Measurement of the coefficient of friction for the capstan equation.

[KC]

As always, i specifically find the capstan effect and it's reciprocal pulley effect in ALL linear force fed into 180 arc(s).
>>the 2 effects are opposite extremes of arc powers, increase in capstan effect is a decrease in pulley effect etc.
The off host crossing provides friction but more pass thru than Crossed on host, and doesn't self cross as Round Turn belay may.
The Muenter SPart is stable self centering flow not seen in the other 2 basic compound Turn forms.
On a carabiner or hook can get even more effect than if flatter across like spar, w/o sides furthermore being pressed/molded around SPart more so  than in flat region .
.
Forms mentioned to me are of the BackHand Turn genesis, to include
Half-Hitch as not quite, Timber as extended but single bearing.
.
How could a Muenter not work?
it is as an incomplete Round Turn, not 3 arcs unless fold back towards input around host.
But it does have Turn on host and turn around SPart as more frictions and control of SPart more by biting into it.
>>i think stronger w/ Round Turn around SPart so is pulled more along than across SPart.
>>But with single turn style on tree have used Muenter as a Swig to tighten it's SPart across it
>>then slide foundation of knot around too far the other way and draw swig again.
.
Muenter controls by friction as any other. Round Turn is uncrossed with 3 arcs, Clove base/Crossed Turn is 3 arcs + ON HOST cross
>>Backhand Turn types have 1 arc on host, and 1 crossing off host
>>to me not as complete to the others until a 3rd arc around host bring Bitter End back beside input SPart, like Cow with last leg not reeved thru.
ABoK points out a few times that BackHand Turn gives a double bearing, BUT only requires 1 pass.
Of course Muenter is 1 leg thru base form of BackHand Turn, while Cow is dual leg thru, 1 leg bearing.
Girth is 2 leg bearing.  Typical choker sling/Bale hitch is 2 leg bearing, but self adjusting.
.
Capstan effect answer in any 180 arc(s) build, includes Bowline collar, but to very minimal extent.
>>for the collar is not very loaded to press the issue/empower the effect
>>most  hold comes from Half Hitch, especially on rawest side going into collar, then yes some reduction to less around arc collar.
>>but not much, as there is not much powering it to induce the effect.
.
i hope i have already explained how linear fed force to arc control, is a conversion w/loss
>>linear can only use part of tensions for friction and then compounds by distance.
>>arc can use all of tensions, cosine and sine for friction BIG game change, then compounds by DEGREE
Arc can finesse cosine and sine together as one to MUCH greater force control, than linear.
>>host mounts should be round /arc NOT square/rectangle etc of linear faces

[struktor]

Munter hitch (#206)

Does such a substitute scheme make sense?
Equivalent diagram fig.1
Equivalent diagram for equations.fig.2 fig.3

Sample results:
μ_r=μ=0.1 
α=β=57.1⁰
T1/T4=2.5

[DDK]

For the Munter hitch, starting at the loaded end of the rope, the tension in the rope decreases each time the rope curves.  For the sake of clarity, let me mention what is obvious; the rope curves at those points it is in contact with another surface and also obvious, the tension is constant for the straight sections of the rope.  The amount of decrease in the tension at each of these points of contact is given by the Capstan equation using the coefficient of friction between the two surfaces in contact and the angle of contact.
 
The above explanation is something of a simplification in that the first curve from the loaded end of the rope is at a collar which due to the extent of its wrapping produces "internal normal forces" which would need to be taken into account for a complete analysis.  In Physics, this is known as the "Stacked Blocks" problem.

Independent of the above simplification which ignores these internal forces, however, a rigorous comparison of the braking in the Munter hitch can be determined.  Using agent_smith's image of the hitch as reference, define "T_Lower" as the tension in the free end of the rope in the Munter hitch when the free end is in the "Lower" position.  Raising the free end into the "Upper" position essentially produces an additional curve and contact in the rope which, due to the Capstan effect, will produce an additional reduction in the tension of the rope in the Upper position, "T_Upper". 

