using maths in making bows

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yeoman
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using maths in making bows

#1 Post by yeoman » Mon Aug 20, 2007 10:48 am

Here's a quick explanation on how a narrow handled bow and a longbow can both have equally distributed stress, owing to the fact that stress is related to the radius of curvature and thickness, rather than each factor on its own. This thread is a split from a recent thread.

To lead us into this, I will open with an analogy for those who aren't maths savvy (like me) and who prefer to see comparisons to explain concepts (also like me.).

Think of the bow's limb as the path ridden by a cyclist on a pushbike.
Think of the speed of the cyclist as the thickness of a bow's limb.
Think of the curve the bike can turn as the bend of the bow's limb
Think of any skidding or sliding the bike does as set or breakage if the cyclist falls.
Think of the angle that the cyclist needs to lean over when turning as the amount of stress in the bow's limb.

Now we've got that out of the way, I can explainerise:

A cyclist travelling in a straight line can go as fast as they want (a straight beam can be as thick as it wants)

When the cyclist goes around a corner (the limb bends) the cyclist must lean over into the curve (the limb experiences stress.)

If the cyclist is going very fast (very thick limb) there is a limit of curve that the can turn (a limit to the bend). Usually, this curve is quite large, even if the cyclist leans over a lot.

If the cyclist is travelling slowly (thin limb), the cyclist can travel a much tighter curve for the same amount of lean, or travel the same curve and lean a lot less.

A cyclist who is travelling just above stall speed can almost ride around themselves in circles without much leaning at all. (think of wood shavings)

Picture a particular cyclist travelling at a particular (high) speed and leaning over a certain amount. As the cyclist slows down, the curve which they travel will get tighter and tighter until the bike stops.

If this could be mapped, or seen from above, the path of the bike would likely look like an elliptically tillered bow limb.

If the cyclist was to travel at a constant speed (same thickness) and wanted to lean a consistent degree (equal stress) his path (tiller shape) would be circular.

Hopefully that is clear enough for most of you. I can clarify if need be.

Now onto the mathsy part. For this example I will use a 66" bendy handle longbow and a 62" rigid handle pyramid bow. This assumes that the cross section of both bows is rectangular.

Image

Image

For any given piece of wood, the maximum amount of bend it can take is related to its thickness, the stress and the stiffness. The formula for this is

R/T=E/2S

Where
R=radius of curvature
T=thickness
E=modulus of elasticity
S=stress

As a beam is bent, it is subjected to progressively more and more stress (duh). It is most advantageous for a bow's limb to bending just to the point where set starts to appear. This is known as the elsatic limit. Lower than this and the bow is overbuilt, more than this and the bow experiences cast-robbing set.

As the stiffness of a beam is constant regardless of dimension, as is the desired stress, the formula can be written

C=R/T, where C is a constant found by dividing the stiffness by twice the allowable stress.

Thus, if the design of a bow is prepared, where the radii of curvature are known for many points along the limb, the maximum thickness allowable for that radius can easily be calculated.

NOTE: no rigid handle is ever truly 'rigid'. All wood bow handles bend somewhat, but not always noticeably. Thus, a 'stiff' handle will still have a radius of curvature, though it will be very large. Given below is a table showing the radius of curvature along a pyramid bow, and a bendy handle longbow.

Image

For the handle of a pyramid bow to be stiff, it must increase in thickness, i.e. the radius must increase. This cannot be an abrupt change, as wood will break at the junction. Rather, a smooth transition is required. If this ratio of thickness, radius and C is maintained throughout the handle and is not added to, i.e. made thicker, then the entire bow will experience even stress. Notice how for the limb, the radius of curvature is the same, due to the circular tiller. This means that the thickness will remain the same throughout the limb. Most bowyers, however, will overbuild a handle, meaning it will experience less stress than other parts of the limb. That's fine, there's nothing wrong with that, but this 'article' is supposed to be showing how equal stress is achieved

To find the width of the limb, a fancy schmancy formula is used using all the values thus discussed, and more. It involves knowing the distance from the string in a perpendicular line to the bow limb, the tension in the string, desired draw weight and so on. Of interesting note is that if the bow is required to be twice the draw weight, the whole bow will be twice as wide, including the handle.

