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[spincakes]

I've been diving into the spinning top science and optimization process for a bit now, and am slowly starting to grasp all the parameters involved. My (simplified) understanding now is that:


• We have a top with x AMI. With a certain stem diameter and grippiness we can spin that up to y RPM. This then gives it a kinetic energy of z Joules.
  - Depending on the scrape angle, stem height, weight and center of gravity, it might be easy or hard to spin the top this fast.
• The top then looses energy due to air resistance (which seems most significant), and tip contact resistance.
  - Surface speed, -smoothness, -shape and -area matter for air resistance. Imbalance increases this as well.
  - Weight, and tip geometry, -material, -imbalance and -forehead oiliness matter for tip resistance.
  - Both are proportional to rotational speed. Tip resistance linearly, air resistance exponentially.
  - I'm neglecting resonances etc. here for now.
• The top will spin until it reaches the critical speed, which is determined by the center of gravity, AMI, TMI, and tip shape(?), then fall.

First question, what am I still missing here?

Jeremy showed me a great calculator that predicts critical speed (and spin time based on a time constant). I feel with estimations of spinning resistance we should be able to somewhat predict the behavior tops that are still being designed, so that's what I'd like to do. I'd be super happy if I could 3D print a large (light) top, and have an idea how long it would spin for before doing so for the first time. For this I'm looking for data.

I feel contact resistance should be 'relatively' simple. Mass x speed x 'magic number discernable from experiments'. I encountered some vacuum experiments here, that should at least give some ballpark figures? Of what would be achievable?

Air resistance seems a lot more complex. Thin disks seem approachable with the von karman disk math? With enough data I feel we should be able to at least be able to make some kind of prediction?

I'd love to hear your thoughts on this!

[Jeremy McCreary]

You're coming up to speed fast!

Wish I could be that optimistic about quantifying the resistances. For starters, tip resistance is probably more complicated than simple Coulomb sliding friction. At the very least, you'd have to bring in Hertzian contact theory.

Worse yet, Iacopo's shown photos of the divots his spike tips can drill in a hard support in a single run. How do you quantify that source of tip resistance??

True, von Karman found an exact solution of the Navier-Stokes equation for the braking torque encountered by a smooth infinite disk spinning in otherwise still air. Empirically, his formula is highly accurate for thin disks with thickness/radius < 10% or so. The edge effects von Karman ignored are apparently very small for disks that thin, but most tops are thicker.

Von Karman got an angular speed exponent of n = 1.5 for his torque, which is a significant departure from a true exponential decay with n = 1 exactly. No such thing as a decay time constant when n isn't 1.

Where speed exponents have been extracted from real SDCs on this forum, I've seen values from 1.0 to 1.5. In looking at the tops involved, I can't make sense of which top goes with which exponent.

Many physicists and engineers have taken a run at the spinning top over the last century. Most don't even recognize air resistance as a major source of spin decay — at least not out loud.

I've read a lot of that literature, and I've never seen an attempt to predict spin time. Despite the fact that spin decay is one of the most obvious things about real tops. Since physicists and engineers tend to pick their battles, I think that's telling.

[spincakes]

Wish I could be that optimistic about quantifying the resistances. For starters, tip resistance is probably more complicated than simple Coulomb sliding friction. At the very least, you'd have to bring in Hertzian contact theory.

Would I need to? My plan would be to do so empirically, gather RPM x Time data from a number of tops in a vacuum. Knowing the AMI I could calculate the actual power losses at each rotational speed, and hopefully find a trend between mass, speed, and energy lost. Tip specifics would matter a lot of course, but gathering data from 'high end' tops I'd ideally get numbers of what would be achievable. I wouldn't need to quantify it from science up, but from experiments downwards.

Where speed exponents have been extracted from real SDCs on this forum, I've seen values from 1.0 to 1.5. In looking at the tops involved, I can't make sense of which top goes with which exponent.

So many questions. I hope to get a grasp at all this, at least enough to work with it for optimization.

If the tip resistance would end up being proportional with mass, it should be possible to take a fixed shape top, measure the air resistance (by subtracting it from a measurement in a vacuum), and optimize it mathematically to keep adding mass (and thus AMI) before tip resistance starts taking over...

[Jeremy McCreary]

I wouldn't need to quantify it from science up, but from experiments downwards.

