Tiller shape vs front profile

[mmattockx]

Semantics. Bow limbs are not, and should not be, strained evenly.

It is not at all semantics. A bow limb is simply a cantilevered beam in bending. The equations defining the strain, stress and deflection of that situation have been well defined and known for a century at least.

If we divide the bow limbs into many sections, the total stress for each section should be quite different. It should be much higher in the fade section than in the tip section, etc. That's the mantra.

But if we look at the unit stress of each section, by dividing the total stress in each section with the mass units in the section, it should be same for all sections. All "wood" in the limb should be evenly and equally strained. This is the underlying principle for the mantra and no set tillering.

You seem to misunderstand stress, strain and load/force. Stress and strain are related by the Young's Modulus (also known as the elastic modulus) and if the strain is constant as you say, then the stress is also constant. I have no idea what you mean with 'total stress' and 'unit stress'. Stress itself is a unit measurement, defined as a force per unit area.

Mark, it’s not that I think it isn’t possible but that it isn’t optimal for bows to be strained equally along their whole length.

I would agree with that, considering stiff tips and the like. I think it would be more precise to say the strain should be equal for the working portion of the limb which is what I was thinking of when I said the whole limb.


Mark

[scp]

It appears people are getting confused about what "even"stress or strain means now. Bow limbs are stressed unevenly depending on the location of measurement.

"Where Energy is Stored in a Bow Limb
Engineer David Dewey computed the numbers for several dissimilar bow
designs. .... Here’s the breakdown for a uniform-stress D-longbow:
Inches from center - energy
0 — 7”....................30%
7 – 14”.....................27%
14 – 21”.....................22%
21 – 28”.....................15%
28 – 35”.......................6%"
(TBB4)
 
Each sections should be stressed differently according to the location. What should be "even" is not the total stress on each sections, but the unit stress of each mass units in each sections.

The easiest way to see this is to look at a pyramid front profile bow. If the thickness stays same for the whole limb, each sections should have the relative widths as expected in the chart. The fade section should be 5 times wider than the tip section. That makes the unit stress "even" for whole limb. That's what we call an "evenly" stressed or well tillered bow.

This is what the mantra is trying to express. It can be an adjustment of width, thickness or both. "Uniform stress" does not mean each sections are stressed the same amount. They should be stressed in proportion to it position. This is why all wooden bow limbs should be tapered in one way or another, unless the material is so strong like fiberglass as to make the tapering almost irrelevant, but not quite.

I am talking about something that's almost too obvious to most people to make them incapable of bothering to see anything. But I see one person who is interested.

[BowEd]

scp.....The guideline you've shown is something I've always kept in mind while making bows and has proven to be sound information.While making more and more bows it has become second nature.It has worked for me.

[RyanY]

It appears people are getting confused about what "even"stress or strain means now. Bow limbs are stressed unevenly depending on the location of measurement.

"Where Energy is Stored in a Bow Limb
Engineer David Dewey computed the numbers for several dissimilar bow
designs. .... Here’s the breakdown for a uniform-stress D-longbow:
Inches from center - energy
0 — 7”....................30%
7 – 14”.....................27%
14 – 21”.....................22%
21 – 28”.....................15%
28 – 35”.......................6%"
(TBB4)
 
Each sections should be stressed differently according to the location. What should be "even" is not the total stress on each sections, but the unit stress of each mass units in each sections.

The easiest way to see this is to look at a pyramid front profile bow. If the thickness stays same for the whole limb, each sections should have the relative widths as expected in the chart. The fade section should be 5 times wider than the tip section. That makes the unit stress "even" for whole limb. That's what we call an "evenly" stressed or well tillered bow.

This is what the mantra is trying to express. It can be an adjustment of width, thickness or both. "Uniform stress" does not mean each sections are stressed the same amount. They should be stressed in proportion to it position. This is why all wooden bow limbs should be tapered in one way or another, unless the material is so strong like fiberglass as to make the tapering almost irrelevant, but not quite.

I am talking about something that's almost too obvious to most people to make them incapable of bothering to see anything. But I see one person who is interested.

Seems like you’re misinterpreting the data. A true pyramid bow from fade to a point at the tip will form an approximate arc of circle tiller regardless of width at the fade. The force will change though. If you make a theoretical bow that is 5” wide at the fade and 1” wide at the tip it will certainly be stiffer toward the tip. The difference in energy storage is more likely related to changes in leverage as you get further from the tip. That is my interpretation. You state that it was for differing bow designs so a pyramid bow would just be one of them.

You’re using the terms energy storage and stress a lot and energy storage doesn’t have to correlate with stress. If you have two bows of the same tiller shape and same set, they will store the same energy even if differing width profiles. The stresses in the bow limb will be different due to where the mass is along the limb.

