Author Topic: What is a knot ? Gordian knots.  (Read 28264 times)

SS369

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Re: What is a knot ? Gordian knots.
« Reply #15 on: December 18, 2011, 08:23:29 PM »
Hi Derek,

I have at my disposal a few ropes that go to a minimum encircling diameter and no further by hand. This particular rope > http://www.bluewaterropes.com/home/productsinfo.asp?Channel=Recreation&Group=&GroupKey=&Category=Ropes,%20Gym&CategoryKey=&ProdKey=41 will only close down to an open space approx. half the rope's original diameter.

Perhaps more within a highly loaded knot? Hard to say.

Quite a few of the static and dynamic climbing ropes I have will not close down much more than the relaxed diameter. A 5/8 inch "bull rope" I own won't make the closure to one diameter, though I suspect it is possibly due to the age of the rope and my own ability to stress it enough, maybe.

Now checking, some of the harder accessory cords (2.5mm and fairly stiff) I have perform very similar to their larger brothers and sisters. Although checking how small the opening is is difficult.

I had suspected the 5.5mm Titan (Dyneema) cord to resist closure, but it does so very tightly. This surprises me in that this particular cord fails a large percentage of knots that I tie to evaluate. The knots generally come undone in loading.

Materials and their construction methods have, in my own experience, a huge impact on the choosing of a particular knot and the further designing/modifying of newer and known ones to make them work with the available or chosen media.

SS


xarax

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Re: What is a knot ? Gordian knots.
« Reply #16 on: December 18, 2011, 11:46:38 PM »
   Of the cordage I have thus far worked with, I have yet to come across one that cannot be turned around a diameter of zero. 

  I guess that, if a rope is forced to bent around a very small diameter, its cross section will be flattened a lot...so it will not remain a physical rope that " retains its topology, and has (approximately) constant length and cross section, i.e. it is (substantially) flexible only along its lengthwise dimension/direction."
   Loaded beyond a certain point, any material can be forced to take any form...but :   
   1.   Its physical properties will change : "Above a certain stress known as the elastic limit or the yield strength of an elastic material, the relationship between stress and strain becomes nonlinear. Beyond this limit, the solid may deform irreversibly, exhibiting plasticity." We still have a physical object, of course,  :), but we do not have the same physical rope we had - to say the least, because we can say that we do not even have a "rope" any more...
   2.   Around the area of maximum curvature, its cross section will be deformed (flattened) substantially, so it will not retain even an approximately constant cross section. The fact that, even after the extreme bend,  the object remains in one piece, does not imply that it remains one and the same object it was at the beginning - before it was bent so much - any more : It better be considered as a compound object, made by ropes of various cross sections, that happen to remain physically connected together, the one after the other.  I believe that we are talking about physical ropes that "are flexible only along their lengthwise dimension/direction", otherwise we enter the area of completely elastic or plastic deformable media, an altogether different zoo.
   Try to measure the round and the flattened cross sections of ropes that have been forced to bent around a minimum diameter. Are those dimentions  approximately the same, to the point we can still call the deformed object a physical rope "of (approximately) constant cross section "? 
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DerekSmith

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Re: What is a knot ? Gordian knots.
« Reply #17 on: December 19, 2011, 12:06:01 AM »
Hi Scott,

While I have a 1" towing line that will happily close to zero at the very lightest of pressures, I have also found up a piece of old (hard) 12mm kernmantle which even under a fair proportion of its working load will not close below 5mm.

The differences seem to be related to flexibility.  The kernmantle refused to deform, so the outside was quickly placed into tension, so to close it more would have meant extending the outer fibres and although this is a dynamic rope, it is not a bungee cord, so I was putting the outer fibres into a real load bearing situation (and loosing).

All the cordage which will close to zero seems to be readily able to change shape such that cord from the inside radius flows easily to the outside radius, preventing an tension from building up.

So already, I have to amend my statement slightly as one of my cords will not close to zero.  But it is an interesting process to consider why...

Derek

SS369

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Re: What is a knot ? Gordian knots.
« Reply #18 on: December 19, 2011, 12:31:29 AM »
Hi Derek.

Yeah, it is an interesting process to consider.

