Compound bow dynamic model: Park 2009 study

Park builds a mathematical model of a single-cam compound bow, both static (the force-draw curve the archer feels) and dynamic (how the arrow is driven after release), and applies it to an Archery Research AR34 shooting a 23.48-gram Easton Protour 470. Over the opening quarter of the power stroke the arrow picks up speed slowly, because the limbs' energy first goes into spinning up the wheel and cam; acceleration then rises steadily and, just before separation, shows a sharp peak of 892 g as the limbs hand their energy back. The arrow spends nearly 60% of its time on the string covering only the first 20% of the stroke, while string and arrow are still close to the archer's face and body. The model gives 17.5 milliseconds from release to separation, and photographs measured between 17.0 and 17.25. The calculated exit speed is within 0.1 m/s of the measured one, and the calculated force-draw curve follows the real one closely.

Varying the parameters in the model, a higher effective limb mass makes the final peak higher and later, more wheel and cam inertia lengthens the slow opening phase, and a heavier arrow evens out the push. For this bow the author puts the minimum desirable arrow mass at about 25 grams. String and cable tensions climb at the end of the stroke to many times the peak static draw weight.

The model follows Hickman's approach for the longbow: limbs treated as rigid arms pivoting on an elastic hinge, an inextensible string, a rigid riser (its measured flex is under 1.5 mm) and a bow symmetrical about the arrow. The radii of the cam's three tracks were measured through the draw with a purpose-built jig, to within 0.5 mm, and described with sine and cosine functions or, for the cable track that changes quickly near the valley and wall, with splines over 21 points.

Almost all the geometry and the masses (string, cables, peep, D-loop, cable slide, arrow) were measured directly. The limb spring constant was chosen so that the calculated peak draw weight matched the measured one, and the bow's virtual mass, which absorbs friction and other losses, was derived by measuring the speed of arrows of different mass. The stored energy is shared among arrow, limbs, wheel and cam, strings, cables, peep, D-loop and cable slide, each at its own speed, and accelerations are obtained by differentiating the speeds numerically. The check uses the force-draw curve, the exit speed and the arrow's position over time measured photographically, to a precision of 0.25 ms.

The model is checked against one bow and one arrow. The exit speed matches the measured one partly because two parameters, the limb spring constant and the virtual mass, were fitted to measurements on that very bow; the most independent check is the time on the string. The effects of limb mass, cam inertia and arrow mass come from the model, not from tests on different bows.

The step from physics to accuracy is a hypothesis. The idea that a sharper acceleration peak makes a bow harder to shoot accurately is Hickman's, and Park takes it up conditionally, supporting it with his own experience of different bows and with the preferences of top archers, without any group measurements. The 25-gram minimum is likewise described by the author as subjective. The model stays in the vertical plane of the bow and ignores arrow flex. The paper acknowledges support from Hoyt and Easton.

For compound archers the study gives a clear picture of the shot: a slow start while wheel, cam and limbs gather speed, then a violent final push just before the arrow leaves the string. What happens in the first milliseconds, with the string still near the face, affects how the arrow leaves: according to the author, accuracy suffers badly if the string touches the face or body at that stage. The author concludes in favour of light cams and limbs with a low effective mass, with no weights or dampers added to the cams or near the limb tips.

On arrows, the author suggests choosing, among those that group well, the heaviest that still allows aiming at the longest competition distance, because it evens out the push. This is a deduction from the model and from his experience: the study does not test it on groups, so it remains a reasonable hypothesis. The paper does not cover the recurve, which has no cams, although the reasoning on effective limb mass comes straight from Hickman's model of the simple bow.