Analysis of a test fall

Force v.s. time data for drop tests using climbing rope has been collected and made available on the net by Hal Murray (murray@pa.dec.com). I find analyzing this data is fascinating, even though it has little relevance to my use of rope in climbing. The fact is that the UIAA and some newer organizations have developed practical tests that provide a high level of confidence that ropes, as used in climbing, will do their part help heep us alive and healthy - understanding why ropes behave this way is not necessary for climbers.

Still, I'm a nerd and can't resist fiddling with data. So here goes...

Data for several experiments are available. Most of these include prusiks, force limiters, or other devices that mask the rope's properties. I've picked the first data set for analysis. The commentary supplied with this data reads:

	SamplesPerSecond 2500
	DateTime 2/12/89 11-04
	NumberOfSamples 5000
	Peak 1450
	Info 10 ft, fall factor 1, old climbing rope, good condition

This data is graphed in green below with force in pounds on the y axis plotted against sample number on the x axis. Total elapsed time is 2 seconds.

The red line is position data computed by estimating the mass, m, of the dropped object, and the time when it was dropped, t₀=0, then updating its position, y_t, and velocity, v_t, each 1/2500th of a second with the rule:

	a_n = f_n/m + g
	v_n = v_n-1 + a_n / 2500
	y_n = y_n-1 + v_n / 2500

where f_n is the nth force (shifted depending on starting time). Ignore the units for the position graph - distances were scaled to fit on the same graph as force. The total distance shown is approximately 1.5 meters and roughly corresponds to rope stretch.

Very nice, so far. The forces look reasonable, and we see a nicely damped oscillation as the mass comes to rest. The force curve is a bit noisy along the initial rise and we could speculate about the cause (is it knot tightening or other internal rope-friction releasing?).

To learn more, I plotted force vs. distance fallen (rope stretch). Force in pounds is on the y axis, and distance in meters on the x axis.

I think this plot is way cool, but keep in mind that the position of the inner spirals can be moved by assuming alternate initial conditions. I chose initial conditions that resulted in a sensible position vs. time plot. Unfortunately, I don't have a record of the initial conditions used to generate the position plots.

Anyway, I think it is fair to conclude that this plot is reasonably close to reality for our purposes. Here are some observations:

There is a slow ramping up at low forces on the first bounce: Ropes behave differently for loads that generate low forces (top rope falls) and those that generate high forces. Due to the inelastic nature of climbing rope, I wouldn't try extrapolating TR fall curves from this data.
Some inelastic stretching occurs: Notice how each loop is shifter to the right - allowing more total stretch for a given force on each successive bounce.
Ropes act like springs during elongation, but not contraction: The graph is essentially linear for each stretching phase.
The effective spring constant increases with each bounce: Each linear successive region is steeper.
There is a small error in my initial conditions: The data should spiral inward without the last segment crossing the previous bounce. Correcting this would shift the graph slightly to the right, but otherwise not appreciably affect it.
The noise in the force during the first bounce seems to be mostly positive, increasing the force above what a simple rope model might suggest. [Either that, or the simple model I'm imagining is wrong!]

Comments, discussion, etc. are welcome, but not spam! I can be reached at cline+@cs.cmu.edu.

Ken Cline