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Stability Analysis of Time Periodic Delayed Differential Equations with HP Time Finite Elements

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Stability Analysis of Time Periodic Delayed Differential Equations with HP Time Finite Elements
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GARG, NITIN ( Author, Primary )
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2008

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Degrees of freedom ( jstor )
Differential equations ( jstor )
Eigenvalues ( jstor )
Error rates ( jstor )
Finite element analysis ( jstor )
Mathieu function ( jstor )
Perceptron convergence procedure ( jstor )
Polynomials ( jstor )
Simulations ( jstor )
Vibration ( jstor )

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University of Florida
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Copyright Nitin Garg. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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STABILITYANALYSISOFTIMEPERIODIC DELAYEDDIFFERENTIALEQUATIONS WITHHPTIMEFINITEELEMENTS By NITINGARG ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2004

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Copyright2004 by NitinGarg

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IdedicatethisworktomyparentsDr.PremGargandDr.KrishnaJindal, andtomybrotherDeepak.

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ACKNOWLEDGMENTS IwouldliketothankDr.BrianP.Mannforgivingmeanopportunityto studyanddevelopmyskillsetunderhisguidance.Igreatlyacknowledgehis timeandsupportDr.Mannhasspenttohelpmeprogressinmyresearch.His desireforexcellenceandperfectionprovidesinspirationtoallstudentsandwill continuetodosoinfuture.IwouldalsoliketothankDr.JohnC.Ziegertforhis commitmentinshapingmycareer.IwouldalsoliketoextendmythankstoDr. TonyL.Schmitz.Hisrelentlesspursuitforperfectionisagreatmotivationfor hisstudents.IthankmycolleaguesScottDuncan,MichaelTummond,AmitD. Jayakaran,UmeshTol,AbhijitBhattacharyya,ChrisZahner,VadimJ.Tymianski, MohammadH.Kurdi,DaveBurton,DukeHughes,RaulZapata,MikeKoplow, andJenniferTateforbeingagreatsupportduringmystayatMTRC/Nonlinear DynamicsLab. Thanksgotomythesiscommittee,Dr.BrianP.Mann,Dr.JohnC.Ziegert, andDr.TonyL.Schmitzfortheirtimeandeort. FinallyIwanttothankmyparentsandmybrotherfortheirlovingsupport. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFTABLES.................................vii LISTOFFIGURES................................viii ABSTRACT....................................xi CHAPTER 1INTRODUCTION..............................1 1.1Motivation...............................1 1.2ThesisOrganization..........................2 2STABILITYOFTIME-PERIODICDELAYEDDIFFERENTIALSYSTEMSWITHPARAMETRICEXCITATION..............4 2.1Introduction..............................4 2.2TheDelayedDierentialMathieuEquation.............6 2.3StabilityPredictionfromDynamicMaps..............7 2.3.1MultipleElementsMapFormulation( h-version )......8 2.3.1.1Multipleelementmonodromyoperator........10 2.3.1.2Mapstabilityfromcharacteristicmultipliers.....12 2.3.2SingleElementStabilityPrediction( p-version ).......12 2.3.2.1Singleelementmonodromyoperator.........14 2.3.2.2Stability.........................16 2.3.3InterpolatedPolynomials...................18 2.3.4TheDampedMathieuEquation(DME)...........21 2.3.4.1DMEsingleelementmonodromyoperator......22 2.3.4.2Floquettransitionmatrix...............24 2.4ErrorAnalysis.............................25 2.5SummaryandConclusions......................29 3SIMULTANEOUSSTABILITYANDSURFACELOCATIONERROR PREDICTIONWITHHPTIMEFINITEELEMENTSMETHOD..30 3.1Introduction..............................30 3.2ModelDevelopment..........................32 3.2.1EquationsofMotion......................32 3.2.2ModalEquationsofMotion..................35 v

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3.2.3CuttingForces.........................36 3.3Analysis................................40 3.3.1FreeVibration.........................42 3.3.2VibrationDuringCutting...................43 3.3.2.1Predictingmotionduringcutting: p-version .....43 3.3.2.2Singleelementmonodromyoperator.........45 3.3.2.3Predictingmotionduringcutting: hp-version ....46 3.3.2.4Multipleelementmonodromyoperator........47 3.3.3Stability............................49 3.3.4SurfaceLocationError....................50 3.4Convergence..............................50 3.5InterpolatedPolynomials.......................56 3.6StabilityExperimentalVerifcation.................59 3.7SummaryandConclusions......................64 4SUMMARY..................................67 4.1CompletedWork...........................67 4.2FutureWork..............................68 REFERENCES...................................69 BIOGRAPHICALSKETCH............................74 vi

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LISTOFTABLES Table page 3{1ModalparametersforafourDOFmillingsystem-1...........54 3{2ModalparametersforafourDOFmillingsystem-2...........54 3{3Cuttingpleasures.............................59 3{4Modalparametersfor12.75[mm]endmill................60 3{5Modalparametersfor19.05[mm]endmill................64 vii

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LISTOFFIGURES Figure page 2{1Multipleelementstabilitypredictionsfromcharacteristicmultipliers ( )trajectoriesinthecomplexplane..................11 2{2ValidationofTFEApredictionsusingEulertimedomainsimulation..13 2{3DelayedMathieuequationstabilitychartfortwodampingcoecient values:1)solidlineindicates =0 : 1;2)dottedlineindicates = 0 : 2.....................................16 2{4Comparisonofstabilityboundariesbasedonminimaltime.......17 2{5ThreedimensionalstabilityregionsfortheDDMEwithrespectto parameters , b ,and .Shadedregionsrepresentstableparameter space...................................19 2{6Fifthorderinterpolatedpolynomialsplottedasafunctionofthenormalizedlocaltime............................21 2{7StabilityboundariesofdampedMathieuequation............24 2{8Gradientplotofeigenvalues( )in vs. parameterspace.......26 2{9Convergenceofeigenvalue( )totrueeigenvaluewithanincreasein polynomialorder.............................28 3{1Spatialrepresentationofmachinetoolstructureatdiscretelocations alongthetool..............................33 3{2Lumpedparametermodeldepictingmultipledegreesoffreedomof thetoolineachcompliantdirection..................34 3{3Schematicrepresentingforcesinup-millingintheplaneofthetooltip.37 3{4Schematicrepresentingregenerationofwavinessofthetooltipinthe planeofmotion. Z representsthedirectionperpendiculartothe machinedsurface.............................39 3{5Schematicrepresentationofelementdiscretizationshownbydividing thetimeincut( t c )intoanumberoftimefniteelementstoforma discretelinearmap,andapproximatingthefreevibration( t f )usingstatetransitionmatrix.......................41 viii

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3{6Schematicrepresentingthestateofunder-cutandover-cutinthecuttingplane.................................51 3{7Percentageerrorreductionwiththeincreaseinnumberofelements forvariousinterpolatingpolynomialorders.Solidlinerepresents convergenceofcubicHermiteswithlinearweightingfunctions.Dottedline,dashedline,anddashedlinewithdotsrepresentconvergenceofthird,ffth,andseventhorderpolynomialsusingGalerkin approachrespectively..........................53 3{8StabilityandsurfacelocationErrorpredictionforafourdegreeof freedommillingsystem.TheparametersareshowninTable3{1. PlotErepresentsthesurfacelocationerroratadepthofcut b = 2[ mm ]andplotErepresentssurfacelocationerrorat b =4[ mm ]..55 3{9Poincaresection,1/tooth,andsimulationdataforcasesA,B,C,and DshowninFigure(3{8).Stablecuttingprocessesareshownincases A(n=11400 rpm;b =2 mm ),B(n=17300 rpm;b =2 mm ), C(n=34800 rpm;b =2 mm ),andD(n=28500 rpm;b =4 mm )...56 3{10Stabilitylobesforafourdegreeoffreedommillingsystem.TheparametersareshowninTable3{2.SolidlinerepresentsTFEApredictionsandbrokenlinerepresentsthestabilitypredictionfromEuler simulation................................57 3{11TFEAconvergesinlargeSLEregionsasnumberofelementsisincreased. ,dashedline,diamonds,andlinewithdotsrepresentSLE predictionusing8,12,25,and50elementsrespectivelywith h-version TFEA.SolidlinerepresentsSLEpredictionsfromEulersimulation.DotsrepresentSLEpredictionusing hp-version TFEAwith 5elementsand7 th orderpolynomials..................58 3{12Fifthorderinterpolatedpolynomialsplottedasafunctionofthenormalizedlocaltime............................60 3{13ExperimentalvalidationofTFEAstabilitypredictionsforthe12 : 75[ mm ] carbideendmilldescribedinTable(3{4).Thesymbol represents astablecase, representsanunstablecaseand . representsborderlinestability..............................61 3{14Poincaresection,phasespace,and1/toothdataforcasesA,B,C, andDshowninFigure(3{13).Casesshowninthisfguredepictan instabilityalsoknownasripbifurcation.Unstablecuttingprocesses areshownincasesA(n=15000 rpm;b =0 : 5 mm ),B(n=15000 rpm;b = 0 : 76 mm ),C(n=15000 rpm;b =1 mm ),andD(n=15300 rpm;b = 1 : 5 mm )..................................62 ix

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3{15ExperimentalvalidationofTFEAstabilitypredictionsforthe19 : 05[ mm ] carbideendmilldescribedinTable(3{5).Thesymbol represents astablecase,symbol representsaclearcutunstablecaseand . representsborderlinestability......................63 3{16Poincaresection,phasespace,and1/toothdataforcasesE,F,G, andHshowninFigure(3{15).Unstablecuttingprocessesareshown incasesA(n=16200 rpm;b =2 : 54 mm ),B(n=16200 rpm;b = 3 : 66 mm ),C(n=17300 rpm;b =2 : 54 mm ),andD(n=17300 rpm;b = 3 : 66 mm ).CasesEandFresultfromanunstableHopfbifurcation andcasesGandHarearesultofripbifurcation...........65 x

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfllmentofthe RequirementsfortheDegreeofMasterofScience STABILITYANALYSISOFTIMEPERIODIC DELAYEDDIFFERENTIALEQUATIONS WITHHPTIMEFINITEELEMENTS By NitinGarg December2004 Chair:BrianP.Mann MajorDepartment:MechanicalandAerospaceEngineering Theresearchpresentedinthisthesisfocusesonp-versionandhp-versiontime fniteelementanalysistopredictthestabilityoftimeperiodicdelayeddierential systems.Applicationofthistechniquetomillinghelpsinsimultaneouslypredicting stabilityandsurfacelocationerrorformultipledegreesoffreedom.Thismethodis capableoffasterconvergencerateswhichmakesitapotentialtoolforwebbased machining-analysisapplications. xi

