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Validation of the mathematical model

Mathematical models may be beautiful by themselves and the way to solve them interesting, but they should mimic the mechanical action of the bow and arrow closely if they are used in the design of a bow or a sensitivity study .

We checked static action by comparing the measured weight of a replica of one of the longbows found on the recovered Mary Rose with calculated values. The Mary Rose was Henry Vlll's warship which sank in 1545 in The Solent, a mile outside Portsmouth. She was recovered in 1982 with 139 yew longbows. Tests with these bows have demonstrated that while it is possible to string and draw the bows to 30 inch, considerable degradation within the cell structure of the wood has prevented a realistic assessment of the original weight. A replica was made by Roy King, bowyer to the Mary Rose Trust. Prof. P. Pratt, Imperial College of Science & Technology London, measured all parameters which are required to calculate the mechanical performance of a bow. The weight of this replica was also measured. It compared very well with the predicted value calculated with the mathematical model (differences within 1%).11 These results imply that if a good estimate of the original modulus and density can be obtained, the original mechanical performance of the longbows can be calculated from the dimensions of these recovered bows.

Data obtained with the test set-up described extensively elsewhere, permitted a comparison of predicted and measured arrow velocities. The dynamic action of bows could be checked in this way. We used a modern bow made of maple in the core and glass fibres embedded in strong synthetic resin at both sides of the core. All the essential parameters listed above were measured. We measured the density and elastic modulus of both the wood and the fibreglass and at a number of stations along the limbs the shape of the cross-sections. The results were used to determine the bending properties of the limbs. Finally the elastic modulus and the mass of the string were measured.

The predicted weight was too high and therefore a knockdown factor was used for the bending stiffness of the limbs, so that the calculated weight became equal to the measured value. The predicted amount of energy stored in the bow by drawing it from the braced situation to full draw, differed only slightly from the measured value. The measured efficiency was a few percent below the calculated value. In the model internal and external damping are neglected. This explains part of the discrepancy .

Classification of the bow

The classification of the bows we use is based on the geometrical shape and the elastic properties of the limbs. The bows of which the upper half is depicted in Figure 1 are called non-recurve bows. In the model the bow is assumed to be symmetric with respect to the line of aim.

Static deformation

So we need to deal with only one half of the bow. These bows have contact with the string only at their tips. In the case of the static-recurve bow, see Figure 2, the outermost parts of the recurved limbs (the ears) are stiff. In the braced situation the string rests on stringbridges. These string-bridges are fitted to prevent the string from slipping past the limbs. When such a bow is drawn, at some moment the string leaves the bridges and has contact with the limbs only at the tips. In a working-recurve bow the limbs are also curved in the 'opposite' direction in the unstrung situation, see Figure 3.

Static deformation shape of the AN-bow

The parts of a working-recurve bow near the tips, however, are elastic and bend during the final part of the draw. When one draws such a bow, the length of contact between the string and limb decreases gradually until the point where the string leaves the limb coincides with the tip. The string remains in that position during the final part of the draw. Elsewhere 5 we dealt with the statics (before arrow release) of these three types of bow. We studied the dynamics (after arrow release) of the non recurve bow 6 the dynamics of the static recurve bow and finally that of the working recurve bow. 8
In the model the action of a bow and arrow combination is fixed by one point in a high dimensional parameter space. Representations of different types of bow used in the past and in our time form clusters in this parameter space. We study the performance of different types of bow and start with a straight-end bow described by Klopsteg.1 This bow is referred to as the KL-bow. The shape of the KL-bow for various draw lengths is shown in Figure 1a. The AN-bow represents another non-recurve bow, the Angular bow found in Egypt and Assyria. The shape of the unstrung bow, shown in Figure 1b, implies that in the braced situation the limbs and the string form the characteristic triangular shape. We consider two static-recurve bows, one from China, India and Persia, to be called the PE-bow, and one which resembles a Turkish flight bow, to be called the TU-bow. The shapes of these bows for various drawlengths are shown in Figure 2. One of the working-recurve bows, to be called the ER-bow, possesses an excessive recurve. It resembles a bow described and shot by Hickman.1

Static deformation of PE and TU-bows

The other working-recurve bow is the modern one which was used for the validation of the model 10. This bow shown in Figure 3b, is referred to as the WR-bow.

Three quality coefficients for these types of bow are shown in Table 1. These coefficients are defined for equal weight, draw length and mass of the limbs. Moreover the mass of the arrows and strings were the same for al I reported bows. This makes an honest comparison possible. Unfortunately the stiffness of the string of the WR-bow is about twice that of the other bows. The static quality coefficient q measures how much recoverable energy is stored in the fully drawn bow. It is defined as the additional deformation energy stored in the elastic limbs and string by drawing the bow from the braced into the fully drawn position divided by the weight times the drawn length. The efficiency is the kinetic energy transferred to the arrow divided by the just mentioned additional deformation energy. So, it is the part of the available amount of energy which is transferred to the arrow as useful energy. The third quality coefficient v is proportional to the initial velocity. The constant depends only on the weight, draw length and mass of the limbs.

Static deformation shape of the ER-bow Static deformation shape of the WR-bow

The static quality coefficient is 1 when the draw-force is uniformly equal to the weight for all draw lengths for a fictitious bow with no fistmele. Just as the efficiency, this coefficient gives the actual value relative to a basic, characteristic value. The results show that in practice q is slightly smaller than 0.5 except for the ER-bow with the extreme recurve.


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