Bow and arrow interaction: Kooi 1998 study

No archers, no arrows shot: this is a piece of calculation. Bob Kooi, a biologist and mathematician in Amsterdam, takes up the model with which Ryszard Pękalski had described the arrow's movement during launch, rebuilds it in full because the original description was incomplete, and changes three things in it.

The first is the bow. Pękalski treated it as a linear spring, with force proportional to the draw beyond the braced position, and this, Kooi writes, cannot be done: because of limb inertia the dynamic force-draw curve differs appreciably from the static one, and the difference matters precisely in the first few milliseconds. In its place goes the dynamic curve computed by his own 1991 model of the working-recurve bow.

The second is the pressure button, which becomes an elastic support working in one direction only, with a spring of 622 newtons per metre, and the contact force between arrow and grip appears explicitly in the equations.

The third is the start. Pękalski needed to place the nock out of the bow's median plane by an estimated amount, minus eighteen millimetres, which amounts to a torsion of the bow about the vertical axis; in the new model that device is not needed, because the nock's transverse movement arises by itself from the bow's transverse elasticity as the string slips off the fingers.

The predictions, compared with the high-speed film published by Pękalski in 1990, are three. The arrow comes free of the button before it leaves the string, that is the lateral contact ends while the push is still going on. The contact is not continuous: just after release it drops to nothing for an instant, then resumes, then ends. And the string releases the arrow at roughly the moment the nock crosses back over the bow's plane, which takes the tail past the grip without striking it.

With the new model the shaft flexes more than with Pękalski's, and the reason is the initial normal force, that is the longitudinal compression of the push, which the earlier model left out.

The arrow is treated as a beam vibrating in the horizontal plane, described by the classical beam equation, and its transverse movement splits into two parts, a vibration and a rotation about the nock. Vertical movements are avowedly neglected. The system that comes out is not linear, because of the contact problem, so it is not solved by Fourier's method as Pękalski did but by finite differences, as described in Kooi and Sparenberg in 1997.

The boundary conditions are where the two models part. In Pękalski the arrow's point is constrained as in a hinge, that is its transverse position is held at zero for the whole launch, and the moment the arrow comes free of the grip is defined as the instant the point's transverse velocity is at its maximum, with that velocity then held constant. In the new model the point carries the head's mass as a concentrated mass and moves, and the separation is not defined in advance: it comes out of the calculation, when the force in the button's spring goes to zero.

The parameters are one single combination, Pękalski's of 1987: an Easton X7 1714 arrow in 7178 aluminium, 0.67 metres long, external diameter 6.75 millimetres, wall thickness 0.356, mass per unit length 0.02 kilograms per metre, flexural rigidity 0.209 newtons per square metre. The bow is a Hoyt Pro Medallist T/D of 1.68 metres and 15.5 kilograms, with a stiffness of 342 newtons per metre, transverse elasticity 270, braced height 0.225 metres and a draw of 0.645, for a force of 143 newtons at full draw: that calculation holds, 342 times 0.420 makes 143.6. The distance between point and grip at full draw is 25 millimetres.

Two numbers are not measured but estimated by fitting: the longitudinal and transverse efficiencies, both set at 0.75 for the reference combination. Pękalski had recovered them with non-linear least-squares regression, obtaining 0.76 and 0.71, plus the eighteen millimetres of the nock's offset.

The verification, as the article itself describes it, is the comparison of two descriptions of the same shot: the curves drawn from high-speed film, shot at between a thousand and two and a half thousand frames a second on a member of the Polish national team and on a shooting machine, and the curves coming out of the simulation, drawn every two milliseconds from release. No new measurement was collected for this article.

It is a model, and the comparison with reality rests on a single shot. The experimental curves are the ones Pękalski published in 1990, read off a figure: one film, one archer, one bow and arrow combination. There is no second set of data, no repetition, and for this article no measurement was taken.

