arigger said:
Thanks,
BTW does anyone have a reliable source for the K value of wire rope? The best I could find was http://www.latchandbatchelor.co.uk/rope-info/rope-data/" .
I answered your other questions first (see below) then looked at this. I've left my answers below because I felt you might be interested in understanding how it all works. However looking at these numbers (although I'm not totally convinced about some of them and they've misprinted the equations) it seems that, unless you are considering very heavy loads (10,000 metric tons plus), these kind of ropes can't be treated as elastic, since there will be significant permanent deformation on loading.
In this case I think the best way to determine the tension (short of actually using a commercial tension meter) would be to determine the angle \theta by measuring the deflection from horizontal caused by the load (e.g. rig a string tightly between the anchor points and then measure the vertical deviation from this of the loaded rope). If this deflection is only small then I don't think you're going to get an accurate answer though --- commercial tension meters may be the only solution.
The derivation I gave relied on being able to determine tension from rope extension which simply doesn't work if there is significant permanent deformation on loading.
One other possibility is prestressing the rope to eliminate future permanent deformations (something that the website you quote mentions), in this case you
might be able to use the original method I suggested depending on how successful the prestressing was and on whether you could determine an accurate value of k for the rope.
Are there any engineers (or more practical physicists than me!) who can offer any other advice?
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arigger said:
The first thing I'm running up against is the l0 vs. lp. How are those two values substantially different? If I pull a line from point to point that should be the pretension length regardless of the amount of tension I apply.
Yes, but as you apply tension, you are pulling on the rope: thus the
amount of rope between the two anchors decreases.
I'm afraid I mathematically simplified things a bit in my derivation. I was imagining a piece of rope that, when pretensioned,
exactly fit between the two anchors (which were a distance l_{\rm p} apart). Thus when the pretension is released (i.e. the rope is released from one of the anchors) the rope elastically contracts to some length l_0 that is actually
less than the distance between the two anchor points.
Of course in reality this isn't possible. However it is still easy to calculate the correct values for l_{\rm p} and l_0 for the more realistic case as follows:
- Attach the rope to the anchor points, pretension it, and make it fast.
- If the tension is enough to make the rope flat then, as you say, l_{\rm p} will just be the length of rope between the anchor points (i.e. the length of that part of the rope under tension).
- Use some kind of marker to mark the position on the rope of the start and end anchors.
- Release the tension in the rope and measure between these two markers: this new, shorter, length is l_0.
To be accurate this requires that the rope be perfectly elastic (i.e. there is no permanent deformation from pretensioning it)
and that you have a very clean way of attaching it to the anchor points (e.g. some kind of clamp or similar). If you are just knotting the rope it would be very difficult practically to determine precisely where the tension carrying part of the rope ends. Similarly if your anchors actually consist of pulleys and the rope is then made fast at some distance from these pulleys you would either need to rederive the solution taking this into account or instead formulate the answer in terms of the pretension (see below).
Lastly note that k must be calculated for the
untensioned length of rope l_0.
arigger said:
If l0 is the slack length then how does that affect the parts between anchors at pretension?
l_0 is the slack length of
that part of the rope which when pretensioned exactly spans the gap between anchors (i.e. l_{\rm p}).
l_0 is being used solely to determine the value of the pretension (the more pretension you apply the more the rope extends and so the greater the difference between l_0 and l_{\rm p}). If you
already know the value of the pretension then you can reformulate the problem
without using l_0 as follows:
- Rewrite equation 2 as T=k\Delta l+T_0 where T_0 is the value of the pretension.
- Replace all the remaining l_{\rm p} terms with D (the distance between anchor points).
- Rework the derivation along the same lines as before.
This has the advantage that it will also work for the cases where the rope is not actually made fast at the anchor points (e.g. the anchor points are just pulleys) but at some fixed distance from them,
provided k is calculated for the entire tension carrying length of the rope not just that part of the rope lying between the pulleys.
Let me know if you need help with this. Also, if you want to apply this to a real situation, it would be good to know exactly how you are anchoring the rope...
arigger said:
I had never heard of a quartic function and can't make sense of the answers I get when I run my numbers on a quartic function calculator I found online. I guess I was expecting a single answer.
The formulae from the general solution for the quartic will give four answers.
However, all but one of these should be unphysical (i.e. negative or complex tensions) so you should be able to pull the correct one out.