Twin-cam compound bow and cam design: Tiermas 2016

The author builds a mathematical model of the compound bow with two identical cam systems (twin-cam) in which the cam shape is no longer approximated by circles and levers but described point by point. On the round-wheel compound from his earlier paper the new model gives the same force-draw curve as the old one, with differences below 3 thousandths of a newton. On a commercial bow, a Hoyt "Smoke", the calculated curve follows the measured one closely. While drawing, the measured forces are always slightly higher than while letting down, because of friction in the wheels and hysteresis in the limbs, string and cables, and at full draw the calculation drifts a little from the data, probably because the string and cables stretch and the cam measurements carry small errors.

The author then uses the model as a virtual test bench: by trial and error he reshapes the cams to get a fuller curve under two constraints, the same full draw and the same peak force. With the modified cams the energy stored up to a draw of 0.764 m rises from 96.5 to 106.5 joules, about 10% more, mainly because the curve rises earlier; the starting draw, on the other hand, gets shorter. Under these constraints the theoretical limit would be a rectangular curve.

The two cams on each wheel, the string cam and the cable cam, are described in polar coordinates with cubic spline functions, and the limb with the modified Hickman model: a rigid rod that bends at a single point. The non-linear equations are solved numerically for 2,000 rotation positions of the wheel. The bow is assumed to be symmetric, with no cable guard, and with inextensible string and cables lying in one plane; the author notes that real twin-cam bows are not always symmetric.

For the Hoyt Smoke the cams were photographed and their radii measured on the enlarged photograph, allowing for the string and cable grooves. Axle-to-axle distance, distance between the limb bases and limb angle were measured (102.0 cm, 42.0 cm and 23 degrees); the effective limb length and hinge position were derived from measurements of the limb tip, and the limb spring constant (1,135 N/rad) was chosen afterwards so that the calculated curve matched the measured one as closely as possible. The cam data had to be smoothed, adding and removing points, because a single stray point is enough to make a cam non-convex and stall the calculation. The energy was computed in two independent ways, which differed by less than 0.1%.

Here too the model is static, as the author repeats: it does not describe the shot, arrow speed or the dynamic behaviour of the wheels and limbs. There is only one test on a commercial bow, and the limb spring constant was chosen precisely to make the calculated curve fit the measured one, so the match is not an independent prediction. The cam shapes come from an enlarged photograph and from data adjusted by hand, and reducing the limb to a hinge is an approximation. The improved cams exist only on paper: the paper does not report that they were built or tested, and the author warns that their minimum radii would need checking for material strength. The 10% gain in energy is one of many possible examples, found by trial and error, not the outcome of an optimisation. The research was funded by a private Finnish company, Byro Energiatekniikka Oy.

For a compound archer the value of the study lies in one idea: the force-draw curve, with its rise, peak and valley, is written into the shape of the cams, and a model like this one lets you see on paper what happens when the cams are rotated or the limb angle is changed, before touching the bow. The gain reported, 10% more energy at the same peak and draw length, comes from a curve that rises faster, in other words from a more aggressive bow to draw; and more stored energy says nothing, on its own, about how comfortable or accurate the bow is at full draw.

For everyday tuning there is nothing to apply: the simulated changes concern people who design or rework cams, and the model ignores the dynamics of the shot, which is exactly where an archer feels the difference.