Ah, forgot that I'm talking to a practicing engineer. Love the idea of pulling optimization guidelines from good spin-down data on some carefully selected reference tops.

We owe our vacuum SDCs mostly to Iacopo and Bill Wells (also an engineer). Iacopo's still have a little convex-downward curvature, meaning that the residual total resistance still has some speed dependence. Bill's have been more linear.

And as I recall, Bill's shown a pretty linear relationship between residual total resistance and tip load with a very clever magnetic deweighting rig.

Would be worth giving all the tops and tips involved in our accumulated vacuum work a very close look.

Question is, how deep a vacuum do you need to make air resistance truly negligible? In a von Karman disk, the aerodynamic braking torque is proportional to the product of air density and sqrt(kinematic viscosity). You can make a significant dent in the density with a garage vacuum rig but would need accelerator-level vacuum to make any dent in the viscosity.

So many questions.

Tell me about it! In the top science biz, the questions seem to multiply a lot faster than the answers. But your empirical approach could definitely help.

[Iacopo]

Iacopo's still have a little convex-downward curvature, meaning that the residual total resistance still has some speed dependence.

I think that the speed dependence is related to the wear out of the contact points, which is more relevant at higher speed.
Not to the curvature of the spinning surface...  I don't even think that there is more tip friction because of the little concavity, compared to a flat surface, because the tip is in contact only with a too little portion of it, maybe 0.01 mm, maybe less, and that portion is practically flat, even if the radius of curvature of the spinning surface is only about 4 mm.

Question is, how deep a vacuum do you need to make air resistance truly negligible?

As I demonstrated in my test, an inexpensive double stage vacuum pump is more than sufficient for this kind of tests, there is still some residual air drag, but really negligible:

http://www.ta0.com/forum/index.php/topic,5248.msg55751.html#msg55751

[Jeremy McCreary]

Iacopo's still have a little convex-downward curvature, meaning that the residual total resistance still has some speed dependence.

I think that the speed dependence is related to the wear out of the contact points, which is more relevant at higher speed.
Not to the curvature of the spinning surface...  I don't even think that there is more tip friction because of the little concavity, compared to a flat surface, because the tip is in contact only with a too little portion of it, maybe 0.01 mm, maybe less, and that portion is practically flat, even if the radius of curvature of the spinning surface is only about 4 mm.

Sorry, I was talking about the slight curvature remaining in your vacuum spin decay curves (SDCs) — not in the support surfaces used.

Any such SDC curvature must mean that total residual braking torque still has some speed dependence. It would not be consistent with tip resistance due solely to simple sliding friction at the tip contact patch, as that kind of frictional torque would have no speed dependence. In the absence of detectable air resistance , the resulting SDC would then be perfectly straight — i.e., no curvature at all.

So what does the slight remaining curvature in your vacuum SDCs mean?

1. Could mean that you really did eliminate all detectable air resistance, and that tip resistance really is speed-dependent after all. In which case, something more complicated than simple friction must be at work.

2. Could also mean that the residual air resistance in your vacuum rig wasn't negligible after all.

3. Maybe something in between.

No. 3 seems the best bet to me, with the truth probably closer to No. 1.

[Jeremy McCreary]

We talked earlier in another recent thread about using the speed drop in a DC electric motor to compare total resistances among tops. We also talked about isolating tip resistance by measuring spin decay curves in vacuum.

Well, there may be a way to use the motor method to compare tops just on air resistance — with all tip resistance eliminated!

At 1:54 below you'll see one of my crazier tops driven upside down in a motorized rig.



With nothing in contact with the tip but air, any speed drop in the motor at a given voltage — relative to the rig's no-top speed at the same voltage — would be due SOLELY to air resistance.

Brushed permanent magnet DC motors are well-suited to this method, as their torque vs. speed curves are highly linear, and the line endpoints (stalled torque and no-load speed) are easy enough to measure. Furthermore, the lines shift upward in a predictable way as voltage is increased without changing slope.

[Iacopo]

I was talking about the slight curvature remaining in your vacuum spin decay curves (SDCs)

What slight curvature...?

These are the curves of my top, (155 grams), in the vacuum, with lubricant, (green numbers), and without lubricant, (red numbers):



This is a heavy top, (656 grams), with lubricant:



This is a light top, (104 grams), with lubricant. Light tops tend to have less tip friction variability:



So, what slight curvature ?  The curvatures that you see are due to the tip friction that changes during the spin, especially without oil and for heavier tops. 