The tables above also show the resulting widths for the limbs got from the formulas. Both bows have the same draw length and weight, and are made from the same theoretical piece of wood, which is actually close to a real test sample I did. The bendy handle bow, owing to its increased radius of curvature toward the grip, does not need to be widened as much as the pyramid bow to experience the same stress. Notice how at the very middle of both bows the dimensions are nearly the same, but the bendy handle bow is much narrower in the inner half of the limb. Looking at the two tiller shapes, I'd say that the pyramid, rigid handle bow was the 'C' shape, and the bendy handle bow the 'D' shape.

Image

That's a pretty rushed explanation; I hope it's clear to at least some of you out there. If it's like mud to everyone, I'll try again be taking more time to explainify myself.

Cheers,

Dave



PS: actual wood smaple can vary greatly in their mechanical properties to those published in the books. Other smaples may be exactly the same.

The MoE is quite simple to calculate. So too is the MoR, which uses most of the same data as the stiffness test. If you (whoever has read this far)
want, I will write about how to calculate the MoR and elastic limit of a sample of wood, using two different methods.
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#2 Post by GrahameA » Mon Aug 20, 2007 11:09 am

Dave

Great post and I we had a rep' sytem I would have given you some.
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#3 Post by yeoman » Mon Aug 20, 2007 11:12 am

rep?
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my head hurts!

#4 Post by wysper » Mon Aug 20, 2007 11:19 am

That was great.
I understood most of it... would have to read it again to make sure LOL

Rep is Reputation. For each good post from someone you can give them rep points, kind of fun, and if used correctly you can pretty quickly see who puts thought into their posts and gives good info..

couse.. you can give rep to funny sarcastic bu$$ers too.. so the system can have some flaws :lol:

but you would have got some for sure for the effort gone into that post.

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#5 Post by Steven J » Mon Aug 20, 2007 4:01 pm

Thanks Dave,

I appreciate your response to my earlier request. I will need to take some time to digest this and will get back to this post shortly.

Steve

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#6 Post by Dennis La Varenne » Tue Aug 21, 2007 12:48 am

Yeoman,

Now we're talking!!! I will have to read it a few times for it to sink in properly, but it will.

I have read up on MoE being stress/strain in a wood mechanics site, but I need to find a simple way to work it out on given samples like you do. Same goes for MoR.

If you have the sums for MoE and MoR, can you put them up too pleeeease?

I have been using Hickman's formula to lay out a couple of bows already based on known MoR (elastic limit). You have to do the sums for every increment along the bow, but it reveals some interesting width:thickness ratios - much less than I would have thought. I continued the calculations into the flares and handle of a rigid handle flatbow and the figures made sense there too.

I think the Hickman formula is usable for any kind of rectangular section bow, and, if an allowance could be found which allowed for a semicircular belly, it could be applied to ELBs as well. I thought of doing it for an ELB of deep rectangular section and guessing an allowance for the belly camber (not very reliable) or the other idea is to lay out an ELB with a width of perhaps 0.75% of the intended width, because the outer part of the limb width doesn't do much work really. The load is concentrated toward the highest point of the back and belly camber as you know.

I think that a bit of experimenting by varying the numbers for the width could result in something reliable.

Yum, yum!

Dennis La Varenne
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#7 Post by yeoman » Wed Aug 22, 2007 11:32 am

Hi all,

glad I could be of some assistance to those seeking to enahnce their mathsyness.

Dennis, within the next couple of days I will post the formulas for calculating stuff.

A quick note on Hickman's method:

While his formulas are pretty much correct, he does have one major flaw in his computations. He defines the bending moment as the forcexdistance from the tip, which is actually not the case.

Rather, the correct way to calculate the bending moment is the distance from the bow segment to the string along a line which is perpindicular to the string, multiplied by the tension in the string. This I know is terribly complicated to understand, but soon I will upload pics of what I mean, which will make it infinitely easier to understand.

Calculating stresses in an ELB is easy. I will post that too ASAP.

Must dash.

