Vibration of carbon arrows: Rieckmann 2012 study

The study tests two ways of computing how a competition arrow vibrates and compares them with direct measurement on three types of Easton ProTour shaft, the 380, the 420 and the 470, all cut to 710 millimetres. The first bending mode comes out at 82.3, 80.3 and 78.4 hertz with the finite difference method and 82.7, 81.8 and 77.6 hertz with the finite element model; measured with a laser vibrometer it gives 83.3, 80.2 and 78.3 hertz. The order is the one an archer would expect: the stiffer the shaft, the higher the frequency, and between the 380 and the 470 there are about five hertz, roughly 6%.

Both models sit within 2% of the measured value for the first mode, better than the 3% the authors had set themselves as a target. The finite element model goes up to the eighth bending mode and keeps pace there too: for the ProTour380, for instance, it computes 249, 498, 819, 1,218, 1,700, 2,256 and 2,875 hertz from the second to the eighth mode, against the 248, 490, 795, 1,185, 1,663, 2,230 and 2,856 measured, with a discrepancy that on this shaft never exceeds 3%, while on the ProTour420 the computed fourth mode sits 5% above the measured one. The same model shows that at low frequencies the behaviour of the shaft is dominated by bending and that the other cylindrical modes, that is the circumferential, torsional and breathing ones, matter little. The nodes of the first mode are measured too, at 142 and 636 millimetres from the nock for the 380, along with the modal loss factor, which stays below 0.012: with damping that small the measured resonance frequency and the computed natural frequency are to all intents the same.

The shafts are the ones used in competition, an inner tube of AL7075-T9 aluminium 3.572 millimetres in outer diameter and 0.1524 thick, wrapped in carbon and epoxy, with an outer diameter that changes along the shaft. The elastic modulus of the carbon was not available from the manufacturer and the authors derived it indirectly: they reproduced the static spine test on the computer, that is the arrow resting on two supports 711 millimetres apart with an 880 gram mass at the centre, and adjusted the modulus until the computed deflection matched the catalogue figure (for the ProTour380, 9.652 millimetres, that is 380 thousandths of an inch), arriving at 222 gigapascals along the fibre direction. The properties in the other directions are estimated from similar materials in the literature. Nock, fletches and point were weighed (1.580 grams for the nock with the fletches, 7.776 grams for the point) and modelled as concentrated masses with their respective moments of inertia, applied at their centres of gravity.

The two models differ in ambition. The finite difference method treats the arrow as an Euler-Bernoulli beam in sixteen segments of 44.375 millimetres, with time steps of 25 microseconds, and computes only the first mode, because deforming the shaft into the shape of the higher modes is awkward: the shaft is flexed as it would be under its own weight and then released. The finite element model, built in ANSYS Workbench 12.1 with eight-node shell elements in two layers, 549 divisions along the 710 millimetres, 5,490 elements and 16,490 nodes, computes every mode up to the eighth in three dimensions.

The measurement uses a Polytec PSV-400 laser vibrometer in a three-head configuration, on 25 points along a line of the shaft. The shaft is suspended horizontally on light elastic cords placed at the nodes of the first mode, so as to add neither mass nor damping, and is excited without contact by two acoustic sources: a compression driver with a 25 millimetre outlet, suitable between 1 and 4 kilohertz, and an 80 millimetre loudspeaker, better for the first four modes between 50 hertz and 1 kilohertz. Three specimens of each of the three shaft types were prepared and tested.

The arrow hangs in mid-air, with no bow and no moving air. The test condition, known in mechanics as free-free, is that of a shaft vibrating on its own: there is no string pushing it, no pressure button touching it, no air drag and no spin from the fletches. That is why the authors close the paper by naming the extension of the models to the shot and to free flight as the next step.

The most delicate figure is the elastic modulus of the carbon, which comes not from a measurement of the material but from a calibration: it is the number that makes the calculation match the deflection the manufacturer states for the static spine test. The transverse properties are estimated from similar materials rather than measured, and nock, fletches and point become two concentrated masses; the authors note that the small steps in thickness and those simplifications may explain part of the discrepancy. The finite difference model works in two dimensions and on the first mode alone, and the mode shape it computes diverges by up to 3% towards the point, with the nodes displaced by 5 millimetres. The shafts are three types of a single model, new and in one length only, with one point mass: nothing says how the picture changes with a shorter arrow, a heavier point or a damaged shaft, which is the case the authors say they want to reach. The manufacturer supplied the arrows, which is stated in the acknowledgements. Finally, the idea that performance depends on the resonance frequencies of the shaft, with which the paper opens, is taken from the literature: no score and no group size is measured here.

The figure that matters for anyone tuning is the gap between one spine size and the next. Three shafts of the same model, cut to the same length, with the same nock, fletches and point, vibrate at 83.3, 80.2 and 78.3 hertz: between the 380 and the 470, which in the manufacturer's charts are two clearly separate steps, there are five hertz, and between two neighbouring sizes two or three. That is a small quantity, and it explains why the behaviour of the arrow as it leaves the bow changes gradually rather than in jumps when the spine is changed by one step.

The other useful point is that the calculation works. If the first bending mode can be predicted to within 2% from the dimensions of the shaft and the masses of the components, then two configurations can be compared on paper without building the arrows: that is the purpose the authors state, namely replacing part of the trial-and-error work in front of the target with computation. Two cautions remain. The first is that here the arrow vibrates freely and does not fly: the figures describe the shaft, not the shot, and dynamic spine on the bow also depends on the pressure button, the string and the point. The second is that no part of this study measures accuracy: a more accurate prediction of the frequency is not, on its own, a prediction of the score.