Some newbie Qs

Dan Cobb said:
Just out of curiosity, is the T-Method (T for tension) for calculating mechanical advantage widely known amongst arborists? I've taught it in rescue classes for years, but can't recall a mention of it here.
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I use it regularly In course when discussing the principles. I learned it from Bruce Smith from On Rope.

I do not know of any others using it in regularly. Most likely because as arborists we tend to use “set” systems of known MA and or, the system in the tree is so over engineered with so many variables from rope angle to anchor flex, trying to quantify MA would be pointless.

Tony
 
Dan Cobb, your guess at 100 lbs and 75 is not far off for a typical pulley. The tension ratio is 1.2 and if you used 100 lbs and 83 lbs that makes a 1.2 ratio. The 1.2 ratio is confirmed for arborist rigging blocks.

I think on rough bark the tension ratio was 2.0, so you'd get 100 lbs climber and 50 lbs on the basal leg for a total of 150 lbs at the TIP. Going from memory here on the 2.0 ratio.

Tony, quantifying MA gets kind of discouraging when you include the tension ratio of real ropes and pulleys. The losses compound. I found a rope or rescue or caver web page that realized the same thing and presented worked out calculations. Using tension ratios simplifies everything and makes the math easy. In the simple math for multi pulley systems, just multiply in a tension ratio for every 180 degree turn the rope makes around a pulley. Every 180 degree rope on pulley turn generally ups your MA.
 
In the Deviation diagram, I'm presuming the rope with figure 8 knot and carabiner represents a typical loop runner with carabiner, on a not shown overhead tree branch.

Looks like the math is the vector sum of two equal rope leg tensions, resolved to a loop runner direction of exactly bisecting the two rope legs angle. For this to work the rope over the carabiner has to act as an impossibly perfect pulley.

Now to get all Neil Degrasse Tyson on it. The tension ratio of a rope going 180 degrees through a biner is around 2, so depending on the loading or motion history of the climber rope, the two legs of rope coming out of the carabiner could be 100% and 50% loads, not both 100%. They wouldn't reach this extreme of difference as the angle of rope bend over the carabiner would be less than 180 degrees. This and the friction of rope on biner would make the loop runner not exactly (or even remotely) bisect the angles of the rope legs. So now the size and direction of the resultant force are totally different.

A better chart for natural redirects, for the enthusiastic student, would be to replace the loop runner and carabiner with a tree branch crotch. There would be two steps in the calculation. The initial TIP would no longer have 180 degree contact (would have less) and the bollard equation would reduce its friction contribution. Then there would be a bollard equation with another different contact angle (amount of wrap) at the branch crotch. Solve the three rope tensions (basal, intermediate, climber weight(easiest one!!) and voila solve the resultant vector forces at the main TIP and the redirect.

Spidey, this one deserves your artwork for a diagram. Dan, not excluding you if you want a kick at the cat. Your hand sketch was pretty good.

By the way,, the tension ratio on rough bark was 2.5 not 2.0. So much for memory. 100 lb climber / 2.5 = 40 lbs basal tension, total 140 lbs at TIP. Kenny, the mu for 2.5 was 0.3 and the mu for 2.0 was 0.2 in the bollard equation. mu 0.2 is probably an average tree bark.
 
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Tuebor said:
I always explained it by having you imagining the two of us hanging on each end of a rope that is looped over a limb. We would both have to weigh the same to stay in balance and keep one from sinking to the ground and the other rising up in the air. The limb would be supporting our combined weight. Now substitute the base anchor for one of us. The base anchor has to "pull" a load equal to the climber's weight on the other end, and so the limb is still loaded to twice the weight of the climber.
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The way you explained that just made the concept “click” in my brain. Thank you. Outstanding, sir!!
 
Bart_ said:
Every 180 degree rope on pulley turn generally ups your MA.
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I've always used the rule of thumb that only moving pulleys generate mechanical advantage. Gets a little cloudy when dealing with complex (as opposed to compound) MA systems.

You can always throw in pulley efficiency with the T-method if desired. (And vector math if you want to account for angles.) Example for a typical "5:1" system. (The 87% efficiency is just a random number in the range of typical pulley efficiencies.)
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Depending on the topography, I may flip mine around so I can pull it by walking downhill instead of uphill, which makes it easier to pull at the expense of even less MA.
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I guess a tension ratio of 1.2 means 83% efficient. Real rope on a real pulley runs 1.2 ratio.

2nd diagram, 4:1 system should put 400 lbs onto load
With a tension ratio of 1.2 the numbers go 100, 83 (87), 68 (76), 57 (66), 47 (57)
100 + 83 + 68 + 57 + 47 = 359 lbs at anchor
83 + 68 + 57 + 47 = 255 lbs onto load
1.2x1.2x1.2x1.2 = 2.07
100 / 47 = 2.13 i.e. nuked by 4 turns around pulleys

A planned 4:1 became a 2.5:1 doh! And that's with a favourable 1.2 pulley, smaller diameter portable kit type pulleys would be worse.

first diagram, 5:1 system should put 500 lbs onto load
reverse the roles of the numbers, you get 359 lbs onto the load
A planned 5:1 became a 3.6:1 doh!

