Wooden Bow Lethargy

by Dick Baugh (Jan. 29, 2016)

More on Bow Design
Which would you rather shoot, a high tech bow with training wheels, cams, space age materials, sights and blinding arrow speed or a bow that looks like it had been shot by one of Robin Hood's Merry Men? Or something in between? Archery is many things to many people. The battle between esthetics and performance will never end. The objective of this article is to compare the published arrow speed of a contemporary English longbow, a hickory flatbow designed for highest arrow speed and a longbow with solid carbon fiber limbs, all with similar draw weight. The hickory flatbow, optimized for arrow speed, is 80 inches (203 cm) long and the other two are about 72 inches (183 cm ) long. Finally I'd like to discuss the reasons for the difference in performance.

Comparing Measured and Computed Performance of Some Bows
There is plenty of published arrow speed data available for different bows. Let's look at arrow speed for some different bows. We have also done computer modeling of these bows. The computer model includes the dimensions and draw weight of the bow and the typical mechanical properties (density and elastic modulus) of the wood being used but not the internal damping of the wood. Internal damping is very difficult to model because it depends in some strange way on the strain level in the limbs and how quickly the archer draws and releases the arrow. Essentially internal damping is what is left after all the other sources of loss have been factored in. I'm confident in the computer model because its results agree pretty well with a much more detailed model used in Bob Kooi's PhD thesis.

What arrow speed should one expect for a 50 pound bow shooting a 500 grain arrow? In an ideal world with a massless stretchless bowstring, a brace height of 7 inches, a draw length of 28 inches, a linear force-draw characteristic and all of the potential energy of the limbs going into kinetic energy of the arrow,

V_arrow = sqrt[(28-7)*50*32.17*7000/(12*500) = 198.5 ft/sec

In your wildest dreams! Let's compare that number with the real world. The computed speed is based on a computer model using the published density and elastic modulus of the wood but not internal damping.

Bow A: A Chris Boyton High Performance Longbow. At over 70", 13/16ths of an inch wide at the arrow pass, draws 45# @ 28 in, 6 in brace height and has 3 laminations of Ipe (or Ironwood ) and is backed with Tonkin cane. Data is from http://www.archers-review.com/bow-reviews/. Model A is a solid Ipe longbow, semi ellipse limb cross-section, 150 grain bowstring with the same length, width at riser, draw weight and brace height as Bow A. Density = 64 lb/cuft, elastic modulus = 3.14E6 psi and a 10 inch rigid handle riser but internal friction is absent.

Bow B: A pecan flatbow, 79.25 in nock-to-nock pulling 48 lb @ 28 in, brace height = 5 in from the back, 1 11/16 wide at riser, 0.25 in wide at the tips, 18 inch rigid handle riser, 500 grain arrow speed = 174 ft/sec. Data from The Traditional Bowyer's Bible, page 124. Model B has same dimensions as Bow B with the limb thickness adjusted for 48 # @ 28 in, assumed mechanical properties of pecan: density = 41.2 lb/cuft, elastic modulus = 1.73E6 psi.

Bow C: A Decabow F-01 Slimline solid carbon fiber longbow 71 in long pulling 49.39 lb @ 28 in, brace = 6.24 in. 187.46 ft/sec with a 439 grain arrow. Data from http://www.bowdyno.com/en/measured/bowdyno/43. Model C is a solid carbon fiber longbow with the same dimensions. Density = 96.77 lb/cuft, elastic modulus = 17.7E6 psi.

Bow A was designed to look like a classic wooden longbow whereas bow B was designed for maximum arrow speed. Maximizing arrow speed means lowering compressive strain by using a rectangular limb cross section instead of semi-elliptical and making the limbs longer to increase the radius of curvature.

Bow C achieves excellent performance by using modern materials that have relatively little internal friction manifested by the relatively small difference between computed arrow speed and measured.

Sources of Loss
Why don't wooden bows have greater arrow speed? There are several factors that prevent a bow from transferring 100% of the potential energy stored in the limbs by pulling the arrow back to full draw into kinetic energy of the arrow. I would like to list and discuss several causes of loss plus actual arrow speed data taken on real bows, both wooden and carbon fiber.

