1

PhysicalMechanismofIce-StructureInteraction1

XuJi*,ErkanOterkus**2

*DepartmentofNavalArchitecture,OceanandMarineEngineering,UniversityofStrathclyde3100MontroseStreetGlasgowG40LZ,UnitedKingdom4

Phone:+44(0)14154840945e-mail:x.ji@strath.ac.uk6

**DepartmentofNavalArchitecture,OceanandMarineEngineering,UniversityofStrathclyde7100MontroseStreetGlasgowG40LZ,UnitedKingdom8

Phone:+44(0)14154838769e-mail:erkan.oterkus@strath.ac.uk10

11

ABSTRACT.Toobtaintheeffectofvelocityandstructuralnaturalfrequency(structuralstiffness)onice12

failure,anextendeddynamicVanderPolbasedsingledegree-of-freedomice-structureinteractionmodel13

isdeveloped.Threebasicmodesofresponsewerereproduced:intermittentcrushing,frequencylock-in14

andcontinuouscrushing.Furtheranalysisonphysicalmechanismofice-structureinteractionispresented15

onthebasisof feedbackmechanismandenergymechanism,respectively. Internaleffectandexternal16

effectfromiceandstructurewerebothexplainedinthefeedbackbranch.Basedonreproducedresults,17

energyexchangesatdifferentconfigurationsarecomputedfromtheenergyconservationusingthefirst18

lawofthermodynamics.Ageneralconclusiononthepredominanttypeofvibrationwhentheicevelocity19

increasesduringtheinteractionprocessisforced,self-excitedandforcedineachthreemodesofresponses.20

Iceforcevariationsalsoshowsthatthereismoreimpulseenergyduringthelock-inrange.Moreover,ice-21

induced vibration (IIV) demonstrates an analogy of friction-induced self-excited vibration. Finally, the22

similarity between strain-stress curve and Stribeck curve shows that static and kinetic friction force23

variationsareattributedtoiceforcecharacteristic,andcanbeusedtoexplainthelowereffectivepressure24

magnitudeduringcontinuouscrushingthanthepeakpressureduringintermittentcrushing.25

KEYWORDS:Physicalmechanism; Ice-structure interaction; Ice-inducedvibrations;VanderPolequation;26

Frequencylock-in.27

28

NOMENCLATURE29

L Correlationoricefailurelength(m)

H Icethickness(m)

0v Referencevelocity(ms-1)

, v Y Icevelocity(ms-1)

2

d Segmentwidth(m)

D Structuralwidth(m)

n Numberofsegments

0K Referencestructuralstiffness(kNm-1)

K Structuralstiffness(kNm-1)

nω Angularnaturalfrequencyofstructure(rads-1)

nf Naturalfrequencyofthestructure(Hz)

0f Referencefrequency(Hz)

M Massofthestructure(kg)

C Structuraldampingcoefficient(kgs-1)

X Structuralacceleration(ms-2)

X Structuralvelocity(ms-1)

X Structuraldisplacement(m)

F Iceforce(kN)

T Time(s)

A Magnificationfactor

σ Icestress(kPa)q Dimensionlessfluctuationvariable, a ε Scalarparametersthatcontrolthe q profile

iω Angularfrequencyoficeforce(rads-1)

if Icefailurefrequency(Hz)

B Coefficientdependingoniceproperties

Y Icedisplacement(m)

Y Iceacceleration(ms-2)

ε Strainrate(s-1)

rv Relativevelocitybetweeniceandstructure(ms-1)

λ Dimensionlesscoefficient

maxσ Maximumstressatductile-brittlerange(kPa)

, d bσ σ Minimumstressatductileandbrittlerange(kPa)

, α β Positiveandnegativeindicestocontroltheenvelopeprofile

tv Transitionicevelocityapproximatelyinthemiddleoftransitionrange(ms-1)ξ Dampingratio

tWΔ Incrementalworkdonebyiceforce

mEΔ Incrementalchangeofmechanicalenergyinthestructure

dEΔ Incrementalheatdissipatedbythestructuraldamping

pEΔ Incrementalchangeofpotentialenergyinthestructure

kEΔ Incrementalchangeofkineticenergyinthestructure

3

sf Frequencyofstructuraldisplacement

sµ Staticfrictioncoefficient

kµ Kineticfrictioncoefficient

1. INTRODUCTION1

Ice-structure interactiondrewpeople’sattentionsince theoilexplorationandexploitation inCook Inlet,2Alaska,1962.Largevariationsoficeproperties(Peyton,1966)andlargeamplitudeice-inducedvibration(IIV)3phenomenon(Blenkarn,1970)werenoticedanddiscussedfromthedatacollectedfromthisarea.Although4globalwarminghadanegativeeffectoniceresearchactivitiesduringthepastdecadeortwo,thereisan5increasinginterestonthepossibilityofusinganewrouteintheArcticOceanfromFarEasttoEuropeandoil6andgasexplorations inthisarea.Hence, it isessential todesignshipsandoffshorestructureswhichare7resistanttopossibleiceimpactsonthestructure.However,becauseofthecomplexnatureoficeandlimited8full-scaledata, icemodels andexperiments showdifferences (Sodhi, 1988)whichmakes the ice related9researchstillachallengingarea.10

Even after aroundhalf a century, thebasic physicalmechanismof the severest vibrations during ice-11structureinteraction,IIV,isstillnotfullyunderstood.Blenkarn(1970)andMäättänen(2015)arguedthatit12wasthereasonofself-excitedvibrationbecauseofthenegativedampingtheory.Ontheotherhand,Sodhi13hasforcedvibrationfromresonanceopinionasinSodhi(1988)becausenegativedampingexplanationisnot14rigorous.Besides,Sodhi(1991a)foundthatenergyisalwaysdissipatingintoice,whichrulesoutthechance15ofnegativedampingtooccur.16

