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Author Topic: Size matters — just not the way I thought  (Read 191 times)

Jeremy McCreary

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Size matters — just not the way I thought
« on: September 15, 2021, 01:51:47 AM »

Dagnabbit, wrong again!

But air resistance ... may well favor smaller tops. Many tops have disk-like rotors. Consider, then, the swirling air flow induced by a thin smooth spinning disk of radius R.
The resulting aerodynamic braking torque generated by the disk faces grows with R4. Empirically, the braking torque from the disk's edge seems to be negligible up to thicknesses approaching R/10.

Made 2 big mistakes here:
1. Lost sight of the fact that what a player experiences during spin-down is the top's deceleration rate α, not the underlying braking torque.
2. And when you expose all the contributing factors involving R in a disk this thin, you find that α actually improves with absolute size as scaled by R.

Good to know, as keeping α to a minimum at operating speeds is a powerful way to boost spin time.

von Karman's disk
As an oversimplified but useful model of a common top genre, consider the "von Karman disk" — a smooth "thin" disk of radius R and axial length L spinning at constant speed ω about a fixed axis in otherwise still air. Per von Karman, the steady-state air resistance the disk encounters is then

QA = kA ωn R4,

where QA is the aerodynamic braking torque about the spin axis, and kA is a constant depending only on ambient air properties. For the speed exponent, von Karman got n = 1.5.

But just how "thin" does this disk have to be? Per many, many studies, von Karman nailed it as long as the disk's lateral aspect ratio λ = L / R is under 10% or so. Meaning that aerodynamic edge effects are negligible for disks this thin. Only the faces count.

Deceleration rate
Graphically, a top's deceleration rate α is just the slope of its spin decay curve — a plot of ω over time. Mathematically,

α = dω/dt = Q / I3,

where Q < 0 is the net braking torque about the spin axis, and I3 is the disk's axial moment of inertia (AMI). For a von Karman disk subject only to air resistance, Q = QA.

Absolute size and AMI
The AMI of a von Karman disk is

I3 = ½ M R² = ½ (ρ V) R² = ½ π ρ R5 λ,

where M is the disk's mass, ρ is its uniform mass density, and V = π R² L = π R³ λ is its volume. We can now expose all the ways that absolute size, as scaled by R, affects the disk's deceleration rate:

α = kA ω1.5 R4 / (½ π ρ R5 λ) = k ω1.5 / R,

where the new constant k = 2 kA / (π ρ λ) is effectively fixed at design time.

So while air resistance grows very rapidly with increasing absolute disk size, the more practically important deceleration rate actually improves!

Absolute size and critical speed

If a von Karman disk were made into a top with a stem and tip of negligible mass, the square of its critical speed for stable sleep would be

ωC² = 4 M g H (I1 + M H²) / I3²,

where g is the acceleration of gravity, H is the top's CM-contact distance, and I1 is the disk's central transverse moment of inertia. To a very good approximation,  I1 = ½ I3 in a von Karman disk.

After exposing all the ways that R affects critical speed and simplifying, we're left with

ωC² = 4 g η / R   4 g (4 η³ + η) / R

where η = H / R is relative CM height -- another fixed top proportion. Note the cubic depence on η, and the complete lack of dependence on ρ and M.

This result can only be good for spin time, as increasing absolute size now improves both deceleration rate and critical speed — at least in a von Karman-like top.

Absolute size and spin time in real tops
So far, we've been scaling absolute size by max radius R. This turns out to be useful for real tops with other simple shapes as well, as the moments of inertia then scale as R5 G, where G is a purely geometric factor depending only on fixed top proportions like λ and η above.

On hard, smooth surfaces, these tops will generally lose most of their speed early on, when Q ~ QA. Of course, they'll also be subject to tip resistance, and aerodynamic edge effects generally won't be negligible.

But in light of the von Karman model above, seems likely that spin time would increase with absolute size in most tops — at least to a point. That's certainly the trend I've seen in my own tops, and Iacopo's reported the same trend in his.
« Last Edit: September 15, 2021, 03:30:27 PM by Jeremy McCreary »
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ortwin

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Re: Size matters — just not the way I thought
« Reply #1 on: September 15, 2021, 03:23:05 AM »

....
where η = H / R is another fixed top proportion. Note that ρ and M are nowhere to be found.
...

