Arrow model for recurve archery: Williams 2025

A mechanical engineering research project at Bristol, written by an author who has been shooting for eight years, rebuilds Park's model of lateral arrow flex during and immediately after the power stroke in Python, and uses it to produce arrow selection tables. There is a single criterion: the right shaft is the one that, once clear of the bow, has stopped rotating about its own centre of mass in the horizontal plane, which is to say it does not fishtail.

Solving the model for one combination after another, the author prints two new tables, one for Easton X7 and one for X10, in steps of 4 lbf of draw weight and half an inch of arrow length, from 20 to 52 lbf and from 25 to 32 inches. For the X10 the recommended shafts run from spine 1000 down to spine 325. Set against Easton's own charts, the new ones point to weaker shafts, by as much as four sizes, and never more than one size stiffer; the largest departures fall on long arrows for the X7 and on low draw weights for the X10. In most cells the residual rotation rate in free flight stays below 20 degrees per second, which the author treats as correctable by small adjustments of draw weight and pressure button. The half inch step earns its place: in many rows half an inch more arrow changes the recommended shaft.

The finding the author puts first concerns non-parallel shafts: a shaft that is stiffer in the static bending test can behave, when shot, like a weaker one.

The model is the one Park published in 2013 for the lateral plane: the arrow is an Euler-Bernoulli beam whose mass and stiffness vary along the shaft, loaded end on by the string and resting against the pressure button. The equations of motion are discretised by finite differences and solved in Python 3.11, with the shaft divided into 32 segments, a time step of 14 microseconds and a window of 35 milliseconds; above a 20 microsecond step the solution goes unstable. No solution algorithm existed in the literature, and the author publishes his own in an appendix.

The bow's dynamics are not modelled. In their place sits a high order polynomial fitted to the dynamic draw force curve derived from Kooi's model, rescaled for draw weight and draw length, with arrow mass left out. Lateral movement and rotation of the bow are set to zero. The nock's movement through the release is imposed as a diagonal path described by two parameters, the forward travel and the maximum sideways displacement, both fixed at the mean values Park and Logan measured on sixteen national team archers. For the parallel X7, Young's modulus and geometry come from the static spine; for the non-parallel X10, the mass and stiffness profiles come from a model by Park himself. Draw length is assumed to be the arrow length less an inch and a half, and the remaining components are those recommended in the Easton catalogue.

The model has not been checked against a single real arrow. Validation consists of a comparison with Park's model on identical inputs, and the author calls it qualitative: the two solutions resemble each other and the free flight rotation rate, the quantity on which shaft choice turns, comes out similar, but differences remain, which he puts down to the bow dynamics being handled crudely and to the boundary condition at the nock, whose parameters he had to estimate. No archer has shot the new tables, and he says so.

Shaft choice rests on one quantity, the residual rotation in the horizontal plane; the vertical plane is set aside because the author regards it as fixable with a bare shaft test. Everything that varies between archers, the release above all, enters as the mean of sixteen people: the same table row serves anyone at that draw weight and arrow length. Pressure button settings and point weights are held at catalogue values, and the assumption that departures can be absorbed by small adjustments remains an assumption.

The work is a third year individual project on an engineering degree, published in the University of Bristol's student research journal, and the peer review field in this record stays empty. In the X10 table one cell carries an X7 size by mistake (2312, at 44 lbf and 32 inches).

For anyone choosing shafts, this is a calculated second opinion rather than a measurement. It says that the manufacturer's charts, at the extremes, tend to recommend shafts that are too stiff, and that the gap widens for long arrows in the X7 range and for low draw weights in the X10 range, by as much as four sizes. It also says that half an inch of arrow length can move the choice, which a chart in whole inch steps cannot show. What remains is that neither column of numbers has been verified by shooting, and that the author explicitly asks for his own to be validated.

The point that outlasts the tables concerns barrelled shafts such as the X10. The static test measures how far a shaft bends when it rests on two points and is loaded in the middle, but it is a poor guide to how that shaft behaves under the push of the string, because mass and stiffness vary along the arrow. Two shafts with the same number printed on them can react differently, and a shaft that is stiffer in the static test can turn out weaker when shot. This is why dynamic spine, read off the flight of the arrow and off bare shaft tuning, tells you more than the number on the tube.