[Don W]

Ryan, you wrote "If you make a theoretical bow that is 5” wide at the fade and 1” wide at the tip it will certainly be stiffer toward the tip. "

That doesn't make any sense to me. Can you explain what you mean?

[RyanY]

Don, I’m just going off of what scp said about that energy storage in the limb correlating to width in a pyramid bow. But for the theoretical bow, if a true pyramid, tapering to a point, would produce a circular arc, then if the tips are wider than a point there will be excess mass along the whole limb stiffening the outer limbs progressively more toward the tip. This is assuming there is no thickness taper.

[RyanY]

Post from Woodbear Sep 6, 2006

“Some clarification of stress and strain is needed here.

Tom has correctly defined strain, it is a measure of the geometric deformation (stretch or compression) of the back & belly wood of the bow arm. I find it easiest to think of it in terms of percentage. That is a strain of 1% means the wood is 1% different in length compared to the original length of the wood without any applied force. 1% compression means the belly is only 99% as long as it is when no force is applied. 1% tension means that the back is 101% as long as without applied force.

Think about a short section of the bow arm say 100mm long, 20mm thick, and bending with a radius of 1000mm. Assuming that the cross section is rectangular, the length of the wood on the back is 101mm, and the length of the wood on the belly is 99mm long. The surface strain is 1% in both tension, and compression.

The relationship between stress and strain is stress equals strain multiplied by the elastic modulus (stress = strain * MOE). Strain is a unitless ratio, and stress has the same units as elastic modulus (MOE), and is measured in force per unit area (i.e. g/mm^2).

In the 100mm section of bow arm for example, let the wood be Yew with MOE of about 800,000g/mm^2, and the stress will be 8,000g/mm^2, or 1% of the MOE. Since the MOE is the same through out the wood (ignoring knots & defects etc.) equal stress and equal strain must both happen at the same time.

A bow with a perfectly circular tiller, and a thickness taper, can not have uniform stress or strain. Stress and strain will be highest where the wood is thickest. To obtain uniform strain and stress, the radius of curvature of the drawn bow must be proportional to the thickness of the bow arms through out the bow. A pyramid bow with circular tiller, and uniform arm thickness has uniform strain. An ELB with a thickness taper, can have uniform stress and strain if the tiller is elliptical, with the bend radius proportional to the thickness of the arms.

What Tim was talking about as stress, should properly be called bending moment, or bending force in the bow arm. The bending moment at any point on the bow arm can be found by measuring the distance from the location on the bow arm, to the string (at full draw for the maximum value experienced by the bow.), and multiplying by the string tension. The distance must be measured at right angles from the string to the bow arm location. It is clear that the distance from the string to the nock is zero right at the nock. The maximum distance from a point on the bow to the string will be from the center of the handle to the string. Half way down the arm from the center to the nock, there will be only about the distance between the string and bow arm, and consequently only half the bending moment. This is not exact since the bow arm is not straight but bent in an arc. A convenient way to see this is to take a photo of the bow on the tiller stick at full draw, and then measure the distance from the bow arm to the string all along the bow. The bending moment is proportional to the distance from the string, since the string tension is the same every where in the string.

Dave”

[RyanY]

Post from woodbear Sept 6, 2006

“
The question has been asked a number of times as to where the energy is stored in the drawn bow. Being curious about the answer, I adapted a bow design spread sheet to compute theoretical energy stored along the length of the bow arms.

The bow arm is divided up into 5 sections from the center to the tip. The first 20% I called handle. This includes the fades too as they are usually not fully bending when present. From 20% to 40% is inner bow arm. From 40% 60% is mid arm. From 60% to 80% is outer arm. Finally the tip area is from 80% to 100% of the bow arm length. There is a mirror image bow arm on the other side of the center of the bow. The computed bows are symmetric.

It is interesting to note that for an ELB like bow, the location of maximum energy storage is dead center in the bow handle. This even for an elliptically tillered bow. It makes sense, equal stress, and more wood equals more stored energy.

The comparison Indian style paddle bow is also working handle, but not fully stressed in the handle. I reduced the handle stress by about 10%. The handle is also significantly narrowed, both of which make for less bend in the handle. Even so, a substantial part of the energy of the draw is stored in the handle.

It seems that even though the bend is of long radius, and the movement of material is small, a very large part of the propulsion of arrow comes from energy stored in the handle of the bow. I have not computed the numbers for a stiff handle bow (yet), but I suspect that even on what we think of as a non-working handle bow, the handle is contributing as much energy as the tip portion of the bow.


Wood elastic modulus for Yew used for both bows. Energy is in Ft-Lb. for one arm, total the five energies, and multiply by 2 for the other arm, and you have the total draw energy.

