# An Approach to the Study of Ancient Archery Using Mathematical Modelling

By Kooi, B. W.; Bergman, C. A. | Antiquity, March 1997 | Go to article overview

# An Approach to the Study of Ancient Archery Using Mathematical Modelling

Kooi, B. W., Bergman, C. A., Antiquity

The archer's bow is a machine whose purpose is to impart stored energy effectively and accurately to propel the arrow. A mathematical modelling of different bow types shows how their engineering characteristics define their performances.

Introduction

One way of studying ancient bows is to make replicas and use them for experiments (Pope 1974; Miller et al. 1986; McEwen et al. 1991). In the present paper, the emphasis is on a different approach, the use of mathematical models which permit theoretical experiments using computers to gain insight into the performance of different types of bows. The use of physical laws and measured quantities, such as the specific mass of materials, in constitutive relations yields mathematical equations. In many cases, the complexity of the models obtained does not permit the derivation of the solutions by paper and pencil. Computers can then be used to approximate the solution; however, even this procedure will often necessitate simplifications. Essential detailed information may be missing, or assumptions need to be made to keep the model manageable. In that case, the model has to be validated by comparison of predicted results with actual measured quantities to justify the assumptions; fortunately, replicas can be employed for this purpose.

Mathematical models must accommodate all factors that determine the action of the bow; these are the 'design parameters'. Calculations are possible only if all the parameters are known. Descriptions of bows in the literature are often incomplete, so that comprehensive evaluation becomes impossible.

Theoretical experiments with models deal to a large extent with the influence of the design parameters on the performance of the bow. This presupposes a definition of good performance that fits the context of interest. Flight shooters are only interested in a large initial velocity, whereas target archers want a bow to shoot smoothly and accurately.

In the 1930s, bows and arrows became the object of study by scientists and engineers (Hickman et al. 1947; Klopsteg 1987). They performed experiments to determine the influence of different design parameters. Hickman made a very simple mathematical model for flatbows. Their work had a major impact on the design and construction of modern bows and arrows.

In the next section, the principles behind Kooi's mathematical model are highlighted (a full description of the model is beyond the scope of this paper; the reader is referred to Kooi & Sparenberg 1980; Kooi 1981; 1983; 1991; 1994). The model developed is much more advanced, so that more detailed information is obtained; it gives a better understanding of the action of rather general types of bow, including both ancient and modern examples.

This model was validated by a comparison of the measured initial velocity of an arrow shot with a modern bow with the predicted value (Tuijn & Kooi 1992). As part of the Mary Rose project (Hardy 1992), the measured weight of a medieval longbow replica was correlated with the predicted value. In both cases the predicted values matched actual measurements.

The aim of the present paper is to use the model to evaluate the performance of bows past and present. The function of the siyahs or 'ears' of the Asiatic composite bow is examined, along with the reasons for the differences in the performance of the longbow and the Turkish bow in flight shooting.

Mathematical modelling

In general, the bow proper consists of two equal elastic limbs, often separated by a rigid middle part, the grip. Not all design types follow this idealized description. Many Native American bows, for example the short flat-limbed examples from California, bend through their grips. The c. 2-m long Japanese bow, with the grip positioned at a point approximately one-third along its length measured from the lower nock, bends with an asymmetrical profile. Such design variations must always be considered before applying a mathematical model to the study of bow function and performance. …

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