Compound nocking point path: Park 2010 study

James Park computes the path taken by the nocking point in the vertical plane as the bow is drawn and then as the string comes forward again. The answer, in general, is that it is not a straight line perpendicular to the string's rest position, and the deviation can be written in a single formula that holds for compound, recurve and longbow alike.

Several consequences follow from the formula. If the arrow pass sits above the geometric centre of the bow, as it almost always does, the nocking point rises as the bow is drawn, and rises more the higher the arrow sits above that centre. The deviation falls as the bow's axle-to-axle length increases, while the unbraced limb angle matters little. On a bow built with identical cams top and bottom, the measured path is well approximated by a parabola with its turning point at the brace height.

The part that concerns bow design lies in the degrees of freedom of the cams. Park reviews the four configurations in use and counts how many asymmetric degrees of freedom each offers: two symmetric for the twin cam, three asymmetric for the single cam, one symmetric and three asymmetric for the hybrid, three symmetric for the binary. Only asymmetric degrees allow the path to be straightened, so only single-cam and hybrid bows can do it; twin-cam, binary, recurve and longbow have none.

Figures measured on real bows give the scale of the effect. On a single-cam bow of a reputable make, 0.865 metres axle to axle, the uncorrected path would have a computed maximum deviation of 21.8 millimetres; the same cam, corrected by the manufacturer through the relative radii of the two string-side tracks, gives a measured path within 1.5 millimetres of the perpendicular straight line. It is the best of the many bows the author has measured.

Finally the dynamic path, which is what the arrow actually experiences, combining the geometric path over time with the rotation of the bow during the power stroke. The rotation is small compared with the geometric component, but it is always present and cannot be cancelled. On the single-cam bow examined, the bow rotates with the upper limb moving away from the archer, because of the different inertia of cam and wheel; on a hybrid cam the rotation goes the other way, from the reaction of the string pushing the arrow. The resulting dynamic path takes the nocking point first below the straight line, then above it more rapidly, and finally keeps it almost constant over the last part of the power stroke.

The core is geometric. Park writes the lengths of the string from both ends to the nocking point, at rest and at draw, and from them derives the distance of the nocking point from a line perpendicular to the string at rest. The final formula contains the height of the arrow above the geometric centre, the active limb length, the limb angle unbraced and at draw, and the string extension as it unwraps from the cams; it does not contain the cam radius or the setback of the grip, which is why it also holds unchanged for bows without cams.

The assumptions are stated: identical limbs set at the same angle, identical and synchronised cams, a nocking point above the geometric centre, limb tips moving on a circular path, and equal deflections of the two limbs. The last is the weakest, because a nocking point off centre makes the string meet the two limbs at different angles. Park tests it by measuring several compound bows of different configurations: upper and lower limb deflections matched to within half a degree over the full draw. In the worst case, a short axle-to-axle bow at competition draw weight, the stiffness measured at the nocking point exceeded 8400 newtons per metre at full draw; the string-limb angles were 44 and 46 degrees, giving a difference in deflection of 0.22 degrees and a maximum error of half a millimetre in the computed path.

The dynamic part is added to the geometric one. The bow's rotation about the point of contact with the hand comes from the moments of the push on the arrow, which passes above the grip, from the weight of the bow applied at a centre of mass displaced towards the target, and from the difference in angular momentum between the bottom cam and the top wheel, which on a single-cam bow have different masses and inertias. The position of the arrow over time comes from a dynamic bow model the same author published the previous year.

There are three comparisons with reality, all presented as graphs. The computed geometric path is set beside the one measured on a twin-cam bow, with the agreement described as reasonable; on a binary cam the measured path rises monotonically as the formula predicts; on the single-cam bow the path is measured after the manufacturer's correction. The computed rotation was checked with accelerometers attached to the riser.

This is modelling work with limited experimental checking. The bows measured are not listed, the data appear only as graphs, and the agreement between calculation and measurement on the twin-cam bow is described in words, as "reasonable", with no quantified discrepancy. The number of bows on which the equal-deflection assumption was tested is not given, only the result: within half a degree.

The equal-deflection assumption is explicitly weaker for recurve bows and longbows, which have softer limbs and, near the brace height, string-limb angles far from a right angle. The numerical check on the error, half a millimetre, was made on a compound bow: for bows with a string attached directly to the limbs there is no equivalent, and the formula remains applicable with a caution the author states but does not quantify.

The dynamic part rests on another model by the same author for the position of the arrow over time, so it inherits that model's assumptions, and it treats the bow as very stiff in the vertical plane, which the author justifies for a compound and not for a recurve. The unequal limb deflection caused by a centre of mass in front of the hand, documented elsewhere, is not modelled.

The most important point, though, is that nothing here reaches the target. There is no measured group, no score, no link between path deviation and arrow dispersion. The author is explicit on one thing: he makes no judgement about what the ideal nocking-point path would be, only about the designer's opportunity to modify it. The consequences for arrow flex and rotation are expected on mechanical grounds, not observed. The work acknowledges support from Hoyt and Easton Technical Products.

A compound archer will find here the mechanics of a problem they know as a symptom: arrows landing low when the bow is pulled harder against the stops. If one cam reaches its stop before the other, the nocking point rises over the last part of the draw, the riser tilts with its top moving towards the archer, the sight is in effect higher and the arrow hits low; and since the lagging cam keeps rotating the harder the bow is pulled, the point of impact moves with how firmly full draw is reached. The remedy, Park writes, is cam timing, adjusted through the cable lengths.

The second point concerns the nocking point on the string, and here the author's guidance depends on the configuration. On a twin-cam or binary bow no asymmetric degree of freedom is available to straighten the path, so the deviation can only be reduced by bringing the nocking point closer to the geometric centre of the string, within the clearance the riser allows. On a single-cam or hybrid bow the manufacturer has already been able to straighten it, and the position the design intends is the one level with the launcher mounting hole.

The third is a point about choosing a bow: the longer the bow axle to axle, the less the nocking point deviates from the straight line. It follows that on a longer bow it is also easier to keep the rear of the arrow from touching the launcher.

This record should also be read for what it does not say. There is no measurement of how many points a deviation of 20 millimetres rather than 1.5 costs, because nobody shot for score, and Park himself declines to say what the ideal path would be: his work concerns the path a bow has and the scope for changing it. One useful order of magnitude does remain: between a corrected single cam and the same cam without correction there are about twenty millimetres of vertical travel of the nocking point, and the contribution of bow rotation during the power stroke is small by comparison.