Can I build a bow of X length, Y width, and Z draw length at 50#?

[LittleBen]

We've all probably seen this question asked roughly 1000 times, so in the spirit of sharing I'm going to divulge my method (read secret) in answering this question.

All bows, experience the highest stress at the limb surfaces (i.e. back and belly).
Therefore:
(Assumption #1) most if not all failures are due to overstressing the absolute outer surface first.

(assumption #2) pretend for a minute that all the energy is stored at the surface (back and belly).

(Assumption #3) All of the limb does an equal amount of work or the proportion of the limb which does the work is the same between the two designs being compared.

If these assumptions are true then limb surface area is proportional to maximum energy storage at failure of the bow (basically).


So what does it all mean?

It means if you've built or know of a hickory board bow thats 64" long with 8" handle (i.e. 56" of bending limb) and the limb is 2" wide at the fades and 1" wide at the tips (for easy math), then the limb surface area is 56in*((2+1)/2))in = 56in*1.5in = 84in^2.
Lets say that bow draws 50# @ 28" with an 8" brace height and stores approximately 50#/2 * (28-8)in = 1000#*in of energy. so for every 100#*in of energy you need 8.4in^2 of limb surface area.

So then you ask the question like this:
How wide does a 60" hickory bow with a 10" handle section have to be to draw the same weight at 26"?

50#/2 * (26-8)in = 900#*in of energy (approximately); therefore 9*8.4in^2 of limb area = 75.6in^2 of limb area.

Lets then assume the tips will also be 1" wide ... how wide do the fades need to be?

60"-10" = 50" bending limb,

75.6in^2/50in = 1.5inch. so the average limb width needs to be 1.5" approximately. For a pyrimidal limb that would be 2" fades tapering to 1" tips. or nearly 3" wide fades tapering to very thin (say 1/4") tips.


So thats basically it.

Ao another simple example If you cut limb length in half, you have to double the width assuming the same profile and so on an so forth.

Making it simpler, if you wanna build yourself another bow at the same draw length and draw weight as a bow you already have of the same material, and the tip width and limb profile is the same then you can simplify this all to:

Limb length 1 / limb length 2 = limb width 2 / limb width 1.

or (limblength1/limblength2)*limbwidth1 = limbwidth2

so for two bows with the same 8" handle, the first 68" long and 1" wide, and the second 58" long and unknown width:

(60"/50")*1" = limbwidth 2 = 1.2"
so the shorter bow needs to be 1.2" wide.


Hope that makes sense and someone can make use of it. This is the method I use all the time, and I successfully adjusted the width for a 58" bows based on a 68" bow just this month. It works.

There's already a mass formula so a part of me dreams this will become the surface area model or something like that.

Cheers,

Ben

[KellyG]

Ok I am a little scared I understand the math. (could not replicate it or remember the formula but I follow it.) But it give me a headache.

[LittleBen]

Sometimes I go a little crazy and my mind just explodes onto the page. Maybe I'll edit that to be more clear and simple later. Glad it makes sense to someone else though.

I started thinking about it because you get so many opinions on proper dimensions, I wanted a way to compare them.

[inksoup]

that's so cool Ben...
now i know how to calculate my bows... thank you.

but this is just for hickory i think. and it is related to its MOR and MOE values.

and i think we need to find a way to include those values in to this calculation.

here is some explanation:  http://www.wood-database.com/wood-articles/bow-woods/

so

( MOR / MOE ) * 1000 = gives us some human readable value. if it is something over 10.xx then wood it good to go.

ok how to include this into above calculation.
well i will work on this.

best.

[4runr]

I hate school!

[Roy from Pa]

Can I build a bow of X length, Y width, and Z draw length at 50#?

NO....  

[LittleBen]

Ink, those are not real numbers for hickory bows just an example of how to do it. You can use this to compare any two bows of the same material. I use it for hickory backed ipe tri-laminate bows. You can get into MOE and MOR, but at that point we're getting into much more serious calculation and probably super tiller is a good idea since it already has solid formulas built in.

If I want to compare two different woods I generally just use their SG (density) to do so. But MOE and MOR are useful too obviously.

[LittleBen]

Haha Roy you Scrooge.

[canopyboy]

Ben, you've got me thinking...

I will say that I think your method only works with bows of roughly the same design and the same thickness (and profile thereof.)

I typically use a beam theory method that takes into account thickness (bow weight goes up by width to the first power, but thickness to the third power.) But I'm still limited to bows of similar style and materials to ones I've made or have data on.

Now, imagine you combine a strain energy approach with a beam theory basis? You could start with surface strain energy where most of the work is being done, but in reality it would be more accurate if you integrated through the neutral axis.

Next, consider that each material (wood species, glass, carbon, sinew, etc) has it's own capacity to absorb strain energy before permanent deformation of failure. Each material also has it's own rate of strain in response to tension or compression (or shear.) If we could roll that up we'd be able to size a bow between wood species or look at the difference of changing glass thickness or core materials.

Just something to think about...

[LittleBen]

Sounds like grad school all over again.

I think you're right on with the limitations of my method. Definately doesn't work well for vastly different designs, and doesn't tell you proper thickness, really it just gives some idea whether your bow will explode or not, and a rough approximation of what width to use to prevent explosion.

[Roy from Pa]

Must be something strange in the water down there in Virginia..  

[razorback]

Roy, the question was can you build one, not can you draw one.  

[canopyboy]

Originally posted by Roy from Pa:
Must be something strange in the water down there in Virginia..      

Yup, barley and hops! I understand there may even be some in that watery stuff you drink.

No other way to get through engineering school. Most of the stuff they try to teach you can only make sense when drunk. Try it, then re-read Ben's post and I'll bet you'll think it makes perfect sense. If it doesn't, move up to stronger beer...

   

[LittleBen]

Absolutely hilarious

[inksoup]

oh hell yes... i have got my engineering degree by the help of lots of beers and raki.

there should be a correlation with lamination and power that you gain with it. problem is how to formulate it. probably there is a way. maybe it is to with multiplying or adding the mor an moe values and including the strength of glues used.  

maybe we can find it if we have some values on...

[halfseminole]

Strangely enough, I've done flow prediction analysis of untested designs and never had a drink.  

I get the formulas too.  I think they're most applicable to wood bows, not fiberglass.  MOE makes a real difference there, just as it won't predict sinew backing or horn composites.  But enough of us make wood bows that this is supremely useful.  I know I could have used this in the sixteen pages of notes I made for the bow I need to post sometime soon.

[mikkekeswick]

To be honest with wood bows I've found it's easier to make them a little wide at rough out stage then use the mass principle to tiller the width to required drawlenght/weight and strain.
I think working out a width/length ratio and reducing straight to that width is not a good idea.
Afterall woods properties vary so much from one sample to the next that only bows made from the same board would be comparable.

[LittleBen]

Good point mike. Hadn't considered that.

Now that I think about it , since I've been buying pretty large boards of ipe, I've made a lot of bows from that single board and we're yet to see what happens with different boards.