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[Aerobie]

I've been looking at some math to compare spins of different sizes of tops and differing launch twirl RPM.

First of all, most of my twirler spin durations are proportional to launch twirl RPM^0.63.

Next, I wanted to account for top diameter and weight.  This led to the following:

       Merit = Seconds / (RPM^0.63 x Diameter^0.5 x Weight)

If diameter is in mm and weight is in Kg, merit for my better tops and twirls is about 20 ish.

My record twirler computed as follows:
Diameter = 57.15mm
Weight = 0.145 Kg
Best ever spin time = 26:05 or 1,565 seconds for a twirl of 860 RM
Merit for this twirl = 1,565 / (860^0.63 x 57.15^0.5 x 0.145) = 20.2

A 1,587 RPM twirl of 16:25 duration with a smaller top (38mm, .082Kg) had a merit of 18.7

This might be used to predict spin seconds ~ 20 x RPM^.63 x Diameter^.5 x Kg

Try this with some of your twirls. 

Alan

[Iacopo]

Here are some data of my tops.  All the following tops have an external, non recessed tip, and they have been spun by a single twirl.  They all have a brass flywheel.
Weights in kilograms and diameters in millimeters.

Top Nr. 25, toroidal flywheel, ruby tip, glass spinning surface.
Diameter = 54.9
Weight = 0.187
Best spin time = 23'56", started from 1238 RPM.
Merit = 11.6

Top Nr. 25, toroidal flywheel, carbide spiked tip, carbide spinning surface.
It is the same top as above, but with a carbide tip instead of the ruby tip.
Diameter = 54.9
Weight = 0.187
Best spin time = 24'38", started from 1234 RPM.
Merit = 12.0

Top Nr. 26, squared edge flywheel, carbide spiked tip, carbide spinning surface.
Diameter = 59.9
Weight = 0.107
Best spin time = 26'20", started from 1162 RPM.
Merit = 22.3

Top Nr. 27, squared edge flywheel, carbide spiked tip, carbide spinning surface.
Diameter = 52.0
Weight = 0.156
Best spin time = 25'40", started from 1220 RPM.
Merit = 15.5

Top Nr. 28, squared edge flywheel, carbide spiked tip, carbide spinning surface.
Diameter = 47.0
Weight = 0.216
Best spin time = 20'10, started from 1194 RPM.
Merit = 9.4

Top Nr. 29, toroidal flywheel, carbide spiked tip, carbide spinning surface.
Diameter = 59.0
Weight = 0.119
Best spin time = 29'38", started from 1310 RPM.
Merit = 21.1

[Iacopo]

The higher starting speed of the top Nr. 29 is due to the long stem.
The long stem makes the top easier and more stable to spin.

I think your formula is too punishing for the heavier tops;
the tops with the higher merit are the lightest ones, and the worst are the heaviest ones.

[Jeremy McCreary]

...most of my twirler spin durations are proportional to launch twirl RPM^0.63.

Good to have you back, Alan! That's a remarkable result. How many tops in your sample? What were they like in terms of shape and construction?

[Aerobie]

Hello Iacopo,

Considering that launch RPM is part of the formula, I think linear weight is theoretically appropriate.  I note that your toroidal wheels have lower merit than your square edged wheels.  Of course a square edged wheel has greater rotational inertia than a toroid of equal diameter and weight.

Here are some merits.   All are cylindrical, except "toroidal" OD.

                               Average    Best
Diameter     Grams     Merit      Merit
25.4mm       43           18.6      19.2
31.5            58           17.9       19.5
38.1           104          14.6       15.2
50.8            78           18.3       20.2
50.8           105          16.6       18.4
57.1           158          15.7       17.2
57.1           183          10.7       12.8   "toroidal" OD
57.1           114          13.2       15.9
76.2           105          15.4       16.4

One could argue that heavier tops are penalized by the formula.
Sometimes a top spins longer than expected for its launch RPM, yielding a merit closer to 20.
 

Best,
Alan

[Iacopo]

I note that your toroidal wheels have lower merit than your square edged wheels.  Of course a square edged wheel has greater rotational inertia than a toroid of equal diameter and weight.

