I recently went down a deep rabbit hole analyzing the physics of flywheel shooters and found an issue with previous work on the subject. I’ve documented my findings here. Here’s the abstract:

This paper describes a new mathematical model for the physics of a hooded flywheel shooter, derives equations for the initial wheel velocity and the ratio of the initial wheel velocity to the final wheel velocity, and compares the results to previous work. The analysis predicts that the wheel speed lost during a shot is half of what is predicted by previous work. A formula for the shooter’s mechanical efficiency is also derived and shown to be less than 50%.

The assumption of rolling contact is a big one. Our 2022 shooter def didn’t operate in that range, there were distinct warm spots on the ball after every launch. I’d be surprised if our 2021 shooter was in the nonslip range, it certainly made plenty of powder.

I’ve had our students do fine with an empirical model - our first prototype has a few different ranges of adjustability, they tune in what they like, and then we cut final plates. For an initial cut at gearing, I run 1:1 with a CIM, but with space for a belt or geartrain so we can go up or down if we want to.

Empirically, running faster surface speeds and getting into the slip regime delivers more achieved projectile range. I haven’t seen the efficiency loss of slipping result in worse range performance.

Note that neither distinct warm spots on the ball nor the existence of powder is evidence that the ball was slipping when it exited, only that it was slipping at some point. My analysis doesn’t assume that the ball was never slipping, only that it had stopped slipping by the time it exited. Of course, it is still possible that your shooters had balls still slipping on exit, and something else it going on that caused the exit velocity to vary with wheel speed.

Very nice paper. I’ll agree with others that the assumption of non-slipping is probably incorrect in many FRC applications, where the projectile is only being accelerated for a few milliseconds. But since that assumption only matters as the projectile leaves the shooter, it’s probably close enough.

I’m a bit confused on the efficiency section. If we look at the system of the wheel and projectile, the only outside force acting is the friction with the hood. Is the suggestion that energy lost through that force makes up more than 50% of the total energy lost by the flywheel with each shot? Intuitively, that seems pretty high to me.

The only big thing this paper is missing IMO is application to FRC. Deriving these equations is great, but how would a team use them to actually design a flywheel shooter? An example from a recent shooting game would be amazing, and even more so if you can then compare the theoretical results with some empirical testing.

There is also friction between the wheel and the projectile. Energy lost in the form of heat at that interface will be much larger than any heat generated at the interface between the hood and the projectile because of the much larger initial in surface velocity. That is the source of the warm spots on the balls that @s-neff referred to. I do agree that the energy lost is higher than I initially expected. That’s why I included it in the analysis.

I hope to add suggestions related to that to a future version of the paper, but as a first pass, how about:

Determine maximum exit velocity (v_p) needed based on proposed shooting angle and distance to target.

Choose a wheel radius (r_w) substantially smaller than the projectile radius (r_p) so that the overall shooter size is driven primarily by the size of the projectile.

Use the equation for final wheel rotational velocity to determine the required final wheel speed (\omega_{wf} = 2v_p/r_w).

Choose motor(s) and gearing so that the final wheel speed corresponds to the something close to the peak of the power curve. This ensures that the wheel will be able regain it’s lost speed nearly as quickly as possible.

Solve the wheel velocity ratio equation for the moment of inertia of the wheel (I_w) when the wheel velocity ratio (\omega_{wi}/\omega_{wf}) is, say, 1.5. With that moment of inertia the motor will stay in the high power region of the motor curve while recovering.

Achieve the desired moment of inertia by choosing the shooter wheel’s mass (m_w) and shape (k_w). An additional flywheel can also be added and gearing it to spin faster than the shooter wheel can create a higher effective moment of inertia in a more weight-efficient fashion. This is not covered in the paper.