Inthisstudy,beforeprovidingfurtherexplanationsoficemechanics,anextensionofJiandOterkus(2016)17model will be introduced first. This model is based on substituting an empirical parameter to include18structural stiffnessand ice velocityeffects. Then, a seriesof reproducednumerical resultsbasedon the19experiments done by Sodhi (1991b) will be presented. Finally, physical mechanism of ice-structure20interactionprocessandiceforcefrequencylock-induringIIVwillbediscussedfrombothMäättänenand21Sodhi’spointofviewbyanalysingnegativedampingphenomena,energyexchangesandstressvariations22basedonthereproducednumericalresults.23

2. MODELDESCRIPTION24

2.1Icefailure/correlationlengthparameter25

Icefailurelengthisanidealisedconceptfornumericalcalculationbasedonthedamagezoneorcrushing26zoneconcept inexperimentaltests. Icefailure lengthistakenasaconstant1/3of icethickness inJiand27Oterkus(2016),whichmeansicefailsatacertainlengthificethicknessdoesnotvary.However,itranges28from1/2to1/5oficethicknessaccordingtothetestsbySodhiandMorris(1986)coveringanareawhenit29isusedfor ice forcepredominant frequencycalculations. It is thereasonthat icedamagezonebecomes30smallerbyincreasingicevelocity(Kry,1981,Sodhi,1998).Atlowicevelocity,thereisalargedamagezone31with radial cracks along from the contact area.When the velocity reaches high level, the damage zone32becomesmuchsmallerwithonlymicrocracksneartheinteractionsurface.33

Sodhi(1998)proposedanotherconcept,i.e.correlationlengthparameter L ,todescribethesizeandthe34amountofdamagezoneoficeinrelationtoicevelocity.Heproposedanequationtoestimatethisparameter35intheformof 0/ ( / )( / )L H v v d H= ,where 0v isareferencevelocity, /d D n= isthesegmentwidth,n is36

4

thenumberofsegmentsusedas1,3,5or7tocontrolthestructuralwidthD ,and /d H ratioisinthe1-31range. It canbeseen that thecorrelation length isdecreasingwith increasing icevelocity. In thispaper,2structuralwidthisconsideredasonewholesegment,i.e. 1n = .Thus,theequationisusedintheformof3

0/ ( / )( / )L H v v D H= .Assuming / 2D H = ,then 0/ 2 /L H v v= whichisdoneundertheassumptionthat4theicefailurelengthhasarelationshiptoicethicknessandnottostructuralwidth(Sodhi,1998).5

Inadditiontoicevelocity,structuralstiffnesshasalinearrelationshipwithicefailurelengthiniceforce6frequencycalculation(Määttanen,1975,SodhiandNakazawa,1990).ExperimentaltestconductedbySodhi7(1991b)alsoshowsthisrelationship.TheseareTest63,Test66andTest67underalmostthesameconditions8exceptthatthestructuralstiffness ineachtestwas3230,1710and890kNm-1,respectively.As listedin9Table1,theicevelocity,icethicknessandstructuralwidtharenearlythesame.Time-historyplottingofice10forceinSodhi(1991b)showedthattheaveragemaximumvaluewasaround15kNinalltests.However,the11frequencyshowedanapproximatelylinearlydecreasingtrendwhenthestructuralstiffnessdecreases,with12thevalueof3.33Hz,2.17Hzand1.25Hz,respectively.13

Table1.TestconfigurationsfromSodhi(1991b)14

TestNo.

Icevelocity(ms-1)

Icethickness(m)

Structuralstiffness(kNm-1)

Structuralnaturalfrequency(Hz)

Structuralwidth(m)

Fig.No.inSodhi(1991b)

63 0.0411 0.027 3230 11.68 0.05 Fig.3a66 0.0411 0.0275 1710 8.50 0.05 Fig.2b67 0.0412 0.027 890 6.13 0.05 Fig.3b110 0.1031 0.03 2700 10.68 0.05 Fig.5203 0.1452 0.024 1130 6.91 0.05 Fig.2c

15

Followingtheeffectoficevelocityonicefailurefromcorrelationlengthparameterandbyaddingupthe16linearstructuralstiffnesseffect,thegeneralformoficefailurelengthcanbewrittenas17

0 02L v KH v K

= (1)18

where 0K is the reference structural stiffness and K is the structural stiffness.Constant valueof 2 also19

satisfiestheratioofstructuralwidthtoicethicknessintherangeof1.67and2.08aslistedinTable1.Ifthe20massofthestructureremainsthesame,thenaturalfrequencyofthestructurewillbeproportionaltothe21structuralstiffnessas /n K Mω = ,where nω istheangularnaturalfrequencyofthestructureandM is22

themassof the structure.Besides, the ISO19906:2010 tends touse the structuralnatural frequency to23definethehighesticevelocitythatlock-inconditioncanoccur, v nv fγ= ,where 0.06vγ = m.Hence,(1)can24berewrittenintheformof25

0 02n

v fL Hv f

= (2)26

bysubstitutingthestiffnesswiththefrequency,where 0f isthereferencefrequencyand nf isthenatural27frequencyofthestructure.28

2.2Mainequations29Inthisstudy,governingequationsand icestress-strainraterelationshipareadopteddirectly fromJiand30Oterkus(2016).UnliketheexperimentdonebySodhi(1991b)inwhichanindenterispushedthroughice31

5

sheet, in the current numerical model ice is moving against a stationary structure. The ice-structure1interactionmodelistakenasamass-spring-dampersystemasillustratedinFig.1.Thereare“internaleffect”2and “external effect” regarding to ice and structure. Internal effect corresponds to ice own failure3characteristicandisrepresentedbyVanderPoloscillatorwithoutforcingterm.Ontheotherhand,external4effectcorrespondstostructuraleffectsincludingstructuraldisplacementandstructuralvelocity,i.e.relative5displacementandrelativevelocitybetweeniceandstructure.TheVanderPoloscillatorandicestain-stress6function are coupled as ice force variations. Relative velocity is considered in ice strain rate-stress7relationship and the right-hand side of Van der Pol oscillator. Relative displacement between ice and8structureisalsoconsideredwithintheoscillator.Compressivestressresultsinicedeformationandwhen9thedeformationexceedsthenaturalicefailurelength,icefailureoccurs.Therefore,icewillfailunderboth10internalandexternaleffects.ThemodelconsistsofequationofmotionandVanderPolequation,i.e.11

