Lead Climbing Fall Impact Force and Fall Factor

If you have ever wondered where the term 'Fall factor' comes from and why it is important this video will make it clear. Some background in physics may be beneficial!

youtube.com/watch?v=AMHo6zp…

Great post, make more!

Excellent! You are great stick figure drawer.

Fun to see a follow up with some hypothetical constant values and your force calculation.

I enjoyed this - especially how it all comes together in the last 15 seconds.

Your handwriting with a mouse is better than mine with a pen. Cool video, thanks for sharing.

Cool! I always had it in my head that if you doubled the fall factor, you doubled the peak load. But this shows otherwise! Since the h/l term (fall factor) is inside the square root, you would have to quadruple the fall factor to double the peak load. Nice, that's good news for us!

Fantastic

Nick Crews wrote:Cool! I always had it in my head that if you doubled the fall factor, you doubled the peak load. But this shows otherwise! Since the h/l term (fall factor) is inside the square root, you would have to quadruple the fall factor to double the peak load. Nice, that's good news for us!

Yeah, I, uh, was thinking, uh...the same thing. :)

Great stuff! Thanks for posting this.

Nicely presented, but unfortunately wrong. You don't have the correct potential energy term, because after falling 2H the climber falls an additional amount, which is not insignificant, because of rope stretch.

The correct equations have been known for a long time; a reference (with credit for the original work) is at 4sport.ua/_upl/2/1404/Stand….

I was actually more concerned that the climber was going to hit the ground since it appears that the rope isn't going through the piece of pro!!

rgold wrote:Nicely presented, but unfortunately wrong. You don't have the correct potential energy term, because after falling 2H the climber falls an additional amount, which is not insignificant, because of rope stretch. The correct equations have been known for a long time; a reference (with credit for the original work) is at 4sport.ua/_upl/2/1404/Stand….

Right you are and thanks for the link! I plan on correcting my original when I get the chance and also making a followup looking at some set fall examples and seeing how common equipment ratings stack up!

rgold wrote:Nicely presented, but unfortunately wrong. You don't have the correct potential energy term, because after falling 2H the climber falls an additional amount, which is not insignificant, because of rope stretch. The correct equations have been known for a long time; a reference (with credit for the original work) is at 4sport.ua/_upl/2/1404/Stand….

I would argue the paper rgold linked is also wrong. The problem is both JadeMonkeyPhysics and the paper that rgold linked assume the wrong physical model.

The model you two have assumed is that the rope acts as a single elastic spring a.k.a. an un-damped harmonic oscillator. In reality the rope has a sort of internal "friction" (not to be confused with the "dry" friction of the rope running through the carabiners) that adds a "dashpot" to the system. The internal friction may come from literal friction of the rope fibers rubbing against each other as the rope elongates, literal friction of the sheath rubbing against the core, and/or just an inherent material property of the nylon. This means the rope system is more of a standard linear solid a.k.a. a damped harmonic oscillator.

The evidence for this can be seen in a few places. First, a series of technical papers written by Ulrich Leuthausser sigmadewe.com/index.php?id=…(start with Physics of Climbing Ropes - Part 1). Second, you can look at the real data published by rope manufacturers. If you calculate the spring constant from the static elongation and then use rgold's formula to calculate impact force the formula drastically under predicts the actually measured impact force. Third, watch a climber take a lead fall that is relatively vertical (falls with big swings would mask the effect). There is no large and pronounced bouncing (a.k.a. oscillating) like a bungee jumping line. There is a hard and sudden stop with almost no bouncing and that is due to the damping from the internal rope "friction".

JadeMonkeyPhysics, you will also want to consider that the spring constant "k" for a rope is going to change (ie be a function of) the length of rope out. Look at it like springs in series. I like how rgold handles it using relative stretch. A rope will a constant unit length spring constant but you will have to multiply it by the amount of rope out to find the actual spring constant for that particular fall.

Fascinating stuff! I glad to see I'm not the only nerd on here so I hope the discussion continues.

Mike Slavens wrote: I would argue the paper rgold linked is also wrong.

The paper is not wrong, it just makes a lot of simplifications, not always made clear. I am quite sure that Asst. Prof. R.Goldstone can solve all appropriate differential equations with all appropriate non-linear rope energy dissipation models while accounting for friction on carabiners with modified capstan equation, but that would make this little white paper totally unaccessible to the median mp.com reader.

Mike Slavens wrote: This means the rope system is more of a standard linear solid a.k.a. a damped harmonic oscillator.

RGold's paper has been floating around for quite some time. I don't remember whether it's mentioned in the paper itself or not, but in one of these many discussions he explicitly stated that the harmonic oscillator was left undampened for simplification/accessibility (as it avoids differential calculus). That said, I'd be psyched to read a white paper that handles "all appropriate differential equations with all appropriate non-linear rope energy dissipation models while accounting for friction on carabiners with modified capstan equation"

amarius wrote: The paper is not wrong, it just makes a lot of simplifications, not always made clear. I am quite sure that Asst. Prof. R.Goldstone can solve all appropriate differential equations with all appropriate non-linear rope energy dissipation models while accounting for friction on carabiners with modified capstan equation, but that would make this little white paper totally unaccessible to the median mp.com reader.

