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# Pulley system composition – a systematic approach

Tuomas Pöysti 2020

This article builds on the basic theory of pulley systems and partly on my method of analyzing pulley systems. The English texts are quite recent as I’m writing this, but they are based on older texts and ideas. I’m constantly (or at least in recurring bursts) learning more about pulley systems, and this text contains some newer ideas. I’m afraid it’s a bit academic one, and the balance between compactness and understandability may be a bit clumsy.

## A change of direction?

This is something I realized way too late: The fact that a pulley system is pulled upwards or downwards (i.e. if it implements a change of direction) should not be treated as an attribute of the system.

Consider the “mystery” pulley system on the left:

It is ideally a 3:1. It is pulled upwards by force 1, it hauls load 3 and thus the anchor must see reaction 2 in order to balance the forces. No matter what is inside the mystery box, if we turn it upside down between load and anchor, the result will be an ideally 2:1 pulley system that is pulled downwards. Thus:

There is an even number of pulley systems, half of which are direction changing. Any upwards pulled pulley system S can as well be seen as a direction changing one S’ with one unit smaller ideal MA.

This means that it is sufficient to study pulley systems with no change of direction. This is part of the definition (in progress) of ordinary pulley systems.

In addition, the one unit smaller MA

MA’ = MA – 1

is not only an ideal approximation, but holds for calculated effective MA’s, too. The missing term 1 is the actual input force, not affected by pulley efficiencies.

## Pulley system elements and their allowed connections

For deeper analysis, I define pulley system parts as elements. Elements have a direction, and depending on element, some other properties.

The restrictions concerning connections between different elements are mostly premises. Their function is to further specify the definition of ordinary pulley systems.

• MA of ordinary pulley systems is an integer greater than one
• Every pulley adds MA in an ordinary pulley system
• Positions of all element are fully defined with respect to each other in an ordinary pulley system.

All of these points are not yet fully covered by the restrictions (there will be examples later). At this point, the restrictions are based on a very intuitive reasoning, and of course are subject to change as I learn more.

### Element’s direction

Most of pulley system’s elements are meant to move. Ideally, all elements and movements are vertical.

• If an element moves upwards, it’s direction is positive.
• In other cases it is sufficient to say the element is non-positive.

For lack of better symbols, I tag non-positive elements with a minus sign. The only reason for omitting term “negative” is to make the anchor element fit the set of non-positive elements.

### Anchor

In some texts, I have used term ground, but it might be misleading, since this element is up by definition. Pulley systems are defined to haul loads upwards, against gravity, towards the anchor.

Anchor is a stationary element. It also serves as the frame of reference when it comes to moving directions.

Anchor and all elements attached directly into it are non-positive by definition.

The load is hauled upwards, and thus positive by direction.

### Input element

The input element of an ordinary (not direction changing) pulley system is a positive element. It can be thought to stand for the “pulling hand”. The input element is or is connected to a single positive rope element.

### Pulleys

Pulleys are connected to exactly three elements each:

1. One positive rope element at sheave
2. One non-positive rope element at sheave
3. One element at pulleys attachment point

Pulley may be in one of two orientations U (upwards) and D (downwards):

A U pulley is always connected to a positive element at the attachment point, a D pulley is connected to a non-positive one, respectively.

Thus, a U pulley’s attachment point may be connected to:

• Any point of a positive rope element

and a D pulley’s attachment point may be connected to

• Anchor
• Any point of a non-positive rope element.

### Rope elements

All pulley systems have more than one rope elements. What is a single rope in real life, shuttling between pulleys of a simple pulley system, is in my terminology a set of rope elements.

A rope element is any rope leg, a straight portion of rope. If a rope runs through a pulley, it is said to consist of two rope elements, each connected at sheave of the pulley. As said earlier, one of the elements must be positive and the other must be non-positive.

Both ends of a rope element must be connected to some other element. It is possible (in a limited manner) to connect other elements to any point of a rope element.

If a rope element is positive,

• It’s lower end can be connected
• at sheave of a U pulley
• to any point of another positive rope element
• It’s upper end can be connected
• at sheave of a D pulley
• to the attachment point of a U pulley

Quite symmetrically, if a rope element is non-positive,

• It’s lower end can be connected
• at sheave of a U pulley
• to the attachment point of a D pulley
• It’s upper end can be connected
• to anchor
• at sheave of a D pulley
• to any point of another non-positive rope element

### An example

Hopefully an example helps:

This Z-rig is divided into three rope elements R1 … R3, two pulleys, load, anchor and input elements. Once again, all elements exist ideally in the vertical dimension only; the rope elements R2 and R3 are tilted for clarity.

In this example, R1 and R3 are positive rope elements, as they are connected to load and input element. R2 must be non-positive, since each pulley must have one positive and one non-positive element at it’s sheave. It’s a matter of taste if P2 is connected to a middle point of R1 or directly to load. Both are permissible by the conditions laid out earlier.

## Up to three pulley systems

Let’s do some exercises with these tools and try to find out what kind of ordinary pulley systems are possible using up to three pulleys.

### One pulley

According to the conditions about a pulley’s connections, there must be exactly one positive rope element and one non-positive element. Something must be connected to anchor, load and input element.

