Calculating Pyramid Bow Limb Width Taper

[bvalentine002]

I want to pose a trignonmety question to help design bows.

When building a pyramid style long bow, what is the optimum rate of width taper that will place equal stress across the entire limb thus delivering low individual stress points and a circular tiller? Is there a mathematical formula for calculating this?

Obviously there are a lot of variables so I propose the following assumptions:
1. The limb material is equally stiff (no weak spots), flat (no crown or reflex/deflex), dense (no soft spots), and thick (no taper from fade to tip). This would most likely apply to board-bows or glass laminated bows with non-tapering cores laminations.

2. I am not concerned about compression or tension failures. I am just interested to know what the width taper would have to be to avoid thickness tapering.

3. An example rate of width taper would be something like 1.5" of taper over 30" of limb length or .05" per inch. Thus, a limb that has 20" of working length (like a kids bow) would need to have 1" of total taper and a 40" working limb (like a long ELB) would need to have 2" of total taper.

If a taper can be found, it should be easy to build the starting dimension of a pyramid limb by applying the taper to the working limb length and adding the desired tip width. Using the above example, a 30 working limb with 1/2" tips should be 2" wide at the fades.

-Brett

[Rangeball]

Brett, maybe you covered this in your assumptions, but I think at some point you'd have to take wood type into account, perhaps develop a wood specific formula.

The acceptable ratio is likely to be very different depending on the wood used, such as osage/ipe vs. red-oak/cherry, etc.

[Pat B]

The only true pyramid bow I made is hickory backed red oak. 66" long, symetrical, 4" handle area, 2 1/4" at the fades and tapering to 3/8" tips...so 66-4=62 divided by 2 = 31" long limbs going from 2 1/4" to 3/8".
   I think the wood the bow is made of will have a big affect on this formula also.  Pat

[Jeremy]

The true pyramid bows I've made have been around 2-3/4" at the fades, tapering to 3/8" tips, 64" ntn, 8" handle/fade area.  Anything under that starting width and you have to thickness taper to get the circular tiller and avoid the inner limbs doing all or the majority of the work.

[badger4x]

Brett, Thats really a very good question, I have made a lot of pyramid bows and have given a lot of thought to how I could figure the optimum thickness tasper. I have a weight formula I use that might help, lets assume that all dense woods will end at 3/8 tips and all medium dense woods will end at 1/2" tips. Now in reality I don't favor a circular tiller, I prefer an elyptical tiller and stiff tips. But just for the sake of a formula we will go with circular tiller. If you are making your bows between 62" and 68" long and you are sure the mc of the wood is below 8% you can safely figure that 16oz of wood will give you a 50# bow and that you should add 1oz of wood for every 5# of draw weight at 28". This may mean starting your fades at 1 1/4" or it may mean starting them at 2 1/2". Obviously the thickness of the woods determines how much a piece of wood can bend and the width determines how much it will bend in relation to other parts of the limb. I will usually start off with a pyramid shape and thin modify it a bit by side tillering. seems when I am done the mass weight of the bow almost always comes out very close to what I listed above regardles of the wood species, from poplar to ipe. Steve

[bvalentine002]

It seems to me that an even levevl of stress across the full length of the limb is mathematical and doesn't depend on the material.

The optimal ratio may be too narrow for certain wood (cherry), or wider than necessary for others (osage). Thus the actual taper can be modified.

I'm not that concerned about hitting a weight either. Assuming we can come up with an optimal taper, varying the thickness (stack height)should be the best way to chcange weight.

Keep 'em coming...
Brett

[Tim Clark]

To a point, Brett, I agree... however, w/r/t the thickness assumption that you just stated for changing the weight -- I'm not sure. At least as far as I see it, a wood such as osage would have to be made much thinner to hit weight with that assumption, whereas a wood like cherry would have to be proportionally thicker to hit that same weight (all else being held as constant, that is), and therein lies part of the problem, again as far as my own experience with cherry (and of course this is only for example, I am not hanging you on wood or word choice) being that thick, it would cause serious problems on the belly, most likley resulting in failure, and that failure most likely beginning with severe crysalling.

Not based on fact, but thoughts only. So... let's get it tried, then true? I have a lot of black cherry boards that I've been itching to make into bows... we'll see. But first let's get some good ideas on what an appropriate ratio(s) would be. Hey, I'm game! (just don't shoot me!)

[Rangeball]

If I remember correctly TBB #2 recommended a 4:1 ratio for a try pyramid bow.  Not sure how you figure the length though...

[bvalentine002]

Tim,
I see your point. The thickness of the wood would change the distance from the center of the lamination. This increases the amount of compression that the outer-most layer of the belly is under. Thus, as the thickness increases, the weight would too, but the compressive forces would be so great that they would crush the wood.

I think what we may be finding through this discussion is that there some woods that cannot be made into pyramid bows with equal limb thickness. Maybe we should reframe this for Glass-Lam bows and stronger wood.

