Pocketed flywheel dimensions

We want to add a pocketed flywheels(aluminum or brass) to our shooter.

Our shooter specs:
2x Neo direct drive.
2x 4" x 1" WCP Solid roller wheels
1x 4" x 2" WCP Solid roller wheel

Is there any math that is used to determine how the flywheels should be pocketed?
What is the diameter of your flywheel? Is it the same as the shooter wheels, or smaller?
What are other teams doing for their pocketed flywheels?

What’s the goal of the pockets overall?

I’m no mechanical engineer, but generally I imagine that pocketing reduces the total weight of the flywheel, while still keeping the mass that remains far from the center of mass (keeping moment of inertia up). It’s just an extra variable you can play with to achieve a specific moment of inertia, with constraints on physical dimensions or weight.

I think the more informative question might be to ask what sort of moment of inertia you’re looking for.

This could be derived easily from a requirement on spoolup time (how fast from stopped to shooting speed do you want your motor?).

The more meaningful, but far more complex way to derive it, is to ask how much RPM pull-down per shot is acceptable. Most teams determine this through experimentation, not first-principles math, as the answer depends heavily on ball quality, wheel and backplate material, ball compression, etc.

Assume the pocketed flywheel looks something like this. A thick outer section and a thin center, with a bearing hole in the center. The bolt circle is ignored in the calculation, as well as any radius on the transition from thin to thick sections.

To get the moment of inertia, you can use the equation I = \tfrac{1}{2} \rho \left[ b_1 \left( r_1^4 - r_0^4 \right) + b_2 \left( r_2^4 - r_1^4 \right) \right] where \rho is the material density, b_1 and b_2 are the inner and outer thicknesses, and r_0, r_1, and r_2 are the bearing, transition, and outer radii, respectively.

 
Actually calculating how much MoI you need is a much harder problem. The bottom line is there’s no good way to tell how long it will take to spin up a certain MoI flywheel with certain motors other than prototyping. There are just too many hard-to-measure variables at play. However, if you know your launch velocity and a minimum acceptable rpm that the wheel can drop to, you can use conservation of energy to figure out what MoI you need to propel the ball at the right velocity (with a fudge factor for compression) and still stay above the minimum. You’d be solving the equation \tfrac{1}{2} I \omega_{ss}^2 = \tfrac{1}{2} I \omega_{min}^2 + \tfrac{1}{2} m_{ball} v_{ball}^2 \cdot k for I, where k is the fudge factor. You can guess at k, but it will also be highly dependent on your specific shooter configuration.

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