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# Pulley system analysis

Tuomas Pöysti 2020

As I wrote in the introductory article about pulley systems, pulley systems are commonly named after their ideal mechanical advantage (MA) value, which on the other hand is usually very different from the effective MA value.

## The T-method

To find out both values for a pulley system, the first thing to do is to analyze it. The basic analysis, which only yields ideal MA, is commonly called the T-method.

Start by drawing a schematic picture of the pulley system. Label each pulley, for example P1, P2 etc. (it would be better to use subscript numbers, but I have chosen to go with this notation, because the internet is not ready for subscripts). These labels also stand for their efficiency values for brevity. Left: schematic picture of a Z-rig. Right: line tensions concerning a pulley with efficiency P. In this notation, the arrow shows which leg is pulled (the input force).

Then consider how a pulley-rope friction affects the rope tension. The traditional T-method uses letter T instead of F, but I prefer F (probably because it is common in physics). If the rope on the primary side is pulled at force F, a pulley with efficiency number P will eat some of the force, leaving P*F or PF to the secondary side. Notice: efficiency is a value between 0 … 1. This also applies if you prefer percents; 80% = 0.8.

To calculate the ideal MA first, let’s just assume P1 and P2 are both 1, that is, the pulleys have no friction. Starting from the input force, study what tension each element of the pulley system sees. Each pulley doubles the tension, the rope has the same tension on both sides of each pulley. At the rope grab or tractor two tensions unite, so they are summed up. The T-method. It does not make any difference if T or F is used as label to force (or tension), since the first thing is to forget it and use only it’s multiplier number (1 to start with).

The analysis tells us that the ideal MA of the Z-rig 3:1. Another noteworthy detail: the anchor ideally sees twice the input force, 2F, although the load is 3F. The upward forces (input force F and anchor reaction 2F) combined must match the downward forces.

## Enter efficiency

Let’s factor the efficiency in; stop assuming that all P’s are 1. All that changes is one has to multiply the rope tension by P every time the rope runs around pulley with efficiency P. The expressions get a bit long sometimes, but the actual matter is quite simple.

The output is Z-rig’s calculated effective MA: 1+P2+P1P2. For example, if both P1 and P2 were 80%, the calculated MA would be

1 + 0.8 + 0.8*0.8 = 1 + 0.8 + 0.64 = 2.44.

or 2.44:1. For 50% “pulleys”, such as bare carabiners, we get

1 + 0.5 + 0.25 = 1.75 or 1.75:1.

## The building blocks of MA

Some notes on the example:

• The expression 1+P2+P1P2 involves three terms
• One of the terms is 1
• The value of rest of the terms is less than one each
• Even for 80% pulleys, term 1 covers over 40% of the total MA

If we return to the ideal world for a second, three terms makes absolute sense, since if we take P1=P2=1, the expression becomes 1+1+1=3, just as in the ideal T-method. And considering pulley efficiency, each term varies between 0 … 1, as efficiency varies between 0 … 1. With the exception of term 1, of course.

Let’s study another pulley system, an upside down “piggyback” 4:1, which consists of two consecutive 2:1 pulleys and is ideally a 3:1.

Expression of it’s effective MA is

P1 + P2 + P1P2

No term 1 this time. This is related to an inherent feature of this pulley system: it is direction changing, the input force is pulled downwards.

It makes sense if you think about it: since the input pull is to the “wrong” direction, it takes at least one turn at a pulley for the force (or energy) to get to the load and do something useful. That is, every bit of the energy is taxed by at least one of the pulleys. It seems that in case of Z-rig, in contrary, one of the three “paths” through the pulley system avoids the tolls of pulley friction.

Next example is a nice 5:1, possibly my favorite among pulley systems. A complex 5:1, which can be seen as variation of Z-rig, where the rigid (1:1) connection between P2 and tractor is replaced by a 2:1.

The MA expression is:

1 + P2 + P3 + P1P2 + P2P3

Five terms, one of which is 1, as expected. Let’s carry out a thought experiment: for each term, imagine all of the pulleys not included in the term having efficiency of absolute zero – like they had been dipped in cement and let dry, not spinning, the rope stuck in. Then, figure out how the pulley system would act.

Rationale for this is: mathematically the lack of a variable means independence of that variable. So if a term does not include efficiency of a pulley, the efficiency does not affect to that term.

Term 1 represents the energy path (red) that is independent of all three pulleys. If All the pulleys were glued completely stuck, the pulley system would still act as a rope. And a rope is essentially a 1:1 pulley system.

Terms P2 and P3 represent a 2:1, term P1P2 a 3:1 and term P2P3 a 4:1 pulley.

I still do not know if the MA’s of these subsystems make any difference, but at least for me this analysis helps understand how each of the five units of MA depend on each pulley’s efficiency.

## Making it systematic

Mathematically, term 1 is of degree 0, P2 and P3 are of degree 1 and P1P2 and P2P3 are of degree 3. The 5:1 example does not have any third degree terms, but some three pulley systems may have also term P1P2P3.

All possible terms for a four pulley systems are listed on the bottom row of this table:

The middle row shows their respective degree and on top row is shown an example of each terms value with the assumption that all pulleys are 80% efficient. You may as well appoint different efficiency values to each pulley and calculate each combination’s value to the respective cell.

Let’s analyze some pulley systems. The most important thing is to carefully document each pulley system’s schematic picture and especially which label refers to which pulley.

I’m not going through the analysis here, just jumping to the results. Instead of writing each expression, they are presented here by pointing out their terms among all possible candidates.

With a little spreadsheet skills, you may quite easily develop this into a calculator, which automatically updates effective MA values when you change each pulleys efficiency value. Adding new pulley systems is a matter of documenting the system and pulley labels, analyzing and adding a row to your spreadsheet.

If you found this interesting, read my other articles including this one published in Richard Delaney’s RopeLab in 2019.