After reading this, I came up with a random amt-ballista design. I should note that locally we've always handled siege and large weapons as special case weapons with their own safety inspection and rules, and are usually built with the permission and knowledge of the local champion and only used in special battlegames when the siege weapon is present, which may or may not resemble regular battlegames with the addition of a siege weapon.
If you read my bow topic in another thread, that technique could be used, drill two holes in a big stick, mount the stick to a support piece, bend the stick with a metal wire string, and for a ballistae probably aim for either 35 lbs or whatever the champion approves, probably with a greater minimum usage range. I think the weapon only needs one actual operator to be considered siege, and may have the additional 2 operators doing something reasonably useless, like holding onto handles on the sides of the long bent piece to transport the whole thing around quickly, possibly while also carrying extra ammo. Ammo for a large "bow" can be made from golf clubs or any sword core, and can be padded as per a sword with a bigger emphasis on the stabbing tip, and possibly a reduced emphasis on the shaft and nock. Fletchings are optional, as our arrows don't typically reach speeds where small wings will achieve any relevant effect on flight.
I would suggest making a smaller bow first, since smaller is usually easier to make, although I've seen some pretty easy improv siege weapons, including one that was basically a large rubber band with two people holding it by either side and a third person standing in the middle and pulling back on it to elastically launch throwing dagger sized projectiles.
Elastic bows work pretty well, and may even be historically accurate, since they did have rubber-like vines and saps in medieval times, to the best of my knowledge. (May be inaccurate) Elastic bows can also be mounted to a cross to form a large crossbow, with the tension power being stored in the elastic string instead of the flexing primary shaft. I sometimes use a split tube or half tube to guide the projectile, although that technically works on an elastic or shaft powered bow.
In terms of what you shoot, in addition to bolts I've seen siege grape shot work well too, usually launching a good number (or even a smaller number) of spellball-like projectiles. Actual spellballs work too, since it's usually pretty obvious that they're not actual spellballs and are siege ammo due to the way they're fired.
Trebuchets work and aren't that hard to make, just a long swinging pendulum-like arm with a long and short side, weight the short side and put the projectile on the long side. Extra moving parts means a bigger safety hazard, primarily for the operator or anyone standing very close to the weapon, although that is a hazard for any large or powerful siege weapon and would be pretty safe it the release momentum was around that of a 35 lb amtgard bow. A movement blocker at the apex of movement to halt the swinging motion enables the projectile to be released from the bucket at the end.
I should note that the primary purpose of medieval siege weapons was not typically to target individuals, they were used to knock down large walls and gates that were creating tactical choke points, occasionally knocking walls down onto the armies they were besieging. One way to simulate this with amtgard is to have an invisible rules defined siege weapon and tactical walls, and there's some rules defined way for the invading team to use the fake siege weapon to destroy the gate, walls, or whatever and allow access past the chokepoint, as well as removing the range increase advantage for archers stationed on high walls and towers, potentially forcing them to retreat or die as the wall or tower collapses. They were also used to attack walled cities, flinging burning or momentum weapons into the city at ranges great enough that the wall archers cannot return fire, nor could the city soldiers leave the city due to losing their wall advantage and losing to the invaders. Firing firey or large bolts into the city would help destabilize the internals of the city, provoking surrender, leaving the walls for aggression and losing the wall defense, or simply weakening the city for a longer siege.
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