Measurement of the coefficient of friction for the capstan equation.

[struktor]

Hello  agent_smith,

B=Rope-on-metal friction , more then 180deg angle of contact.

In the absence of friction, it can reach 180deg + 90deg = 270deg .

180deg + beta

0 < beta < 90deg

[agent_smith]

Thanks 'struktor',

The contact angles in my real-world image of a #206 Munter hitch were (obviously) approximations.
I wasn't really intending to say 179.134 degrees, or 189.3256 degrees (or whatever) etc.
The 180 degree U turns in my photo image are just approximations.

Be that as it may, is it within your scope of math knowledge to be able to directly apply the capstan equations directly to the Munter hitch photo image?
Or is this outside of your scope of knowledge? (its a genuine question - not an insult!).

I will have my new load cell late October, and could verify your calculations.

The coefficient of friction for rope-on-rope, and rope on metal is something you would have to work out?

Note also that under load, modern synthetic rope flatten a little bit and so there is more surface area contact (particularly at the rope on metal carabiner interface).

I suppose that if you knew the tension force at the load cell, you could work backwards and derive the coefficient of friction?
There will be static friction to initially overcome and then once the rope is moving, it is now kinetic friction?

I can easily measure the load while the system is in equilibrium.
Not sure how I would measure load at the brake end of the rope while it is moving? Maybe attach the load cell to the free end of the rope, hold it in my hand and then allow the rope to flow through the Munter hitch (and observe the LCD screen display)?

Summary:
What do you think 'struktor' - can you apply the math direct to the real world image of the Munter hitch and try to predict the force on the brake hand end of the rope?
Make your math predictions with calculations directly applied to the Munter hitch image (rather than abstractly on computer generated imagery).
Then I could check your predictions once I have my load cell...

Assumptions:
[ ] Rope would  be EN892 Beal 'Joker' 9.1mm diameter Link: https://sport.beal-planet.com/en/mountain-line/1418-5132-joker-91mm-gd.html#/14-color-blue/58-length-50m
[ ] Carabiner has radius of 5.0mm
[ ] Contact angle (rope-on-rope): Approx 180 degrees / Pi radians)
[ ] Contact angle (rope-on-carabiner 'first' point): Approx 180 degrees / Pi/2 radians) - feel free to be more precise!
[ ] Contact angle (rope-on-carabiner 'second' point): Approx 90 degrees (variable)
[ ] Mass held by Munter hitch belay = 100kg

Challenge accepted?

[struktor]

Thank you, agent_smith,

Need someone with FEA software (eg. ANSYS).
Examples of the MES program:
  https://youtu.be/3Y0EIBbXHi8
  https://youtu.be/ne9fxOG7f8A
http://personal.strath.ac.uk/andrew.mclaren/Steven%20Welsh%20Technical%20Paper%202013.pdf

I use a large simplification.
How does rope-to-rope friction work?
They can be simplified by adding an insert?
The figure shows a rope friction insert.

[KC]

Nice eye candy, what are you drawing with please Struktor?
i L-earn from drawing too, is all geometry..
And note that drawing tools basics are always a flat linear face or radial for 2D objects;
as these are important concept differences all across the board into drawing as knotting etc.
Thanks so much for links too!
.
In reply #30 tho, i do look at only the 180 arc opposing the pull from opposite side of the host as the only radial friction component that uses all tensions for seating to host to give then controlling frictions, nips and grips(w/oppsing180).  Using both the byproduct of deformation as side force and the force holding primary against Load too, as 1 for seating forces to host.
But i shy from calling The SPart (that for me ends halfway around the circle where 180 starts) and the pull from Bitter End seize as radial friction but rather more linear parts especially viewable here where 1 end pull towards Load if each of these legs.  But in in any case more linear rope part if endpoints in opposing directions.  Pure axis inline not a harsh point for there is no cross axis resistance, so directional claiming just a directional axis (vertical vs horizontal etc.) not a big point as would be in rigids that do resist on the cross axis.
.
Uniqueness of 180 arc would be that both ends, as center apex does, as in fact the whole component does
>> pull and work all as 1 in the same unique direction, as no other form does. 
>>giving structural arcs to bridges and ropes etc. thru this geometry.

