Calculating Compound M/A
We recently had a request for additional information beyond what was shown in our “Theory of Mechanical Advantage” video by Chief Instructor Dennis O'Connell. The reader would like to know more about calculating compound mechanical advantages.
First of all, a simple mechanical advantage (MA) is quite easy to calculate as long as you follow a couple of basic rules.
MAs are generally expressed in numeric ratios such as 2:1, 3:1, 4:1, etc. The second digit of the ratio, or the constant "1" represents the load weight. The first digit, or the variable 2, 3, 4, etc. represents the theoretical factor that we divide the load weight by, or inversely multiply the force we apply to the haul line.
I say theoretical as these calculations do not take into account frictional losses at the pulleys and resistance to bend as the rope wraps around the pulley tread. So a 3:1 mechanical advantage would make the weight of a 100-pound load feel like 33 pounds at the haul line, but we do lose some advantage due to those frictional losses. An even more important consideration is the fact that we multiply our hauling effort by the variable, which is important to understand when we think about the victim or an on-line rescuer that has become fouled in the structure. This is also important when considering the stresses on the haul system including the anchor, rope, and all components in the system.
We also need to pay attention to the amount of rope that must be hauled through the system to move the load a given distance. If we are using a 4:1 MA and need to move the load 25 feet, we need to pull 100 feet of rope through the system (4 X 25 feet = 100 feet).
To calculate a simple MA, remember this: if the anchor knot is at the load, it will be an odd mechanical advantage (3:1, 5:1, 7:1, etc.). If the anchor knot is at the anchor, it will be even (2:1, 4:1, 6:1, etc.) “even/anchor-odd/load.” And if you count the number of lines coming directly from the load, you will determine the variable (remember not to count the haul line if it passes one final change of direction pulley). For instance, if the knot is at the anchor and there are four lines coming from the load, this will result in a 4:1 simple MA. And if your haul line is being pulled away from the anchor, that only means you have created one final change of direction which oftentimes is done to allow the addition of a progress capture device (ratchet), or simply to make it a more convenient direction of pull. But this 5th line, called the haul line, does not come directly from the load. It comes from the final directional pulley to the haul team and is not to be counted in the simple MA ratio. We would call this set up a 4:1 MA with a change of direction (CD).
Calculating compound MAs is also quite easy. Compound MAs (sometimes called a stacked MA) simply means we are attaching a second MA to the haul line of the original MA. When we do so, we multiply the first digit of the original MA by the first digit of the second MA. If you attach a 2:1 MA to the haul line of a 4:1 MA (2 X 4 = 8), you end up with an 8:1 compound MA. Keep in mind that we have added even more frictional losses into this system, but it is still a pretty powerful MA.
There are potential benefits as well as potential penalties when using compound MAs. One benefit includes using less gear when stacking MAs. For instance, to build a simple 6:1 MA, you will require at least five pulleys, and if you want a final CD, that would require one last pulley for a total of six pulleys. If you decide to build a 6:1 compound MA, you can get away with as few as three pulleys by attaching a 2:1 MA to the haul line of a 3:1 MA. If you wanted one final CD, you would again add one more pulley for a total of four pulleys. The obvious advantage is that fewer pulleys are required, but hidden in there as another advantage is fewer pulleys for the rope to wrap which translates to less frictional loss and bend resistance.
Another benefit to stacking MAs may be the reach you need to attach to the load. If the load is 25 feet away from the anchor and you are using a 6:1 simple MA, you will need at least 150 feet of rope, plus some extra to tie the anchor knot, and some spare to wrap over the final directional - if you use one. If the load is 50 feet below the anchor and you want to stick with the simple 6:1 MA, you are looking at a minimum of 300 feet of rope.
So, what if we send a 3:1 MA down from the anchor to the load 25 feet below and attach a 2:1 to the haul line of the original 3:1 to build a 6:1 compound MA?
Well, in this case we would need 75 feet of rope plus some extra for knots for the original 3:1, and two times the length of the compounding MA throw. Throw? What the heck is throw? Throw is a term we use when we have a limited distance between the compounding MA anchor and where we can safely attach the compounding MA to the haul line of the original MA.
