Well, now that we've cleared up The Great AMI Misunderstanding of 2021 (for which I'm as guilty as anyone), I'd like to share the formulas I use to estimate the AMI, "specific AMI" (AMI per unit mass), and axial radius of gyration of a flywheel approximating a hollow cylinder. These are handy formulas for any topmaker, as many tops can be usefully decomposed into a series of coaxial cylinders for AMI estimation purposes. The simple rule: AMIs about the same axis add, while specific AMIs and radii of gyration do not.

I'll use my von Braun space station top's big yellow flywheel as an example. Here's the flywheel in the "Top A" variant with no fairings.

This hollow cylinder has outer radius

*R* = 84 mm, inner radius

*r* = 71 mm, and inner/outer "radius ratio"

*q* =

*r* /

*R* = 85%. Its mass

*M* = 114 g accounts for 83% of Top A's total mass and 69% of fully faired Top D's.

**AMI: **Since the flywheel's density is fairly uniform, and the inner gear teeth are negligible, its AMI

*I*_{3} is easily and reliably estimated using the formulas

*I*_{3} = ½

*M* (

*R*² +

*r*²) = ½

*M* *R*² (1 +

*q*²) = ½

*π* *ρ* *L* *R*^{4} (1 -

*q*^{4}),

where

*ρ* is mass density in kg/m³, and

*L* is the cylinder's axial length in m. I prefer the versions with radius ratio

*q*, as they highlight the roles of max radius and relative wall thickness. (Turns out, it's often useful to scale top dimensions by max radius. Then you're left with a bunch of proportions and one very conspicuous measure of absolute size.)

After converting millimeters to meters and grams to kilograms, this gives a flywheel AMI of

*I*_{3} = 6.9e-4 kg m²

This figure represents a whopping 96% of Top A's estimated total AMI and 84% of fully faired Top D's.

**Specific AMI: **The flywheel's specific AMI

*J*_{3} is just

*J*_{3} =

*I*_{3} /

*M* = ½

*R*² (1 +

*q*²) = 6.1e-3 m²

Adding spokes and a core to make Top A bumps AMI from 6.9e-4 to 7.1e-4 kg m² but

reduces specific AMI to 5.2e-3 m². And adding 2 disk fairings to turn Top A into Top D further reduces it to 5.0e-3 m². With each step, the structure as a whole becomes less mass-efficent.

**Axial radius of gyration: **The flywheel's axial radius of gyration

*K*_{3} is just

*K*_{3} = sqrt(

*J*_{3}) =

*R* sqrt[(1 +

*q*²) / 2] = 93%

*R* = 0.078 m

Specific AMI and axial radius of gyration are strictly geometric measures of mass distribution. In its own way, each gauges how much AMI a top gets out of the mass it has. The larger they are, the lower the critical speed. The axial radius of gyration is also useful in AMI measurement with a trifilar pendulum.

Not only that. The relative axial radius of gyration

*K*_{3} /

*R* = sqrt[(1 +

*q*²) / 2] is something you can learn to eyeball in a top. For example...

**Handy fact: ***The axial radius of gyration of a hollow cylinder always lies within its wall.* This example is no exception. In flywheels with large radius ratios, this fact gives you a way to eyeball

*K*_{3}. Mathematically,

*R* >

*K*_{3} ≥

*r***Another handy fact: **These formulas also apply to a

solid cylinder. Just set

*r* = 0 or

*q* = 0 as needed.