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CHAPTER1 INTRODUCTION 1.1 Motivation Thefnalgeometryofmostmechanicalpartswithdesireddimensionalaccuracy,surfacefnish,andsurfacequalityisobtainedbymachining.Someofthe importantmetalremovaloperationsareshaping,turning,drilling,broaching,boring,andmilling.Theeciencyoftheseprocessesismeasuredbymetalremoval rates,machinetime,cycletime,andtoolwear.Understandingofthedynamicsof theseprocessesplaysakeyroleinincreasingtheireciency.Whilevariousmethodsareusefulinpredictingthenatureofthemillingprocess[1{4],thedesirefor moreecientmethod,suitableforwebbasedandsoftwareapplications,toanalyze anyradial-immersionmillingprocessforstabilityandsurfacelocationerrorledto thisresearch. Timefniteelementanalysis(TFEA)iscurrentlytheonlymethodforpredictingmillingstabilityandsurfaceplacement.Thismethodwasfrstappliedtoan interruptedturningprocessbyHalley[5]andBayly etal. [6].Theauthorsmatched anapproximationforthecuttingmotionobtainedusingasinglefniteelementto theexactsolutionforfreevibrationtoobtainadiscretelinearmap.Thecharacteristicmultipliersoreigenvaluesofthemapwereusedtodeterminethestability ofthesystem.Comparisonswithexperimentaltestsshowedstrongagreementfor smallfractionsofthespindleperiodincut( ).However,diminishedcorrelation wereshownforlargervaluesof( ).ThiswascorrectedbyBayly etal. [6]bydividingthetimeinthecutintomultiplefniteelementsintime.Thissolutionmethod developedbyBayly etal. [6]forsinglemodeinterruptedturningprocesswasextendedbyMann[7]tomultiplemodemilling.Earlycomparisonwithexperimental 1

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2 resultshaveshownstrongagreementforthecaseofasinglemodeineachdirection. However,asthedegreesoffreedomwereincreased,alargenumberofelementswere requiredtoattainaconvergedsolution.Althougharbitrarilyincreasingthenumber ofelementswillresultinconvergedsolution,adramaticincreaseincomputational timeoftenaccompanieslargeelementcomputations.Thisapproach,wheretheconvergenceisobtainedbyincreasingthenumberofelements,isknownas h-version TFEA.Theanalysispresentedinthisthesisextendsthesolutionmethoddeveloped byMann[8]to hp-version TFEA.Theabilitytoachieveconvergencebyincreasing thenumberofelementsaswellastheorderoftheapproximatingpolynomials, makesthisnumericalprocesscomputationallylessexpensive.Fasterconvergenceof stabilityboundariescanbeobtainedformultipledegreeoffreedommillingsystems. 1.2 Thesis Organization Chapter2investigatestwodierenttemporalfniteelementtechniques,a multipleelement( h-version )andsingleelement( p-version )method,toanalyze thestabilityofatime-varyingsystemwithadelayinthecontrolfeedback.A representativeproblem,knownasthedelayeddierentialMathieuequation,is chosentoinvestigatethecombinedeectoftime-delayandparametricexcitation onstability.Adiscretelinearmapisobtainedbyapproximatingtheexactsolution withaseriesexpansionoforthogonalpolynomialsconstrainedatintermitantnodes. Characteristicmultipliersofthemapareusedtodeterminetheunstableparameter domains.Additionally,thedescribedanalysisprovidesanewapproachtoextract theFloquettransitionmatrixofsystemsgovernedbytimeperiodicdierential equations(i.e.,parametricexcitationwithoutdelayedfeedback). Chapter3investigatesthestabilityofmillingprocesswith p-version and hp-version ofTFEA.Inthe p-version approach,alinearcombinationofhigher orderorthogonalpolynomials,coupledwith C 1 continuity,isusedtoapproximate theexactsolutionforthecuttingmotionwithasingletimeelement.Asymptotic

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3 convergenceoftheapproximatesolutiontotheexactsolutionisobtainedbysimply increasingthepolynomialorder.Theprocessofincreasingthepolynomialorder whilekeepingthenumberofelementsfxedisknownas p-convergence .The hpversion techniqueutilizesthefastconvergingpropertiesof p-version andrefnement propertiesof h-version toapproximatetheexactsolutionoverthecuttingperiod. Theconvergence,inthiscase,canbeobtainedbyincreasingboththepolynomial orderandnumberofelements.Thisprocessislesscomputationallyexpensiveand betterconvergencecanbeobtainedinashortertime.Theconvergenceobtained byincreasingbothpolynomialorderandnumberofelementsisknownas hpconvergence . Thefnalchaptersummarizesthecontributionmadefromthisresearchand outlinesopportunitiesforfuturework.

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CHAPTER2 STABILITYOFTIME-PERIODICDELAYEDDIFFERENTIALSYSTEMS WITHPARAMETRICEXCITATION 2.1 Introduction Thestabilityofsystemsgovernedbytime-periodicdierentialequations isimportanttovariousfeldsofscienceandengineering.Forinstance,recent literaturehasdescribedapplicationsinhighspeedmilling[1,6,9{12],quantum mechanics,structuresunderoscillatingloadsandrotatinghelicopterblades[13]. Someofthemethodsavailableforstabilityanalysisoftimeperiodicsystems withouttimedelaysareHill'smethod[14,15],Floquettheory[16{22],and perturbation[14,23].Morerecently,Sinha etal. [24,25]haveusedChebyshev polynomialstoanalyzethestabilityandcontrolofthesesystems.Theeectof timedelayoncontrolstabilityhasbeenexaminedbyYangandWu[13]who studiedtheeectofatimedelayonstructuraldynamics. TheDelayedDampedMathieuEquation(DDME)providesarepresentative systemwithbothdelayedfeedbackcontrolandparametricexcitation.Forinstance, inhighspeedmillingprocesswhereperiodicityiscausedbynumberofactive teethincut,andtime-delayisduetotheregenerationofwaviness.Someother areasincludequantummechanics,structuresunderoscillatingloads,rotating helicopterblades,turbineshafts,etc.Mathieu[26]usedthisequationwithoutthe time-delaytostudytheoscillationsofaellipticmembrane.BellmanandCooke[27] andBhattandHsu[28]bothmadeattemptstolayoutthecriterionforstability usingD-subdivision[29]methodcombinedwiththetheoremofPontryagin[30]. InspergerandStepanhaveusedasemi-discretizationapproach,whichisapplicable forthecombinationofseveraldiscreteandcontinuoustimedelays,toexamine 4

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5 thestabilityoftheDDME[31{34].Theuseoforthogonalpolynomialstosolve systemswithparametricexcitationwithouttimedelayhasbeenadoptedbymany authors.Forinstance,Chang etal. [35]studiedtheresponseoflineardynamic systems.SinhaandChou[36]andSinha etal. [19]usedorthogonalpolynomials toinvestigatethebehavioroftime-periodicdierentialequations.Orthogonal polynomialsareusedbecausetheydecouplethesuccessivesolutionsaspresented inLindlbauer[37]andhenceincreasetherateofconvergence-therebymakingthe processlesscomputationallyexpensive.Inthischaptertheorthogonalpolynomials areusedtodeterminethestabilityofasystemwithparametricexcitationcoupled withdelayedfeedbackcontrol. ThischapterinvestigatesthestabilityoftheDDMEusingtemporalfnite elementanalysis[5,38].Asetoforthogonalpolynomials,constrainedfor C 1 continuityisusedtoobtainadiscretelinearmapbyapproximatingtheexact solution.Characteristicmultipliersofthemapareusedtodetermineunstable parameterdomains.Twodierentapproachesareusedtoformulatedynamic mapsthatdescribethesystemevolution:1)Multipleelementmethod:The minimaltimeperiodofthegoverningequationisdividedintoanumberoftimeelements.Anapproximatesolutionisthenobtainedasalinearcombinationof interpolatedpolynomialsovertheentiresetoftimeelements.Thistechnique employscubicHermitepolynomialsastrialfunctionstoapproximatetheexact solution.Asymptoticconvergenceoftheapproximatedsolutiontotheexact solutionisobtainedbyincreasingthenumberoftime-elements.Thisisalsoknown as h-convergence .2)Singleelementmethod:Alinearcombinationofhigherorder orthogonalpolynomials,coupledwith C 1 continuity,isusedtoapproximatethe exactsolutionoveraminimaltime-periodofthegoverningequationbyusinga singletimeelement.Asymptoticconvergenceoftheapproximatedsolutiontothe exactsolutionisobtainedbyincreasingthepolynomialorderwhilepreservingthe

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6 desiredproperties.Convergenceobtainedbyincreasingtheorderwhilekeepingthe numberofelementsfxedisknownas p-convergence .Therateofconvergencein p-version isfoundtobemoreecientascomparedto h-version . 2.2 The Delayed Dierential Mathieu Equation ThedelayeddiferentialMathieuequationcanbewrittenas ~x ( t )+ _ ~x ( t )+[ + cos( !t )] ~x ( t )= b~x ( t )Tj/T1_2 11.955 Tf11.955 0 Td( ) ; (2.1) wheretheequationhasaperiodof T =2 =! ,atimedelayof =2 , isthe dampingcoecient, + cos( !t )istheparametricexcitation,and b isthefeedback gainofthesystem. BeforelookingatthestabilityofdelayeddierentialMathieuequation (DDME)itisimportanttoknowvarioustypesofbifurcationsthatoccurin parameterspace.Thenextsectiondescribesthepredictedbifurcationsthatoccur fromaquasi-staticvariationofacontrolparameter.Bifurcationisdefnedasthe suddenchangeinthebehavioroftheoscillatingsystemasaparameterpasses throughacriticalvalue,calledabifurcationpoint.Forinstance,considerputtinga penintheverticalpositionandreleasingit.Mostofthetimeitwillfallononeside ortheother.Theexplanationisthatthepenwaspreparedtoanunstableposition. Dependingupontheposition(initialcondition),thepenfellononesideorthe other.Similarlystockmarketbuildsintoanunstablepositionovermonthstoyears beforetheoccurrenceofcrash.Thisisseeninthebuildupofthetrajectoriesof suchvariablesastheprice,volume,andvolatilityasafunctionoftime.Thetime evolutionofthesevariablesindicatethatthepatternsarenotstableandthatan instabilityisripening.Thisdefnesabubble.Forinstance,higherthebubblebuilds up,themoreunstablethemarketbecomes,untilitreachesthepointwhereany newsoreventcouldtoppleit.

