SENIOR THESIS

SUBMITTED AS PART OF THE APPLIED PHYSICS PROGRAM

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF BACHELOR OF SCIENCE IN PHYSICS

Spring Loaded Camming Device for rock climbing:

analysis and optimization

Phillip Anuta

DEPARTMENT OF PHYSICS INDIANA UNIVERSITY

5/8/09

Supervisor: Signature Typed Name: Co-signer: Signature Typed Name:

2

Contents

Abstract 3

Introduction 4

MathematicalTheory 7

Results&Discussion11

Conclusion 13

References 13

3

Abstract

ThepurposeofthispaperistoexaminetheprofileoftheSpringLoadedCammingDevice

fromamathematicalandphysicalstandpointofview.Usedinrockclimbing,this

removabledeviceisplacedintoacrackorpocketintherockandhasauniqueself‐locking

propertythatpreventsthecamfromcomingout,especiallyduringafall.Becauseofits

uniquedesign,aSLCDhasarangeofwidthsitcanproperlyholdin,reducingtheoverall

amountofgearaclimbertakeswithhim.ThesuccessoftheSLCDderivesfromits

logarithmicshape,wherethecamhasthesameratioofpullingforcetofrictionalforce.

Thisisparticularlyinterestingbecausethisratioisindependentofthewidthofthecrack

andhowmuchthecamlobesareretracted/expanded.Inordertoproduceamaximum

frictionalforce,whileincreasingtherangeofthecam,theoptimumcamminganglefound

rangesbetween13°and16°.

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1.Introduction

Theworldofclimbingandmountaineeringwasrevolutionizedwiththeintroductionofthe

SpringLoadedCammingDevice.Rockwallsthatwereoncethoughtofasun‐climbable

havenowbeenascendedwiththeaidofthislittlemechanicaldevice.SLCDsareusedina

specifictypeofoutdoorclimbingknownastraditionalclimbing,ortradclimbing.The

objectiveisforapartytoascendarockfacebyplacingtheirownprotectivegearinthe

wall,thenattemptingtoremoveitwhenthepassageiscomplete.

Figure1.InpartA,nothingprotectstheclimberfromfallingtotheground.InpartB,gear

isintermittentlyplacedinthewalltoprotecttheclimberfromhittingthegroundwhen

falling.

Whatseparatestradclimbingfromotherstylesisthattherecanbeahigheramountofrisk

involvedsincetheclimberisrelyingsolelyonhisowntoolsandplacementofthesetoolsin

thewallforsafetywhenascending.SportClimbinghoweverinvokesdifferentmeansof

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ascending,whereholesaredrilledintherockwallandanchorsarepermanentlyplaced

priortotheclimb.Intradclimbing,thedesignoftheremovablegearcorrespondsto

specificfeaturesontherockfacewherethegearcanbeplaced,incracks.Continuous

verticalcracksinrockwallsarethereforesoughtafterintradclimbingandcanbefoundin

limestone,sandstoneandgranitecragsallovertheworld.Theseverticalcrackscomeinall

shapesandsizes,wheresomearebigenoughtofitone’sbodyin,andothersonlyfingers

canfitin.Whenpreparingtoascendaroute,theclimbermusttakegearwithhimthatwill

fitaccordinglyinthecrack.Becausethewidthofthecrackcanvarydependinguponits

locationonthewall,aproblemcanarise.Aclimberwouldhavetotakeaplethoraofgear

withhimontheascenttofitineverypossiblecrackwidth.Excessgearcanbecomevery

heavyandtheclimberwouldnotwanttospendthetimediggingthroughmanydifferent

piecesofgeartofindtheonethatfitsproperlyinthecrack(allthewhileheisholdingonto

thewallwithonehand!).Thus,thereisademandforasinglepieceofprotectionthatcould

potentiallyfitintoarangeofwidthsandberemovedeasily.InCalifornia’sYosemiteValley

duringthemid‐1970s,rumorsbegantospreadofanewdevicedevelopedbyRayJardine

thatallowedeffortlessprotectionplacementinamatterofseconds.Thedeviceknownas

‘Friends’,ortheSpringLoadedCammingDevice(SLCD),savedvastamountsofenergy

expenditureandtimewhenplacingtheprotectionintherockwall.

