Primitive Archer

Simk has very good questions! I’ve been thinking about the same things, but I don’t have definitive answers either. Hopefully we can find some together. But here are my thoughts.

I don’t like string angle as a parameter, because it doesn’t have a precise definition. To define an angle, you need two intersecting lines. One of them is obviously the string, but what exactly is the other one? For example, in model 4, what is its string angle? Or model 5? Or with a recurve bow that has a circular arc at the tip – where do you draw the tangent to define the string angle? Sometimes string angle correlates with energy storage – smaller string angle → more energy storage – but I wouldn’t say they are always explicitly connected. Or at least, you shouldn’t focus too much on string angle, because it isn’t the parameter you should be looking at.

Why do shorter bows store less energy? First we should specify what we are comparing. By “shorter bow” I mean a bow that is short relative to the draw length. For a fixed draw length, a shorter bow must be drawn proportionally farther, and that leads to less stored energy.

You should think of a bow as a lever (or two-lever) system. It has a fulcrum and two lever arms. When drawing the bow, the effective lever arm length is decreasing. If we exaggerate a bit, think about a braced bow where the draw force direction is almost orthogonal to the limbs (or to the string, which transfers the force to the limbs), and then compare that to the extreme situation where the limbs are bent so much they are nearly parallel to the draw direction. At brace height, the limbs act like long lever arms; at full draw, they act like very short ones, approaching zero. Thus a bow acts like a variable-ratio lever, because its effective lever arm length changes throughout the draw.

Now remember that lever arm length affects the force needed: a long lever arm gives more mechanical advantage and therefore requires less force. Because of this, a longer-limb bow has more mechanical advantage near full draw than a short-limb bow. With a fixed draw length, when a short bow is drawn to full draw, its lever arms are shorter than those of the longer bow, which means it reaches higher draw force sooner. On the draw-force curve you will see this as the curve rising sharply – this is stacking – and stacking results in less stored energy overall.

But the most important point is that a short-limb bow’s limbs simply cannot bend much more near full draw. The lever system of the bow determines how the limbs bend and thus how they store energy. Therefore, short limbs cannot store additional elastic potential energy at the end of the draw.

The string applies the draw force to the limb tips, bending the limbs. This bending is what stores the energy. The limbs store elastic potential energy just like a stretched rubber band. The more you bend the limbs, the more energy is stored. The work done in drawing the bow is “force × distance”, and that is equal to the bow’s potential energy at full draw.

In physics, when you do work—like lifting a weight—the weight gains potential energy exactly equal to the work done. In the same way, the bow’s potential energy at full draw is exactly the work you have done in drawing it. You can calculate that potential energy by integrating the draw-force curve, i.e., by calculating the area under the curve.

Thus, you do work on the bow by drawing the string, which acts on the limbs, which act like levers and bend the limbs, which stores energy. The lever-arm behaviour determines how the draw force is applied to the bending of the limbs, which ultimately store the energy like springs.

In short, a bow is a complex system of energy-storing springs that also act as variable-ratio mechanical levers.

Here is small comparison of short and long bow. Straight, normal front profile, taper rate 0.004 (evenly distributed stess).

I sometimes bring my shooting machine to the bow fair in Eigeltingen and request all bowmakers to bring their fastest bows. Rules are: 40# bow +/- a few #. I then measure exact drawweight and pick the arrow that brings the bow closest to 8gpp. Then shoot it on the machine. The results were published in the German "Traditionell Bogenschiessen" Mag.

In 2023 we wanted to test just "longbows" according to IFAA which also includes D/R profiles as long as they fully stretch out at brace. The "winning" bows were a classic 66" Longbow made from yew, maple and bamboo and a D/R glassbow which both measured same speeds. Wooden bows are brown, glassbows blue.

you are right Badger - I messed that up - the short bow does store less energy but is more efficient.
But why does it store less energy? String angle?
And why is it more efficient? Limb mass, limbtravel and inertia?
And why does the d/r in Tuomos experiment store more energy than the straight limbed bow? string angle?
Now: In practice we often see that short bows are faster — especially with light arrows — than the longer ones: is that  maybe because limb mass and limb inertia (efficiency) matters more than stored energy?
If that is the case,  when trying to make a fast bow, one should maybe focus more on efficiency than energy storage? How to optimize these contradictory facors?
Given that more stored energy usually means heavier and slower limbs, how do we find the sweet spot where arrow speed is maximized?
It must be where increasing stored energy starts to cost you too much efficiency by increasing  limb mass and inertia.
Now where is that?

Interesting idea thinning the outside of those turns. Makes sense.  I’ll have to keep that in mind when I take on some of the snakey staves I have collected. Sharp looking bow!
Mike

Nice work, Brian. Form and function.