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7 Ingeneral,alinearperiodicordinarydierentialequationcanbewrittenas, _ ~x ( t )= A ( t ) ~x ( t ) ; (2.2) where x isan n dimensionalstatevector,matrix A ( t )istimeperiodicwitha period T ,whichistheminimalperiodofequation(2.2).Equation(2.2)hasa fundamentalsetof n -linearlyindependentsolutions, x i ,where i =1 ; 2 ;::::;n: This impliesthat x i ( t + T )mustbealinearcombinationof x i ( t ),or ~ X ( t + T )= ' ~ X ( t ) ; (2.3) where ' isa n n Floquettransitionmatrix[23].Eigenvalues( m , m =1 ; 2 ;:::;n ) oftheFloquettransitionmatrix,alsoknownascharacteristicmultipliers,are uniqueforagivensystemandcanbecalculatedfrom det( I )Tj/T1_1 11.955 Tf11.955 0 Td(' )=0 : (2.4) Thesystemisstableonlyifallcharacteristicmultipliers( m )areinmodulus lessthanunity.Dependinguponthemannerinwhichthecharacteristicmultiplier leavestheunitcircle,thethreedistinctroutestoinstabilityare:1)Acharacteristicmultiplierpenetratestheunitcirclethrough+1(realaxis),resultinginone ofthefollowingthreebifurcations: transcritical,symmetry-breakingandcyclicfold bifurcations .2)Acharacteristicmultiplierleavestheunitcirclethrough-1(real axis),resultingina period-doubling bifurcation.3)Apairofcomplexconjugate characteristicmultipliersexitstheunitcircleawayfromrealaxis,resultingina Hopf or Neimark bifurcation. 2.3 Stability Prediction from Dynamic Maps Systemsgovernedbytime-delaydierentialequationsdonothaveaclosed formsolution[6,39].Inordertopredicttheirbehavior,anapproximationtothe exactsolutionisrequired.Timefniteelementanalysis(TFEA)isatechnique

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8 toexaminethedynamicbehaviorofasystemwithtimedelay,suchasdelayed dierentialMathieuequation(2.1).Characteristicmultipliersofthemapareused todeterminethestabilityofthesystem. 2.3.1 Multiple Elements Map Formulation ( h-version ) Toapproximatethesolutiontoequation(2.1),theminimaltimeperiod( T ), where T = min ( ; 2 =! ),isdividedintoanumberoftimefniteelements.The approximatesolutioncanthenbewrittenas x ( t ) 4 X i =1 a n ji i ( j ( t )) ; (2.5) where i ( j ( t ))arethecubicHermitepolynomials,ortrialfunctionsdefnedin equation(2.6)and j ( t )isthelocaltimewithinthe j th elementofthe n th period. Coecients a ji arecalculatedbymatchingtheinitialdisplacementsandvelocities ofeachelement. 1 ( j )=1 )Tj/T1_0 11.955 Tf11.955 0 Td(3 j t j 2 +2 j t j 3 ; (2.6a) 2 ( j )= t j " j t j )Tj/T1_0 11.955 Tf11.955 0 Td(2 j t j 2 + j t j 3 # ; (2.6b) 3 ( j )=3 j t j 2 )Tj/T1_0 11.955 Tf11.955 0 Td(2 j t j 3 ; (2.6c) 4 ( j )= t j " )Tj/T1_5 11.955 Tf11.291 16.85699 Td( j t j 2 + j t j 3 # : (2.6d) where t j = T=E and E isthenumberofelementsintowhichminimaltime T is divided.Substitutionoftheapproximatesolutionintoequation(2.1)givesanerror

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9 e rr ( t ), 4 X i =1 ~a n ji i ( j ( t ))+ 4 X i =1 ~a n ji _ i ( j ( t )) +[ + cos( !t )] 4 X i =1 ~a n ji i ( j ( t )) )Tj/T1_1 11.955 Tf11.955 0 Td(b 4 X i =1 ~a n )Tj/T1_3 7.97 Tf6.58701 0 Td(1 ji i ( j ( t ))= e rr ( t ) : (2.7) Theresidualofweightederrorissettozerowhichgivestwoequationsper elementtosolveforthecoecients( a ji ).Thepurposeofweightingtheresidual error( e rr )istoselectfrominfnitepossiblesolutions(
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10 Thecontinuityconditionofequation(2.10)holdstrueforcoecientsatthe beginningandendofeachelement.Theconditionrelatesthestatesattheendof thepreviousperiodtothebeginningofthenextperiod. 2.3.1.1 Multiple element monodromy operator Coecientsoftheassumedsolutioncanberelatedtothoseofthe previousperiodbyarrangingequations(2.8)and(2.10)intoamonodromyoperator matrixofsize(2 E +2) (2 E +2).Theexpressionfortwoelementscanbewritten as 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 100000 010000 N 1 11 N 1 12 N 1 13 N 1 14 00 N 1 21 N 1 22 N 1 23 N 1 24 00 00 N 2 11 N 2 12 N 2 13 N 2 14 00 N 2 21 N 2 22 N 2 23 N 2 24 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 11 a 12 a 21 a 22 a 23 a 24 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000010 000001 P 1 11 P 1 12 P 1 13 P 1 14 00 P 1 21 P 1 22 P 1 23 P 1 24 00 00 P 2 11 P 2 12 P 2 13 P 2 14 00 P 2 21 P 2 22 P 2 23 P 2 24 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 11 a 12 a 21 a 22 a 23 a 24 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n )Tj/T1_4 7.97 Tf6.58701 0 Td(1 ; (2.11) where N j pi = Z t j 0 h i ( j )+ _ i ( j )+[ + cos( !t )] i ( j ( t )) i p ( j ( t )) d j ( t ) ; (2.12)

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11 Figure2{1:Multipleelementstabilitypredictionsfromcharacteristicmultipliers ( )trajectoriesinthecomplexplane. P j pi = b Z t j 0 i ( j ( t )) p ( j ( t )) d j ( t ) ; (2.13) Equation(2.11)takestheformofadiscretelinearmapthatcanbewrittenas A ~a n = B ~a n )Tj/T1_5 7.97 Tf6.586 0 Td(1 or ~a n = Q ~a n )Tj/T1_5 7.97 Tf6.58701 0 Td(1 (2.14) where Q isthemonodromyoperator.Thecoecientsoftheassumedsolution ~a n representthevelocityanddisplacementatdiscretepointsintime.Thisprovidesa dynamicmapoverasingletimedelay.

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12 2.3.1.2 Map stability from characteristic multipliers Characteristicmultipliersofthetransitionmatrix Q determinethe stabilityofthegoverningequationbaseduponwhethertheyresidewithinthe unitcircle.Thesystemisstableforagivensetofparameters( ;;!;b )ifallthe characteristicmultipliershavemoduluslessthanone.Infgure(2{1-A),asolid linerepresentsthestabilityboundary( =1)foragivenparametercombination of and .Theperiodofequation(2.1)is2 , =0 : 1and b =0 : 01.Theregion markedasstablehaseigenvalueslessthanoneandtheregionmarkedunstable haseigenvaluesgreaterthanone.Varying from0 )Tj/T1_0 11.955 Tf12.341 0 Td[(1andkeeping constantat 1.04,trajectoriesofcharacteristicmultipliersexhibitatranscriticalbifurcation(see fgure(2{1-B)).AFlipbifurcationisobservedwhen isheldconstantat1and variesfrom0 : 4 )Tj/T1_0 11.955 Tf12.17999 0 Td(0 : 7asshowninfgure(2{1-C).Stabilitylobesfortheparameters mentionedabovewerecheckedbyEulersimulationasshowninFigure(2{2). 2.3.2 Single Element Stability Prediction ( p-version ) Althoughincreasingthenumberofelementsresultsinconvergedstability boundaries,adramaticincreaseincomputationaltimeoftenaccompanieslarger elementscomputations.Therefore,theapproachdescribedintheprevioussection wasmodifedbyincreasingtheorderofapproximatingpolynomialswhileholding thenumberofelementstoone.Therevisedapproximatesolutionbecomes, ~x ( t ) s X i =1 ~a n i i ( ( t )) ; (2.15) where s isthetotalnumberofhigherorderinterpolatedpolynomials( i ( ( t )).The interpolatingpolynomials,describedlaterinsection2.3.3,areoforder p where p = s )Tj/T1_0 11.955 Tf11.955 0 Td(1 : (2.16) Substitutionoftheapproximatesolution,equation(2.15),intoequation(2.1) leadstoanonzeroerror e rr ( t )

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13 Figure2{2:ValidationofTFEApredictionsusingEulertimedomainsimulation. s X i =1 ~a n i i ( ( t ))+ s X i =1 ~a n i _ i ( ( t ))+[ + cos( !t )] s X i =1 ~a n i i ( ( t )) )Tj/T1_4 11.955 Tf11.955 0 Td(b s X i =1 ~a n i )Tj/T1_3 7.97 Tf7.08501 0 Td(1 i ( ( t ))= e rr ( t ) : (2.17) Theintegralofweightederrorissettozerowhichprovidestwoequations linearinthecoecientsoftheassumedsolution.Weightingfunctions( i ( ( t ))) werechosentobethesameasinterpolatedpolynomials, i ( ( t ))= i ( ( t )) ; (2.18)

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14 i =1 ; 2 ;:::::;s: ThisintegralformulationisknownastheWeightedGalerkinmethod [40].Theresultantequationcanbewrittenas, Z t j 0 " s X i =1 ~a n i i ( ( t )) i ( ( t ))+ s X i =1 ~a n i _ i ( ( t )) i ( ( t )) +[ + cos( !t )] s X i =1 ~a n i i ( ( t )) i ( ( t )) )Tj/T1_1 11.955 Tf9.298 0 Td(b s X i =1 ~a n i )Tj/T1_5 7.97 Tf7.08501 0 Td(1 i ( ( t )) i ( ( t )) # d ( j ( t ))=0 ; (2.19) where t j ,theintegrationtimeforthesingleelement,isequaltotheminimaltime period( T ). Thecoecientsfromfrsttwotrialfunctionsonthefrstelementrepresentthe velocityanddisplacementatthestartofeachperiod.Theycanberelatedtothe coordinatesattheendofthepreviousperiodby 0 B @ a 1 a 2 1 C A n = 0 B @ a s )Tj/T1_5 7.97 Tf7.08501 0 Td(1 a s 1 C A n )Tj/T1_5 7.97 Tf6.58701 0 Td(1 (2.20) 2.3.2.1 Single element monodromy operator Sinceinitialandfnalstatesofthesystemcanbespecifedintermsofasingle polynomialcoecient,asimplisticmappingtothenextperiodcanbewrittenwith theunitymatrixandtheidentitymatrix. Thecoecientsoftheassumedsolutioncanberelatedtothoseoftheprevious periodbyarrangingequations(2.19)and(2.20)intoamonodromyoperatormatrix

