A Miracle of Bowyery

by Dick Baugh
December 22, 2014

 

How a bow really works
Thousands of years ago some inventive caveperson tapered both ends of a long stick, tied the ends together with a piece of string and used it to shoot an arrow. The rest is history. A stickbow is a remarkable way of transferring the potential energy stored in a pair of bow limbs weighing around 13 ounces (.369 kilograms) to an arrow weighing only around 1 ounce (500 grains = .0324 kilo). And we all take it for granted. First the archer uses his/her muscle power to slowly pull the arrow back to full draw, storing potential energy in the bow limbs. Upon release the potential energy stored in the bow limbs starts to be transformed into kinetic energy (1/2 * M * v^2) of the arrow plus bow limbs. In the latter part of the shot more potential energy is being transferred to kinetic energy of the arrow but amazingly the kinetic energy of the limbs is also being transferred to the arrow. The bow owes a lot of its incredible efficiency to one simple feature: the bowstring enables the translation of a very small motion of the bow limbs into a larger motion of the arrow. Therein lies the clever design.

Different bowyers tout their particular bow design philosophies and claim great results but was it because of good design, a superior piece of wood or because they avoided checking arrow speed with a chronograph? Tacit in the rest of this article is the assumption that highest arrow speed for the same strain level and arrow mass/draw weight ratio is the appropriate figure of merit. There is also the battle of arrow speed versus esthetics. Do you want a bow that is butt-ugly but fast, a bow that is beautiful or one that looks historically authentic? Handshock, the jolt that your bow hand receives when the arrow is released is not considered. The only fair arrow speed experimental comparison of one bow design over another is to build them both from identical wood specimens, build them to the same strain level and compare their arrow speeds. In other words the only fair comparison is between two bows that are equally close to breaking. Fair, honest comparisons of anything are hard to do. That's why it costs over a billion dollars to test a new drug. Dr. Placebo has a pretty high cure rate.
What can a computer tell us about "primitive" bow design? We can make more honest comparisons between different designs made from exactly the same material. We can give quantitative answers to quantitative questions instead of the qualitative seat-of-the-pants answers we hear today.

Edisonian versus Newtonian science
Scientists often like to contrast the Edisonian method with the Newtonian method. Edison was an inventive genius with very limited education in mathematics or physics but he had an intuitive feel for what was needed to make a technical contribution to society. The story says that he tried over a thousand different materials before picking the right one for light bulb filaments. Newton, on the other hand, examined some obscure data on the orbits of the planets and used that data plus advanced mathematics to derive the laws of gravitational attraction and the laws of motion of masses acted on by forces. Incidentally, he had to invent differential and integral calculus to get the job done. Most stickbowyers are the Edisonian type. Try this, try that, keep the shooters and put the splinters in the woodpile. The earliest attempt at putting some real science into bowyery was done in the 1930s principally by Klopsteg, Hickmen and Nagler and published in Archery-the Technical Side. The Edisonain approach to bowyery is characterized by a set of rules espoused by a very well respected bowyer who will remain anonymous. His mantra rules are in boldface, (my comments in parens):

The mantra:
Make inner limbs wide or long enough for virtually no set. Less-strained wood has lower hysteresis. Inner-limb wood stores most of the energy, So make inner-limb wood be less strained. This done by making inner-limb wood wider that typical, or longer and less bending.
(That is exactly equivalent to having a longer rigid handle section. How can inner wood store most of the energy if it doesn’t bend much. We need quantitative data to either prove or disprove this concept.)

Make mid limbs wide enough for little let. Set in the inner limb projects our to larger string follow than in other parts of the limb. So keep set especially low in this area. By, as above, wider, or longer/less-bending inner limbs.
(Again, show me the data.)

Make outer limbs and tips narrow enough for lowest possible mass. The certain, but not fully understood, effect of tip and outer-limb mass on efficiency. The lighter the tips and outer limb the more efficient the bow, 'Eiffel-Tower' slimming possibly the best way to accomplish this.
(Be more specific. Get quantitative data.)

Let outer limbs bend only slightly. Arc of circle outer limbs shorten the bow at full draw, increasing stack, decreasing arrow speed per draw weight. Keep outer limbs fairly stiff. Limb vibration indicates energy not available to the arrow. Stiffer outer limbs reduce limb vibration, and solve #4 at the same time.
(Same.)

