Round-wheel compound bow revisited: Tiermas 2017 study

The author goes back to his earlier model of a compound bow with two round wheels and adds one possibility: that the string eccentric and the cable eccentric have different centres. With the model he explores how the force-draw curve changes when four wheel dimensions are varied, and he introduces two static quality coefficients: one for the curve alone, which equals 1 for an ideal rectangular curve, and one for the whole bow, which also takes the riser geometry into account. The reference compound, taken from one of his earlier papers, has a curve coefficient of 0.619.

How far the centre of the cable eccentric sits from the axle mainly sets the let-off: at 16.7 mm it reaches about 80%, but with a cable eccentric of 19.9 mm radius that value may already be out of reach for material strength. The offset of the string eccentric, by contrast, moves the peak and changes the slope of the curve. By changing only the wheels and the limb stiffness, the author builds two virtual bows with the same peak force as the starting bow (223.9 N) and almost the same full draw (67.1 cm against 67.3): the better one stores 73.1 J against the starting bow's 61.7, 18.5% more, with a whole-bow coefficient 18.8% higher, but the holding weight at full draw rises from 73.1 to 88.3 N. Finally the author derives a limiting curve, that of a traditional bow with the same limbs and riser, which no round-wheel compound can exceed in the early part of the draw.

The paper is purely computational: no new measurements were taken on a bow. It starts from the parameters of the round-wheel compound measured in an earlier paper (axle-to-axle 102.1 cm, limbs of 38.9 cm, cable eccentric of 19.9 mm radius), replaces some equations of the old model with ones for offset centres, and computes the force-draw curve for 2,000 wheel positions. The limb is still the modified Hickman model, a rigid rod bending at one point, with its own spring constant.

The wheel parameters were varied one at a time (how far the centre of each eccentric sits from the axle, and the two starting angles), then combined by trial and error for the two virtual bows, with the limb spring constant changed as well. The calculations were checked by comparing two ways of obtaining the energy and by recomputing the same curves with the author's twin-cam model, with negligible differences.

It is a static model, as the author states plainly: it ignores the masses of the modified wheels and limbs and the limb material, so it cannot say whether the improved bows would shoot an arrow faster. Neither virtual bow was built, and some combinations of dimensions are at the limit of material strength, as the author acknowledges for the wheel giving 80% let-off. The 18.5% gain comes at a price: in the better bow the holding weight at full draw rises from 73.1 to 88.3 N, so the let-off drops, and the curve comes to resemble that of some single-cam bows. The quality coefficient is the author's proposal, rewarding energy at a given peak force, not a measure of how pleasant or accurate a bow is to shoot. Finally, the type of bow studied, with round wheels, is now a minority compared with cam-driven compounds.

The study is of interest to anyone who wants to understand where let-off and curve shape come from in a compound, more than to the everyday archer. It sets out clearly what each wheel dimension moves: the position of the cable eccentric decides how much the weight drops at the end of the draw, and the offset of the string eccentric decides where the peak falls and how steep the rise is. The comparison of the three bows also shows the basic trade-off: at the same peak force and draw length, more stored energy comes from a fuller curve, and in the example this cost about 15 N more to hold at full draw.

None of these conclusions concerns the accuracy or actual speed of arrows, which the model does not compute. For someone choosing a bow, the useful point is that stored energy and comfort at full draw pull in different directions; the model puts numbers on it, but only for the static case.