Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math...
Transcript of Rope Behavior - ocw.mit.edu · March 6, 2006 ES.255 Rope Behavior Presentation 10. Silly Math...
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Rope Behavior
Dave CusterES.255
Spring 2006
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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
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where the energy goes
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The Simple Model
Based on Wexler, 1950
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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:
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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.
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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
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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
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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
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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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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
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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
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ES.255 Physics of Rock ClimbingSpring 2006
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