Using the Capstan equation,

T_Lower = T_Upper * e ^μφ     or

T_Upper / T_Lower = e ^-μφ

where we have defined:

μ :	Coefficient of Friction for the Rope/Carabiner Interface
φ :	Additional Angle Swept By the Rope in Contact with the Carabiner (in radians)

For example, lets say that the additional curve for the Upper position wraps an additional angle on the Carabiner of about 90 degrees, that is, /2 radians, and the coefficient of friction is 0.1.  The tension required to support the same load in the Upper position is only 85% that of the Lower position.  Conversely, but in this case, approximately (due to the internal forces), if the tension in the free end of the hitch is maintained as one raises the free end, the load that could be supported in the Upper position would be 117% of that supported in the Lower position.  If the hitch was just able to slip at the Lower position, raising it to the Upper position should stop the slipping.

[agent_smith]

Nice work DDK.
Finally a post that communicates ideas and concepts in a more coherent and understandable way.

All thats needed now is to apply your math directly to the image I have attached.

Please consider that not all visitors to the IGKT forum are math geniuses. Being able to clearly communicate complex ideas so that the layperson (or non scientific) public can easily comprehend is the hallmark of genius.

Anyhow, the next step is to apply the math directly to the image - so the layperson can comprehend it.
I hereby attach an image of the #206 Crossing hitch for that purpose... (#206 is more widely known as the Munter hitch to rock climbers).

[KC]

To me, especially as less  math has been called for;
i simply count the 180 arcs on the mated surfaces as the att_frict research paper states..
1x against host and another time against SPart
.
My standard rule of thumb is 3 arcs for lowering, as like Round Turn, or same crossing self to Crossed Turn.
Here we have only 2 in Munter until fold back around host to be kinda Cow with only 1 leg thru off host crossing and other out,
>>but input and output are now laying next to each other.
But, 3arcs is probably my log reference and reduced that to 2 arc Munter that slides easier.
.
The other quantity here i see, another fave.
The self centering aspect of Munter lent to the SPart by the surrounding rope parts.

As it gets tensioned, these parts are more rigid  and form would be rope guides as like we carved into the round host slightly to maintain positioning.  i see this 'beauty'  in Constrictor and Bag etc. as well.
A Round Turn can cross self in powering and seize, Crossed Turn base is already crossed, but decidedly control side crossing over load side..
Munter/Backhand bases can't cross self to seize hard like the Round Turn and Crossed Turn bases when only 1 leg thru off host crossing.
i do recognize some more friction for the just not bottom side, but also these sides crushing increasing rope contact area in like what happens in ring or hook host with double bearing.

[DerekSmith]

The Guild is a forum on knotting. And so, we?ll keep it as such.
Any mathematical concepts presented here should be knot related and applicable to the use of knots/rope.
Understanding the forces that knots use or will see can be eye opening to the understanding and perhaps better utilization. That is what the ?Knotting Concepts and Explorations? Board is for.
That said, posts of pure knot theory or math don?t truly apply. Very many of the members have little interest in the formulas, unless they can actually apply them.

So, unless I get a huge amount of feedback to the contrary, I?ll not add a child board for ?Mathematical Concepts and Explorations? at this time.

Please keep the discussions knot related.

SS369

Hi Scott, I hope that you and yours are surviving this crazy lockdown.

As for feedback, I doubt that a child board on knot math will attract much attention.  But, understanding the concepts behind knot functionality and exploring novel ways to quantify the parameters involved is certainly of interest to me at least, and clearly a few others that frequent here.

The OP was introducing us to a novel and delightfully simple way of determining cf for whatever cordage we are working with.  The supporting math was, I believe, simply there to confirm the validity of the methodology.

As for cf being math, I think it is no more math than bringing breaking strain or pulley mechanical advantage into a discussion, but for those of us who study 'what makes a knot tick', cf is of far greater importance than the simplistic  tests of loading to destruction - yet both revolve around using numbers.

