Bow and arrow dynamics, elastic string: Marlow 1981

Marlow takes Hickman's mathematical model of a longbow, which treated the string as inextensible, and reworks it with an elastic string. With a 25-gram arrow, a 7-gram string and a bow of 39 pounds at 28 inches, the rigid-string model gives an efficiency of 91.5% and an arrow speed of 56.0 m/s; with the elastic string, efficiency drops to 78.1% and speed to 52.8 m/s. The difference arises because, when the string stretches, the arrow separates while string and limbs are still moving: at that moment 11.0% of the energy is in the moving limbs and 9.8% in the string, and a further 1.9% remains stored as elastic energy. The arrow leaves the string after 16.1 milliseconds, 1.94 cm beyond the string's rest position, with the string stretched by 5.3 cm in total.

The elastic model also gives a string tension at separation of 369 newtons, roughly half the rigid-string figure, with the peak tension coming after the arrow has gone: the author sees this as the reason a breaking string usually still sends the arrow to the target. Air resistance on limbs, string and arrow is put at 1.4% of the stored energy, and hysteresis losses in the materials at under 3%. That is far too little to account, as had been the usual explanation, for the gap between theoretical efficiencies above 90% and measured ones, which the author places between 60 and 85%.

This is a theoretical paper. The equations of motion are derived by Lagrange's method for a system of distributed masses (arrow, string and limbs), reduced to an equivalent point-mass model and integrated numerically. The bow is Hickman's: rigid limbs pivoting on an elastic hinge, with energy proportional to the square of the deflection. All motion is in the plane of the bow and symmetrical about the arrow, so the archer's paradox is left out, and the string is assumed to stay straight.

The ten parameters are typical values chosen by the author: a 20-13 aluminium arrow, 70 cm long and weighing 25 grams; a 10-strand Dacron string of 7 grams, with an elastic constant of 7,000 N/m taken from another author's measurements on a number of strings; limbs of 93 grams and 48 cm, worked out from the density of a piece of laminated wood and fibreglass limb; and a bow spring constant of 620 N/m, fitted to force-draw curves of typical bows. The comparison with reality relies on historical efficiency figures and on the author's own measurements from around 1960, about 80%, which were never published, plus two rules of thumb on string strength reported by others.

The weak point is one the author states himself: there are almost no data to test the model against. The measured efficiencies he cites are largely unpublished or reported without a method, and nothing was measured on the bow being modelled. The model treats the string as a spring that stretches all at once, yet a transverse wave takes about 3.9 milliseconds to travel half the string, a sizeable share of the 16 milliseconds of the shot, and the author accepts that this simplification should be revisited. He also calls his air resistance estimate crude.

The bow is a simplified longbow with a Dacron string from more than forty years ago. Modern strings stretch far less, which is why later work on compound bows goes back to treating them as inextensible: in a present-day bow the share of energy lost to string stretch is probably smaller than estimated here. The archive copy is a scan with no text layer, so the text used for this record was obtained by optical character recognition, which handled the prose and tables well but not every equation.

For archers the interest lies in the overall picture: not all the energy put into drawing the bow reaches the arrow, and some stays in the limbs and string, which keep moving after the arrow has gone. That is the energy that sets the bow and string vibrating after the shot and drains away into the air, the materials and the archer's body; the author reckons the vibration dies out in about a second. The study explains why, on the same bow, a string that stretches less can send the arrow off faster, but it gives no figures that apply to a modern recurve or compound, whose materials and geometry are very different.

Two details are worth knowing. One is that in the model the arrow leaves the string slightly beyond brace height rather than exactly there, because the string is stretched at that moment. The other is that string tension peaks after separation, so a string near its limit tends to fail once the arrow has already left. The appendix, written for physics teachers, suggests simple experiments, such as measuring the force-draw curve with a spring balance and a ruler, or arrow speed with a ballistic pendulum.