Finding a Solution to a Vexing Problem

[yokebutt]

Hello everyone,

I've been searching for a solution to a rather vexing problem for a while now, (vexing because I'm just a simple-minded redneck boatbuilder) and since it looks like a mathematical problem to me, perhaps someone here could help me make sense of it. (Wich unfortunately requires making sense of my writing, most likely the greater obstacle)

A little background; I'm investigating a novel technique to manufacture composite tubing, specifically, using a hollow braid under tension as a method of compacting a composite laminate during it's cure. But what I can't figure out is how the tension of the braid correlates to the amount of "squeeze" exerted on the laminate.

A hollow braid has an even number of strands, half of which forms a right-hand helix, and the other half forms a left-hand helix, the righties and lefties are also interlaced to keep the braid from falling apart. It's the same thing as the classic "chinese finger-trap" i.e, when you stretch it out, its diameter contracts, and when you push it together the diameter expands.

So, by taking a length of the braid and slipping a mandrel (a straight piece of aluminium tubing in this case) into it and pulling on its ends, the braid should cinch down on the mandrel. Since the force of the cinching action is expressed as pressure per unit area, it seems like I'll need the total area of the mandrel in the calculation. But beyond that I'm stuck, do I then base it on the curvature (radius?) the strands describe going through the helix? Or do I try to figure it out based on the angle between mandrel and strands somehow?

Help, I'm way out of my comfort-zone here!

Yoke.

P.S. The braid consists of dry carbon (graphite) fibers and is as soft and flexible as any other textile fabric before it's processed with epoxy resin.

P.P.S. Stretch of the fibers should be too small to matter, and I have some tricks to deal with the friction issues.

 

[Kino]

Sorry, I'm no mathematician, but I've been involved in Engineering a good few years...

If I understand correctly, if the analogy of a "chinese finger-trap" is accurate then you need to know how the inside diameter changes as the length (or tension) increases. This would surely depend on how the braid was manufactured.

You then need to know how the pressure increases as the inside diameter (or length) changes. I think you may only be able to achieve this by experiment. Perhaps you could tighten the braid around a water-filled balloon and measure the pressure rise?

Then, having obtained a graph of tension v pressure you could formulate an approximate equation.

 

[PhilG]

yokebutt:

I tend to agree with Kino that determining the pressure experimentally is probably the best way to go. However, it is also a good idea to have a simplified mathematical model that gives you a ballpark answer. Here is a formula that I think might work:


Pressure = (Total applied tension) / (Radius of curvature of braid strands)

This is pretty much what you were saying in your post. It sort of makes sense as it is, but it can also be derived mathematically. Since the strands make helix curves around the tube, the formula for the radius of curvature R is

$$R = a(1 + \frac{1}{4\pi^2 a^2 n^2})$$

where "a" is the radius of the tube and "n" is the number of wraps (around the tube) one strand makes per unit length of tube. If it's easier to measure the angle a strand makes with the tube, "n" can be figured out from this.

One more thing, since there are two sets of intertwined helices, you might need to multiply the above formula for pressure by two. On the other hand, since there may be small gaps in the braid, the actual pressure might be less by some factor. I don't know if these will compensate exactly. So even for a mathematical model, there should be some factor between, say, 0.5 and 2 multiplying the above formula for pressure.

edit: oops, the formula I gave for pressure isn't right. It should have been

pressure = nT / R,

where "n" is the turns/unit length for one strand as described above, T is the applied tension, and R is the radius of curvature. I should have noticed that from the dimensions: [pressure] = [force] / [area] = [force] / [length]^2.