While long ago I argued strongly for the dissolution of 'Chit Chat' into classifications that at least allowed us to focus on broad areas that we found particularly interesting, there is equally the opposite nonsense of subdividing our field into such minutiae that you finish up with virtually every post in its own child board.  I appreciate that this is not what Mark is suggesting, but as we have seen virtually zero evidence of Knot Mathematics on this forum, I think the proposed new child board will remain desolate for some time to come.

Meantime, I would suggest that as there are so few numerically pertinent posts floating around, we all consider the mantra of -

' If it doesn't interest you - IGNORE IT '

Derek

[SS369]

Good day Derek.

All is so far, so good here, thank you for asking and I hope the same for you and yours! I am enjoying outdoor pursuits and lots of in-my-shop time.

A separate board would also create a whole lot of work to find and move the relevant posts.

What I ask, for the sake of those interested, is to add a ?teaser? in the thread title and to bring it home to knotting within the thread.

I personally find it all interesting, when I can relate it to my world, knots and rope use. Otherwise, for me, it goes to that Quantum Place. ;-)

Just keep it all germane to what the Guild is about.

SS

[agent_smith]

My personal (and likely lone viewpoint) is that the IGKT is first and foremost a knot forum.

At its grass roots level, it is about knots and exploring aspects directly related to both existing and newly discovered knots.
The math is fine - but, it should be applied to a knot structure (rather than existing as an abstract concept).
It comes down to what the IGKT is purporting to be.

We've got visitors to this forum who do not speak English as a first language. We also have posters who don't construct their sentences and paragraphs (ie their narrative) in a coherent and easily understandable way.

Anyhow, its a complex mix of people and competing interests.

I would like to think that it makes sense to apply the math to an image of a real knot - rather than existing as an abstract proposition. I've given examples of ideal candidates for this - eg... the #206 Crossing hitch is a candidate for applying the math directly to the image. And so is the 'Tensionless hitch' (#2047).

Full disclosure:
I am not a moderator. I am simply attempting to provide a viewpoint.
I believe in striving to be a good communicator - which means posting content that is coherent and easily understandable - even by the layperson. I also believe in supporting posted concepts with quality images that are real knots.

[KC]

On the other hand, math can be it's own border crossing shared language amongst it's users.
A common language made millenniums ago.
It can decode what is going on, even internally where can not touch nor see.
i try to apply to modules as single item to focus on, then expand a group of those modules to a full blown knot.
.
The use of radians is more accurate, it actually took the stars  ~365.25 nights to circle the Earth to circle, only rounded to 360.
That lends towards PI and more math.
.
Some of the math does actually blur by me sometimes not being formally trained where i wander;
but i never put that on the author.
My instincts once tuned in, have served me miraculously well; kept me alive and that includes knot insights where my math doesn't go.
.
The site is about knots, i focus on working, load bearing knots.  All load bearing structures are math.
Especially if want to decode what is going on in what stages.
Math is how we really pick stuff apart.
.
If show something per a knot and not abstractly, then mite stay 'hooked' to that knot.
Abstraction blurs by the stuff should not let eye stop at, to see the underlying commonalities in many.
Abstraction sometimes is like an X variable that can fit many things, without having to repeat the pattern 1000s of times for one lesson.
>>to  then be pummeled and buried by lesson or raw memorize of see persisting pattern
>>finding pattern, is leveraged learning, covering points did not individually stop at.
Many things work like a computer game/program or even car.
>>many models, looks, by changing outer skin; but still the same mechanix ruling the item under that skin.
Math is a language of science, oft what we use to make something a science; perhaps knots nothing different.
.
On of the best things i ever did was figure out how to tell angles as shown in hash/second/minutes marks, and then immediately know sine/cosine and some tangents.  Started with math, real work to figure, using at work as had to.  But eventually became constant companion that was there on the fly immediately decoding accurately whatever looking at, immediately becoming more familiar how everything works in everything, to these understandings being second Nature.  Became light play to use, even at work, feeling so good to accurately command and predict seamlessly with what was going on, and how it related to all else/where else could carry the lessons of that day to.
.
So yes, i give you math as others before, as the truest sharing; keys to all that.
"Euclid's Elements has been referred to as the most successful and influential textbook ever written."
>>was/may still be the 2nd most printed editions of any book(wiki)

[struktor]

Measurements without calculations.

