A damper against bow tremor: Mizuno 2017 study

A mechanical engineering group at the Tokyo Institute of Technology tried to reduce bow tremor during aiming by fitting the bow with a small dynamic absorber tuned to each archer's own frequency. The work starts from a geometric calculation: if the sight, one metre from the eye, moves 1.74 millimetres from tremor, at 70 metres the deviation becomes 122 millimetres.

The model represents the arms at aim as a spatial system with six degrees of freedom, two at each shoulder and one at each elbow, with muscles described by Hill's model, which combines muscle stiffness, viscosity and contractile force with tendon stiffness. The bow is a spring between the two wrists. Tremor is excited by the trains of nerve impulses, whose interval is normally distributed: in this model it is that irregularity which produces the tremor.

In the measurements, three seated subjects holding the bow at draw showed two dominant peaks, around 8 and 14 hertz. The amplitude changed from one repetition to the next, but the two peaks remained, and the direction with the largest amplitude was the one the authors call Y. For one subject the model was fitted to peaks at 4.6 and 12.8 hertz, an indication of how much variation there is between people.

The absorber is a beam with a mass at its tip, designed by fixed-point theory: in the two units built, a beam of 23.5 and 26.5 millimetres in length, 16 millimetres wide and 3 millimetres thick, with a 60 gram tip mass, tuned to 9.30 and 8.70 hertz; the Young's modulus of 70.3 gigapascals and the density of 2700 given in the table point to an aluminium alloy.

The results on the two subjects do not point the same way. For the first, the model predicted a 70% reduction in the 8 to 10 hertz band and the test gave 35%; for the second, the model predicted 43% in the 9 to 10 hertz band and the test gave 80%. The authors note that theory and experiment match for neither subject, and conclude on the average of the two cases, which is about 50% both in the calculation and in the measurement.

The model treats the trunk and legs as fixed, the upper arm and forearm as rigid bodies, the right wrist as a lumped mass and the left wrist together with the bow as a single mass; the bow is a spring between the two. The shoulder is a universal joint with two degrees of freedom, the elbow a hinge with one: six degrees in all for the two arms. Each degree of freedom carries a muscle described by Hill's model, with stiffness, viscosity, tendon stiffness and contractile force. That force arises from nerve impulses whose period follows a normal distribution about a mean set by the force the aiming posture demands: it is this randomness that excites the system.

The delicate step is the stiffness of the shoulder muscles, for which no database exists. The authors derive it in two stages. First they estimate the exponent of the force-extension law on three muscles for which published data exist, the biceps brachii, the pectoralis major and the sartorius, and take the average. Then, with that exponent fixed, they leave the cross-sectional areas of the shoulder muscles and tendons as unknowns, four parameters, and identify them with a genetic algorithm by fitting the computed spectrum to the measured one between 3 and 20 hertz, on each subject's two dominant peaks. The resulting areas are then expressed as a linear function of the maximum force the person can produce in the aiming posture, so that stiffness can be derived from upper arm and forearm length, body weight and maximum force, without repeating the vibration measurement.

The measurements were made on seated subjects, to remove the sway of the trunk and legs, which appears at low frequencies. Three accelerometers were mounted on the bow's sight; sampling was at 500 hertz for 5 seconds, repeated several times on each of the three subjects. The amplitude on the three axes was then combined into a single value.

The absorber was built for two further subjects, D and E, and uses the first bending mode of a beam. Calculation and measurement are compared on the sum of amplitudes in the band where the calculation predicted an effect: 8 to 10 hertz for the first subject, 9 to 10 for the second.

Five people in all: three for the tremor measurement and two for the absorber test. On those two, calculation and measurement do not meet. Where the model predicted a 70% reduction the test gave 35%, where it predicted 43% the test gave 80%. The authors say so, and then rest their conclusion on the average of the two cases: averaging two large discrepancies of opposite sign gives a number that describes neither archer. That 50% should therefore be read as the order of magnitude of an effect seen on two people, not as the performance of the device.

The model contains several stated approximations. The trunk and legs are fixed, which is why the subjects drew the bow seated: a condition that is not the shooting line. The bow is a spring, antagonist muscles are absent, spinal reflexes are excluded by choice. The stiffness of the shoulder muscles is not measured but identified by fitting the model to the data, which makes the good agreement of the curve less informative than it looks; the exponent of the force-extension law is the average of three muscles, one of which, the sartorius, is in the leg.

The tremor itself is not stable: the amplitude changed with every repetition on the same subject, and the two dominant peaks vary between people, with one case fitted at 4.6 and 12.8 hertz rather than the roughly 8 and 14 quoted in the text.

No arrow was shot. Tremor is measured at aim, not at release, and there is no score, no group and no measurement of how much of that oscillation reaches the target: the figure of 122 millimetres at 70 metres is a geometric projection, not an observed group. Finally the absorber adds 60 grams of mass at the tip of the bow, and the authors criticise stabilisers precisely because their weight brings fatigue and therefore tremor, yet fatigue was measured neither with nor without the device. The work is a Japanese conference paper, without peer review.

The most valuable thing in this record is not the absorber: it is the tremor spectrum. With three accelerometers on the sight, in three archers seated at full draw, the oscillation concentrates in two peaks, around 8 and 14 hertz, and the amplitude changes with every repetition. These are high frequencies compared with the slow sway of the body, and they are the reason the authors argue that adding mass with stabilisers acts on the low frequencies and does little here.

On the device itself the record authorises no practical conclusion. Two archers, two beams with 60 grams at the tip, tuned to 9.30 and 8.70 hertz, and measured reductions of 35 and 80% in the chosen band: an effect shows, but its size is not predictable, because the calculation missed the prediction in both cases, in opposite directions. The device is not on the market and is not a piece of legal equipment.

The question that matters is left open. Nothing was shot: in this work there is no link at all between tremor measured at aim and where the arrow lands, and the 122 millimetre deviation at 70 metres is the result of a proportion rather than a measured group. What a reader can take away is that the tremor of the bow hand has a frequency structure, that it is individual, and that it is unstable even within the same person from one repetition to the next. The rest is a laboratory prototype.