but no visitor has ever gotten one to stay up by hand. Heck, I couldn't do it at first, either. Scrape angle isn't the problem, and neither is excessive AMI or drag. Instead, these tops suffer from critical speeds (due to unfavorable TMI/AMI ratios) that most users just can't reach by hand.



With the right starter, however, they're lots of fun. The best part: Since they're my planets, I get to be in charge of everything! 

[Aerobie]

My longest twirl was achieved with a 2.25" (57.15mm) diameter, 145g spinning on a .5" (12.7mm) ball.  An 860 RPM twirl ran for 26:05.   But the lube may have been "just right" because a 923 RPM twirl ran 24:27 and a 909 RPM twirl ran 25:28.  With balls one can have too much lube.  As I've mentioned before, I lube with forehead skin oil, wiped hard on the mirror with my thumb.   This top falls at about 133 RPM.

I think 145g may be optimum for a 2.25" top and my personal twirl ability.  I haven't measured the rotational inertia of my tops, but they are all fairly similar geometry with brass or bronze rims and lightweight hubs.

For a long time I thought the tip drag of big balls was not a worry.  I even went larger to achieve no topple.  But lately, I'm seeing longer twirls with smaller balls (.25" and .312").  Iacopo's tips are so sharp that his tip drag must be very low.  What's their topple RPM?

As I've mentioned before, the no-topple tops don't spin quite as long because of their big balls.

Today I've been looking at RPM decay rate.  Surprisingly, it's a fairly constant 8% to 10% per minute over the entire twirl period for my better 2" and 2.25" tops.  I expected a more rapid decay at high speed due to aero drag.  But it appears that my twirl ability doesn't get me into the speed range of substantial aero drag.

Alan

PS  A comment on the electric motor curve.  I would presume that the torque is not affected by RPM, but twirl torque greatly affected by RPM.

[Jeremy McCreary]

Today I've been looking at RPM decay rate.  Surprisingly, it's a fairly constant 8% to 10% per minute over the entire twirl period for my better 2" and 2.25" tops.  I expected a more rapid decay at high speed due to aero drag.  But it appears that my twirl ability doesn't get me into the speed range of substantial aero drag.

You're describing a classic exponential decay. (See Wikipedia page.) By all accounts, tip friction is independent of speed and hence couldn't possibly cause such a decay. So if aerodynamic drag isn't responsible, what is?

...but twirl torque greatly affected by RPM

Totally agree and said so several times.

PS  A comment on the electric motor curve.  I would presume that the torque is not affected by RPM.

It's well-known that the torque of a brushed permanent magnet DC motor (the most common kind) decreases linearly with speed, and not just in theory. (See Wikipedia page.) Said that several times, too, and in fact, that's exactly why I suggested this kind of motor as a rough analog offering some potentially valuable insights. Not an exact model of twirling, just a well-understood system with some interesting parallels.

The curves I posted earlier are against time, not speed, but the decline in motor torque with speed still shines through.

[Iacopo]

Wow, if that 70 mm goes all the way to the tip, you have at most 5 mm of lateral wiggle room at the stem! That's pretty tight at high torque.

Approximately 70 mm is the stem alone, for my latest tops. 
All the way to the tip is about 80 mm.
4 degrees for 80 mm should be the same as 8 degrees for 40 mm, as for lateral wiggle room at the stem.
Sometimes I scrape, spinning hard, but not so often.

Thanks for posting the motor spin-up curves, they were interesting to think about.

[Iacopo]

Iacopo's tips are so sharp that his tip drag must be very low.  What's their topple RPM?

It is 165-175 RPM in my latest tops, when they are perfectly balanced.

In my experience spiked tips generally have less friction than ball tips, but then also it depends very much on the materials both of the tip and the base.  If you still have the tungsten you bought, if you polish it, I think it could be an interesting spinning surface.  Glass for spinning surfaces is good but not excellent. 

