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Pulley system basics

I’m into pulleys and pulley systems. If you read my articles about them, you might find me accidentally diving a bit deep sometimes, but I try to keep this text as simple as possible.

Pulley systems are like bicycle gears. They are able to exchange between speed and force. According to basic principles of mechanics, if a weight of 100kg is lifted one meter, the energy spent is the same than if a weight of 50kg was lifted two meters. The same goes with pulley systems; it is “allowed by physics” to have a system that lifts a 100kg weight one meter by pulling two meters of rope using a force that corresponds to 50kg.

Or inversely, to lift 100kg one meter by pulling half a meter and using a force that corresponds to 200kg. The real life pulley systems are typically of the former type, they reduce speed and increase force.

The ratio of increased force is called mechanical advantage, MA. The “pull 50kg lift 100kg” example is said to have MA 2:1 or simply 2.

Pulleys and MA

Pulley systems are based on pulleys. A pulley is a freely spinning sheave that allows a rope to take a (180 degree) turn without too much affecting the tension of the rope.

Some pulleys

This means that if a tension is put to the rope on one side of the pulley, the other side ideally assumes the same tension. And if the rope takes a 180 degree turn around the pulley, both of these tensions pull the pulley to the same direction. Thus, by pulling the rope by force F, the pulley sees resultant force 2F.

The rope is pulled tight at side A. Ideally, side B assumes the same tension through the pulley. Thus, by pulling only one rope with force F, the pulley ideally sees resultant force 2F.
A schematic version of the same pulley, using my typical notation

This also means that if the pulley needs to drag a load rightwards a distance of one meter, two meters of rope need to be pulled on one side of the pulley. This is the basic principle stated earlier; if you gain on force, you will lose on distance.

MA or not?

Another consequence: only certain pulleys actually add MA in pulley systems. The simplest pulley system (pictured above, usually called “V”) consists of only one pulley, which ideally doubles the force, making the ideal MA of this pulley system 2:1. See image below: Pulley system A is 2:1 “V”, whereas B merely changes the pulling direction.

Pulley system A actually adds MA whereas B only changes pulling direction

Almost all pulley systems more sophisticated than 2:1 “V” need at least one pulley that only serves the function of direction change. To add MA up to 3:1, one direction changing pulley may be added:

This is probably the most important single pulley system there is, at least when it comes to climbing and rope work. The gray thing is a rope grab, but it does not make any difference how the lower pulley is attached to the rope – or directly to the load.

Z-rig or simple 3:1 has one pulley that actually adds MA (the lower one), while the upper pulley just changes the direction of the rope and applies the rope tension once more to the load. This is how simple pulley systems work: they have a set of moving pulleys and corresponding set of stationary direction changing pulleys, which simply manifold the rope between two points, so that the force also manifolds – ideally.

Ideal or not?

The constant usage of word “ideal” here is because real life pulleys are far from being able to actually double force. A pulley-rope combination may have a surprising amount of friction; 10% or 20% of the force may be lost in a single 180 degree turn around a pulley. This is one of the things I have studied and written about. See other articles for more information.

Pulley inefficiency of course causes pulley system inefficiency. Again, read more of my articles for a deeper dive. At this points it is enough to distinguish three categories of pulley system MA:

  1. ideal MA
  2. calculated (effective) MA
  3. measured (effective) MA

Ideal MA supposes there’s no friction. The more there are pulleys in the pulley system, the more miserably wrong the value usually gets. Even if the pulleys are relatively efficient.

Calculated MA uses pulley analysis and empirical data of rope-pulley friction to calculate an estimate of real life MA.

Measured MA means that the pulley system is actually built and the output force – input force ratio is measured using tension load cells (force meters, quite like weighing scales).

