Preface: I'm not an engineer or mathematician, but work in a math-heavy field. So, please point out any errors in my though process here if you find any! If I'm wrong here, your input will help make me a better bowyer, if I'm correct in my reasoning, this may help myself and others.
So I've been doing a lot of reading, thinking, and math (read geometry) on the commonly discussed issue of symmetric/asymmetric/positive/negative/equal tiller. It seems to come up relatively often, and bowyers tend to fall in one camp or the other, but I haven't really seen any data-driven explanations. The following things I've read often or seem like common conclusions:
- Some people don't really worry about tiller symmetry, just about how the bow balances and shoots.
- Nobody really goes for a negative tiller.
- Some folks prefer a slightly shorter bottom limb rather than an a positive tiller.
- Some people always go for a slight positive tiller because the bottom limb is working harder
The last point is the one I've been most curious about, since I've read that countless times without an explanation of WHY and HOW MUCH the bottom limb is working harder/more stressed. So I finally think I've come up with a mathematical explanation as to why the bottom limb is working harder, and why I'll probably strive for a slightly positive tiller for all my future builds.
The reasoning came after reading Dr. D. N. Hickman's "General Formulas for Static Strains and Stresses in Drawing a Bow" from the 1930 November issue of 'Ye Sylvan Archer'. I've drawn up my own diagrams for symmetric and asymmetric tillers to actually visualize the differences myself, but I'll stick to posting Dr. Hickman's diagrams here unless someone wants to see my own.
Image from:
https://www.archerylibrary.com/books/hickman/archery-the-technical-side/hickman/general-formulas-for-static-strains-and-stresses-in-drawing-a-bow/So, there's a lot going on here, but we can, for most part, ignore a lot of this. One of the most important values to a bowyer here is
f, as it is the force applied by the limbs - the force that flings the arrow! Now, reading through Dr. Hickman's relationships and conclusions, we find
f = CA, where the force applied by the limbs a direct linear realtionship to the angle 'A' multiplied by a constant 'C' which represents the material characteristic of the limb that allows it to store energy (ie. draw weight). In order for a limb to do work equal to
f, and equal force must be applied to the limb at full draw.
We want both limbs to do the same amount of work, meaning the force they apply needs to be equal, otherwise one limbs is indeed stressed more than the other. The only variables we have to control
f are the angle 'A' and the constant 'C', which I pose is a variable because we can physically adjust a limb's draw weight.
A problem arises then, when the placement of the draw force on the string, where 'P' intersects 'S', is no longer perfectly centered on the bow's unbraced length or the string's unbraced length. The problem is almost always present because most of us place the arrow pass 1" to 2" above center on a bow os symmetrical lengths. Raising this point on a bow whose limbs are symmetrical in length and draw weight forces the arc 'N' of the top limb to become longer than that of the bottom limb. Since the arc 'N' is directly related to angle 'A', the force
f applied by the top limb is larger than that of the bottom limb.
f = CA
In order to fix this, we need to change the angle 'A' or the varable 'C' for one of the limbs so that the force
f of each limb is equal. Our options for doing so are:
- Lengthen the top limb, effectively decreasing angle 'A' for the top limb. This is pretty much impossible to do if you've already cut the stave to length.
- Weaken the top limb (reduce its draw weight), effectively reducing variable 'C' for the top limb.
- Shorten the bottom limb, effectively increasing angle 'A' for the bottom limb.
- Strengthen the bottom limb (increase its draw weight), effectively increasing variable 'C' for the bottom limb. This is impossible to do outside of gluing on a back/belly lam.
So, the only real solutions to the problem are leaving the bottom limb stiffer than the top limb, or making a bow with the bottom limb shorter than the top limb.
I believe this also explains the tillering principle/opinion I've read regarding tillering so that the limbs "arrive home" at the same time - signs of poorly timed limbs are significant hand shock and difficulty tuning. Since the time it takes for a limb tip to arrive at the brace position is dependent on the distance traveled (arc 'N') and the speed it moves (dependent on the force applied to it, which is force
f), tillering based on matching these variables should result in a well-timed bow.
Anyway, thanks for reading through my reasoning if you've made it this far! I'd appreciate any feedback!