Archer's paradox: Kooi 1997 study

Kooi and Sparenberg simulate on a computer how an arrow flexes as it leaves a modern recurve, the phenomenon known as the archer's paradox, and compare the results with the arrow shapes filmed by Pękalski with a high-speed camera, using the equipment of one of Poland's best archers. The arrow, an 18.8-gram aluminium Easton 1714X7, bends three times before the nock passes the pressure button: first with the point and especially the nock pushed to the left and the middle to the right (for a right-handed archer), then the opposite way as the string reaches brace height, at around 0.015 seconds, then again like the first time, which helps the fletching clear the bow. The arrow touches the button twice, once briefly and lightly just after the release and once for longer and with more force.

In the model, the arrow parts from the string at the moment the nock crosses the bow's centre plane again, not after a long spell in that plane as Pękalski's model predicted. Changing only the shaft diameter, a weaker arrow (15/64 inch) slaps against the button, while a stiffer one (21/64) clears it with more room but leaves the string with a harder sideways jolt at the nock. The energy of the arrow's sideways motion peaks at 0.05 joules, against the 30.2 joules stored in the drawn bow.

The arrow is a slender elastic beam with a mass at each end, the nock and the point, sliding without friction over the rest and held on one side only by the pressure button, represented as a preloaded spring. The string's push along the arrow is not recalculated: it comes from the dynamic force-draw curve of a 68-inch, 30-pound Greenhorn Comet TD 350 recurve, calculated in the 1991 paper and rescaled to the 66-inch, 34-pound Hoyt Pro Medalist used by Pękalski. The release is described as a sideways movement of the nock of 2.29 mm while the string slides 3.5 mm over the fingers; the nock movement and a coefficient for the bow's lateral elasticity (0.68) were tuned to fit the model to the film data.

The pressure button is a commercial one whose characteristics were measured: a spring constant of 622 N/m, a preload of 3.3 N and 7 mm of travel reached at 7.65 N. Shaft stiffness is the manufacturer's figure, 2.6 N·m², higher than the 2.088 N·m² used by Pękalski. The bow is held still, as on a shooting machine, and is centre-shot. The equations are solved by finite differences, with 32 segments along the shaft and time steps of 0.01 milliseconds.

The comparison with experiment rests on a single film, taken from Pękalski's thesis, and two model parameters were tuned precisely to fit that film: the agreement between calculated and filmed shapes is therefore partly built in. The thrust curve also comes from a different bow from the one filmed, rescaled to have the same energy. The bow is held still and centre-shot, with no archer's hand and no stabilisers, which the authors leave to future work.

The diameter test changes the arrow's stiffness and weight together, so it does not separate the two effects, and it covers only three shafts. The arrow's motion is studied only in the horizontal plane and only until the nock passes the button: the rest of the flight and the point of impact are not considered. In the introduction the authors also mention an estimate of the draw weights of medieval longbows (over 450 newtons for 60-gram arrows) as a possible application of the model, but the study does not recalculate it.

For a recurve archer the study gives a mechanical description of what tuning is trying to achieve. The arrow does not leave straight: it bends several times and, if the spine is right, snakes around the button without its tail touching it; with a shaft that is too weak the nock slaps against the button, with one that is too stiff it clears but takes a harder sideways jolt as it leaves the string. The button is not a passive detail: the arrow presses against it twice, and its spring becomes part of the motion.

The model puts the start of the oscillation down to the string's sideways movement over the fingers at the release, which depends on technique. These remain simulation results on an idealised machine, with one bow and one reference arrow: they explain the reasoning behind spine charts and button adjustment, but they do not replace tuning tests on the range.