A simpler TackleClamp hitch.

[X1]

The reason that more wraps improves the ability of a hitch to withstand lengthwise pull is not so complicated as you make it seem

   Correct. It is much more complicated than it seems...
   The ( tiny, a few degrees ) difference in the angle between a less elliptical, almost circular round turn and a more elliptical, elongated one, and the axis of the pole, cannot explain the great difference in the resulting friction forces. On the contrary, sometimes the more elliptical round turns are more efficient than the less elliptical ones - as it happens in the case of ABoK#1755-#1756, for example. Hitches that work when their round turns are very elongated ellipses, lose their gripping strength and start to slip alongside the pole when the same round turns are brought closer together, and become less elliptical. Ashley, in Chapter 22, shows many other hitches where the round turns are tied so that they are deliberately tied in the form of obliquely oriented in relation to the axis of the spar, elongated ellipses. In fact, only a few round turns and only a few hitches in this chapter are tied so that the plane of the round turn is at a right angle with the axis of the pole.
   I have tested hitches where the angles between the long axes of the ellipses of the round turns and the axis of the poles were equal ( as far as I could measure them, with my limited means ). Ceteris paribus, the hitches with the greater number of round turns were always tighter than the ones with the fewer round turns.
   It is true that the original TackleClamp hitch, and the simpler locked, one-and-a-half and Double Cow hitches presented here, are meant to increase the tensile forces induced into the round turns during a pre-tensioning stage. As the standing ends are secured against any slippage, these tensile forces are accumulated, effectively stored, and finally added to the forces induced by the lengthwise pull itself.  However, I do not know which portion of the greatly enhanced gripping power of those hitches is due to the greater magnitude of this sum of gripping forces, and which is due to the smaller eccentricity of the round turns during the lengthwise pull. Another effect that is not often taken into account is the friction between the round turns, which, in a multi-wrap friction hitch, should also play a significant, perhaps major role.

[kd8eeh]

I just realized something: the reason is simply that there is more tension squeezing the pole with each wrapping.  if there are two wraps, then you have 2 times the tension in total squeezing the pole, while if you have ten wraps, there are 10 ropes, all equally tight, squeezing the pole.  that means that the normal force is proportional to each wrap, so every wrap adds more friction linearly.  That makes a whole lot more sense. (although it's sad, i really liked the other physics problem, as it involves some interesting math)

I also just noticed there is no facepalm emonicon.

[X1]

the normal force is proportional to each wrap, so every wrap adds more friction

   No, for many reasons... Let me mention just two :
   1. The total normal force depends on the force with which you pull the two ends of the friction hitch . You can not multiply this force by just multiplying the number of wraps - if you will ever manage to do that, you will discover a means to produce infinite force (on the wraps) by a finite force (on the ends) - that is, perpetual motion - or something very close to it...
   2. The tension is not constant along the length of the rope - because, as you move away from the  standing end(s), the friction forces consume a considerable portion of the tensile forces. This is the "capstan effect", which makes the life of a sailor much easier - he just wraps his tensioned ropes two to four times around a drum, and he is able to counterbalance the great forces of the elements.
   Friction is not such a simple thing as it might look - and the complexity of the construction of the ropes we use does not make it simpler ! I had also mentioned an effect that is not taken into account, although it might play a major role : the friction forces between adjacent wraps. I have seen that one multi-wraps Clove hitch is much more efficient than many common Clove hitches with the same total number of wraps, and I believe that this is due to this effect : the oblique riding turn squezes the round turns the one upon the other, and this locks the accumulated tensile forces within the "coil tube" of the hitch.

[kd8eeh]

Additive friction is likely not the only reason that multiple wraps is more secure, but still I do believe it helps in the same way as before.  Your first argument, that this finite force yeilds an infinite force as the number of wraps goes to infinity, does not cause a problem because the normal force is just a force.  There is no work done by the normal force, and no energy associated with it, so we don't break physics by saying this.

The second point is valid, but as most hitches may be tied, you wrap the pole while the rope is under tension, therefore you can minimize this effect.  Also, I have found a zigzag hitch is much more effective for this type of wrapping, as it produces friction between the rope and itself, in addition to with the pole.

