Symmetrical vs. Asymmetrical Tiller - Geometry Heavy

[Alex C]

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!

[Gordon]

Shorter and slightly stiffer bottom limb is how I mostly design my bows. Seems to work out fine. But thanks for confirming that design principle with some math!

[Hamish]

I tiller the bow to look balanced at full draw. I also take into account if the limbs also have a similar amount of set, with a preference with no more than 1/4" more on the upper limb. If my bow does both these things, I can be confident that I've done the best job that I can. It works no matter what method you use, whether its uneven length limbs, equal limbs, or having a longer fade on the top, and equal limbs. It also works no matter which release method you use, but you need to tailor that by where you put your rope on the tiller tree.

That being said I find a longer upper limb works well for me on bows that bend through the handle, mainly due to balance in the hand, and stress point of using a Med' release.  Max of 2" longer upper limb, which is obtained by having the arrow pass 1" above centre. I also prefer this on stiff handled bows, I usually have the upper limb only 1" to 1&1/2" longer.  This is mainly due to tradition, or copying a specific existing design. Nearly all antique bows that I have seen have a longer upper limb, or at least a longer upper fade but equal working limbs with a stiff handle. The equal length bows only became popularized in the mid 1930's, and 40's, when the previously mentioned physics theorists, got into bow design.
Equal limbs are quicker to layout, and have the advantage that you can change which limb you want for the top during the tillering process, if the stave starts to act unpredictively. In practice I have rarely done this, as  once I worked out which limb was going to be the top limb, I've never needed to change.

[Jim Davis]

There are so many factors that it is hard to keep track of them.

In my thinking, the part about the limb tips arriving back at brace height at the same time is the most important, no matter which factors are used to achieve that.

However, as long a bow doesn't behave inconsistently shot to shot, accuracy is totally up to the archer. Japanese Kyudo bows being one extreme example of bows of unequal limb length.

It is more unsettling to me that with three-under nocking, at release the force center jumps from the center of the three fingers to a point at the arrow nock, a couple of inches higher. But a huge percentage of archers successfully use three-under.

I think that the bottom line is that if the archer does the same thing every shot, the arrow will fly the same every shot.

[Selfbowman]

I’ve also pondered this with my limited math. I learned to put my arrow pass 1-1/4 inch above center of bow both limbs being the same from a glass bowyer years ago . Been doing it ever since. This is on a stiff handle bow. Now that being said that way has broken lots flight records. Yes can’t say it’s better cause I’ve never tried different but once. Did not see any difference making the limb longer. Also where do you hang your scale on the string when on tiller tree. I center the bow on the tree and have a line straight down from the center. I hang my scale on that line. I have noticed that scale will go to the side of the line when one limb is stronger. It goes to the stronger side . Ive been playing with lining the scale about 1/4-3/8 toward my arrow pass when tillering. The bottom line for me is on wood bows they settle in during brake in to the way the archer holds the bow. Glass bows no.

[Mo_coon-catcher]

I don’t worry about exact numbers too much anymore as long as the bow balances. Here’s a bow I did a couple years ago with a 10” difference in limb length and it shot about the same as any other bow. It felt wierd as it would tilt when drawn. I made sure limbs were balance by the angle of the string to limb tips and approximate curve they were making. It wasn’t a light bow either at 66# at 27”. It did end up popping a splinter about 3-4” from the bottom limb tip. But I got a solid year of shooting and a kill from it first. And for arrow pass I just center up my hand in the handle and shoot. If I make a dedicated spot I mark it 1/4” above my knuckle. No idea what the actual number comes to.

Kyle

[Alex C]

There are so many factors that it is hard to keep track of them.

In my thinking, the part about the limb tips arriving back at brace height at the same time is the most important, no matter which factors are used to achieve that.

However, as long a bow doesn't behave inconsistently shot to shot, accuracy is totally up to the archer. Japanese Kyudo bows being one extreme example of bows of unequal limb length.

It is more unsettling to me that with three-under nocking, at release the force center jumps from the center of the three fingers to a point at the arrow nock, a couple of inches higher. But a huge percentage of archers successfully use three-under.

