When I need to do a lot of math on an arm joint and don’t feel like doing it all manually (which is… all the time), I use John V-Neun’s mechanical design calculator.
The versions just include FIRST motor specs from that year, which you don’t need for VEX. The sheet of interest is the “Rotary Mechanism” sheet, which you can use to design a rotary gearing system for a rotary joint (such as an arm).
The specs for VEX motors are:
Spec Voltage - 7.2 (you must also change “applied motor voltage” to 7.2v for vex batteries)
Free Speed - 100 - This is the same for both 269 and 393 motors
Stall Torque - 0.96 for 269, 1.53 for 393
Stall Current - 2.6 for 269, 3.6 for 393
Free Current - 0.18 for 269, 0.15 for 393
If you are using multiple motors, add their stall torques together in the stall torque cell, but leave the rest of the cells alone. This will show you the estimated current draw for one motor. If you mix 269 and 393 motors, this current will be off, and you will have to calculate it on your own, but you can use it as a guideline.
You can plug in your gear reductions to the table (Driving and Driven gears), using 1 and 1 for unused stages (if you have less than 4 stages of reduction).
Applied motor voltage should be vex battery voltage - 7.2
Lever length - It’s the length from the pivot axle to the point mass of the arm, which is not necessarily equal to the length of the arm. For simplicity, I would use the length of the arm from the pivot.
Gearbox efficiency - I leave it at 70%, maybe this is too conservative, but it seems fine.
Applied Load - This is the weight of the point mass. If you counterforce correctly (with rubber bands or latex tubing), this should only be the weight of game pieces plus a little extra for safety margins (e.g. for 2 game pieces, use 1.5lbs for two barrels + a little extra). If you don’t counterforce enough, you will need to increase the design load as the motors will be lifting more of the arm weight.
The spreadsheet will give you information about rotational speed loaded and unloaded, as well as stall load. A rule of thumb says the stall load should always be 4x or more the applied load you estimated earlier. You can also look at the current to see if you will trip breakers, although that is very stall dependent (how much you hold the load in place with the motors after reaching the target position).
There are also a few other sheets for other mechanism types, I’ll let you play around with them on your own.