using maths in making bows

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yeoman
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#31 Post by yeoman » Mon Nov 05, 2007 7:08 pm

Well, the obvious question is, are there any piccies?

Now that I've finished, printed, bound and handed in my thesis, I'll be able to write the next installments in this thread.

Dave
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#32 Post by Dennis La Varenne » Mon Nov 05, 2007 7:41 pm

Dave,

That is a very fair question and I haven't got a satisfactory answer other than to take some and post them.

I will get onto it shortly. There are two bows so far. There are also a couple of others that I have helped with on Ozbow who may want to post some pics if they do their jobs well.

Dennis La Varenne
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#33 Post by Dennis La Varenne » Thu Nov 08, 2007 3:03 am

Dave,

I have attached a Word document with pics which you can download and read to save a longish post.

Have a good read.

Dennis La Varenne
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Dennis La Varénne

Have the courage to argue your beliefs with conviction, but the humility to accept that you may be wrong.

QVIS CVSTODIET IPSOS CVSTODES (Who polices the police?) - DECIMVS IVNIVS IVVENALIS (Juvenal) - Satire VI, lines 347–8

What is the difference between free enterprise capitalism and organised crime?

HOMO LVPVS HOMINIS - Man is his own predator.

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#34 Post by yeoman » Sun Dec 09, 2007 9:28 am

OK, here it finally is: the next step in using maths in making bows.

I last left you all hanging on the end of a wood bending test, the purpose of which was to find the mechanical characteristics of a beam of wood.

Now that we know (hopefully) how to find these, we can move on. Designing a bow using maths is kinda the opposite to tillering one ‘analog’. We need to design the tiller of the bow first, then get dimensions which will give that tiller shape.

To start with, I’ll talk you through the process of designing a pyramid bow. This is because anyone can do it with a compass and ruler, pyramid bows are efficient (within certain limits), pyramid bows are easy to make anyway, and people generally like pyramid bows.

Later, we’ll move on to elliptically tillered bows and reverse tapered bows.

I will also assume for the time being that the cross section of the bow is perfectly rectangular. Later I will broach the subject of different cross sections like the ELB ‘D’ section.

The first step is to decide how long you want your bow to be. For this example, I will ‘make’ a bow which has 28” limbs from fade to nock, with an 8” rigid handle, giving a total length of 64” ntn. When drawing these things, it is better to use larger pieces of paper and draw larger plans, as this increases the accuracy of your measurements. Alternatively, you can use (if you can use it) a CAD program which is extremely accurate. Google ‘DeltaCad’ for a free, easy to use CAD program.

With a piece of paper, accurately draw to scale a straight line which represents one limb, and half of the handle of the bow. Using separate colours may help. By the way, this process will design half the bow…the other half will be exactly the same.

FIRST PICTURE
Image

From B -> C, mark a point which is 25% of the limb length away from B. In this case, the mark goes at 7” from the fade. With a compass, trace a circle which has a radius equal to D -> C with D at the centre. This arc describes the path of the tip of a pyramid bow during brace and draw.

Now, draw a line parallel to A-C on the same side as the arc you drew, at a distance which you like your brace height. In this example, I’m using a brace of 5” from the back of the bow. Only draw the line as far down as the bottom of the half handle, and as far up as it needs to be to touch the arc. This represents half of the string length required to brace the bow at that brace height.

SECOND PICTURE
Image

This is going to take a few pictures, as it does get a bit complex if I put it all in one diagram.

From point B, draw a horizontal line out to the right for quite some distance, line B-F.

Draw a straight line from B to E.

Mark the exact centre of line B-E.

From this point, draw another line, perpendicular to B-E, until it intersects line B-F.

For the sake of clarity, line B-E will be erased for the next picture, as will the green arc.

THIRD PICTURE
Image

Again with the compass, draw an arc with a radius equal to the length G-B with the centre at G. You should notice that this arc will pass through points B and E. This arc is the perfectly circular tiller shape of the bow at brace…the brace height that you chose. This step is not completely necessary, but it is cool to do.

FOURTH PICTURE
Image

Nagler, Klopsteg and Hickman found that at full draw, the tips of a bow are pretty close to halfway between nock and bow, pretty much regardless of brace height. This, we can demonstrate in the next step. From A, draw a line perpendicular to A-C, and nearly as long as line B-G. Put a point at the desired draw length. In this example I am using 28 inches.