Cheers again,

Dave
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#8 Post by Dennis La Varenne » Wed Aug 22, 2007 5:39 pm

Dave,

Actually, that does make sense and shouldn't be too difficult to calculate. Up to date, I have not put up Hickman's formula (other than referring to it) so it didn't begin to look like I was contradicting you.

The two are ways of arriving at a similar end, albeit one perhaps a bit more accurate than the other. Either should yield a decent working bow if reasonable care is taken in the making of the bow.

You practical experience in building your bows to formula supports the contention I made in my essay that new/untried woods can be tested by building scale models of bows because the forces on the limbs are the same as for a full scale bow. Once the maths are understood, it becomes obvious. That is the beauty of it.

Anyway, thanks for the PM, too.

Dennis La Varenne
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#9 Post by yeoman » Thu Aug 23, 2007 1:22 pm

Okay, it’s nearly time for us to do some bend testing, but first, let me explainerise what I said in my last post.

In the 30’s, Hickman began experimenting and theorising about the formulas and mathematics of bow limb engineering. One component of these calculations is the bending moment, or the amount of bending force acting on any particular part of the bow limb.

He calculated the bending moment to be the distance from the tip to the section, multiplied by the force applied. Therefore, if the section under question was .2m from the tip with a force of 10 newtons, the bending moment would have come out as 2Nm.

However, in later years other mathematicians and engineers took up the challenge of bow maths. Nagler was one of them, and he found that the bending moment was actually the product of something else. NOTE: Hickmans calculations are correct only is the defection is very small relative to the beam’s length. A bow, however, bands quite far.

Below is a picture of a Hickman bending moment and a Nagler bending moment. Nagler’s method is a little more difficult to measure and calculate, and is greatly aided by either a large scale drawing of the bow plan, or a CAD program. However more difficult it is, it is much more accurate, which can only be considered a Good Thing. I use deltaCAD, which is free, and works extremely well for what I need.

Image

Now we can get onto the MoR and bending stress calculation, almost.

For those not in the know, here are some definitions:

MoE (modulus of elasticity): this is the measure of how stiff a beam is, i.e.: how hard it is to cause a deflection.

MoR (modulus of rupture): this is the bending stress at which the beam fails, whether it be as simple as a splinter raising on the back or a spectacular explosion of splinters.

Elastic limit: This is the bending stress at which point set begins to emerge.

Strain: this is a figure, given as either a percentage or a decimal, and denotes how much elongation or compression the surface of a beam experiences when under a load. A bent bow’s back will stretch, its belly will compress. More on strain later.

Stress: A term used to define the pressure of compression or elongation in a bent beam.

So now we may finally get onto doing bend tests.

To do a bend test, you need a beam which is uniform in cross section, and a decent length, with the same grain orientation that will be used in the bow. I try to choose boards which are a little longer than I will need for the bow so I can cut a bit off the end and do a bend test.

Making the beam the right thickness for its length is important. If your beam is 19mm thick but only 30cm long, applying enough force is going to be difficult. Likewise if the beam is 5mm thick and 2m long, it will sag under its own weight before you apply any force.

I generally use a beam about 20mm wide, 5-10mm thick, and 400mm long.

You will need two supports which will not budge, a bucket or something similar, some dumbbell weights or sand/water with a set of scales.

See below for a picture of the setup. Also present is another bit of wood to which I’ve taped a piece of paper to measure deflection and set.

Image

Draw yourself up a table with three columns: load, deflection and set. Be consistent in your measurements so you don’t confuse yourself later.

Mark the exact centre of the beam, and hang your bucket from here. Starting with a small amount of force, say 2kg, load it into the bucket, and measure the deflection of the beam at the centre. It is better to load and unload a few times to exercise the wood before measuring the deflection.

Image

Take the weight off and observe to see if there is any set (I hope not at this stage!). Again, wait a few moments before seriously looking for set, as the beam will ‘creep’ back into place.

Then, with increments of maybe 1kg at a time, repeat the above process until set develops. This set will likely be only a couple of mm. Mark the set on the paper IN A DIFFERENT COLOUR, as well as recording it on the table which you would conscientiously have been filling out all the while.