2 2

( ) ( )

( 1) ( )ii i

MX CX KX F T ADH q aBq q q q Y XH

σωεω ω

⎧ + + = = +⎪⎨

+ − + = −⎪⎩

(3)12

inconjunctionwiththeicestress-strainraterelationshipgivenin(4).In(3), X isthedisplacementofthe13structure,the“dot”symbolrepresentsthederivativewithrespecttotimeT , A isthemagnificationfactor14adjustedfromexperimentaldata,D isthestructuralwidth,q isthedimensionlessfluctuationvariable,a 15andε arescalarparametersthatcontrolthelowerboundoficeforcevalueandsaw-toothiceforceprofile,16respectively, 2i ifω π= istheangularfrequencyoficeforce, B isacoefficientdependingoniceproperties17andY isthedisplacementofice.18

19

Fig.1.Schematicsketchofdynamicice-structuremodel.20

2.3Icestress-strainrateequation21

Itisfoundandprovedthaticeuniaxialstressorindentationstressisafunctionofthestrainrateasshown22inFig.2(Blenkarn,1970,MichelandToussaint,1978,Palmerandothers,1983,SodhiandHaehnel,2003).23Thestrainrateisdefinedby /rv Dλ ,wherethedimensionlesscoefficientλ variesfrom1to4andD is24thestructuralwidth(YueandGuo,2011).Itcanbeexpressedbytwoseparatedimensionalpowerfunctions:25

max

max

( )( / ) , / 1

( )( / ) , / 1d r t d r t

b r t b r t

v v v vv v v v

α

β

σ σ σσ

σ σ σ⎧ − + ≤⎪= ⎨

− + >⎪⎩ (4)26

where maxσ is themaximumstressatductile-brittlerange, dσ and bσ aretheminimumstressatductile27andmaximumstressatbrittlerange,respectively,α and β arepositiveandnegativeindicestocontrolthe28

Ice

6

envelopeprofile,respectively,and tv isthetransitionicevelocityapproximatelyinthemiddleoftransition1range.2

3

Ice Ductile Ductile-brittle Brittle

Intermittentcrushing

Frequencylock-inContinuouscrushing

Structure Quasi-static Steady-state RandomFig.2.Strainratevs.uniaxialorindentationstresscorrespondingtoicefailureandstructuralresponsemode.4

2.4Parametervalues5

Eachparameterinthe(3)and(4)isdeterminedfromthetestsconductedbySodhi(1991b)andsummarised6inTable1.Intheequationofmotion, 0.22A = isthemagnificationfactoradjustedbyanyoneofthecases7todeterminetheupperboundoftheiceforceexceptfortheTest.110whichis0.15. 2a = issettoassume8thatall forcewilldroptozeroaftereachcycleof loading. 0.05 mD = and 600 kgM = are fromthetest9

configuration. 0.1ξ = isnotgivenbutfoundinFig.5ofSodhi(1994).IntheVanderPolequation, 4.6ε = is10

adjustedforbetterforceenvelopebehaviour. 0.1B = iscalibratedbytheresultsfromTest.63andTest.67.11

Intheicestress-strainrateequation, 8800 kPadσ = becausethemaximumpressureis8.8MPafromTest.12

64.InTest.66,theeffectivefailurepressurevariesfrom8-13MPaunderintermittentcrushing.Therefore,13

max 13000 kPaσ = isusedformaximumvalueand 0.35α = .InTest.67,thepressureis4.4MPa.Therefore,14

4400 kPabσ = and 6β = − exceptforTest.203athighvelocity,i.e. 1700 kPabσ = fromthedataprovided.15

Transitionicevelocityissetto 10.05 m stv−= becausehighvelocityrangedescribedbySodhiwasabove0.116

ms-1andthemiddlevalueisestimated.Intheicefailurelengthequation, 10 0.03 m sv −= isusedassuggested17

bySodhi(1998). 0 710K = kNm-1, i.e. 0 5.47f = Hz, isadjustedbytheresultsfromTest.63andTest.67.18

Summaryofallparametervaluesislistedbelow:19

7

1 10 0

ma

0

x

0.05 m, 600 kg, 0.1, 0.22 (0.15 for Test.110), 2;4.6, B 0.1, 0.03 m s , 710 kN m , ;

8800 kPa, 4300 kPa (1700 kPa for Test. 203), 13000 kPa, 0.35, 6, 0

5.47 Hz

d b t

D Mf

A av K

v

ξεσ σ σ α β

− −

= = = = == = = =

= = = = = − =

=1.05 m s .−

1

2

3. RESULTS3

Basedontheexperimentalresults,reproducedresultsgeneratedfromthenumericalmodelareshownin4Fig.3,whichareinaprettygoodmatchwiththosefromexperiments.Inthesefigures,recordsofvariables5are plotted with respect to time, such as ice force, displacement of the structure, acceleration of the6structure and structural displacement with respect to the ice. Relative displacement is calculated by7subtractingthestructuraldisplacementfromicedisplacement.Positivedirectionofthestructuralmotion,8i.e.thesamedirectionasicemotion,isintheoppositedirectionwithrespecttoSodhi’sresultssincetheice9isassumedtomoveagainstthestructureinthemodelwhereastheindenterissettomoveagainsttheice10sheetintheSodhi’sexperiments.11

Fig.3(a),Fig.3(b)andFig.3(c)showresults forTest.63,Test.66andTest.67, respectively,whereall12parametersarethesameapartfromthestructuralstiffness.Thevaluesofstiffnessare3230,1710and89013kNm-1,respectively.Amplitudeandfrequencyoficeforceanddisplacementarealmostthesameasthose14inexperiments,reachingat15kN,witharoundfourteen,eightandfivecyclesoficeloadinginfourseconds,15respectively.Althoughtheicefailureforcesareapproximatelythesame,theicefailurefrequencyineach16figureshowsacleardependenceonthestiffness,becausestructuralstiffnesshasalinearrelationshipwith17icefailurelengthascontrolledby(1).18

