Arrow leaving a compound bow: Park 2011 study

The compound bow was this literature's gap: Pękalski, Kooi, Sparenberg and Zanevskyy had studied the arrow leaving a recurve bow, nobody the arrow leaving a compound, where the release is mechanical, the cables pull sideways against their guard and the nocking point moves according to the cam design. James Park, an engineer at Monash, takes Kooi and Sparenberg's method, changes all the initial and boundary conditions, and does the thing those two works promised without doing: he measures.

Photographs with a timed flash, one per shot, every two milliseconds from release, with the bow mounted in a shooting machine. A resolution of about two tenths of a millimetre, the camera's linearity checked, and the pixel limit used as the error bars in the figures. Three Easton ProTour shafts, the 470 recommended for that bow and the two stiffer ones, 420 and 380, all 0.715 metres with a 7.8-gram tungsten point, a pin nock and three 38-millimetre vanes. The bow is an Archery Research AR34 single cam, the rest a Trophy Taker TT1 blade whose stiffness was measured at 368 newtons per metre.

The vertical plane's numbers: the arrow comes off the blade at 14 milliseconds, leaves the string at 17, clears the bow at 20. The flexing is small in the first part of the power stroke and greatest just before exit, with an amplitude of a few millimetres against the ten or fifteen expected on a recurve. Entering the airstream the angle of attack is minus four tenths of a degree and the rotation about the centre of mass about a hundredth of a degree per millisecond. In the flexing, besides the fundamental, there is a clear second harmonic, which can give the shaft an S shape.

Two results matter more than the others. The first: the arrow's motion is governed by the nocking point's path, not by the string's longitudinal force, and this is not asserted but shown by elimination, holding the force constant and seeing that nothing changes, then changing the nocking point's path and seeing that everything does. The second: the three stiffnesses behave almost alike, in both planes and with the bow twisted too, which was then checked on the target at thirty-five metres with the shooting machine.

One degree of bow twist, which a badly placed hand produces easily, weighs more than the variation in cable tension, and it is not symmetrical: anticlockwise the arrow flexes far more and goes left, clockwise it flexes little but takes an angle of attack of minus four tenths and goes right.

The arrow is an inextensible Euler-Bernoulli beam, split into thirty-two segments of twenty-two millimetres with concentrated masses and with mass and stiffness varying along the shaft, solved by explicit finite differences with a time step of a hundredth of a millisecond. The improvement over Kooi and Sparenberg's model is that the shaft is not uniform: the ProTour has an aluminium core under the carbon and tapers towards the point, and besides, the shanks glued inside the shaft, the point's above all, stiffen it considerably at the ends. The vanes' mass too is placed at the right point along the shaft rather than added to the nock. Gravity is included, and amounts to 0.12 millimetres of static deflection.

The boundary conditions: while the nock is on the string it follows the nocking point's path in both planes, with zero moment since it can rotate freely; the point is free and with zero moment; the rest is a movable condition at a known position along the shaft, modelled with the measured stiffness vertically and a very high stiffness horizontally, because the blade flexes in one plane only.

The two inputs that settle everything come from two earlier papers by the same author: the dynamic force curve, different from the static one because limbs and cams store kinetic energy and return it at the end of the power stroke with a peak just before the arrow leaves, and the nocking point's path, vertically according to the cam design and horizontally according to the varying forces on the cables, computed with and without bow twist.

The numerical check is done and declared: redoing the calculation with sixteen segments and a time step of five hundredths of a millisecond changes the position by at most six hundredths of a millimetre. And the shaft model is checked against an independent measurement, the free-flight resonant frequency taken with a stroboscope: 75.2 hertz computed against 74 measured for the 470, 77.5 against 77 for the 420, 80.0 against 80 for the 380, with the rear node at fourteen and fifteen centimetres from the nock against the fifteen measured, and the front node at sixty-five centimetres in both cases.

The photography: a darkened room, the shutter open for two seconds, the shot and the flash inside that window, an adjustable delay up to thirty milliseconds in half-millisecond steps. The flash is triggered by a microswitch on the release's cocking lever. One image per shot, which requires the behaviour to repeat, and the repeatability was verified to within the pixel limit.

Finally the three spine sizes were tested on the target: twelve arrows per size, each shot three times at thirty-five metres with the shooting machine, with only the sight setting changed for the different masses.

This is the soundest work in its line, and that is worth saying before the limits: it has the model and the measurements, the convergence check, an independent verification of the shaft model with a stroboscope, the sensitivity analysis carried out rather than promised, and the main prediction checked on the target. That said, there are five things to know.

The first: one bow measured, one. The AR34 single cam. The binary-cam case, from which comes the article's most repeated recommendation, that a nocking-point path rising towards the braced position favours clearance while a flat or falling one does not, is computed only: the article says so, the figure is the computed and not the measured behaviour, and all the other parameters stay those of the AR34. So the difference between single, hybrid, dual and binary cams is a model prediction on a mixed parameter set.

