Fun with Target Arrow Modeling
Archery is a sport with as many variables to shooting as there are archers. One of these variables is labelled as "forgiveness" with respect to an arrow. Anecdotally, arrow forgiveness is the characteristic of an arrow shaft, point weight, nock weight and fletching that provides a robust response to slightly varying inputs. My curiousity lies in what physical characteristic can the archer apply to an arrow to achieve forgiveness. In order to apply some science to the sport, I went a ahead and looked at the first mode of a target arrow shaft.
An arrow shaft is fundamentally a hollow beam and vibrates according to Eulerian theory. For our purposes, the first mode, or the mode with two nodes and one anti-node is most relevant to the study. Two flavors of arrow shaft are analyzed; a Gold Tip XXX carbon fiber shaft and an Easton X7 (or 2712) made from high zoot aluminum. Right away you can imagine that the carbon fiber shaft should have significantly different vibration characteristics, or should it? :) Since the mass per unit length of each arrow is different, and the Young's Modulus is different by an order of magnitude or so, the expectation is that one arrow will be very different in terms of vibration frequency and node positions.
It turns out that even though the first mode frequencies are quite different, the node positions are very similar with respect to the applied point mass. Upon first inspection, it looks like trying to match the tip node with the rest contact point is optimistic. Inspecting the shortened XXX shaft, the node position stays relative to the tip so one could shorten the shaft on a heavier tip and iterate to a match. For example, if I shorten my 29.25" XXX to 27.5", I can get the tip node very near the blade tip of the rest. This alignment is supposed to promote improved grouping due to the arrow shaft deflection being minimized at the rest at the shot break when the arrow is loaded by the string. I will test this out by trimming three arrows to the shorter length and group tuning them at 50 yards. In the meantime, absorb the data for what it is, which is a relative comparison through virtual means using correlated deflection data (spine) and applying it to a normal modes analysis.
One interesting outcome I observed came about when using my iPhone 6 to record a shot using slow motion. At 240 frames per second, it is still hard to observe arrow oscillations though one can see a hint of it if advancing frame by frame in Quicktime. The surprise came when both first and second mode oscillations can be seen in the front stab. I've felt it thrum before but I didn't expect what I saw. It does enlighten one to how the new Fuse Carbon stabs came about.Slo mo video
|Arrow Shaft||Length||Mass (gr)||Nock (gr)||Point (gr)||Total (gr)||Tip Node (mm)||Distance between Nodes (mm)||Freq (Hz)|
|Gold Tip XXX||31"||288||29||125||442||80||553||102|