As I've mentioned before, the no-topple tops don't spin quite as long because of their big balls.

Apart from higher friction, there is another reason why very large balls make for shorter spins;
they worsen the distribution of weight of the top, adding weight to the top, (which increases tip friction), without substantially increasing the rotational inertia of the top.  This worsens spin decay.  For longer spins the countrary is needed, less weight with more rotational inertia, which is obtained concentrating the largest possible part of the weight to the farest outside of the top.

Today I've been looking at RPM decay rate.  Surprisingly, it's a fairly constant 8% to 10% per minute over the entire twirl period for my better 2" and 2.25" tops.  I expected a more rapid decay at high speed due to aero drag.  But it appears that my twirl ability doesn't get me into the speed range of substantial aero drag.

Maybe percentages calculated in this way could be misleading.  If you simply look at RPM lost per minute, you see that they are much higher at high speed than at slow speed.   

[Iacopo]

I have made a slow motion video while spinning this top, and made the following graph from it:


   


I expected something different but the strongest acceleration happens in the second half of the curve, and not at the beginning.  Maybe it's because the fingers are not in optimal position at the beginning of the twirl, so the torque is weaker at the beginning. 
The graph then gives the sensation that the limit is not simply in the power of the hand by itself, but very much in the short duration of the spinning action, which ends after only half a second, when the top is still in full acceleration.

[Aerobie]

Nice work (as usual) Iacopo.

I concur with your comment on finger position.  I just tried very slowly twirling some tops and noting my finger position.  I start with thumb tip and finger tip on the stem, then as I twirl, my thumb and finger roll on stem bringing the stem closer to the root (and the muscle) of my thumb and finger and thus to a location of greater strength.

I also tried reversing the process above.  But the RPM is lower.  Perhaps with practice??

Best regards,

Alan

[Jeremy McCreary]

I have made a slow motion video while spinning this top, and made the following graph from it:

Our first finger-powered spin-up curve (SUC)! Our premiere experimentalist strikes again!

My twirls of the high-AMI, high-drag lime and black top I showed earlier also last ~0.5 sec.

I expected something different but the strongest acceleration happens in the second half of the curve, and not at the beginning.  Maybe it's because the fingers are not in optimal position at the beginning of the twirl, so the torque is weaker at the beginning. The graph then gives the sensation that the limit is not simply in the power of the hand by itself, but very much in the short duration of the spinning action, which ends after only half a second, when the top is still in full acceleration.

Aside from the downward concavity during mid-late twirl, this SUC does bear some resemblance to the SUC of a high-torque DC motor loaded only by AMI. But if the concavity is real and not just an artifact of finger position, your finger torque-speed curve must be nonlinear with the steepest slope when finger-stem contact was lost.

I like your idea that the short duration of finger-stem contact may have something to do with this SUC shape. Off to twirl some tops with some new things to look for...

[Iacopo]

I also tried reversing the process above.  But the RPM is lower.  Perhaps with practice??

I spin with my right hand and I find very natural to spin clockwise.
I suppose that spinning clockwise (with the right hand) makes for a stronger twirl than spinning counterclockwise, because, while spinning clockwise, the index finger flexes, in the other way instead it would have to extend, so different muscles are involved.  Flexor muscles are generally more robust than extensor ones.

if the concavity is real and not just an artifact of finger position,

I observed my hand while spinning.  In the beginning the thumb seems moved by the extensor muscles, (which are weaker), then, it becomes moved by the flexors, (which are stronger). Maybe this too explains the concave curve.
When, at the beginning of the twirl, the thumb is behind the index finger, I feel I have not much power in that position.  When the thumb goes at the side of the index finger, at that point I feel it becomes more powerful.
Also I agree with Alan when he says:

I start with thumb tip and finger tip on the stem, then as I twirl, my thumb and finger roll on stem bringing the stem closer to the root (and the muscle) of my thumb and finger and thus to a location of greater strength.