AlthoughthereareslightlydifferentvariationsoftheSLCD,figure2showsasketch

ofatypicalcamwiththefour‘lobes’relaxedintheoutermostposition.Whenthetriggeris

pulled,thelobesofthecamretractasseeninfigure3,decreasingthewidthofthecam.

Whenplacedinthecrackandthetriggerisreleased,thespringsinsidethecampushthe

lobesoutwardagainstthecrackwalls,holdingthecaminplace.Itisthespringsinthecam,

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pushingoutward,thatholdthecaminplace.Thecamcannotbepulledoutofthe

placementunlessthetriggerisagainpulledandthelobesretract.Iftheclimbertakesafall

onthecam,thedownwardforceonthecamistranslatedoutwardintoaverylarge

frictionalforcebetweenthecamlobesandthewallsofthecrack.TheseSLCDsare

extremelyusefulherebecauseeachindividualSLCDhasarangeofwidths;soonecam

couldpotentiallyfitseveraldifferentcrackwidths.

Figure2.SpringLoadedCammingDevice.

Groovesareetchedalongtheoutside

ofthelobes,resultinginahigher

coefficientoffriction Figure3.Retractingthelobesbypullingthe

trigger,thecamcanbeplacedindifferent

widthsofcracks

ThedesignoftheSLCDisquiteelegant,consideringifitisplacedproperlyinasolidcrack,

itissupposedtoresistaforceofseveralkilonewtonsproducedfromtheclimberfalling.

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2.MathematicalTheory

TheshapeofaSLCDisalogarithmicspiralasseeninfigure5.Theradiusvariesasa

functionof

€

θ asseenintheequation:

€

r = aebθ (1)

whereaisasizingconstantandbisthegrowthconstantwhichcontrolstheseverityofthe

spiral’scurvature.Thisgeometricshapehasaveryuniquepropertythatisattributedto

thesuccessofcammingdevices.Asseeninthefigurebelow,thereexistsarelationshipat

thepointofintersectionbetweentheradiusrandthetangenttothecurveT.Theangle

€

α

betweenthesetworemainsconstantas

€

θ varies.

Logarithmicspiral

Figure5.Thetangentline(T)representsthesurfaceofthecrackwherethecamlobesare

touching.Angles

€

α andφremainconstant,independentof

€

θ .

Anotherangle

€

φ thatalsoremainsconstantregardlessof

€

θ isofparticularinteresttous.

Thiscammingangletellsustheangularseparationbetweenthewall/camcontactpoint,

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andtherotationalaxisofthecamwherethedownwardforceisapplied.Thefollowing

derivationshowshowthecammingangleisonlydependentofthelogarithmicspiral

growthconstantb.

€

tanφ =drrdθ

→

€

tanφ ⋅ dθ =drr

€

tanφdθ =1rdr∫∫ →

€

tanφ ⋅θ = ln r( )*

*given

€

r = aebθ ,acanbeomittedsinceitisonlya

scalingconstantanddoesnotaffectthe

curvatureofthespiral

€

ln r( ) →

€

ln eb⋅θ( ) = b ⋅ θ →

€

tanφ ⋅ / θ = b ⋅ / θ

→

€

φ = arctan(b)

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Thefollowingfiguregivesaforcediagramofwhatexactlyhappenswhenthecamisloaded

properly.

N=normalforcewall

exertsoncam

Figure7.ForceDiagram

Usingtherelationship ofSLCDcorrectlyloaded inacrack.wesee

€

F1 = F2x ⋅ tanφ

Alsoknowingthefrictionalforce:

€

Ff = µ ⋅ F2x whereµisthecoefficientoffriction

→

€

Ff ≥ F1

→

€

µ ⋅ / F 2x ≥ / F 2x ⋅ tanφ →

€

µ ≥ tanφ ifcamholds

Theforceoftheclimberpullingonthecamisrepresentedby

€

F1andthefrictionalforce

opposing

€

F1isshownby

€

Ff oneachsideofthecam.Forthecamtohold,thefrictional

forcemustbeequalinmagnitudetotheforceexertedbytheclimber.Theamountof

outwardforcethecamproducesisdirectlyrelatedtothecammingangleφ.Given

€

µ ≥ tanφ

fromfigure7,and

€

tanφ = b fromfigure6,weseethat

€

µ ≥ b .Thistellsusthatthecoefficient

€

N = F2x

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offrictionbetweenthelobeofthecamandtherockwallmustbegreaterthanorequalto

thegrowthconstantofthecammingdevice.