A possible analogy that I have thought of is the bending of flexible pipe. Some use a tubing bender with formed members that match the desired radius. Some use the tubular spring that goes around the outside. I have used sand in the tube for many years and it works very well.

The sand (kern) aids the tube (mantle) retain its shape throughout the bending process. The core won't compress much with the cover becoming more inflexible as the forces come to bear against each other.

Most of the towing lines I have seen and the one that I own are loose weave construction and will flex or torque down to very small dimensions. I wonder why they are designed that way instead of a more solid affair as in climbing ropes. To dissipate heat? To keep the fibers as straight as possible?

And what about webbing, both tubular and non, do they conform to the constraint? The forces in a tape knot must really be compounded.

SS

DerekSmith

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Re: What is a knot ? Gordian knots.
« Reply #19 on: December 19, 2011, 12:34:20 AM »
   Of the cordage I have thus far worked with, I have yet to come across one that cannot be turned around a diameter of zero. 


   1.   Its physical properties will change : "Above a certain stress known as the elastic limit or the yield strength of an elastic material, the relationship between stress and strain becomes nonlinear. Beyond this limit, the solid may deform irreversibly, exhibiting plasticity." We still have a physical object, of course,  :), but we do not have the same physical rope we had - to say the least, because we can say that we do not even have a "rope" any more...


The cordage which takes easily to a zero diameter turn does so without "irreversible deformation" - it easily and readily returns to its linear shape when put back under linear tension and will do so repeatedly without seemingly have sustained "Plastic deformation"

Quote

   Try to measure the round and the flattened cross sections of ropes that have been forced to bent around a minimum diameter. Are those dimentions  approximately the same, to the point we can still call the deformed object a physical rope "of (approximately) constant cross section "?

No, the dimensions are not the same - the cordage has responded to being turned around a zero diameter and has 'shape shifted' to accommodate the turn.

Therefore the criterion that the cordage is only cordage if it is circular section is an inappropriate constraint, because cordage is flexible in this aspect to varying degrees dependant upon the construction of the cordage.

Derek

xarax

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Re: What is a knot ? Gordian knots.
« Reply #20 on: December 19, 2011, 01:19:31 AM »
The cordage which takes easily to a zero diameter turn does so without "irreversible deformation"

   How do you know it ? I mean, you have not examined each one fiber under electronic microscope... to be sure that, after a severe turn, there are not some parts of some fibers that have been deformed locally - and will never return to their previous physical characteristics - have you ? This local deformation is called "fatigue', and it will deteriorate the physical properties of any rope, especially when you force them to turn around such extreme curvatures.
   When a rope of a circular cross section is bent, the fibers that are forced to cover the longer outer paths are subject to extreme elongations. Are you sure that those elongations would remain within the limits of elasticity region in each and every fiber, and no fiber would be stressed enough to reach the plasticity region ?
   I say that, to make a turn of minimum or zero diameter, a rope has to be physically deformed, and/or change cross section beyond normal, i.e. flattened to a point where it is not a rope anymore, it is a stripe or whatever, and certainly it is not the rope it was before it was forced to such a tight turn.

the criterion that the cordage is only cordage if it is circular section is an inappropriate constraint, because cordage is flexible in this aspect to varying degrees dependent upon the construction

  I have not said that it should be only of circular cross section, I said that it should be of (approximately) constant cross section. Also, that it should be flexible only in the lengthwise dimension/ direction. Otherwise, any piece of elastic material could have been considered a rope, and we do not want this, right ?
   I would be glad if you could help us here, to define what a physical rope is, so that we exclude any piece of any plastic or elastic material whatsoever, that just happens to be spatially elongated. 
« Last Edit: December 19, 2011, 01:23:03 AM by xarax »
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DerekSmith

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Re: What is a knot ? Gordian knots.
« Reply #21 on: December 19, 2011, 11:47:20 AM »
Hi Derek.

Yeah, it is an interesting process to consider.

A possible analogy that I have thought of is the bending of flexible pipe. Some use a tubing bender with formed members that match the desired radius. Some use the tubular spring that goes around the outside. I have used sand in the tube for many years and it works very well.