Next, we need to find the point at which the tip of the bow will be at full draw. To do this, use the compass to draw a circle which has a radius equal to the length of the brace string we found earlier (1/2 string, really). Put the centre of this arc on the full draw mark. The point of intersection between this arc and the bow tip movement arc is where the tip will sit at full draw. This point is point ‘I’.

FIFTH PICTURE
Image

Now, similarly to what we did earlier when drawing the braced limb, we do the same here. Draw a straight line from B to I. Halfway along this line; draw another line perpendicular until it intersects with line B-F.

Using the compass, draw an arc with a radius equal to J-B, with the centre at J, so that the arc touches both B and I. Also, draw a straight line from H to I, which represents the string.

This is the shape of the bow at full draw. In this example, the tip moved 12.9 inches to get to full draw of 28 inches. Nearly halfway.

Taking out the now superfluous lines, arcs and points, we have the SIXTH PICTURE, which should be a geometrically accurate representation of the bow which you desire to make. With practise, this process takes only a few minutes.

Image

Now, there is a problem or two with what I’ve just told you. Mostly, this process does not take into account the string stretch. When we draw the bow at brace, we draw the bow when the string is experiencing the very most tension. Thus, this measured length of string is how long the stretched and finished string should be. Perhaps counter-intuitively, the string is under the least strain when at full draw (apart from being unstrung of course :wink: ). As such, the length of the string actually decreases somewhat at full draw.

This change in length, however, is not very significant, and may be regarded as a measuring, percentile or rounding error, and should not change the end product too much. A slight variation in the timber between the tested sample and the proper stave will probably account for more variability than the string stretch factor.

That’s probably enough for the time being. Practise it a few time with different variable of limb length, draw length and brace or handle length.

I’ve tried to break it down simply, but if there are any questions you are all most welcome to ask.

Have fun,

Dave
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#35 Post by yeoman » Tue Dec 11, 2007 5:43 am

No more interest?
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#36 Post by BenBow » Tue Dec 11, 2007 8:27 am

yeoman I've read parts of this thread and most of it is over my head. Have you looked at the Supertiller spreadsheet and how it might fit with what your explaining here?

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#37 Post by Jaydo » Tue Dec 11, 2007 11:12 am

Yeoman, although a lot of this i dont quite understand yet, i do get it hopwever and maybe after some more practicle experience i might understand it better.

great information here and really helpful to people who are looking at making bows but dont quite understand the theory behind it
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#38 Post by Dennis La Varenne » Tue Dec 11, 2007 12:15 pm

Dave,

Lots of interest from me, but lots of digesting first.

Does DeltaCad come in Mac????

Dennis La Varenne
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Have the courage to argue your beliefs with conviction, but the humility to accept that you may be wrong.

QVIS CVSTODIET IPSOS CVSTODES (Who polices the police?) - DECIMVS IVNIVS IVVENALIS (Juvenal) - Satire VI, lines 347–8

What is the difference between free enterprise capitalism and organised crime?

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#39 Post by GrahameA » Tue Dec 11, 2007 1:24 pm

Hi Dave

Much interest. What we need to do is condense it into a book/pdf. Project for next year.
Grahame.
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#40 Post by Dennis La Varenne » Tue Dec 11, 2007 2:11 pm

Dave,

I printed out your post and read it at work here during my lunch. It is pretty straight forward now that I have perused it.

I found I was making one small blue in my own scale drawings about the placement of the point of the bending arc (D in your drawing). I had it right on the fadeout rather on a point 25% out toward the tip as you have.

When I checked my source again, it corroborates your drawings. Mine were just a bit less bendy than yours because the radii were longer. No matter though. All's well now.

Dennis La Varenne
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Have the courage to argue your beliefs with conviction, but the humility to accept that you may be wrong.

QVIS CVSTODIET IPSOS CVSTODES (Who polices the police?) - DECIMVS IVNIVS IVVENALIS (Juvenal) - Satire VI, lines 347–8

What is the difference between free enterprise capitalism and organised crime?

HOMO LVPVS HOMINIS - Man is his own predator.