To more clearly demonstrate the effect i am talking about, consider a system of pullies.  The rope wraps around the top pully, then around the bottom pully, then back around the top, forming a complete loop.  For these purposes, we may assume both ends are loaded (equivilant to pulling both ends of the hitch).  In this case, it is obvious that it will be easier to lift the bottom pully if there are more wraps; and there is less tension required to keep the bottom pully in equilibrium.  So, with one wrap, inducing some amount of tension on the ends will yeild less force upward on the bottom pully than if there are more wraps and the same tension.  Friction would put a limit on this effect, but this is the effect i am describing when you wrap a rope many times around a single pole; the bottom of the pole may be treated as the bottom  pully.

I devised an experiment to test this, and it was true.  The experiment involved putting two identical caribeaners next to each other, with there gates on the same side, but hinging opposite directions, so that you don't have to account for the leverage the rope may have with only one caribeaner.  Then i wrapped the rope around this once, and tied the ends together around a weight that was not quite heavy enough to cause the gates to open.  Then i put the same weight on the same carribeaners, but with extra wraps, and the gates oppened.  Granted, it's a little crude, but it's the best i can do with what i had sitting around.

[X1]

  ...[that] this finite force yields an infinite force as the number of wraps goes to infinity, does not cause a problem.
  There is no work done by the normal force, and no energy associated with it, so we don't break physics by saying this.

   No, you just break the pole !  When you find that something is infinite, think that there are only two such things - and, regarding the first one, we are not so sure . 

I devised an experiment to test this, and it was true.

   You could just tie a 4 and a 6 wraps Double Cow hitch on the same pole with opposing standing ends, pre-tension and then load them with equal forces, and see which slips first...

   ( The many wraps around a pole is a friction complex system, it is not a frictionless simple mechanism ! You can not describe, explain and predict the peculiar characteristics of the former, by the simple mechanics of the later. I believe that what you have described is an attempt to reduce the working of the multi-wraps friction hitch to that of the many-pulleys block and tackle system. After Archimedes, nobody can deny the notion of the mechanical advantage, with the help of which, "if we would have had a sufficiently long beam of a lever, we could move the earth..." The same can be said with the system of many pulleys you describe - in fact, with any compound system of the 6 simple machines ( lever, wheel and axle, pulley, inclined plane, wedge, screw ). What is not taken into account in those simple machines is the energy loss due to friction  ( between the rope and the pole/pulley, and between the adjacent round turns), and the deformations of the rope and/or the pole ( change of the length of the rope or of the cross section of the pole ). So, you wish to describe, explain and predict a mechanism based on friction, by mechanisms where friction is not taken into account... When you add friction, you will see that there " every wrap [DOES NOT] add more friction linearly ", and that every linearity disappears. That does not mean, of course, that if you add another wrap, you will not get more friction, and that an infinite number of wraps would not be attached on the pole with an "infinite" force !  It means that, starting from the outer wrap, every new wrap is offering less friction than the previous one, and it is dissipating the tensile force it receives from the previous and transmits to the next one more, to the point the addition of another wrap on the friction hitch would be practically pointless. The same is true with a block and tackle system of many pulleys. It is no co-incidence that we never see more than 3 or 4 pairs of pulleys ( threefold or fourfold purchase ).)

[kd8eeh]

That is all true, and this is why the best (or at least most of the best) hitches for lenghtwise pull are tied with the rope under continuios tension, so that you can make it very tight as you tie the knot and don't encounter this problem afterward

.

When you find that something is infinite, think that there are only two such things - and, regarding the first one, we are not so sure . 

What do you mean by this?

[X1]

What do you mean by this?

   Forget it, it was just a joke, which loses its lightness if it is taken seriously.
   The meaning was that, when our calculations predict the appearence of an actually infinite number, when it is obvious that, in nature, we do experience only potentially infinite quantities, it is our calculations that we should blame, not nature ! 
   ( The procedure of somehow "cuting off" the infinite quantities, so we get the observed finite ones, is called "re-normalization" - for good reason. An infinite quantity is an ab-normal thing...)     