The sand (kern) aids the tube (mantle) retain its shape throughout the bending process. The core won't compress much with the cover becoming more inflexible as the forces come to bear against each other.
SS

Scott, I like your analogy of considering a sand filled pipe.  However, I think the kernmantle taking a non zero diameter turn is an almost opposite model.

Consider the soft copper pipe to be filled with a bunch of high tensile steel wires.  The pipe is as you say, the braided (mantle) while the wires are the (kern).  The cores of some ropes are braided, giving the rope bend flexibility, but my climbing rope is full of a bunch of aligned straight monofibres - like a copper pipe filled with steel wires.

Now we bend them.

In the case of the sand filled pipe, the pipe is ductile, so it responds to the bending forces by thickening on the inside of the bend and thinning on the outside of the bend.  The sand is essentially incompressible, but it can flow, so as the inner curve volume shrinks, the outer curve volume expands and the sand flows / shifts but maintains an approximate round pipe section.

If on the other hand we try to bend the pipe filled with steel wires, the tube stretches easily, but the wires inside it refuse to stretch.  Because the pipe is full of wires, the tube keeps the wires in their places - the inner wires cannot move or 'give' their excess length to the outer wires that require extra length to accommodate the larger outer radius.  We are struck immediately with the fact that the pipe full of thin wires is now behaving as if it were a bar of solid iron.

In some climbing ropes the core fibres are given a light braiding, in others like my kernmantle, the core fibres run straight and parallel from one end of the rope to the other, and although the individual fibres are much more elastic that the iron wires in the coper pipe, the effect is the same  as we try to bend the rope, the outer braid constrains the fibres to remain 'in their place', so the outer fibres remain running all the way around the outer radius of the curve.

These fibres are more elastic than the iron wires, so they stretch under the tension imposed by the larger outer radius.  But the amount of stretch required to expand the outer radius  to accommodate the turn quickly escalates to the working load of the fibres.  If I were to use a mechanical system to continue to tighten the curve, eventually I would overload the tensile limit of the outer fibres and they would fail.

A rope however, with a laid or braided core is able to allow the outer tension to be 'fed' from the inner fibres that are under compression,  the flow of the braid from one side of the cable to the other is responsible for giving us the amazing flexibility of cordage so critical in its usefulness  not only to pass round pulleys, spars etc, but also to pass around the cordage itself and engage in the aspect we are all here for - the creation of knots.

Derek

xarax

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Re: What is a knot ? Gordian knots.
« Reply #22 on: December 19, 2011, 03:01:04 PM »
I have also found up a piece of ... kernmantle which, even under a fair proportion of its working load, will not close below 5mm.

   That is the kind of ropes I use most of the times... because they tend to remain of constant, circulat cross section, so they "write" nicelly on the camera s image sensor.

  The differences seem to be related to flexibility.  The kernmantle refused to deform, so the outside was quickly placed into tension, so to close it more would have meant extending the outer fibres and although this is a dynamic rope, it is not a bungee cord, so I was putting the outer fibres into a real load bearing situation
  All the cordage which will close to zero seems to be readily able to change shape such that cord from the inside radius flows easily to the outside radius, preventing an tension from building up.

  That was exactly what I have said . However, I would like to comment on the first sentence, " The differences seem to be related to flexibility". Of course, but why are mantle's ropes not flexible enough to bend around small diameter curves ? Because the fibers of the core are not allowed to slide freely inside the sheath. If we could imagine an array of flexible parallel fibers, enclosed loosely inside an outer soft envelope, that rope would be a mantle's rope, yet it could bend around as small a diameter as one single fiber would allow it to do, i.e. almost zero.
   It would be interesting if one could actually make such a rope - by enclosing nylon monifilament fishing lines inside a most flexible tube, for example - and see if it can be forced to bend around almost a zero diameter, or not...

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xarax

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Re: What is a knot ? Gordian knots.
« Reply #23 on: December 19, 2011, 03:34:04 PM »
  the bending of flexible pipe. I have used sand in the tube for many years and it works very well.

   When I was young, I remember I had made a modern rocking chair, by bending steel tubes...There were two ways I knew that could secure an almost circular cross section alongside the bent tube. Besides sand, one could fill the tube with pine resin, or any other liguid material that could be pressed to flow and fill the tube, yet solidify at the end of the process. I had used both methods, and, at the end, I made my chair using a normal tunbing bender ... :)

And what about webbing, both tubular and non, do they conform to the constraint?

