The limb deformation of the compound bow

Meccanica (2017) 52:1475–1483 DOI 10.1007/s11012-016-0485-0 The limb deformation of the compound bow M. Tiermas Received: 24 February 2016 / Accepted: 3 July 2016 / Published online: 11 July 2016 Ó The Author(s) 2016. This article is published with open access at Springerlink.com Abstract A model of the archery round-wheel compound bow with a more realistic limb deformation is introduced. The model is based on the former compound bow model of the author with the supposition that the elastic limb deformation is consistent with the Euler–Bernoulli equation. The Young’s modulus of an example bow limb is determined. The model is compared to the experimental data and to the former model which uses the modified Hickman approximation for the bow limbs. The effects of the shape of the limb on the force-draw curve of the compound bow are also considered. The model presented here may be used when designing or adjusting the limbs of the compound bow. An approximation for the path of the limb tip for the straight limb of uniform bending stiffness for small deformations is also derived. Keywords Compound bow  Elastic deformation  Force-draw curve  Eccentric wheel M. Tiermas (&) Department of Physics, University of Helsinki, Helsinki, Finland e-mail: 1 Introduction The compound bow, which was invented by Claude Lapp in 1938 [1], has become a popular instrument among the bow hunters and sportsmen. Yet, the mechanical action of the compound bow has not been investigated until quite recently. In the earlier studies [2–5] concerning compound bows the limb of the bow has been treated as a straight rod, which bends only on one point, which is either at the bottom of the limb or at some distance from the limb bottom. This approximation is the motivation of this paper. Hickman introduced a simple model of the traditional long bow in [6], where he also noticed that the limb bends in the arc of circle. If the length of the limb is L, he also showed that the path made by the limb tip is an arc of the circle whose radius is 3L/4 and whose center is located at a distance of 3L/4 from the tip of the undeflected limb. This idea combined with the Hooke’s law for the force acting on the tip of the limb is the basis of his long bow model. The Hickman model is confirmed by Tuijn and Kooi [7], and its modified version is also included in the compound bow model by the author [5]. The Hickman model was overtaken by Kooi in [8], where the deformation of the limb of the bow with or without recurve is thoroughly investigated. As a result, the initial shape and also the possible length-dependent bending stiffness of the limb can be taken into account. This kind of elastic limb bending is more realistic and also offers a possibility for further 123 1476 Meccanica (2017) 52:1475–1483 generalizations. The aim of this paper is to insert this elastic bending in the former model of the compound bow presented in [5]. eccentric system is fixed on the other end of the limb. Let us call the axle point T ¼ ðxðLÞ; y(L)) of this eccentric system as the tip of the limb, as noted in Fig. 1. Let us also introduce the angle 2 Formulation of the problem k ¼ x  xU At first, we shall follow the theory of the static deformation of the bow limb as introduced by Kooi [8]. Let us assume that the limbs of the compound bow are inextensible, they are symmetric as in [5, 8], and can be represented by an elastic line of zero thickness. Then it will be sufficient to consider the upper part of the bow only. Let Ltot be the total length of the limb measured from the riser end (the bottom) of the limb to the axle point along the limb. As a part of the limb is inside the modified part of the riser (the limb pocket), let L be the length of the free (elastic) limb measured from the point where the limb touches the riser to the axle point along the limb. Let us assume that the part of the limb which is inside the limb pocket is straight and rigid. This rigid part of length Ltot  L is also seen in Fig.1 as the line GO. Let O ¼ ðxð0Þ; yð0ÞÞ be the point where the free limb meets the limb pocket. Let us call the point G in Fig. 1 as the bottom of the limb, whereas the point O may be called as the bottom of the free limb. The where x is the angle of the infinitesimal segment of the limb with respect to the vertical line and xU the angle of the infinitesimal segment of the limb without the string (undeflected limb) with respect to the vertical line. There are also two geometric conditions, dx ¼ sinðk þ xU Þ dl ð2Þ dy ¼ cosðk þ xU Þ dl ð3Þ where x and y are the Cartesian x- and y-coordinates of the infinitesimal segment of the limb measured from the bottom of the free limb towards the tip of the limb, and l the length coordinate along the limb measured from the bottom of the free limb with length L towards the tip of the limb. The boundary conditions at l ¼ 0 are kð0Þ ¼ xð0Þ ¼ yð0Þ ¼ 0 123 ð4Þ Let us assume the Euler–Bernoulli equation for the elastic line is valid. With the help of geometrics and the Euler–Bernoulli equation, the following equation can be derived, W Fig. 1 The upper part of the compound bow in drawn position with related variables and forces. The size of the wheel system is exaggerated. In balance, the supporting force K prevents the limb from moving. The force K acting on the limb tip T is the sum of components F s , F cl and F cu . The line segment GO is the rigid part of the limb inside the limb pocket, whereas the curve OT is the free limb. Note that the angle a is here negative ð1Þ dk ¼ K ððb  xÞ cos s  y sin sÞ dl ð5Þ where W is the bending stiffness of the limb, K the absolute value of the total force K acting on the tip of the limb, s the angle between K and the vertical line, and b the distance between the bottom of the free limb and the point where the line that goes via the tip of the limb and is parallel to K meets the vertical line that goes via the bottom of the free limb. For the details of Eqs. (1)–(5), the reader is kindly asked to examine the paper [8], where the deformation of the conventional bow (with or without recurve) is presented elaborately. Now our problem differs from [8], for the unknowns in the group of Eqs. (1)–(5) are K and b. Also we must find s and the respective limb tip coordinates yL ¼ yðLÞ and xL ¼ xðLÞ. In [5], it was supposed that the limb bends as in Hickman model with a slight modification, which is Meccanica (2017) 52:1475–1483 1477 quite different case as here. However, we can still use the compound bow model introduced in [5] in order to find the limb tip y-coordinate and the angle s, for these values can be calculated without the Hickman assumption. From Fig. 1 we find that the y-coordinate of the limb tip with respect to the bottom of the free limb is  e g yR ¼  þ ðLtot  LÞ cos hU ð6Þ 2 2 where e is the distance between the axle points of the upper and the lower limbs, g the length of the riser measured from the bottom of the upper limb to th (...truncated)