Compound limb asymmetry: Zanevskyy 2012 study

The main earlier model of the twin-cam compound, Park's (2009), starts from the assumption that both limbs bend in the same way. Here the author builds a mechanical model of the drawn bow without that assumption and looks at what changes. From 12 bow parameters (limb and riser lengths, cam radii, limb stiffness, angles, positions of the grip and the nocking point) the model derives a further 20 values: limb and cam angles, string and cable forces, and the components of the draw force.

For the bow used as an example, at full draw the upper limb comes out more deflected than the lower: 0.124 against 0.110 radians, a difference of about 12% relative to the mean, while the symmetrical model gives 0.117 for both. Other quantities change little: the vertical component of the force by 2.1%, the vertical position of the nocking point by 1.5%, the string and cable forces by 0.1-0.2%. Compared with a photograph of the same bow drawn by a hanging weight, the asymmetrical model comes closer to the measured angles (0.125 and 0.106 radians): it is out by 0.001 radians on the upper limb against 0.008 for the symmetrical model, and by 0.004 against 0.011 on the lower. The largest discrepancy concerns the load, 98.1 newtons measured against 104.9 calculated, or 7%. The model also calculates that, compared with a virtual recurve with the same peak draw force, this compound stores 75% more energy.

The model treats each limb as a rigid bar hinged to the riser with a spring of constant stiffness, and each cam as a profile whose radius varies with angle along a cosine curve (characteristic radius 33 mm, coefficient 1.3). The equilibrium equations for cams, limbs and string form a non-linear system with no explicit solution, solved numerically with the Newton-Raphson method in MathCAD; the programme is given in an appendix. The asymmetry comes from the geometry: the length of string below the nocking point is longer than that above it (parameter D, half the difference, is 40 mm) and the grip does not lie on the line through the limb pivots. For the comparison with the symmetrical model the author forces the limbs to match by artificially adjusting their stiffness by 0.89 Nm.

For the experimental check, he hung a compound bow (serial number 028815P) by its grip, loaded the nocking point with a weight to a draw of 0.738 m and photographed it with a digital camera about 4 metres away, perpendicular to the plane of the bow. The coordinates of the key points were read in pixels in MS Paint, using a ruler on the riser as the scale (0.761 mm per pixel); the error due to the image is estimated at around 1%.

The check covers a single bow, in a single position, drawn by a weight and photographed at rest, and the author says as much: the model still needs generalising. The cam profile is an idealised cosine curve rather than that of a commercial cam, and the limbs are rigid bars on a spring rather than flexing beams, with stiffness assumed constant. The model is static: it describes the bow held at full draw, not what happens during the release, when differences between the limbs could affect the motion of the nocking point.

The 12% refers to limb angles, not forces, which stay almost the same in both models, and it depends on the geometry of that particular bow, especially on how far off centre the nocking point and grip are. The 7% discrepancy on the load is larger than any difference between the symmetrical and asymmetrical models in the results table, limb angles included (6.4% and 5.6%): the case for the asymmetrical model therefore rests only on the direct comparison with the measured angles. Reading coordinates with a drawing programme and a ruler is a simple method, and the 1% uncertainty accounts only for perspective distortion.

The idea that the two limbs of a compound work identically is a simplification: with the nocking point above the centre of the string and the grip offset, the upper and lower limbs bend by different amounts at full draw, even though the forces involved hardly change. For compound archers this gives a mechanical basis to an idea familiar to anyone who tunes: the position of the nocking point and the grip changes the way the two limbs work, and the bow does not work symmetrically even when the limbs are identical.

The study does not, however, say how to adjust a bow, measures no effect on arrow flight or score, and does not deal with the recurve. The author offers the MathCAD programme as a tool for coaches and archers without a mathematical background, but using it requires 12 precise measurements of the bow, including limb stiffness, which is hard to obtain without laboratory equipment.