The second: in the horizontal plane the measurement has a gap, and it is in the worst place. The article writes that between ten and fourteen milliseconds no convenient position was available for the camera. The maximum flexing arrives shortly before the arrow leaves the string, at seventeen, so the missing window is the climb towards the maximum. In the same plane the images of the early and the late phases are taken from two different positions, from above and behind for the first, from above and in front for the second, so as not to have the bow and the string in front of the lens: a practical necessity, but it means the geometry of the measurement changes inside one series.

The third: the agreement between calculation and measurement gets worse exactly where it is needed. The article is explicit, the agreement is excellent below fourteen milliseconds and after that there is a small difference, and that difference falls on the maximum flexing and on the exit condition, the two numbers the angle of attack depends on. The author also says where he thinks the problem lies, in the small differences between the computed and the real nocking-point path and secondarily in the lumped stiffness due to the point insert and the taper. It is an honest diagnosis, and it remains a limit.

The fourth concerns the judgements of agreement, which are all verbal. Excellent, satisfactory, very good, a small difference, superior to the accuracy obtained by Kooi and Sparenberg: none of these is a number. The error bars in the figures are the camera's pixel limit, which measures resolution and not the discrepancy between prediction and measurement, and a mean discrepancy is reported nowhere. The comparison with Kooi and Sparenberg's accuracy in particular is made by looking at the two figures.

The fifth, and for the register the most important, concerns the strongest practical conclusion. That the three stiffnesses give equivalent groups was tested, but the result is given without numbers: one reads that the groups were small and of similar sizes, in any case within a forty-millimetre circle, and that the flight had been judged excellent, in brackets, subjectively. There is no diameter for each of the three groups, no dispersion, no statistical comparison. Thirty-six shots per size on a shooting machine are a good test, and the numbers would have been enough to make it checkable.

To this are added two details that narrow the conclusion. The three shafts tested are the recommended one and two stiffer: none softer, that is none in the direction where the clearance problem is expected, so the tolerance cannot be called symmetrical. And the three shafts have different masses, 23.0, 23.7 and 24.4 grams, while the article itself explains that a heavier arrow makes the nocking point's path less abrupt and so reduces flexing and angle of attack: across the three tests stiffness grows together with mass, and the two effects are not separated.

Finally, some confirmations are by eye. That the centre of mass goes left of the shaft's line at full draw was not photographed, because it would have needed an excessive depth of field, and was inspected visually. The string's contact with the face or the body is not modelled, and the article assumes it resembles bow twist. The shooting machine, which is the right choice for isolating the equipment, removes the archer precisely while the most useful result concerns an archer's error.

Of all the mechanical work on archery, this is the one you come away from with the most to do, because the author measures what he computes and then translates it.

Clearance over the blade. With the nock two millimetres above the blade's unloaded position, the tail passes cleanly, and that is verified photographically. On a binary-cam bow, in the model, four millimetres are needed. Separation from the blade happens at fourteen milliseconds, predicted at 13.83, and the arrow leaves the string at seventeen: so contact with the rest ends well before exit, and what matters is that the tail, and in particular the leading edge of the vanes, does not find the blade in its way.

The blade's lateral position is not critical. Moving it two millimetres changes the behaviour little, and changes mostly the angle of attack entering the airstream, from plus one tenth to minus two tenths of a degree. Together with the result in the vertical plane this leads to a conclusion worth keeping: the rest's position does not settle the bow's accuracy, so long as the arrow clears. It is the opposite of what one hears at the field.

Stabilisers shift the tune. The nocking point's path depends on the position of the bow's centre of mass, hence on the stabiliser setup: changing them means the arrow leaves differently, and may make it necessary to reset the blade's height. Anyone who changes rods and weights and then finds contact where there was none has the explanation here. In the same direction, a higher braced position helps clearance.

Bow twist is the enemy. One degree, which a badly placed hand produces without effort, weighs more than the variation in cable tension and is not symmetrical: anticlockwise on a right-handed bow means far more flexing and the arrow left, clockwise means little flexing, an angle of attack of minus four tenths and the arrow right. It is not constant twist that does the damage, it is its variation from arrow to arrow. And the string's contact with the face or the body, the article says, is to be treated the same way and at the same price, which also concerns anyone holding at full draw for differing lengths of time.

On spine, the news is a freedom. Between the recommended size and two stiffer sizes, the behaviour at exit and the group at thirty-five metres do not change appreciably, not even with the bow twisted. It means that someone shooting a compound with a mechanical release can choose a shaft for other reasons, for instance wind drift or speed, instead of staying inside a narrow range as on a recurve. Two cautions: the shafts tested are the recommended one and two stiffer, never a softer one, and the stiffer ones were also heavier, while the article explains that weight by itself reduces flexing. So the freedom towards the stiff side is documented, the one towards the soft side is not.

The last pointer is a simple one: all else equal, a heavier arrow makes the nocking point's movement less abrupt and reduces flexing and angle of attack, so it is preferable. It remains to be weighed against speed, which the article does not put in the balance.