• These can be connected to anchor:
• A D pulley’s attachment point
• The upper end of a non-positive rope element
• These can be connected to load:
• A U pulley’s attachment point
• The lower end of a positive rope element
• These can be connected to input element:
• A positive rope element

The last one is the key: as the input element needs to be positive, there are no other solutions than a single U pulley, the simple 2:1 “V”.

It’s upside down version is of course a 1:1, a mere 180º deviation. That is, the options are one ordinary pulley system and it’s upside down counterpart.

### Two pulleys

Now there are more options. But again starting with the requirement for positive rope element connected to the input element, there has to be a U pulley with the input element as it’s positive rope element. This is common for all ordinary pulley systems.

Now that there is another pulley to come, there will be other options than to connect the non-negative rope element to anchor and the pulley’s attachment point to load. But they are quite limited.

It is possible to connect:

• The upper end of the non-positive rope element
• at the sheave of a D pulley
• to the anchor
• The attachment point of the U pulley
• to the upper end of a positive rope element

At this point, it helps to figure out only the possible connections between the pulleys. We can figure the other connections later. That is, there are only two options left, two branches of investigation:

As for the other connections, there is only one solution for each branch:

In addition, there of course are their upside down versions, a simple 2:1 with a change of direction and a downwards pulled 3:1.

### Three pulleys

The third pulley certainly increases the degrees of freedom and thus the number of “investigation branches”, but things are surprisingly manageable, though. I try to make it comprehensible without getting lost in the details.

Starting with the two branches of the two pulley case, and systematically enumerating all allowed ways of connecting the third pulley, we end up with seven branches in the next step, out of which six are unique (compare 1c and 2b):

#### Branch 1a

This is a very good example of how important it is to carefully process all possible connections, not just the obvious ones. To me, at least, 1a seems a clear case of simple 4:1 and nothing else – that is, the rightmost U pulley should be connected to the load. But it is possible to connect it to the positive rope element, too.

The rest of the connections are unambiguous: there are only load an anchor elements to connect to. We are left with two pulley systems, the simple 4:1 and “Z on V” 6:1:

#### Branch 1b

This is an interesting one, too. I have to admit I never before came up with the pulley system that seems to be the only outcome of this branch:

Not that this 5:1 was anyhow practically useful, but I have been interested in impractical pulley systems, too, and this was a new one to me for some reason. This means that the systematic approach works!

#### Branches 1c, 1d and 2b

As said, 1d and 2b are identical, so lets discard 2b. The outcome is a set of six pulley systems, two of which are identical (1c2 and 1d2):

Now we encounter some problems with my rules about connections: they allow some invalid pulley systems. 1c4, 1d1 and 1d3 do not fulfill the requirements of ordinary pulley systems as I’d like to define them. Besides they are either stupid, non-working or both.

But the rest four (or three unique) pulley systems are already familiar:

• 1c1 is “the crevasse”, 5:1
• 1c2/1d2 is what I and Petzl have called “double mariner”, 7:1
• 1c3 is again “Z on V” 6:1

#### Branch 2a

Also this branch produces a non-working pulley system (2a1), that has a similar problem to 1d3. It seems quite simple to create new rules to prohibit this kind of connections, but I don’t want to be too quick.

The other solution is a feasible pulley system, the “V on Z” 6:1.

#### Branch 2c

This is a simple one. The non-positive rope elements could be connected to each other in various ways, but the end result is anyway the same as if they all were connected to anchor – the “V on V on V” 8:1:

## Discussion

A week ago I was convinced that 1) there are exactly six “ordinary” pulley systems that use three pulleys and 2) that I can convince others, too, by systematically studying pulley systems’ anatomy. But a bit paradoxically, I was slightly discouraged by the fact that the systematic method seemed to work and introduced a pulley system I was unaware of.

Now I’d say I’m curious if there are more “ordinary” pulley systems with up to three pulleys than:

• One pulley: one system
• Two pulleys: two systems
• Three pulleys: seven systems

That is, ten systems in total. And If we count their upside down versions as “decent” pulley systems, there are another ten systems.

The number of possible pulley systems seems surprisingly low to me. Before actually trying to “collect them all”, even up to three pulleys, I somehow thought there would be substantially more options – no idea why.

The fourth pulley probably makes a big difference here. What changes between two and three pulleys is the possibility to connect rope elements to other rope elements (in practice, use rope grabs on rope legs between two pulleys). There will be vastly more branches in the investigation schemes of four pulley systems.

That leads us to using a computer program to enumerate the candidates. In order to do that, the restrictions or conditions for element connections should be a lot more precisely defined. For example, I have quite intuitively skipped the idea of connecting an attachment point of a U pulley to the positive rope element that is connected at the same pulley’s sheave. A computer will not automatically do that, because it does not have the gift/burden of human thinking.

The more precise conditions will prevent such loops and hopefully other useless connections, such as in the cases of 1c4, 1d1 and 1d3 above. Or then it has to be done an uglier way, simply pruning the non-viable solutions by hand.

For me, one of the biggest problems of this article is the absence of definition of “ordinary pulley system”, not to mention “pulley systems” in general. Hopefully I’m some day able to fix these problems.

Finally, let’s summarize what I think are the possible pulley systems consisting of up to three pulleys. I’m very interested in seeing more, because that way I’m able to enhance my methodology!