Thequestioncannowbeaddressed,howdoesoneproperlychooseacamming

angle(orgrowthconstantb)?Twoissuesofgreatimportancearedirectlyrelatedtothis

cammingangle:therangeofthecamandtheholdingpower.Figure8visuallyshowsthe

affectbhasontherangeandcammingangleφ.

Figure8.Threespiralswithvaryinggrowthconstantb.Therangecanbeseenasthe

distancebetweenwherethecurvesoriginate(θ=0°)andwhereitends(θ=360°).The

associatedcamminganglesareincludedaswell.

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3.Results&Discussion

Inlookingbackatfigure7,whenthecammingangleφincreases,lesshorizontalforceF2xis

producedfromF2whichdirectlyreducesthefrictionalforceFf.Ifthecoefficientoffriction

(µ)betweenthecamlobesandtherockfaceisknown,thiswillallowustocalculatethe

maximumcammingangleavailablebeforeFf=F1.

Figure9.Plotoffrictionalforcevs.cammingangle.Thecoefficientoffriction(µ)istaken

tobe0.38.Thefrictionalforceandthecamrangeiscalculatedforacamwith2lobes.

Inexaminingfigure9,twoboundarieswilllimitthechoiceofcammingangle.Astheangleφ

increases,thefrictionalforceapproachestheboundaryconditionFf≤F1.Although

theoreticallyFfcanbeequaltoF1,,itisverydangerousrealisticallybecausethechanceof

thecamslippingisveryhigh.Thislimitisseenwherethecriticalforceintersectsthe

frictionalforce,whichhappensat20°.Thisdeterminestheupperboundaryforchoosingφ,

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wherethelowerboundaryhastodowiththenormalforceproducedonthecrackwall.In

thelowerrangesofcammingangles,thenormalforcethecamproducesontherockwallis

tremendouslylarge.Whenφis7°,thenormalforceproducedfromonelobeisabout20kN,

or3timestheamountofthedownwardforce.Ifwetaketheamountofcamsurfacearea

touchingthewalltobeabout¼squareinch,(whichiscommon),thiswouldproduceabout

496,422,504pascals ontherockwall,whichwouldcertainlypulverizeorcracktherock(or

deformthecam),whichwouldseverelyincreasethelikelihoodthecamwouldpullout.The

ratioofdownwardforcetonormalforceismorecommonlyseentobe2:1.Higherratios

thanthisincreasethechanceofdestroyingtherock.

Sincecamsareusedinseveraldifferenttypesofrock,thecoefficientsoffrictionof

eachrockmustbetakenintoconsideration.Theinteractionbetweengranite/

limestone/sandstoneandaluminumhasaµofaround0.35.Thetypewiththelowest

rock/aluminumcoefficientoffrictionwouldthereforedictatethemaximumcammingangle

thedevicewouldhave.

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4.Conclusion

Thepointinwhichthefrictionalforceandthecamrangecurvesintersectinfigure9

givesusthebestapproximationforchoosingthecammingangle.Atthispoint,boththe

rangeandfrictionalforceareoptimized,around14°.Notsurprising,theworld’sfour

largestcammanufacturersusethefollowingcamangles:

Table1.Differentcamminganglesproducedbymanufacturers

Manufacturer Cam angle (degrees) Metolius 13.25 Wild Country 13.75 Black Diamond 15 Colorado Custom Hardware 16

ThesmalleranglesbyMetoliusareconsideredthesafest,buttheygiveupabitofrangefor

theextraholdingforce.CCHhasamuchhighercamangle,whichgivesanadvantageofa

widercammingrangebutlessholdingpower,givingitastaticallyslightlyhigherchangeof

pullingthantheothers.

References

[1]www.metoliusclimbing.com

[2]www.blackdiamondequipment.com

[3]Cox,StevenM.Mountaineering:TheFreedomoftheHills,7thEdition.2003

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