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#41 Post by ed » Tue Dec 11, 2007 5:09 pm

Still interested. I am working on a couple of bows already planned and then will give this a try. May have to wait until sometime in the new year though. DeltaCad is trailware now and so downloaded CadSTD instead and will see how that works out.

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#42 Post by ed » Tue Dec 11, 2007 5:48 pm

A question,
I gather that the use of an arc from 25% from the fade to map the arc of the tip of the bow is due to the limb not bending like a stiff beam at a hinge at 'B'. The question is, how do you come by the figure 25%? And how much can this vary from bow to bow?

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#43 Post by yeoman » Wed Dec 12, 2007 7:19 pm

Ed: that's correct. If it were a hinged beam itwould be an arc with the centre at the fade.

the 25% figure was calculated by some seriously complex maths which I can't quite follow, but is definately accurate. If the bow is straight limbed, and the tiller is circular or slightly elliptical, then the 25% figure will be correct regardless of bow length, draw length or fade length.

Grahame, I'm hoping to put this stuff into a book or series of published articles at least, sometime in the next year. It'd have better explanations, and better pictures too.

I'm not sure if DeltaCAD comes in Mac or not. google 'free Mac CAD' and see what happens.

All: I'm glad that there's still interest. Remember, if there's things you think are explained in too confounded a way, holler and I'll try to re-explain.

HOMEWORK: try drawing a few bows with the above directions, either freehand or on computer, and post them here.

Dave
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#44 Post by archangel » Wed Dec 12, 2007 7:38 pm

Dave,

Congratulations! This could become one of those "classics" for bowyers everywhere. As both an aspiring bow-builder and maths teacher, I have followed your thread with great interest and dowloaded DeltaCAD as you suggest - handy little program for sure.
HOMEWORK: try drawing a few bows with the above directions, either freehand or on computer, and post them here.
I am going to try and apply the software using my (just-finished) ELB. I'm curious to see how close it comes, now there is an optimum model to compare with the real thing. After this what is the Next Step ... how to apply this to building an asiatic composite, with its complex curves?? :roll:
"The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low, and achieving our mark." (Michelangelo Buonarroti)

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#45 Post by yeoman » Thu Dec 13, 2007 5:44 am

That's a little more complex...it may take a little time to figure that one out. :)

I think the next step is to figure out how to do self-recurves or something like that.

Benbow: just saw your post. I have used supertiller, in fact that's what inspired me to write my own bow design program. I know the guy who wrote supertiller and he's a good buddy of mine. Just for fun, when I get bow dimesnions out of my program, I'll plug them into supertiller and yes, the results are quite similar.

Quick poll, if money were not an issue, what would people prefer: a PDF or a published book?

Dave
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#46 Post by BenBow » Fri Dec 14, 2007 11:10 am

yeoman I have another called "Selfbow 10arc rev070326.xls" that has a bend test in it as well as the limb cross section and many other interesting things but I haven't had time to play with it yet.
Alan is great about sharing knowledge as you are too. Thanks for this thread.

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#47 Post by yeoman » Fri Dec 14, 2007 5:55 pm

The Author of 10arc is also a Dave, and another great engineer. He's also helped me a lot with my own bow-making spreadsheet. 10arc and supertiller are great tools, and put mine somewhat to shame.

Dave
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#48 Post by BenBow » Sat Dec 15, 2007 11:19 am

Well I am sure enjoying the fruit of you math types work. Thanks!

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#49 Post by Dennis La Varenne » Sun Feb 10, 2008 6:00 pm

Dave,

I have digested all of the presentation so far and am awaiting the next instalment as are the others above I expect.

Amy more coming? Don't lose heart because you don't get a lot of responses. It is technical stuff and it scares off a lot of people, but there are those of us who, although we may have to wade through it a bit, still find it interesting and have started applying it to our bowmaking with good results.

Dennis La Varenne
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Have the courage to argue your beliefs with conviction, but the humility to accept that you may be wrong.

QVIS CVSTODIET IPSOS CVSTODES (Who polices the police?) - DECIMVS IVNIVS IVVENALIS (Juvenal) - Satire VI, lines 347–8

What is the difference between free enterprise capitalism and organised crime?

HOMO LVPVS HOMINIS - Man is his own predator.

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Re: using maths in making bows

#50 Post by yeoman » Thu Jul 31, 2008 6:41 am

TTT

I am about halfway through writing the next installment in this dull thread.

