The mathematical model of the bow predicts draw weight well and efficiency less well

The summary today

Comparisons between model and real bow are few, almost all reported in words or in a graph, and on draw weight they point in different directions. The first, in 1969, sets the calculated draw curve beside that of a commercial recurve: a simple approximation, normalised on the measured curve, follows it very closely up to 6.5 inches of draw, and the full model, calibrated on one measured point at 11 inches, agrees well enough in the author's judgement. The Dutch group that built the classic bow model reports that on a replica of a Mary Rose longbow the predicted draw weight came within 1% of the measured one; on a modern maple and fibreglass recurve, by contrast, it was too high, and the model was brought into line with the measurement by a correction factor on limb stiffness. On a compound too, in 2009, the calculated force-draw curve follows the measured one, but with the limb stiffness chosen so that the peak force matches. On efficiency the picture is clearer: models with an inextensible string predicted more than 90% against the 70 to 85% measured, an analysis from 1981 puts the gap down to the energy left in the limbs and string as the arrow leaves, and on the Dutch recurve, even after the correction, the measured efficiency stayed a few percentage points below the calculated value. The three-link model of the Japanese bow, in 1997, reproduced the pattern of string tension, torsion and compression without matching the values. The half about efficiency holds, because the models put it higher than the measurement; the half about draw weight holds for the longbow, while on the recurve and the compound the model has to be tuned to the real bow. All of this rests on a handful of bows, with no stated sample size.

Also phrased as: "bow simulation"; "mathematical model of the bow"; "calculated bow efficiency"; "draw weight predictions"; "reliability of bow models"

Evidence

• 1969
  For Equipment tests Published later

  Analytical model of the working recurve, with the calculated draw curve set beside one measured on a commercial recurve: the simplest approximation, normalised on the measured curve, follows it very closely up to 6.5 inches of draw, and the full model, normalised on an experimental point at 11 inches, gives an agreement the author judges sufficient, even though some geometric parameters of the real bow differ from those used in the calculation. The comparison is graphical only, on one bow, and the model is calibrated on the measurement at one point.

  Ballistics of the Modern-Working Recurve Bow and Arrow · Burton G. Schuster

• 1969
  Context Model Published later

  Under the same model's assumptions, an inextensible string and no hysteresis loss, efficiency depends only on how the energy is shared among arrow, limbs and string; once the string's mass is counted it becomes 100/(1 + m/3M) per cent, as if a third of the string's mass were added to the arrow, in line with an experimental finding from 1937. The calculated efficiency is not set beside that of a measured bow.

  Ballistics of the Modern-Working Recurve Bow and Arrow · Burton G. Schuster

• 1981
  For Model Published later

  Analysis of bow and arrow dynamics with an elastic string: earlier models, which treated the string as inextensible, predicted efficiencies above 90%, against the 70 to 85% measured around 1960. On the longbow modelled by Hickman the elastic string brings the calculated efficiency down to 78%, against 92% with an inextensible string, and air resistance accounts for less than 2% of the total energy: according to the author, the gap between theory and measurement is explained not by air and hysteresis, as it used to be, but by the energy still left in the limbs and string when the arrow leaves. It is a calculation, compared with other people's measurements.

  Bow and arrow dynamics · W. C. Marlow

• 1997
  For Equipment tests Published later

  Validations of the mathematical bow model taken from earlier work: on a replica of a Mary Rose longbow the predicted draw weight differed from the measured one by less than 1%; on a modern maple and fibreglass bow, after the limb stiffness had been corrected, the stored energy differed little from the measurement and the measured efficiency was a few percentage points below the calculated one, partly because the model ignores internal and external damping.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 1997
  Against Equipment tests Published later

  On the same modern maple and fibreglass bow the predicted draw weight was too high: to make it match the measured value, a correction factor was applied to limb stiffness. On the recurve, then, draw weight was not predicted well without tuning the model to the real bow; the error is given in words, without figures.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 1997
  Context Equipment tests Published later

  A Japanese bow on a launching rig with strain gauges, compared with a model of three rigid links and two elastic hinges: string tension, grip torsion, compression and bending moment follow the same qualitative trends but do not match in value, and the calculated bending moment at the centre of the arrow is about 1/6 of the measured one. The authors conclude that the model describes the contact between arrow and bow differently; regional journal.

