More than two wheeled linear shooter and the math behind it?

Does anyone know the math behind the linear shooter design and if more than two wheels would help boost the distance of the shooter?

Assuming our parts are in, I was going to try a three wheel shooter this weekend just to see if it made a difference. So I might have an answer Saturday evening.

Get a camera (preferrably high speed) to watch the act of shooting.

Our team is using a flat belt of material to accelerate the disk in a linear shooting design; as such, we must simply make the shooter long enough to accelerate the disk completely. This is very quick process.

To model this with more accuracy, we think of the forces acting upon the disk. The disk experiences forward acceleration due to rolling, which is based on the angular acceleration from the belt, which is related to the torque of the motor.

The motor accelerates the belt based on the moment of inertia of the belt system, which is deterined experimentally; consider the motor and belt as a single system, with the motor just being the active torque on the belt.
The belt accelerates the disk (and the disk accelerates the belt) based on the moment of inertia of both the disk and the belt - use conservation of momentum.
The angular speed of the disk is related to its rate of travel along the shooter, assuming no slipping.

(1) T[belt] = I[belt] * a[belt] ; relation of torque of belt to angular acceleration.
(2) L[belt] = I[belt] * w[belt] ; relation of angular momentum of belt to angular velocity.
(3) L[disk] = I[disk] * w[disk] ; the above for the disk, assumed to be zero as w[disk] will be zero before the collision.
(4) L[belt] + L[disk] = L[belt + disk] ; conservation of momentum (might be unnecessary?)
(5) T[belt+disk] = I[belt+disk] * a[belt] ; now the motor is accelerating the belt and disk.
(6) T[belt] = T[disk] = T[belt+disk] ; they’re equivalent (redundant, but note this).
(7) a[belt+disk] = T[belt+disk] / I[belt+disk] ; acceleration of the system.

Okay I’m going to stop because I’m running out of time and I think I messed up, but now you need to find out how long the (possibly variable) force must be applied to reach equilibrium (running speed, when the disk is fully accelerated) and the distance of this act. Then, you take the angular acceleration of the belt, integrate to turn it into angular velocity, then you need to turn that into tagentital velocity, integrate for position, and see if what you get is feasible (time to equilibrium <= acceleration possible in your shooter’s distance.

Feeding the disk into the mechanism with the belt already spinning will cause slippage. This probably is not an issue; if the time it takes to accelerate the disk is 1/2 (probably much less) of the total time it takes to shoot, approximately 1/2 (probably much less) of that would be spent slipping in my estimate. If you wish, you could model the force during the time spent slipping as related to the difference in tangenital (as opposed to angular) velocity of the disk and belt.

F = k * (v[belt] - v[disk]) ; where k is a constant.

If your mechanism is not a flat belt, but wheels, the approximation should be the same. At first I thought the varying distance to the wheels from the center of mass of the disk would cause varying force/acceleration, but this shouldn’t be the case (I think). Bumping as the disk transition wheels would probably have more of an effect on the acceleration of the disk.

Here are a few rules of thumb:

For a 2-wheel shooter, you want to spin the second wheel about 35% faster than the first.

There’s a point above which if you increase the wheel speed, the speed of the frisbee will actually decrease (due to excessive slipping).

Greater moment of inertia in the wheels allows them to maintain their speed as a frisbee passes through.

Use closed-loop speed control (like bang-bang) to improve spinup and recovery performance.

You want the frisbee to roll on the guide rail, not slide. Do not use a low-friction surface for the guide rail.

Experiment with wheel material, frisbee compression, and frisbee placement (in the axial directiom) relative to the wheel.

Out of curiosity, where did you get the 35% figure from?

It’s a place to start, and then experiment from there. If you assume there’s no slipping and you want each wheel to contribute the same amount of kinetic energy to the frisbee, then you get V2=V1√2. If the first wheel is slipping too much and the second wheel is not, you want to slow the first one down and speed up the second one, and vice versa. Then you’ll have a speed ratio where you’ve got about the same amount of speed headroom in both wheels.

We will be testing this theory. Currently first wheel is geared 30% less than the second. Also will be giving a not insignificant initial velocity to the frisbee before it enters the shooter wheels

IMO, the focus for solving the slipping issue should initially be directed toward maximizing the friction factors, not RPM.

We are concentrating urethane materials with better properties to match the task being in contact with the disk edge. We are also concentrating on having two points of wheel contact from slightly above and below the peak of the disk edge curve, to better capture the disk and maintain better consistency of motion.

Once we feel these items are well handled, we will then further tune with RPM adjustments.

-Dick Ledford