For those three of you who are interested, I highly recommend going through the thread again to try to reacquaint yourselfves with the theories and methods used so far. Sorry to say, it isn't really going to be getting any easier.

Dave
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Re: using maths in making bows

#51 Post by Steven J » Thu Jul 31, 2008 8:32 am

Dave,

I am looking forward to the next installment. It is great to see this thread come up on top again.

Steve
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Re: using maths in making bows

#52 Post by looseplucker » Thu Jul 31, 2008 12:23 pm

I'm waiting for it too. I regard myself as very much a rank apprentice, but the use of maths I have found to be both fascinating and rewarding. Likely I will not get to the points that Dave and Dennis are at - but from my perspective as a newbie this is good stuff. Certainly has been eye opening :idea:

What it really means to me is that if I do a bow by the maths (and I've done one already on the hickman formula) I am much better placed if I do one ''analogue" to understand why and how the timber is behaving the way it is.

Keep it coming Dave - and Dennis.
Are you well informed or is your news limited?

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Re: using maths in making bows

#53 Post by yeoman » Thu Jul 31, 2008 5:32 pm

Ok, my keen audience, the time has finally come to read the next chapter in how to use maths to make a bow. So you three, listen up. Hopefully I can pick up where I left off and not leave too many people behind. If this is your first time reading this thread, I recommend going back and reading from the beginning. If you have read up to here, go and get a drink, as we may be here for a while.

I’ve just read up to where I did my last instructional post. I’m glad I don’t have to type and draw all that again!

Before I go on with the how-to, let me say a couple of words about the how-shouldn’t.

There have been posts recently in which people have used formulas set out by Hickman in the 30s and 40s to make some flatbows. They have, by all accounts, had quite some success and I commend them for trying to have a go at mathematically producing a bow. However, there are some shortfalls in Hickman’s methodology.

Firstly is Hickman’s definition of a bending moment. I’ve already discussed this here: http://ozbow.net/phpBB3/viewtopic.php?p=45003#p45003 on page 1. While he is somewhat correct, he is also mainly wrong. He was right in that that method can describe the bending moment, but it has to be on a VERY RIGID BEAM which deflects only a small amount. A very small amount. I’m talking a wrench, pry bar, etc.

On a beam which deflects a large amount, the Nagler approach is the more correct one. Thus, we will use the Nagler method to create a bow with even stress along the length of the limb. That’s why we had to learn how to draw a bow, because having the geometric drawing in front of us is integral to making a bow using the Nagler method.

Nagler discovered that the bending moment at a given point of a bow is the tension in the string, in newtons, multiplied by the distance from that given point to a location on the string, such that this line is 90 degrees to the string. You may well have a confused puppy look on your face so just have a look at the bow on the right here to see what I mean: Image

Something else pivotal in the bow design process which Nagler made mention of is that for a given draw length, limb length and wood type, there is a given thickness which will give optimum bending stress. To put it another way, he stated that for a bow to perform maximally, it must be thin enough not to take much set, yet thick enough that it can be as narrow as possible to save on mass. This was a Good Thing.

The formula is given as follows:

R/T=E/2S where:
R=radius
T=thickness
E=MoE
2=2
S= stress desired/required in the bow limb

The formula is given this way so that you can easily change it for thickness or radius. There’s another function for this manner though but we’ll get to that later.

When using this formula it’s important to use the same units consistently, such that R & T are the same unit and E & S are too. This means that the E number should have an extra 3 zeros on the end of it compared to the S. For example, if I looked in Bootle and got the mechanical properties for grey Ironbark, I’d find the MoR to be 181Mpa and the MoE to be 24Gpa. For the formula then the numbers would have to be 181 and 24000 respectively. We don’t have to worry about putting all the zeros on the end as long as they are in the same unit. There is no difference between 10 divided by 1 and 1000 divided by 100. It’s the ration we’re after.

OK, so what do we do for the S value? Using the rated MoR of a given timber will in fact make a bow which when fully drawn will experience the MoR. The MoR is when the structural integrity of the wood fails. That is, if you make a bow to experience MoR at full draw, it will break/crack/explode/etc.

This is where your bend test comes in. The bend test was used to test your individual piece of timber to find its mechanical properties, as published figures are not always accurate. All wood is different.