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15 ofsize(2+ s ) s .Theexpressioncanbewrittenas 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 100 ::: 00 010 ::: 00 N 11 N 12 N 13 :::N 1 s )Tj/T1_4 7.97 Tf6.586 0 Td(1 N 1 s N 21 N 22 N 22 :::N 2 s )Tj/T1_4 7.97 Tf6.586 0 Td(1 N 2 s . . . . . . . . . . . . . . . . . . N s 1 N s 2 N s 3 :::N ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 N ss 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 1 a 2 a 3 a 4 . . . a s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 ::: 10 000 ::: 01 P 11 P 12 P 13 :::P 1 s )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 1 s P 21 P 22 P 22 :::P 2 s )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 2 s . . . . . . . . . . . . . . . . . . P s 1 P s 2 P s 3 :::P ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 P ss 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 1 a 2 a 3 a 4 . . . a s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n )Tj/T1_4 7.97 Tf6.58701 0 Td(1 ; (2.21) where N ii = Z t j 0 h i ( ( t ))+ _ i ( ( t ))+[ + cos( !t )] i ( ( t )) i i ( ( t )) d ( t ) ; (2.22) P ii = b Z t j 0 i ( ( t )) i ( ( t )) d ( t ) : (2.23) Adiscretelinearmap,describedbyequation(2.21),canbewrittenas A ~a n = B ~a n )Tj/T1_4 7.97 Tf6.586 0 Td(1 or ~a n = Q ~a n )Tj/T1_4 7.97 Tf6.586 0 Td(1 ; (2.24) wherethemonodromyoperatorisgivenby Q = ( A T A ) )Tj/T1_4 7.97 Tf6.586 0 Td(1 A T B : (2.25)

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16 Figure2{3:DelayedMathieuequationstabilitychartfortwodampingcoecient values:1)solidlineindicates =0 : 1;2)dottedlineindicates =0 : 2. 2.3.2.2 Stability Asdiscussedintheprecedingsection(2.3.1.2),theconditionforasymptotic stability(seeFigure2{3andFigure2{5)isthatthecharacteristicmultipliers( i ) ofthetransitionmatrix Q shouldsatisfythefollowingcondition, max j i j < 1 ; (2.26) where i =1 ; 2 ;:::::s .Figure(2{3)representsthestabilityboundaries( =1)of delayedMathieuequationfordierentvaluesof .Solidlineenclosesthestability regioncorrespondingto =0 : 2,anddottedlinecorrespondingto =0 : 1.Asthe valueofdamping( )increasestheamountofstableparameterspacealsoincreases.

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17 Figure2{4:Comparisonofstabilityboundariesbasedonminimaltime. Inbothcases =0,periodequalstimedelay( )=2 andconvergencewas obtainedusing7 th orderpolynomials.Figure(2{5)isathreedimensionalgraphical representationofstablilityin( b;; )parameterspace.Theregionenclosedwithin thesurfaceisstableandthesurface( =1)representsthetransitionfromstableto unstableregion.Here =0 : 2,timedelay( )=2 and ! =1. DelayeddampedMathieuequation(2.1)isasecondorderperiodicsystemwith adelayproportionalto b .Thecoecient + cos( !t )isatimeperiodicfunction withperiod T p =2 =! and isaconstanttimedelayinthesystem.Tofndthe boundsofproportionalcontrolthatstabilizethesystem,itisrequiredthat T p [41{43].Thisleadstoconvergenceofstabilityboundariesatalowerpolynomial

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18 orderwhichmakestheprocesslesscomputationallyexpensive.Basedontheabove discussion,alongwiththeassumptionoftimedelay,threecasesarise:1)Thetime delay isequaltotheperiodofthesystem T p ;2)Thetimedelayislessthan timeperiod;3)Thetimedelayisgreaterthantimeperiod.Assuming ! =3,the systemstabilityiscorrectlyanalyzedfor 2 = 3.Figure(2{4)representsstability boundarywithrespecttotwodierentvaluesoftimedelay( )keeping ! constant. Here ! =3whichmakesperiodoftheequation(2.1)equalto2 = 3.Thesolid linedepictsthestabilityboundarywhentimedelay( )isequalto2 = 3,andthe dottedlineiswhentimedelay( )equals2 .Since,inlattercase,thetimedelayis greaterthantimeperiodthestabilityboundary(dottedlines)isincorrect.Region A isunstableandregion B isstableaspredictedbyEulersimulation(notshown here);andthecorrectstabilityboundarycurve(solidline)wheretimedelay( )is equaltothetimeperiodoftheequation.Moreover,ittook11 th orderinterpolated polynomialstoobtainconvergencewhenthetimedelaywas2 ascomparedto7 th orderpolynomialsforatimedelayof2 = 3. 2.3.3 Interpolated Polynomials Forconvergenceofasecondorderdierentialsystem,thetrialfunctionsor interpolatedpolynomials,mustsatisfyatleasttwoconditionsC 0 continuity, andcompleteness[44].Apolynomialiscompletetodegree p ifallthepowers from0(theconstantterm)to p inclusivearepresent.Completenesswasensured byincreasingthedegreeofthepolynomialswhilepreserving C 0 continuity.The singleelement approachrequiresanelementthatexceedsaminimal C 0 continuity condition,(i.e.,anelementwith C 1 continuitytoobtainbetterconvergence). Todrivepolynomialwith C 1 continuity,thepolynomialanditsderivativewere interpolatedateachinter-elementnode.Thisrequirestwoboundaryconditionsat

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19 Figure2{5:ThreedimensionalstabilityregionsfortheDDMEwithrespectto parameters , b ,and .Shadedregionsrepresentstableparameterspace.

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20 eachinter-elementnode.Thesolutiontakestheform s X i =1 i ( ( t ))=1 ; (2.27a) s X i =1 _ i ( ( t ))=1 ; (2.27b) where ~ isaninterpolatedpolynomial.Therelationshipbetweenthepolynomial order( p )andthetotalnumberofpolynomials( s )is p = s )Tj/T1_0 11.955 Tf11.955 0 Td(1 : (2.28) CubicHermitepolynomials,equation(2.6),representasetoflowestdegree polynomialswith C 1 continuity.Theexpressionforasetof5 th orderinterpolating polynomialstakestheform 1 ( )=1 )Tj/T1_0 11.955 Tf11.955 0 Td(23 ( t ) t j 2 +66 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(68 ( t ) t j 4 +24 ( t ) t j 5 ; (2.29a) 2 ( )= t j " ( t ) t j )Tj/T1_0 11.955 Tf11.955 0 Td(6 ( t ) t j 2 +13 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(12 ( t ) t j 4 +4 ( t ) t j 5 # ; (2.29b) 3 ( )=16 ( t ) t j 2 )Tj/T1_0 11.955 Tf11.955 0 Td(32 ( t ) t j 3 +16 ( t ) t j 4 ; (2.29c) 4 ( )= t j " )Tj/T1_0 11.955 Tf9.298 0 Td(8 ( t ) t j 2 +32 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(40 ( t ) t j 4 +16 ( t ) t j 5 # ; (2.29d) 5 ( )=7 ( t ) t j 2 )Tj/T1_0 11.955 Tf11.955 0 Td(34 ( t ) t j 3 +52 ( t ) t j 4 )Tj/T1_0 11.955 Tf11.955 0 Td(24 ( t ) t j 5 ; (2.29e) 6 ( )= t j " )Tj/T1_2 11.955 Tf11.291 16.85699 Td( ( t ) t j 2 +5 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(8 ( t ) t j 4 +4 ( t ) t j 5 # : (2.29f) Figure(2{6A)andfgure(2{6B)representthedisplacementandvelocity constraintsinlocaltimerespectivelyforffthorderinterpolatedpolynomials ( p =5 ;s =6).Localtimeisdividedintothreenodeswhichincludetwoboundary nodesandoneintermittentnode.Thesumofdisplacementandvelocityateach

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21 Figure2{6:Fifthorderinterpolatedpolynomialsplottedasafunctionofthenormalizedlocaltime. nodeisone.Constraininginterpolatedpolynomialsinthisfashionhelpsmaintain completeness, C 1 continuityandorthogonality. 2.3.4 The Damped Mathieu Equation (DME) Mathieu'sequation-inabsenceoftimedelay-canbewrittenas ~ x ( t )+ ~ _ x ( t )+[ + cos( !t )] ~x ( t )=0 : (2.30) Stabilityandinstabilityofnonlinearsystemscanbetestedbyanapproximate procedurewhichleadstoalinearequation,calledthevariationalequation.The stabilityortheinstabilityoftheoriginalsystemresolvesitselfintothequestionof theboundednessorotherwisethesolutionsofthelinearequation.Forinstance,

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22 thevariationalequationfortheundamped,forced,pendulum(aformofDung oscillator)isMathieuequation(2.30).Thereforetheregionsofstabilityinthe planefortheoriginalsystemwillbetheextensionofthosefortheMathieu equation. Substitutionoftheassumedsolution,equation(2.15),intothegoverning solution,equation(2.30),leadstoanonzeroerror e rr ( t ) s X i =1 ~a n i i ( ( t ))+ s X i =1 ~a n i _ i ( ( t ))+[ + cos( !t )] s X i =1 ~a n i i ( ( t ))= e rr ( t ) : (2.31) ApplyingtheGalerkinresidualmethod(seesection2.3.2),wehave Z t j 0 " s X i =1 ~a n i i ( ( t )) ~ i ( ( t ))+ s X i =1 ~a n i _ i ( ( t )) i ( ( t )) +[ + cos( !t )] s X i =1 ~a n i i ( ( t )) i ( ( t )) # d ( ( t ))=0 : (2.32) Thecontinuitycondition,equation(2.20),isimpliedtorelatethecoecients atthebeginningandendofeachperiod. 2.3.4.1 DME single element monodromy operator TheMonodromyoperatorfordelayedMathieuequation,seeequation (2.30),isaspecialcaseofequation(2.21)where P ii =0.Theexpressioncanbe writtenas

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23 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 100 ::: 00 010 ::: 00 N 11 N 12 N 13 :::N 1 s )Tj/T1_3 7.97 Tf6.586 0 Td(1 N 1 s N 21 N 22 N 22 :::N 2 s )Tj/T1_3 7.97 Tf6.586 0 Td(1 N 2 s . . . . . . . . . . . . . . . . . . N s 1 N s 2 N s 3 :::N ss )Tj/T1_3 7.97 Tf6.586 0 Td(1 N ss 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 1 a 2 a 3 a 4 . . . a s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 ::: 10 000 ::: 01 000 ::: 00 000 ::: 00 . . . . . . . . . . . . . . . . . . 000 ::: 00 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 1 a 2 a 3 a 4 . . . a s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n )Tj/T1_3 7.97 Tf6.586 0 Td(1 ; (2.33) where N ii isgivenbyequation(2.22). Adiscretelinearmap,describedbyequation(2.33),canbewrittenas A ~a n = B ~a n )Tj/T1_3 7.97 Tf6.586 0 Td(1 ; (2.34) where Q isgivenbyequation(2.25).Forstability,allcharacteristicmultipliers oftransitionmatrix( Q )shouldhavemagnitudeoflessthanoneforagivenset of ;!; .Infgure(2{7),asolidlineindicatesthestabilityboundaryofdamped Mathieuequation(2.30)for =0 : 1anddottedlinefor =0 : 2.Periodofthe equation(2.30)inboththecasesin2 andconvergencewasobtainedusing9 th orderpolynomials.