I didn't mean to insult Mr/Dr Goldstone or to demean his mathematical ability and apologize if it came off that way. My phrasing could have been better selected. You are correct in that the paper has nothing wrong with it mathematically. The equations are correct and his derivation is accurate. And, as Mr/Dr Goldstone states at the end of his paper, this set of equations has been commonly used for a long time.

JadeMonkeyPhysics has shown an in depth knowledge of the math through his video so I took the discussion a bit further. I also wanted to spur some discussion around those simplifications. I think we should at least advertise that there are significant simplifications in those formulas and the implication of those simplifications.

I have a pile of papers and links on my hard drive that use more sophisticated models, so I am far from unaware, and I made it clear in my exposition of the Wexlar result that it was a substantial (perhaps over-) simplification.

Although the model is crude, it only requires a cheap scientific calculator to produce answers. The same would be true if one used a single damped harmonic oscillator formulation, but information not available on the web would be needed for the constant of proportionality in the damping term. After that, all models rely on systems of differential equations and, in addition to using parameters that can't be just looked up, also require some type of mathematical software (or programming of an Excel spreadsheet) in order to get answers out of them.

To call any of these models "wrong" is easy enough to do but is not necessarily useful or enlightening. In some sense, no mathematical model of a physical process is "right," they all involve simplifying assumptions. In the case of rope models, almost all rely on the damped harmonic oscillator as a component of the model, in spite of the fact that, for example, there doesn't seem to be a clear physical argument for why viscous damping is the correct form of damping.

JMP's model is "wrong" in a quite different sense, one that isn't comparable to the general wrongness of all mathematical models. He has a potential energy term that doesn't match what happens physically. If the missing quantity was of a smaller order of magnitude than the other quantities, than ignoring it could be part of an approximate modeling process, but this is not the case; climbing ropes stretch substantially (on the order of 30% for the UIAA impacts) and so ignoring the contribution that extra distance makes to the potential energy term is "wrong" in a fully objective sense.

With all due respect Dr. Gold, making a substantial/over simplification can be objectively wrong. Particularly in this case where the simplification adds a substantial error and is done to avoid math that is difficult for the average person but not that difficult for mathematicians and engineers. The simple spring model makes a substantial simplification by ignoring the damping effect and JMP makes another substantial simplification by ignoring rope stretch, so I struggle to see how one is objectively wrong and the other isn't.

Yes philosophically all mathematical models have inherent flaws but they are typically considered accurate if the error is within 2% of actual measurement which is the case for many models using hand calculations. In this case the model/formula has an error on the order of 30%. This is even more concerning considering the error isn't conservative. This model/formula drastically under predicts the impact force and yet it is generally accepted as correct.

I agree that calling something wrong is easy enough, and I would go one step further and almost call it somewhat inflammatory if its not useful or enlightening. However in this case I think it is very useful and enlightening. The simple spring model perpetuates the wrong understanding that impact force is only dependent on fall factor not the actual height of the fall. Lower fall factor but long falls can generate a higher impact force than short falls but with a higher fall factor because of the damping. Accurately calculating impact force (at least to <2% error) becomes very useful/enlightening when objectively analyzing things such as if a 6kN piece is rated for lead falls or just aid. Its also useful when analyzing gear failing on lead falls; is there something wrong with the gear or the placement, or did the fall simply exceed the strength rating of the piece. What about things like can I buy bolts only rated to 3000-lbs and not 5000-lbs. Maybe we need to push UIAA and rope manufactures to test and publish more data points to understand if viscous damping is accurate enough or if another model needs to be used, or at least be able to solve for all the terms with viscous damping.

I do admit that regardless of any modeling, calculations, computer programs, or Excel sheets no one is going to bring a tape measure on lead and say "oh shit I'm 6.2 meters past my last C3 with 18.7 meters of rope out, for the love of god some calculate my impact force to see if that cam is going to hold" so maybe its not that useful. But if we can show that you at least X feet of rope out for every 5' you need to lead past a C3 and we save someone from a broken leg because of it I would think is worthwhile.

Mike Slavens wrote:Yes philosophically all mathematical models have inherent flaws but they are typically considered accurate if the error is within 2% of actual measurement which is the case for many models using hand calculations.

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What may be considered an accurate estimate depends on its intended use.

Mike Slavens wrote: In this case the model/formula has an error on the order of 30%. This is even more concerning considering the error isn't conservative. This model/formula drastically under predicts the impact force and yet it is generally accepted as correct.

If you fit the spring coefficient--which you should, because real ropes are not characterized by a single number--the error is not necessarily conservative.

Agreed that the SLS model predicts higher impact forces for longer falls with the same fall factor, but in practice there are many other aspects that make longer falls more dangerous, some of which are quite difficult to model.

The usefulness of Wexler's equation is in showing that even short falls may produce high forces (if the fall factor is high). Using it to conclude that large falls are harmless if the fall factor is low is misguided.

Where does the 2% come from? Acceptable error varies a good bit by discipline/goal.

Hey guys,

I'm wondering if anyone happens to have some test data available that we could use to see just how far off these models really are.