We want to choose a bending stress which is less than the MoR, but not so small that the resulting bow will be the size of a ski to have a decent draw weight. For this example, let’s look at a bend test I did with a nameless piece of timber.

Here’s the dimensions:
W-27mm
T-3.8mm
L-304
Yep, it was a small bit.

Here’s some of the bend test data:
Load (kg)-----deflection (mm)--------set (mm)
9------------------21-----------------------0
10-----------------22.5--------------------0
11------------------24.9-------------------0
12------------------27---------------------2

I stopped the bend test when set got to 2mm. Why’d I do that? Because that much set was enough. For a 28” draw bow, the tips will, regardless of bow length, move about 14” back from the handle. That is, the tips will sit about ½ way between your arrow nock and bow handle. This is relevant. For this bend test, 2mm of set equals 7% of the deflection under load.

No matter what the dimensions of the board, if it were to experience the same stress as it did in the bend test, the set taken would be 7% of the delfection, whether it was a diving board or a matchstick.

7% of 14 inches is about 1 inch. 1 inch of set is pretty good for a bow. Steve Gardner, one of the world’s most prolific bowyers (10,000+ under his belt a couple of years ago) set a world flight record with a selfbow which had 1.5” set. So, we want to make a bow which experiences enough stress to induce about 1” set. The bending stress in this example in the bend test was 137 megapascals. That is the number we will be using for ‘S’ in this part of the example. The MoR doesn’t matter. Bows break at MoR.

Incidentally the MoE for this bit of timber was 20663 mPa. This will be the E value in our example.

We want rhe limbs of our bows to have even bending stress throughout their length when drawn. The reasons for this are simple, but think of it this way: ropes should be as uniform as possible for durability. Similarly, the bending stress along a limb’s length must be evenly distributed to increase the durability of the bow. Also, if one section is overworked, it will take too much set. If it is under stressed, it will have too much mass. Both of these negatively affect bow speed and user joy.

I didn’t test the sample to MoR. However, I have indeed tested quite a few samples to MoR and made what I think is an interesting observation. On many of the better woods that I have tested, the optimum stress to put into the wood is some 60-66% of the MoR of the wood. I’m not saying that this is a constant, or that it’s a rule. However, if anyone asks, you heard it first from me, eh? ;-)

So anyway now we have our E and S values:
E-20663
S-137

So let’s start some actual BOW MAKING!!! Almost.

The first thing we are going to do is figure out how thick our bow’s limb is going to be. This is where we pull together what I said about there being an optimum thickness for each degree of curvature in a bow limb and how to use the R/T=E/2S formula. And also as I said, to begin with we will make a pyramid bow. Because it’s easiest.

Below is a picture of a pyramid bow that I drew up in CAD. It is tillered so that the limbs bend in circular arcs, with a rigid handle. The bow was drawn according to the instruction I posted in another thread.

Image

Each limb is 27.5” long. The radius of curvature of each limb is 26”. As this bow has the same radius of curvature along the length, and as for each radius of curvature there is an optimum thickness for each wood, this bow will have limbs which are the same thickness along their entire length.

So, we have our formula, R/T=E/2S but we need to know only the thickness. The rearranged formula is thus T=2RS/E I know that formula almost spells a naughty word for bottom. Mathematician s have fun too you know!

Let’s finally substitute some values.
T=unknown
E=20663
S=137
Thus:
T=2RS/E
T=(2x26x137)/20663
T=0.34 inches

The thickness of this bow’s limbs along their entire length (once outside the fades of course) will be .43 inches, and when braced and drawn to the specifications in the picture, will experience an even stress of 117mPa (provided the width taper is right).

If you have done your own bend tests, try substituting your own values in. If not, play around with some values from the table I posted somewhere which lists mechanical properties of about 300 different timbers. Here it is: http://ozbow.net/phpBB3/viewtopic.php?f=33&t=3329

Now that we’ve done this, we can figure out how to find the width-profile of this bow to make it whatever draw weight we desire.

That’s enough for one thread. I’ll post another one in a moment.

Dave
Last edited by yeoman on Wed Aug 06, 2008 7:08 pm, edited 1 time in total.
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Re: using maths in making bows

#54 Post by yeoman » Thu Jul 31, 2008 7:50 pm

Time for the next post. I told you it wouldn’t take long. In this thread we will be figuring out how wide to make our bow’s limb now that we know how thick to make it.