  Although it might have been considered as a unnatural limitation, I have not included webbing in my attempted definition of a physical rope. I have made this decision on purpose. I tend to think (just a naive though, that is) that the knots made by flexible tape-looking strips ( elognated, rectangular cross section) work in a altogether different way than the knots made by flexible tube-looking, round ropes ( almost circular cross section ) - , and while I have recently came to learn some things about the various mechanisms utilized by the later, I am a 1000% ingnorant of what on earth is happening with the former. Perhaps some more experienced knot tyers in this forum could help us here... How do the knots made by flexible strips really work ? What are the differences with the knots made by round ropes ?
   
« Last Edit: December 19, 2011, 03:52:15 PM by xarax »
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xarax

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Re: What is a knot ? Gordian knots.
« Reply #24 on: December 19, 2011, 04:08:07 PM »
If ... we try to bend the pipe filled with steel wires, the tube stretches easily, but the wires inside it refuse to stretch.  Because the pipe is full of wires, the tube keeps the wires in their places - the inner wires cannot move or 'give' their excess length to the outer wires that require extra length to accommodate the larger outer radius.  We are struck immediately with the fact that the pipe full of thin wires is now behaving as if it were a bar of solid iron.

  Noope ! If the wires can slide inside the tube, relatively to the inner surface of this tube and relatively to each other, we essentially have a wire rope, where the individual fibers are not connected together by each one of them making helical turns around the bunch of the others - the usual way of the wire ropes-, but by being inside the envelope/mantle of the steel tube. While the whole compound rope is forced to bent, the wires of the inside paths can remain as they were, even be compressed a little bit, while the wires of the outer, longer paths can stretch and also can slide relatively to the others. The cross section can remain round, and the flattening can be minimized.
« Last Edit: December 19, 2011, 04:10:46 PM by xarax »
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DerekSmith

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Re: What is a knot ? Gordian knots.
« Reply #25 on: December 22, 2011, 07:44:45 PM »
If ... we try to bend the pipe filled with steel wires, the tube stretches easily, but the wires inside it refuse to stretch.  Because the pipe is full of wires, the tube keeps the wires in their places - the inner wires cannot move or 'give' their excess length to the outer wires that require extra length to accommodate the larger outer radius.  We are struck immediately with the fact that the pipe full of thin wires is now behaving as if it were a bar of solid iron.

  Noope ! If the wires can slide inside the tube, relatively to the inner surface of this tube and relatively to each other, we essentially have a wire rope, where the individual fibers are not connected together by each one of them making helical turns around the bunch of the others - the usual way of the wire ropes-, but by being inside the envelope/mantle of the steel tube. While the whole compound rope is forced to bent, the wires of the inside paths can remain as they were, even be compressed a little bit, while the wires of the outer, longer paths can stretch and also can slide relatively to the others. The cross section can remain round, and the flattening can be minimized.

Indeed Xarax,

If the wires slide inside the tube, then they do not need to be stretched, they only need to conform to the curve.

If we follow this with a theoretical tube bend around zero diameters, then you would see the inner wires (the wires near the tight bend) protrude from the tube, because the part of the tube they were in has now squished in on itself.  About a third the way out, the wires would be flush with the ends of the tube, and right out at the outer edge of the tube, the wires would have retreated inside the tube, because they have remained the same length while the tube has been stretched around the outside of the curve.

The question you posed back in post #22 can be answered if you take a very short piece of kernmantle with parallel core threads, lightly tape the  sheath to stop it from unravelling, then bend it around a zero diameter turn, the inner fibres will protrude, while the outer fibres will run inside the sheath.  If these fibres were prevented from slipping, then they would have had to have stretched by the amount they shrank into the sheath, and to stretch them, they would have had to have been put under extreme tension, possibly even beyond their breaking point (dependant upon their intrinsic elasticity).  In a non sliding core, the inability to stretch the fibres enough would resist the closure of the bend to a zero diameter.

