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

*I*_{3} 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

*I*_{3} and

*I*_{1}, center of mass (CM) to contact distance

*H*, and critical speed

*ω*_{C} 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

*r*²,

*A* = 4 π²

*R* *r*,

*V* = 4 π²

*R* *r*², 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* =

*s*² = 4

*r*²,

*A* = 16 π

*R* *r*,

*V* = 4 π²

*R* *r*², 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)

*r*²,

*A* = (36 / √3)

*R* *r*²,

*V* = (6 √3) π

*R* *r*², and

*B* = 2 /

*r* yet again!

**Results**Using

*r*_{circle} = 3.33 mm,

*r*_{square} = 2.95 mm, and

*r*_{triangle} = 2.59 mm, I dialed in

*I*_{3} = 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.