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Rope Behavior Dave Custer S.255 Spring 2006

Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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Page 1: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

Rope Behavior

Dave CusterES.255

Spring 2006

Page 2: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.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

Page 3: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.255 Rope Behavior Presentation

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where the energy goes

Page 4: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.255 Rope Behavior Presentation

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The Simple Model

Based on Wexler, 1950

Page 5: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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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:

Page 6: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.255 Rope Behavior Presentation

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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.

Page 7: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.255 Rope Behavior Presentation

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KinematicsGraphs(simplespring)

Page 8: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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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

Page 9: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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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

Page 10: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.255 Rope Behavior Presentation

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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.

Page 11: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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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

Page 12: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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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

Page 13: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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AttawayAdmonition(s)

Gear Placement

Anchor Placements

Page 14: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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Statistical Analysis of Test Facility Data: Expected Error

Page 15: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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Page 16: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

March 6, 2006 ES.255 Rope Behavior Presentation

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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)

Page 17: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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Effects of HumiditySee: A. B. Spierings, O. Henkel, and M. Schmid. Water absorption and the effects of moisture on the dynamic properties of synthetic mountaineering ropes. International Journal of Impact Engineering 2005.

Effects on:Drops Held: See Fig. 2Force: See Fig. 3Elongation: See Fig. 4

Page 18: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

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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 Zanantoni et al., UIAA SafeCom minutes

Page 19: Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math Tricks with Rope Hangtag Info. The ratio of heat to spring energy: ()() uiaa uiaa uiaa

MIT OpenCourseWarehttp://ocw.mit.edu

ES.255 Physics of Rock ClimbingSpring 2006

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