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24 Figure2{7:StabilityboundariesofdampedMathieuequation. 2.3.4.2 Floquet transition matrix Equation(2.33)describesadiscretedynamicalsystemormapthatcanbe writtenas ~a n = Q ~a n )Tj/T1_5 7.97 Tf6.58701 0 Td(1 ; (2.35) where Q isgivenby Q = ( A T A ) )Tj/T1_5 7.97 Tf6.586 0 Td(1 A T B ; (2.36) andthedimensionsof Q area s s squarematrix.Equation(2.35)canbe

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25 rewrittenas 2 6 6 6 6 6 6 6 6 6 6 4 a 1 a 2 . . . a s )Tj/T1_3 7.97 Tf6.58701 0 Td(1 a s 3 7 7 7 7 7 7 7 7 7 7 5 n = 2 6 6 6 6 6 6 6 6 6 6 4 00 ::: 0 c 1 c 2 00 ::: 0 c 3 c 4 . . . . . . . . . . . . . . . . . . 00 ::: 0 ' 11 ' 12 00 ::: 0 ' 21 ' 22 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 a 1 a 2 . . . a s )Tj/T1_3 7.97 Tf6.586 0 Td(1 a s 3 7 7 7 7 7 7 7 7 7 7 5 n )Tj/T1_3 7.97 Tf6.58701 0 Td(1 (2.37) where c i ( i =1 ; 2 ; 3 ;::: )aretheconstanttermsin Q , a 1 ;a 2 ;:::::;a s )Tj/T1_3 7.97 Tf6.586 0 Td(1 ;a s arethe coecientsand n istheperiod. ' 11 , ' 12 , ' 21 , ' 22 aretheelementsoftheFloquet transitionmatrix( ' ).Thereforefromequation(2.37)wehave ' = 2 6 4 ' 11 ' 12 ' 21 ' 22 3 7 5 : (2.38) 2.4 Error Analysis Theerroranalysisaccountsforthediscretizationerrorandassumesallother formsoferrorareabsent.Discretizationerroriscausedbyusingafnitenumber oftrialfunctions ( ( t ))andadiscretesetofcoecients a i inassumedsolutionto approximatetheexactsolutionto x ( ( t )). Thecompletenessandcontinuityaretwominimumconditionsrequiredto closelyapproximatethesecondordersystemwithalinearcombinationoftrial functions.Themeasuredenergyerrorprovidesametricforcloseness,where EnergyError = Z domain E ( ( t )) D [ E ( ( t ))] dt 1 = 2 ; (2.39) and E ( ( t ))= ~x ( ( t )) )Tj/T1_4 7.97 Tf18.632 14.944 Td(s X i =1 ~a n i ~ i ( ( t )) ; (2.40) and D isthedierentialoperatorforthegoverningdierentialequation.Inphysicalapplications,theexpression E ( ( t )) D [ E ( ( t ))],orsimilarly ~x ( ( t )) D [ ~x ( ( t ))], typicallycorrespondstoenergydensity.Thepolynomialsolutionswillconvergeas

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26 Figure2{8:Gradientplotofeigenvalues( )in vs. parameterspace. p approachesinfnity[44].Theeigenvaluesconvergeatthesamerateastheglobal energy.Therateofconvergenceofglobalenergyisoftheorderof O ( E 2( p )Tj/T1_3 7.97 Tf6.586 0 Td(m +1) ), where2 m istheorderofgoverningdierentialequation, p isthepolynomialorder,and E isthenumberoftimeelements.Thisisbecauseinphysicalsystems theeigenvalueiseithertheglobalenergyortheratioofglobalpotentialenergy toglobalkineticenergy[44].Extrapolationisoneofthemostcommonmethods toestimateconvergence[40].Theassumptionunderlyingthismethodisthata solutionisalwaysavailableateachrefnementstep.

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27 Thischapterdiscussestwodierentapproachestoextrapolateconvergence: 1)Twopointextrapolation:Asalreadydiscussedeigenvalueshavethesamerate ofconvergenceasglobalenergy[44].Thereforeconvergenceofanapproximated eigenvaluetoexacteigenvalueisonecriterionforminimizationoferror.The exponentialextrapolationcanbewrittenas est = ext (1 )Tj/T1_1 11.955 Tf11.955 0 Td(e ap ) ; (2.41) where est isanextrapolatedeigenvalue, a isanunknownconstantand p isthe polynomialorder.Initially,tosolvefor a , est istheeigenvaluecorresponding tothemaximumeigenvaluegradientatalowpolynomialorder.Figure(2{8) representsaparticularcaseofdelayedMathieuequation(2.1)where =0 : 2, periodis2 and ! =1.Thirdorderinterpolatedpolynomialsareusedtopredict 15 : 83correspondingtothemaximumgradient.Therefore,inatwopoint extrapolation, est =15 : 83.Thecorrespondingexacteigenvalue, ext ,isfound usingEulertimemarching(simulation).Thesystemwassimulatedusingthesame valuesofparameters( , , b )correspondingtomaximumeigenvaluegradient,and theFloquettransitionmatrix( ' )wascomputedlookingbackonetimeperiod.The maximumeigenvalueof ' is( ext ). ext = max j ( I )Tj/T1_1 11.955 Tf11.955 0 Td(' )=0 j : (2.42) 2)Threepointextrapolation:Thisapproach[40,45]requiresathreeelement solutioncorrespondingtopolynomialorder p , p )Tj/T1_0 11.955 Tf11.979 0 Td[(1and p )Tj/T1_0 11.955 Tf11.98 0 Td[(2.Theexactsolutionof k ~x ( t ) k 2 canbeestimatedbysolving k ~x ( t ) k 2 )-222(k ~x ( t ) p k 2 k ~x ( t ) k 2 )-222(k ~x ( t ) p )Tj/T1_4 7.97 Tf6.58701 0 Td(1 k 2 = k ~x ( t ) k 2 )-222(k ~x ( t ) p )Tj/T1_4 7.97 Tf6.58701 0 Td(1 k 2 k ~x ( t ) k 2 )-222(k ~x ( t ) p )Tj/T1_4 7.97 Tf6.58701 0 Td(2 k 2 log( p )Tj/T1_7 5.978 Tf5.756 0 Td(1 =p ) log( p )Tj/T1_7 5.978 Tf5.756 0 Td(2 =p )Tj/T1_7 5.978 Tf5.756 0 Td(1) ; (2.43) where k ~x ( t ) k 2 representstheenergyformof ~x ( t )and p isthepolynomialorder. Consideringeigenvalues( )havethesamerateofconvergenceasglobalenergy,

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28 Figure2{9:Convergenceofeigenvalue( )totrueeigenvaluewithanincreasein polynomialorder. equation(2.43)canberewrittenas 2 )Tj/T1_1 11.955 Tf11.955 0 Td( 2 p 2 )Tj/T1_1 11.955 Tf11.955 0 Td( 2 p )Tj/T1_2 7.97 Tf6.58701 0 Td(1 = 2 )Tj/T1_1 11.955 Tf11.955 0 Td( 2 p )Tj/T1_2 7.97 Tf6.586 0 Td(1 2 )Tj/T1_1 11.955 Tf11.955 0 Td( 2 p )Tj/T1_2 7.97 Tf6.586 0 Td(2 log( p )Tj/T1_7 5.978 Tf5.756 0 Td(1 =p ) log( p )Tj/T1_7 5.978 Tf5.756 0 Td(2 =p )Tj/T1_7 5.978 Tf5.756 0 Td(1) ; (2.44) where istheexacteigenvaluecalculatedfromatimemarchingalgorithm.Once isknown p canbecalculatedusing(2.44). Bothtwopointandthreepointextrapolationtechniqueswerefoundtohave closeagreementwiththepredictedeigenvalues.Infgure(2{9)dotsrepresent 3-pointextrapolationtechniqueanddotted-linewithcirclesindicates2-point extrapolationwhereassolid-linedepictsthetrueeigenvalueascomputedfromEuler simulation.

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29 2.5 Summary and Conclusions Inthischapter,weanalyzethestabilityandtheeectoftimedelayonthe controlofthesystemsgovernedbydelayeddierentialequationswithperiodic coecients.ThedelayeddierentialMathieuequationisanalyzedforstability usingtwodierenttechniques:1)asingleelement(p-version)timefniteelement method;and2)amultipleelement(h-version)timefniteelementmethod.Inboth casesthetruesolutionisapproximatedusingasetofinterpolatedpolynomials. Stabilityofthesystemisdeterminedbytheeigenvaluesofthediscretelinear map.Thelaterapproachusesonlyonetimefniteelementbutasetofhigher orderorthogonalpolynomialstoobtainconvergence.Ontheotherhand,hversionusescubicHermitepolynomialsandanincreasingnumberofelements toensureconvergence.Therateofconvergenceofthep-versionisveryhigh.A strongconvergenceisobtainedwhen p isincreasedascomparedtoincreasingthe numberofelementsinh-version.Howeverduetosymbolicmanipulationp-version becomesmorecomplicatedandcomputationallyexpensiveasthepolynomialorder isincreasedbeyondacertainvalue. Resultsobtainedfrombothapproacheswereverifedusingtimedomain simulation.Thedevelopedtemporalfniteelementapproachisshowntobea powerfulandrexibleapproachtothesolutionofequationswithperiodiccoecient andtimedelays.