The residual air pressure in the vacuum chamber, no more than 0.5 mm hg, produced an air drag that is no more than 1/300 of that at the normal air pressure.  What is the sense of becoming obsessed about that 0.3% error, when we are dealing with an impredictable tip friction variability which is normally far larger ? 

[spincakes]

As I demonstrated in my test, an inexpensive double stage vacuum pump is more than sufficient for this kind of tests, there is still some residual air drag, but really negligible:

http://www.ta0.com/forum/index.php/topic,5248.msg55751.html#msg55751

This thread and these experiments are a wealth of information! Do you have the 'raw' data somewhere? SDCs and mass / AMI? I'd love to process it with the rest I'm gathering.

With nothing in contact with the tip but air, any speed drop in the motor at a given voltage — relative to the rig's no-top speed at the same voltage — would be due SOLELY to air resistance.

There would still be the extra bearing/friction losses from the added mass (and any forces from wobble) in the system right?

These are the curves of my top, (155 grams), in the vacuum, with lubricant, (green numbers), and without lubricant, (red numbers):

All from the same top - that's an insane difference. Are the numbers related to the order in which they were recorded? Are they all on the same base? Curious how wear plays a role in this - I saw the microscope images in the other thread, quite significant.

This is a light top, (104 grams), with lubricant. Light tops tend to have less tip friction variability:

Light tops it is then. O:)

My next plan is to 3D print 3-5 tops with the exact same shape, but with different densities / masses, and see how the measured losses compare. Either this gives me sensible correlating data, mass vs losses, or it'll show large 'random' variations, that will give me other things to optimize. If the data does correlate I should be able to split the air from the tip resistance, and hopefully estimate the spin time for this particular shape when further changing mass / AMI / CG etc.

[Jeremy McCreary]

I was talking about the slight curvature remaining in your vacuum spin decay curves (SDCs)

So, what slight curvature ?  The curvatures that you see are due to the tip friction that changes during the spin, especially without oil and for heavier tops. 

The residual air pressure in the vacuum chamber, no more than 0.5 mm hg, produced an air drag that is no more than 1/300 of that at the normal air pressure.  What is the sense of becoming obsessed about that 0.3% error, when we are dealing with an impredictable tip friction variability which is normally far larger ?

Perhaps I should have been clearer. In my mind, a spin decay curve (SDC) is a raw plot of angular speed vs. time. The slope of an SDC at any time t is proportional to the total braking torque (TBT) acting on the test top at that time.

"Total" here means the net torque from all resistances present — regardless of their physical causes. There's no way around this.

A straight line in an SDC has a constant slope indicating a constant TBT, as you'd expect from, say, pure sliding friction at the tip in the complete absence of air resistance. Such friction has no speed dependence.

Any departure from a straight line in an SDC points to a speed-dependent TBT. A sagging (convex-downward) SDC — the kind we see in air — has an ever-decreasing slope, indicating an ever-decreasing TBT, as you'd expect when the TBT is dominated by, say, air resistance or the viscous torque from tip lube.

The last linear plot you showed is not an SDC. It's a plot of TBT vs. speed, and whatever the cause, the TBT nearly doubled between between 200 and 1,300 RPM. If that plot is based solely on speed vs. time data, then the underlying SDC must sag. No way around this.

May we see that raw SDC, please?

These are all kinematic observations. They don't settle the underlying dynamics, meaning that they don't by themselves identify the torques making up the TBT. The dynamics are a matter of interpretation based on what we know or theorize about the various resistances involved.

[Jeremy McCreary]

With nothing in contact with the tip but air, any speed drop in the motor at a given voltage — relative to the rig's no-top speed at the same voltage — would be due SOLELY to air resistance.

There would still be the extra bearing/friction losses from the added mass (and any forces from wobble) in the system right?

Ah, forgot about the bearing resistances in the rig. Tops still wobbling at terminal speed in the rig might have to be excluded.

Since the experimental observation would just be the difference between the rig's terminal speeds with and without the test top, perhaps there would be a way to tare out the effect of the top's weight on the rig bearings.

[Iacopo]

A straight line in an SDC has a constant slope indicating a constant TBT...

Ah, sorry, you are saying about the spin decay.  You are right, of course.
The TBT is never constant but generally increases with speed, so the spin decay curves can't be straight.

