Rope Behavior - MITweb.mit.edu/sp255/www/produce/SP255_rope_behavior.pdfMarch 6, 2006 SP.255 Rope Behavior Presentation 6 Contributions to terms in the Wexler equation mg T mg MF mgh

Rope Behavior

Dave CusterSP.255

Spring 2006

March 6, 2006 SP.255 Rope Behavior Presentation

2

Overview• Simple models

– Very simple– Wexler

• More complications– Damping– Carabiner friction– Belayer behavior

• What you can do with the simple models– Estimate forces and times– Figure out how often to place gear– Evaluate ropes– Test testing laboratories

• Experimental results– Mägdefrau data– Belay and sharp edge tests– Humidity


3

where the energy goes


4

The Simple Model

Based on Wexler, 1950


5

The Wexler Equation

The maximum tension in the rope (the Wexler equation):

⎟⎟⎠

⎞⎜⎜⎝

⎛++=⎟

⎟⎠

⎞⎜⎜⎝

⎛++= ff

gmMmg

Lh

gmMmgT

cc

211211max

2

21 kymgymgh =+

kkmghgmmg

k

kmghgmmgy

2

212

214 22

22

++=

++=

Conservation of energy dictates that the climber’s gravitational potential energy before the fall is equal to the spring energy stored in the rope after the fall:

Solve for y and ignore the imaginary root:


6

Contributions to terms in the Wexler equation

( )mgMFmgT

kymgymgh

211

221

++=

=+

Friction over the topcarabiner increases therope modulus.

Belayer behavior and damping reduce thequantity under the radical sign.


7

KinematicsGraphs(simplespring)


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Friction Over the Top Carabiner

The dependency of the friction coefficient on mass, velocity, diameter, rope coating, and temperature has not been investigated


9

Pavier Model & DampingSpring in series with spring/dashpot parallel combo

Provides general idea of damping coefficient

Produces close match between model and experiment

Matches with the observation that climbing ropes are not far from critical damping/morethan half the energy is lost in each cycle

No model for why this works

35 kN

20 kN3 kNs


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Silly Math Tricks with Rope Hangtag Info

The ratio of heat to spring energy:( )

( )( ) ( )uiaauiaauiaa

uiaauiaa

h

ssh

F

F

UU

εε

εγ

×−×+××

×==

m 8.221m 8.2m 6.4m/s 8.9kg 80

m 8.221

2

The hang tag provides the force and % rope extension. The length of ropeand the fall height are defined by thetest standard.


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GROMF Conditions

Table #: Generic, run of the mill fall (GROMF) characteristics

quantity symbol value UnitsMass of climber mc 80 KgAcceleration of gravity g 10* m/s2

Rope modulus M 24000 NRope length L 30 MFree fall height H 2 MFall factor ff 1/15

Spring constant of rope k=M/l 800 N/m


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GROMF EstimatesApproximate GROMF results based on modeling the rope as a simple spring

quantity symbol value units

Time of free fall tf 0.6 sTime from rope engagement to dead-point tδ 0.1 sTime of rope stretch (total) tr 1.2 sTime, top to bottom of fall (tf +tr/2) 1.2 sRope stretch ymax 3.2 mTotal fall height (free fall height + rope stretch) h+ ymax 5.2 mVelocity at the end of free fall v0 6.3 m/sVelocity at dead-point vmax 7.1 m/sMaximum deceleration amax 30 m/s2

Frequency (angular) ω 3.2 s-1


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AttawayAdmonition(s)

Gear Placement

Anchor Placements


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Statistical Analysis of Test Facility Data: Expected Error


15


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Experimental Verification:MägdefrauData

Maegdefrau Datasqrt fall factor vs.

anchor load

0

2

4

6

8

0.00 0.50 1.00

sqrt fall factor

anch

or lo

ad (k

N)

Single Rope

Rope Pair

Maegdefrau DataLoad Rate vs. sqrt F/l

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 10.00 20.00 30.00 40.00

load rate (kN/s)

sqrt(

F/l)

(m-1

/2)


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Experimental Verification:CAI Data

C. Zantoni et al, CAI,UIAA Meeting Minutes2005


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Experimental VerificationCAI Belaying Forces w/ Slip



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Experimental Verification:CAI Fixed Point Anchor Force Data

C. Z

anto

niet

al,

CA

I,U

IAA

Mee

ting

Min

utes

2005


20

CAI Results from Experiments w/ Sharp Edge & Energy Absorption



21

Effects of Humidity: Force


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Effects of Humidity: Drops Held

(Again, from A.B. Spierings, et al. in the International Journal of Impact Engineering)


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Effects of Humidity: Elongation


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Bibliography/References• M Pavier Experimental and theoretical simulations

of climbing falls • O. Henkel, M. Schmid, A.B. Spierings Water

absorption and the effects of moisture on the dynamic properties of synthetic mountaineering ropes

• A Wexler, The theory of belaying• UIAA, Standard 101, dynamic ropes• S Attaway, Rope System Analysis• C Zantoni et al., UIAA SafeCom minutes

http://www.blackwell-synergy.com/doi/abs/10.1046/j.1460-2687.1999.00010.x

http://www.blackwell-synergy.com/doi/abs/10.1046/j.1460-2687.1999.00010.x

http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V3K-4HCMSFC-4-H&_cdi=5733&_user=501045&_orig=browse&_coverDate=10%2F21%2F2005&_sk=999999999&view=c&wchp=dGLzVlz-zSkWW&md5=b0b3e87ba0643abe0b7707b687460edf&ie=/sdarticle.pdf






http://www.losalamos.org/climb/xRopes.pdf

http://www.losalamos.org/climb/xRopes.pdf


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Questions?

Rope Behavior - MITweb.mit.edu/sp255/www/produce/SP255_rope_behavior.pdfMarch 6, 2006 SP.255 Rope Behavior Presentation 6 Contributions to terms in the Wexler equation mg T mg MF mgh

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