Rope Drag Simulation

In rock climbing the rope drags you down. If there are a lot of bends in the rope, it REALLY drags you down.

I wanted to know how rope drag evolves under different circumstances, so I made a program that simulates rope drag for climbing routes on a vertical wall.

On the first attempt I assumed the rope path to be completely straight between points.

The program starts at the first point and builds to the last point. At each point the drag calculation is:

And that worked pretty well.

One complexity I wanted to add was that ropes droop between supported points.

That matters because the bend angles change, and therefore the friction forces are different.

And a sideways path would show zero drag added if the rope path was assumed straight, but in reality the rope droops, and the bend angles aren't zero.

The shape of this droop is special because it follows a catenary path, not parabolic, or anything else. This is exactly the shape of least potential energy for an idealized rope/chain between two supported points. Interestingly, inverted catenaries create the strongest arches.

This path is also interesting because it also has implications with calculus of variations.

The shape of least potential energy results in the same horizontal component of tension at every point.

With the horizontal force and the rope density, the shape of a catenary is completely defined.

With the catenaries, the drag calculation at every point looks like:

Which seems like a never ending circle.

1. Different rope path

2. Different angles

3. Different friction

4. Different rope path

5. Different angles

6. Different friction

7. Different rope path

8. Different angles

9. Different friction

10. Different rope path

11. ...

Which seems like iterations.

I think seeing these catenary paths converge is beautiful.

A realistic path for the rope is also cool.

A perhaps tricky part when iterating is solving for the catenary path of the rope from the current point to the next point. The updated estimation for drag from the previous iteration defines the shape of the catenary curve.

But where do the points fit on the curve?

One interesting thing is that if the points are too far spaced out the weight of the rope will iteratively droop down to negative infinity under its own weight, just as it would in reality if something didn’t stop it. Divergent result:

But doesn’t this all heavily depend on how you calculate the friction at each bend? Yes. Yes it does. The Capstan equation is commonly used for climbing ropes and carabiners from what I could find. I mean, it says so on Wikipedia and this MIT thing, so there is that.

• en.wikipedia.org/wiki/Rope_drag

• ocw.mit.edu/courses/experimental-study-group/es-255-physics-of-rock-climbing-spring-2006/lecture-notes/MITES_255S06_rope_behav.pdf

Compared to the general equation for friction, the Capstan equation basically just accounts for the extra surface area as the rope bends around something. I didn't think the surface area would increase like the equation represents, so I did a garage experiment that was not scientific in any way.

"Multiplier" is what the drag before the bend is multiplied by to get to the drag after. That just normalizes things so you can compare across different loads.

(drag before) * (multiplier) = (drag after)

This is how the equations compare. 'Multiplier' on y axis. Bend angle in degrees on x.

The experiment was almost not rigorous in any way. Didn't calibrate instruments, low sample sizes, no uncertainty consideration, etc. Inferences made here are loose at best… I don't think there were any confounding variables I didn't reasonably control for though. (humidity)

My interpretation is that friction with climbing ropes and carabiners is probably closer to what you would expect with the general equation, FF=μN, which assumes constant surface area contact. I thought it was interesting that the equation I saw being used most often, the capstan equation, had by far a greater error from the experimental results, and bends the wrong direction overall. I think this is likely due to the advertised limitations on rigidity or stiffness.

All the results below were calculated with a modified version of the experimental data trendline equation. Below ~2 lbs I found there is slightly more friction than the trendline suggests. Without the correction, the rope seems to droop to infinity at much shorter traversing distance than in reality at small loads. My instruments were not optimal for that range. I did a little more careful testing later and think there is some small load dependence at small loads, and I think the corrected equation better reflects reality.