Don’t stop there though. Just because you know the elastic limit, doesn’t mean you are finished. I keep going till the beam breaks, whether splinter, severe chrysals, or breakage.

By the end of the process you should have a nice long table of data showing the force, deflection and set of the bending beam.

Now it’s time to come inside and use these figures and some formulas to get some more numbers.

This process will be made easier if you do it in an excel sheet.

You first need to calculate what’s known as the Moment of Inertia (there are lots of Moments in engineering, and should not be confused with the ‘moment’ that you’re told to wait in the car while a parent or partner nips into a shop for four hours). The MoI is a measure of the cross section if the beam, and is calculated thus:

MoI = (b x t^3 ) / 12 where:

b - width of the section
t - thickness of the section

You only need to calculate this once, as your beam is uniform in cross section.

Let’s say I have a beam 25mm wide x 10mm thick. The MoI would be:

MoI = (25*10^3)/12
= 25*1000)/12
=2083.3333333333333333forever.

The next thing to calculate is the Elastic Modulus, and is another cross sectional figure. It should not be confused with Modulus of Elasticity however.

The formula is:

Zx=(b*t^2)/6

So, with the same dimensions given above, the Zx would be:

Zx = (25*10^2)/6
= 25*100/6
= 416.666666666666666666666666

Next, you need to find the bending Moment at each load applied to the beam. The formula for this is:

Bending moment = (F*L)/4

Where F= force in newtons and L=length of your beam.

With these three formulas in hand, you can apply them to your own measurements and arms yourselves with those two important numbers.

By using an excel spreadsheet, both the bending stress and stiffness can be calculated at once. Here are the formulas for MoE and bending stress (remember, MoR is a bending stress):

MoE=(F*L^3)/(48*d*MoI)

Bending Stress = Bending Moment/Zx

Where L=length of beam
d=deflection

I always do my bend tests with linear distance measurements in mm, and force in Newtons. This ensures that the MoE figure and bending stress are given as megapascals. Using inches and pounds will give PSI.

For practise, here are some results from a bend test for you to do some calculations with:

Width: 31.2mm
Thickness: 19mm
Length: 1208mm

Load-------deflection------set
60-----------18------------0
65-----------23------------0
70-----------64------------0
75-----------71------------5

I’ll post the answers in the next post to curb the temptation of cheating :-P


I’ve now done 4 pages of typing in a word doc, si I should probably leave it there for a couple of days before putting more maths in the thread.

Enjoy,

Dave
Last edited by yeoman on Wed Aug 06, 2008 6:59 pm, edited 1 time in total.
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#10 Post by yeoman » Fri Aug 24, 2007 1:12 pm

What, is no one interested anymore? :D
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#11 Post by Dennis La Varenne » Fri Aug 24, 2007 1:28 pm

Dave,

I'm doing a few sums still. I'm still interested.

Another couple of questions -

How does one convert buckets of weight to Newtons?

Are you saying that Bending Stress is the same as MoR?


Dennis La Varenne
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#12 Post by GrahameA » Fri Aug 24, 2007 1:45 pm

Dennis

Operating under earths gravity

1kg = 9.8 Newtons
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#13 Post by Dennis La Varenne » Sat Aug 25, 2007 8:46 pm

Dave and Grahame,

Even if this thread doesn't get a lot of visitors, I really believe that it is one of the fundamentals in bowmaking. All bows behave according to this kind of mathematics even if most of the Ozbow members do not understand it necessarily.

It should go up on the Traditional Crafts section perhaps. It was worked out toward the end of the wood bow era before the Second World War and well before the compound age by 30+ years.

What is important is that Dave is teaching the method to making best performing bows of any particular design by understanding the stresses involved and working within those limits - even elliptical section and D-section bows.

Those of us who are beginning to understand the maths can help anyone else who can do the first part of Dave's testing using the bucket and making accurate observations.

This really is seminal stuff.

And Grahame, thanks for the conversion. I found this out on Wikipedia. It is a bit obvious really once you are shown it.

Anyway, Dave has give me some homework and I haven't done it yet. Life has a habit of interrupting the good things, and I don't want this thread to fizzle out.

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.

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

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#14 Post by yeoman » Mon Aug 27, 2007 11:35 am

Thanks for the encouraging words.