AccelerationinFig.3(a)showsaroundfiftypercentbiggerthanthatinthetestbecausetherearesome19dropsduringloadingphases.Moreover,therearesomedropsaroundthepeakvalues,whicharerealistic20sincethestressvarieswiththerelativevelocity.Whentheforcereachesthemaximum,therelativevelocity21willbecomezeroleadingtheforcetodroptothecorrespondingstressvalue.Inaddition,thisdifferencemay22bethereasonoffilteringprocess inexperimentaldatathateliminatessomesharppeakfluctuationsand23lowerstheaccelerationamplitudesignificantly.24

IfwetaketheFig.3(c)asanexampleinwhichforceanddisplacementareplottedmoreclearly,thereis25almostnoaccelerationofthestructureduringloadingphaseinthedetailedplot.Attheinstantoficefailure,26the structure is in themaximum excursion place and the potential energy stored in the spring is then27transformedintokineticenergy,movingthestructurebackwardsagainsticemotion.Theinteractionforce28thendropstoamuchlowerlevelsuddenlyataround1/3ofthevalueattheinstantoffailure.Itisthereason29thatcrushediceisincontactwithandextrudedbythebackwardsmotionofthestructure.Aftertheextrusion,30icelostcontactwiththestructureforashorttimewhiletheforceisalmostzero,whichiscalledasseparation31phaseaccordingtoSodhi.32

It canbenoticed that there is somesuddenunloadingof ice force inFig.3(a)and (b)becauseof the33externaleffectappearance.Itoccurswhenthestructureismovingbackwardswithrespecttoicemotionat34first.Thecompressivestressbetweeniceandstructureresultsinicedeformationandfailureoccurswhen35thedeformationexceedsthenaturalicefailurelength𝐿cantolerate.So,relativedisplacementissubtracted36bytheamountof𝐿additionallyleadingtoanegativevalue.Moreover,thisexternaleffectoccursafterthree37cyclesofloadinginFig.3(a)andfourcyclesinFig.3(b),respectively.Duetothespacelimit,simulationsare38

8

allruninsixteensecondsbutplottedinfoursecondsonly.Thereasonthatthereisnosuchexternaleffect1inTestNo.67isbecausestructuralstiffness,i.e.structuralnaturalfrequency,hasanimpactonthelock-in2conditionrangeastheISO19906:2010revealed.Asdefinedin(2),thehigherthestructuralstiffness,the3smaller the ice failure length.Therefore, lock-in conditionwilloccurevenunder the same icevelocity if4structuralstiffnessishighenough.5

Fig.3(d)showsasteady-statevibrationofthestructurewithafrequencyclosetoitsnaturalfrequency.6Theiceforceshowsalmostexactlythesameperiodic“spike”likeloadingenvelopeprofileasthatinthetest,7inwhichthemaximumamplitudeofforceisaround11kNandafrequencyof9Hzapproximately.Itcanbe8noticedthatrangeandamplitudeofstructuraldisplacementandaccelerationaredifferentfromthoseinthe9test. There are two reasons for this difference because of ice force. The first reason is that ice force is10controlledtodroptozeroduringeachcycleofloading.Thesecondisonlythemaximumvalueisconsidered11and predicted. Similar difference can also be found in Fig. 3(e), which shows the continuous crushing12behaviour under high ice velocity. Ice force ismatching at around 2.5 kNwith the test and no obvious13vibrationofthestructureisfound.14

15

Fig.3(a). Fig.3(b).16

9

1

Fig.3(c). Fig.3(d).2

3

10

1

Fig.3(e).2

Fig.3.Timehistoryof iceforce,structuraldisplacement,acceleration,andicedisplacementwithrelative3displacementatdifferentstructuralstiffnessesandicevelocities.(a)Test.63,K=3230kNm-1,v=0.0411ms-41.(b)Test.66,K=1710kNm-1,v=0.0411ms-1.(c)Test.67,K=890kNm-1,v=0.0412ms-1.(d)Test.110,K=27005kNm-1,v=0.1031ms-1.(e)Test.203,K=1130kNm-1,v=0.1452ms-1.6

4. PHYSICALMECHANISM7

AccordingtothedefinitionfromDenHartog(1947):8

• Inself-excitedvibration,thealternatingforcethatsustainsthemotioniscreatedorcontrolledbythe9motionitself.Whenthemotionstops,thealternatingforcedisappears.10

• Inforcedvibration,thesustainingalternatingforceexistsindependentlyapartfromthemotionand11persistsevenwhenthevibratorymotionisstopped.12

Sodhi(1988)discussedice-inducedvibrations(IIV)asforcedvibrationbecauseiceforcepersistswhenthe13structure ispreventedfrommoving.However, it isafactthat icefailedatacertainamountwithcertain14

11

frequency(Neill,1976,SodhiandMorris,1986,Kärnäandothers,1993,Sodhi,2001).So,thesituationwill1bedifferentiftheanalysisoficefailureprocessisrestrictedtojustonesinglefailurecycle,whenthereisno2feedbackcomingfromthestructure.Thereisatimeintervalafterthefirstamountoficefailsandbefore3anotherpieceoficeapproaches,whichisaprocessforicefragmentstoberemoved.BothSodhi(1988)and4Määttänen(2015)pointedoutthisprocessandnamed“clearingprocess”and“gap”,respectively,sinceice5forcewill vanish if thestructure is stopped frommoving.Therefore, it isnota forcedvibrationbutself-6excitedvibrationinthissituation.7

Depending on the number of ice elements in the system of interest, these twomechanisms can be8converted.AsimilarstatementcanalsobefoundinDing(2013).Ifthestudiedsystemisextendedtoamacro9scale,theexternalexcitationcausedforcedvibrationwillbeconvertedintotheinternalexcitationthatleads10toself-excitationvibration,andviceversa.Takingiceasanexample,iftheiceandstructurearecoupledas11onedynamicsystem,theIIVshouldbelongtoself-excitedvibrationcategory.Onthecontrary,ifthestructure12isisolatedfromtheiceandisconsideredasonedynamicsystem,thefluctuatingforcecausedbyicefailure13would be an external excitation and IIV should be categorised as forced vibration. Therefore, there are14internalandexternaleffectsfromthestructurepointofview.15