The comparison, moreover, is by eye. The article writes that setting the two figures side by side suggests the new model follows the experimental data better at the bounds of the intervals of time and space, and the abstract speaks of satisfactory conformity. There is no measure of the discrepancy, no sum of squares, no interval, and no criterion stated in advance for deciding which of the two models sits closer. That the new one wins is plausible, because it contains mechanics the other lacks, but the proof offered is the superimposed drawing.

There is also a detail that makes the comparison less clean than it looks: the two models do not have the same number of free parameters. Pękalski had three fitted to the data, the two efficiencies and the nock's initial offset; Kooi has two, because he does away with the third. Removing a parameter and improving the fit is a strong result, but the fit is not corrected for that difference, and without a numerical index it cannot be.

The release is not in the model. It is the thing that varies most from archer to archer and from shot to shot, and in the equations it enters only as an initial condition on the nock. The article says so and proposes the way out: experiments on each archer would be needed to measure how the string leaves the fingers and how the bow hand moves, and from there a reference release could be defined. Until that exists, the model describes a machine's shot better than a person's.

The vertical plane is excluded by choice. The rest, the vertical contact and the vertical component of the paradox stay outside, and with them the clearance problems an archer actually meets when the nock or the vanes brush the rest.

One combination only, and from 1987. An aluminium shaft on a recurve of that era: no carbon, no compound, no recent limb geometry, and a single value of button stiffness. This is the gap that weighs most in practical terms, because the button is precisely what the article adds to the model and precisely what the archer adjusts: with one value only, 622 newtons per metre, there is no seeing what changes in the shot when that spring gets harder or softer. The same holds for spine: the softer and stiffer shafts discussed, 5.95 and 8.33 millimetres in diameter, are Pękalski's calculations with the old model, not recalculations with the new one.

One calculation in the table does not close. The text defines the arrow's total mass as the sum of shaft, head, fletching and nock, and the table gives a total of 18.5 grams with 13.5 for the shaft, 4 for the head, 1 for the fletching and 4 for the nock, which make 22.5. The first three, instead, make exactly 18.5: the nock is outside the total, despite the definition, and later in the boundary conditions its mass is avowedly neglected. Those 4 grams, moreover, are a lot for a real nock, which usually sits around a gram.

Finally, the point that matters for the register: accuracy is not in this work. The sentence by which the arrow's movement around the grip improves the accuracy of the shot is an aside, not a result: in the article there is no score, no dispersion, no comparison between better and worse centred shots. And the practical promises, that is tuning bow and arrow by simulation, running the programme on a low-cost personal computer and identifying the most important parameters with a sensitivity analysis, are all in the future tense. None of the three is carried out in the article.

This work is not used to choose a spine, and should not be read as though it were. It is used to understand what happens in the ten or fifteen milliseconds while the arrow is still attached to the bow, and that understanding changes how tuning is approached.

Three things, in order of usefulness.

The first is the sequence of separation. The arrow comes free of the button before it leaves the string, and the tail passes the grip after it has already gone, at roughly the moment it crosses back over the bow's plane. That means the button is not working throughout the launch: it does its work in the first part, and by the time the arrow leaves it is no longer touching. Anyone looking for a mark on paper or on the rest to find where the arrow touches is looking at a contact that ends before the arrow goes.

The second is that the contact is not continuous. Just after release the arrow comes off the button for an instant, then presses again, then leaves. A button too hard or too soft does not merely shift the average force: it changes the moment that sequence closes, and so the position of the tail as it passes the grip.

The third is Pękalski's criterion, which Kooi says may remain useful: a bow and arrow combination is well chosen when the arrow leaves the string at roughly the instant the nock crosses back over the bow's plane. Put that way it is a criterion of geometry rather than of a chart, and it is the mechanical reason why a shaft that is too soft and one that is too stiff both fail: they are not in the right phase at the moment of separation. What remains is the way of noticing, which on the field is still the bare-shaft test, the clearance check and the group on the target.

What this article does not give, and it is worth knowing before looking for it: no number to take to the bench. One combination computed, one value of button stiffness, an aluminium shaft from the late eighties, no carbon arrow, and a release that is not in the equations. The promise of tuning by simulation, which the article makes twice, is still waiting for somebody to keep it.