Yes, I thought the same, this is the reason why I specified what shape have the flywheels.
Using the radius of gyration instead of the diameter would be more accurate, but I understand that this last one is easier to measure.

The fact that my toroidal wheels have lower merit than my squared edges (cylindrical) ones is misleading, in fact by the time I collected data that show that toroidal wheels spin a bit longer than cylindrical ones.

For example, my tops Nr. 26 (cylindrical) and Nr. 29 (toroidal) have similar diameter and weight,
and the toroidal one has lower merit in spite of the fact that it spins longer, (29'38" vs. 26'20");
it could be thought that the toroidal one spinned longer just because it was started from a higher speed and no other reasons, but reality is that the toroidal one spins longer even at parity of starting speed, (up to about 28'10" started from 1160 RPM).
Your formula rewards the cylindrical wheel, partly because, as you noted, it overestimates the rotational inertia in toroidal wheels, and partly because this cylindrical wheel is lighter, and your formula rewards lighter tops.

Also I made an experiment which showed that a wheel with rounded edge seems to have lower air drag than a wheel with squared edge, (at the tested speeds):

Considering that launch RPM is part of the formula, I think linear weight is theoretically appropriate.

I think it depends on the exact meaning you give to the term "merit" and what you are looking for.
If you want a light top that spins for long, it could be ok.  You are not simply looking for the top that spins longer, rather, with this formula,  you are looking for a compromise between the lightest top and the longest spin.

[Jeremy McCreary]

Still unclear as to what this figure of merit means at a practical level. If the merit of top A is greater than that of top B, what does that tell us?

Clearly if A and B have the same diameter and mass and start at the same launch speed, the merit will be proportional to spin time. But here, we're comparing tops with significant differences in diameter, mass, and launch speed.

[Aerobie]

Still unclear as to what this figure of merit means at a practical level. If the merit of top A is greater than that of top B, what does that tell us?

Clearly if A and B have the same diameter and mass and start at the same launch speed, the merit will be proportional to spin time. But here, we're comparing tops with significant differences in diameter, mass, and launch speed.

My intent with this figure of merit is to answer the question, "How long does this top spin, after correcting for its size, weight, and launch RPM?" because when evaluating the merit of design parameters I rarely have two tops of identical size and weight available for comparison.  Examples of design parameters are, aspect ratio (diameter / thickness), or perimeter shape (square, rounded, or toroidal).

Note that I've posted merit for tops ranging over a factor of 3 in diameter and a factor of 4 in weight, yet merits are pretty similar. 

Alan

[Aerobie]

I'm surprised that my 57mm tops spin longest when twirled, because one could argue that larger tops would spend more of their spin time in the lower speed range, where aero drag is lower.

I've been assuming that twirl energy (RPM^2 x inertia) is constant for tops of different diameters.  But I've just realized that I've not verified that.  Perhaps I can review launch RPM for tests of various diameters and see if this is true.  Stay tuned.

[Iacopo]

I've been assuming that twirl energy (RPM^2 x inertia) is constant for tops of different diameters.  But I've just realized that I've not verified that.  Perhaps I can review launch RPM for tests of various diameters and see if this is true.  Stay tuned.

This is something I observed in my tops, and I can say that the energy I can put in a top with a single twirl of my fingers is relatively constant, even when the rotational inertias are quite different between them.

Here are some data;

Top Nr.     Moment of inertia           Energy
                      kg-m²                       joule
3                  0.000003                     0.09       
1                  0.000006                     0.11
10                0.000064                     0.51         
8                  0.000045                     0.52     
18                0.000063                     0.53   
13                0.000068                     0.57
12                0.000069                     0.59 
6                  0.000304                     0.61     
15                0.000076                     0.67
20                0.000142                     0.70 
14                0.000067                     0.71

Apart from the first two super light tops, all the others receive, with a single twirl, a quantity of energy which ranges from 0.5 to 0.7 joule.

The giant top Nr. 6 receives 0.6 joule, so it is not substantially different from the others tops.

