Beyond its first bending mode the arrow vibrates in higher modes, the second at about three times the first

The summary today

That the arrow also vibrates in higher modes has been measured, in the laboratory and by a single research group; how much it matters for the shot has not. The most direct test is a 2012 laser vibrometry study of competition carbon arrows with an aluminium core, complete with nock, fletching and point: the first eight bending modes run from 80.2 to 2777 Hz in one model and from 83.3 to 2856 Hz in another, and the second, at 241 and 248 Hz, is three times the first. The same paper shows that only bending modes matter for the arrow's motion, that the two still points of the first mode sit about a fifth of the length from the nock and 7 cm from the point, and that three shafts of the same model agree within 0.4 Hz on the first mode but not on the eighth, where the spread rises to 12 to 16 Hz. In the same year, footage at 4000 frames per second of the arrows of 16 elite archers finds the first mode in free flight between 80 and 105 Hz and, in most shots, a small second-mode component, visible as an S bend especially early in the power stroke, while after the arrow leaves the bow the first mode dominates; the calculation for another shaft gives 74 and 228 Hz, and the same calculation repeated in 2013 for the vertical plane comes back with the same figures. All these measurements come from a single research group, and the only outside paper, from 2015, measures the first mode at 84.8 Hz but derives the second, 272 Hz, from the model alone, and writes that higher modes are probably unimportant in archery. Seeing the second mode on video takes at least a thousand frames per second.

Also phrased as: "arrow flexing in an s shape"; "second bending mode of the arrow"; "arrow vibration frequency"; "arrow vibration nodes"; "archer's paradox in slow motion"

Evidence

• 2011
  For Model Published later

  Model of a compound arrow during the power stroke: the bending has a substantial second-harmonic component in both planes, clearly visible in the shaft 18 ms after release, and in some cases it produces an S shape. The paper does not give the frequency of the second mode.

  The Behaviour of An Arrow Shot from a Compound Archery Bow · J L Park

• 2011
  Context Model Published later

  Model of the free flight of an arrow shot from a compound: in flight the Protour 470 flexes at its own resonant frequency, 77.1 Hz, about once for every arrow length travelled, with an amplitude estimated at 5 to 10 mm on a compound and 10 to 15 mm on a recurve. It deals only with the first mode and says nothing about the higher ones.

  Arrow behaviour in free flight · J L Park

• 2012
  For Equipment tests Published later

  Free-free laser vibrometry on carbon arrows with an aluminium core, complete with nock, fletching and point, three shafts per model: first eight bending modes measured at 80.2, 241, 478, 775, 1152, 1618, 2165 and 2777 Hz (ProTour 420) and at 83.3, 248, 490, 795, 1185, 1663, 2230 and 2856 Hz (ProTour 380); the second mode is three times the first.

  Modelling the three-dimensional vibration of composite archery arrows under free–free boundary conditions · Marianne Rieckmann, James L Park, John Codrington, Ben Cazzolato

• 2012
  Context Equipment tests Published later

  In the finite element model, torsion, circumferential deformation and expansion of the tube turn out negligible: up to the eighth mode only bending matters. On a 710 mm shaft the still points of the first mode sit 139 to 144 mm from the nock and at 636 to 642 mm, within 5 mm between models and measurement and barely moving with spine. Three shafts of the same model agree on the first mode with a standard deviation of 0.4 Hz, which rises to 12.5 and 16.5 Hz at the eighth.

  Modelling the three-dimensional vibration of composite archery arrows under free–free boundary conditions · Marianne Rieckmann, James L Park, John Codrington, Ben Cazzolato

• 2012
  For Archers Published later

  Footage at 4000 frames per second of the X10 and ACE arrows of 16 elite recurve archers: the first mode in free flight runs from 80 to 105 Hz, 13 arrows out of 16 between 80 and 87 Hz; in most cases a small second-mode component is visible, as an S bend, especially early in the power stroke, while after the arrow leaves the bow the first mode dominates. The second-mode frequency, 225 to 300 Hz, is taken from the literature rather than measured, and the component is judged by eye.

