Compound bow limb deformation: Tiermas 2016 study

The paper deals with how the limbs of a round-wheel compound bow bend, and how much it matters to describe this well when predicting the force-draw curve. In his earlier model the author treated each limb as a rigid rod hinged at a single point (a modified version of Hickman's approximation); here he replaces it with an elastic limb that curves along its whole length, following the Euler-Bernoulli equation. On the same bow measured in the earlier paper, the calculated curve matches the measured one when the Young's modulus of the limb material is set at 33.75 GPa, and the difference from the old model is tiny: under 0.8 newtons on the curve, under 0.3 mm on the position of the limb tip and on the draw.

The second part tries three limb profiles, straight, curved and recurved, with the same stiffness. Once the forces are brought to a common scale the curves look very alike: the curved limb gives the least steep initial rise and the highest force at full draw, but the differences stay small. Thickness, width and material also mainly shift the height of the curve, and to some extent the full draw, rather than its shape. A recurve bow behaves the other way round, because the string lies along the limbs and the working part of the limb changes during the draw; in a compound the force always acts at the wheel axle. Finally, for a straight limb of constant cross-section, the author shows that under small deflections the tip travels along a circular arc whose radius is 5/6 of the free limb length.

It is a theoretical study with a single experimental check. The equations for the elastic limb, taken from the theory of static deformation of traditional bows, are solved numerically (a Runge-Kutta method plus an iterative search for the zeros) and inserted into the round-wheel compound model the author had published earlier. The test bow is the one from that paper: the width and thickness of the limb were measured with callipers at several points to give its stiffness along the length, and the part held inside the limb pocket (8.5 cm) with a steel rule. The Young's modulus was not measured: it is the value that gives the best least-squares fit between the calculated curve and the force-draw curve measured while drawing and letting down.

The calculations were checked by comparing the energy from the area under the curve with the energy stored in the limbs (difference under 0.15%) and, for a straight uniform limb, against a known solution based on elliptic integrals (under 0.1%). For the comparison of profiles the author fixed the bow's dimensions, including free limbs of 30.4 cm out of a total length of 38.9 cm, and brought the forces to the same scale with correction factors.

The model is static: it describes the bow held still at each draw length, not the shot, so it says nothing about arrow speed, vibration or dynamic efficiency, as the author himself points out. The experimental check covers a single round-wheel bow, a type of compound that is uncommon today, and a Young's modulus obtained by fitting the curve also absorbs the errors in the other measurements, so the close match with the data was partly built in. The limb is treated as a line with no thickness, of uniform material and rectangular cross-section, which is reasonable for a limb made of one material and less so for laminated limbs. The straight, curved and recurved profiles are hypothetical limbs, not commercial ones, and they are compared after a normalisation of the forces chosen by the author. The author also notes that with very low stiffness values the calculation becomes unstable.

For compound archers the study changes nothing in everyday shooting, but it clarifies a design point: in a compound's force-draw curve the wheels are in charge, not the shape of the limbs. Thicker, wider or stiffer limbs make the bow heavier, raising the curve, and may change the draw length a little, but they leave its profile almost untouched; even moving from a straight limb to a curved or recurved one, at the same peak force, makes little difference. According to the model, then, the rise and the valley depend far more on the wheels than on the limbs.

The author adds a caution about strongly recurved limbs on a compound: the tips can end up working in the opposite direction to the useful one, to the point of buckling. It remains a theoretical observation, which the paper does not check on a real bow.