While there is a variety of common wisdom regarding the effects of mechanical advantage and friction on suspension lift lines, the only published charts I'd ever seen attempting to quantitatively illustrate the subject were speculative and presented conclusions seemingly defying the laws of physics.
I've finally had some time over the past few weeks to explore this issue in more depth, first developing a mathematical model of the system, and then performing some experimental measurements to test the predictions of the model and determine the relevant coefficients of frictions. What follows are my conclusions from that work.
For background on the type of lifting setup which is discussed herein, please see this illustrated article.
Let us imagine you are lifting a 100 lb bottom in a single-point suspension -- or, alternately, that you are lifting 100 lb worth of the body weight of a heavier bottom (for example, lifting their hips while their chest is statically supported). If you are pulling down on the lift line coming from the ring, how hard do you have to pull to keep them moving upward?
|1:1||159 lb||159 lb|
|2:1||97 lb||119 lb|
|3:1||78 lb||103 lb|
|4:1||70 lb||99 lb|
|5:1||65 lb||96 lb|
|Downward pull for 100 lb load under ideal conditions|
The above figures are the output of my mathematical model using experimental friction data for hemp rope under the most ideal conditions I could achieve. The ratio in the first column indicates how many lines go up from the bottom -- even numbers mean the lift rope had to start at the ring. The second column indicates lifting force if you use a carabiner to reduce friction where the support line attaches to the harness; the third column if you have rope-on-rope friction at the bottom.
There are two main conclusions we can draw from this immediately, which remain true even in situations where the exact friction is slightly different:
1) If you do not use a carabiner to reduce friction at the bottom, the only good option is a 3:1 configuration, and the amount of force needed to lift will be similar to the weight of the load.
2) Even if you do use a carabiner to reduce friction, ratios beyond 3:1 have rapidly diminishing returns, and the lifting force needed will always be at least half the weight of the load (and usually more than 2/3rds of it).
I mentioned that the numbers I present in the above table are for the most ideal setup; I discovered a number of factors which seemed to influence the effects of friction on lifting force:
Angle of Force / Rotational Freedom
In order to get consistent results, I discovered that I had to put swivels on both the ring and the load, so that each side of the pulley system was free to rotate into its preferred orientation.
On the ring side, this effect was sometimes quite dramatic, and seemed to be related in part to the thickness of the ring; using a door-knocker style trapeze ring about 1" thick at the bottom, pulling at angles where the rope wasn't naturally coming through the carabiner facing me increased the force needed in a 3:1 lift by up to 15%. With a carabiner, the effect was around 5%.
In a 3:1 lift with carabiners on both sides, the effect of the load side of the pulley not being in its ideal orientation seemed to also be around 5%; and it's important to note that the rope's preferred orientation for the pulley was not in nice neat formation the way you'd run it, but was usually with a fair bit of twist in the lines between top and bottom. Since you can't anticipate how much your rope will want to twist under load, nor would you necessarily want to encourage/allow that twist anyway, it's safe to assume this effect always contributes some extra friction under real-world conditions.
When I first ran through my tests with 8mm hemp, I didn't find any significant difference in friction for 3:1 lifts using a thick ring at the top vs. a carabiner at the top. However, in testing a variety of rope types, it seemed that for some ropes the thicker ring did produce more friction (up to 10% in 3:1), and interestingly the effect was most dramatic with the narrowest rope.
This is an important result, especially in combination with the observation above about thick rings increasing the effect of off-angle lifting, because some members of the community have put forward the idea of using thicker rings as a safety measure to reduce the chance of rope breaks. While I would love to see more investigation aimed specifically at studying this issue, based upon these very limited data my conclusion would be:
3) Use of a ring a lot thicker than your rope diameter may increase the forces involved in a lift by up to ~20%; thick rings should not be relied upon to reduce the risk of rope breaks, and may actually increase them.
Friction of Neighboring Lines
In both 3:1 and 5:1 lifting configurations, I compared the force required when all ropes were run through the same biner in the same direction, when ropes were run through the same biner in reversing directions, and when each span was given its own dedicated biner. My observations were:
4) In a 3:1 lift, using two different biners for the two passes through the ring makes no significant difference. Going in reverse directions through a shared biner increases required force by around 4-8%. If using a thick ring, prioritize bringing the lift line through the ring in the direction facing you over matching the direction of the first pass.
5) In a 5:1 lift, required force is increased around 10% if lines through shared biners do not go in the same direction. Force may be reduced a further 10% by using separate biners for each pass, however only if they are separately supported and well spaced, for example in a rigging plate. Chaining additional biners off the biner for the previous rope negates any advantage.