Very discouraging.

And the tension ratio for a biner on an alpine butterfly or probably also for rope on rope for serious gear minimalists is 2.0. So the numbers would go 100, 50, 25, 12.5, 6.25 lbs
50 + 25 + 12.5 + 6.25 = 93.75 lbs from putting 100lbs into a 4:1 system!!! 2nd diagram
for first diagram 100 + 50 + 25 + 12.5 + 6.25 = 193.75 lbs from putting 100 lbs into a 5:1 system!!!
Moral of that story always use pulleys, not biners on a bight.


Thanks for the illustrations :)

edit - Dan, easy way for mechanical advantage - count how many strands of rope leave the moving body, easy to see 4 and 5 in the diagrams. doesn't matter if from a pulley or fixed, rope is pulling
 
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Actual pulley efficiencies vary quite a bit. A few manufacturers provide figures. A good bit of testing has been independently performed. Tests below were with Blue Water 11mm static rescue rope.Screenshot_20210703-005923_Drive.jpg
 
I still like the T Method. Probably because I'm a rescue instructor and math geek. It works for any possible arrangement whether simple, compound or complex. Counting strands is not accurate for all systems.
 
Betcha bluewater static rope is parallel lay core kernmantle. The main tension loss doesn't come from the pulley bearing'r rating which is where the bushing vs bearing distinction is used for marketing purposes, ok a tiny bit as bushings bite a bit, but the flexure of the rope e.g. rope dia vs pulley dia, pulley groove dia and rope construction. So to me the Bluewater chart is a small snippet, but misses the direct applicability to arborist ropes. IMO Neat chart though.

I got my tension ratio 1.2 on tachyon, I think also stable braid and it agreed with Donzelli's measurements on an arborist rigging block.

1/0.95 = 1.05 tension ratio. Have a bit of difficulty seeing that would really happen. However pulleys are vastly superior to grinding rope on bark..
 
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I used to make and sell pulleys like this when I was flying RC model sailplanes. Maybe I should fire up the lathe and mill again. A 4" diameter wheel with the right profile would be pretty gentle on the rope, and it would be easy enough to beef it up to life support strength.

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nice work. rotary table for lightening holes on pulley?

On this whole rope bend pulley tension loss thing one can be humbled by harkening back to the nuclear industry's first remote handling arms. To get force fidelity, they used very thin, wider steel band "ropes" operating on drum pulleys. Virtually no hysteresis, kept steel in elastic region, just torque loss from precision roller bearings. I think as a kid I might have gotten to see one in person but was too young and naive to appreciate what it was.
 
Winchman said:
Thanks. The rotary table was also used to mill the outside of the side plates. Sometimes I'd mount it vertically to mill a round groove in a pulley, too. You can see more of my machining and welding projects at: https://app.photobucket.com/u/winchman
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You do some very fine work, sir! I consider myself a moderate fabricator, but you’ve got me beat by far! I am quite impressed!

I see you too are into old airplanes. Were you actually on-site to take the crash photos yourself?
 
Thanks again. The old airplane pictures were sent from a friend in SC. It's really sad to see the old planes being lost or neglected.

I managed to get a ride in an AT-6 shortly after getting my PPL in 1996. It was a real treat to fly almost all but the takeoff and landing. Didn't have any problem at all with the stick controls, even for a couple loops and rolls.

My wife and I had a Cessna 172 for a while, but I gave up flying after she developed a medical condition that made flying really uncomfortable for her. She had taken enough training to be able to fly and land the plane, and she was really good at navigation. It just wasn't much fun without her in the plane, but we do have some good memories.
 
A metal worker after my own heart. Wish I had a rotary table etc etc but machine tools are like that. Always something cool that would enable doing "fill-in-blank". Similar to gear OCD in a way. I machine small pulleys on my mill via stub split/expanding shafts in a collet after boring the pulley center. Never had a lathe.
 
A friend who is a gunsmith ran across a lightly used 10x24 lathe with all the accessories and a bunch of tooling and instruments at an estate sale in the late '80s. He knew I was interested, bought it, and sold it to me for a small fraction of what it was worth. It's got thousands of hours on it now, and it's still going strong.

One of my most interesting lathe projects was this hollow aluminum dodecahedron.
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First I had to figure out the inside and outside dimensions of the blanks to get the wall thickness down to 1/8" in the thinnest places. Then I had to design and build the fixture to fit in the lathe chuck. After all that math and geometry, the actual machining was pretty easy, but tedious and repetitive. Here's what it looked like in motion:
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More pictures here: https://app.photobucket.com/u/winchman/a/07897bfd-a4c5-4f05-bf3e-f5ae5fafe919

I got it right on the first try, and I only made one. It's not very useful, but it sure was fun to make.
 
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