Arrow in the vertical plane, recurve: Park 2013 study

Park applies to the vertical plane the flexible-arrow model he had already used for the horizontal plane, to see what a recurve arrow does in the first milliseconds after release and which nocking point height gives it the cleanest launch. The starting point is geometric: the arrow pass is not at the centre of the bow, so the nocking point rises as the bow is drawn. On his bow the calculation gives 4.8 mm of rise at full draw, and a draw board measurement 5.2 mm, with an error of half a millimetre.

In the model the arrow stops pressing on the rest after 0.7 milliseconds; the highest force on the rest, 0.14 newtons, is at full draw before release. With the nocking point 1.62 mm above the rest (measured at brace) the arrow leaves with no rotation in the vertical plane and a residual angle of attack of 0.03 degrees. Rotation and angle of attack cannot both be brought to zero, as an earlier rigid-body model had already found, but the angle left over here is several orders of magnitude smaller. Vertical flex of the arrow is about a tenth of the horizontal flex.

The setting is forgiving: moving the nocking point by 1 mm changes the arrow's rotation by about 8 degrees per second, with the angle of attack staying under 0.2 degrees, and going to the next spine moves the best height by only 0.2 mm. On the shooting machine, at 20 metres, fletched and bare shafts landed at the same height within the aiming error, less than 20 mm; in the paper test, from 0.8 to 12 metres, point and fletching passed less than 1 mm apart vertically. With the nock raised by 10 mm the tear at 2 metres was 40 mm, a flight the author says no archer would accept.

The arrow is treated as a flexible beam split into 32 segments, with mass and stiffness varying along the shaft as in an X10 tapered at both ends, and the equations of motion are solved with a time step of 0.01 milliseconds. The string force comes from a recurve bow model published by others, scaled to the bow tested; a third of the string's mass is added to the arrow's. The path of the nocking point during the power stroke is taken to be the same as during the draw, a quadratic curve. The bow does not move or rotate: the high-speed footage Park had shot with Logan at the London 2012 test event had shown no movement above half a millimetre before the arrow left the string. Air drag, lift and friction on the rest are left out.

The bow is the author's own: a Hoyt Nexus with a 25-inch riser and medium 990TX limbs, 68 inches overall, a draw force of 166 newtons, a 220 mm brace height and the arrow pass 40 mm above the bow's centre. The arrow is a 600-spine X10, 705 mm long, with a 7.78 gram point, 21.65 grams in all and a speed of 57.7 m/s. The checks were run on a shooting machine, with the string pushed to the left just after release to imitate it leaving the middle finger: fletched against bare shaft at 20 metres, and a paper test at six distances.

It is a model of a single bow and a single arrow, the author's own, checked with a machine rather than with archers. Park himself points out that the model assumes the string is held exactly at the nocking point, whereas an archer holds it with three fingers, and that it does not consider a bow twisted by the hand. He adds that high-speed video shows the string often leaving the middle finger last, and names this as the area the model should handle in more detail. The 1.62 mm figure belongs to the model's geometry: the paper does not say how it relates to the measurement taken with a bracing square, so it should not be read as a height to copy.

The machine tests show that the result is consistent with the model, not that they confirm it in detail: at 20 metres the gap between bare and fletched shafts was below the aiming error, estimated at under 20 mm, which is a wide threshold. The rest force and the rigid-body assumption of the earlier model are both discussed, but only the first was tested in the model (removing it moves the best height from 1.62 to 1.60 mm).

For anyone tuning a recurve the study offers two points. The first is that in the vertical plane only one setting matters, the nocking point height, because the nock's rise during the draw comes from the bow's geometry and the archer cannot change it. The second is that this setting is forgiving: a millimetre either way makes little difference to how the arrow leaves, and changing spine by one size moves it by a few tenths of a millimetre. An error of a centimetre, on the other hand, shows up straight away on paper.

The study also gives a sense of timing: the arrow leaves the rest in under a millisecond, so what sets its vertical launch is mainly the path of the nock on the string, not contact with the rest. According to the author, the paper test and the bare shaft test are simple ways to check the vertical plane, because there is only one variable to adjust. All of this assumes an arrow already well tuned horizontally and a clean release: the differences that come from the fingers and the bow hand stay outside the model.