I am absolutely certain that this does not depend on the residual air drag, and by reasoned demonstration, not on a hunch.

1. Could mean...  that tip resistance really is speed-dependent after all. In which case, something more complicated than simple friction must be at work.

Could this be the guilt ? This is a carbide spinning surface at the microscope after one of my tops spun on it for many hours.
The spiked tip digged the microscopical concavity.



Wear out is always present in the contact points, not only with conical sharp tips, but also with ball tips, even if more difficult to observe, but we know that there are tops with ball tips that do not topple at the end of the spin, because the wear out produced a tiny flattened area in the tip.

[Jeremy McCreary]

The TBT is never constant but generally increases with speed, so the spin decay curves can't be straight.

I am absolutely certain that this does not depend on the residual air drag, and by reasoned demonstration, not on a hunch....
Could this be the guilt ? This is a carbide spinning surface at the microscope after one of my tops spun on it for many hours.
The spiked tip digged the microscopical concavity.



Wear out is always present in the contact points, not only with conical sharp tips, but also with ball tips, even if more difficult to observe, but we know that there are tops with ball tips that do not topple at the end of the spin, because the wear out produced a tiny flattened area in the tip.

Exactly, and some combo of wear, abrasion, drilling, and Hertzian sinkage at the contact is my prime suspect for speed dependence in tip resistance — at least in the absence of excess lube.

If so, tip resistance is way more complicated than simple sliding friction. For one thing, material properties beyond a constant coefficient of sliding friction would have to be involved. And even if you knew how to apply these properties, you may not have reliable values for them.

This problem is begging for the kind of empirical approach @spincakes is proposing.

Whatever the residual resistances, the constant slope in your torque-speed curve indicates a pure exponential decay at that vacuum level.

A decay process with two exponential decays superimposed is called "biexponential". In a top in air, we might have one exponential decay for air resistance and a slower one for tip resistance.

I've tried to fit biexponential decays to measured SDC data on the forum — including that for your silver top (Nr. what?) — with limited success. But too many unknown parameters to do this in any rational way.

Will have to dive deeper into biexponential decay now.

[Iacopo]

Do you have the 'raw' data somewhere? SDCs and mass / AMI? I'd love to process it with the rest I'm gathering.

All the data available are in the thread. The data of the tested tops are written in the photos of those tops, before the graphs.

All from the same top - that's an insane difference. Are the numbers related to the order in which they were recorded? Are they all on the same base? Curious how wear plays a role in this - I saw the microscope images in the other thread, quite significant.

Yes, the spins are numbered chronologically.  All on the same carbide base, the tip too was made of carbide.
It would be very interesting to see what it happens exactly in the contact points areas during the spin. I suppose that the surface of the contact points is not smooth because what we normally call "carbide" is not really pure carbide, but powder of carbide, (tungsten carbide in my case), embedded in a matrix of a softer metal, like cobalt, which is supposed to wear out more easily. The arrangement of the protruding carbide particles from the matrix could determine how much friction there is between the contact points, and, as this arrangement changes, because of the wear, the friction too might change, for this reason.

...I should be able to split the air from the tip resistance,...

I have little time for these experiments now, but I think I have an interesting way for to measure the air drag, with the motor, but a different set up.  I would add to the motor a simple torque meter, composed of two tiny strings, (phishing line?), side by side. The motor axis stays vertical and the top stays hanging on the rotor through the strings. When the motor is off, the strings are straight and parallel.  When the motor is on, after its speed and that of the top stabilize, the strings are no more parallel but twisted, because of the air drag on the top.  A photo could show how much twisted are the strings, and from that the air drag could be deduced.
The weight of the top should be considered in the calculations.  The diameter and the lenght of the strings could be chosen based on the weight and dimensions of the top. I believe that this set up would be more accurate than that with the motor alone.

[Iacopo]

If so, tip resistance is way more complicated than simple sliding friction.

We have to consider that the contact point areas are really little.
If we suppose a contact point with a diameter of 0.01 mm, (but with carbide tips on carbide bases it can be even littler), its area would be less than 0.01 x 0.01 = 0.0001 mm².
It's just one part out of 10.000 parts by which an already tiny square millimeter is divided !
Even if the top weighs only 100 grams, that weight is all on that microscopical area, and the pressure on the contact point areas must be extremely high.
No surprise that there is abrasion !