Round-wheel compound bow model: Tiermas 2015 study

The paper proposes a mathematical model of one specific kind of compound bow: the type with two identical round wheels mounted off centre at the limb tips. Each wheel has two concentric grooves, a larger one for the string and a smaller one for the cable, and turns about an axle that does not pass through its centre; this geometry is what produces the familiar compound force-draw curve, which rises steeply, peaks and then drops into the valley at full draw. For each wheel position the model calculates the draw length and the force on the arrow.

Tested on a real 55-pound compound, the model follows the measured force-draw curve closely, with a slight departure near full draw that the author puts down to stretch in the string and cables, which the calculation leaves out. The curve measured while drawing always sits slightly above the one measured while letting down, because of friction in the wheels and hysteresis in the limbs, string and cables. The path of the limb tip is also well described as an arc of a circle, provided the classic parameter is changed: the radius is 0.598 times the limb length, instead of the three quarters of the older model devised for traditional bows. An internal check on the calculations, working out the stored energy in two independent ways up to the full draw of 0.68 m, gives differences below a millionth of a joule.

The model is geometric and static. It assumes a symmetric bow, inextensible string and cables that do not slip on the wheels, cables in the same plane as the string (the cable guard is ignored) and limbs that bend according to Hooke's law, like a rigid rod hinged at a single point. The equations have no analytical solution and are solved by iteration.

The bow was a 55-pound JahtiJakt. Axle-to-axle distance (102.1 cm), riser length (38.1 cm) and limb length (38.9 cm) were measured with a steel rule; the groove radii (2.68 and 1.99 cm) and the offset of the axle from the wheel centre (1.37 cm) from enlarged photos of the wheels; and the angles with a marine navigation protractor. Values are the mean of the upper and lower halves of the bow, which did not differ appreciably. The hinge position was found by fitting an arc of a circle to measured positions of the wheel axle with the limb flexed. The force-draw curve was measured with a Hanson 8910 bow scale, calibrated with 2.5 kg discs up to 27.5 kg, drawing the bow slowly with a winch to just past full draw and then letting it down, while a video camera recorded the scale and the rule. Limb stiffness (1032 N/rad) was not measured directly: it is the value that gives the best fit between model and measured curve.

The model was tested on a single bow, with round wheels, a configuration that is now uncommon: current compounds use cams of other shapes, in single, twin or hybrid set-ups, which the model does not cover. Limb stiffness is derived from the same curve the model then reproduces, so the good agreement with the force-draw curve is partly built in: it shows that the geometry is well described, not that the model can predict the curve of a bow that has never been measured. The curve appears only in a chart, with no table of values or numerical measure of the fit.

The model is static: it says nothing about arrow speed, efficiency, vibration or the vertical movement of the nocking point, and it ignores the cable guard and stretch in the string and cables. The work was funded by a Finnish company, A-Pipe Oy, with no further detail on its role.

For compound archers there is no direct practical effect: the study does not deal with aiming or with tuning a modern cam bow. It does explain in mechanical terms where the shape of a compound's force-draw curve comes from, namely the combination of string groove, cable groove and off-centre axle, and it shows why a bow scale reads slightly higher while the bow is being drawn than while it is being let down. According to the author, anyone designing or modifying wheels of this kind has a model accurate enough to predict what changes when the axle is moved or the groove radii are altered.