@Jeremy: Not that I would have digested all that by now, but still I want to start the discussion on the role of ρ.

ρ might not be found directly, but in the kind of endurance tops that I fool around with, "external tip top", in comes into play through H . Smaller
ρ makes the disk or flywheel thicker thereby moving CM upwards.
That means it still makes sense to go for platinum and such.
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ta0

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Re: Size matters — just not the way I thought
« Reply #2 on: September 15, 2021, 10:13:57 AM »

I checked the first part, but I still need to look at the critical speed stuff.
It's interesting to see that in spite of the very large R4 dependence of the drag, the deceleration still decreases with R because I3 increases as R5.
Now, if you were making tops cutting them from a board or a sheet, so the thickness is constant, the deceleration due to drag would be independent of the radius of the top.
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Jeremy McCreary

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Re: Size matters — just not the way I thought
« Reply #3 on: September 15, 2021, 11:29:28 AM »

NB: Corrected an error in the final critical speed formula. No effect on my conclusions.
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BillW

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Re: Size matters — just not the way I thought
« Reply #4 on: September 15, 2021, 01:16:09 PM »

Dagnabbit, wrong again!
Jeremy, this is all so good I am going to print it out so to study more thoroughly.
Thank you for defining all symbols.

However, I have made it through your excellent post all the way to your second line:

"1. Lost sight of the fact that what a player experiences during spin-down is the top's deceleration rate α, not the underlying braking torque."
Newton's second law, F=ma, when applied to rotating objects, becomes τ=Iα. So the top's deceleration rate α is directly proportional to the top's total  braking torque τ. I don't think you made a mistake.

More later...
« Last Edit: September 15, 2021, 01:22:03 PM by BillW »
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Jeremy McCreary

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Re: Size matters — just not the way I thought
« Reply #5 on: September 15, 2021, 03:04:26 PM »

Dagnabbit, wrong again!
Jeremy, this is all so good I am going to print it out so to study more thoroughly.
Thank you for defining all symbols.

However, I have made it through your excellent post all the way to your second line:

"1. Lost sight of the fact that what a player experiences during spin-down is the top's deceleration rate α, not the underlying braking torque."
Newton's second law, F=ma, when applied to rotating objects, becomes τ=Iα. So the top's deceleration rate α is directly proportional to the top's total  braking torque τ. I don't think you made a mistake.

Many thanks for the kind words. But I did screw up -- by failing to think absolute size all the way through to actual top performance in play. The performance being limited most noticeably by spin decay.

Like every airplane, every top has its own "envelope" of speeds and motions and spin times attainable in play. Deceleration rate has a lot to do with that envelope.
« Last Edit: September 15, 2021, 03:09:34 PM by Jeremy McCreary »
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ta0

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Re: Size matters — just not the way I thought
« Reply #6 on: September 15, 2021, 03:48:30 PM »

"1. Lost sight of the fact that what a player experiences during spin-down is the top's deceleration rate α, not the underlying braking torque."
Newton's second law, F=ma, when applied to rotating objects, becomes τ=Iα. So the top's deceleration rate α is directly proportional to the top's total  braking torque τ. I don't think you made a mistake.
In Jeremy's case, the moment of inertia was also varying when changing R, so in that sense the acceleration (deceleration) was not proportional to the torque when varying R.
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Jeremy McCreary

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Re: Size matters — just not the way I thought
« Reply #7 on: September 15, 2021, 04:31:32 PM »

Yes, everything important here varied with absolute size as scaled by R.

The central question -- which I could have made clearer -- was about net effects of air resistance and absolute size on spin time in a top type with a reliable formula for the air resistance. And to my knowledge, the von Karman top is the only game in town.
« Last Edit: September 15, 2021, 05:13:12 PM by Jeremy McCreary »
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BillW

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Re: Size matters — just not the way I thought
« Reply #8 on: September 15, 2021, 06:04:11 PM »

In Jeremy's case, the moment of inertia was also varying when changing R, so in that sense the acceleration (deceleration) was not proportional to the torque when varying R.
I should have said: For the same rotational moment of inertia ... a top's deceleration rate α is directly proportional to the its total  braking torque τ.