Fromtheicepointofview,thereareinternalandexternaleffectstoo.Internalfromiceistheoriginalice16failure characteristic. It is thebehaviour under the assumption that ice fails against a theoretically rigid17structure,whichmeansthereisnovibrationorfeedbackcomingfromtheexternalstructure.Butstructure18doesvibrateinrealcases.Hence,thestructuralvibrationoroscillationeffectistheexternaleffecttotheice19failure.Backinnumericalmodelling,theexternaleffectisrepresentedinthestressvariationsandtheforcing20termontheright-handsideofVanderPolequation.Ifthestressisaconstantandtheforcingtermiszero,21then icewill failunderpurely internaleffect.Otherwise,relativevelocityandrelativedisplacementfrom22structurewillbeeffectiveinicefailureandleadtodifferenticeforceandstructuralmovementbehaviours,23i.e.,thegeneralthreeresponsesshownintheFig.3(c),3(d)and3(e).24

Furtherexplanationscanalsobegivenfromtheicepointofviewinanalysingicefailure.Iftheicevelocity25is slow, ice forcewill drop to almost zeroafter each failurewhichmeans that time interval for clearing26processisrelativelylong.Ifthevelocityincreases,thetimeintervalwillconsequentlybecomeshorter.Atthe27intermediatevelocityrange,icefailsunderthestructuraloscillationeffectbesidestheoriginalfailurenature28whichmeansthatwhenthestructuremovesbackwardsagainsttheice,compressivestressarisesdueto29externalstructurefeedbackandleadtoicedeformationandfailuremorepredominantly.30

4.1Reasonsoflock-in31

Eventhoughiceforcefrequencytendsto increasewith increasing icevelocity, itstill locks inaroundthe32structuralnaturalfrequencybecauseoftheexternalstructuraloscillationfeedbackeffectandrelatedstress33variations.Thechangeofstressdependsonthestrain,i.e.relativevelocitybetweeniceandstructure.When34thestructureandicemoveinthesamedirectionduringIIV,therelativevelocityislowandthevalueofstress35ishigh.Theincreasingiceresistancedeceleratesthestructuremovingwithrespecttoiceandresultsinice36failure frequency laggingbehind theoriginally supposed frequency if thestructure is consideredas rigid37without vibration. The lagging frequency is called hysteresis frequency. Once ice failure occurs, ice and38structurestartmovinginoppositedirectionsandtherelativevelocitybecomeshigh.Hence,lowicestress39value reduces the ice resistance and accelerates the structurebackwards against ice (Määttänen, 2015)40exertingmuchlowericeforcevalueuponthestructure.Moreiceelementfailuresresultinhighericeforce41

12

frequency than the originally supposed frequency. At the same time, during each interaction with the1structure the oncoming ice sheet will also decelerate the structural velocity and raise the stress value2becauseoflowerrelativevelocity.Oncethestructuraldecelerationandaccelerationoscillationprocessare3inastablefeedbackconditionwhenrestoringforceisequaltotheiceforce,thestructurewillbeinaself-4excited oscillation condition. Therefore, the structural natural frequency will be predominant of the5oscillationandiceforcepredominantfrequencylocksinthestructuralnaturalfrequency.6

WhenicevelocitiesareabovetheIIVlock-inrange,relativevelocitywillstayathigherrangeresultinginvery7smalltimeintervalforthestructuretogivefeedbacktoice.Consequently,iceoriginalfailurecharacteristic8becomespredominantandiceforcefrequencywillincreasewithincreasingicevelocity.Atthesametime,9theresponseamplitudeisdecreasingwithlowericestressleadingtothestructurevibratingatthenatural10frequencywithsmallamplitudeorevendiminish(Huangandothers,2007).11

4.2Damping12

Dampingcannotbenegligiblewhenthestructure isoscillating,especially in IIV.YueandGuo(2011)and13Kärnäandothers(2013)noticedthatvibratingfrequenciesduringIIVareslightlybelowthenaturalfrequency.14Itisthereasonwhendampingforceislargecomparedtothespringorinertiaforcesthatdifferfromthe15naturalfrequencyappreciably(DenHartog,1947).16

Mathematically,self-excitedvibrationoccurswhenthereisanequilibriumbetweendampingenergyand17externalexcitationenergyduringafullcycleofvibration(Ding,2013)sothatthestructurevibratesbyitself18without input fromexternal excitation energy to themechanical system. In otherwords, the excitation19energy is totally dissipated by damping, i.e. zero net input energy in a cycle of vibration. Therefore, an20alternativewayofunderstandingforthismechanismisbyaddingthenegativedampingtofreevibration21(DenHartog,1947).Itisthereforecalled“self-excited”becausethereisnoexternalexcitationduringacycle22ofvibration.23

Negative damping, in essence, is used as an external source of energy to increase the amplitude of24vibration. As the characteristic of decreasing ice stress with increasing loading rate, Blenkarn (1970)25proposediceforceasafunctionofrelativevelocityandexplainedtheincreasedvibrationamplitudeinIIV26asnegativedampingtheory:27

( )MX CX K XF vX =+ −+ (5)28

Forsmallmotions,forcingtermcanbewrittenas:29

( )( ) ( )X X F vF v F vv

∂− = −∂

(6)30

Hence,(5)becomes:31

( ) ( )FMX C X KX F vv

∂+ + + =∂

(7)32

Sodhi(1988)expressedsomedisagreementeventhoughhethoughtnegativedampingwasrealisticunder33someconditions.Generalreasonwasthattheforcingtermwasnotonlycontrolledbyrelativevelocitybut34alsorelativedisplacement,time,etc.Therefore,amoredetailedrelationshipbetweeniceforceandrelative35