Rate of Lift
In performing my tests, I attempted to lift the load at the slowest constant speed I could maintain, and measured the force to maintain that lift without acceleration. In the real world, some initial acceleration is required, and it's common to continue accelerating the bottom for a substantial portion of the lift. The scale I was using was not responsive enough to capture rapid changes in force during acceleration, but a few basic measurements showed that forces were substantially higher if I lifted the way I normally would in practice:
6) Forces during a typical lift may be at least 20% higher than the minimum force needed to maintain upward motion.
I suspect that the number may actually frequently be higher than 20%, but I'd need better tools to measure it, or to differentiate between the effects from static friction and those from acceleration.
Type of Rope
Slippery rope of course has less friction and allows these systems to operate more efficiently; however the difference isn't as much as you might expect. In a 3:1 lift, the difference between the slipperiest rope I've ever used for support lines (filament polyester) and the least slippery (a jute/dyneema hybrid) was only about 20% -- both in setups with and without rope-on-rope friction. Among different natural fiber ropes, the differences were much smaller, and frequently overwhelmed by other sources of variance.
7) While using synthetic rope may reduce lifting forces up to 20%, differences in friction between typical natural fibers are not very significant.
More Realistic Lifting Force
The table at the top is based on the best conditions I could create for testing; in real-world use, your setup will likely be imperfect and some of the factors I list above will come into play to increase friction. Here's the same table, with the friction constants increased 10% to create a more realistic/pessimistic estimate:
|1:1||169 lb||169 lb|
|2:1||106 lb||134 lb|
|3:1||87 lb||119 lb|
|4:1||78 lb||116 lb|
|5:1||74 lb||114 lb|
|Downward pull for 100 lb load under realistic conditions|
These numbers actually better reflect my real-world experience, which is that I may need to lift myself off the ground to rapidly lift a bottom of similar or even slightly lower weight than myself, and that single-point lifts without a carabiner at the bottom are generally impossible unless you substantially outweigh the bottom or push up on them.
Because the largest amount of lifting tension is lost in the first pass through the ring, redirecting your downward pull to an upward force, there is a tremendous advantage to pulling up from the bottom instead of down from the ring. Compare to the above table the following numbers for force required when lifting up:
|1:1||100 lb||100 lb|
|2:1||63 lb||79 lb|
|3:1||51 lb||70 lb|
|4:1||46 lb||68 lb|
|5:1||44 lb||67 lb|
|Upward pull for 100 lb load under realistic conditions|
Of course, it may not be as ergonomic to lift that way, depending on your position; lifting up requires less force, but you may also have less strength available in that direction. I find it frequently works well to pull up with one hand while pulling down with the other.
Holding / Lowering Force
I did not extend the mathematical model to include holding/lowering force, but did do a few measurements of the minimum force necessary to hold a load once lifted:
|3:1||10-15% load||0-3% load||(may jam)|
|5:1||1-2% load||0% load||(jams)|
We can conclude from this:
8) There may be some advantages in line control to a 5:1 lift when using a carabiner at the bottom with a heavy load. There is never an advantage in control going over 3:1 with rope-on-rope friction at the bottom; you only increase the chance of jams.
One of the most surprising things I discovered during my experimentation is that these types of systems are extremely sensitive to minor details of the setup. While I could get consistent numbers for any given exact setup, simple changes like swapping the position of two pieces of hardware or clipping a carabiner in a different position would often change the results by 5-10%. Sometimes I could find no reasonable explanation for these effects.
A particular mystery is that my attempts to directly measure friction of a single pass over a biner in a 1:1 lift gave results for natural rope around 32-33%, but all my other measurements only align properly if I use a figure of 37% in the model. I was unable to account for this discrepancy by adjusting the model to treat the first pass as having lower friction (which seemed plausible since the angle of contact could be smaller); nor does it seem sufficiently explained by friction of neighboring ropes, given the apparent lack of significance that had in 3:1 setups.
You can see all my data in this spreadsheet; the first tab has the math, the second tab has the measurements.
I conducted these tests by standing on a cheap electronic bathroom scale while pulling down on the lift line. I was actually fairly surprised how reproducible my readings were -- I'd guess the accuracy was around +/- 1 lb. The load was a bunch of 20 lb sandbags, which my scale confirmed to each be within 1 lb of their nominal weight. An additional ~2 lb worth of gear was used to clip together all the sandbags. All carabiners used were Black Diamond Rocklocks.