Line deviation and pulley efficiency, part II

Tuomas Pöysti 2020

Earlier I did a little empirical study to compare one 180º deviation with two 90º deviations using a similar pulley. I thought it would be nice to have some data points on other angles than those two. To be exact, I never even measured a single 90° deviation, but calculated a hypothetical value from case of two consecutive deviations.

Materials

I included four different pulleys:

• Rock Exotica Omniblock 2.6 (66mm sheave)
• Rock Exotica Omniblock 2.0 (50mm sheave)
• Petzl Pro Traxion (38mm sheave)
• Rock Exotica Omniblock 1.1 (28mm sheave)

Of course this does not isolate sheave diameter as a parameter, because the pulleys are different in other ways, too – especially when it comes to bearings. I stick to my firm stand that bearings (actual rolling ones) have a negligible effect on pulley efficiency, though. This does not mean that there couldn’t be other significant differences, maybe from an unexpected source such as groove geometry.

I used my standard method and Beal Access Unicore 10.5mm. The deviation angles are due to locations of trees on the common yard of the apartment house I live in. That is, I measured the angles afterwards. For some reason I did not include the case of 180° tests in the original set, so I ended up adding it later in a different location (my balcony “lab”) and on the next day.

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Results

I measured these values:

or as plotted against deviation angle:

Needless to say, I did not actually measure the “0º” values. They surely should be 100%.

Notes

First off, there’s probably something wrong with the 180° values. As said, they were measured subsequently in a different location. I carried out my normal “calibration” test before each session: when the cells were connected in series, I got 1.0036 and 1.0017 “efficiencies”, respectively (the expectation being 1.000). Thus, the 180° values are prone to errors of magnitude of 0.17% due to this inaccuracy and the rest up to 0.36%. But only to assess the accuracy limits of the two-cell method. The potential error in the values may be caused by numerous other things, like different rope part etc.

The overall shape of the plot is interesting. It seems that after certain threshold, increasing deviation angle does not affect pulley efficiency. According to this data, the threshold angle could as well be 90º or 130º.

In the earlier article linked above, I hypothetically calculated that a single 90º deviation using a pulley with 28mm sheave would yield something like 86.4%. This is very close to the measured value of 86.9%. The decimals are sheer noise, though.

How about the resultant force hypothesis, then?

The force pressing the rope against the sheave is the resultant of the force vectors of each rope leg:

Let’s assume the force loss at a pulley (with a sheave of given size) is proportional to the resultant force:

L = k*R,

where

L = 1 – efficiency.

Thus,

k = L/R,

so L/R should be constant.

The resultant magnitudes for deviation angles of this study are:

• 180º: R = 2.0
• 145º: R = 1.91
• 96º: R = 1.46
• 60º: R = 1.0

L/R values for each sheave size look like this:

The values are not constant enough to get me excited. The curves are somewhat linear, but not horizontal. I’m still convinced there are other factors as well, at least.

Edit: about “count of changes in curvature hypothesis”

After one night’s sleep, some extra thoughts.

For a long time I have had an idea about number of deviations being a key parameter with respect to pulley efficiency – or descender braking power.

For example, according to my hypothesis, the capstan equation would not explain the friction between a rope and a rappel rack, because it merely factors in the cumulative angle at which the rope is in contact with the device. This is not to say the capstan equation is not valid when there are only a few points where the curvature of the rope changes (when the rope enters the capstan drum and when it leaves it).

We have already seen that two 90º deviations add more friction than one 180º deviation. But what would it look like if we assumed this is not the case, that it does not make any difference to the total efficiency whether there are one 180º, two 90º, six 30º or 180 one degree deviations?

Let’s assume there’s a total deviation of 180º, which is divided into N smaller deviations of equal angle D:

N = 180º/D

Further, let’s assume that each deviation has equal efficiency e, and total efficiency E is their product

E = e^N and thus

e = E^(1/N).

-> e = E^(1(180°/D)) = E^(D/180º)

Using a capstan like accumulated total efficiency E = 0.3 shows the exponential nature of the function:

But in case of decent pulleys (E = 0.85), the function is practically linear:

Although we approached this function from another direction, it is essentially an approximation of what I measured in the study. Each small deviation has efficiency e, no matter if there are N in a row (or on a circle…), and thus this is an approximation of pulley efficiency as a function of deviation angle.

But it surely is a bad approximation, since the reality is something like this:

There might be other explanations as well, but to me the most interesting question is if this has something to do with the fact that the approximation discards any “fixed costs” a pulley might have with regard to efficiency.

The changes in the rope’s curvature (or bend radius) as it enters and exits the pulley’s sheave or capstan drum is something each deviation has the same amount, regardless of deviation angle. But this is just imagination, what we need is more experimental studies!