Can we accurately predict the shape of a cable system with multiple unknowns?

[Cyrus]

I am studying the statics of cable systems and I find it to be very interesting. I REALLY like statics. Such an interesting subject, anyways.
My first question is this:
When we have a system of cables supporting weights, say like traffic lights, we can assume the cables to be straight line segments if the cable is of negligible weight. But in all the examples, we had known dimensions, in order to determine the tensile forces in each cable. Is it possible to say given an arrangement of ropes, and suspended masses, this will be the angle, and that will be the tension. Can we predict how the system will look as an end result? I say no, as there are now too many unknowns. I ask this because I wonder how a shipping company that is concerned with rigging would go about designing a system where the can know how the pulleys and cables will be positioned as not to interfere with one another.
Question number 2:
Lets say we have a cable that has a significant weight. The proof of that is readily simple for a given continuous loading; however, how would we derive an equation to describe the curve of the cable given a discontinous change in mass, Like tieing a heavy rope to a lighter rope somewhere along the length.
Question number 3:
How about if we have a cable that has a distributed load w.r.t x and a distributed load due to its own weight, w.r.t. arc length s. How would we go about describing that curve? I would say superposition, assume that the cable sags under its own weight, and then find that curve. Then find the curve if the weight is negligible and then add the two resulting curves to get the new total deflection curve. But this is based on a hunch, and I want to prove it formally.
Question number 4:
What if either of the two types of distributed forces also has a concentrated force somewhere along the way? How could we adjust the proofs to account for that?
These are very difficult to anwser, so if I figure any of it out ill post the solution. If you already know the solution or can provide me a website that has one, I will forever be in your debt .
Thanks,
Cyrus

 

[Cyrus]

For the first problem, I am inclined to say its some sort of an optimization problem where you get the minimum potential energy for all the suspended masses in question, but still, too many variables and unknowns. I guess it must be trial and error in the real world application to get a specific configuration that suits ones needs in terms of spatial distributions. I will try to get some string of significant mass and load it to see if it responds accordingly.

 

[Astronuc]

It's not clear to me that the first problem statement leads to an indeterminate system. Clearly the net force in a static system is zero - there is not motion, and not acceleration. Both ends of a cable are fixed, and there is some tension.

One certainly needs to now the linear mass of the cables.

What unknowns does one expect?

Presumably one fixes the lengths of the cables, and the positions of the anchors, or ends of each cable, and what ever weight (load) is borne.

 

[Cyrus]

The unknowns are the direction and tension of the cable, because I am not telling you what they are, I am asking you what they will be, but I am not giving you any geometry of the situation other than the boundary conditions and the length of each segment of the rope and the mass, and the length between boundaries.

 

[Cyrus]

Hmmmm, one thing I wonder about also. Suspension bridges are built with suspenders to support the cable, yet it is assumed that the roadway gives a uniform distributed loading on the cables via the suspenders. But there are a finite number of them. Clearly if n=1, then the curve will be cut into two line segments, forming a V shape, now matter what the loading conditions. I wonder how big does n have to be to be "close" to the assumption.

 

[Cyrus]

Hmm, if we assume that when loaded, the line segments of the rope take on a linear slope, then we are dealing with a series of line segments. And this is exactly an extended trapezoidal rule approximation on the interval L, where L is the length of the entire cable span in the x direction. Then the error will be about equal to:
$$\frac{- (L)^3}{12n^2} \bar {f''}$$
where:
$$\bar {f''}$$
is equal to the average of $$f''(x)$$ in a < x < b .
So the error decreases with 1/n^2, in plain terms, very fast, which justifies the use of a small number of suspenders, and explains why we see things like the golden gate bridge, which has a small n, yet a graceful and curved cable shape.
Lets assume a simple parabolic curve of the uniformly loaded cable, then f(x)=x^2, f'(x)=2x, f''(x)=2
so the average value of f''(x) is simply 2. and the length of the span is L, so the approximation is good to:
$$-L^3/ 6n^2$$
And that seems fair. A very long span with a small n, means a gross underapproximation, which I can see intuitively. A small L, means a much better approximation for a smaller value of n. That makes reasonable sense as well.


Does that sound at all reasonable?

You can see the very large value of n on the picture, but a large value of L as well. The fact that it arches so much shows that the parabolic function has a very very low average value for f''(x), helping to blow down the value of the error induced by the large span L.

 

[Astronuc]

The main cables of a suspension bridge are very stiff - large cross section. Traditional suspension bridges are not very common now - cable-stayed designs have more or less replaced them.

Cable-stayed: http://www.pbs.org/wgbh/nova/bridge/meetcable.html

http://www.matsuo-bridge.co.jp/english/bridges/basics/cablestay.shtm

http://en.wikipedia.org/wiki/Cable-stayed_bridge

This is a somewhat interesting case -
http://en.wikipedia.org/wiki/Eastern_span_replacement_of_the_San_Francisco-Oakland_Bay_Bridge

 

[Cyrus]

My trapezoidal analysis does not apply I found an error in my reasoning, *$^& word. Grrrrrrrrrr. :-(

 

[Cyrus]

I showed the problem to a guy at work (He's a monster), and I think he was able to solve it using what I started. He used a jacobian and optimization of potential energy as I had suspected needed to be done. Ill go back and try and get him to explain to me the anwser. Its a big mess of equations, something along the order of 9+ systems of nonlinear equations to solve. I think this is a purely numerical solution, via Newton-raphson method for vectors. If I can figure that out, then I can predict what the cable will look like given descrete loads along its length because the load is actually suspended by fininte numbers of risers.

The reason why I wanted to solve this question is because let's say I have a bridge. I know what shape my rope will make given a uniformly distributed load. But because I have descrete numbers of risers, I will have only an approximately distributed loading condition. So I wanted to know how much error I would get using n number of risers, and how many risers it woudl take to get very close to a uniformly distributed loading condition on the suspension cable. The reason being, if I am building a bridge, I don't want too many, becuase of aerodynamics, cost, weight etc, but I don't want too little either, because my loading is not uniform along the cable, and it may snap. I wanted to do an analysis to see what would be a "GOOD" number of stringers, and find out how "GOOD", "GOOD" really is.

I will test out the equations at home on a riggin system to verify that the anwser actually makes some physical sense, or its back to the drawing board.