  Dynamics of Japanese Bows and Arrows : Analysis of "Tsunomi" with Link Model and Experiment · Koji Kobayashi, Keiji Kyogoku, Tsunamitsu Nakahara

• 2009
  For Equipment tests Published later

  Static and dynamic model of an Archery Research AR34 single cam compound, with almost every parameter measured on the bow: the calculated force-draw curve closely follows the measured one, valley and wall included, but the limb spring constant is chosen precisely so that the calculated peak draw force matches the measured one. The exit velocity also falls within 0.1 m/s of the measurement, but the author points out that this agreement is expected, since the virtual mass is derived from measured velocities. The comparison of the curves is graphical only, on one bow.

  A compound archery bow dynamic model, suggesting modifications to improve accuracy · J L Park

• 2009
  Context Model Published later

  Model of bow and arrow vibration for a modern Olympic bow with commercial parameters: treating the string as inextensible changes the first natural frequency by about 0.05% (132.293 against 132.358 s⁻¹), and the calculated arrow vibration period, about 18.4 ms, agrees with the 18 to 19 ms of joint bow and arrow motion seen in high-speed video. It is a comparison with a single experimental figure.

  Modeling of the Archery Bow and Arrow Vibrations · I. Zaniewski

Status history

1. 1969 Reported The first comparison between a model of the recurve and a real bow: a simple approximation, normalised on the measured draw curve of a commercial recurve, follows it very closely up to 6.5 inches of draw, and the full model, calibrated on one measured point at 11 inches, agrees well enough in the author's judgement. On efficiency the model is not compared with a measurement: with an inextensible string and no losses it derives efficiency from the sharing of energy among the parts alone. One bow, a graphical comparison and a calibration: there is something in favour of the half about draw weight, and it is limited.
2. 1981 Reported An analysis with an elastic string shows that models with an inextensible string overestimate efficiency: they predicted more than 90% against the 70 to 85% measured, and on Hickman's longbow the elastic string takes the calculation from 92 to 78%. According to the author the gap comes not from air and hysteresis but from the energy left in the limbs and string as the arrow leaves. It is a calculation compared with other people's measurements, so it does not move the status, but it gives a reason for the second half of the claim: reported.
3. 1997 Contested In the same year two groups set their models against real bows. On a replica of a medieval longbow the predicted draw weight comes within 1% of the measured one, but on a modern recurve it is too high and has to be corrected with a factor on limb stiffness; after the correction the measured efficiency stays a few points below the calculated value, partly because the model ignores damping. The three-link model of the Japanese bow reproduces the pattern of the forces measured with strain gauges, but gives a bending moment at the centre of the arrow of about a sixth of the measured one. On draw weight this brings the first case against, and on a recurve, where the agreement of 1969 had also come only after calibrating on a measurement: with cases on both sides and only a few bows, the studies point in opposite directions, and the status is contested.
4. 2009 Contested Two papers add comparisons with real bows without changing the picture. On a single cam compound the model closely follows the measured force-draw curve, valley and wall included, but the limb stiffness is chosen so that the peak force matches, and the exit velocity agrees by construction: another agreement reached after calibration, as on the 1997 recurve. A model of the vibration of an Olympic bow calculates an arrow period of about 18.4 ms, in line with the 18 to 19 ms seen in a high-speed video, but it concerns vibration, not draw weight or efficiency. On draw weight the picture stays divided: contested.

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• 22 studiesHow much energy passes from bow to arrow: models and measurements since 1937
• 29 studiesThe archer's paradox in research: from 1987 to today

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See all claims on equipment and tuning