MoR is the same a bending stress. It is simply the bending stress that was experienced by the beam at the point where it failed. It is difficult to measure if the beam suddenly explodes, far easier if the beam simply chrysals or splinters.

Dave
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#15 Post by Dennis La Varenne » Mon Aug 27, 2007 1:47 pm

Dave et al,

I did some looking up of Modulus of Rupture on a few sites (there are lots of them) and it seems that the MoR figure is taken at that point where the graph of bending load to strain begins to flatten out - where large scale plastic deformation begins to occur rather than at the point of breakage if I understand things correctly.

Does this sound right to you?

I have been away for the weekend, Dave, so I still haven't done my homework. Mea culpa.

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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#16 Post by greybeard » Wed Aug 29, 2007 6:10 pm

Hi Dave,

Your thread is indeed interesting reading.
Could the data you obtain from bend tests be used to key in a draw weight at a given draw length if sufficient statistics have been kept on previous bows.

Daryl.
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#17 Post by Coach » Wed Aug 29, 2007 8:06 pm

WOW ,,, Looking at all this Maths is enough to put one off making a bow ,, I should have paid more attention to my Maths teacher :shock:

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#18 Post by Poppy » Wed Aug 29, 2007 8:22 pm

Just finished reading this post, My head hurts :wink:

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#19 Post by Hood » Wed Aug 29, 2007 8:24 pm

Poppy wrote:Just finished reading this post, My head hurts :wink:
Well stop head butting the wall :wink: :lol:

Mine hurt as well so your not alone. :lol:
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#20 Post by greybeard » Wed Aug 29, 2007 8:41 pm

Hi Coach,

All the number crunching does make it look like a daunting task. I think Dave is doing a great job qualifying and or quantifying the stresses experienced in a wooden bow (from a milled board).
Although it may be a more exacting process I don't think I will ever replace the joy of crafting a bow from a bush billet (with all it's inconsisentices) using the tiller board, spokeshave, sand paper and a gut feeling.

Daryl.
"And you must not stick for a groat or twelvepence more than another man would give, if it be a good bow.
For a good bow twice paid for, is better than an ill bow once broken.
[Ascham]

“If a cluttered desk is a sign of a cluttered mind, of what, then, is an empty desk a sign?” [Einstein]

I am old enough to make my own decisions....Just not young enough to remember what I decided!....

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#21 Post by GrahameA » Thu Aug 30, 2007 7:18 am

Good Morning

Personally, the maths is not that heavy. All you need do is plug your numbers into the formula. The hard stuff is working it all out in the first place.

Dave, I reckon you have done a brilliant job and would encourage you to do more.

Darryl, you are right about the hand crafting. Selfbow making has a lot of 'art' in it but the maths give you some idea what you can expect. And provides an excellent excuse for when it does not work out.
Last edited by GrahameA on Thu Aug 30, 2007 12:01 pm, edited 1 time in total.
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#22 Post by Steven J » Thu Aug 30, 2007 11:37 am

As others have said,

Dave, we are all finding this most interesting and given more time will also find it useful. I suppose that many of the readers here also have families as I do and find getting away to the shed for lengths of time to do all these tests is a little infrequent.

I encourage those that do not feel confident with maths to do their best in understanding this an seek help if need be. As I also encourage my students at school, there is no problem with struggling to understand a concept - we must learn to enjoy the struggle. Anything that has taken effort is always enjoyed and appreciated more.

Thanks again Dave. You have invested a lot of time to understanding this and then spent more time in reproducing it for us here.

Steve

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#23 Post by yeoman » Fri Aug 31, 2007 11:35 am

Cheers all,

I'm trying to ake it as comprehensible as possible. I barely scraped through maths in highschool myself, believe it or not.

Daryl, calculating figures from previous bows is a pretty difficult task, for so many reasons. I may be misunderstanding your question though.

I agree that taking a split stave and crafting a well made bow is an exhilirating experience. Doing the same with a milled board is similarly so.

However, another form of exhiliration is doing bend tests, seeing working properties emerge, calculating figures, machining out a lump of wood and have it perform to within a couple of percent of the calculated figure. It's amazing, really. Usually maths hates me, and I it.

