Arrow spine and shaft selection: Kuch 2025

Sixty Easton X10 shafts of spine 325 were measured all the way round their circumference on a purpose-built bench, assembled into arrows and then shot by a machine, five times each in random order, indoors at 70 metres. One arrow was damaged after the first session, so the analysis covers 59 arrows.

The mean vertical position on the target is explained to 47% by two things: the mass of the assembled arrow (standardised coefficient minus 0.525, p below 0.001) and the mean spine (0.248, p equal to 0.032). The heavier the arrow, the lower it lands. The horizontal position is explained to 19% by the mean spine (minus 0.378, p equal to 0.004) and by the amplitude with which the stiffness varies as you go round the shaft (0.325, p equal to 0.012): the stiffer the shaft, the further left; the more the stiffness varies round the shaft, the further right.

Straightness and diameter enter neither model, with all the shafts inside the manufacturer's tolerance of plus or minus 0.025 millimetres. Nor does the angle between the reference vane and the stiffest point of the shaft, against what the archers and coaches involved expected.

For the closeness of the arrows to each other, which is a different question from where each one lands, the best combination of parameters is mass, mean spine and the spatial frequency of the stiffness variation. Some shafts had a different and lower frequency from the others, which the authors read as possible manufacturing defects.

The figures on the material. The assembled arrows weighed 27.250 grams plus or minus 0.057, with 0.282 grams between the lightest and the heaviest; the shafts alone spanned 0.104 grams and the points 0.149, which is three and five times the mass-matching tolerance the manufacturer states, 0.032 grams per dozen. The mean deflection measured was 8.290 millimetres against the 8.255 that the 325 label implies. Vertical spread on the target was 8.8 centimetres, inside the 12.2 of the ten ring.

The measuring bench has two position-controlled actuators, one in the central column performing the three-point flexural test over a fixed span of 71.12 centimetres and one at the side to rotate the shaft about its own axis. A Keyence optical sensor, accurate to two thousandths of a millimetre, measured diameter and straightness and tracked the displacement of the loaded section; an HBM force sensor, accurate to 0.02 newtons, gave the two forces from which the stiffness is derived. Measurements were taken about every sixteenth of a turn for at least four complete rotations, 64 readings per shaft, ten minutes each.

A sine wave was fitted to the series of 64 readings, so each shaft is described by four numbers: mean spine, amplitude, spatial frequency and phase. The parameters were estimated with a Fourier transform and refined with a non-linear weighted least-squares regression.

The shafts were cut to about 81 centimetres and assembled by a national-level archer, with one vane always at the label position, so that the nock orientation was the same for all of them. The shooting machine was bolted to the floor, with a Hoyt Formula XI riser, Velos Formula limbs, a front stabiliser and a clicker; three mechanical fingers held the string to reproduce the most common release technique, and the draw weight was 231 newtons at a draw length of 71.12 centimetres. The test was indoors, with the target at 70 metres and graph paper to read off the coordinates of the impacts, changed and replaced in the same position after each session.

The machine's repeatability was checked first: one randomly chosen arrow, shot thirty times, gave a median distance from the group centre of 1.3 centimetres and a 95% probability of landing within 3.5 centimetres. Then five sessions were run with all sixty arrows in random order; each arrow's position across the five sessions varied by 2.2 centimetres plus or minus 1.1 horizontally and 2.0 plus or minus 1.0 vertically.

Two multiple linear regressions with a bidirectional stepwise procedure, driven by the Bayesian information criterion, built the models for vertical and horizontal position, with checks on residual normality, homoscedasticity, collinearity and independence. For the closeness between arrows, all 1,711 possible pairs were compared: for each of the 255 subsets of the eight parameters, the normalised Euclidean distance between the pair's mechanical characteristics was compared with their distance on the target, and the subset with the lowest root mean square error was kept.

The authors state the most important limitations themselves. One bow, one setting, one spine category, and one of the stiffest: stiffness, mass and diameter go together, so the result does not transfer to more flexible arrows, which are also lighter, and the authors write that it is unclear how the two effects would offset each other. Nor can it be extrapolated how the vast number of possible bow settings would interact with the arrows' characteristics.

The sine model describes the stiffness of a shaft resting on a bench; it does not describe what happens in flight, where the arrow buckles under the push of the string and its stiffness changes along its length. Natural frequency and bending modes in free flight did not enter the models.

Straightness and diameter most probably came out as irrelevant because they barely varied: all sixty shafts were inside tolerance. The centre of gravity could not be used at all, because the measurements came out identical. And the models leave most of the horizontal spread unexplained, 81% of it, so the stiffness figures account for a minor part of where the arrows go sideways.

On vane position the result is a negative in a single configuration. The authors recall the coaches' practice of rotating the nock by 120 degrees until the arrows group, and conclude that the effect of nock rotation remains to be determined rather than that it does not exist.

Finally, the machine. Removing the archer is the point of the experiment, but it also removes the thing that would make the result apply to a person: the finger release, the bow hand, the posture. How much of these coefficients survives when an archer is shooting, the study does not say.

For anyone matching arrows, this study proposes an order. Mass first, because it was the strongest predictor of height on the target and because the mass spread inside one batch, 0.282 grams between the lightest and the heaviest arrow, is three to five times what the manufacturer's stated tolerance would lead you to expect; pairing heavy points with light shafts and vice versa, the authors calculate, would bring that spread down to 0.027 grams. Then spine. And then, if there is any way to measure it, the shape of the stiffness variation round the shaft, which is what this paper adds to the two usual criteria.

What the study removes from the list is just as useful. With shafts inside tolerance, straightness and diameter predicted nothing; and neither did the position of the vanes relative to the stiff side of the shaft, which is a widely held belief among archers and coaches. On that last point the authors' caution is worth keeping: they measured one angle, in one configuration, and the phase of the fitted sine waves was distributed at random, a sign that the manufacturer applies the label without reference to the stiff side.

The last figure is the one that puts the rest in proportion. With a machine bolted to the floor, indoors, with no wind and no archer, the sixty arrows spread over 8.8 centimetres vertically, and each individual arrow moved 2.0 to 2.2 centimetres from one session to the next. That is the floor the equipment leaves before anyone picks up the bow. The ten ring is 12.2 centimetres across, and the inner ring 6.1.