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CHAPTER3 SIMULTANEOUSSTABILITYANDSURFACELOCATIONERROR PREDICTIONWITHHPTIMEFINITEELEMENTSMETHOD 3.1 Introduction Machininginstabilitiesareaprimaryfactorthatlimitstheproductivityof highspeedmilling.Thedegreeofstabilityofagivenstructureisexpressedby thelimitingvalueofthechipwidthatwhich chatter beginstooccur.Chatteris defnedasaformofunstable,self-excitedvibrationwhichoccursduetoavariation inthechipthickness[1,46].Theremovaloftheundulatingsurfaceproducedby theprecedingtoothwiththecurrenttooth,referredtoasregenerationofwaviness, leadstoavariablechipthickness;whichcausesrelativevibrationsbetweenthe toolandtheworkpiece.Forthegivendynamicsofthesystem,dependingonthe selectedchipwidth,thevibrationscandiminishforstablecutting,orquicklygrow forchatter.Itisimportanttoavoidchatterbecausepost-bifurcationcuttingwill leavechattermarksonthemachinedsurfaceandcauseapoorsurfacequality.In someextremecases,itmaycausebreakageofthetoolormayalsoinrictdamage onthemachine.Evenintheabsenceofchatter,theremaybeasteadystate dierencebetweenthemachinedsurfaceandthecommandedsurface.Thisis knownassurfacelocationerroranditisduetotherelativemotionbetweenthe toolandworkpiece[47]. TheinitialinvestigationsofmachininginstabilitybyTlusty,Tobias,and Merrit[48],ledtothedevelopmentofanimportantanalyticaltool,thestability lobediagram,whichallowsthemachinisttopicktheappropriatecombination ofcontrolparameters,chipwidthandspindlespeedforstablemachining.The dynamicsofthemillingprocesscanbeexpressedintheformoftime-periodic 30

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31 delay-dierentialequations.Whilethetypicalapproachistoconstrainanalytical solutionstolinearmodels,ithasbeenshownthatnonlinearitiescanplayan importantrole[6,49].Forinstance,dependingupontheinitialconditions,thesame cuttingexperimentcanexhibitdierentresultswhenperformedmultipletimes. Thisindicatesthepresenceofmultipleattractors,towhichthesolutionconverges dependinguponthestartingconditions.Thiscanonlybecapturedwithnonlinear forcemodelsbecauselinearforcemodelswillonlyshowonesteadystatesolution irrespectiveoftheinitialconditions. Inmostoftheinitialinvestigations,millinganalysiswereperformedunderthe assumptionofacontinuouscuttingprocess[1,46].Sinceitisknownthatmillingis aninterruptedcuttingprocessbecauseeachteethentersandleavestheworkpiece, theassumptionofacontinuousprocessisclearlyanapproximation.Whiletime marchingcanbeusedtocapturetheinterruptednatureofthemillingprocess, theexplorationoftheparameterspacebytimedomainiscomputationallytaxing [1,3,4]. Thischapterpresentsageneralizedapproachtosimultaneouslyinvestigate stabilityandsurfacelocationerrorofamultipledegreeoffreedommillingsystem. Thesolutiontechnique,calledTemporalFiniteElementAnalysis(TFEA),formsan approximatesolutionbydividingthetimeincutintoafnitenumberofelement(s). Theapproximatesolutionduringcuttingismatchedwiththeexactsolutionfor freevibrationtoobtainadiscretelinearmap.Eigenvaluesofthemapareusedto determinethestability,andfxedpointsofthemapareusedforpredictingsteady statesurfacelocationerror. Thepreviousresearchusing h-version TFEA[6{8,50{52],cansometimeslead tolargecomputationtimes.Forinstance,thischapterpresentsacasestudyofa 4degreeoffreedom(DOF)millingsystem,thatrequires450minutestoachieve aconvergedsolution,inlargesurfacelocationerrorregions,whentheparameter

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32 domainisdividedinto19628points.Thischapterextendstheapproachdeveloped byMann etal. [7,8,52]toamoreversatile p-version and hp-version ofTFEA. Inthe p-version approach,alinearcombinationofhigherorderpolynomialsis usedtoapproximatetheexactsolutionoverthetimeinthecutwithasingle timeelement.Asymptoticconvergenceoftheapproximatedsolutiontotheexact solutionisobtainedbyincreasingthepolynomialorderwhilepreservingthe desiredproperties.Convergenceobtainedbyincreasingtheorderwhilekeeping thenumberofelementsfxedisknownas p-convergence .In hp-version boththe polynomialorderandthenumberofelementscanbeincreasedsimultaneouslyto approximatethetruesolution.Convergenceobtainedbysimultaneouslyincreasing theorderoftheinterpolatingfunctionandthenumberofelementsisknownas hp-convergence .Thisdrasticallyreducesthecomputationaltimebyreducing thenumberofelementsrequiredtoachieveconvergence.Forinstance,forthe same4-DOFcaseconvergenceisachievedinlessthan105minutes,reducingthe computationaltimetoapproximately1 = 4.Ingeneral,theparameterdomaincanbe analyzedforstabilityandsurfacelocationerrorbyusingalowerresolution.Inthis casetheresultscanbeobtainedinamatterofafewminutes. Stabilitypredictionsfromthevarioustimefniteelementanalysistechniques arecomparedtotheexperimentalcuttingtestsperformedusinghalf-inchand three-quarterinch,tworute,carbideendmills.Astrongcorrelationwasfound betweentheanalyticalpredictionsandexperimentalresults. 3.2 Model Development 3.2.1 Equations of Motion Amachinetool,justlikeanycontinuoussystem,hasinfnitedegreesof freedom.Thesignifcanceofadegreeofafreedomisdeterminedbythedominance ofthemode.Thedominantmodehasalargereectonthedynamicsofthe machinetool.Toreducethecomplexityofthemodel,consider r degreesof

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33 Figure3{1:Spatialrepresentationofmachinetoolstructureatdiscretelocations alongthetool. freedom,oneateachof r dierentlocationsalongthetoollengthineachdirection whichimpliesatotalof2 r degreesoffreedom.Aschematicdiagramofamachine toolwith r degreesoffreedomisshowninFigure(3{1). Inmatrixform,theequationsofmotion,inspatialcoordinatescanbewritten as 2 6 4 M xx M xy M yx M yy 3 7 5 2 6 4 ~x ( t ) ~y ( t ) 3 7 5 + 2 6 4 C xx C xy C yx C yy 3 7 5 2 6 4 _ ~x ( t ) _ ~y ( t ) 3 7 5 + 2 6 4 K xx K xy K yx K yy 3 7 5 2 6 4 ~x ( t ) ~y ( t ) 3 7 5 = 2 6 4 ~ F x ( t ) ~ F y ( t ) 3 7 5 ; (3.1)

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34 Figure3{2:Lumpedparametermodeldepictingmultipledegreesoffreedomofthe toolineachcompliantdirection. where M x;y , C x;y and K x;y arethematricesrepresentingspatialmass,damping, andstinessrespectively.Figure(3{2)isaschematicrepresentingalumpedmass, stiness,anddampingmodelofthecuttingtoolineachcompliantdirection. Vectors ~x ( t )=[ x 1 ( t ) x 2 ( t ) :::x r ( t )] T and ~y ( t )=[ y 1 ( t ) y 2 ( t ) :::y r ( t )] T representlocal displacementineachdirectionalongthelengthofthetoolduetotheforces ~ F x ( t ) and ~ F y ( t )thatareactingonthetooltip.

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35 3.2.2 Modal Equations of Motion Totransformtheequationsofmotionfromspatialcoordinatestomodal coordinates,theassumptionofproportionaldampingisapplied C = M + f K ; (3.2) where C iswrittenasalinearcombinationof M and K or,inotherwords, proportionaldampingexists.Neglectingdamping,theeigenvaluesolutionineither directioncanbefound.Analyzing ~x directiontoshowtheprocess,itisassumed that ~x = ~e t ; (3.3) andthecharacteristicequationbecomes M 2 + K ~ =0 ; (3.4) where 2 r = )Tj/T1_2 11.955 Tf9.298 0 Td(! 2 n;r and ! n;r isthenaturalfrequency. Normalizingthecoordinatesatthepointoftheapplicationoftheforce,that is,atthetooltip,itcanbeshownthat 1 ; 2 ;:::r = r 1 ; r )Tj/T1_5 7.97 Tf6.586 0 Td(1 1 ;::::::::::::; 2 1 ; 1 T 2 1 ;:::; 2 r : (3.5) Themodalmatrixin ~x direction P x canthenbewrittenas: P x =[ 1 ; 2 ; 3 ;:::::::::; r ] : (3.6) similarlylet P y =[ r 1 ;r 2 ;r 3 ;:::::::::;r r ] : (3.7) Theequationofmotioninmodalcoordinatescanbewrittenas: M q ( t ) ~q ( t )+ C q _ ~q ( t )+ K q ~q ( t )= ~ F q ( t ) ; (3.8)

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36 where M q = 2 6 4 P T x 0 0 P T y 3 7 5 2 6 4 M xx M xy M xy M yy 3 7 5 2 6 4 P x 0 0 P y 3 7 5 ; (3.9) C q = 2 6 4 P T x 0 0 P T y 3 7 5 2 6 4 C xx C xy C xy C yy 3 7 5 2 6 4 P x 0 0 P y 3 7 5 ; (3.10) K q = 2 6 4 P T x 0 0 P T y 3 7 5 2 6 4 K xx K xy K xy K yy 3 7 5 2 6 4 P x 0 0 P y 3 7 5 ;and (3.11) ~ F q ( t )= 2 6 4 P T x ~ F x ( t ) P T y ~ F y ( t ) 3 7 5 : (3.12) Theconversionofspatialcoordinatestomodalcoordinatesdecouplesthemodal equationsandshowsthebeneftthatsystemidentifcationonlyneedstobeapplied atthetooltip. 3.2.3 Cutting Forces Theexpressionforthecuttingforcesactingonthetooltip,basedupon circulartoolpathapproximation,ineachcompliantdirection,isgivenby F x ( t )= )Tj/T1_0 11.955 Tf11.291 0 Td([( F t ( t )cos ( t )+ F n ( t )sin ( t ))] ; (3.13) and F y ( t )= F t ( t )sin ( t ) )Tj/T1_4 11.955 Tf11.955 0 Td(F n ( t )cos ( t ) : (3.14) where F t ( t )and F n ( t )arethetangentialandnormalforcecomponentsrespectively. Figure(3{3)isaschematicrepresentationoftheforcesactingonthetooltipinthe planeofmotion.Summationofcuttingforcesover N teethgives

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37 Figure3{3:Schematicrepresentingforcesinup-millingintheplaneofthetooltip. F xp ( t )= )Tj/T1_2 7.97 Tf16.14 14.944 Td(N X p =1 g p ( t )[ F t p ( t )cos p ( t )+ F n p ( t )sin p ( t )] ; (3.15) and F yp ( t )= N X p =1 g p ( t )[ F t p ( t )sin p ( t ) )Tj/T1_1 11.955 Tf11.955 0 Td(F n p ( t )cos p ( t )] : (3.16) where F t p ( t )= K t b! p ( t )+ K te b; (3.17)