Dennis,

Yes, that diagram can be explained as being the MoR, though not all woods will give a graph like that. Mostl;y you will get that with steel beams or reo concrete and stuff like that.

More maths will come next week.

Dave
https://www.instagram.com/armworks_australia/

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#24 Post by Dennis La Varenne » Wed Sep 05, 2007 11:21 am

Dave,

I have worked through the problems you set and have a set of figures for each of them but they don't look right.

Later, for the benefit of everyone, I will post them here along with my workings so everyone can see how I went about it and you can show where I went wrong with them. I think that will be of help to everyone.

I want to understand the maths involved because it explains what is going on, and apart from faults in the wood, allows a fair degree of predictability of outcome relative to design. It also allows for optimum designs to be done for a given wood sample - optimum meaning obtaining the best performance from the sample without overstressing it.

Regards all,

Dennis La Varenne
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.

Dennis La Varenne
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#25 Post by Dennis La Varenne » Wed Sep 05, 2007 2:25 pm

Dave,
Here are my workings of the various formulae you provided. the wood sample is -
31.2mm wide;
19mm thick;
1208mm long.

The supplied loading figures being –
LOAD DEFLECTION SET
60 N 18mm 0mm
65 23 0
70 64 0
75 71 5

Modulus of Inertia –
MoI = (b.t^3)/12

= 31.2x19x19x19 / 12

= 214 000.8 / 12

= 17 833.4 (?units)


Elastic Modulus (Zx) –
Zx = (b.t^2)/6

= 31.2x19x19 / 6

= 11 263.2 / 6

= 1 877.2 (?units)


Bending Moment -
BM = (F.d)/4

= 75x71 (at (d) from table) / 4

= 5325 / 4

= 1 331.25 (?units)


Modulus of Elasticity -
MoE = (F.L^3)/(48.d.MoI)

= 75x1208x1208x1208 / 48x71x17 833.4

= 132 209 318 400 / 60 776 227.2

= 2 175.3459 (?units)

Bending Stress (BS) –
BS=BM/Zx

= 1 331.25 / 1 877.2

= 0.709 167 909 (?units)

I don't understand whether or not I was supposed to do the sums for each of the rows of figures of the load data or not.

I don't know what uints each of the equations gives. The load/deflection table did not tab into colums as I intended. Anyway, everone can refer to your original table above.

Dennis La Varenne
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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yeoman
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#26 Post by yeoman » Wed Sep 05, 2007 4:23 pm

Hi Dennis,

I don't have my working with me right now, but provisionally, I think they look OK.

I'll post my results tomorrow, and some explanation of the units.

Sorry to rush but I have to dash.
You are dead on with your comment about predictability and getting maximum returns from the wood. It's exactly why Nagler, Hickman and Klopsteg did it in the first place.


Dave
https://www.instagram.com/armworks_australia/

Bow making courses, knife making courses, armour making courses and more:
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yeoman
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#27 Post by yeoman » Thu Sep 06, 2007 3:11 pm

OK, here's the next official installment:

Units of measure:

MoI: As the formula calculates a figure based on width and thickness, the unit will be something to do with the same unit of distance measurement. As the formula (in part) is bxt^3, and since we know that when multiplying pronumerals the integers are added, and the unit we measured the distance in is mm, the unit will be mm^4. This doesn't actually matter, but is nice to have in the back of the mind.

MoE: This is a measure of stiffness which is calculated by physical dimension, displacement and force, among other things. In the end, the easiest way to represent this figure is with a unit of pressure, the Pascal. Or rather, since the numbers are so large, we use Mega Pascals, or mPa.

MoR: This is also measured in mPa, but will usually be 3 zeros less than the stiffness.

Bending Moment: as this is calculated from a force and a distance, the unit is Nmm, like Nm which is used to measure torque in an engine drive shaft.

Now to check your calculations Dennis:

MoI: completely correct, a big star, except for the fact that MoI is 'moment' of inertia, not 'modulus'.

Zx: again, spot on.

