Stabiliser and bow vibration: Zaniewski 2012

A simulation of a recurve bow fitted with a multi-rod stabiliser during the common motion of string and arrow, that is the first 15.8 milliseconds of the shot. The geometry is that of a commercial system: a 760 mm central rod, two 420 mm side rods on an adjustable mount, a 250 mm short rod, with end weights of 43, 37 and 28 grams.

During the shot the central rod bends upwards and its free end moves by about a millimetre, in the opposite sense to the rotation of the riser, which it therefore partly offsets. The side rods go through roughly two cycles, with the first peak 4.6 milliseconds after release, the trough at 9.4 and the second peak at 12.7, always at the same instants whatever their position. The amplitude, by contrast, depends strongly on position: at maximum separation the combined displacement is 1.3 times that of parallel rods at the same inclination, and 2.4 times at zero inclination.

The dissipation decrement of the side rods comes out between 0.24 and 0.29, where in vibration theory 0.1 already counts as high: within three or four cycles, that is between 28 and 34 milliseconds, the amplitude falls by an order of magnitude. Since dissipated power goes with the square of the amplitude, separating the rods further and flattening their plane raises dissipation by 1.7 to 5.6 times compared with parallel rods. The closing comparison: without a stabiliser the kinetic energy of riser and limbs during the shot is 79% greater than on the same bow with one fitted, across seven full cycles of oscillation during the motion.

Lagrange equations of the second kind for the bow plus rods, with the rods treated as cantilevered Euler-Bernoulli beams carrying a mass at the tip, and the resulting Cauchy problem solved by Runge-Kutta in Mathematica's NDSolve. The plane considered is the bow's vertical plane of symmetry, and the motion is internal only, from string release to the arrow leaving the string.

The numbers for the example bow come from the author's earlier work: limbs 531 mm long and 107 grams each, a riser virtual arm of 342 mm, a 6.9 gram string, and an arrow 783 mm long weighing 22.4 grams. The side rods move on two angles, the inclination relative to the central rod, from 0 to 17 degrees, and the separation between them, from 0 to 35 degrees, and the simulation is run on the four extreme combinations. For each one the author reports the displacement of the free end at the three turning points, split into normal and tangential components.

The calculation method is checked against a known case: with tip mass and bow inertia set to zero, the computed eigenvalues match those of the textbook cantilever, and for a single rod with values typical of a bow the relative error is 0.21%.

There is no archer in the model. The bow is pinned at a fixed point, the hand and arm have neither mass nor give, and the author puts this first among the open problems: the model needs completing with the properties of the shooter's body. Nor is there any measurement, no accelerometers on the rods and no comparison with a real bow, and the only check is internal, between the calculation method and a textbook solution. The numbers hold for one bow and one stabiliser, with one arrow and one draw weight, and we do not know how they would move elsewhere.

The extra energy dissipated is never converted into anything an archer measures: there is no link to the point of impact, to group size or to scores. The claim in the introduction that recoil scatters arrows is carried over from another paper rather than shown here, and the same goes for the idea, cited in the same line, that the sensation of recoil produces a reflex just before release.

The model stops when the arrow leaves the string, leaving out what most archers call bow vibration, meaning what is felt afterwards; the author names this as his second open problem. In the text the displacement of the central rod is given once as about 1 mm and a few lines later as 1.2 mm, and the first peak of the side rods is printed as 4.6 seconds instead of 4.6 milliseconds.

The study speaks to anyone setting up a stabiliser, and it makes one quantitative claim about an adjustment: all else equal, side rods set wider apart and less inclined flex more and therefore dissipate more energy, by as much as 5.6 times compared with parallel rods. That is a calculation, not a field test, and it says nothing about groups or scores: energy lost in the rods is energy that does not reach the bow hand, and the link between that hand and the point of impact stays outside the model.

Two numbers do help frame what an archer feels. First, everything the stabiliser does while the arrow is still on the string takes less than 16 milliseconds, and in that time the tip of the central rod moves about a millimetre, not a visible sweep. Second, the oscillation of the side rods decays by an order of magnitude in around three hundredths of a second, so by the time an archer feels it in the hand the arrow has long gone. In the model the stabiliser works mainly in opposition: the riser rotates one way and the central rod bends the other.