13

velocityneedstobejustified.Becauseofthisreasonrelativedisplacementeffectisalsoconsideredinthe1currentnumericalmodelduring ice forcecalculation.AsSodhimentioned,aplotof ice forcevs. relative2velocitycanbemorepersuasive.Forinstance,Fig.4istheresultfromTest.110inwhichiceforcecanbe3taken as a function of relative velocity. It is clear that the slope, / rF v∂ ∂ , is sometimes positive and4

sometimesnegative.Thenegativevaluecanleadtonetnegativedampingin(7)whichwillthenleadthe5system tounstable condition. Theblue line is the first cycle of loadingwhichhashigher negative slope6comparingwithotherstableconditionssuchastheredlineinacycleof“spike”likeiceloadingundersteady-7statevibrationandothersinblackdashedlines.8

9

Fig.4.Relativevelocityvs.iceforceinfourseconds(blackdashed):thefirstcycleofloading(blue)and“spike”10likeloading(red).11

4.3Energyconservation12

Itiscertainthatenergyisalwaysconservativeduringice-structureinteractionprocess.Theultimatereason13ofincreasedstructuralvibrationisduetomorenetenergyinputtostructure.Forceincreaseswhen“negative14damping”occursandmoreenergytransmitsintothestructureleadingtoincreasedvibrationamplitude(Den15Hartog,1947).NotethatthenegativedampingproposedbyBlenkarn(1970)isactuallymovingtheexternal16structuralvelocityrelatedforcingtermfromtheright-handsideoftheequationofmotiontotheleft-hand17side.18

Sodhi (1991a) presented further evidence that there is no real negative damping based on his19experimentalresultsgiveninSodhi(1991b).Instead,hefoundthatthecumulativeworkdonebytheindenter20isanon-decreasingfunction.Accordingtothefirstlawofthermodynamics,frictionalanddampingforceof21thestructurearealwaysdissipatingenergy intoheat.Therefore,apositivedamping forcedoesnegative22

14

work.Inthepresentmodel,theenergyrelationshipsatisfiesthefollowingequationduringtheinteraction1process:2

t m dW E EΔ = Δ + Δ (8)3

where tWΔ istheincrementalworkdonebyice, mEΔ istheincrementalchangeofmechanicalenergyin4

thestructureand dEΔ istheincrementalheatdissipatedbythestructuraldampingsincefrictionbetween5

ice and structure is not considered in themodel. The incremental change ofmechanical energy in the6structurecanbewrittenas7

m p kE E EΔ = Δ + Δ (9)8

where pEΔ and kEΔ arethe incrementalchangeofpotentialenergyandkineticenergy inthestructure,9

respectively.Thecomputationof tWΔ betweenanytwoinstantsoftimeisthroughmultiplyingtheaverage10

forcebythecorresponding incrementalstructuraldisplacement, i.e. F XΔ .Thechangeofkineticenergy11

andpotentialenergyareobtainedfrom 20.5M XΔ and 20.5K XΔ ,respectively.Theenergydissipateddueto12dampingcanbecomputedfrom(8).Thecumulativeformandintegralformof(8)and(9)are13

t dW PE KE EΣΔ = + +ΣΔ (10)14

0 0 0 0

T T T T

dFdX KXdX MXdX dE= + +∫ ∫ ∫ ∫ (11)15

Eachparameterin(10)isshowninFig.5atfourdifferenttestssuchastotalcumulativeenergydoneby16

iceforce( tWΣΔ )inred,potentialenergy( PE )inpurple,kineticenergy(KE )ingreen,mechanicalenergy17

ofthestructure( PE KE+ )inblack,anddissipatedenergyduetodamping( dEΣΔ )inblue.Itcanbenoticed18

thatmostoftheenergyarisesfromiceforcedissipatedbythedampingofstructurewhichisdifferentfrom19whatSodhi(1991a)found,i.e.energysuppliedbycarriagewasdissipatedmostlybytheindenter.Because20inthetestsofSodhi(1991b),anindenter,i.e.structure,wasattachedtoacarriagetomoveagainsttheice21inabasin.Tosimplifythemodellingandunderstanding,iceismovingtowardsthestructureinthecurrent22numericalmodel.Therefore,subtractingtheworkdonebycarriagefromindenterintheexperimentisthe23sameworkdonebytheiceforceinthepresentmodel.24

Accordingtothenumericalresults,therearethreedifferentenergyexchangecharacteristicsbecauseof25threedifferenttypesofice-structureinteractionsatdifferenticevelocitieswhichcapturethegeneralsimilar26patternwiththoseinSodhi(1991a).Fig.5(a)and5(b)showthefailureunderintermittentcrushing.During27eachcycleof loading, thestructuremoveswith the increasing force, resulting in largedisplacementbut28relativelysmallvelocityofthestructure.Energysuppliedbyiceismostlystoredinthestructuralspring.After29thefailureofice,thestoredpotentialenergyisthentransferredtokineticenergyleadingtothebackwards30movementof thestructureandextrusionof ice.Dampingmechanismconsumes theenergyall the time31whenthevelocityofthestructureincreasesandreachesabalanceconditionwiththeenergyfromiceforce32attheendofeachcycleoficefailure.Fig.5(c)showstheenergyexchangeatsteady-statevibration.From33theenlargeddetailofthedashedbox,itcanbeseenthatpotentialandkineticenergyareinasinusoidal34exchangerelationshipsimilartowhatSodhi(1991a)found.Afterthefirstfewcycleofloading,theenergy35suppliedbyiceforceinonecycleisequaltotheenergydissipatedbythedamping.Duringthecontinuous36

15

crushingathighvelocity,asshowninFig.5(d),mostofthemechanicalenergyareintheformofpotential1energyremainingataroundaconstantandkineticenergyremainsatalmostzero.Becausethestructureis2pushedbyicetoarelativelystaticposition,itvibratesatmuchloweramplitude.3

