Arrow flexing in free flight: Park 2011 study

Park modelled an arrow's flight in the horizontal plane as that of a flexible beam and compared it with real shots from a compound bow fired by a shooting machine. A Protour 470 with X vanes, leaving the bow at 78.9 m/s with an angle of -0.6 degrees and a rotation of -45 degrees per second about its centre of mass (the bow was set up on purpose to exaggerate the effect), swung about its centre of mass with a period of about 21.8 metres, or 0.28 seconds. The swing died away with a time constant of 0.34 seconds, about 25.8 metres of flight. Meanwhile the arrow flexed at 77.1 Hz, roughly once per arrow length travelled, and the heavy point strayed less than 3 mm from a straight line.

With Titeflight vanes 24% larger the period was 12% shorter, at 19.4 metres, and damping was stronger; even with the largest vanes in common use, 125 mm long, the model predicts that the arrow still swings before settling. Without vanes the arrow stayed stable but swung with a period of about 80 metres: at 15 metres the model placed it 44 mm right of the fletched version, and the machine put it 45 mm across. The accuracy test is the most interesting part: with the bow set to give a large lateral swing, or one of half that size, the machine's groups at 35 metres were the same, with every arrow inside a 15 mm circle.

The arrow is treated as an Euler-Bernoulli beam, flexible but inextensible, divided into 16 segments, with mass and stiffness varying along the shaft and the masses of nock, vanes and point lumped at the right places; the equations are solved by finite differences in steps of 0.05 ms. Lift and drag are calculated for point, shaft and vanes with coefficients from the aerodynamics literature (for the vanes, from a study of rowing blades). The initial conditions, meaning speed, angle, rotation and flex as the arrow leaves the bow, come from measurements; the measured initial drag was 0.93 times the arrow's weight.

For comparison the arrow was shot by a machine from an Archery Research AR34 through sheets of paper at several distances up to 35 metres, measuring the tear between point and vanes to about 2 mm; successive shots gave the same tear within 2 mm. Some compound archers from the Australian national team repeated the paper test at short range. Accuracy was assessed with machine groups at 35 metres at two different swing amplitudes.

The model is compared with a single arrow, in two vane configurations, and a single compound bow; what the author says about the recurve is a reasoned extension, not a test. Only the horizontal plane is covered, with no wind. The comparison with measurements is graphical, on paper tears, with no measure of agreement, and with the larger vanes the measured damping was a little stronger than calculated. The number of arrows in the groups is not given, and the finding of no difference holds within the machine's aiming precision and for swings repeated identically: it says nothing about an archer, whose error changes from one arrow to the next.

The tests with national team archers are described as broadly in agreement, without data, and the author himself does not rule out that they were twisting the bow. The estimate of errors of about 0.1 metres at 50 metres when the archer twists the bow comes from another paper by the same author.

For anyone setting up a bow the study has two consequences. The first concerns paper tuning: since the arrow almost always leaves with a little rotation about its centre of mass and then swings with a period of about 20 metres, a tear read at a single distance can mislead. The author suggests looking at it at no fewer than two distances, about a quarter of a period apart, for example 1 and 6 metres. The second is that a lateral swing that is always the same, on a compound shot with a mechanical release, did not enlarge the machine's groups: in the author's view the lateral position of the rest is not critical, provided the arrow behaves the same way on every shot.

The focus therefore moves from the arrow to the archer: what opens up groups is variability, such as twisting the bow differently from shot to shot or the string touching the face, rather than a swing that repeats itself exactly. For the recurve the author notes that bare-shaft tuning works when the initial angle and rotation are small, but presents this as a question still to be studied.