Bow and arrow vibrations: Zaniewski 2009 model

A mathematical model of the vibrations of the bow and arrow system in the vertical plane, built with nocking point height in view. The author treats bow and arrow as a kinematic chain: limbs as rigid beams hinged to the riser, with a spring standing in for their bending, an elastic string, and the arrow as an Euler-Bernoulli beam free at both ends with an oscillator attached at the nock. He writes the energies, derives the Lagrange equations and finds the natural frequencies by setting the determinant of the system to zero, in MathCAD.

Two findings. First, the arrow's natural frequencies fall as longitudinal acceleration rises, and at certain values one vibration mode becomes unstable, meaning the angle between arrow and flight axis grows without coming back. In dimensionless terms the exact solution puts these thresholds at 14.79 and 56.47; the three term approximation gives 14.56 (1.6% error) and 65.68 (16.3%), the two term version finds only the first, 15.42 (4.3%), and an arrow treated as a rigid rod finds neither. Second, with the three term approximation the error on the frequencies stays around 2% at both low and high accelerations, close enough for engineering work.

For a bow and arrow of average size the first two circular frequencies come out at 132.3 and 1,730 radians per second; ignoring string elasticity shifts the first by 0.05%, so the string can be taken as inextensible. The one comparison with reality is a period: for a 712 mm arrow the model gives the two node mode a period of 18.4 milliseconds, and the duration of the common motion of bow and arrow on a high speed film lies between 18 and 19 milliseconds.

The asymmetry of the bow in the vertical plane, with the arrow sitting above the hand, is put at 3 or 4%, and on that basis the author sets it aside and works with a symmetrical scheme, having first shown that the small oscillations of the two limbs are equal and opposite. There are three generalised coordinates: the rotation of the riser, the oscillation of the limbs and the displacement of the nock.

The arrow is treated first as a rigid rod, with a virtual mass concentrated at the nock, and then as an elastic beam. At that second stage the vibration mode is not derived exactly but assembled by hand, as the sum of a linear function and two sinusoids, one whole and one half; the choice is judged by comparing the frequencies it yields with those of the exact solution the same author published in 1999, and he reports the error case by case for three different shapes.

The numbers for the example bow come from manufacturers' catalogues: limbs of 95.3 grams, a virtual arm of 434.3 mm, limb stiffness of 120.78 Nm, a string branch stiffness of 14,000 N/m, and an arrow of 9.1 grams, 700 mm long. For the closing comparison the arrow on the film is 712 mm long, with a 14.3 gram shaft and a distributed bending stiffness of 4.76 Nm², giving a dimensionless acceleration of 5.28.

There is no measurement anywhere. The study is calculation, and its only purchase on data is the comparison between one computed period, 18.4 milliseconds, and the duration of the common motion of bow and arrow read off a high speed film. That film, in the reference list, is a 1992 demonstration recording by an accessory manufacturer, not a study, and the comparison concerns a single number. The arrow's acceleration, given as around 600 to 700 times gravity, is likewise cited from the author's earlier work rather than measured here.

The scheme is symmetrical by choice, the limbs are rigid beams with a spring standing in for their bending, and the archer is absent: the bow hand and the fingers never enter the model, and the bow is pinned at a fixed point. The model stops at the push of the string and says nothing about what happens once the arrow has gone.

Above all, the stated aim is vertical tuning, that is a zero angle of attack for the arrow, yet the study never reaches a procedure. It delivers a calculation method and the observation that approximate vibration modes are good enough, not a nocking point height for a given bow. The paper also has a long editorial history: received in 2005, revised in 2007, published in 2009, and later reissued by the journal under a second DOI, which is the one recorded here.

For an archer the study changes nothing directly: it does not say where to put the nocking point or how to tune a bow. What it leaves is one useful idea and one caution.

The idea is that during the milliseconds in which the string is pushing it, the arrow is not a rigid rod but a beam loaded end on, and a beam loaded end on, past a certain load, bends and does not come back. The author calculates that threshold, and shows that the arrow's natural frequencies are not the ones measured on a stationary arrow: under the acceleration of the power stroke they drop. Testing shafts by resting them and making them ring measures something other than what happens when they are shot, which is the same reason static spine and dynamic spine do not coincide.

The caution concerns the use of models. Here a calculation arrives at a vibration period that matches the duration of the common motion of bow and arrow seen on a film, and that single match is the whole of the work's contact with data. A model that gets one number right is not thereby trustworthy on all the others.