Kyudo bow dynamics: Ohtsuki 2019 study

The study builds a mathematical model of the yumi, the long asymmetric bow of kyudo, and compares it with measurements taken on a real bow. The model splits the bow into nine sections, each with its own curvature and stiffness, and works out how each section bends as the string is drawn, without the simplifications that only hold for small deflections. On the test bow the calculated and measured shapes agree well, both for the displacement of the two tips and for their angle.

The authors then change the initial curvature of some sections and look at what happens to the draw force. The force-draw curve is not a straight line: force rises quickly at first, more slowly in the middle and then climbs again towards the end. The more the central sections curve outwards, the heavier the bow and the more marked this three-stage pattern becomes; the more the sections between grip and tips curve inwards, the lighter the bow. For the same draw, force rises almost in proportion to the initial curvature, but the effect fades as the draw gets longer. During the draw the tip of the short side, below the grip, moves further than the tip of the long side, which rotates while bending along its whole length. A final section simulates the bow's recovery after release with a 35 g arrow.

Each section of the bow is treated as a flexible beam capable of large deflections, using elastica theory: the equations are solved with elliptic integrals and the Newton-Raphson method. The model assumes the material stays elastic, the bow does not stretch and there is no shear deformation; the bow pivots freely about the grip, and the string, shorter than the bow, is pulled at one point with a perpendicular load.

The test used a commercial shihan-yumi, a short Japanese bow rated at 4.2 kg, 164 cm long, with a 157 cm string. The bow was held so that the bamboo node at the grip (nigiri-bushi) acted as the centre of rotation, and the load was applied with weights through a cable and pulley. Tip displacements were read off a millimetre grid stuck to the bench, and tip angles off a protractor marked in half degrees. The lengths, curvatures and stiffness of the nine sections were measured on the bow itself. On this bow the parts above and below the grip were in a ratio of about 6 to 4, whereas on a standard-length yumi the ratio is about 2 to 1.

The shooting simulation reduces everything to three masses: the bow's mass of 270 g split between the two tips (162 g on the long side, 108 g on the short side) and a 35 g aluminium arrow at the nocking point. The equations of motion are solved step by step with the Runge-Kutta method, starting from a draw set in dimensionless units.

This is a static model checked against a single bow, and a short one: the shihan-yumi has different proportions from a standard yumi, so the results may not carry over unchanged. The experimental check covers only the shape of the bow under load, using simple tools such as a millimetre grid and a protractor; the force-draw curve and the effects of initial curvature are calculated, not measured on different bows. All results are given in dimensionless quantities and charts, so no values in kilograms or centimetres can be taken from them for comparison with a real bow.

The dynamic part is even more schematic: a massless bow with two weights at the tips, the arrow as a point, and no friction or vibration of string and arrow. What comes out is a sequence of shapes, with no figures for arrow speed or energy. The directions "outwards" and "inwards" are the paper's own and refer to the shape of the bow. The paper is in Japanese, with the abstract and captions in English.

For recurve or compound archers the study changes nothing in practice: it concerns a different bow, with a shape no Western bow has. For kyudo practitioners it offers a mechanical account of how draw force changes during the draw: it does not rise evenly, but has a stiff opening stretch, a softer middle and a final build-up, and how marked this is depends on the initial curvature of its sections. The short side moving more than the long side fits the yumi's asymmetry. The model is meant for people who design or study bows and does not say which shape shoots better; the references to bow-hand technique, such as uwaoshi and tsunomi, stay in the introduction and notes, with nothing measured.