See below for an image of half the previously described bow, with some green dots. I have in this picture divided the limb up into 10cm sections. Seven of them. More is better, but seven will do.

Image

An interesting feature of a bow’s width profile is that it is always proportional. If a given optimum bow has, at a particular draw weight fades 50mm wide and tips 10mm wide, then a bow with twice the draw weight will have fades 100mm wide and tips 20mm wide. Actually, ideally a bow’s tips should have a width of zero but I’ll save that for when we have the chat about the tooth fairy and the easter bunny.

Another peculiarity, this time of only pyramid bows, is that the length of lines we use to calculate the bending moment (see the line marked ‘distance to string’ on the pic below) are representative of the relative widths of each segment of the bow. I know, you must have the confused puppy face again by now so it’s time for another piccie.
Image

If I draw a line from each little green circle to the string, so that the line I draw is at right angles to the string, then I’ll get a diagram like the following:

Image

If I then draw a straight line the same length as the length of the limb, and split it into the same number of sections that I split the limb into, and redrew each purple line at its corresponding point on the straight line (wait for it) I will get a diagram as follows:

Image

The thin blue line is a straight line between the widest section and the end of the bow limb. This is to illustrate that the limbs of a pyramid bow do NOT taper in straight lines but rather bulge slightly. This diagram also goes some way to demonstrating that in an ideal world, the tips of bows would have zero width.

The ratio of the length of these lines is the same as the width profile of a finished bow. Of course, the proportion of the length of these lines are not also in proportion to the length of the limb!!

From here, there are a couple of ways to proceed. If you wanted to make a bow of the length, draw length and whatnot of the one I have drawn, you can stop reading before the end of this post and go and do it. What you do is this:

Figure out how thick you need the limb to be.
Draw your width profile such that the fades are 2” wide, and the rest of the sections are to scale. To make it easy for you here are the numbers calculated for you:

Distance from fade (mm) --------width (mm)
0----------------------------------------50
100-------------------------------------44.6
200-------------------------------------38.9
300-------------------------------------32.2
400-------------------------------------24.8
500-------------------------------------16.8
600-------------------------------------8.6
700 (tip) ------------------------------8.6

So draw that out, make up a handle/fade configuration, cut it out, and cut some nocks in. Measure the draw weight at full draw (see note below).

Divide the weight you want by the weight you have. For example, if you had 65lb but wanted 44, divide 44 by 65, which equals 0.67.

Multiply the width of each section by this number, draw new lines on, then carefully plane down to those new lines. This could potentially leave your tips as narrow as 3-4mm. If you’ve not made many bows, let no part of your bow be narrower than say 8-10mm. This means having slightly wider tips proportionately, but never mind. Having a bow that shoots is a Good Thing.

There you go. Go to it and make yourselves some pyramid bows.

NOTE I skipped a large bit there. When I say measure the bow at full draw, I didn’t mention that you have to carefully brace the bow and slowly bring it up to full draw, just like you would need to on a normally tillered bow. Make sure you check for stiff spots or hinges. Proceed as you would with a normal bow but tiller VERY carefully if it needs any.

Right, for those of you who are still awake, and still here, let’s move on. The others will be back in a bit when they’ve conquered that beast.

To figure out exactly how wide the limbs need to be for your intended draw weight, we need to use some more formulas. I know, I can hear you groaning. I’ll try to keep this as painless as possible.

First, you have to know the angle between the arrow (which is the direct line of force) and the string (which is the means by which the force is transmitted to the bow tips).

For the 62” bow pictured, it’s easy. I’ll tell you. It’s 61 degrees. If you’re drawing your own bow, you’ll have to figure it out yourself.

With this, we need to figure out how much tension will be in the string at full draw, at our intended draw weight. For this example, I will use a figure of 44lb. This is 20kg, just for you young spring chickens. Knowing the kg is good here. This is so you can more easily convert your draw weight into a draw FORCE proper, in newtons. To get newtons, multiply your draw weight by 9.8. Thus, our 44lb bow will have a draw force of 195.7 Newtons. Who picked my mistake? The force we need is actually HALF of that number I just told you. This is because we are actually calculating HALF a bow, which we will make two of to make whole bow. So HALF of 195.7 is 97.85 Newtons.