Derek

xarax

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Re: What is a knot ? Gordian knots.
« Reply #26 on: December 23, 2011, 04:26:35 AM »
  Right. So, why it is exactly the kermantle static climbing ropes ( those with parallel fibers) the ropes that DO NOT ,bend around small diameters ? What prevents the parallel fibers from sliding along each other ? The outer woven sheath is, most of the times, very flexible, and not so tightly woven, I think, that could induce great friction forces between the fibers of the core, and so prevent them from sliding.
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Dan_Lehman

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Re: What is a knot ? Gordian knots.
« Reply #27 on: December 23, 2011, 08:22:49 PM »
  Right. So, why it is exactly the kermantle static climbing ropes ( those with parallel fibers) the ropes that DO NOT ,bend around small diameters ? What prevents the parallel fibers from sliding along each other ? The outer woven sheath is, most of the times, very flexible, and not so tightly woven, I think, that could induce great friction forces between the fibers of the core, and so prevent them from sliding.

1) (Rock)climbing ropes DO NOT have parallel fibres!  Parallel
fibres are used for minimal stretch; climbing ropes have
hard-laid core cords (in equal balance between Z/S-lay)
(though long ago one --Rocco, of Spain?-- used some braided
core).

2) IMO, the amount of material packed per-length in such
ropes impedes their bending; the sheath is relatively tight
around the core.
(Caving's infamous PMI original, "pit" rope, is maybe the toughest
I've found --though I have some tough laid marine rope--, with
reluctance to bend to 2dia. (in 11mm rope), let alone that 5mm!!)
Relative tightness in the rope, and lesser flexibility, can be
seen as help in resisting ingress of foreign material (dirt),
and abrasion resistance.

Rockclimbing ropes have some uniform measure of flexibility,
which IIRC is determined by setting an overhand knot with
a given weight and then measuring the size of the hole in
its belly with a graduated conical device.


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xarax

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Re: What is a knot ? Gordian knots.
« Reply #28 on: December 23, 2011, 09:13:53 PM »
Parallel fibres are used for minimal stretch

   AND for maximum strength, in a given diameter and weight. I have many kernmantle ropes that have parallel fibers - I do not know if they are sold as "climbing ropes", or not. ( I use them only because they are stiff, and keep their round cross section even when bend around tight curves. )
 
the sheath is relatively tight around the core.

  Yes, but is it tight enough to squeeze the fibers and prevent them from sliding alongside the bunch ?

Rockclimbing ropes have some uniform measure of flexibility, which IIRC is determined by setting an overhand knot with a given weight and then measuring the size of the hole in its belly with a graduated conical device.

  Thank you for this information, but I wonder why they use an overhand knot -where there is some additional friction between the tails - and not just a single 360 degrees bight, a single nipping loop ?

« Last Edit: December 23, 2011, 09:14:31 PM by xarax »
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xarax

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Re: What is a knot ? Gordian knots.
« Reply #29 on: December 26, 2011, 08:58:18 PM »
many kernmantle ropes ...have parallel fibers - I do not know if they are sold as "climbing ropes", or not. ( I use them only because they are stiff, and keep their round cross section even when bend around tight curves. )

     Here is the dis-section of one of them. However, in this rope, the parallel fibres are placed in 6 individual bunches, and each bunch is firmly wrapped inside its own "tube" by a plastic tape spiral tube ( See the first attached picture). Then, around those 6 bunches, there is another  tightly woven sheath - not the final, outer one,  but an inner, second layer sheath, much more tightly woven than the outer one. So, I guess that, by those three successive envelopes outside them, the parallel fibres are held together tightly enough, and they can not slide relatively to each other.
   In most of the permanently ropes I have dissected, the fibres are first woven into laid or braided sub-cords - 3 to 13 (?!)  of them - , then those laid or braided sub-cords are placed in tight parallel bunches inside the outer woven sheath. ( See the second attached picture).  So I can see how the individual fibres do not slip, but I do not understand how the laid or braided bunches of those fibres do not slip...because the outer sheath does never seem to be so tightly woven. Perhaps the "grooves" of the surfaces of the laid or braided sub-cords have a role to play in this...
« Last Edit: December 26, 2011, 09:37:11 PM by xarax »
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anything