The round-wheel compound bow model revisited: a new extension

Sports Eng DOI 10.1007/s12283-017-0225-2 ORIGINAL ARTICLE The round-wheel compound bow model revisited: a new extension Marko Tiermas1 Ó The Author(s) 2017. This article is published with open access at Springerlink.com Abstract An extended offset-eccentric model of an archery twin-round-wheel compound bow is derived. Varying some parameters of the model, the respective effects on the calculated force–draw curve are considered. Two static quality coefficients for the compound bow are introduced. It was found that the twin-round-wheel compound bow can be designed to be more energetic with the help of the model. For a bow with some modifications 18.5% increment of energy was calculated. Also a theoretical limit for the force–draw curve of the compound bow is concluded. Keywords Compound bow  Force–draw curve  Eccentric wheel 1 Introduction The force–draw relation of the archery bow is one of the main interests of the serious archer. Indeed, the force–draw (FD) curve of the bow not only determines the energy which is stored in the limbs and transferred mainly to the kinetic energy of the arrow, but also the experience of drawing, aiming and releasing the bow. Compared to conventional (or traditional) bows, the compound bow (a bow with pulley systems at the tips of the bow limbs) offers greater possibilities to manipulate the FD curve due to the more complex bow configuration. The simplest type of compound bows, the symmetric twin-round-wheel compound bow, is presented in Fig. 1. There are only a few researches concerning the compound bow. The first investigations of the compound bow including a model of the asymmetric single-cam compound bow were presented by Park [1, 2]. In [3] Zanevskyy has introduced an asymmetric model for a special type of compound bow with centric cable eccentrics. A model for a more usual round-wheel compound bow is presented in [4], and the static deformation of the limbs of this kind of bow is studied in [5]. A detailed model for the twin-cam compound bow is introduced in [6]. While the most effective compound bows nowadays in use have cam systems that differ markedly from circular, the compound bow with eccentrics has still a special role. Compared to non-round cams of the newest compound bows, the round eccentrics can be manufactured by simpler means. Although the mathematical model including round eccentrics (with or without the extension of this paper) is as well far from trivial, it is conceptually more simple and numerically more robust than models including non-round cams. The aim of this paper is to develop the original round-wheel compound bow model of paper [4] further and to check the possibilities of improving the efficiency of the round-wheel compound bow. The idea of offset between the cable and the string eccentric centres is combined with the original model, as this offers more options to modify the FD curve of the bow. The round-wheel compound bow model with this extension may be called briefly as offset-eccentric model. 2 Offset-eccentric model & Marko Tiermas 1 Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, 00560 Helsinki, Finland Let us consider the round-wheel compound bow in case of the cable and the string eccentrics of the upper wheel have different centres, as in Fig. 2. The respective upper part of the bow is presented in Fig. 3, from which we notice that M. Tiermas Fig. 1 A typical twin-round-wheel compound bow in the initial position and its upper wheel system (Ref. [4], https://creativecom mons.org/licenses/by/4.0/) Fig. 3 The upper part of the compound bow in the initial (1) and drawn (2) positions. Note that in position 2 e is negative. The cables and the cable eccentric are left out from the figure for clarity (Ref. [4], replaced symbols dR , e and e0 , https://creativecommons.org/licenses/ by/4.0/) the upper string eccentric and the point where the string touches the upper string eccentric. Further, from Fig. 3 we also conclude that e s cos u ¼  dR sin e þ R sin u; ð2Þ 2 where e is the distance between the upper and the lower axle point. From Eqs. (1) and (2) we get he i e 0  dR sin e0 þ Rðe0  e þ uÞ cos u  2 2 ð3Þ þ dR sin e  R sin u ¼ 0: The cable and the string eccentrics are firmly attached to each other, so the rotating angle is the same for both eccentrics, e0  e ¼ a0  a; Fig. 2 The wheel system of the upper limb of the round-wheel compound bow with different eccentric centres when the bow is drawn. Note that e is here negative s ¼ s0 þ Rðe0  eÞ þ Ru e0 ¼  dR sin e0 þ Rðe0  eÞ þ Ru; 2 ð1Þ where s is the length of the straight half-string, s0 the value of s in the initial position, e the angle between the horizontal line and the line that connects the centre of the upper string eccentric and the upper axle point, e0 the value of e in the initial position, R the radius of the string eccentric, e0 the distance between the upper and the lower axle point in the initial position, dR the distance between the axle and the centre of the string eccentric, and u the angle between the horizontal line and the line that connects the centre of ð4Þ where a is the angle between the horizontal line and the line that connects the centre of the upper cable eccentric and the upper axle point, and a0 the value of a in the initial position. Now, with a fixed a the unknown e can be solved from Eq. (4), when u can be obtained from Eq. (3) with the Brent–Dekker method [7] for example. After this, s can be calculated from Eq. (1). Moreover, from Fig. 2 we see that the lever arms of the string and the cable tensions are ds ¼ R  dR cosðu  eÞ; dc ¼ r þ d cosða  dÞ; ð5Þ where r is the radius of the cable eccentric, d the distance between the axle and the centre of the cable eccentric, and d the angle between the horizontal line and the line that connects the centre of the upper cable eccentric and the point where the straight cable contacts the upper cable eccentric. The round-wheel compound bow model revisited: a new extension The draw is here defined as the distance from the midpoint of the string to the vertical line that connects the riser ends (or bottoms) of the upper and the lower limbs. According to Fig. 3, the draw is D ¼ ð1  AÞL sin hU þ AL sin h  dR cos e þ R cos u þ s sin u; ð6Þ where L is the length of the limb (measured from the bottom of the limb to the axle point along the limb), A the ratio between the length of the supposed elastic portion of the limb with respect to L, h the angle between the vertical line and the line that connects the upper axle point and the supposed hinge point of the limb, and hU the angle between the undeflected bow limb and the vertical line. The offset-eccentric model can now be formed from the original round-wheel model by replacing equations (8)– (11) and (13) of paper [4] with Eqs. (1)–(6). By choosing dR ¼ d and e0 ¼ a0 the offset-eccentric model is simplified into the original model. 3 Results As one measure of eva (...truncated)