[kd8eeh]

It's an infinite limit, not actually an infinite quantity.  Those happen all the time in nature.

In the end, let's leave it at friction is complicated.

[X1]

   The distinction between actual and potential infinity I mentioned quantities is a diferent thing.
   " In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (Elements, Book IX, Proposition 20)." (1) Im can assure you that the magnitude of the normal forces is NOT ( never ever ) infinite...Heraklas knew it !  (2)

1. http://en.wikipedia.org/wiki/Infinity
2. http://en.wikipedia.org/wiki/Heraklas

[xarax]

  Pictures of the Double Cow hitch, on a monochromatic rope.
  Tied on a 11 mm climbing rope, and pre-tensioned by hand as tightly as possible, by pulling the ends the one after the other.
  Rock solid, no sign of any release after the pre-tensioning, good-looking : what more can one demand of a tight hitch ?

[DerekSmith]

what more can one demand of a tight hitch ?

Ease of tying might come top of answers to that question, followed closely by intrinsic security (because just 30 seconds of flogging and it fell to pieces)

[xarax]

what more can one demand of a tight hitch ?

Ease of tying might come top of answers to that question, followed closely by intrinsic security (because just 30 seconds of flogging and it fell to pieces)

   If I judge from the top of your answer to my question, you have never tied it properly... because it can be set up in-the-bight in a few seconds ( literally ), and it can be tightened hard in a few more. However, I will suppose that you had not seen the attached pictures ( illustrating how it can be tied in-the-bight ), or that you may find difficult, indeed, to tie two Cow hitches, the one next to the other, in-the-end... Also, I will not question WHICH ONE ( =1 ) other hitch is even remotely as tight as this hitch, and can also be tied in-the-end AND in-the-bight so easily and quickly, because I know you do not like comparisons...   So, I will answer the bottom of your answer : 300 seconds of flogging ( measured with my watch - yours may run slower... ), had not ANY noticeable effect on the grip. In fact, its only problem is that it so tight and irresponsive to ANY flogging, that it should better be slipped always, even if one does not tension it very hard.
   Frankly, Derek, it seems to me you have lost something ( regarding knots...) - but I hope you have gained something else ( regarding something else ).
   The good thing with superb knots, like this, is that they do not need to be advertised ( like the mediocre knots - the fake, so-called "Sailors" hitch, for example, or the fake so-called "Zeppelin" loop ) - so I will say nothing else in favour of the Locked Double Cow hitch. When this old generation of knot tyers, and the rocks in our heads we carry, will become dust, this hitch will occupy the place in the knotting toolbox it deserves.

[xarax]

   This tread was meant to present "a simpler TackleClamp hitch", and that is exactly what the Locked Double Cow hitch is. It is not as tight as the original TackleClamp hitch, and it even uses more material - but it can be tied veeery easily in-the-end AND in-the-bight. Its overall shape and structure is very simple, and very nice, too ( once one has tied it just a few times, it is almost impossible to forget it... provided he remembers the Cow hitch, of course  ). Even its mere name, by itself, is able to describe the knot quite sufficiently and accurately ( so, it is NOT irrelevant or misleading, as it happens with the names of most other hitches...). Now, the original TackleClamp hitch can tied easily, ( starting from the Clove hitch) only if we are allowed to use both ends ( i.e. the ends of a relatively short rope - because, to follow this easy to remember and execute method described at (1), we have to tuck both of them once ). To tie the TackleClamp hitch in-the-end, one has to remember the mnemonic picture, and follow the sequence of moves described at (2), which are not-so-easy to memorize, indeed. ( Dan Lehman, for once, was right on this.. ). To tie the TackleClamp hitch in-the-bight, it is even more difficult (3), and this was the reason I tried to find something tight but easier to remember how to tie in-the-bight, in the first place -  a replacement of the mediocre Pile hitch, in short... With the Locked Double Cow hitch, I believe I was lucky to succeed in this, and to meet a most useful and beautiful hitch, able to withstand a lengthwise pull. I do dare to describe it as a superb knot, although I always measure my words most carefully, when it comes to superlatives...
   Of course, an even simpler, two-wrap tight hitch, is the Locked Single Cow hitch, which is also TIB ( See (4), and the attached pictures ) - another superb knot - and I still wonder why Ashley, who came so near to it with the "similar", yet less tight, well-known ABoK#1683 (5), had not tied it 70 years ago...