To find the tension in the string, put your draw force (in newtons) into this formula:

Tension = draw force/(COS angle) you need a scientific calculator to figure out what cos of the angle is. Why? Just cos. The calculator built into Microsoft windows has a cos function. Change the view to ‘scientific’.

So back to our example, if this bow were to draw 44lb the string tension would be about 201.8 N. Remember this number.

The formula for figuring the width of a bow’s section is given as:

W=(6dt)/(S(T^2))
Where:
W= width
d= distance to string
t= tension
S= desired stress
T= thickness

Now don’t be scared by the formula. It’s not that bad. The ‘distance to string’ is that same annoying thing I keep going on about. From your bow section where you want to calculate the width, draw a line to the string, such that it is a right angles to the string.

It’s for this reason that drawing in a CAD program is really good. You can also do it on a massive sheet of paper. The bigger the better. Can get fiddly though. To make it easy for you, here are some measurements for the bow in this example:

Section-----distance to string (mm)
1 (fades)-----------------582.5
2--------------------------520.1
3--------------------------453.6
4--------------------------375.9
5--------------------------289.7
6--------------------------196.9
7--------------------------99.5

Of course you’ll get far more satisfaction measuring up a bow you drew yourself.

So, using the above numbers, formulas and mechanical properties, we get some dimensions as follows:

Distance from fade------width
0-----------------------------------67
100--------------------------------59.9
200--------------------------------52.2
300--------------------------------43.3
400--------------------------------33.3
500--------------------------------22.7
600--------------------------------11.5

Just out of interest, compare these figure with the table I put in above showing the relative widths of the ‘distance to string’ lengths above. Pretty darn close in proportion. Not exact, which is a pain, but when you’re dealing with four variables and four decimal places, these things are bound to happen.

Why was it so wide? Well, there's a couple of reasons. Firstly, is the short length. The shorter you make a bow the wider and thinnner it must be to reduce string follow.

Secondly, the figure I used for the allowable stress may have been too low. It is possible that perhaps in my bend test I should have carried on a little longer to get a bigger set and a bigger bending stress. Doing this would have resulted in a thicker bow, which would automatically have made it narrower. So in your own bend tests and bow-dimension calculations, play around with how much bending stress you use and see how much it affects the final dimensions. To get a narrower bow, you mayhave to put up with slightly more set.

That’s pretty much it for pyramid bows. As long as you undermestand what I’ve written, you can pretty much make whatever pyramid bow to whatever specification you like.

I’ll give you some time to read through all this before we move onto the real toughy- elliptically tillered bows.

NOTE: this thread is about using maths to make bows, not about how to work wood. While I didn’t say at any point that you should pencil in your lines, cut to just outside them and then sand down to them, this is of course best practise. It may also be worth adding just a couple of mm to the thickness and width for the first few, just as insurance.

Ciao,

Dave
Last edited by yeoman on Wed Aug 06, 2008 7:21 pm, edited 1 time in total.
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Re: using maths in making bows

#55 Post by Steven J » Thu Jul 31, 2008 10:50 pm

Many thanks Dave,

Is there any chance of getting this put into a PDF? I am having issues with the width of the page and get a little frustrated trying to read a document of this complexity.

Steve
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Re: using maths in making bows

#56 Post by Dennis La Varenne » Fri Aug 01, 2008 2:58 am

Dave,

This is very good stuff and shows up some of the shortcomings of using the Hickman equation which I have been using to date. Looseplucker and I have been having lots of emails on the matter, and he with you, I believe. Even though I was using the MoR number, I was having success with what I was making, but in the past few days, after going back to Robert Elmer's Target Archery, I should have been using the elastic limit of the wood, and the importance of doing your own bending tests.

Your diagram showing how the width tapers actually 'curve' inwards to the tip rather than take a straight line in order to hold maximum stress is well shown in Paul Klopsteg's diagram (pp 189 in my edition) in Elmer's book. I have attached a copy of it below.

Anyway, I will have a good read and reread and do some practice at blueprinting and then the real thing.

I second Steven J's proposal on putting the lot into a PDF when you get to the end of the instructional part of the topic.