You can scale the measuring lever in the coefficient of friction.

a/b = e ^{μ*(1+2*m)*π}
(L-b)/b = e ^{μ*(1+2*m)*π}
L/b-1 = e ^{μ*(1+2*m)*π}
b = L/(1+e ^{μ*(1+2*m)*π})

for m=0                                 fig. scale
b = L/(1+e ^μ*π)

for m=1                                 fig. scale (blue)
b = L/(1+e ^μ*3*π)

L- The total length of the measuring lever
L=a+b

[agent_smith]

Hello struktor,
Hope you are well.

Just asking a question here...
(They say humans are doomed because we ask questions).

With regard to your post at reply #24; was there any particular knot that you had in mind to apply your excellent work to?
It would be great to see the math concepts applied directly to specific knots/hitches (particularly for readers who don't have a strong math background).
And when I use the term 'math concepts' - it also includes math equations.

Or is your post intended only for a select group of individuals (just asking so its clear)?

[struktor]

Hello,

I am a retired engineer.
This has an impact on my technical problem solving.

Knot Testing System (# 206).
The model was made in the FreeCad program.
The bearing can be repositioned to change the angle.

The weight F should be much heavier than the lever (beam).
The sum of the weights is shifted by the error value.

[KC]

Please, though just Muenter/BackHand Turn Base type shown specifically,
is there Crossed(on or off host) or Uncrossed Turn single or group of forming a Bend or Hitch where this would not apply?
.
i think  this capstan and pulley effects are in ALL linear force fed (thru SPart of Hitch & Bend) controlling arcs of knot internals;
and also see same in ALL rigging; as concepts of rope mechanics engulfing ALL linear force rope usage,
of which Bends and Hitches are a small island/segment inside.
.
Once again, as defining contrast, separating Round Binding as a radial induced force
>>from host expansion evenly diffused all around w/o favored direction
>>from inside controlling arcs domain
vs. focused linear directional external force into controlling arcs of Hitches, Bends, Rigging etc.
Round Binding is all controlling arcs, evenly loaded until nip(s) w/o direction.
>>has no SPart for linear input during USAGE, perhaps a linear feed/SPart in setup, but not usage.

[DerekSmith]

Hello,

I am a retired engineer.
This has an impact on my technical problem solving.

Knot Testing System (# 206).
The model was made in the FreeCad program.
The bearing can be repositioned to change the angle.

The weight F should be much heavier than the lever (beam).
The sum of the weights is shifted by the error value.

This is ingenious - BUT - in understanding the result, don't you first have to quantify the cf for cord on cord and cord on stator?

Have you run this technique in real life, and if yes, have you observed any anomolies as load causes plastic deformation of the cord?

Also, you have given us a most ingenious method for studying the impact of knot structure on locking due to 'cogging' which is complex because the cord actually rotates as well as sliding.

Oh happy knot nerd times...

Derek

[agent_smith]

Breathing life back into an old topic:

The #206 Crossing hitch (aka Italian hitch / Munter hitch) is a good knot structure to directly apply math.
The humble Italian/Munter hitch is used routinely around the world as a belay system (ie it is used to arrest falls).

Refer to attached images.

Perhaps a keen mathematical wizard could apply the math directly to the images, with step-by-step calculations?

The top image is a proposed test rig to measure load at the tail (brake-hand) end of the climbing rope.
Changing the angle of the brake-hand with respect to the carabiner will affect the degree of grip strength required to arrest fall.

The pulley is not ideal - having approximately 5% loss of efficiency.
Could also substitute a carabiner with 5mm bending radius (which is realistic) but adds another variable.

I'm still waiting for my new linescale 3 load cell...at the moment I cant measure force.
Maybe a math wiz could make predictions - and then I'll check with load cell later?

...

Math is good

And practical application to a real-world 'knot' is even better.