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Author Topic: Flywheels and fairings and spokes, oh my!  (Read 7234 times)

ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #75 on: March 05, 2021, 02:44:53 PM »

Once evolution took a few further steps in the right direction and the flywheel provides good results ( spin times), I might get T-shirt with something corresponding to this one:

Please order an extra for me!
 
So what is your guess? What will the last flywheel contour on our T-shirt look like?


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ta0

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Re: Flywheels and fairings and spokes, oh my!
« Reply #76 on: March 05, 2021, 02:58:35 PM »

The ideal shape of the flywheel is a very interesting subject.
The ideal moment of inertia is given by the power of the fingers.
Once selected the material, the density is the other design constraint.
I wanted to figure out what would be the toroid with the smallest area if the moment of inertia and density are fixed.
I turns out that the area decreases as the radius is larger (notice that it gets thinner to keep I constant). It is inversely proportional to the square root of the radius.

However, the total air drag torque is the the torque per square cm integrated over the area. This torque is equal to the radius at each point times the drag force per square cm. And we saw that this force increases with speed. If it increases linearly, that's also proportional to the radius at the point.

So, it seems that the gain due to area with larger radius, 1/sqrt(R), is overwhelmed by the increase in torque by unit area, at least proportional to R2.
If this reasoning is correct, a fat small flywheel beats a thin large flywheel, even before considering the drag of the spokes. I was expecting the reverse  :o

But air drag is not the only factor. Weight affects tip friction and there is also the effect on critical speed. A very complex optimization problem.

My current guess for the ideal flywheel is a squarish toroid, with bottom and top surfaces slanted outwards (so it's taller on the outside).
« Last Edit: March 05, 2021, 03:38:54 PM by ta0 »
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ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #77 on: March 05, 2021, 04:38:29 PM »

The ideal moment of inertia is given by the power of the fingers.

Is that so? I don't see this directly. Probably has been discussed here many times before. Maybe you could point me to one of those discussions if it can't be made clear in two sentences or so.
Also does that not also depend on the spinning technique? Two fingers of one hand, two fingers of two hands (like you start a stemless one), a Frisbee style throw, a handle like you would spin a curling stone....  They all should give  different max possible power/force/strength/speed and also different ratios of those factors that could be relevant here.

Sorry up to this point I am not really making helpful contributions for the theory, only asking questions. ...

Tip friction I would ignore at this stage I think it is of less importance. Maybe only 1/5th  or so as important as air drag. Main argument: vacuum experiments of Iacopo and others.
For low critical speed we have to bring the center of mass as far down a possible, right?
In the design I favor at the moment (see below), looks, maybe by chance maybe not, a bit like the Kemner Dynamo top. There are some practical things that influences parts of the contour: I want a few flat millimeters on the upper surface for the "laser balancing method". The outer vertical line should be straight for a good two hand two finger start if stemless. There should be enough material (~8 mm) across so I can put the setscrews in that I want so badly. Same thing in vertical direction for  dynamical balancing.





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ta0

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Re: Flywheels and fairings and spokes, oh my!
« Reply #78 on: March 05, 2021, 06:52:23 PM »

Yes, it depends on the spinning technique.
There is no gearbox between your fingers and the top, so ideally you would design the top for a moment of inertia at which there is maximum transfer of power. From Iacopo's measurements on the order of 0.0001 kg m2, for a single twirl, but it depends on the person and technique.
« Last Edit: March 05, 2021, 06:55:20 PM by ta0 »
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Jeremy McCreary

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Re: Flywheels and fairings and spokes, oh my!
« Reply #79 on: March 06, 2021, 09:09:51 PM »

The ideal shape of the flywheel is a ... very complex optimization problem.... My current guess for the ideal flywheel is a squarish toroid, with bottom and top surfaces slanted outwards (so it's taller on the outside).

You could be right!

Battle of the toroids
ta0 got me thinking about "toroidal" finger tops with various toroids for flywheels. So I did a spreadsheet experiment taking advantage of the centroid theorem of Pappus to calculate toroid surface areas and volumes. Assumed that each top would have the same low-mass "core" (combined stem+hub+tip assemblies) but then promptly ignored both the core and how the toroids might attach to it.

Also like ta0, I settled first on a flywheel AMI I3 known to give good release speeds with a single twirl at low vertical ground clearance G without scraping. My choice was Simonelli Nr. 33's AMI of 8.02e-5 kg m². Flywheel mass M then became a derived figure of merit, as smaller M would mean less total weight on the tip contact would mean less tip resistance.

Toroids à la Pappus
To make a toroid, you pick a closed plane figure called a "generator" and an axis in the same plane but outside the generator. Then you put the generator into circular orbit about the axis with its centroid at constant "major radius" R, yielding an orbital circumference of C = 2 π R. The generators I examined were the circle, square, and equilateral triangle.