4

Fig.5.Timevs.totalenergy(red),potentialenergy(purple),kineticenergy(green),mechanicalenergy(black)5anddampingenergy(blue)atdifferenttestconfigurations.(a)Test.67,K=890kNm-1,v=0.0412ms-1.(b)6Test.63,K=3230kNm-1,v=0.0411ms-1.(c)Test.110,K=2700kNm-1,v=0.1031ms-1.(d)Test.203,K=11307kNm-1,v=0.1452ms-1.8

4.4Stressandforcevariations9

Since 0.0412 m s-1 and 0.1452 m s-1 in Sodhi (1991b) was defined as intermediate and high velocity,10respectively, estimated rangeof ice velocity at the test condition isbetween0.01m s-1and0.165m s-111approximately.FivesetsoftestsfortheconfigurationsinTable1wereconductedexceptthaticevelocities12wereusedfrom0.01ms-1to0.165ms-1at0.005ms-1intervals.Histogramsoftimehistoryplottingofstress13values and ice force values ineach setof tests show similarpatternwitheachother as the ice velocity14increases.Twosetsoftests,Test.67andTest.110,arechosenasshowninFig.6(a)andFig.6(b),respectively.15Duringeach0.005ms-1 icevelocity, timehistoryplottingofstressand ice forcepointsarecountedand16accumulated at 0.725MPa and 0.5 kN intervals, i.e. 0.3625MPa and 0.25 kN from both sides of each17histogram,respectively.Rangesofstressandforcearedeterminedbytheminimumandmaximumvaluesof18them. Each histogram amplitude is then normalized into the relative amplitude by the maximum19

16

accumulatedamplitudeineachsetoftests.Stressvariationsshowatrendfromlowvelocityhighstressto1highvelocitylowstress.Iceforcevariationsshowaconcentrationataroundintermediatevelocitiesrange,2i.e.IIVrange,whichmeansthereismoreintegralofforceoverthesametimeinterval,i.e.impulseenergy,3appliedtoincreasethemomentumofthestructure.Thissituationoccursattheconditionwhenstressvalues4are relatively evenly distributed. These patterns may be worthwhile to be noticed from an energy5conservationpointofview.6

7

Fig.6(a).8

9

Fig.6(b).10

Fig.6.Histogramofstressandiceforcevariationsatdifferenticevelocities(a)Test.67,K=890kNm-1.(b)11Test.110,K=2700kNm-1.12

17

4.5Typeofvibration1

ItisdebatablethatwhetherIIVisforcedvibrationfromresonanceorself-excitedvibrationfromnegative2dampingasintroducedearlier.Onethingiscertainthatforcedvibrationdoesnotneedaninitialcondition.3However,self-excitedvibrationneedstobetriggedbyforcedvibrationanditneedsenergyfromanexternal4sourcetosustain(DenHartog,1947).5

Nomatterwhich typeofvibration,bothof themhas the following relationship, i s nf f f= = , inwhich6

frequencyof iceforce,structuraldisplacementandstructuralnatural frequencyareallequal.Commonly7speaking,resonanceisaspecialtypeofforcedvibrationwhichoccurswhentheexternalexcitationfrequency8isequaltothestructuralnaturalfrequency.Forinstance,makingoneforktovibratefirstwillcauseanother9identicalforktoautomaticallyvibrate.So,itiscontrolledbytheexternalsource.Iftheexcitationfrequency10is different from the structural natural frequency, resonancewill disappear.On the other hand, in self-11excitedvibration,vibratingfrequencyisequaltothenaturalfrequencyandtheexcitingforceshouldbea12functionofthemotionvariables,suchasdisplacement,velocityoracceleration(Rao,2004).13

InIIV,vibrationisself-excitedpredominantlybecausefeedbackfromexternalstructuremakesmoreeffect14onicefailureandicewillaffectthestructureinreturn,likelock-inphenomenon.Thereasonoftheincreased15vibrationamplitudecanbeexplainedbythe“negativedamping”theoryinanalternativewaybykeepingin16mindthatthedampingisnotnegativeinreality.Theincreasedvibrationisattributedtomoreenergyinto17thestructurewhenthereishighericeforceandthestressvalueisconcentratedmainlyinthehigherrange18ofstressvalues.19

For vibrations below and above the IIV range, structure vibrates at forced vibration mechanism20predominantlybecausethetimeintervalbetweeneachcycleofvibrationissoshortthatthereisnotimefor21thestructuretogivefeedbacktoice.So,theinternalicefailurecharacteristicmakesmoreeffectwithlower22stressandiceforce.23

4.6Friction-inducedvibration24

Frictionalforceisanotherimportanteffectthatbuildsuptheiceforcecharacteristic.Withtheformationof25microcracksinsideicewhenitisinteractingwithstructure,thestaticiceforceisbuildingupatthesame26timewiththetrendofslidingupanddownmotionalongtheverticaldirectionof interactionsurface.Ice27failureoccurswhenthecrackpropagatestillacertainlevelthatcannotholdtheforceperpendiculartothe28interactionsurface.At thesametime, themaximumstatic frictional forcealong theverticaldirectionof29interactionsurfaceisreached.30

Afterthefailure,iceforceshowsatypicaltransitioninstantfromstaticfrictiontokineticfriction,which31canbefoundatallicevelocityrangesinthetestsfromSodhi(1991a),suchasthecreepdeformationatvery32lowicevelocityinTest.64,theextrusionphaseatintermittentcrushingunderintermediatevelocityinTest.33204andtransitionfromintermittentcrushingtocontinuouscrushingathighvelocityinTest.206.34

Sodhi(1991b)foundthateffectivepressureduringcontinuouscrushingisinanorderofmagnitudelower35thanthepeakpressureduringintermittentcrushingfrombothfull-scaleandsmall-scaleexperiments.Itcan36alsobeexplainedthatkineticfrictionoccursathighervelocitiesleadingtomuchlowerpressure.Sukhorukov37(2013)foundthatthemeanvalueofstaticandkineticfrictioncoefficientsoficeonsteelare0.50and0.1138

ondrysurface,and0.40and0.09onwetcondition,respectively,asshowninTable.2,where sµ and kµ are39

