I whipped up a quick spreadsheet to calculate the maximum theoretical ball velocity from a wheeled (single and double) shooter and the associated wheel speed drop.
It’s easy to estimate shooter power by calculating the free load surface velocity of the wheels (and divide by 2 in the case of a single-wheel shooter) and ignore the speed drop due to the transferred energy. For our prototyping efforts I wanted a better way to calculate the theoretical maximum performance of a shooter in order to evaluate prototype efficiency.
The assumptions this mathematical model makes:
The surface velocity of the ball and the surface velocity of the wheel match (no slip) when the shot is released.
No energy loss due to friction (the main source of inefficiency in a shooter AFAIK).
No energy loss due to heat from compressing the ball.
No energy added by the motor(s) during the shot.
For single-wheel shooters, the ball being compressed does not affect the circumference of the ball (i.e. the relation between linear velocity and angular velocity of the ball).
Double-wheeled shooters have the wheels spinning at the same velocity.
Let me know if this is useful to you or if I made some big mistake.
EDIT: I’m still a little fuzzy on how link sharing works in Google Drive. I think I have it set up correctly so you can copy the spreadsheet to your own drive in order to edit it. Let me know if you can’t.
Oh, certainly. These calculations are only for determining the strict limits defined by physics, not predicting real-world shooter performance.
The only objection I would make is that motor power is a minor effect. The contact time in most shooters is on the order of (low) tens of milliseconds. I think it’s in the neighborhood of .02-.03sec on our prototype. In that time a CIM, at peak power, can only add ~7J of energy to the system. Depending on the mass of your flywheel, most of that energy will go to spinning up the wheel. And the CIM will rarely be running at peak power in the prototypes that most teams build.
Even if the amount of power added during that time is small, the motor back EMF will help counteract any forces that try to slow the wheel (e.g. it acts as a generator).
Back EMF decreases as motor speed decreases (given a constant voltage). The motor acts like less of a generator at lower speeds.
Back EMF decreasing does increase the torque the motor produces, but that’s still governed by the amount of power the motor can produce at a given speed.
A CIM slowed down to 90% of free load speed will produce ~121W. Which means in the .02-.03sec it takes for the ball to pass through the shooter, the CIM will only add ~3J to the system. Most of that energy will end up in the flywheel (assuming it’s mass is much larger than the ball, which it should be).