So what we need is an experiment or two... or three.
Do we want to compare spin times for thin disks of same moment of inertia I but different R?  I'm ready to spin! :D
« Last Edit: September 15, 2021, 06:06:22 PM by BillW »
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Jeremy McCreary

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Re: Size matters — just not the way I thought
« Reply #9 on: September 15, 2021, 06:07:58 PM »

Now, if you were making tops cutting them from a board or a sheet, so the thickness is constant, the deceleration due to drag would be independent of the radius of the top.

That's correct. And I could have chosen to scale the absolute size of the von Karman disk by thickness L instead of radius R.

However, von Karman ignored thickness and got away with it in sufficiently thin disks. Now, there are several competing formulas for air resistance in thicker disks as a function of both R and L. But experimental support is shakier, and I'm still not sure which one best applies to tops.

When it comes to calculating mass properties, max radius again emerges as the most convenient way to scale absolute size in all common top shapes with tractable mathematical descriptions.
« Last Edit: September 15, 2021, 06:30:00 PM by Jeremy McCreary »
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Jeremy McCreary

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Re: Size matters — just not the way I thought
« Reply #10 on: Today at 01:40:33 AM »

So what we need is an experiment or two... or three.
Do we want to compare spin times for thin disks of same moment of inertia I but different R?

Been thinking about how best to test the above relationships among absolute size, deceleration rate, and critical speed with some real disk tops. But in the end, I think your time and effort would be better spent on the experiments you're reporting in the Tip friction and air drag measurements thread.

Bottom line: If these relationships are in error, it's probably due to a stupid math error on my part. The rest is on pretty firm ground.

Recall that the focus was on a smooth, thin "von Karman disk" (VK disk) of radius R, axial length L, and lateral aspect ratio λ = L / R ≤ 10% spinning about a fixed axis in still air. Forgot to mention that the VK disk also operates at Reynolds number

Re = ω R² / ν < 300,000,

where ω is the angular speed in rad/s, and ν = 1.75e-5 m²/s is the kinematic viscosity of air at room temperature and pressure (RTP).

von Karman's air resistance formula
Not much point in testing this part, as many studies have confirmed the accuracy of his formula for the air resistance (aerodynamic braking torque) acting on such a VK disk:

QA = kA ω1.5 R4,

where QA is the aerodynamic braking torque about the spin axis, and kA = -9.94e-3 kg/m²√s is a negative constant with the same value for all VK disks operating at RTP.

Can you make real VK disks?
Yes! No problem making circular disks with λ ≤ 10%, of course. And an unusually large and fast disk with R = 0.125 m and ω = 314 rad/s (3,000 RPM) still qualifies at Re = 280,000.

Is a VK disk inertially "thin" as well?
Yes! This is tantamount to asking if the VK disk's central moments of inertia obey

I1 = ½ I3,

to a sufficient approximation, where I3 is the axial moment, and I1 is the transverse moment through the CM. It's easy to show that for λ ≤ 10%, the error here is at most 0.33%.

Is it possible to make a real VK top?
You can come pretty close. A "von Karman top" (VK top) has a VK disk for a rotor and a slender stem and tip of much smaller mass. Something akin to a koma with a single thin metal rod for both stem and tip. To a good approximation, the mass properties of the top are then those of the VK disk.

The well-established formula for the square of the top's critical speed is then

ωC² = 4 M g H (I1 + M H²) / I3²,

where M is the top's mass, g is the acceleration of gravity, and H is the top's CM-contact distance.

Given this foundation, the relationships I gave should test well if my triple-checked math is correct.



But if you really want to test these relationships, I suggest the following approach:
1. Stick to λ ≤ 10% and Re < 300,000 to keep the von Karman formula applicable.
2. Vary R while keeping the product (ρ λ) constant — say, with 3 different VK test tops.
3. Measure spin decay curves (SDCs) for all 3 tops.
4. Estimate the SDC slopes at high speeds so that QA makes up most of the total braking torque.
5. Check these slopes against the deceleration rate

α = k ω1.5 / R,

where ρ is the mass density of the VK disk. The constant k = 2 kA / (π ρ λ) will then be the same for all 3 tops.

This experimental program shouldn't be too hard to implement with your skills and equipment. You won't need the critical speed.

« Last Edit: Today at 02:02:49 AM by Jeremy McCreary »
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