[struktor]

Thank you KC,

I am currently using FreeCad.
https://www.freecadweb.org


Good programs cost a lot.
But they can also do a lot.
Finite Element Modeling of Tight Elastic Knots
https://arxiv.org/pdf/2010.09109.pdf

[struktor]

Angle of friction.
https://en.wikipedia.org/wiki/Friction#Angle_of_friction

The rope slides down the vertical rod.
It will stop before it reaches a right angle.
This is due to the angle of friction.
The same will happen when the rod is replaced with a rope.

[DerekSmith]

@structor

Hi, I have just tried this form of cf test rig with a slight variation,  Instead of a vertical rod I clamped an 11mm DMM aluminium bena horizontally.  I passed a length of 550 nylon paracord around the flat end and took this horizontally to a load.  The cord had 180 degree surface contact.

I then moved the load end horizontally until it just started to slide and marked the angle.  I then moved the load in the other direction until again it just started to slide in the other direction, and again marked the angle.  The angle was reasonably constant with various loads with a total swing of 26mm at a distance of 100mm from the contact point on the bina - or += 13mm from the perpendicular  This made the tan of the angle to be 0.13, i.e. 7.4 degrees.

Now the web reference you gave stated  tan θ = μs , or 0.13  where previously using the capstan test system I had arrived at a μs value for 550 on polished aluminium to be ca 0.09, so sort of closeish...

BUT -   although the cord load made little difference to the slip angle, the ark of contact certainly did.  Of course, this is directly in line with knots such as the prusik , the VT where multiple turns rapidly escalate the grip.

So, do you have any idea how we are supposed to incorporate the number of radians of contact into the sliding angle method?

Derek

[mcjtom]

You may be interested in this Knotting Matters report:

https://forum.igkt.net/index.php?topic=7281.0

Part of it tries to determine the static friction coefficient of a 3 mm polyester cord on a bamboo spar (turns out to be some 0.15 +/- 0.03, see Table 2 and the graph before it).

The method uses the same Capstan model but employs regression analysis to find the most likely/plausible value of the friction coefficient for this pair of materials based on a range of contact angles (wraps), a range of working loads, and the measured rope tension reductions resulting from those arrangements.

One useful consequence of doing it this way is that it allows for estimating the realistic uncertainty of the computed friction coefficient value, which turned out to be relatively low in this experiment (given the simplicity of the equipment and procedure used).

I think that using similar techniques to estimate friction of rope-on-rope within knots will be problematic - there are papers describing such attempts in the textile industry.  It's complicated...
 

[mcjtom]

Summary:
What do you think 'struktor' - can you apply the math direct to the real world image of the Munter hitch and try to predict the force on the brake hand end of the rope?
Make your math predictions with calculations directly applied to the Munter hitch image (rather than abstractly on computer generated imagery).
Then I could check your predictions once I have my load cell...
.
.
.

Challenge accepted?

Figure 2 in the same paper above shows the general results for a Munter-type construct as well.  It turns out in this setup that adding a Munter hitch was worth about one extra round turn (360 deg) on the spar in friction terms.

It is possible to roughly estimate the friction coefficient of rope-on-rope within the turn in Munter construct using the data in the paper (by splitting the friction into three parts: rope on spar prior to Munter turn, the Munter turn, and rope on spar following the Munter turn).  For the cord used in the experiment, it seems to be somewhere in the 0.40 - 0.45 range, which is, not surprisingly, about 3 times higher than the coefficient of friction between the same cord and the bamboo spar surface.

Measuring (or guessing) the friction coefficients between a particular rope and a carabiner and rope-on-rope can probably lead to a fair estimation of the tension reduction in a simple Munter on carabiner (including the effects of angle changes of the free/holding rope end - to increase or reduce the friction) but the model will likely fall apart when applied to more complex knots.

[struktor]

There are better specialists than me.
https://www.sciencedirect.com/science/article/pii/S2352431622001080

[mcjtom]

Awaiting a tomograph, this little calculator may work as a first approximation of the braking-hand-side tension reduction when repelling with a Munter on a carabiner.  Values in the white fields can be changed at will.


For it to even work within reason, the friction coefficient values need to be realistic for the particular rope/carabiner pair (best if measured).


Would love someone to verify it in the field.  Mark?