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A more general model is obtained when the motion of the limb is described as a vibrating beam undergoing large displacements. The mathematical model is formed by a set of coupled partial differential equations with initial and boundary conditions. With modern working-recurve bows the phenomenon that the string lies during a part of the draw along the limb, see Figure 3, makes the mathematical treatment much more complicated. The place where the string contacts the limb has to be determined as part of the solution. Problems of this type are called free and moving boundary problems for the static and dynamic case respectively. The description of such a model is beyond the scope of this paper. The reader is referred to [5,7-10]. To show the merits of such a model, the calculated and measured force-draw curves of the Greenhorn bow are compared in Figure 8. Since the predicted weight of the bow was too high, a knockdown factor was used for the bending stiffness of the limbs, such that the calculated weight became equal to the measured one. Such an adaptation is usual in this kind of modelling, since the size of different parts of the bow, especially the thickness of the thin layers of fibre-glass cannot be measured precisely. The comparison of measured and predicted values is summarized in Table V. The predicted efficiency is about 2 % too high. In the model no internal or external damping is taken into account. This may explain part of the discrepancy. These results indicate that the model is good enough to use it for, for example, sensitivity analysis as part of the design process of bows.

Comparison of predicted and measured static force-draw curve
Fig.8: Comparison of the predicted and measured static force-draw curve of the GH bow, after matching of predicted and measured weights (O= predicted values).

Schuster [11] made a model for the working-recurve bow too. He made two unrealistic assumptions, namely that the working-recurve is in the form of a circular arc which unrolls along an initial tangent and that this is the only part of the bow that 'works'. The advantage of Schuster's model is that the mathematical treatment is simpler.

Table V. Comparison of experiment and model calculation for the GH bow. Effective weights have been matched. Hysteresis has not been taken into account. Arrow mass 18.1 g; string mass 6.55 g and string stiffness 260 N per % of elongation (Dacron B50).
MeasuredPredicted
Eff. weight [N]126.9126.9
W [J]32.1632.9
Kin. energy [J]22.5723.9
Eff. [%]70.272.9

5 Conclusion

For teaching purposes our measuring method is very suitable. If it is possible to install some kind of rack to attach the bow, aiming is no longer necessary, for the coils can be placed precisely in the trajectory of the arrows. Our rack was found in a storeroom for cast-off equipment. Some minor adaptations were sufficient to make it a shooting device. The electronic circuit is rather simple. The transient recorder might be a problem to get, but it is advisable to use one during the start of the measurements to be sure to have signals of the right sign. Care is needed to ensure that the arrow point is always magnetized on the same pole of the permanent magnet.

Marlow [6] already pointed out that experiments on archery can be done on different levels and in a great variety. In the appendix of his paper he indicated several possibilities. If one should want to skip the experimental phase, video productions are now available from companies of teaching materials [12], but accurate measurements as described in this paper are not (yet) possible with this kind of production. With our method small effects can be detected, such as the differences arising from string material and stabilizer configuration.

Marlow also mentioned several possibilities for applications of mechanics. For model calculations we discuss different models of different degree of complexity. The Lagrangian method reduces the entire field of statics and dynamics to a simple procedure. In this way the dynamics of the bow and arrow is a beautiful illustration of several notions of mechanics.

Acknowledgements

Firstly we want to express our gratitude to our student, Laurens de Lange, who invested, after finishing his free experiment, much more time in these measurements. Further we thank his colleague archer Pim Pouw, who was very interested in this experiment, and spent a lot of time in the realization of the set-up and the measurements. Also we thank dr C. Prins, who kindly put at our disposal some masses and a precision balance.

References


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