As a side note, I’m not very fond of the discourse of pitting theory against practice. If a theory does not match with practice, it is probably not the theory’s fault, but the one’s who applies an unsuitable theory. As far as my studies among pulley system theory and practice go, calculated MA may be a very useful approximation. To enhance calculated MA values, one needs high quality empirical data as an input and error estimation tools. So I find measured MA value a less pragmatic of the two, something that is useful while studying pulley systems, whereas skillful calculations are the way to go when making decisions in practical situations.

Ideal MA is accurate in one sense: using a pulley system with MA 3:1, if one does a 10 meter hoist, one truly has to pull (at least) 30 meters of rope through the pulley system. Besides that, pulley systems are named after their ideal MA’s, so we call the Z-rig a “3:1” even if it’s effective MA is around 2.

Simple and beyond

Simple pulley systems

As said earlier, simple pulley systems are merely a manifold of rope between two points, made as frictionless as possible using pulleys.

A simple pulley system. If the load is on the left side and the right end is anchored, this is a 5:1. Otherwise this is a 4:1 with a change of direction.

There’s an interesting threshold for the maximum effective MA of a simple pulley system, no matter how many pulleys is used. Usually, using ropes and pulleys typical to climbing and rope access, this threshold is around 5:1. But using suboptimal pulleys, the “hard limit” might be as low as 3:1.

Simple pulley systems also waste a lot of pulleys to gain MA. Each added MA unit takes an extra pulley. To build a simple 4:1, one needs three pulleys.

Complex pulley systems

In literature, non-simple pulley systems are usually divided to complex and compound, but I find this classification useless. Compound should be a subcategory of complex, if anything.

Compound pulley system consists of two or more pulley systems that are put one after another.

An example of a complex (compound) pulley system, 3:1 is used to pull another 3:1, ideally producing MA 9:1

The essential point is that any non-simple pulley systems are complex. If you want to call consecutive pulley systems also compound, go ahead. But all non-simple pulley systems have one thing in common: Whereas simple pulley systems consist of pulleys that are stationary with respect to either the anchor or the load, complex pulley systems (according to probably the most popular definition) have also pulleys that move at different rate than either of those.

For example, in the picture above, pulley C is the odd one. B and D are stationary with respect to the anchor and A moves along the load, but C moves three times the rate of A. This ratio is called collapse ratio, and it tells how soon the first pulley to hit the limit of it’s moving range is going to do that. In case of the example above, the collapse rate of C (and thus the whole system) is three; the load can be hauled one third of the length of the whole pulley system, at best, before we have to do something to the collapsed red pulley system. Which leads us to the final topic of this article.

Collapsed 9:1 complex pulley system

PCD and resetting

Even simple pulley systems collapse eventually. Besides that, it is nice to have a ratchet of some kind in the pulley system so that the hauler does not need to hold the rope tight all the time.

This ratchet is usually called progress capturing device, PCD. It is usually a special kind of pulley that simply won’t let the rope to move other than in certain direction.

Petzl Pro Traxion, a progress capture pulley. When the toothed cam is activated, the rope will only move rightwards.

If this kind of pulley is used in place of pulley B in the earlier example, the hauler may let the rope go at any point, especially when the red subsystem has collapsed, and the PCD will hold the load.

PCD (pulley B) is holding the load, only one leg of rope in the pulley system is under tension. Notice the PCD symbol.

Now, if we also have some arrangement at pulley C, with which it can be reattached to the blue rope closer to pulley A, we have reset the pulley system and are ready to pull again until the next collapse. This is exactly where rope grabs come handy (remember earlier picture?).

This specific pulley system would also benefit from a rope grab at pulley A, so that when the blue subsystem collapses, pulley A can be moved back towards the load. This is actually more typical example of rope grabs in pulley systems. David Fasulo appropriately calls this tractor in his book Self Rescue.

The thing to take home here: the load is held by PCD when resetting or otherwise not pulling and by the tractor while pulling.

An example of fully resettable 3:1 pulley system, a Z-rig. The whole rope may pass through this pulley system during several resets.

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