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38 and F n p ( t )= K n b! p ( t )+ K ne b; (3.18) where ! p ( t )istheinstantaneouschipthickness, K t ,and K n arethecutting coecientsalongthetangentialandnormaldirectionrespectively, K te and K ne are theedgecoecients,and g p isaswitchingfunctionbetween0and1.Depending onwhetherthe p th isinoroutofthecut,theswitchingfunction,switchesbetween 0and1.Instantaneouschipthickness, ! p ( t ),isafunctionofregenerationinthe complianttooldirection,feedpertooth, h ,andthecutterrotationangle, p ( t ). Itcanbewrittenassumofthecommandedchipthickness, h sin p ( t ),andchip thicknessduetothesurfaceleftbytheprevioustoothonetimeperiodago.This isalsoknownastheregenerativeeect.Figure(3{4)showstheregenerationeect byun-wrappingthesurfacegeneratedbytheprevioustoothanditseectonthe commandedchipthickness.Notethatthewavegeneratedbytheprevioustoothis outofphasewiththemotionofthecurrenttoothcausingtheinstantaneouschip thicknesstodeviatefromthecommandedchipthickness.Theexpressionforthe instantaneouschipthicknesstakestheform w p ( t )= h sin p ( t )+[ x 1 ( t ) )Tj/T1_1 11.955 Tf11.955 0 Td(x 1 ( t )Tj/T1_1 11.955 Tf11.955 0 Td( )]sin p ( t )+[ y 1 ( t ) )Tj/T1_1 11.955 Tf11.95599 0 Td(y 1 ( t )Tj/T1_1 11.955 Tf11.955 0 Td( )]cos p ( t ) : (3.19) Here =60 =N n[s]isthetoothpassingperiod,nisthespindlespeedgiven in[rpm]and N isthetotalnumberofcuttingteeth.Substitutionofequations (3.17-3.19)intoequations(3.15)and(3.16)gives 2 6 4 F xp ( t ) F yp ( t ) 3 7 5 = N X p =1 g p ( t ) b 0 B @ h 2 6 4 )Tj/T1_1 11.955 Tf9.299 0 Td(K t sc )Tj/T1_1 11.955 Tf11.955 0 Td(K n s 2 K t s 2 )Tj/T1_1 11.955 Tf11.955 0 Td(K n sc 3 7 5 + 2 6 4 )Tj/T1_1 11.955 Tf9.298 0 Td(K te c )Tj/T1_1 11.955 Tf11.955 0 Td(K ne s K te s )Tj/T1_1 11.955 Tf11.955 0 Td(K ne c 3 7 5 + 2 6 4 )Tj/T1_1 11.955 Tf9.298 0 Td(K t sc )Tj/T1_1 11.955 Tf11.955 0 Td(K n s 2 )Tj/T1_1 11.955 Tf9.298 0 Td(K t c 2 )Tj/T1_1 11.955 Tf11.955 0 Td(K n sc K t s 2 )Tj/T1_1 11.955 Tf11.955 0 Td(K n scK t sc )Tj/T1_1 11.955 Tf11.955 0 Td(K n c 2 3 7 5 2 6 4 x ( t ) )Tj/T1_1 11.955 Tf11.955 0 Td(x ( t )Tj/T1_1 11.955 Tf11.955 0 Td( ) y ( t ) )Tj/T1_1 11.955 Tf11.955 0 Td(y ( t )Tj/T1_1 11.955 Tf11.955 0 Td( ) 3 7 5 1 C A ; (3.20)

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39 Figure3{4:Schematicrepresentingregenerationofwavinessofthetooltipinthe planeofmotion. Z representsthedirectionperpendiculartothemachinedsurface. where s =sin p ( t ), c =cos p ( t ). Substitutionofmodeshapesfromequations(3.6)and(3.7)intoequation (3.20)transformstheforcesfromspatialcoordinatestomodalspace.Theexpressiontakestheform 2 6 4 F qx ( t ) F qy ( t ) 3 7 5 = b N X p =1 g p ( t ) 0 B @ h 2 6 4 )Tj/T1_1 11.955 Tf9.298 0 Td(K t sc )Tj/T1_1 11.955 Tf11.955 0 Td(K n s 2 K t s 2 )Tj/T1_1 11.955 Tf11.95599 0 Td(K n sc 3 7 5 + 2 6 4 )Tj/T1_1 11.955 Tf9.299 0 Td(K te c )Tj/T1_1 11.955 Tf11.955 0 Td(K ne s K te s )Tj/T1_1 11.955 Tf11.955 0 Td(K ne c 3 7 5 1 C A + : b N X p =1 g p ( t ) 0 B @ 2 6 4 )Tj/T1_1 11.955 Tf9.298 0 Td(K t sc )Tj/T1_1 11.955 Tf11.955 0 Td(K n s 2 )Tj/T1_1 11.955 Tf9.298 0 Td(K t c 2 )Tj/T1_1 11.955 Tf11.955 0 Td(K n sc K t s 2 )Tj/T1_1 11.955 Tf11.95599 0 Td(K n scK t sc )Tj/T1_1 11.955 Tf11.955 0 Td(K n c 2 3 7 5 1 C A 0 B @ 2 6 4 1 ::: 1 ; 0 ::: 0 0 ::: 0 ; 1 ::: 1 3 7 5 2 6 4 ~q x ( t ) )Tj/T1_1 11.955 Tf11.894 0 Td[(~q x ( t )Tj/T1_1 11.955 Tf11.955 0 Td( ) ~q y ( t ) )Tj/T1_1 11.955 Tf11.894 0 Td[(~q y ( t )Tj/T1_1 11.955 Tf11.95599 0 Td( ) 3 7 5 1 C A ; (3.21)

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40 Substitutingequation(3.21)intoequation(3.8)gives M q ( t ) ~q ( t )+ C q _ ~q ( t )+ K q ~q ( t )= b N X p =1 g p ( t ) 0 B @ h 2 6 4 )Tj/T1_3 11.955 Tf9.298 0 Td(K t sc )Tj/T1_3 11.955 Tf11.955 0 Td(K n s 2 K t s 2 )Tj/T1_3 11.955 Tf11.955 0 Td(K n sc 3 7 5 + 2 6 4 )Tj/T1_3 11.955 Tf9.299 0 Td(K te c )Tj/T1_3 11.955 Tf11.955 0 Td(K ne s K te s )Tj/T1_3 11.955 Tf11.955 0 Td(K ne c 3 7 5 1 C A | {z } 2 r 1 + : b N X p =1 g p ( t ) 2 6 4 )Tj/T1_3 11.955 Tf9.298 0 Td(K t sc )Tj/T1_3 11.955 Tf11.955 0 Td(K n s 2 )Tj/T1_3 11.955 Tf9.298 0 Td(K t c 2 )Tj/T1_3 11.955 Tf11.955 0 Td(K n sc K t s 2 )Tj/T1_3 11.955 Tf11.95599 0 Td(K n scK t sc )Tj/T1_3 11.955 Tf11.955 0 Td(K n c 2 3 7 5 | {z } 2 r 2 r 0 B @ 2 6 4 1 ::: 1 ; 0 ::: 0 0 ::: 0 ; 1 ::: 1 3 7 5 2 6 4 ~q x ( t ) )Tj/T1_3 11.955 Tf11.895 0 Td[(~q x ( t )Tj/T1_3 11.955 Tf11.955 0 Td( ) ~q y ( t ) )Tj/T1_3 11.955 Tf11.895 0 Td[(~q y ( t )Tj/T1_3 11.955 Tf11.955 0 Td( ) 3 7 5 1 C A : (3.22) Inshortequation(3.22)canbewrittenas M q ~q ( t )+ C q _ ~q ( t )+ K q ~q ( t )= b K c ( t )[ ~q ( t ) )Tj/T1_3 11.955 Tf11.895 0 Td[(~q ( t )Tj/T1_3 11.955 Tf11.955 0 Td( )]+ b ~ f o ( t ) ; (3.23) where K c ( t )= N X p =1 g p ( t ) 2 6 4 )Tj/T1_3 11.955 Tf9.298 0 Td(K t sc )Tj/T1_3 11.955 Tf11.955 0 Td(K n s 2 )Tj/T1_3 11.955 Tf9.298 0 Td(K t c 2 )Tj/T1_3 11.955 Tf11.955 0 Td(K n sc K t s 2 )Tj/T1_3 11.955 Tf11.955 0 Td(K n scK t sc )Tj/T1_3 11.955 Tf11.955 0 Td(K n c 2 3 7 5 ; (3.24) and ~ f o ( t )= N X p =1 g p ( t ) 0 B @ h 2 6 4 )Tj/T1_3 11.955 Tf9.299 0 Td(K t sc )Tj/T1_3 11.955 Tf11.955 0 Td(K n s 2 K t s 2 )Tj/T1_3 11.955 Tf11.955 0 Td(K n sc 3 7 5 + 2 6 4 )Tj/T1_3 11.955 Tf9.298 0 Td(K te c )Tj/T1_3 11.955 Tf11.955 0 Td(K ne s K te s )Tj/T1_3 11.955 Tf11.955 0 Td(K ne c 3 7 5 1 C A (3.25) 3.3 Analysis Thepresenceofatimedelayintheequationofmotionpreventsaclosed formsolution.Tounderstandthebehaviorofthesystem,thetruesolutioncanbe approximatedwithTFEA.Threedierentmethodscanbeappliedtoapproximate thesolution;1)MultipleelementmethodwithcubicHermitepolynomials( hversion ),2)Singleelementapproachwithhigherorderorthogonalpolynomials ( p-version ),and3)Multipleelementtechniquewithhigherorderorthogonal polynomials( hp-version ).

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41 Figure3{5:Schematicrepresentationofelementdiscretizationshownbydividing thetimeincut( t c )intoanumberoftimefniteelementstoformadiscretelinear map,andapproximatingthefreevibration( t f )usingstatetransitionmatrix. Alloftheabovemethodsdividethetimeinthecutintotimeelement(s)toform adiscretelinearmapwhichapproximatescuttingmotionsolution.Figure(3{5) representstheprocessofdividingtimeintodiscretetimeelementstocreatea dynamiclinearmap.In h-version theconvergencetothetruesolutionisobtained byincreasingthenumberofelements;in p-version convergenceisobtainedby increasingthepolynomialorderwhilekeepingthenumberofelementstoone. Asthenumberofelementsisincreasedinthecaseof h-version ,thesizeofthe globalmatricesincreaserapidly,makingitcomputationallyexpensive.Rapid convergenceisobtainedinthecaseof p-version withtheincreaseinthepolynomial