I say "Rain makes apple sauce." Let's examine these claims in detail. First of all there is absolutely nothing quantitative about them. How much stiffer or less strained should I make the inner part of the limbs? How much does that add to arrow speed? He says "Keep outer limbs fairly stiff." Somehow there must be a compromise between stiff outer limbs and low mass outer limbs. You can't have both

Furthermore, he claims "Mantra" bows way outshoot Hickman and Klopsteg bows, because H and K weren't aware of a few factors and their implications when creating their model.
(Maybe Mantra bows way outshoot Hickman and Klopsteg bows because the bowyers mis-applied or didn't understand H & K's design rules.)

Another respected bowyer who holds many "primitive" flight records says "8 in stiff handle and fade area with an additional 4" either side of the fade barely flexing. Outer 10" of the limb and tips mostly stiff." Other published experts say essentially the same thing about bow design. The inner and outer thirds of each limb should be stiff. The middle third should do most of the bending. What does the computer say?

Linear theory versus nonlinear theory:
Hickman and Klopsteg claim in Archery-the Technical Side that all parts of the limb should be equally strained and close to breaking. This is a "linear theory" that says that strain in one part of the limb is as bad as strain in any other part of the limb. The "mantra" design treats different parts of the limb differently. I refer to that as a "nonlinear theory". Which is best?
The "mantra" bowyers are all in the same embarrassing situation as today's theoretical physicists. The boffins have quantum mechanics to explain the workings of tiny things like atoms and electrons, Newtonian physics to explain the workings of gravity and heavy objects moving at slow speeds and general relativity for great distances and speeds near the speed of light. The bowyers, on the other hand have one rule for the inner third of a bow limb, another rule for the middle third and something else for the outer third. The physicists admit their shortcomings and are striving to develop a unified theory that covers the whole shebang whereas you guys just wave your hands and talk a little louder. I'm not criticizing the results, just the reasons behind it all. The world is full of people who do the right things for the wrong reasons. Computer modeling can answer some of these questions.

Actual examples
In order to compare bows with different draw weights the standard criterion is arrow speed for an arrow weight of 10 grains per pound of draw weight at a draw length of 28 inches. There is a lot of data on 50 # @ 28 in bows shooting 500 grain arrows. As a point of reference imagine a bow with a massless, stretchless string and a brace height of 6 in pulling 50 # @ 28 in and a perfectly linear force-draw curve. If it were 100 % efficient it would send a 500 grain arrow at 203.2 ft/sec. In our wildest dreams! Let's look at some real life examples and compare them with some computer simulations.
Experimental data has been taken on:
a. DECABOW carbon fiber-wood longbow, 71 in (NTN), 49.4 lb @ 28 in. 187.5ft/sec @ 8.89 grains/# (decabow.com website)
b. Steve Gardner flatbow, 64 in nock-to-nock (NTN), 50 lb @ 28 in, osage backed with hickory, some reflex. 176 ft/sec @ 10 grains/# This bow was selected because we have its dimensions in great detail. A very well performing specimen. (personal communication)
c. Aidrian Hayes Longbow (72 in NTN?), ipe-lemonwood-bamboo, 50 lb @ 28 in, 158 ft/sec @ 10 grains/#
d. Chris Boyton Traditional English Longbow, hickory & lemonwood, 72 in NTN, 45 lb @ 28 in, 153 ft/sec @ 10 grains/#
e. Neil Harrington Longbow 72 in NTN, material?, 47 lb @ 28 in, 157 ft/sec @ 10 grains/#
(c, d, & e from the archers-review.com website)

From this we see that the carbon fiber longbow gives pretty good speed, a well designed osage-hickory bow is not bad but the three English longbows, built by professionals are drop dead beautiful but not great performers. Is it because of excessive hysteresis or excessive mass in the limbs? Maybe both. Hickman & Klopsteg (H & K) extol the virtues of rectangular limb cross section as giving less strain and limb mass for the same draw weight relative to the D-shaped cross section beloved by the longbowyers. The battle between esthetics and performance continues.