Dennis La Varenne
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Re: using maths in making bows

#57 Post by yeoman » Fri Aug 01, 2008 6:37 am

I sure can Steve. However it's a bit unusual, as the text on my screen is only as wide as the screen itself...I don't have to scroll sideways at all.

Dennis, what're you doing up at 3am? :shock:

The fact that you've been using the MoR of the timber for the equations shows just how short the Hickman formulas come in adequacy.

Excellent diagram and text there Dennis, should help people out some.

I was thinking of writing this up with more explanations, more diagrams and real world examples, and publishing it as a book. However I don't think the 5-6 copies that would sell would make it a viable concept.

Dave
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Re: using maths in making bows

#58 Post by Dennis La Varenne » Fri Aug 01, 2008 11:46 am

Dave,

I now realise that using the MoR value was wrong. When I went back to my copy of Target Archery, even Elmer uses the Hickman equation using the elastic limit of Yew in his example (I forget which page now.).

The Nagler method, albeit more long-winded takes care of two things which the Hickman does not, and these are -

1. How to maintain optimum stress on the limb by the inward curvature of the limb width toward the tips, and
2. Maintaining optimum safe stress on the limb based on the string angle as you are about to explain which allows for different draw lengths.

Hitherto, I have made the presumption that the Hickman equation was based on the elastic limit at a 28" draw length. I have no reference for that assumption, but it gave reasonably good results just the same.

I allowed for any difference in draw length by adding a % of the S number for draw lengths < 28", and subtracting it for draw lengths > 28" so that maximum stress was arrived at at whatever drawlength desired. This seemed to work reasonably well also.

However, I now realise that using the MoR would have resulted in a bow which would throw lighter than intended and that I achieved my results by allowing a cautionary bit of extra thickness when it came to the actual build - quite apart from bringing my bows very close to breaking (none did thankfully). Perhaps surprisingly they developed very little string follow, certainly less than 20mm. But, this was by accident more than design I now believe even though it worked out well.

As you know, the ^3 relationship in regard to thickness requires only the merest extra thickness to put on a lot of draw weight.

As you also say, this extra thickness allowance is still prudent because wood is not homogenous and there will always be weak spots which will want to give more than stiffer wood on either side and extra thickness needs to be left anyway. Good bowyery skills are not circumvented by maths alone.

The main issue of your thread is that I, and others, will now be able to be able to make more predictably better performing bows by blueprint so long as our bowyery skills do not let us down.

My ambition is still to be able to make the narrow rectangular section selfbows of modest mid-40 lbs draw weight which were used in Elmer's day and which could hold on the gold or even well below it at the 100 yds in the old York round using the under-chin target anchor if the wood was good quality.

They were flat shooters by anyone's standards, even today, and they were made under the influence of Hickman, Klopsteg and Nagler's mathematical efforts in understanding the mechanics of wooden bows. Very few modern trad archers even know of these blokes existence or contribution to archery.

Dennis La Varenne
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Re: using maths in making bows

#59 Post by looseplucker » Fri Aug 01, 2008 12:16 pm

Dave -

Great thread but the piccies dont resolve on my 'puter - I'll email you off forum.

I have followed this pretty assiduously and had a great time with the emails between ourselves and with Dennis (leading to some very interesting calculations based on the material by Elmer and the limit of elasticity). If not for a few chores this weekend and a gig, then I know what I would be doing 8)

Anyhow - this is right up there for me as the most informative and helpful threads on a post I have ever seen - and that is not to take away from the other brilliant stuff you can find here or on other similar forums. You've put an incredible amount of work into this, and your willingness, and Dennis' to share with the rest of the blokes on the forum is fantastic.

For mine, novice though I am, for this site Jeff, may I nominate it for thread of the year?
Are you well informed or is your news limited?

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Re: using maths in making bows

#60 Post by longbow steve » Fri Aug 01, 2008 12:50 pm

My ambition is still to be able to make the narrow rectangular section selfbows of modest mid-40 lbs draw weight which were used in Elmer's day and which could hold on the gold or even well below it at the 100 yds in the old York round using the under-chin target anchor if the wood was good quality.
Hi Dennis, I would love to see this, is there record of arrow weights and timber used in the bow? I have seen selfbows that rival my lam bows but the claims above seem to rival modern recurve technology.
Great thread despite my skepticism of the above claim :D . Steve

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