How the Japanese bow bends: Ohtsuki 2017 study

The study describes how a Japanese bow, the yumi used in kyudo, bends as it is drawn, using elastica theory, the mathematics of large deflections in a thin flexible beam. A real yumi is asymmetric, built from layers of bamboo and wood, and held below its centre; the authors reduce it to a straight, symmetric beam held in the middle, with a string shorter or longer than the beam and a load pulling on the centre of the string.

The equations, solved with elliptic integrals, give the curves linking the load to the movement of the bow tip, the deflection of the point where the string is drawn, the angle at the tip, the bending moment, the curvature and the stress at the surface of the material. The curves shown in the graphs, for the movement and angle of the tip and the deflection at the drawing point, share the same shape: at first a small load produces a large movement, then the movement grows less and less even as the load rises steeply. Measurements on a strip of plastic, with strings of different lengths, follow the theoretical curves closely.

The authors link this behaviour to something experienced kyudo practitioners report: pulling the string at the nock with ever greater force, the distance between the grip and the string does not grow in proportion. The paper gives no numerical values for a real bow: all the results are dimensionless and shown in graphs.

The calculation deals with only half of the beam, thanks to its symmetry, and rests on a few stated assumptions: an elastic material, a beam that does not stretch, cross sections that stay plane (the Euler-Bernoulli assumption) and no shear deformation. The parameter separating the cases is the ratio between the length of the string and that of the beam. Below 1 the beam is already bent and the string is taut, as in a strung bow; at 1 the beam is straight and the string just taut; above 1 the string is slack.

For the check, the authors used a strip of PVC 17 mm wide, 300 mm long and about 0.71 mm thick, with an elastic modulus of 3.40 GPa, and strings from 210 to 450 mm long. They hung weights from the centre of the string and read the movement of the tip and of the string's centre with a light pointer against graph paper fixed behind the strip.

The model removes exactly the features that set the yumi apart from other bows: the asymmetry between upper and lower limbs, the grip below the centre, the layers of different materials, and the curvature and cross section that vary along the bow. The authors acknowledge it is an approximation, and the paper does not estimate how much the results would change for a real bow.

The analysis is static only: it describes the shape of the bow under a steady load and says nothing about the release, which the authors put at 20 to 30 milliseconds, or about the energy passed to the arrow. The experimental check uses a strip of plastic 30 cm long, not a bow, and the agreement with theory is described in words, with no measure of the error. The link with what practitioners feel is the authors' interpretation. The text is in Japanese with an English abstract; the equations cannot be read in the extracted text, so this record relies on their description.

For anyone who shoots, in kyudo or with an Olympic recurve, the study offers no practical guidance: it describes the statics of a simplified model and is mainly of use to people who design bows or want to calculate how they behave. The principle it yields is that in the model the bow stiffens as it is drawn: for each extra unit of load the drawing point moves less and less, so each extra centimetre needs more force than the one before.

According to the authors, this gives a mechanical explanation for something people who shoot the yumi already feel, but the model does not say at what draw length the stiffening becomes significant for a real bow, or whether it changes the shot. On arrows, spine and launch speed the paper says nothing.