Questions to be answered:
a. For a given nock to nock length and maximum strain level which is faster, a bow with a long rigid handle riser or short riser?
b. Do stiff outer limbs or circular arc bending give greater arrow speed?

In order to compare the design rules of H & K with those of the Mantra bowyers the computer will be used to compare the arrow speed of different designs, all made with the following constraints: use osage orange with elastic modulus = 1.689E6 psi & specific gravity = 0.855 obtained from www.wood-database.com, maximum compressive strain = 1.063 %, tip width = .25 in, rectangular limb cross section, constant thickness unless otherwise noted, trapezoidal width profile from riser to tips, draw weight = 50 lb @ 28 in, brace height = 6 in, stretchless bowstring weighing 105 grains and nock to nock length = 66 in, symmetrical in that upper and lower limbs assumed identical. Note that hysteresis (limb internal friction) is being ignored. With the same compressive strain in all examples the effects of hysteresis should be the same in the examples.
The effects of rigid riser length:

Note that width at the riser was adjusted to give the same strain level with the different riser lengths. The conclusion is that for the same compressive strain arrow speed doesn't depend much on the rigid riser length.

What if we keep the width at the riser fixed at 1.5 inches and vary the riser length, adjusting the thickness (constant throughout the length of the limbs) to get 50 # @ 28 in? Then the maximum strain will be greater with longer rigid risers. The results:

Now we see that for a constant width at the riser there is a substantial increase in arrow speed AND strain as we increase the length of the rigid handle riser. I claim that's why stiff inner limbs give greater arrow speed. Essentially you are making the outer part of the limbs store more energy per unit mass and work harder.

How about "Let outer limbs bend only slightly. Arc of circle outer limbs shorten the bow at full draw, increasing stack, decreasing arrow speed per draw weight. Keep outer limbs fairly stiff. Limb vibration indicates energy not available to the arrow. Stiffer outer limbs reduce limb vibration, and solve #4 at the same time." Let's compare three different 66 inch bows, all with more or less the same maximum compressive strain, 12 in rigid riser and 0.25 inch tip width shooting 50o grain arrows:

a. constant thickness limbs so that ends are somewhat stiff
b. thickness tapered outer extremities so that limbs bend in circular arcs
c. outer extremities 50 % thicker than near riser, a la Mollegabet or Holmegaard

What happens with wider (0.5 in) tips?
d. 0.5 in tip width and constant thickness for very little curvature in outer part of limbs
e. 0.5 in tip width, thickness tapered for circular arc curvature

Conclusion: Thicker extremities (example c) reduce arrow speed and thickness tapering of extremities to get a more circular arc increases arrow speed for the same maximum compression strain. This disagrees with the claims of the mantra bowyers that stiff outer limbs increase arrow speed.
What about longbow performance?
Longbows look cool but why do they perform so poorly? Let's examine some typical longbow designs made from the same material as the bows described above, all 6 feet long, 0.5 in. tip diameter, 6 inch brace height, bending through the handle, pulling 50 lb @ 28 in. and shooting 500 grain arrows.
f. circular cross section and bending in a more or less circular arc, similar to that of the Mary Rose bows.
g. Osage longbow dimensions taken from Essentials of Archery, semi-ellipse cross section.

Overall conclusions:
The mantra bowyers are half right and half wrong. Having a stiffer inner part of the limbs increases arrow speed but it does so at the cost of greater compressive strain in the middle part of the limb. Contrary to the claims of the mantra bowyers having stiffer limb outer extremities by making them thicker and heavier reduces arrow speed. A better set of design rules: use narrow width tips and adjust the thickness for circular arc bending. Yes, for a given full-draw weight circular arc tillering stores less energy (doesn't have as 'fat' a force-draw curve) than tillering for stiffer outer portion of the limbs but the resultant lower mass in the outer extremities results in greater arrow speed. Not all bowyers will agree with these claims but that's just fine with me. I also would greatly appreciate getting more detailed dimensions of your favorite straight limbed stick bow plus resultant arrow speed at as close to 10 grains per pound of draw weight @ 28 in. Compressive strain is very difficult to measure but is a key parameter in determining bow performance. It's probably easier to model on a computer than it is to measure.

E-mail your comments to "Richard A. Baugh" at richardbaugh@att.net