18

the staticandkinetic frictioncoefficients, respectively.Althoughvaluesof steelon icehave lower static1frictioncoefficient,theyarestillmuchhigherthanthekineticfrictioncoefficients.2

Table2.Staticandkineticfrictioncoefficientsfromice-steelexperiments(Sukhorukov,2013).3

Slidingconfiguration

Surfacecondition

sµ kµ

Iceonsteel Dry 0.50±0.12 0.11±0.02Iceonsteel Wet 0.40±0.05 0.09±0.02Steelonice Dry 0.43±0.09 0.12±0.03Steelonice Wet 0.36±0.09 0.13±0.04Icestressvariations in IIV isverysimilar to frictionalcoefficientvariations in friction-inducedvibration4

when considering relative velocity only as shown in the three-region Stribeck curve in Fig. 7. A typical5exampleistheself-excitedvibrationofabowedviolinstring.Thebowandstringaremovinginthesame6directionat firstwhenthebowdrags thestringaside.Thecoefficientof friction ishighbecauserelative7velocityislowandpotentialenergyisstoringinthestring.Whenthemaximumstaticforcecannotholdthe8restoring force fromstring, thestringwill slipbackreleasing theenergy tokineticenergy in the formof9backward velocity. The decrease in coefficient of friction yields lower frictional force, which will then10accelerate the velocity further to a certain level. However, the coefficient will raise again due to the11coefficientcurveshowninFig.7andthesystemwillbeinastablefeedbackconditionwhenrestoringforce12isequal to the frictional force.Lessenergy is lost thanthe inputat firstandthedifference isenoughto13overcomethedampingandsustainthevibration (SchmitzandSmith,2011), liketheenergyexchange in14steady-steadystateshowninFig.5(c).Thesoundofvibrationsisproducedatitsnaturalfrequencysinceit15determinestherestorationofthestring.16

17

Fig.7.Stribeckcurve.18

Slidingvelocity

Coefficientoffrictio

n

19

5. CONCLUSIONS1

To obtain the effect of velocity and structural natural frequency (structural stiffness) on ice failure, an2extendedmodelbasedonthepreviousworkofJiandOterkus(2016)wasdeveloped.Aseriesofvalidation3caseswereconductedandcomparedwiththeresultsfromSodhi(1991b)whichshowthetypicalthreekinds4ofresponse;intermittentcrushing,lock-inandcontinuouscrushingandbothnumericalandexperimental5resultsareinagoodagreementwitheachother.6

Physicalmechanismduring ice-structure interactionprocess under different velocitieswerediscussed7basedonthegeneralbranchoffeedbackmechanismandenergymechanism,respectively.Internaleffect8andexternaleffectfromiceandstructurewerebothexplainedinthefeedbackbranch.Energyexchangesof9eachtypeofenergywerereproducedandcoincidedwithanalysesinSodhi(1991a).10

ReasonsoftheincreasedvibrationamplitudeduringIIVandlock-inwerepresentedanddiscussedstarting11fromthedisagreementbetweenSodhiandMäättänenandanalysesweregivenfrombothperspectives.In12addition,astudyonthestressandforcevariationsinfullrangeofvelocitiesshowedthattherewasmore13impulseenergyduringIIVrangewhichcanbeanexplanationfortheincreasedvibrationamplitudefromthe14energypointofview.15

IIVisinaresonanttypeofself-excitedvibrationbecausethestructuraleffectismorepredominant.Even16thoughnegativedampingisnotnegativeinreality,itcanbeusedtoexplaintheself-excitedvibrationinan17alternativeway.Ageneralconclusiononthepredominanttypeofvibrationduringtheinteractionprocessis18forced,self-excitedandforcedineachthreetypesofresponses.19

Similarvariationsbetweenicestressandcoefficientoffrictionshowsthatthereisalikelihoodtousestatic20andkineticfrictionforcetoexplainthepressuredifferenceathighandlowvelocitiesaswellastheunstable21andstableconditionsduringice-inducedvibration.22

REFERENCES23

Blenkarn,K(1970)MeasurementandanalysisoficeforcesonCookInletstructures.OffshoreTechnologyConference,24OffshoreTechnologyConference.25DenHartog,J(1947)MechanicalVibrations.3rded.NewYork,McGraw-Hill.26Ding,W(2013)Self-ExcitedVibration:Theory,Paradigms,andResearchMethods.SpringerScience&BusinessMedia.27Huang,Y,ShiQandSongA(2007)Modelteststudyoftheinteractionbetweeniceandacompliantverticalnarrow28structure.ColdRegionsScienceandTechnology,49(2):151-160.29Ji,XandOterkusE(2016)Adynamicice-structureinteractionmodelforice-inducedvibrationsbyusingvanderpol30equation.OceanEngineering,128:147-152.31Kärnä,T,AndresenH,GurtnerA,MetrikineA,SodhiD,LooM,KuiperG,GibsonR,FenzDandMuggeridgeK(2013)32Ice-InducedVibrationsofOffshoreStructures–LookingBeyondISO19906.22ndInternationalConferenceonPortand33OceanEngineeringUnderArcticConditions(POAC),Espoo,Finland,1673-1684.34Kärnä,T,MuhonenAandM.S(1993)Rateeffectsinbrittleicecrushing.Conf.PortOceanEng.underArcticConditions.35(POAC´93),Hamburg,59-71.(2).36Kry,PR(1981)Scaleeffectsincontinuouscrushingofice.ProceedingsoftheFifthInternationalAssociationHydraulic37ResearchSymposiumonIceProblems,27-31.38Määttanen,M (1975) Experiences of ice forces against a steel lighthousemounted on the seabed, and proposed39constructionalrefinements.Proc.oftheThirdInt.Conf.onPortandOceanEngineeringunderArcticConditionsPOAC,40Fairbanks.41Määttänen,M(2015) Ice inducedfrequency lock-invibrations-Convergingtowardsconsensus.Proceedingsofthe42InternationalConferenceonPortandOceanEngineeringunderArcticConditions,POAC.43

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