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# Low-efficiency PCD’s

Tuomas Pöysti 2020

In this article I studied how different pulley systems react to varying pulley efficiency, especially at low efficiency levels of carabiners. Mostly, all pulleys were considered equally inefficient.

PCD (progress capturing device) is a special pulley in this sense. There are popular PCD’s that make even worse pulleys than carabiners. These include descenders and EN 15151-1 belay devices like Petzl I’D and Grigri, tube type belay devices with autoblocking mode and such.

For these devices, a general efficiency value 30% is not even a pessimistic one.

I tend to define pulley systems for analysis so that P1 is the most obvious PCD candidate. Let’s study some pulley systems and find out how 30% P1 changes them compared to all-equal pulleys.

## “Crevasse” and “Double mariner”

“All varying” means that efficiencies of all pulleys vary from 0 to 1. “30% PCD” is otherwise the similar, but P1 is constantly 30% efficient.

“Crevasse’s” MA expression is

1 + P2 + P3 + P1P2 + P1P2P3

and “double mariner’s”

1 + P2 + P3 + P1P2 + P1P3 + P2P3 + P1P2P3

Compared to all 80% pulleys, changing to 30% PCD drops the “crevasse’s” MA by 0.4 and “double mariner’s” MA by 1.1, the end results being 3.5:1 and 3.9:1, respectively.

If P1 is an inefficient pulley, the “double mariner” is an especially bad idea. Attaching the blue 2:1 to the rope on the right side of P1 adds second degree term P1P3 and third degree term P1P2P3 which in this case have respective values 0.3*0.8=0.24 and 0.3*0.8*0.8=0.19. The difference of 0.4 between said MA’s is their sum, of course.

## Simple pulleys

The simple 2:1, 4:1 and other systems with even ideal MA are a bit complicated when it comes to PCD, since P1 tends to be what I have elsewhere called a “positive” pulley. It cannot be connected to anchor – rather, these pulley systems work only as “drop loop”, and while P1 is near the load, the system might as well be a “Z on V”, a 2:1 pulled by a 3:1. Unless of course, the haul is short and there’s no need for resetting.

On the other hand, simple systems beyond 5:1 are almost useless in the context of climbing equipment. It leaves us with the Z-rig and 5:1. Their MA expressions, respectively:

1 + P2 + P1P2

1 + P4 + P3P4 + P2P3P4 + P1P2P3P4

The Z-rig has calculated efficiency of 2.44:1 using only 80% pulleys. If P1 is dropped to 30%, the result is 2.04:1, a 16% drop.

The corresponding numbers for 5:1 are 3.36:1, 3.11:1 and 7%. For a change, something that goes at least relatively better with a higher MA simple pulley!

This is simple to explain. Check the MA expressions and see what kind of terms “bad” P1 gets to ruin in each case. In 5:1, it is the fourth degree term with value 0.8^4 = 0.41 to start with. In practice: P1 is “deep” inside the pulley system and only gets to tax what is left after all the other pulleys, and only a small fraction of MA is based on it.

Actually trying a 3:1 with a high friction PCD might be educating. As said, using a 30% PCD and 80% pulley as the “closer to hand” one, something like 2:1 can be expected as the effective MA. It is like having a 2:1 pulley, but at the same time just barely getting the slack pulled through the PCD. All you need to do is to stop expecting for any MA through the PCD, and you’ll be happy!

## The “lab mouse” pulley systems

Just to have some material to play with, let’s take my “lab mouse” pulley systems in.

Their MA expressions in form of a table:

Let’s try yet a different approach. Pulley systems also have efficiency. Naturally, it is the ratio between actual and ideal MA’s – in this case calculated and ideal.

Here are two figures for each pulley system. First, their efficiencies assuming all 80% pulleys. Then, their efficiencies assuming 30% PCD and 80% efficient other pulleys.

B, D, E and F have high MA’s and thus a bit lower efficiencies (they have more high degree terms). To be an 11:1, D is actually quite modest with high degree terms, and it shows in the figures.

Pay extra attention to A and C, though. They have especially few terms involving P1- that is, they are insensitive to P1. In general, none of the pulley systems in this article have first degree term P1, which is a very nice thing considering PCD’s.

These pulley systems are generally more sensitive to P4. Here’s what happens if P4 is 30% efficient and the rest are 80% (the yellow bars):

## Conclusions

It is a nice consequence that the most obvious PCD location is typically a pulley, to which the pulley system is not very sensitive when it comes to efficiency. Maybe it’s not a consequence, though – the PCD works best closest to load, whereas the most efficiency-wise important pulley is usually at the other end of the pulley system.

If the PCD happens to be especially sticky “pulley” such as a descender, pulley system analysis might help avoiding bad decisions.

## 2 replies on “Low-efficiency PCD’s”

Dear Sir,
I find your posts very valuable and I read them even though I barely understand anything, since I have no knowledge of physics and math.
The reason is that I like pulleys and the “magic” of it.
I am better at understanding visually so I implore you to maybe make a video or a post with real pulleys with these suggestions and conclusions about how we are to make the best and most efficient pulley system, or the optimal system according to your knowledge.