Tuomas Pöysti 2020

As I mentioned in the end of this article, the capstan equation cannot generally describe a rope-carabiner or a rope-capstan combination as a pulley. The rope’s internal part of friction is hard to explain where the rope does not undergo any deformation.

## Some notes on deformation

If a rope is wrapped three times around a capstan drum and then forced to slide against it, there seems to be two places where the rope sees a momentary deformation. They are the points where the rope enters and exits the drum. At enter point, the straight rope assumes a constant finite bend radius, and at exit point it straightens. The number of rounds does not make any difference.

Curvature is defined as the reciprocal of bend radius, and despite the fact that a straight curve’s curvature is zero, it still *has* a curvature.

I see no reason to assume that the constantly curved portion of the rope produces any more internal friction than the constantly straight portion. The absolute curvature does not seem to be relevant, but how the curvature changes.

## An empirical experiment

I finally carried out a pulley efficiency test using other than 180º deviation. To be exact, I measured both the typical 180º deviation with a pulley, and then the same deviation consisting of two consecutive 90º deviations.

I don’t have an actual lab, and this was too much for my typical balcony test site, so I headed for the common yard in front of our house.

I used my standard method of measuring pulley efficiency. I repeated six times with both setups. I used a different line material than usually, Beal Access 10.5mm Unicore. The pulleys were Petzl Partner. In the 180º setup, I used both pulleys three times.

The result were:

- One 180º deviation:
**84.2%**(average deviation 0.5%-points) - Two 90º deviations:
**74.7%**(average deviation 0.6%-points)

## Discussion

First off, two consecutive 90º deviations clearly does not equal one 180º deviation when it comes to efficiency.

I did not try to measure an actual 90º deviation at this point, but we can estimate it by assuming that the 2×90º result is the second power of a single 90º efficiency value. Thus, single 90º efficiency would be square root of 74.7%, that is, **86.4%**.

In the future I will measure actual 90º deviations using several pulleys and pulley-like devices, and I’ll be surprised if that value is grossly wrong. Not to say it would be the first time, though.

We have three suggestive data points:

- 180º – 84.2%
- 90º – 86.4%
- 0º – 100%

It will be interesting to empirically fill in the gaps (and of course to directly verify the suggestive 90º value). Thus far the plot looks like this:

Or, expressed as force lost in the pulley:

But how could the deformation theory explain these values? I have two thoughts on that.

Firstly, it might only be a partial explanation. Another factor might be the resultant force the rope legs apply to the pulley (at least between 0º and 180º). It’s magnitude is (as calculated by simple vector math):

which somewhat resembles the force loss plot, but this is not a reason to get too excited.

Secondly, the rope’s internal, energy consuming phenomena must distribute over a certain axial distance. Let’s say they have to do with friction between core fibers. In that case it cannot occur in an arbitrarily short rope segment. Same goes with other models that are based on differing tension over the rope’s section.

This would explain why the consecutive curvature changes may not be arbitrarily close to each other in order to make a difference. Consider these three capstan or bollard cross sections:

We already know that the leftmost one, in form of a two-carabiner toprope anchor, is clearly less efficient than a circular profile (or a single carabiner). The transition from that to circular is continuous, so it’s hard to believe the corresponding change in efficiency comes with any steps, either.

This is mostly speculation, we’ll see as we study it!