After this, things go a little awry, and I know what the problem was. When I originally wrote the table of data, the 'load' was in KG, with the intention being that people get to practise changing from KG to N. It may be a bit confusing, but I refer to the load as the weight that is applied (I load up the beam with kg), and the force (N) is what the weight does.

Sorry for that, I should have written the units in.

However, if the units were Newtons from the start, your calculations would have been spot on. Minus 10 points for me, plus 5 for you for compensation.

Again, sorry for that.

For every deflection and load increase, the bending moment will change, which means that has to be calculated at each interval as well as the MoE and bending stress. The stiffness should not change much if at all throughout the bending test.

So there you go. If I can iron out my wrinkles, and when everyone is ready, we'll move onto the next step. Any questions class?

Very well, off to play lunch. :wink:

Dave
https://www.instagram.com/armworks_australia/

Bow making courses, knife making courses, armour making courses and more:
http://www.tharwavalleyforge.com/

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Dennis La Varenne
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#28 Post by Dennis La Varenne » Mon Sep 10, 2007 8:13 pm

Dave,

Many thanks for correcting my homework. Now I just have to look up 'integers' and 'pronumerals' in my elementary maths books.

At least I am on the right track. Grahame is right. The maths is not very difficult really - just substitution of the numbers into the formulae.

However, did you get your PM about some wood I got hold of???

Dennis La Varenne
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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#29 Post by yeoman » Tue Sep 11, 2007 12:43 pm

A pronumeral is a letter that stands for a value which is substituted in, and as for an integer, I made a mistake there. I should have said INDICE, which is the value of a higher multiplicative power. For example, b^2 has an indice of 2, G^4 has an indice of 4. Sorry about that.

No, I don't think I did get the message. Send again please?

Cheers,

Dave
https://www.instagram.com/armworks_australia/

Bow making courses, knife making courses, armour making courses and more:
http://www.tharwavalleyforge.com/

Articles to start making bows:
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Dennis La Varenne
Posts: 1776
Joined: Sun Sep 07, 2003 10:56 pm
Location: Tocumwal, NSW. Australia

#30 Post by Dennis La Varenne » Thu Nov 01, 2007 10:25 pm

Hello all,

I want to let you know that I have just finished another 3Rivers bamboo backed hickory flatbow blank to intended draw weight and almost perfect tiller without it touching a tiller board.

I used maths to blueprint it before I started and followed the blueprints and it came out almost on the money.

I made the correct allowance for length of draw this time so that the MoR number (132 for Hickory - the bamboo backing was ignored) was calculated to be reached at or just past my 26 inch draw.

The astonishing thing about blueprinting bows I have found with these two is that the tiller is so good right after taking the stave down to the pencil line that there is only the barest minimum of scraping to do to get the tiller right.

My tillering technique these days is to even up the belly string distances on either limb after I have low-braced it. When they are good, I increase the brace height and check it again, stretch it to half draw several times then shoot it from half draw then full draw. Both of them have not needed anything much at all by way of final tillering so far and have not changed at all despite getting a flogging.

This second bow has developed 5/8" string follow from a straight laid blank which increases to 1 1/4" after shooting and which I push out by reverse bending it and holding at about 3" of reflex for about 30 seconds. I have always done this with my wood bows with never a hint of problems and I thoroughly recommend it (despite the naysayers) because it seems to help keep set to a minimum compared to when I did not do it earlier in my bowmaking.

This little 50lb x 70" bow is the quickest shooting 50lb wood bow I have ever made by a long chalk and the little 40lb x 68" is not that far behind with suitably lighter arrows despite its built in Howard Hill 1" deflexed limbs. (Heaven only knows what possessed the man to build bows like that if it is true. There is not a scrap of measurable performance merit in the design despite the hype.)

Having the MoR number reached at full draw or just past it means the wood is working to very near its elastic limit and obviously having a beneficial effect on cast which I have not had before.

Its only problem is slight rotational twisting of the both tips contrary to each other because I foolishly trusted the maker's layout and alignment. But they are not worsening, so I am leaving well enough alone.

I have dashed on a coat of Tung oil so I can shoot it at the Morwell do on the weekend, and I have named it Arcus mathematicus II.

Dennis La Varenne
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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