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would one of these do... ?http://www.hardwareandtools.com/cmpsprg49/
If you want to absorb a shock, why not use shock absorbers?
Another method would be to have an arrangement of a vertical bar on each side with short perpendicular bars joined by their free ends with a rope lacing and a rope "spring" like the driving force in the "Onager" artwork posted by corrado33 at the bottom of each. That would give a reaction against the catapult arm starting soft and getting gradually harder.
Btw. does anyone know how they dealt with varying humidity vs range and precision with such rope "springs"?Natural fiber ropes like hemp and linnen change their characteristica enormously going from very dry to soaked, leather, rawhide and sinew have the same issues in that respect.
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Mechanics and Construction