Shape of "constrained" rod...

backboiler

... or what shape is a simple bow (the "bow and arrow" kind)?

Say I have a straight length of timber and I push the ends together or tie a string between them so that the straight length curves to make a bow, what is this shape? Some conic section?
How would the equation of that line be derived?

LeixlipRed

If you look at a bow in 2d only then its parabolic in shape. In 3-d if you bend a piece of timber it's still a cuboid just curved (there possibly might be a name for such a shape?). If you could give a description of what you're attempting to perhaps we can help more.

backboiler

The 2-D view is what I'm on about. I have no application in mind for this really, just a thought I pulled out of the wind*.
Can you show why you say it's a parabola? I'm talking about down and dirty with force vectors etc., with a supporting "real-world" view, if appropriate. Is the shape dependent on the rigidity of the material or is it defined completely by the lengths of the "rod" and "string" (must try it with a length of rebar steel later to see).

* Not strictly true. I needed a quick to make and easily reproducible form to lay out a curve for a building project. The original plan was to use an arc but the proposed centre of the arc was not accessible so I couldn't draw it directly (yes I could have mirrored it but I didn't). I noticed that the bow I made did not follow an arc but some other shape. Now I'm just curious to know what it is.

LeixlipRed

Just google "parabola" and you'll see what I mean. When you say arc do you mean semi-circular?

backboiler

LeixlipRed wrote: »

Just google "parabola" and you'll see what I mean. When you say arc do you mean semi-circular?

Arc: yeah part of a circle, about 36 degrees, i think it was. Will have a look.

backboiler

LeixlipRed wrote: »

Just google "parabola" and you'll see what I mean.

Don't see anything there but a link from the Wikipedia page to Catenary has a derivation of the kind I'm looking for. That's for a hanging chain under evenly distributed force field. The case I'm talking about is more like two point loads being resisted by the rigid material (isn't it?).

LeixlipRed

I know nothing about physics so you're asking the wrong person. You might get more joy on the engineering forum possibly? I can move the thread there if you want.

backboiler

LeixlipRed wrote: »

You might get more joy on the engineering forum possibly? I can move the thread there if you want.

Well maybe, I don't mind either way. If no-one else has a comment then moving makes sense.

mr potato head

The bow acts like a simply loaded beam.
You could use the Euler-Bernoulli beam theory to get an general curve for the Bow.

There is a bit of a difference with the bow example as the loading comes from a string, but this will give you a curve

I don't know how much maths/mechanics you've done but there are a good few university mechanics tutorials that you might find online to see worked examples.

There is a good site for finding these sorts of equations called efunda.com but unfortunitly its a subscription site. You might find other calculators online that are free.

Here's an example of a calculator from it, 10mm square wood beam, 1m long cantilever with 1N load at the end.



This is just looking at one side of the beam but is mirrored on the other side if the beam is a constant section held in the middle.

Hope this helps, post again if you have questions...

backboiler

Thanks, I'll have a look when I have a bit more time. I wonder if the fact that the load is a variable quantity provided by the rod itself and the result (i.e. the "shortening" in X-axis direction - defined by the string's length) is fixed would mean thavt a different approach is needed.

mr potato head wrote: »

I don't know how much maths/mechanics you've done but there are a good few university mechanics tutorials that you might find online to see worked examples.

Mechanics, not a huge amount but click on the attachment link below for an enemy, once conquered, I've spent a decade trying to forget. :P
It's been a while, Mr. Anton.

coolwings

It is a beam under load.
But the load from the bowstring is not at 90 degrees to the beam mid point, that's why you didn't get an arc.

If the beam was supported at it's midpoint, and two strings are attached one at each end, each with a pull at 90 degrees to the wood/bow, then you will get an arc, assuming the variance of natural materials, choice of straight grain wood etc.

It should be good enough for description of an arc of a circle.

mr potato head

Someone else might know a better equation for a large displacement of beams.

Somewhere else you could look if you have a chance are in some of the papers below, i used some of them in a project about 4/5 years ago but can't remember everything in them... There are some good papers on bow physics, most are on this page.

B.W. Kooi has written a good few papers in the 80s. C.N Hickman and P.E Klopsteg wrote early papers in the 40s onwards.
There are two authors W.C. Marlow and B.G. Shuster who published in the American journal of Physics (haven’t ever been able to find the B.G. Shuster paper)

From memory most of their models deal with the drawing/firing of the bow, but I think there are bits on the unstrung and stringing.

coolwings

backboiler wrote: »

... I needed a quick to make and easily reproducible form to lay out a curve for a building project. ...

Hammer in a peg for support in mid-beam. Locate beam.
Measure back a set distance.
Move parallel to the beam to left and right and hammer in two pegs.
Tie pegs to beam ends with equal length cables, pulling at 90 degrees to the beam and compressing the beam into an arc shaped bow.

Should be an arc, and should be reproducable.