I think that the bottom line is that if the archer does the same thing every shot, the arrow will fly the same every shot.

I suppose limb mass is a third factor not explained by that diagram when it comes to tips arriving at brace height simultaneously.  A limb with significant set or knots, or something may need more mass to hold an equivalent energy (draw weight) as a more perfect limb.  I know nothing about Kyudo bows, but I'd be fascinated to get some measurements of limb deflection, energy storage, draw weight, etc. to see if they follow this concept to an extreme or maybe completely debunk it.

I agree with the three-under shooting - I shoot three-under and I've been thinking about how that changes stress dynamically.  It would seem that to make a bow perform well, the bottom limb must be under more stress at full draw, but then the change in angle 'A' on release must redistribute stress so that both limbs apply the same force.  I wonder if anyone's got input on bows shot three-under not lasting quite as long as bows shot split? Due to an overstressed bottom limb at FD.

I’ve also pondered this with my limited math. I learned to put my arrow pass 1-1/4 inch above center of bow both limbs being the same from a glass bowyer years ago . Been doing it ever since. This is on a stiff handle bow. Now that being said that way has broken lots flight records. Yes can’t say it’s better cause I’ve never tried different but once. Did not see any difference making the limb longer. Also where do you hang your scale on the string when on tiller tree. I center the bow on the tree and have a line straight down from the center. I hang my scale on that line. I have noticed that scale will go to the side of the line when one limb is stronger. It goes to the stronger side . Ive been playing with lining the scale about 1/4-3/8 toward my arrow pass when tillering. The bottom line for me is on wood bows they settle in during brake in to the way the archer holds the bow. Glass bows no.

My thinking is that a longer top limb or a stronger bottom limb independently reach the same conclusion - no difference in the final force applied to/by the limbs.  So those bows with flight records, I'd love to know if the top limb is slightly weaker (even if unintentionally).  Like hanging a weight on both limbs - I'd expect the top limb to deflect farther with the same weight.

The draw line on my tillering tree is about 1" below the arrow pass, so if I tiller to a "symmetrical" tiller, I'm actually making the bottom limb slightly stiffer.  I'm trying to finish up a yew flatbow this weekend and I'll get some pictures/measurements to make sure that's actually what's going on.

Regarding a bow settling in, I'd imagine that is accomplishing the same result as the stiffer bottom limb, it's just happening through set, whether measureable or not.  If you've got a stronger top limb, it's going to be under more stress until it takes enough set so that the force applied to both limbs is equal.  That may be so little set that it's not really measurable, but I'd bet it's there.  That's actually something I'd really like to test!  I've got a PhD thesis to finish up, but maybe come summer I'll make a couple test bows and see if this is actually what happens during break-in.

[Alex C]

I don’t worry about exact numbers too much anymore as long as the bow balances. Here’s a bow I did a couple years ago with a 10” difference in limb length and it shot about the same as any other bow. It felt wierd as it would tilt when drawn. I made sure limbs were balance by the angle of the string to limb tips and approximate curve they were making. It wasn’t a light bow either at 66# at 27”. It did end up popping a splinter about 3-4” from the bottom limb tip. But I got a solid year of shooting and a kill from it first. And for arrow pass I just center up my hand in the handle and shoot. If I make a dedicated spot I mark it 1/4” above my knuckle. No idea what the actual number comes to.

Kyle

That'd be an awesome example to get measurements from! Do you still happen to have that bow? A quick sinew wrap at the splinter might be enought to measure deflection of each limb with maybe only 10-15lbs hanging off the tip.

Also, That's a real pretty bow! Those knots at the handle are just cool...

[RyanY]

Great analysis. Thank you for sharing. Your paper on the yumi design will be due at the end of the semester. 

[Selfbowman]

Yes I think you will find that the top limb will settle in mostly during the build. If you measure the top limb strung about 4-6” from fade you find the measurement will be as much as a 1/4” bigger than that point compared to bottom  limb.