question: twice as big => twice as strong?

[leif]

Let's say we have made a simple pyramid flatbow 50# @28" and we want to put a bow of similar efficiency on a gastraphetes but with double draw length, would doubling it's size in every direction (2x width,2x thickness,2x length) work with the same efficiency?

the only thing I now : it would be 2x stronger due to width and 8x stronger due to thickness but don't know how the length increase will decrease the strength.

[jeffp51]

my math is a little sketchy, but you are doubling in three dimensions, so I think that increases every thing by the cube (factor of 3) instead of the square (factor of 2)  so you wouldn't need to make a bow twice as long to double strength
 

[leif]

don't forget I want to double the draw length 56", which means to avoid stacking/ breaking needs to be 2x as long

[SLIMBOB]

Roughly, you only need to double the width to double the draw weight.  A 2 inch wide bow of some proportions, that draws 50 lbs at 28 inches....2 of those bows held together would draw 100 lbs at 28 inches.
OK...misread the question....interesting question...and I believe the answer is yes.  just a guess.

[Hawkdancer]

Don't tangle with a dude/damsel who can draw 56" !  I ain't got a clue about your premise!
Sounds a bit theoretical, though!
Hawkdancer

[Del the cat]

My take is to look at it piecemeal.
Call the draw weight W
1. Double the width you double the the draw weight... so that 2W
2. Double the thickness, that gives 8 times the draw weight so it's now 16W
3. If you double the bow length and double the draw length..... that's where I'm stumped 
Does it stay the same poundage? That's my guess.

But the real problem is you can't simply double the thickness of a bow and expect it to bend the same amount.
I'd suggest increasing the thickness by a lesser amount is more realistic
E.G 25% would almost double the draw weight (1.25 cubed is almost 2).
Increasing the thickness by 50% would give 3.3 times the draw weight.
So double the width, increase the thickness by 25% and double the length would be my first try and I guess/reckon that would give 200# if scaled from a 50# bow. That would seem a manageable thing to try.
Del

[Hrothgar]

All math aside, a person could go to the lumberyard and purchase a 12' long oak 2x4 and start from there

[Del the cat]

All math aside, a person could go to the lumberyard and purchase a 12' long oak 2x4 and start from there

Good point... easy enough to do a quick test with with some lumber 6' of 2 x 1 compared with 12' of 4 x 2
Del

[Jim Davis]

+1 for Del

It is said that Thomas Edison once saw one  of his engineers struggling with a math problem and asked him what he was doing. The engineer said he was trying to calculate the volume of a light  bulb. Edison said, Why don't you just fill the glass globe with water, then pour it into a graduated cylinder and read the volume?

The math for this  bow question is beyond most of  us and the answer can be found directly by building a bow to the suggested dimensions.

Or, you can, for variety, argue about how many angels can dance on the head of a pin.

[SLIMBOB]

12. Exactly 12.

[George Tsoukalas]

Thank you. I had never heard of one of these and I am of Greek ancestry.
I watched  you tube video just now.
Very interesting.
Jawge

[leif]

Thanks for the replies they made me want to look more into it.
the question is theoretical.  just imagine tillering a 12' 2x4 with hand tools
I really had to get an answer so i did an experiment.
I used a metal ruler instead of boards
below are the pictures of the set up and the data(if you can understand anything)

The results: doubling length will make the spring 8-9 times weaker, of course there are errors in the experiment but it seems that it follows a cube pattern like  thickness does (but weakening instead of strengthening).

so 2x width 2x thickness 2x length =2x8x1/8=2x strong.
but we also have 2x draw length so it goes 2x draw weight (from hooke's law) but this is a big assumption to make.

i guess the resulting F-D curve would be more aggressive 200# @ 56" (100# @ 28") rather than 100# @ 56"
it would store 8x the energy of the normal bow (about time to double the arrow )

I thought that someone would have made a miniature bow and scale it up to man sized, the same principles would apply here.

[Hrothgar]

Leif, kudos for your perseverance and calculations...something I wouldn't do. I hope, for curiosity's sake, someone will try to build this.

[Del the cat]

@ Leif.
Nice work
I like the combination of experimentation and a bit of physics with a pinch of assumption/approximation
Del

[Badger]

+1 for Del

It is said that Thomas Edison once saw one  of his engineers struggling with a math problem and asked him what he was doing. The engineer said he was trying to calculate the volume of a light  bulb. Edison said, Why don't you just fill the glass globe with water, then pour it into a graduated cylinder and read the volume?

The math for this  bow question is beyond most of  us and the answer can be found directly by building a bow to the suggested dimensions.

Or, you can, for variety, argue about how many angels can dance on the head of a pin.

  Several years ago I had to build a bow for a TV show about 24 ft long I think. It was configured slightly different than a bow but was basically a bow that operated a catapult. I built 3 scale models to calculate the increase and it came out very accurate when I was done. I wish I could remember how it worked out but I don't.