Dynamic spine of the arrow: Fish 2019 study

A university course project, posted on arXiv and not peer reviewed, starting from a sensible objection: the spine measurement the standard prescribes hangs 880 grams at the middle of seventy-one centimetres of shaft and reads the maximum deflection, that is it measures a static deformation and ignores everything that happens when the arrow goes. The authors build a rig doing the opposite. The shaft is clamped at both ends on an optical board and a voice coil actuator shakes it at the centre with a random signal, while the encoder inside the actuator reads the displacement to ten microns. From that signal a second-order model with three parameters is estimated, gain, natural frequency and damping.

The rig works, in the sense that the model describes the signal well: variance accounted for is 98.2 per cent with the three-parameter model and 99.4 with the non-parametric one, and adding one or two zeros improves nothing. Seven shafts, three materials: carbon 300, 500 and 600, aluminium 2219 and 1916, cedar 405 and 520, the last two also tested with the grain turned.

Frequency distinguishes the spines in carbon, 40.3 hertz for the 300 and 34.5 for the 500, and in aluminium, 40.8 for the 2219 and 31.6 for the 1916. The comparison with the values expected from factory data, though, holds only for carbon, where the discrepancy sits between minus four and plus two per cent, while on aluminium it breaks: the 2219 was expected at 31.9 hertz and measures 40.8, twenty-eight per cent more, and the 1916 was expected at 27.6 and measures 31.6. The abstract speaks of agreement with manufacturer-rated values, and that agreement holds for one material out of three.

Two neighbouring shafts get confused: this test's carbon 500 gives 34.5 hertz and 486 newtons per metre, the 600 gives 35.3 and 491, so the nominally softer shaft comes out marginally stiffer. The authors say so.

Within one spine the variability is large, from fourteen to twenty-two per cent on carbon's stiffness, and the conclusions are thoroughly honest: it may be production, it may be the way the shaft is clamped each time, it may be errors in the estimates, and telling them apart would need a calibration with known stiffness standards, which was outside the course's scope.

The rig: two clamps at the ends on an optical board, a BEI Kimco voice coil actuator with an integrated encoder, eighty-nine newtons of peak force, six millimetres of travel and ten microns of resolution, joined to the shaft by a three-dimensionally printed clamp enclosing it completely to stop lateral and rotational movement. The first version, with a loudspeaker and a sensor not fixed to the shaft, was abandoned precisely because it added dynamics between actuator and arrow.

The input is a random binary voltage. The force is not commanded but measured, because the back electromotive force reduces the actual current in the coil: a sense resistor in series reads the current, and from it comes the force that enters the identification as the input vector. Closed-loop current control, the authors declare, was outside the scope. The encoder's displacement is the output. Acquisition with a National Instruments controller, switching at twenty kilohertz to avoid current ripple, a control loop limited to four kilohertz, and a double-pole filter with its knee around two thousand hertz to remove switching noise.

Identification goes two ways. The non-parametric one deconvolves the force's autocorrelation from the cross-correlation of force and displacement with a Toeplitz matrix inversion, and uses one thousand five hundred and one parameters. The parametric one fits a second-order model with three parameters by least squares with Levenberg-Marquardt non-linear minimisation. Two variants with one zero and with a pair of conjugate zeros were also tried, justified by the anti-resonances the Euler-Bernoulli beam model predicts, with no appreciable gain in variance accounted for.

The sensitivity analysis was done: each of the three parameters was varied with the other two held fixed, and the variance accounted for responds to all three, so none is superfluous.

From the three estimated parameters come a lumped mass, a damping coefficient and a stiffness, reported in a table as though they were three measurements. They are not: they are the same triple rewritten, because stiffness comes from the DC gain, frequency from the fit and mass from the ratio of the two. That is why checking that the frequency comes back from the root of the ratio of stiffness to mass, which it does on every shaft to within three tenths of a hertz, is not a check but an identity.

The last part is fatigue: one shaft per material, about six hundred thousand cycles, then measured once more, with no range of variation.

The underlying limit is conceptual, and the article does not discuss it. The rig exists to get past a test that does not resemble shooting, and puts in its place another that does not resemble it either. In a shot the shaft is free at the point, pushed at the nock and under a strong longitudinal compression, which as is known lowers the natural frequency and, past a threshold, takes it to zero. Here the shaft is clamped at both ends and shaken at the centre, with no longitudinal load. These are three different problems with three different sets of boundary conditions, and moving from the second to the third does not get closer to the first.

