Bow torque and arrow shaft choice: Pekalski 1990 study

The study builds three tools for studying the archer-bow-arrow system and uses them to answer two practical questions: how much bow torsion matters, and how shafts are chosen for a given bow. The tools are a mathematical model of the arrow's motion while it is still in contact with bow and string, a machine that releases the string in place of the archer, and three series of high-speed cine film, between 1,000 and 2,500 frames per second.

The author divides the shot into two phases: phase 1 runs from the release of the string to the moment the arrow loses contact with bow and string, phase 2 is the ballistic flight. Phase 1 is where all the work concentrates, because it is the only one in which the archer-bow subsystem acts on the arrow and because it sets the initial conditions of phase 2.

The first quantitative result concerns torsion of the bow about the vertical axis. Measured with the machine at 30 metres, the sensitivity of the point of impact to that angle is 15.8 centimetres per degree when torsion is negative and 21.9 centimetres per degree when it is positive: one direction costs 30% more than the other. From this the author derives that, to put every arrow in the ten at 30 metres, the angle of torsion has to be repeated within 0.2 degrees.

The second result comes from the simulation, comparing three Easton X7 shafts of different diameter, the 15/14, the 17/14 taken as reference and the 21/14. With the suitable shaft the nock leaves the string with practically no lateral velocity, at the instant when the bow's transverse force is also zero, and stays within 2% of the reference lateral displacement for 2 thousandths of a second before separation, that is for 15% of the duration of phase 1, during which the arrow covers 20% of its travel. Shafts that are too weak and too stiff leave the string with a lateral velocity: the too weak one, by the position and sign of its velocity, goes on to touch the bow with the fletching, while the too stiff one flexes little, lengthens phase 1, leaves more slowly and is seen on the machine to slip sideways.

From these criteria the author builds a nomogram that, for Easton X7 shafts, derives outside diameter and wall thickness from arrow length and draw weight, across 800 combinations. Compared with the equipment of the 22 best Polish archers, only four fell outside the predicted band, and two women archers with European and World medals were using shafts close to those proposed.

The mathematical model describes three things about phase 1: the arrow's translation along the shooting axis, its transverse vibrations in the horizontal plane and the interaction with the edge of the arrow rest. The shaft is modelled as a prismatic bar, with the transverse vibrations given by a partial differential equation solved by Fourier's method on a basis of Krilov functions; the translation is approximated by a point mass equal to the arrow's mass and a spring representing the bow's longitudinal elasticity. Among the initial conditions is a parameter proportional to the bow's angle of torsion about the vertical axis, which scales the amplitude of the transverse vibrations: it is the parameter that links the archer's technique to the arrow's behaviour.

The machine, which the author calls the DMLA, is made of three parts: a very rigid frame set in concrete; a clamp fixing the bow's handle to the frame with two jaws mounted on three ball bearings, so that the bow can rotate almost without friction about two perpendicular axes and can be locked at any chosen angle; and a release mechanism allowing the string to be fixed at any draw length, released smoothly and practically without friction, and the string's holding point to be moved in all three directions.

The high-speed filming was done on 16 mm film between 1,000 and 2,500 frames per second. The first series films a woman archer from the Polish national team under laboratory conditions, with equipment prepared to make frame analysis more precise and with all the mechanical characteristics measured; the second films a group of Olympic and World Championship medal holders.

There are three verifications. The model was compared with the arrow's motion reconstructed by polynomial interpolation from footage taken with the camera directly above the archer. The motion of the arrow released by the machine was compared, on film at 1,000 frames per second, with that of an arrow released by a national-team archer. And the machine's own accuracy was measured at 30 metres.

For the nomogram, the working hypothesis is that a bow-arrow pairing is well chosen when the model's dimensionless parameters coincide with those of the system taken as reference: rescaling the coordinates and the time, the motion must be identical to that of the reference shaft. The reference is the equipment of a leading Polish archer. The nomogram covers the 10 X7 shafts in the catalogue, with the outside diameter in sixty-fourths of an inch and the wall thickness in thousandths, and assumes for every bow the same ratio between transverse and longitudinal elasticity, 1.3, and the same efficiency, 0.75, where efficiency is the ratio of the arrow's kinetic energy at the start of flight to the work done drawing the bow. The combinations considered are 800: 40 draw-weight values in steps of 5 newtons and 20 draw-length values in steps of 1 centimetre.

