Arrow flexing and drag: Ando 2014 study

The question is whether the shaft bending in flight changes the air resistance on the arrow. The authors launched the same arrows in four ways, from the most rigid to the most ordinary, and filmed all of it with high-speed cameras.

The bending amplitudes measured just clear of the bow show how different the four conditions are. With the compressed-air launcher the shaft does not bend at all: zero amplitude. With a recurve bow on a robot and a finger-like mechanism the amplitude is 1.48 cm, with a mechanical release it drops to 1.22 mm, and with the two top national-level archers it settles at 6.86 mm, which is 2.2 times less than the finger mechanism and 5.6 times more than the release. The frequency, by contrast, is the same in every case, about 86 hertz: the same vibration mode is excited.

The mean drag coefficient is similar in all four conditions, between 2.51 and 2.77 at a Reynolds number of about 2 times ten to the fourth. What changes is its variability: the standard deviation is lowest in the archers' shots, 0.0727, and highest in the compressed-air launches with no bending, 0.267, with the two robot conditions in between. At lower Reynolds numbers the behaviour separates: without bending the coefficient drops to about 1.5, the laminar value, and between 1.3 and 1.8 times ten to the fourth it splits between the laminar and the turbulent value; with the wide bending of the finger mechanism it stays turbulent across the whole range. From this the authors draw the hypothesis that bending of a suitable amplitude makes the boundary layer turbulent and the flight more repeatable.

The trials were run in the indoor hall of the Japan Institute of Sports Sciences and at the Tamagawa ground of the University of Electro-Communications. The shafts are two competition Easton models, A/C/E and X10, with mean diameters of 4.24 and 3.80 mm respectively over a length of 0.626 m, bullet points of the type used in competition and Range-O-Matic Spin-Wing vanes.

The four ways of launching are built to separate the bending from the rest of the shot. The first is a compressed-air launcher, which pushes the shaft without making it bend. The second and third use the same robot with a recurve bow, changing the release: a finger-like mechanism, which produces a great deal of bending, and a mechanical release, which reduces the amplitude. The fourth is shooting by two top national-level archers, seventy-two arrows in all.

The filming is done with two Phantom V310 cameras and one Phantom Miro LC310. The two side cameras, set a known distance apart, give the deceleration and the change of attitude of the arrow, from which the drag coefficient is calculated by the procedure of an earlier paper by the same group. The camera placed directly above the trajectory serves the bending measurement: the displacement of the nock is read frame by frame, a straight mean trajectory is fitted by least squares, and the difference between the real position and the mean one is the bending, from which amplitude and frequency are derived. Each condition was repeated five times to obtain the mean and standard deviation of the oscillation characteristics.

The text is a four-page contribution to the proceedings of a mechanical engineering conference, not a peer-reviewed journal article, and the space constraint shows: the authors state explicitly that, for reasons of length, they present only the results for the A/C/E shaft. Of the X10, which is the model most used by top archers and the thinner of the two, only the opening specifications remain.

There are two archers and they are not described: there is no draw weight, no draw length, no arrow spine, no pressure button setting and no separation point of the string. The seventy-two arrows cover all the conditions, and the statistics on the oscillation rest on five repetitions per condition. The measurements stop at the stretch just clear of the bow: no data covers the rest of the flight towards seventy metres, where the bending decays and where the authors themselves suggest the boundary layer may return to laminar.

The drag coefficient is not measured directly but derived from the deceleration and the attitude between two cameras, and the error budget of that estimate is not reported. The arrow, besides bending, spins about its own axis because of the vanes, and the contribution of that spin is not separated out.

The point to keep in mind is the last one: there is no impact data. The study does not measure where the arrows land and does not compare group sizes between conditions. The step from a lower variability of the drag coefficient to better accuracy on the target is an inference by the authors, declared as such in the conclusion, not a measured result.

This study is about what happens to the arrow after it leaves the string, a stretch of the shot that is usually discussed very little. The figure to remember is that a top archer's shaft, just clear of the bow, bends with an amplitude of about seven millimetres at about 86 hertz, and that the same shaft launched without bending, or launched with a centimetre and a half of bending, shows an almost identical mean drag coefficient but a much more variable one from shot to shot.

No guidance about choosing spine or about how much bending is desirable follows from this, for two reasons. The first is that the study did not vary the arrows' spine: it varied the way the string was released, leaving the shaft the same. The second is that it never looked at the target: the sentence about accuracy is a consequence the authors propose, not a measurement.

One concrete thing remains for anyone shooting very thin arrows over distances where the speed drops: the authors note that at lower Reynolds numbers, with small bending, the boundary layer can stay laminar and the aerodynamic behaviour splits into two different values, and they flag this as the situation to watch. It is a fluid mechanics hypothesis applied to shooting, based on measurements taken two metres from the bow.