1. http://igkt.net/sm/index.php?topic=4224.msg26058#msg26058
    http://igkt.net/sm/index.php?topic=4224.msg26059#msg26059
2. http://igkt.net/sm/index.php?topic=4224.msg26178#msg26178
    http://igkt.net/sm/index.php?topic=4224.msg26331#msg26331
3. http://igkt.net/sm/index.php?topic=4224.msg26364#msg26364
4. http://igkt.net/sm/index.php?topic=4035.msg24785#msg24785
    http://igkt.net/sm/index.php?topic=4035.msg24787#msg24787
5. http://igkt.net/sm/index.php?topic=4441.msg28170#msg28170
    http://igkt.net/sm/index.php?topic=4814.0

P.S. I have described what I mean by a "tight hitch" = "binder", at (6) and (7) :
6.  http://igkt.net/sm/index.php?topic=4155.0
7.  http://igkt.net/sm/index.php?topic=4673.0

[Sweeney]

what more can one demand of a tight hitch ?

Ease of tying might come top of answers to that question, followed closely by intrinsic security (because just 30 seconds of flogging and it fell to pieces)

   If I judge from the top of your answer to my question, you have never tied it properly... because it can be set up in-the-bight in a few seconds ( literally ), and it can be tightened hard in a few more. However, I will suppose that you had not seen the attached pictures ( illustrating how it can be tied in-the-bight ), or that you may find difficult, indeed, to tie two Cow hitches, the one next to the other, in-the-end... Also, I will not question WHICH ONE ( =1 ) other hitch is even remotely as tight as this hitch, and can also be tied in-the-end AND in-the-bight so easily and quickly, because I know you do not like comparisons...   So, I will answer the bottom of your answer : 300 seconds of flogging ( measured with my watch - yours may run slower... ), had not ANY noticeable effect on the grip. In fact, its only problem is that it so tight and irresponsive to ANY flogging, that it should better be slipped always, even if one does not tension it very hard.

Although I am not sure I have a use for this hitch it is easy to tie around a pole etc where the ends are not available by starting with a slingstone hitch (ABoK #272) and taking the ends around and down through the bights on each side (the photos posted by Xarax should be easy to follow from the slingstone hitch point).

Barry

[xarax]

starting with a slingstone hitch (ABoK #272) and taking the ends around and down through the bights on each side

  Thank you, Sweeney. I have met this hitch while I was trying to simplify the TackleClanp hitch, which is a 3 /4 wraps very tight, self-locking hitch / binder ( the most tight I know ) able to withstand a lengthwise pull even in its simplest form ( for more gripping power, one can always tie it with 6 wraps, as shown in (1)). So, although it may seem strange, I had NOT started from the Cow hitch, or any other similar 2 wrap hitch, utilizing a mechanical advantage... In other words, I was trying to find simpler ways to implement the opposing bights locking mechanism mainly, not the leveraged tensioning of the wraps offered by the mechanical advantage due to the zigzag path of the Standing part on the surface of the pole. It was some time after I had already tied and tried this hitch, when I realized that it was just two Cow hitches, the one next to the other... It had happened to me many times, to arrive at India through West India !
   And after I had tied this hitch, exploring the possibility to work with less wraps, I had reduced them to three and finally to two, and I had arrived at the Locked Single Cow hitch, NOT starting from the Cow or the Pedigree hitch, as one would, again, had expected ! I believe that you can follow the indirect, zigzag path of my reasoning, by reading the posts of this thread. I am not clever enough to find the one best way to connect the dots right from the start, but I dare to do the next less dumb thing, to connect the dots in every possible way...

1. http://igkt.net/sm/index.php?topic=4035.msg25143#msg25143