  High-speed video analysis of arrow behaviour during the power stroke of a recurve archery bow · James L Park, Oliver Logan

• 2012
  For Model Published later

  Calculation for a 705 mm X10 600 arrow: first mode at 74 Hz (73.3 Hz by an independent method), second at 228 Hz, typically around 250 Hz. Seeing the second mode on video needs a frame rate well over twice its frequency, at least 1000 frames per second.

  Arrow behaviour in the lateral plane during and immediately following the power stroke of a recurve archery bow · James L Park

• 2013
  For Model Published later

  Model of the arrow in the vertical plane during the power stroke of a recurve: for an X10 600 of 705 mm and 21.65 g the first bending mode falls at 74 Hz and the second at 228 Hz, 3.08 times the first. It is the same calculation, on the same shaft, as the 2012 paper on the lateral plane, repeated for the other plane.

  Arrow behaviour in the vertical plane during and immediately following the power stroke of a recurve bow · James L Park

• 2015
  Context Equipment tests Published later

  A 0.696 m carbon arrow: fundamental mode measured at 84.8 ± 0.1 Hz (85.4 Hz in the finite element model, 97.2 Hz from uniform beam theory), with still points at 0.125 and 0.630 m; the second mode, at 272 Hz with still points at 0.070, 0.365 and 0.670 m, comes from the model alone, and the author writes that higher modes are probably unimportant in archery. Preprint.

  Applications of Physics to Archery · H. O. Meyer

Status history

1. 2011 Transferred The first sign comes from a model of a compound arrow during the power stroke: the bending is not only in the fundamental mode but has a substantial second-harmonic component in both planes, which gives the shaft an S shape 18 ms after release. It is a calculation, not a measurement on the arrow, and it does not say at what frequency the second mode lies: the claim rests on a model. In the same year a model of free flight finds the arrow flexing at its resonant frequency, 77.1 Hz, but it deals only with the first mode.
2. 2012 Reported The measurements arrive together. Laser vibrometry on competition arrows finds the first eight bending modes, with the second at three times the first, 241 against 80.2 Hz; free-flight footage of the arrows of 16 elite archers finds the first mode between 80 and 105 Hz and a small second-mode component, visible as an S bend early in the power stroke; the calculation for another shaft gives 74 and 228 Hz. Only bending modes matter for the motion, and the two still points of the first mode sit about a fifth of the length from the nock and 7 cm from the point. The measurements agree, but the ratio between modes is measured in a single paper and all of them come from the same group: the supporting evidence is limited.
3. 2013 Reported The same author repeats the calculation in the vertical plane for the shaft already studied in 2012: the second mode falls at 228 Hz, 3.08 times the 74 of the first. It is a model of the same arrow and adds no measurements, so the direct evidence is still that of 2012 and the status remains reported.
4. 2015 Reported A paper from outside that group measures the first mode of a carbon arrow at 84.8 Hz, with the two still points at 0.125 and 0.630 m on 0.696 m, but derives the second mode, at 272 Hz, from the finite element model rather than measuring it, and writes that higher modes are probably unimportant in archery. The ratio of about three appears here too, but from a calculation and in a preprint: the picture does not change.

Related

Related articles

• The archer's paradox: the physics of the flexing arrowAt full draw the arrow points to one side and still reaches the centre: how it flexes and clears the riser in the first twenty milliseconds, and how spine matches it to the bow.

Research lines

• 29 studiesThe archer's paradox in research: from 1987 to today
• 17 studiesGrip pressure and bow torque: from 1971 to today

Other claims in this domain

• CorroboratedAlong its flight the arrow loses an appreciable part of its launch speed12 studiesStatus since 2011
• CorroboratedAs it leaves the bow, the arrow flexes and snakes around the riser instead of leaving straight19 studies · 47 archers measuredStatus since 2012
• ReportedA camera-based scoring system misreads some arrows: the ones barely sticking out and the ones that shift the target face7 studies · 2 archers measuredStatus since 2017
• ReportedA chest band with fibre optic sensors measures breathing and mean heart rate with small errors, and an archer can wear it while shooting2 studiesStatus since 2019
• ReportedA compound bow stores between 70 and 100 joules and sheds about two thirds of the force at full draw4 studiesStatus since 2015

See all claims on equipment and tuning