Rangeball,
I'm not sure I understand the 4:1 ratio? Are you talking about width to thickness ratio? I have heard 4:1 when deciding how thick to make the backing and belly wood (such as a belly that is 4x as thick as the backing).

-Brett

[Rangeball]

Brett, I think any bow can be made a pyramid provided it starts wide enough and is long enough.  If I recall correctly the one discussed in TBB #2 was over 3" at the fades.

The ratio I was referencing was width at fades to tip width, if I remember correctly. I may be completely lost though  

[Tim Clark]

oops, actually it's ME that has misunderstood, in that I thought the question pertained to selfbows/backed/unbacked board bows. Different story, I am sure, for glass. Sorry.

[tim baker]

"When building a pyramid style long bow, what is the optimum rate of width taper that will place equal stress across the entire limb thus delivering low individual stress points and a circular tiller? Is there a mathematical formula for calculating this?"


"When building a pyramid style long bow, what is the optimum rate of width taper that will place equal stress across the entire limb thus delivering low individual stress points and a circular tiller? Is there a mathematical formula for calculating this?"

If this is the only goal then a triangular limb of any starting width will yield perfect circular tiller every times if same thickness from limb from grip to tip. Draw weight would be determined by limb width. Thickness by the elasticity of the wood and the length of the bow.

But this is not the best design for a couple of reasons.  Such a bow is evenly strained from grip to tip, so takes equal set grip to tip. Near-grip set yields exaggerated string follow, so tiller should be elliptical. Bend near the tips shortens the bow at full draw, increasing stack, lowering energy storage. So outer limbs should be relatively stiff. Tim Baker

[Jeremy]

Originally posted by tim baker:
then a triangular limb of any starting width will yield perfect circular tiller every times if same thickness from limb from grip to tip.

Tim, I have to disagree with you on that one.  Think of the extremes: A 66" bow, 3/4" wide at the fades tapering to 3/8" tips will have ALL the work beign done in the inner quarter of the limbs.  The forces the limb is under decrease as you travel out toward the tips.  Simple leverage.  Otherwise all the glass bows out there wouldn't need tapers - they're nearly all triangular limbs.  The inner limbs need to be strengthened - either by making them thicker or wider.  Thought I remembered you saying that before.  

[tim baker]

Jeremy: Perfectly truangular limbs, straight sides aiming at a point, will yield evenly strained, arc-of-a-circle tiller if even thickness. Is that the point of disagreement?

[bvalentine002]

Jeremy,
Thanks for responding...you sait it almost exactly as I would have.

To help explain more...if you took the same 66" bow and made the fades 4" wide and the tips 3/8", all the work would be done at the tips (whip tillered) becuase the fades would be too stiff to bend.

I'll try to say it another way: I would like to know what the perfect width at the fades and tip are for a given limb length in order to evenly distribute stress along the entire limb.

My goal in knowing is to avoid points of stress along a limb, thus avoiding set and compression failures in those areas.

Come On...any engineers out there?
-Brett

[badger4x]

One way to think of it is like this any wood that is the same thickness and bending the same amount will be under identical strain regardless of width, if wood is slightly thinner and still bending the same it is not under the same stress, Thickness determines how far wood can bend, width determines how far it will bend, and lots of other factors come in to make a decent bow, Steve

[tim baker]

Jeremy, Assuming uniform limb thickness and triangular fron view, the bow could be any width at the fades, 1" to 10" and stress along the length of the limb will be equal at all points, with circular tiller resulting. That's if the straight sides are aiming at a point at the nocks. This is straigh out of the engineering text books. It's against all intuition, but true. If the sidea are aiming at 3/8" instead of a point then the outer limbs will be a bit stiffer, the midlimb and grip bending slightly more. Tim

[Rangeball]

Tim, assuming one wanted a bow in the pyramid style with even limb thickness, to avoid near grip set could they simply leave the limb out full width for a few inches or so then start the taper?

[tim baker]

Rangeball: If perfectly uniform in thickness, and parallel in width for a few inches out of the grip, that would cause even more set near the grip, because that portion would be narrower than if the sides continued widening all the way to the fades. I hope this makes sense.

If the first few inches out of the grip is parallel then that portion must be a touch thicker, and have a whisper of thickness taper, just as if it was a typical bow design--after all, no part of a limb can know how the rest of the limb is shaped. A pure pyramid isn't a bad design, but performance will rise if tillered more elliptically, to reduce or eliminate near-grip set, the worst place to have set. Tim

[Red Dwarf]

Tim
I have just built a couple of board bows with limbs that are parallel for 70% of their length and then taper evenly to the nocks.
These take their set about 2" into the parallel section (from the nock end). Am I correct in assuming that this is also due to the limb being narrower in that particular area than if the limb had continued to widen?
Could this be avoided by thickening the limb in that immediate area ie. limb thinner either side of the transition area?

Sorry for the slight side-track Brett

Red Dwarf