Recurve bow mechanics: Kooi 1991 study

Kooi builds a computer model of the modern recurve, the bow whose limb tips bend during the last part of the draw, and compares it with five other bows studied with the same method: a straight-limbed bow described by Klopsteg, the angular bow of Egypt and Assyria, two Asian bows with rigid tips (one from China, India and Persia, one Turkish) and a bow with an extreme recurve similar to one used by Hickman. To compare them he brings them all to the same draw length, draw weight and limb mass, and judges them by three dimensionless numbers: how much energy the bow stores (q), how much of it reaches the arrow (efficiency, η) and the resulting arrow speed (v).

The modern recurve has q = 0.434, an efficiency of 0.729 and v = 2.23. The four historical bows sit close by: v between 1.92 and 2.01, efficiency between 0.619 and 0.765. The bow with the extreme recurve stores a great deal (q = 0.810) but passes little of it to the arrow (0.417), so in the end the arrow leaves only slightly faster than from the historical bows (v = 2.08). The author concludes that the shape of the bow, within certain limits, matters less than is often claimed; the improvement in modern bows comes mainly from materials, which store more energy per unit of mass. The model also shows strong string vibration after the arrow leaves, from which the author infers that a recurve needs a generous brace height.

The bow is described as an elastic line (Euler-Bernoulli beam theory) with stiffness and mass varying along the limb, symmetrical about the line of aim and centre-shot, meaning the arrow passes through the plane of the bow. Over the outer part of the limb the string lies against the limb itself, and the point where it leaves the limb shifts during the draw and again during the shot: this is what makes the problem mathematically hard, because one of the system's boundaries moves. The string is elastic and slides over the limb without friction; its mass is lumped at three points, at the two ends and at the nocking point.

Statics are solved with a numerical method the author had already published, dynamics with a finite difference scheme (Crank-Nicolson, then fully implicit once the string touches the limb again) that tracks the string's contact point moment by moment. The arrow is a point mass on the string and the bow is held still. For the comparisons, draw length, draw weight and the mass of one limb are all set to 1; with draw length in centimetres, force in kilograms-force and mass in kilograms, the unit of time corresponds to 0.03193 seconds.

This is theoretical work with no measurements: the author writes that the efficiency calculated for the modern recurve agrees with the literature, but does not show the comparison. The model leaves out internal friction in the materials, air, arrow flex and the bow's movement in the hand, and treats the string in a simplified way. The argument about materials, which opens and closes the paper, is not demonstrated here: it comes from another paper by the same author, cited in the text.

The speed comparison also needs care. In the modern recurve the arrow is lighter, relative to limb mass, than in the historical bows (0.0629 against 0.0769), and by its definition the coefficient v rises when the arrow is lighter: part of the modern recurve's advantage in the table comes from this choice, not from the bow. The six bows are reconstructions with parameters chosen by the author, described in detail in a later paper.

For a recurve archer the most interesting point is a correction of expectations: at equal draw length, draw weight and limb mass, an Olympic bow is not very different from bows of centuries ago in the way it turns energy into speed. What has changed, according to the author, is mainly the materials, which store more energy per unit of mass; this paper, though, takes that from another study rather than demonstrating it. The study says nothing about accuracy or tuning, because it does not consider them.

One observation connects with experience on the range: after the arrow leaves, a recurve's string oscillates a great deal, and the author concludes from this that brace height cannot be low. It is a deduction from the model, not a rule for setting brace height, which remains a matter of tuning bow and arrow together.