Arrow in the lateral plane, recurve: Park 2012 study

A mathematical model of the arrow's lateral flex during and immediately after the power stroke of a recurve bow, built by extending the work of Kooi and Sparenberg to cover today's carbon shafts, in which mass and stiffness vary along the shaft. The model was checked against high-speed video of sixteen world-class archers, against a static measurement of arrow alignment and against photographs of the arrow in flight taken with a shooting machine.

The computed times sit very close to those measured on the archers: the string leaves the fingers at 1.74 milliseconds against 1.9 measured, the arrow lifts off the pressure button at 6.95 against under 8, reaches the peak of its first flex at 5.2 against 5.4, is straight again at 8.4 against 8.6, reaches the peak of its second flex at 12.9 against 12.4, separates from the string at 14.67 against 14.6 and the nock passes the pressure button at 18.5 against 18. The computed frequency of the first flexing mode is 74 Hz and of the second 228 Hz.

The most useful part concerns tuning. Changing spine by one size calls for about 11 N more or less draw force to keep the arrow free of angular rotation, so for a given draw length one or at most two sizes can be tuned satisfactorily. Lengthening the shaft by 10 mm, from 705 to 715, brings the matching draw force down from 164.5 to 160 N. Going from a 120-grain point to a 100-grain one changes almost nothing (164.5 to 165 N), while taking the fletching from 11 grains to 3 brings it to 161.5 N. Moving centre shot from 0 to +2 mm calls for a change from 163 to 166 N, and no pressure button position comes out better than the others. A shaft Young's modulus 1% higher moves the arrow 25 mm to the right at 70 metres. Two arrows of similar nominal spine but different types do not behave alike: an X10 of size 550 needed 175.5 N, while a Protour of size 570 needed 180 N despite the X10 being 4% stiffer.

The arrow is treated as an inextensible Euler-Bernoulli beam with distributed masses, solved by finite differences, but with two departures from earlier models. The first is that mass per unit length and stiffness are allowed to vary along the shaft, as they do in modern arrows: in the worked case an X10 of 705 mm tapered at both ends, with a 7.78 g steel point, a 0.78 g nock and nock pin, 0.71 g of fletching and a 12.38 g shaft, for a total of 21.65 g. The second is that the model follows the arrow after it has left the bow, where Pekalski and Kooi and Sparenberg stopped at the portion inside the bow.

The lateral forces of the string on the fingers were measured rather than estimated by eye, which the author identifies as the point at which earlier models were subjective. The pressure button is modelled as a spring that deflects only beyond a threshold force, with its own moving mass: in the base case a spring constant of 570 N/m, an initial force of 3.5 N and 5 mm of travel. The longitudinal force from the string comes from Kooi's bow model, scaled on brace height, draw length and draw force of the bow under study. The bow is assumed not to rotate but to gain a small lateral velocity, because its mass is more than 150 times that of the arrow and in the high-speed video neither displacement nor rotation was detectable while the arrow was on the string.

The checking was done three ways. First, by comparing the computed times and shapes with the averages from the sixteen archers filmed at 4,000 frames per second. Second, by mounting the bow in a shooting machine and measuring where the arrow pointed at full draw with the sight centred on the target: at 20 metres it was aligned about 220 mm to the right, with an uncertainty of 25 mm, against 226 mm computed. Third, by photographing the arrow in flight with a flash fired by an electronic timer triggered by the release, so that the real shape could be compared with the predicted one. The bow modelled is the author's own, a Hoyt Nexus with a 635 mm riser and medium 990TX limbs, of similar draw force to the women in the filmed group. Each tuning parameter was then varied one at a time, each time looking for the draw force that brings the arrow's angular rotation to zero.

This is a model, and no archer was measured here: the sixteen used for comparison come from another paper by the same author, published the same year from the same footage. The quantitative check on timings is a comparison of means rather than a case-by-case verification, and the one independent measurement of position, the 220 mm at 20 metres, carries an uncertainty of 25 mm, more than 10%.

The tuning figures apply to one specific bow and arrow, the author's own with 705 mm X10 shafts, and he warns that scaling the dynamic force-draw curve of a different bow introduces an error, because that curve varies a little with arrow mass. The 11 N per spine size, the 4.5 N per 10 mm of shaft and the rest should be read as orders of magnitude for similar bows, not as a table.

Several simplifications are stated openly. Aerodynamic drag along the shaft is left out because it is far smaller than the force from the string. The limbs are assumed not to flex laterally, and the author writes that the assumption needs further examination. The path of the string across the fingers could be modelled in more detail. The lateral energy the bow gains is treated as a simple displacement, without the forces this puts on the rear of the arrow. The author also suggests fitting a pressure sensor to the button, since its interaction with the arrow is computed here rather than measured.

Finally, the tuning criterion is a choice made by the model: everything is set to bring the arrow's angular rotation to zero as it leaves, accepting a small residual angle of attack, 0.7 degrees in the base case. That choice comes from an earlier paper by the same author on down-range flight, not from any comparison of scores.

For anyone tuning a recurve, the most concrete thing this model yields is how little room there is in the choice of spine. One spine size calls for about 11 N more or less draw force, and a bow is usually adjustable over about 10% of its nominal draw force: for a given draw length, then, the shaft sizes that can really be tuned number one, or at most two. The author also points out that a shaft 10 mm longer brings the matching draw force down by about 4.5 N, which makes it sensible to start with shafts slightly long and trim them to the match.

The other finding runs against a common habit: changing point mass from 120 to 100 grains is not a tuning tool, because it moves the matching draw force by half a newton. Mass at the rear of the arrow counts for far more, and changing fletching type moves it by three. The point, according to the author, should be chosen for other reasons, that is for wind drift or for speed, depending on the competition format.

On the pressure button the model finds no optimal centre shot setting, and shows that a very light button slightly reduces the effect of release errors but, if it is too light, produces multiple contacts and so larger errors on the target. In no case does tuning cancel out the effect of an inconsistent release.

Two notes on consistency. A shaft Young's modulus 1% higher moves the point of impact by 25 mm at 70 metres, which is why the shafts in a set should stay together and batches should not be mixed. And twisting the bow does not change the tune, but it does move the arrow on the target: with this bow a difference of just 0.13 degrees from arrow to arrow, which is about 1.7 mm in the alignment of the arrow point against the front stabiliser as the archer sees it at full draw, is enough to lose a scoring ring at 70 metres.