Archery and mathematical modelling

by B.W. Kooi

This article was first published in the journal of The Society of Archer-Antiquaries, Volume 34, 1991.
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One way of studying ancient bows is to make replicas and use them for experiments. In the present paper the emphasis is on a different approach, the use of mathematical models. Such models permit theoretical experiments on computers to gain insight into the performance of different types of bow. 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 will not permit the derivation of the solutions by paper and pencil operations. Computers can then be used to approximate the solution. However, even this procedure will mostly necessitate simplifications. Sometimes essential detailed information is missing. In other situations assumptions need to be made to keep the model manageable. In that case the model has to be validated by the comparison of predicted results with actually measured quantities to justify the assumptions. For that purpose, fortunately, replicas can be employed.

Mathematical models must accommodate all quantities which determine the action of the bow. Such quantities are often called 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 consists to a large extent of the research on the influence of the design parameters on the performance of the bow. This presupposes definition of good performance which fits the context of interest. Flight shooters are only interested in a large initial velocity. For target archery, on the other hand, the bows have to shoot smoothly and steadily .

In the 1930s bows and arrows became the object of study by scientists and engineers; see Hickman, Klopsteg and Nagler 1, 2. Their work influenced strongly the design and construction of the bow and arrow. Experiments were performed to determine the influence of different parameters. Hickman made a very simple mathematical model for flatbows. Later Schuster 3 and Marlow 4 also developed mathematical models to describe the mechanical action of a bow. Schuster dealt with the ballistic of the modern, so called working-recurve, bow. Schuster's model has the strange feature that bows appear to have 100% efficiency. Marlow introduced an elastic string on the model in order to explain this discrepancy with reality.

The description of our mathematical model is beyond the scope of this paper. The reader is referred to papers 5, 6, 7. The developed mode is much more advanced, so that more detailed information is obtained. This gives a better understanding of the action of rather general types of bow. Elsewhere 8, 9 we have show how this model can be adapted for the description of the ballistics of a modern bow. The predicted efficiency is smaller than 100% because in this model part of the available energy remains in the limbs and string and is not transferred to the arrow.

This model is validated by a comparison of the measured initial velocity of an arrow shot with a modern bow with a predicted value 10.

As part of the Mary Rose project 11 the measured weight of a replica was correlated with the predicted value. In both cases the predictions were sufficiently good.

The aim of the present paper is to use the model for an evaluation of the performance of bows used in the past and in our time. We try to uncover the function of the siyahs or ears of the Asiatic composite bow and to find the reason for the differences in the performance of the longbow and the Turkish bow in flight shooting

Mathematical modelling

In essence the bow proper consists of two elastic limbs, often separated by a rigid middle part, the grip. The bow is braced by fastening string between both ends of the limbs. After a arrow is set on the string the archer pulls the bow from braced situation into full draw. This completes the static action in which potential energy is stored in the elastic parts of the bow After aiming, the arrow is loosed or released. The force in the string accelerates the arrow and transfers part of the available energy as kinetic energy to the arrow. Meanwhile the bow is held in its place and the archer feels a recoil force in the bowhand. After the arrow has left the string the bow returns to the braced position because of damping.

As stated before, a complete description of the mathematical model is beyond the scope a this paper. An extensive discussion is presented elsewhere. 5, 6, 7, 8, 9. A summary of all important quantities in the model which determine the mechanical action of the bow is listed below.

length of the limbs
length of the grip
shape of the unstrung limbs
shape of cross-section of the Iimbs at all positions along the limbs
elastic properties of the materials of the limbs
specific mass of the materials of the limbs
shape and mass of the ears, if these are present
mass of the horns
draw length
mass of the string
elastic properties the string
mass of the arrow.

These quantities, the design parameters, determine the weight of the bow. In practice the bowyer tillers the bow such that it has finally the desired weight for a particular draw length. The archer on the other hand sets the fistmele by the adjustment of the length of the string.

For flight shooting the initial velocity of the arrow leaving the string is very important. The higher this velocity the greater the maximum attainable distance. The actual distance depend also on the elevation angle (nearly 45 ) and the drag of the arrow in the air. A requirement for target shooting and hunting is that the bow shoots smoothly. It is difficult to translate this feature into mathematics. High efficiency is a good criterion. However, a heavy arrow always yields a high efficiency and, unfortunately so, a small initial velocity and therefore a short distance. Hence, we have a combination of factors. The recoil-force, i.e. the force the archer feels in the bowhand after release, also seems to be important. The way this force changes in time can be calculated with the model, but it cannot be summarised by a single number.

The bow should not exaggerate human error. To assess the sensitivity of the bow, its performance is calculated repeatedly with slightly different values for the design parameters. If the performance depends strongly on a design parameter, the archer has to take care that the value of this parameter is as constant as possible. To achieve this archers need skill besides technique.