I scaled each generator's size by the toroid's "minor radius" r -- the circle's radius for the circle, and the inradius for the polygons. From r, I got the generator's own perimeter p and area a as shown below. And from Pappus, I then got the toroid's total surface area A = C p, volume V = C a, and surface/volume ratio B = A / V = p / a. Moments are a different story, but as long you twist a generator in-plane about its centroid, A, V, and B all stay the same -- even if the generator twists as it orbits! Twisty donuts, anyone?



Adjustable parameters
This handy way of describing toroids left my 3 toroidal tops with 4 adjustable parameters each: Major radius R, minor radius r, uniform density ρ, and with the toroid mounted on the core, ground clearance G. Could have equalized the toroids' AMIs by varying any combination of these parameters, but this time I varied only minor radius r. Every top then had R = 35 mm, ρ = 8,500 kg/m³ (typical brass), and a challenging G = 4 mm.

Each top's maximum radius X, minimum radius x, central moments I3 and I1, center of mass (CM) to contact distance H, and critical speed ωC followed. The toroidal moments for the circular and square generators came from Wikipedia's list. For the triangle, however, I had resort to the convenient but powerful methods of Diaz et al. (2005).

Generator = circle: Here, r is just the circle's radius. Then X = R + r, x = R - r, p = 2 π r, a = π r², A = 4 π² R r, V = 4 π² R r², and B = 2 / r.

Generator = square: The near and far sides parallel the axis. The inradius r gives the side length s = 2 r. Then X = R + r, x = R - r, p = 4 s = 8 r, a = s² = 4 r², A = 16 π R r, V = 4 π² R r², and again B = 2 / r.

Generator = equilateral triangle: Per ta0's vision, the far side parallels the axis. The inradius r gives the side length s = (6 / √3) r. Then X = R + r, x = R - 2 r, p = 3 s = (18 / √3) r, a = (3 √3) r², A = (36 / √3) R r², V = (6 √3) π R r², and B = 2 / r yet again!

Results
Using rcircle = 3.33 mm, rsquare = 2.95 mm, and rtriangle = 2.59 mm, I dialed in I3 = 8.02e-5 kg m², the AMI of Simonelli Nr. 33.
1. Equalizing the AMIs also equalized the masses and central TMIs. Don't understand the latter, but per Pappus, the former happened because generator areas and toroid volumes all came out the same.
2. All masses M were ~65 g, about 15% under Nr. 33's by virtue of using brass instead of tungsten.
3. All maximum radii X were ~38 mm, also very close to Nr. 33's. The minimum radii x were 32, 32, and 30 mm for the circle, square, and triangle, resp.
4. The triangular and square generators produced 29% and 13% more surface area than the circle due to the same variations in their perimeters.
5. The toroids were 7, 6, and 9 mm tall for the circle, square, and triangle, resp.
6. Top CM-contact distances were 7, 7, and 8 mm for the circle, square, and triangle, resp.
7. Top critical speeds were 422, 410, and 460 RPM  for the circle, square, and triangle, resp.

Bottom line
No clear winner here to my eye -- in large part because surface area alone is unlikely to be a good predictor of air resistance. (Wasn't for von Karman's disk, either.) The aerodynamic shear stress on any small part of the flywheel's surface will vary with both surface orientation and distance from the axis. And even if you came up with an expression for that stress, as von Karman did, you'd still have to integrate it over the entire surface of the flywheel. Easy for a thin disk with negligible edge effects, but a daunting task for a toroid.

Surprising findings
Didn't expect that the formula B = 2 / r for surface/volume ratio would be the same for all 3 generators! Also surprised that mass, major max radius, and central TMI varied so little from top to top, even though they were free to do so. As a result, the small differences in critical speed mainly reflected differences in CM-contact distance.

« Last Edit: March 06, 2021, 11:37:29 PM by Jeremy McCreary »
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Playing with the physical world through LEGO

ta0

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Re: Flywheels and fairings and spokes, oh my!
« Reply #80 on: March 06, 2021, 10:56:28 PM »

Didn't expect that the formula B = 2 / r for surface/volume ratio would be the same for all 3 generators! Also surprised that mass, major radius, and central TMI varied so little from top to top, even though they were free to do so. As a result, the small differences in critical speed mainly reflected differences in CM-contact distance.

The same ratio B for the three shapes looked too much of a coincidence. So I thought about it, and found this is a general property for any regular polygon. If you divide an n-polygon into n triangles, each one with base the polygon side (L) and height the inner circle radius (r) the perimeter is n L and the area is n 1/2 L r => B = 2/r.

That the circular and square toroids have similar weight for the same AMI is not too surprising. I'm more surprised about the triangular toroid. Maybe there is a simple reason, if the centroid is at the same distance to the axis. I need to think about it.
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Jeremy McCreary

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Re: Flywheels and fairings and spokes, oh my!
« Reply #81 on: March 06, 2021, 11:08:11 PM »

The same ratio B for the three shapes looked too much of a coincidence. So I thought about it, and found this is a general property for any regular polygon. If you divide an n-polygon into n triangles, each one with base the polygon side (L) and height the inner circle radius (r) the perimeter is n L and the area is n 1/2 L r => B = 2/r.