70# @ 28", Long bow, uniform stress design with tapering thickness and width. (i.e. elliptical tiller)

% of arm from center - description - energy
0-20% - handle area - 9.58Ft-Lb 30%
20-40% - inner arm - 8.55Ft-Lb 27%
40-60% - mid arm - 6.97Ft-Lb 22%
60-80% - outer arm - 4.76Ft-Lb 15%
80-100% - tip area - 1.80Ft-Lb 6%


50# @ 25", Indian style paddle bow uniform stress except with narrowed handle at 90% of working arm stress (i.e. very elliptical tiller, 90% working handle)

% of arm from center - description - energy
0-20% - handle area - 4.44Ft-Lb 24%
20-40% - inner arm - 5.74Ft-Lb 30%
40-60% - mid arm - 4.68Ft-Lb 25%
60-80% - outer arm - 2.96Ft-Lb 16%
80-100% - tip area - 1.02Ft-Lb 5%


I hope this is what you eanted in the way of discussion of bow stress & energy storage.



Graphs of the bow designs with the stored energy overlaid on the back profile drawing.

Dave”

[Don W]

I think the fact that a limb needs to taper from fade to tip is easy to comprehend. I think once you start to study, learn and experience you get that to much stress causes set, and set is not your friend. But there has to be a better way to explain the correlation between profile and tiller and how to get to an efficient (or more efficient) bow. Even if that means creating categories like hunting, target, general, it's for heavy arrows, for light arrows, etc. We seem to interprete speed with efficiency, and speed can be gaged with each situation. So how do I constantly make my heavy hunting arrows go faster? What exact profile do I need to start with and expand from. I think the mass theory started down that road, but never got to completion. But there is probably more to it as well. Sorry for the ramble, but I am really trying to wrap my head around this.

Don...For heavy hunting arrows your power needs to come from your inner limbs.

I take this as meaning a bend through the handle bow is more efficient for heavier arrows.

[scp]

I guess we need to break down the whole analysis into simpler and much smaller pieces. First thing first. Can we all agree that WOODEN self-bow limbs need to be tapered to be efficient? If so, why?

[mmattockx]

The easiest way to see this is to look at a pyramid front profile bow. If the thickness stays same for the whole limb, each sections should have the relative widths as expected in the chart. The fade section should be 5 times wider than the tip section. That makes the unit stress "even" for whole limb. That's what we call an "evenly" stressed or well tillered bow.

A pyramid bow with a circular tiller will have the same bending stress (not 'unit stress') along the whole limb. You are misunderstanding what stress is defined as.

Thanks to RyanY for posting those quotes from David Dewey.


Mark

[scp]

I don't care for studying the engineering  or physics terms at my age. I use ordinary language for everything, even for esoteric philosophical arguments. That said, I am not going to argue against any ordinary language words interpretation by engineers. But I would ignore nitpicking about the terms used here, when the usage should be reasonably comprehensible to any sympathetic reader.

The main issue is why we need to taper wooden bow limbs, especially when the cheap fiberglass bow limbs are just simple slats of same width and thickness all the way from fade to tip. We need to stop thinking that we understand when we have not defined the question clearly enough.

Why do we even need to bother tillering bow limbs? Even one piece bamboo slat bow needs to be tapered. Why? In ordinary language terms, if at all possible, please. Just as in the mantra. We are here trying to make our "primitive" mind acquire enough practical skill or wisdom to make efficient self-bows. Why do we need to taper the wooden bow limbs in a certain way? Do we know?

[RyanY]

We have to be nit picky about terms or there’s no point in having discussions like these if we all mean different things.

In the simplest terms for our meat processors, if we don’t tiller and taper a bow limb it’ll break.   

[scp]

The easiest way to see this is to look at a pyramid front profile bow. If the thickness stays same for the whole limb, each sections should have the relative widths as expected in the chart. The fade section should be 5 times wider than the tip section. That makes the unit stress "even" for whole limb. That's what we call an "evenly" stressed or well tillered bow.

A pyramid bow with a circular tiller will have the same bending stress (not 'unit stress') along the whole limb. You are misunderstanding what stress is defined as.

Thanks to RyanY for posting those quotes from David Dewey.

Mark

With "total stress", I am talking about the stress potential of each section. The question is why the wooden bow limbs need to be tapered in a certain way to avoid set.

[scp]

We have to be nit picky about terms or there’s no point in having discussions like these if we all mean different things.

In the simplest terms for our meat processors, if we don’t tiller and taper a bow limb it’ll break.   

We are not trying to write an engineering dissertation here. We are trying to learn practical skills to make bows with as little set as possible. How well do I understand the internal combustion engine is almost irrelevant for me to learn the skill to drive a car as energy efficiently as possible. Is it more energy efficient to use the break as little as possible? Does it even matter?

Are you sure we understand and agree with the "primitive" mantra we are talking about here?

BTW If we keep on correcting each other in our use of ordinary language terms, we will never agree on anything. If an engineer is still trying to correct Tim Baker in his use of terms, we will never learn anything from that very experienced bow maker.