This study's chain has one declared weak link: the machine replaces the archer, and the evidence that it can do so is qualitative. The author writes that between the motion of the arrow released by the machine and that of the arrow released by a national-team archer there were «no qualitative differences» on film. There are no figures behind that comparison, and on that sentence rest all the results obtained with the machine, the sensitivity to torsion included.

The verification of the model is likewise judged by the author himself in subjective terms: two figures are compared, the motion reconstructed from the film and the simulated one, and the differences are observed to be greater at the edges of the time and space intervals. No measure of error, no statistics. The author also states what the model lacks: the elasticity of the arrow rest's edge, to be added in a next approximation.

What the model covers is phase 1 only, and only in the horizontal plane. Vibrations in the vertical plane, the aerodynamics, the fingers leaving the string and the ballistic flight are absent: for phase 2, the author writes, no model has been developed. This is why the effect of the too-weak shaft touching the bow with the fletching is «verified experimentally only», because it happens where the model no longer reaches.

The two figures on torsion, 15.8 and 21.9 centimetres per degree, come from one bow with one reference setup, at one distance. How many arrows were shot to obtain them is not reported, there are no confidence intervals and there is no repetition: the 30% asymmetry between the two directions rests on two values with no stated uncertainty. The 0.2 degree threshold is computed from those numbers rather than measured on archers, and the author himself observes that until then competitors had no way of controlling that angle: that an archer can repeat it within two tenths of a degree has not been shown.

The nomogram's validation is weaker than it looks. Eighteen archers out of twenty-two inside the predicted band is not an independent check: the band's width is not stated, and the criterion itself is anchored to the equipment of a single leading Polish archer, taken as an approximation of the optimum. The nomogram and the archers' choices derive in part from the same origin, so the agreement is partly circular. The nomogram also assumes for every bow the same elasticity ratio and the same efficiency, measured on the reference bow.

The paper's age remains. The catalogue on which the nomogram is built is that of Easton X7 aluminium shafts, which the author describes as used by almost all the best archers: today top recurve archery shoots carbon and aluminium shafts, so the specific table no longer applies, even if the method behind it stands. Figures 8 and 9, which hold the curves and the nomogram, are not legible in the digitised copy: the values in this record come from the text. Draw weight is given in kilograms force in one place and in newtons in another, and the shafts are written as «15/14 in» or «17/14 in» as though they were inches, when they are the Easton code, sixty-fourths of an inch for the diameter and thousandths for the wall. The abstract speaks of three series of filming, the methods describe two.

This is the first work to put a number on the cost of twisting the bow, and the number is large: at 30 metres, one degree of torsion about the vertical axis moves the impact 15.8 centimetres in one direction and 21.9 in the other. On a 122-centimetre face, one degree takes the arrow from the centre to outside the gold. It is the quantitative content of the otherwise general instruction not to torque the grip.

From there comes the threshold most often quoted: for every arrow to land in the ten at 30 metres, the angle of torsion has to be repeated within 0.2 degrees. It is worth knowing how it arose: it is a calculation based on the sensitivity measured on the machine, not a measurement on archers shooting. It says how intolerant the system is, not how capable an archer is of matching it.

The part on spine explains a mechanism, and explains why tuning has a tolerance rather than a single point. With the suitable shaft the nock leaves the string at the instant when the bow's transverse force is zero, hence with practically no lateral velocity, and that condition lasts 2 thousandths of a second, 15% of the contact phase, during which the arrow covers 20% of its travel. It is a window, not an instant: inside that window small differences between one arrow and the next, or between one release and the next, count for little. With a shaft too weak or too stiff that window is absent, and the nock leaves with a sideways push.

The two directions of error are not symmetrical. An arrow that is too weak leaves and goes on to touch the bow with the fletching; one that is too stiff flexes little, stays in contact longer, leaves more slowly and slips sideways. Losing speed is not a detail: the author notes that a faster arrow spends less time in the air and is less affected by changing air.

One last figure useful as a reference point. The machine, with no archer and with the same equipment, grouped at 30 metres with a standard deviation of 0.9 centimetres, that is practically everything in the ten. It is a measure of how much of the group depends on the bow and the arrows and how much on the person shooting: with ordinary equipment, the kit on its own shoots better than the best archer in the world.