Limb geometry decides what share of the bow's stored energy passes to the arrow

The summary today

The mathematical models of a single author say so, from 1991 to 2007, with a result that runs against intuition: storing more energy is not enough, what matters is how much of it is passed to the arrow. At equal draw weight and draw length, the modern recurve with limbs that work right out to the tips combines stored energy of 0.434 with an efficiency of 0.770, and launches the arrow faster than any other bow compared, with a velocity coefficient of 2.09. Bows with rigid ears store more than straight bows, from 0.432 to 0.491 against 0.395 to 0.407, but pass on less, with efficiencies from 0.619 to 0.668, because when the string leaves the bridges the thrust on the arrow drops suddenly and sets off an oscillation that uses up energy; in the end their arrow goes about as fast as that of a straight bow. Pushing the curvature to the extreme almost doubles the stored energy, up to 0.810, and almost halves the efficiency, which falls to 0.417. As early as 1991 the same author drew a consequence from this that bears on the claim: because the bows that store more pass on a smaller share, at equal draw weight and draw length the arrow's speed changes little from one bow to another, and shape parameters matter less than is claimed. Geometry decides the share of the stored energy that passes to the arrow, while the energy the arrow actually receives changes much less. These are, however, figures from a model: it was checked against the draw weight of a longbow replica, to within 1 per cent, and against the arrow speed of a single modern bow, where the measured efficiency came out a few points below the calculated one. It compares historical types, not the limbs on sale today, and no measurement on equipment has yet compared different geometries with everything else held equal.

Also phrased as: "limb efficiency"; "recurve limbs and arrow speed"; "rigid ears and energy"; "bow efficiency and limb shape"

Evidence

• 1983
  Context Model Published later

  A doctoral thesis on the mechanics of bow and arrow: according to its summary, in theoretical experiments with the model the bow with rigid ears does not turn out to be better in itself than a long straight bow. It is the basis of the later models, and the summary gives no figures.

  On the mechanics of the bow and arrow · Bote Willem Kooi

• 1991
  For Model Published later

  A dynamic model of the modern recurve solved by finite differences, comparing six types of bow: with the same arrow and limb mass, efficiency runs from 0.765 for the straight bow to 0.716 for the angular one, 0.668 and 0.619 for the two composites with rigid ears, down to 0.417 for the recurve pushed to the extreme; the modern recurve, calculated with slightly different masses, reaches 0.729. Limb shape changes a great deal the share of energy that passes to the arrow.

  On the mechanics of the modern working-recurve bow · B. W. Kooi

• 1991
  Context Model Published later

  In the same work the arrow's speed comes out almost the same for every bow, with a coefficient from 1.92 to 2.08 at equal arrow mass, because the bows that store more energy pass it on less well. The author concludes that, within certain limits, the shape parameters matter less than is often claimed, and that the bow's progress comes mainly from materials that store more energy per unit of mass.

  On the mechanics of the modern working-recurve bow · B. W. Kooi

• 1997
  For Model Published later

  A model of six types of bow at equal draw weight and draw length: the modern recurve with limbs that work right out to the tips has stored energy of 0.434, efficiency of 0.770 and the highest velocity coefficient, 2.09. Pushed to the extreme, the recurve reaches 0.810 of stored energy but falls to 0.417 efficiency, against 0.407 and 0.765 for the straight bow.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 1997
  Context Model Published later

  Static recurve bows, with rigid ears, store from 0.432 to 0.491 against 0.395 to 0.407 for non-recurve bows, but transfer less well, with efficiencies from 0.619 to 0.668 against 0.716 to 0.765: the velocity coefficient stays similar, 1.94 to 1.99 against 1.92 to 2.01.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 1997
  Context Model Published later

  The reason lies in the force-draw curve: in static recurve bows, Persian and Turkish in the model, there is a kink at the point where the string leaves the bridges, absent in non-recurve bows, with a sudden drop in the thrust on the arrow and an oscillation that wastes energy. Its amplitude depends on the elastic modulus of the string and on the ratio between the mass of the arrow and that of the ears.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 1997
  Context Model Published later

  The angular bow of Egypt and Assyria does not beat the straight bow on any of the three indices: 0.395 energy, 0.716 efficiency and 1.92 velocity, against 0.407, 0.765 and 2.01.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 2007
  Context Model Published later

  A review chapter by the same author: a bow like the one described and shot in the 1940s, with a very high static coefficient, 0.81, has an efficiency of only 0.417 and a disappointing overall performance. It also cites a Dutch bow with rigid limbs hinged to the grip and a spring, which with a 16 gram arrow had a measured efficiency of 81.8 per cent, more than the 72.9 of the modern recurve with a heavier arrow; it is a figure taken from a 1978 article, not a new measurement.

  Design and materials in archery · B.W. Kooi

Status history

1. 1983 Not studied A doctoral thesis builds the mathematical model of bow and arrow from which the comparisons will start, and in its theoretical experiments it finds that the bow with rigid ears is not in itself better than a long straight bow. The summary gives no figures and does not separate how much energy the bow stores from how much it passes to the arrow: there is no evidence for the claim yet.
2. 1991 Transferred The dynamic model of the modern recurve puts figures on the comparison: with the same arrow, efficiency runs from 0.765 for the straight bow to 0.619 and 0.668 for the composites with rigid ears, down to 0.417 for the recurve pushed to the extreme, so limb shape decides how much of the energy passes to the arrow. The arrow's speed, though, comes out almost the same for all of them, because the bows that store more pass on less, and the author concludes that shape parameters matter less than is claimed. It is a model: transferred.
3. 1997 Transferred A second paper by the same author takes up the comparison of six types of bow at equal draw weight and draw length, and separates the two things: the modern recurve combines stored energy and efficiency, 0.434 and 0.770, and launches the arrow fastest; bows with rigid ears store more than straight bows but pass on less, because when the string leaves the bridges the thrust drops suddenly and an oscillation starts that uses up energy. Pushing the recurve to the extreme almost doubles the stored energy and almost halves the efficiency. It is a model, checked against the draw weight of a longbow replica and the arrow speed of a single modern bow, not against different geometries compared with one another: models alone leave the claim transferred.
4. 2007 Transferred A review chapter by the same author repeats the lesson with an example: a bow with a very high static coefficient, 0.81, but an efficiency of 0.417 gives a disappointing performance. It also cites a bow with rigid limbs hinged to the grip and a spring, with a measured efficiency of 81.8 per cent, but that figure is taken from a 1978 article and is not a new measurement. What remains is the models of a single author, with no comparison of geometries measured on equipment: transferred.

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Research lines

• 22 studiesHow much energy passes from bow to arrow: models and measurements since 1937
• 22 studiesTraditional and historical bows, from models to archers

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