In a former paper we dealt with some aspects of the dynamics of the non-recurve bow, of which the string was assumed to be inextensible and without mass. One aim of this paper is to investigate the influence of the elasticity and mass of the string on the non-recurve bow. The main object is to discuss the dynamics of the static-recurve bow. The governing equations of motion lead to a system of non-linear partial differential equations with initial and boundary conditions. These boundary conditions vary abruptly in the course of the dynamic process. Numerical solutions are obtained using a finite-difference method. The vibratory motion of the bow after the arrow has left the string is described for a clamped bow as well as for a bow shot open-handed.
4.2
Introduction
The bow is a mechanical device to propel a projectile, which is generally an arrow. To that end a string, shorter than the bow, is placed between the tips of the bow. Then holding the middle of the bow in place with the ”bow” hand, the string is drawn with the other hand, the ”shaft” hand. During this, additional energy is stored in the elastic limbs and to a lesser extend also in the string. A part of this energy is, after release, transferred to the arrow.
One way to differentiate between types of bows is to do this on the ground of the shape of the unstrung bow. Then we distinguish between the non-recurve bow, the static-recurve and the working-recurve bow. A recurved or reflexed bow is a bow of which the limbs are in unstrung situation curved away from the archer if he/she holds the unstrung bow just like during shooting. For such a bow it is possible that in the strung conditions the string lies along part of the limbs.
In the case of a non-recurve bow the string has contact with the bow only at the tips in all situations, static or dynamic. The bow with flexible straight limbs (straight-end bow) Figure 4.1.a, but also a bow of which the flexible limbs meet at an angle (Angular bow),
Figure 4.1: Shapes of different types of bows: (a) and (b) non-recurve, (c) static-recurve and (d) working-recurve.
see Rausing [16] and Figure 4.1.b, and even a bow with a slight reflex are non-recurve bows by definition.
The static-recurve bow is a bow which possesses rigid, strongly curved outer parts (ears) of the limbs. In the braced situation the string rests upon the string-bridges, see Figure 4.1.c. These string-bridges are fitted to prevent the string from slipping past the bow. If such a bow is drawn, at some moment the string leaves the bridges and has contact with the bow only at the tips. Because the ears are stiff, they do not deform when the bow is drawn. After release the string touches at a certain moment the string-bridges again before the arrow leaves the string. Some Tartar, Chinese, Persian, Indian and Turkish are static-recurve bows, see Rausing [16], Payne-Gallwey [14], Latham and Paterson [10], Faris and Elmer [4], or Balfour [1].
The entire limbs of a working-recurve bow are flexible. In the braced situation the string has contact with the bow along a part of the limbs near the tips, see Figure 4.1.d. The length of those parts diminishes with increasing draw-length. If the draw-length exceeds a certain value the string leaves the limbs from the tips. After release both phenomena occur in reversed order before arrow exit. Essential difference between the static-recurve and working-recurve is that for the static-recurve the points of contact of string and limbs change abruptly from tip to string-bridge or inversely. For the working-recurve the points where the string leaves the limbs change gradually. Most modern bows are working-recurve bows.
paper we investigate the influence of the mass and elasticity of the string on a non-recurve bow. Further we consider the behaviour of the bow and string after the arrow has left the string. This is done for two different cases, one by which the bow is clamped and the other for a bow shot open-handed. The main object of this paper is the dynamics of the static-recurve bow. The dynamics of the working-recurve bow will be the subject of a forthcoming paper.
In reality a static-recurve bow is very complicated. It nearly always is made of wood, horn and sinew, hold together with glue and protected from the weather by a thin covering of tree bark, lacquer or leather. This holds also to a certain extend for the English longbow, where the different properties of sapwood and heartwood are deliberately put to use. In spite of this, these bows are considered as an inextensible elastic line endowed with bending stiffness and mass distribution, which depend on the properties of the employed materials and structure of the bow. Other assumptions are the symmetry of the bow with respect to the line of aim, the bow is centre-shot and the rigid arrow is released without lateral deflections. Also neither internal or external damping nor hysteresis are taken into account. The absence of damping throws some measure of uncertainty into our calculations especially for what happens after the arrow has left the string.
In Section 4.3 we derive the governing equations of motion of the static-recurve bow. For such a bow a simple lumped parameter model for the string is used. The dynamic process of shooting is divided in a number of time intervals which are bounded by characteristic events. These events are: the string touches the string-bridges again, the bow leaves the bow hand and the arrow leaves the string. During each of these time intervals, of which the length is not known beforehand, we have the same system of partial differential equations, however the boundary conditions are different. The initial conditions are determined for the first time interval by the static fully drawn position and for the succeeding intervals by the end conditions of the preceding interval.