The lightest tops instead are different. There is no way to put much energy into the lightest tops.  The Nr.1  can't receive more than 0.11 joule.  The Nr.3  no more than 0.09 joule.  So, very light tops cannot spin for long.  Since the moment of inertia is low, energy can come only in the form of higher rotational speed, but there is a physiological limit to speed for the fingers.  I have never been able to spin a top to more than 2400 RPMs (it is 40 rounds per second !) with just one twirl of the fingers, and at this so high speed the top has still only 0.09 joule, (the top Nr. 3).


There are two reasons for the variability from 0.5 to 0.7 joule, it's not a totally random variability;
The tops which I can spin with more energy have knurled stems, and their stems are long.
The two my absolute best tops as for the quantity of energy I can put in them, (Nr. 14 and Nr. 20), have the two longest stems of all of my tops.
My tops unable to receive more than 0.55 joule have all a smooth stem and/or a short stem.
   

[Aerobie]

Hello Iacopo,

I greatly enjoy our discussions.

So, if twirl energy input is relatively constant (except for very light tops), then do you agree that bigger and heavier tops would launch at lower RPM  and spend less time in the RPM range dominated by aero drag?   And, if so, why don't tops bigger than 57 or 59mm spin longest?

At high RPM drag, is aero and tip friction.
At low RPM, aero drag tends to vanish, leaving only tip friction.

Best,

Alan

[Iacopo]

do you agree that bigger and heavier tops would launch at lower RPM  and spend less time in the RPM range dominated by aero drag?   And, if so, why don't tops bigger than 57 or 59mm spin longest?

Yes, I agree, in this sense bigger tops have an advantage.

Anyway, bigger tops also have a disadvantage:  at parity of toppling down speed, a bigger top has more energy than a littler top, when they topple down.  So, the larger the top, the larger the quantity of energy lost when it topples down.

In other terms, it could be said that larger tops are more efficient, (they lose less RPM per minute than littler tops), but, on the other hand, they are launched at lower speed, while the toppling down speed is not necessarily lower, (being unrelated to the size of the top), so there are less RPM useful for a longer spin. For very large tops the difference between starting speed and toppling down speed can be very little.

In your formula you are using Merit as a fixed number for to foresee the spin time of a top, but for my taste there is still a bit too much difference between the merits, (ranging from 12.8 to 20.2 in your tops and from 9.4 to 22.3 in the mine).  I believe the reason is because the formula considers that a heavier top always spins longer, but this is true only up to a certain weight, about one hectogram for single twirl tops with non recessed tip.  From there, the heavier the top, the less it spins.
I am not very good at mathematics, but I think that if you can find a way to consider this in the formula, the merits of the various tops would be more similar.

   

[Aerobie]

I have made tops, which don't topple when spun on an ordinary concave mirror.  They stop, still standing - every time.

I achieve this by a combination of good balance, a very low clearance between the bottom surface of the wheel, and a larger ball tip.  (All my tops have a ball tip).  But the increased aero shear of the low clearance and the increased friction of the larger ball erases the benefit of no topple.  So they are nearly my best, but not quite my best.  But perhaps I should apply this to a larger top.

Another interesting parameter is shaft diameter.  It like choosing the best gear ratio for a dragster.  Currently my best diameter is about 6mm, but larger might be preferred for a larger top. 

Considering your impressive spins, I'd love for you to spin some of my tops.  I'll bet you're a stronger spinner than I am.  Let's get together by messenging and I'll send you a top to spin.

Alan

[Aerobie]

The tops which I can spin with more energy have knurled stems, and their stems are long.

This makes sense to me.  When we twirl hard there is some unwanted lateral motion.  The longer the stem, the smaller the tilt angle imparted by that lateral motion.  So long stems are more tolerant of unwanted lateral motion and permit us to twirl harder.

Alan

[ta0]

The tops which I can spin with more energy have knurled stems, and their stems are long.

This makes sense to me.  When we twirl hard there is some unwanted lateral motion.  The longer the stem, the smaller the tilt angle imparted by that lateral motion.  So long stems are more tolerant of unwanted lateral motion and permit us to twirl harder.

Alan

This is a very good observation that hadn't occurred to me before. Thanks!