Damping is the number that does not hold. The estimates run from 0.32 to 0.67. With a damping of 0.32 an oscillation's amplitude falls to twelve per cent in one cycle, with 0.51 to two and a half, with 0.67 to three thousandths: it means the vibration dies before completing a turn. Arrows do not behave like that, and this is known from direct measurement: a shaft in free flight keeps flexing for many cycles, a little under one per metre travelled. What the rig measures is not the shaft's damping, it is the rig's: the two clamps, the printed central clamp and the actuator with its back electromotive force. In the conclusions one reads that it would be interesting to evaluate how an arrow's damping affects performance, that is building on a number belonging to the test equipment.

The lumped mass goes the same way. Aluminium 2219 comes out at 22.7 grams, about the weight of the whole shaft: for a beam clamped at both ends and driven at the centre the effective mass must be a fraction of the total, not the total. The figure almost certainly includes the actuator's moving part and the clamp, and of any subtraction of those masses nothing is written.

The comparison with factory data needs care. The abstract says the estimates agree with manufacturer-rated spine values, but the reference table contains no manufacturer values: it contains frequencies the authors computed by converting factory spine with a lumped-parameter model, that is a prediction of their own. Against that prediction carbon comes back within four per cent and aluminium does not, with a twenty-eight per cent discrepancy on the 2219 and fourteen on the 1916. That discrepancy is not commented on, and in the measured comparison aluminium 2219 comes out stiffer even than carbon 300, while in the prediction it sat eight hertz below.

The fatigue result, finally, is presented as an absence of effect and the numbers say something else. After six hundred thousand cycles all four shafts tested come out stiffer, from plus fourteen per cent in vertical-grain cedar 405 to plus fifty-two in carbon 500, and all four have a higher frequency. Aluminium 2219's damping goes from 0.67 to 0.35, halving. The article calls all this a slight change and adds that the sample is small, but a systematic rise in stiffness in the same direction on four specimens out of four does not look like a null effect and does not look like fatigue, which does not stiffen a shaft. It looks like a difference in mounting between the two sessions, which is precisely the second of the three causes the authors name for the variability within one spine. Read that way, the datum measures the rig's reproducibility and not the shaft's life. Add that the measurement after fatigue is a single one, with no range, compared against a mean that had one.

A few smaller details. The variability given in the table is a range between minimum and maximum, not a standard deviation, and on two rows it drops to half a percentage point, 4.3 newtons per metre on 891.6 in vertical-grain cedar 405, against fourteen per cent for the same shaft turned ninety degrees: a thirtyfold difference in the spread of the same measurement on the same shaft is another pointer at the rig. The life estimate, ten cycles per shot and thirty shots a day, gives five and a half years, but thirty arrows a day is few for a target archer: at a hundred and fifty it becomes one year. And there is no measurement on the target: that too much or too little flexing damages the group is written in the introduction as a premise.

What does remain is that the work states its own boundaries precisely: no closed-loop current control, no calibration with known stiffness standards, a sample too small to generalise. It is not an article selling itself for more than it is.

For choosing arrows this work is of no use, and it does not claim to be. It is of use for two things, one of which really counts.

The one that counts: within one nominal spine the measured stiffness changes a lot. In carbon 300 it runs from 853.6 newtons per metre with a range of 175.3, that is twenty-one per cent, in carbon 600 twenty-two, in aluminium 1916 nineteen. Even if part of that spread comes from the rig, and the authors say they cannot separate it, the practical message for anyone wanting a matched set does not change: the spine printed on the shaft is a category, not a measurement of the shaft in your hand. Weighing the arrows and reading the number on the tube leaves a variable out.

The other: two neighbouring spines can be indistinguishable. This test's carbon 500 and 600 give 486 and 491 newtons per metre, so the nominally softer shaft comes out marginally stiffer. Between adjacent sizes the nominal step can disappear inside the real variability.

What should not be taken home are the damping figures. They run from 0.32 to 0.67, and with values like those a vibration dies in less than a turn, while a shaft in flight flexes for many metres. Those numbers describe the rig, with its clamps and its actuator, not the arrow: anyone looking here for guidance on choosing shafts of high or low damping will not find it.

Nor should the fatigue result be taken home. Literally it says that after six hundred thousand cycles nothing changes, and for an archer that would be good news, that a shaft breaks from wear or a knock long before it breaks from cyclic fatigue. The news may well be right, but it does not come from these data: the four shafts tested all come out stiffer than before, from plus fourteen to plus fifty-two per cent, which is the mark of a different mounting rather than of an absence of effect.

The value of this work, for someone reading a register of evidence, is to have measured something the others take for granted, how alike two shafts of the same model really are, and to have said plainly that answering it would need a calibrated rig. It is an open question put well, not an answer.