In the second part of Section 4.3 we give the equations of a string now considered as a continuum. This mathematical model for the string is used only for non-recurve bows. In this case we get at each time interval two systems of partial differential equations defined on two space intervals, one along the bow, the other along the string. These systems are connected by the boundary conditions at one end of each space interval.
In order to obtain a numerical approximation of the solution of the equations of motion we use a finite-difference method. In Section 4.4 a finite-difference scheme is given for the static-recurve bow. After that, the finite-difference equations for the motion of a string considered as a continuum, are discussed. These equations are, with the exception of the boundary conditions, the same as the equations of motion in the case of a refined lumped parameter model for the string.
In [9] and [8] we introduced quality coefficients to be able to compare the performance of different bows. One coefficient is related to the amount of energy stored in the bow in pulling it from the braced situation to full draw. Another one is the efficiency, which is the percentage of energy put into the bow that is imparted to the arrow. Finally, the muzzle
velocity, this is the velocity of the arrow when it leaves the bow. For flight shooting, a form of shooting with the object of reaching the greatest distance, it is the last mentioned quality coefficient which is important. For target shooting and hunting the efficiency is important and also the smoothness with which the bow delivers its power. In Section 4.5 we re- examine the definitions of these quality coefficients. They are used in Sections 4.6· · · 4.12 in order to judge the performance of a bow.
Hickman [6] and recently Marlow [12] developed mathematical models for a type of a non-recurve bow. They used a model where the elasticity of the limbs is concentrated in two elastic hinges. The mass of the limbs is accounted for by concentrated masses placed at the rigid limbs. Hickman assumed the string to be inextensible. Marlow dropped this assumption and he claims that the results of his elastic string considerations are in reasonable agreement with experiment and remove the long standing discrepancy between theory and practice. However, in Section 4.6 we show that his model can yield unreliable results. It turns out that when the model for the string is replaced by a more realistic one possessing elasticity, however keeping the limbs rigid and rotating about elastic hinges, unrealistic heavy oscillations of the acceleration force acting upon the arrow may occur.
In Section 4.7 we discuss the influence of the strain stiffness and mass of the string of a non-recurve bow. Changing both parameters simultaneously gives us the opportunity to deal with the influence of the number of strands of a string. Increasing this number makes a string stiffer hut also heavier. These effects have an opposite influence on the shooting performance. It appears that there exists an optimum number of strands. At the end of Section 4.7 we compare our results with those obtained experimentally by Hickman in [6]. The vibratory motion of the bow after the arrow has left the string is investigated in Section 4.8. It appears that the tensile force in the string attains its maximum after arrow exit. This maximum force determines among others the number of strands needed to make the string strong enough. In the second part of Section 4.8 we discuss the influence of the mass of the grip when the bow is shot open-hand.
In Section 4.9 we start with a straight-end bow and change some of its parameters one by one. In [8] we followed the same procedure, there we started with a bow described by Hickman in [6]. The bow we are interested in now is more realistic; the string is extensible and possesses mass while the tips of the limb have non-zero mass per unit of length and bending stiffness.
In Section 4.10 we consider again the model of a bow consisting of a grip, two elastic hinges, two rigid limbs and an inextensible string. However, in this case the limbs have a sharp bend, hence the bow resembles a static-recurve bow. It is possible to reveal with this simple model some essential favourable features of a static-recurve bow.
The static-recurve bow is also the subject of Section 4.11. We have no accurate experi- mental information with respect to these bows. The shape of the unstrung bow, as depicted in a number of books and papers shows a large variety. Therefore we deal in Section 4.11 with a few bows which seem to be representative for the static-recurve bow. The lack of detailed information however, makes that we have to be cautious with the interpretation of the results. Yet, it is likely that the performance of a static-recurve bow differs not much from the performance of a long straight-end bow. For a comparable performance, however,
recurve, mentioned in literature. The results obtained in Section 4.9 and Section 4.11 are used to explain the differences in the performances.
In Section 4.13 we check the finite-difference procedure developed in Section 4.4. To that end we consider a vibrating beam with small deflections, hence the linearized theory applies. Numerical solutions by means of our finite-difference method are compared with the results obtained by an analytic method.