In the diagram below you can see the original 3:1 MA extending from its anchor to the load. The added MA, which in this case is a 2:1 has a total throw of 10 feet which requires a little over 20 feet of rope to construct. So, if we add the 75+ feet of rope required for the original 3:1 to the 20+ feet for the added 2:1, we arrive at a bit over 95 feet of rope required for this compound 6:1 MA to reach a load 25 feet from the anchor. This can be two separate ropes, one a bit over 75 feet and a second a bit over 20 feet, or it can be one rope a bit over 95 feet that we can treat as if they were two separate ropes. More on that in a bit.
Remember that we must consider the amount of rope that we need to pull through the system in order to move our load the required distance. So, using a 6:1 compound MA to move the load 25 feet we must pull a total of 150 feet of rope through the system. Whoa, wait a minute! I thought we determined that our total rope needs were only a bit over 95 feet, so how did we come up with 150 feet of rope? One of the disadvantages of compound MAs is the need for resets when the throw is not long enough to move the load the needed distance. So, even though we are using in the neighborhood of 95 feet of total rope, we are pulling the same section of rope through the second MA multiple times.
Well, this is one of the big disadvantages of a compound MA. We need to reset the system multiple times to move the load the required distance. To help envision a reset cycle, let’s assume we have our original 3:1 mounted to an anchor, and 25 feet from that anchor is the 3:1 attached to the load. The haul line of the original 3:1 goes through a final CD, and we have attached a ratchet at that final CD to capture the progress of the loads movement. One option is to find a second anchor and in this case we found one 10 feet away from the final CD of the 3:1. We tie an anchor knot and attach it to that second anchor and route the remaining 20+ feet of rope through a pulley which we attach to the haul line of the original 3:1 with a rope grab. We now have our 2:1 pulling on the haul line of a 3:1 resulting in a 6:1 compound MA. But…… and there’s always a “but,” isn’t there? We can only move the load a bit over 3 feet at a time before we completely collapse the 2:1 and need to reset it for the next haul. Remember, the 2:1 only has a 10-foot travel or “throw” and that distance is divided by 3 as it is pulling on a 3:1 MA. In addition to that, we have pulled about 20 feet of rope through the 2:1 just to move the load a bit over 3 feet. In order to move the load the entire 25 feet we will need to reset the system about 8 times and that is some slow going. Just to point out one option to speed up the haul by reducing the amount of resets needed, if you sent the original MA to the victim as a 2:1 and then stacked a 3:1 MA with 10 feet of throw onto that 2:1, you would still have your compound 6:1 but would only need to do about 5 resets and could do it with a bit over 80 feet of rope.
There are all sorts of options when deciding what type and ratio of MA to use in a rescue effort. You can get pretty creative when building MAs, but be aware that creativity can sometimes lead to crazy. Remember the KISS principle…keep it simple and safe.
If you are overbuilding an MA just to show a cooler way of doing it, you may be missing the point of the job. There is someone in trouble that is relying on you getting them up and out of their predicament, and sometimes we can get a little carried away with our creativity, especially when it comes to MAs. 3:1 Z-rigs are a great option especially with the addition of devices like the Petzl ID or the CMC MPD as your first MA change of direction and progress capture device. Plus, this gives you the ability to convert to a lower with friction control already built in. But you can really complicate things by compounding a second MA onto a Z-rig to get a higher ratio MA. You will soon learn that now you have to perform two separate resets of the haul cams. And, if you are out of sequence in the reset, the haul cam of the second MA will jam into the traveling pulley of that system and stop you in your tracks. There are some tricks to really make the resets for this system go nicely, but that will have to wait for another day.
There are hundreds of variations that you can use for compounding MAs, but once again I caution you to remember KISS. I have my favorites and every once in a while the situation calls for something a little different, and that’s where understanding the advantages and disadvantages of the systems is of great value.
For additional video resources on mechanical advantage as well as other techniques and systems, visit Roco Resources.