## On the Mechanics of some Replica Bows

by B. W. Kooi

This article first appeared in the Journal of the Society of Archer-Antiquaries, volume 36, 1993.

### 1 Introduction

In this paper different approaches to the study of ancient bows are compared. One way of studying ancient bows is to make replica bows for experiments, see Bergman, McEwen and Miller 1,2,3 and Alrune 4. Another way is to make mathematical models. Such models enable us to compare the performance of bows used in the past and in our time. Part of the modelling process is to identify all quantities which determine the action of the bow and arrow combination. Calculations are possible only when all the so called design parameters are known. Descriptions of bows in the literature are often incomplete. For instance artistic representations give a limited amount of information. In other cases parts of recovered bows are missing. Often the researches do not know all the design parameters and are not aware of their importance for a good understanding of the features of the bow al hand. In a former paper Kooi 5 we summarized all the important quantities which determine the mechanical action of a bow. In Table 1 the most important parameters are recalled and a nomenclature is introduced.

Both approaches supplement each other. Simplifications are necessary to keep mathematical models manageable. Therefore these models have to be validated by comparison of predicted results with actually measured quantities. Validation is a very difficult and complex subject. To take full advantage of modelling one does not want to check the model for all possible parameter combinations. So, a limited number of characteristic situations is chosen for which the comparisons are made. If the observed differences are small enough the model is also assumed to hold in other situations. The required degree of agreement between model predictions and measured data is often determined by the experience of the researcher and is by no way objective in practice. For validation purposes replica bows can be employed. On the other hand even simple mathematical models can be of help with the production of test procedures and the design of e experimental mental set-up.

### 2 Description of the replica bows

We consider a number of bows used in an experimental study with replica bows of a variety of types in Bergman 2. These bows are: a replica of a Medieval longbow, a replica of an Egyptian composite bow and a replica of a Tartar bow.

Furthermore we consider a replica of a Mesolithic Elm bow described by Alrune 4. The bow is the Holmegaard bow 7000-7400 B.C. This bow is a flat bow with length 2L=154 cm.

We assume that the replica bows each resemble one of the 'theoretical' bows described in reference 6 and which represent different type of bows:
Longbow: KL-bow (non-recurve bow, Figure I a in reference 5),
Egyptian angular bow: AN-bow (non-recurve bow, Figure 1b in reference 5)
Tartar bow: TU -bow (static-recurve bow, Figure 2b in reference 5).

The half length of the Holmegaard bow is 1.166 times the draw being 66 cm. Therefore we assume that it can be represented by a KL-bow with length L=1.143*|QD|, see Kooi 6. This bow is denoted as the HO-bow.

Finally we study a modern working-recurve bow. We refer to Tuijn and Kooi 7 for a description of the bow, the test set-up and the experimental procedures. In Kooi 8 the mathematical model of this bow, the WR-bow, is described.

For the replica bows described in reference 2 the masses of the bows are known (McEwen, personal communication) and this makes it possible to estimate the effectiveness of the usage of the materials of the limbs. The overall length 2L, the weight E(|OD|) and length |OD| as well as the mass of these bows 2mb are provided in Table 2. In Table 2 in reference 5 we gave the mechanical properties and the energy storage capacity per unit of mass for materials used in making bows. This data can be used to estimate the maximum amount of energy which can be stored in the fully drawn bow.

The actually stored elastic energy in the limbs of the fully drawn bow is denoted by Ab. It is calculated using the mathematical model whereby the actual values for the draw and weight are used. The quotient of actually stored energy per unit of mass of the limbs in the fully drawn bow Ab/2mb and the maximum allowable energy per unit of mass dbv is denoted as the utility coefficient aD. This quantity equals 1 when all material is used to the full extent. In practice it is smaller. We conclude that the materials of the replica bows are used rather well. Observe that the value of dbv for the composite Egyptian and Tartar bow is taken as an average of the values for the materials of which it is made; horn, sinew and wood.

The utility coefficient aD is slightly smaller for the composite bows than for the yew longbow but the product aD* dbv is still higher. This shows that in the fully drawn situation these bows store more deformation energy per unit of mass than the longbow.    