Trebuchet Workshop

A trebuchet is a seesaw built for throwing. A long wooden beam turns on an axle. The short arm carries a heavy box of stones, the counterweight. The long arm ends in a sling: two ropes and a leather pouch holding the stone. One rope is tied to the arm, and the other ends in a loop that sits over an iron prong at the tip.

When the trigger lets go, the counterweight falls and the long arm whips up. First the stone slides along a trough under the frame. Then it lifts off and swings around the end of the arm, faster and faster. When the sling reaches the release angle, the loop slips off the prong, the pouch opens, and the stone flies free.

The bar in the bottom corner shows where the counterweight's energy goes during the swing: into the stone's motion (gold), the beam's spin (blue), the counterweight's own motion (green), lifting the stone and beam (pink), or still waiting in the raised counterweight (gray). At the moment of release, the gold part is the machine's efficiency.

The Range finder scene leaves the machine out. It throws a stone from the ground at any speed and angle you choose, so you can study the flight on its own.

The first trebuchets had no counterweight. In a traction trebuchet, a team of people hauled on ropes tied to the short arm. These machines were used in China by the 4th century BC and are described in the writings of the Mohists. They spread west, and by the 6th and 7th centuries AD the Byzantines, Persians and Arabs used them too.

The counterweight trebuchet appeared around the Mediterranean, in both Christian and Muslim lands, in the 12th century AD. The earliest known description and drawing of one is by Mardi ibn Ali al-Tarsusi, from around 1187. Both fixed and hinged counterweights were built. Counterweight trebuchets reached China with the Mongol armies, during the long siege of Xiangyang and Fancheng (1268–1273).

Don't mix trebuchets up with torsion catapults such as the Greek and Roman ballista and the onager. Those are different machines: they store energy in twisted bundles of rope or sinew, not in a raised weight.

For 25 sieges from history, from Troy to Rome, play Siege.

The model. The machine is described by three angles: the beam's, the counterweight hanger's and the sling's. From the kinetic energy T and the potential energy V of the beam, counterweight and stone, Lagrange's equations, d/dt (∂L/∂q̇) − ∂L/∂q = 0 with L = T − V, give the angular accelerations, solved as a small matrix system. While the stone slides in the trough, it is held on the floor by a constraint whose force is found with a Lagrange multiplier. When that force would have to pull down, the stone lifts off. The sling lets go when its angle to the arm drops to the release angle (0° means the sling points straight out along the arm). Then the stone flies under gravity and, if you choose, quadratic air drag.

The numbers. The beam is 8 m long, a uniform rod of 150 kg, on an axle 4.6 m up; the cocked arm's tip rests 0.6 m above the ground. The hanger is 1.2 m long. Stones are granite balls (2,700 kg/m³), so a 15 kg stone is 22 cm across. Air is 1.2 kg/m³, C_d = 0.47, g = 9.80665 m/s². The swing is integrated with the classical Runge–Kutta method in steps of 0.5 ms, and the flight in steps of 2 ms. Range is measured from the axle to where the stone touches the ground.

What's left out, and why. There is no friction in the axle, hinge or trough before release, the ropes don't stretch, and the frame doesn't move. The masses are points and the beam has the same thickness all along (it is drawn tapered). C_d is held fixed, although NASA notes that a sphere's drag coefficient ranges from about 0.07 to 0.5 with speed and size, and real stones are lumpy. There is no wind and no spin. After release, the beam's swing is slowed by extra friction so it settles; that doesn't change the shot. These choices keep the physics clear, so treat the ranges as good estimates, not predictions for a real machine.

How it was checked. The equations of motion agree with a finite-difference Lagrangian built only from positions, to within 4 × 10⁻⁵ at 80 random states. Before release, the total energy stays constant to within 1 part in a billion of the stored energy, in five very different machines. Vacuum flights match v² sin 2θ ÷ g, the flight time and the peak height to within 2 × 10⁻⁷, and the best vacuum angle comes out at 45.0°. A 15 kg ball dropped from 3 km lands at 116.5 m/s, close to its terminal speed √(g/k) = 117.3 m/s. Cutting the time steps to a fifth changes the range by less than a millimeter.