Welcome to the real world. Several factors prevent 100% of the energy stored in the bow limbs from being transferred to the arrow. Some are easily quantifiable but internal friction (hysteresis or whatever) in wood bow limbs is very difficult to model because it is highly dependent on the strain level. About all we can do is compute the readily measurable losses and then say that what is left is internal friction.

a. Mass of the Limbs After the archer releases the arrow, it AND bow limbs start moving and acquiring kinetic energy. Midway through the shot, when the arrow is not yet released from the bowstring, both the bow limbs and the arrow acquire kinetic energy.. One of the many neat things about a bow is that during the final part of the shot, before the arrow is released from the bowstring, some of the kinetic energy in the bow limbs is also transferred to the arrow. This is not a perfectly efficient process. The details can be analyzed by a computer model but it ain't easy. Bob Kooi's PhD thesis did just that. Lighter weight limbs mean that there is less kinetic energy stored in the limbs that has to be transferred to the arrow as the arrow speeds up and the limbs slow down. Mass distribution matters also. Consequently, bow limbs with narrow, lighter weight tips tend to be more efficient. The Devil is in the details.

b. Stacking and or String Follow Stacking is the situation wherein the bow is initially easy to pull and gets stiff only in the last part of the draw. This happens for bows with excessive string follow and/or very short limbs. Remedies are longer limbs and adding reflex and/or recurving the tips.

c. Bowstring Mass The bow not only has to propel the arrow but also the bowstring. Just before the arrow leaves the bowstring, the part of the string nearest the arrow is going at the same speed as the arrow. The part of the bowstring half way between the arrow and the bow tip is going at half the arrow speed and the part of the bowstring nearest the tip is hardly moving at all. The net result after some elementary physics is that you need to add one third of the bowstring mass to the mass of the arrow, not one half. A typical modern bowstring weighs about 3 grains for every pound of draw weight. For example, a 50 # bow shooting 500 grain arrows would use a 150 grain bowstring and cause a loss in arrow kinetic energy of

1-500/(500+1/3 * 150)] = 9% or a loss of arrow speed of 4.5%

d. Bowstring Stretch This is why Nylon is such bad stuff to use for a bowstring. It has excellent tensile strength but its stretch interferes with the efficient transfer of limb kinetic energy to kinetic energy of the arrow. Again, the end result is difficult to analyze but modern materials such as HMPE make it a very small effect. There is also a tradeoff between bowstring mass and bowstring stretch. The heavier it is the less it stretches. Somewhere there is an optimum-just right. Experiment!

e. Creep If you measure the draw weight at 28 inches and you actually release the arrow at 27.5 inches you've lost something. For a bow with a linear force-draw characteristic, 28 inch draw length and 7 inch brace height, a 0.5 inch forward creep causes an energy loss of about 4.7% or a speed loss of 2.4%. For example, 170 ft/sec for a 28 inch draw and 166 fps for a 27.5 inch draw.

f. Air Resistance on the Bow Limbs This should be inconsequential. The reasons are that the limbs don't travel very far and they move much more slowly relative to the arrow.

g. Internal Friction Saving the worst for last. Herein lies the rub. Call it hysteresis, call it internal friction, call it the Bowyer's Curse. It's only a minor issue with either fiberglass or carbon fiber but it's definitely there in wood.

How do we estimate the internal friction losses in a wooden bow? Internal friction is a major factor in limiting efficiency and arrow speed of a wooden bow. We know that the archer who draws and releases quickly gets greater arrow speed. That says that internal friction gets worse with time held at full draw. We also think that internal friction increases with greater strain level in the limbs. I would say that the only way to get an estimate of internal friction is to compare the experimentally measured arrow speed with the arrow speed computed from a model based only on the elastic modulus and density of the wood plus the mass and stretch in the bowstring. We also know that internal friction, if any, is much less with fiberglass or carbon fiber bow limbs.

Wood is interesting stuff. If you look up the mechanical properties of the sorts of wood used for bows, hickory, red oak, ash, etc. you will see that wood, parallel to the grain, can only be crushed about 0.5% but under tension it can be stretched almost 1%. Wood is stronger under tension than it is under compression. How should the bowyer apply that information to good bow design? This was discussed in Archery-the Technical Side, published in 1947 with articles written in the 1930s by real scientists who were also dedicated bowyers and archers. Their advice for the wooden bowyer:

a. Make the limb cross section rectangular instead of the semi-ellipse espoused by those completely smitten with the English longbow. That will reduce the crushing forces on the belly of the bow.

b. Put equal strain into all parts of the bow limbs. Just like Oliver Wendell Holmes' Wonderful One Hoss Shay. The simplest way to do that is to have limbs of constant thickness with the width adjusted for bending in circular arcs.

c. Find the sweet spot. If the limbs are narrow and short they will weigh less (that's good) but will have more strain (that's bad). If the limbs are wide and long they will weigh more (that's bad) but be under less strain (that's good). The excellent arrow speed achieved by Tim Baker's 80 inch hickory flatbow is partly because the tips are narrow for less mass and partly because there is relatively low compressive strain in the limbs.

Conclusions
Shoot what you like but be creative:
Observer: "What are you doing?"
Master Bowyer: "I'm designing a longbow with a unique shape. I'm even getting celebrity athletes to endorse it."
Observer: "Like who?"
Master Bowyer: "Former NFL stars Bo Jackson and Howie Long. They both like its unique oblong shape. Now I just need a name for it."
Observer: "How about the Bo Long oblong longbow?"
Thanks to Stephan Pastis, creator of Pearls Before Swine.

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