That the circular and square toroids have similar weight for the same AMI is not too surprising. I'm more surprised about the triangular toroid. Maybe there is a simple reason, if the centroid is at the same distance to the axis. I need to think about it.

Nice explanation of the surface/volume ratio formula!

If you or anyone else comes up with other toroid generators of interest with fairly simple mathematical descriptions, happy to add a column to my spreadsheet.

Unfortunately, that criterion pretty much excludes ortwin's  planned flywheel and most of Iacopo's as well.
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ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #82 on: March 07, 2021, 01:49:37 AM »

If you or anyone else comes up with other toroid generators of interest with fairly simple mathematical descriptions, happy to add a column to my spreadsheet.

Unfortunately, that criterion pretty much excludes ortwin's  planned flywheel and most of Iacopo's as well.
Since it will be very late in to the year when I finally will have digested all the calculations you present, I might must as well propose the  generator shape of interest to me right now. In my evolutionary diagramm it would  be just the one before the last one. The right angle triangle. The center os mass should be lower for this one.


There is a link missing  in my evolutionary diagram: the additive combination of the tall and narrow rectangle with the flat and wide rectangle. Maybe this shape could also be of interest and not too complicated for describingit mathematically.
« Last Edit: March 07, 2021, 08:16:54 AM by ta0 »
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Jeremy McCreary

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Re: Flywheels and fairings and spokes, oh my!
« Reply #83 on: March 07, 2021, 02:06:05 AM »

@ortwin: The surface area and volume of a toroid with a right triangle generator will be easy. The moments of inertia will be the hard part.

Assuming you want the hypotenuse sloping upward and outward. Will need a specific slope (e.g., rise over run) to put numbers to it.
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ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #84 on: March 07, 2021, 02:41:11 AM »

@ortwin: The surface area and volume of a toroid with a right triangle generator will be easy. The moments of inertia will be the hard part.

Assuming you want the hypotenuse sloping upward and outward. Will need a specific slope (e.g., rise over run) to put numbers to it.


Isoscelic, is that the word? 45 degrees.
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ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #85 on: March 07, 2021, 05:53:25 AM »

...
My current guess for the ideal flywheel is a squarish toroid, with bottom and top surfaces slanted outwards (so it's taller on the outside).


Not sure I understand exactly what you mean by this. Something like the flywheel contour on the left side of the stem in the sketch below?



 :-\ Not directly visible before you click the pic.  >:( I have to read up one of these days how to properly post the pictures here.
« Last Edit: March 07, 2021, 08:18:43 AM by ta0 »
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ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #86 on: March 07, 2021, 06:11:48 AM »

Will need a specific slope (e.g., rise over run) to put numbers to it.
Can't you just have the results presented as formula where that slope is still a parameter? I guess that is not easily possible, otherwise we could easily differentiate by that parameter and optimize the slope.
Probably same story with rectangles where you leave the height to width ratio open or with an ellipse where you do not specify height to with. 

Could you show me the formula for the critical speed? If I am not too scared by the first look of it, I just might try to understand some of it and draw conclusions.


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ta0

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Re: Flywheels and fairings and spokes, oh my!
« Reply #87 on: March 07, 2021, 08:23:29 AM »

My current guess for the ideal flywheel is a squarish toroid, with bottom and top surfaces slanted outwards (so it's taller on the outside).

Not sure I understand exactly what you mean by this. Something like the flywheel contour on the left side of the stem in the sketch below?
 :-\ Not directly visible before you click the pic.  >:( I have to read up one of these days how to properly post the pictures here.
Close, but the bottom surface slanted the other way. My idea was to restrict a bit the Von Karman pumping on the top and bottom.
But my thinking is evolving . . .

To post the whole image, just replace the thumbnail with a direct link to the complete image. But you don't want to link to a very large image. When you upload it, there is a pencil icon that you can click to edit the size.


« Last Edit: March 07, 2021, 08:31:13 AM by ta0 »
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ortwin

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Re: Flywheels and fairings and spokes, oh my!
« Reply #88 on: March 07, 2021, 08:49:35 AM »


Close, but the bottom surface slanted the other way. My idea was to restrict a bit the Von Karman pumping on the top and bottom.
But my thinking is evolving . . .
My reasoning for slanting it like this, is to bring the CM further down compared to the contour on the right side while keeping the scraping angle and AMI the same.
« Last Edit: March 08, 2021, 03:30:40 AM by ortwin »
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ta0

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Re: Flywheels and fairings and spokes, oh my!
« Reply #89 on: March 07, 2021, 09:53:49 AM »

Can't you just have the results presented as formula where that slope is still a parameter?
Sure. Is your triangle slanted out (taller on the outside) or slanted in (taller close to the axis)?
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