Spreadsheet for Throwing Arm
 

Here is a spreadsheet I made for the throwing arm exit velocity calculations.

You can enter the the radius of the throwing arm, and the motor characteristics, and the angles of the arm when it begins and finishes throwing the ball. Also enter the reduction factor of your motor(Gear boxes and everything combined), the weight of the ball+weight of your throwing arm, and your efficiency (.8 is my starting assumption)

Heres how I set it up:
1)Calculate the force of the vector of the gravity of the ball +arm to overcome
2)Using the torque/speed line for the motor find the torque supplied to the arm at that speed.
3)F=ma, find the acceleration on the ball.
4)assuming the circle is a polygon of n=360 determine the distance needed to travel 1deg of rotation
5) Knowing previous velocity and acceleration and distance, calculate the time to travel the 1 degree
6)Knowing time and distance for this degree, calculate both rotational velocity, and strait line velocity.

Wash rinse and repeat each degree until the release point is reached.

If anyone sees mistakes, or improvements I'd be interested.

Re: Spreadsheet for Throwing Arm
 

Are you assuming the ball exits the throwing arm at a tangent to the rotation of the arm? We've found that, depending on our prototype catapult holds/cradles the ball, this is not always the case, and can have a very significant impact on the speed and trajectory of the ball.

Re: Spreadsheet for Throwing Arm
 

That is the assumption. Of course design choices could change the trajectory. More accurately this shows the speed of the arm at exit velocity, it would be up to the design to determine how that is imparted to the ball.

Re: Spreadsheet for Throwing Arm
 

Here's a good paper on this subject:Rolling Release Catapult

Re: Spreadsheet for Throwing Arm
 

I'm not sure your math is correct there. At the least, your acceleration calculation seems wrong. You're basing acceleration off your "Weight N" and "Force applied". "Weight N" seems to be the vertical force of the object you're launching. Acceleration is based off the total force applied and the mass of the object you're accelerating. In your calculations, it's like the mass of the object changes based on the rotation of the arm.

I'm not sure if there's other problems. I usually do my spreadsheets iterating through time instead of position, so I'm not sure if your time per distance calculations are correct or not.

Re: Spreadsheet for Throwing Arm
 

While the weight of the ball doesn't change, the location of the downward vector relative to the pivot does. For example when the ball is directly below the pivot of the arm gravity is pulling down on the ball, and if you look at the line of this vector it passes through the pivot point, effectively making the lever arm 0, and the torque also 0.

Note: I'm not saying that I'm sure this is right, but that was the logic.