rope physics

[Gary_Yngve]

Let's assume a rope is modeled by an elastic spring.

Let's neglect the force of gravity once the rope starts going tight.

 

The most force in the system occurs when all the kinetic energy is absorbed and transferred into potential energy in the spring of the form .5ks^2.

 

KE = PE

 

mgy = .5ks^2

 

F = ks = sqrt(2kmgy)

 

So assume we have a single rope.

Under a load of 80kg, its maximum impact force is rated at f.

 

As a twin rope, under a load of 80kg, its maximum impact force should be 2*(single rope fall with 40kg), which by the formula above, is 2*sqrt(40/80)*f = 1.4*f.

 

As a double rope, under a load of 55kg and just one strand weighted, its force should be sqrt(55/80)*f = .83*f.

 

Or another way of looking at it: if my double rope has impact force d, then tested as a single rope it should have force 1.2d and tested as twin ropes force 1.7d.

 

Why does all this matter?

It means that if my double rope is rated to 5 kN, as twins, it would be good at 8.5 kN.

 

But where it doesn't make sense is with the Beal Joker.

The Joker is advertised as:

single: 8.2

double: 6

twin: 9.5

 

I don't get it. The numbers don't work out. Is the effect of gravity during the fall or second-order effects really that significant?

[MattStan]

Hmm... physics is definitely not my strong suit, but perhaps it's due to how the Joker's dynamic elongation works? As a single it goes 37% (80kg mass) and 32% as a half (55kg mass), so my uneducated, shot-from-the-hip guess is the Joker's particular elongation characteristics during the half-rope test accounts for the seeming discrepancy.

 

Keep in mind, too, that the testing labs are less than perfect in their implementation of test procedures. I had more than a few manufacturers tell me of how they would send test ropes off the same spool to multiple labs only to receive widely varying results from each lab, and even get notable discrepancies within a single lab day-to-day. That's what killed the edge-resistance test (which was a bogus test anyway) -- rope A passes in labs X and Y, but fails in lab Z.

[Mr_Phil]

Ropes aren't elastic.

 

[Gary_Yngve]

Ropes certainly aren't perfectly elastic, but I'm talking in the physics world of spherical cows. In the regime of climbing falls, the rope should behave pretty elastically -- that's the whole basis of the fall-factor.

 

I'd use a higher-order model and simulate it if I couldn't solve it analytically, but the problem is I don't have the data to discover the system parameters. So I'm stuck assuming ropes are elastic.

[TrogdortheBurninator]

Aren't there a number of ropes now that ae rated as half/twin? You could assume something like a voigt model (easier than a zenner model and I dont think creep effects have any baring on the calculation)

 

This should give you a first order differential equation. Perhaps you could solve for the constants using some of the current twin/half systems, then see if the solution is valid for the Joker.

[JayB]

Bringing up some very painful memories...damped spring-mass systems....taylor series solutions to higher order diff. eq's....the trauma is slowly fading with time.

[Gary_Yngve]

Right Trogdor, it's not hard to write a first-order diffeq incorporating damping

 

m*(x';x") = (0,1;k1,k2)*(x;x') - (0,m*g)

 

k1 can be solved for explicitly given static elongation, as k2=0 then.

 

Once I solve for k1, I can solve for k2 afterward using the max impact force, though with only one point of data, it may not be very accurate. I suppose for the Joker, I'll have three points of data.

 

I'll put it onto the queue and get back to yall in a week.

[TrogdortheBurninator]

That is why I suggested the half/twin double rated systems because at least they give two points and there are multiple ones on the market.

[crackers]

I'll talk to Matt and Jim at Sterling on Thursday and ask them what diff's they use...

[Baltoro]

I fell on my Jokers as twins and they worked. I fell on my Jokers as doubles and they worked. I don't fall on a Joker as a single. I apply the ice leading mentality. In all seriousness I am curious what you guys come up with. I foolishly took chemistry instead of physics and can barely follow your posts beyond the first line or two.

[G-spotter]

Why don't you brain trustees branch out into topology and figure out how a rope can tie an overhand knot in the middle of the rope while there is a climber tied into each end.