Mathematical models and simulations: what they calculate about bow, arrow and archer

What a model does

A model is a set of equations that describes the bow, the arrow or the archer's body and calculates what happens during the shot: the force-draw curve, the arrow's launch speed, the bending of the shaft, the trajectory, the work of the muscles. The inputs are measurements or estimates such as masses and stiffnesses; the outputs are quantities that often nobody has measured. A model lets you change one parameter at a time, but its results are only as good as its assumptions. That is why, in the registry, evidence that comes only from a calculation has its own label, "model", does not count towards the share of claims with evidence in archers, and an equipment claim that rests on models alone stays "transferred", like the one on static and dynamic spine.

Reproducing a measurement is not predicting it

The most useful distinction when reading these studies is between a model that reproduces a measurement and one that predicts it. In the first case a parameter is chosen precisely so that the calculation matches the real bow, and the agreement that follows is partly built in. The 2009 model of a single-cam compound follows the measured force-draw curve closely, valley and wall included, but the limb spring constant had been chosen so that the calculated peak force would match the measured one; the launch speed also fell within 0.1 m/s, and the author warns that this agreement was to be expected, because the virtual mass had been derived from measured speeds. In a prediction, by contrast, the measurement plays no part in the calculation. The duralumin kyudo shafts tested on the bench in 2024 buckled at about 50 and 65 newtons, against 46.8 and 62.2 from Euler's formula, which uses only the length, cross-section and material of the shaft, and 46.7 and 62.0 from a finite element model.

Draw weight and efficiency

The most instructive case is the mathematical bow model of Kooi's group in the Netherlands, which began with a 1983 doctoral thesis. According to the 1997 paper, 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 it came out too high, and the model was brought into line with the measurement by a correction factor on limb stiffness. On efficiency the gap always runs the same way. Models with an inextensible string predicted more than 90%, against 70 to 85% measured, and a 1981 analysis with an elastic string brought the calculated efficiency of a longbow down to 78%, against 92% for the same bow with an inextensible string, and put the gap between theory and measurement down to the energy left in the limbs and string when the arrow leaves. On the same recurve, even after the correction, measured efficiency stayed a few points below the calculated value. This is why the entry on what the bow model predicts is "contested": the half about efficiency holds, the half about draw weight applies to the longbow, and on recurve and compound bows the model has to be calibrated on the bow it is meant to describe.

The same caution applies when a model compares bows nobody has measured, as in the entry on limb geometry: in the 1991 comparison efficiency ranges from 0.765 for the straight bow to 0.417 for the extreme recurve, differences inside the calculation that no equipment test has yet checked.

The arrow: calculated thresholds and measured arrows

In 2001 a model of the bow-arrow system placed the arrow's loss of stability at a dimensionless force of 14.79, against 9.87 in the static buckling test, and calculated that for an average target bow and arrow the peak thrust comes to 12.3, below the threshold. In 2009 a second model states in its conclusions that instability is a feature of the launch, yet the arrow used to check it sits at 5.28. When a model's conclusions and its numbers disagree, the registry follows the numbers. The entry on buckling during the power stroke is therefore "contradicted", partly because a 2015 estimate on a real compound puts the peak axial load at about 60 newtons against a critical load of about 100.

The 2001 paper also finds that two shafts with a static stiffness ratio of 1.38 behave in the shot as if the ratio were 1.47, and in 2012 another model calculated that an X10 550, 4% stiffer than a ProTour 570, needs 175.5 newtons of draw weight to leave without rotation, against 180. Nobody has yet shot the same shafts to compare the static ranking with the ranking in the shot. The history of the archer's paradox models, checked against high-speed footage, is in the line on the archer's paradox.

The compound and the Japanese bow

On the compound, the 2010 calculations follow the path of the nocking point: on a single-cam bow it would stray 21.8 mm from a straight line without correction, while with the cam tracks corrected the measured path stays within 1.5 mm. The rotation of the bow in the vertical plane is calculated and then checked with accelerometers on the riser; the claim that it cannot be cancelled by moving the centre of mass, on the other hand, is an argument from the model, not a test done by moving weights.

On the Japanese bow, where the two limbs deform differently, comparison with measurement shows the limits more clearly. In 2013 a model of rigid bodies with torsional springs reproduced the upper limb, 240 mm measured against 230 calculated, but not the lower one, 180 against 220. In 2019, by contrast, a large-deformation model agreed with measurements taken at nine points along a shihan-yumi.

Air and trajectory

The ballistic calculation is the oldest model in the registry, and in 1933 its agreement with measured shots was stated in one sentence, with no values. A comparison with the point of impact came with a 2015 preprint: with a compound on a shooting machine, the sight marks that put the arrow in the centre, converted into launch angles, fall on the curve of a calculation containing only gravity and air drag, and the entry on the trajectory is "reported". An analytical model from 2023, in a regional journal, shows how much the inputs matter: for an arrow at 60 m/s over 70 metres, misjudging by half how fast it slows moves the impact by 35 to 38 cm, and a speed estimate 3 m/s out moves it by 69 to 80 cm.

Models of the archer's body

In 2015 a musculoskeletal model, built on motion capture of a single archer with a 48-pound bow, calculated that no shoulder muscle goes above a fifth of its maximum, and the entry on shoulder load stays "transferred". The lower-back loads of the thirteen university students in the three papers from 2024 and 2025 were not measured but estimated by ergonomics software from posture reconstructed with inertial sensors: from about 470 to about 890 newtons of compression, against the 3,400 used as a reference limit, with bows from 20 to 32 pounds. The entry on lower-back loads therefore applies to light bows and short distances.

What to check

• which parameters were measured on the bow or arrow and which were taken from the literature or estimated;
• whether a parameter was chosen to make the calculation match the measurement, because then agreement on that quantity is not a prediction;
• what the result is compared with, and whether the comparison has numbers or only a graph and a sentence;
• how many bows, arrows and shots lie behind the check, which is often just one;
• whether the conclusions say the same thing as the numbers.

How design and sample are weighed is explained in How we read a study; what spine means for choosing arrows is in the article on the archer's paradox.

@misc{campagna2026_enresearchmethodsmathematicalmodelssimul,
  author = {Campagna, Matteo},
  title = {Mathematical models and simulations in archery research},
  year = {2026},
  howpublished = {Research, matteocampagna.com},
  url = {https://matteocampagna.com/en/research/methods/mathematical-models-simulations},
  note = {accessed 2026-10-16}
}