This object was made in Tinkercad. https://www.tinkercad.com/things/124l7C6LIuo
This is a catapult designed using gears in mind. Holes are placed on the gears to be able to provide tension to both the string and rubberband sections of the project.
Grooves are placed into the wheels to provide areas to attach rubberbands for traction and to allow for the gears to rotate while the front axle rotates, tightening the strings and rotating the gears, which then tightens the rubber band, bringing the catapult back into launching position. Another rubber band will be strapped to the front of the catapult providing tension and, once released, will allow the arm of the catapult to be pulled forward.
A bar should be placed between the gap where the arm of the catapult lies to allow for a stopping force so that the projectile will launch, this bar could be where the front rubber band can be wrapped around to provide the pulling motion.
A crude rendition of a gear controlled catapult, but one that I am very proud of given the time and patience I placed into making it, I'm very happy with the design results and would love some constructive criticism back! :D