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42 order,howeverpolynomialordercannotbeincreasedbeyondacertainpointdue tocomputationallimitationsingeneratinghigherorderpolynomials.Toovercome thelimitationsofboth h-version and p-version , hp-version hastherexibilityto increaseboththepolynomialorderandthenumberofelements,increasingthe rateofconvergenceandreducingcomputationalexpense.Thischapterlooksatthe stabilityandsurfacelocationerrorpredictionsusing p-version and hp-version of timefniteelementanalysisandcomparestherateofconvergenceforboth.The analysisfor h-version canbefoundinreferences[5{7,51,53,54].Thetimeeach toothspendsinthecutcanbedividedintotwoparts,freevibrationandvibration duringcutting. 3.3.1 Free Vibration Settingtherighthandsideoftheequation(3.23)tozerogivestheequationof motionforfreevibration M q ~q ( t )+ C q _ ~q ( t )+ K q ~q ( t )=0(3.26) Let ~q ( t )and _ ~q ( t )bethetwostatesofthesystem.Writingequation(3.26)in statespaceform 2 6 4 _ ~q ( t ) ~q ( t ) 3 7 5 = 2 6 4 0 I )Tj/T1_2 11.955 Tf9.298 0 Td(M )Tj/T1_8 7.97 Tf6.586 0 Td(1 q K q )Tj/T1_2 11.955 Tf9.299 0 Td(M )Tj/T1_8 7.97 Tf6.58701 0 Td(1 q C q 3 7 5 | {z } 4 r 4 r 2 6 4 ~q ( t ) _ ~q ( t ) 3 7 5 (3.27) Inshortequation(3.27)canbewrittenas 2 6 4 _ ~q ( t ) ~q ( t ) 3 7 5 = G 2 6 4 ~q ( t ) _ ~q ( t ) 3 7 5 (3.28) Thestatesofthetoolfromthebeginningofthefreevibrationtotheendof thefreevibrationcanberelatedusingastatetransitionmatrix b . = e G t f (3.29)

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43 where t f isthetimeoffreevibration. Usingthestatetransitionmatrix b thestatesatthebeginningandendof eachtimeperiodforfreevibrationcanberelatedas 2 6 4 ~q ( n ) _ ~q ( n ) 3 7 5 = b 2 6 4 ~q (( n )Tj/T1_0 11.955 Tf11.955 0 Td(1) + t c ) _ ~q (( n )Tj/T1_0 11.955 Tf11.955 0 Td(1) + t c ) 3 7 5 : (3.30) t c isthetimeincutand isthetoothpassingperiod. 3.3.2 Vibration During Cutting ThischapterlooksattheproblemofapproximatingthetruesolutioninTFEA bytwodierentways, p-version and hp-version .Thetechniqueforthesolution approximationby h-version canbefoundinthereferences[5{7,51,53,54]. 3.3.2.1 Predicting motion during cutting: p-version Althoughincreasingthenumberofelementsin h-version resultsinconverged stabilityboundaries,adramaticincreaseincomputationaltimeoftenaccompanies thelargermatrixsizesobtainedforsystemswithalargernumberofdegreesof freedom.Therefore,theapproachdescribedinthepreviouspublicationswas modifedbyincreasingtheorderofapproximatingpolynomialswhileholdingthe numberofelementstoone.Theapproximatesolutionbecomes[55], ~q ( t ) s X i =1 ~a n i i ( ( t )) ; (3.31) where s isthetotalnumberofhigherorderinterpolatedpolynomials,( i ( ( t )),and ( t )= t )Tj/T1_1 11.955 Tf10.89 0 Td(n isthelocaltimewithinthe n th period.Theinterpolatingpolynomials, describedlaterinsection3.5,areoforder p where p = s )Tj/T1_0 11.955 Tf11.955 0 Td(1 : (3.32)

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44 Substitutionoftheapproximatesolution,equation(3.31),intoequation(3.23)leads toanon-zeroerror e rr ( t ) M q s X i =1 ~a n i i ( ( t ))+ C q s X i =1 ~a n i _ i ( ( t ))+[ K q )Tj/T1_1 11.955 Tf11.955 0 Td(b K c ( ( t ))] s X i =1 ~a n i i ( ( t )) + b K c ( ( t )) s X i =1 ~a n i )Tj/T1_5 7.97 Tf7.08501 0 Td(1 ~ i ( ( t )) )Tj/T1_1 11.955 Tf11.955 0 Td(b ~ f o ( ( t ))= e rr ( t ) : (3.33) Theintegralofweightederrorissettozerowhichprovides s equationslinear inthecoecientsoftheassumedsolution.Weightingfunctions,( i ( ( t ))),were chosentobethesameasinterpolatedpolynomials, i ( ( t ))= i ( ( t )) ; (3.34) i =1 ; 2 ;:::::s: ThisintegralformulationisknownastheWeightedGalerkinmethod [40].Theresultantequationcanbewrittenas, Z t j 0 " M q s X i =1 ~a n i i ( ( t )) i ( ( t ))+ C q s X i =1 ~a n i _ i ( ( t )) i ( ( t )) +[ K q )Tj/T1_1 11.955 Tf11.955 0 Td(b K c ( ( t ))] s X i =1 ~a n i i ( ( t )) i ( ( t )) + b K c ( ( t )) s X i =1 ~a n i )Tj/T1_5 7.97 Tf7.084 0 Td(1 i ( ( t )) i ( ( t )) )Tj/T1_1 11.955 Tf11.955 0 Td(b ~ f o ( ( t )) i ( ( t )) # d ( t )=0 ; (3.35) where t j ,theintegrationtimeforthesingleelement,isequaltothetimeperiod ( ). Thecoecientsfromthefrsttwotrialfunctionsonthefrstelementrepresent thevelocityanddisplacementatthestartofeachperiod.Therelationbetween statesatthebeginningandendoffreevibrationcanbewrittenas 0 B @ ~a 1 ~a 2 1 C A n = b 0 B @ ~a s )Tj/T1_5 7.97 Tf7.08501 0 Td(1 ~a s 1 C A n )Tj/T1_5 7.97 Tf6.58701 0 Td(1 (3.36)

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45 3.3.2.2 Single element monodromy operator Sincetheinitialandthefnalstatesofthesystemcanbespecifedinterms ofasinglepolynomialcoecient,asimplisticmappingtothenextperiodcanbe writtenwiththeunitymatrixandtheidentitymatrix. Thecoecientsoftheassumedsolutioncanberelatedtothoseofthe previousperiodbyarrangingequations(3.35)and(3.36)intoamonodromy operatormatrix.Theexpressioncanbewrittenas 2 6 6 6 6 6 6 6 6 6 6 4 I00 ::: 00 N 11 N 12 N 13 ::: N 1s )Tj/T1_4 7.97 Tf6.586 0 Td(1 N 1s N 21 N 22 N 22 ::: N 2s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 N 2s . . . . . . . . . . . . . . . . . . N s1 N s2 N s3 ::: N ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 N ss 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 ~a 1 ~a 2 ~a 3 . . . ~a s 3 7 7 7 7 7 7 7 7 7 7 5 n = 2 6 6 6 6 6 6 6 6 6 6 4 000 ::: 0b P 11 P 12 P 13 ::: P 1s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 P 1s P 21 P 22 P 22 ::: P 2s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 P 2s . . . . . . . . . . . . . . . . . . P s1 P s2 P s3 ::: P ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 P ss 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 ~a 1 ~a 2 ~a 3 . . . ~a s 3 7 7 7 7 7 7 7 7 7 7 5 n )Tj/T1_6 7.97 Tf6.58701 0 Td(1 + 2 6 6 6 6 6 6 6 6 6 6 4 ~ 0 ~ C 1 ~ C 2 . . . ~ C s 3 7 7 7 7 7 7 7 7 7 7 5 ; (3.37) wheretheundefnedtermsinsidethematricesare N ii = Z t j 0 h M q i ( ( t ))+ C q _ i ( ( t ))+[ K q )Tj/T1_3 11.955 Tf11.955 0 Td(b K c ( )] i ( ( t )) i i ( ( t )) d; (3.38) P ii = )Tj/T1_3 11.955 Tf9.298 0 Td(b Z t j 0 K c ( ( t )) i ( ( t )) i ( ( t )) d: (3.39) C i = b Z t j 0 ~ f o ( ( t )) i ( ( t )) d: (3.40) where i =1: s .

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46 Adiscretelinearmap,describedbyequation(3.37),canbewrittenas A ~a n = B ~a n )Tj/T1_5 7.97 Tf6.586 0 Td(1 + ~ C or ~a n = Q ~a n )Tj/T1_5 7.97 Tf6.586 0 Td(1 + ~ D ; (3.41) where Q = ( A T A ) )Tj/T1_5 7.97 Tf6.586 0 Td(1 A T B : (3.42) isthemonodromyoperator. 3.3.2.3 Predicting motion during cutting: hp-version Thedimensionsofthe N , P ,and C matricesdescribedbyequations(3.383.40)quicklyincreasewithincreaseinpolynomialorder.Itrequiresalmost2hours ofcomputationaltimetogenerate11 th orderpolynomialsaloneona2 : 4 GHz: , Pentium-4computerwith2 Gb ram.Toreducecomputationaltimewhilestillbe abletakeadvantageofincreasedconvergenceusinghigherorderpolynomials,the timeincutisdividedintoafnitenumberofelementsalsoknownastimefnite elements.Theapproximatesolutioncanthenbewrittenas ~q ( t ) s X i =1 ~a n ji i ( j ( t )) ; (3.43) where i ( j ( t ))arethehigherorderorthogonalpolynomials,ortrialfunctionsand j ( t )isthelocaltimewithinthe j th elementofthe n th period.Coecients a ji are calculatedbymatchingtheinitialdisplacementsandvelocitiesofeachelement. Substitutionoftheapproximatesolutionintoequation(3.23)givesanerror e rr ( t ), M q s X i =1 ~a n ji i ( j ( t ))+ C q s X i =1 ~a n ji _ i ( j ( t ))+[ K q )Tj/T1_2 11.955 Tf11.955 0 Td(b K c ( j ( t ))] s X i =1 ~a n ji i ( j ( t )) b K c ( j ( t )) s X i =1 ~a n )Tj/T1_5 7.97 Tf6.586 0 Td(1 ji i ( j ( t )) )Tj/T1_2 11.955 Tf11.955 0 Td(b ~ f o ( j ( t ))= e rr ( t ) : (3.44) Theresidualofweightederrorissettozerowhichgives s equationsperelement. Thepurposeofweightingtheresidualerror( e rr )istoselectfrominfnitepossible

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47 solutions(

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48 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I00 ::: 000 ::: 00 N 1 11 N 1 12 N 1 13 ::: N 1 1s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 N 1 1s 0 ::: 00 N 1 21 N 1 22 N 1 22 ::: N 1 2s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 N 1 2s 0 ::: 00 . . . . . . . . . . . . . . . . . . 0 ::: 00 N 1 s1 N 1 s2 N 1 s3 ::: N 1 ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 N 1 ss 0 ::: 00 000 ::: N 2 11 N 2 12 N 2 13 ::: N 2 1s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 N 2 1s 000 ::: N 2 21 N 2 22 N 2 22 ::: N 2 2s )Tj/T1_4 7.97 Tf6.58701 0 Td(1 N 2 2s . . . . . . . . . ::: . . . . . . . . . . . . . . . . . . 000 ::: N 2 s1 N 2 s2 N 2 s3 ::: N 2 ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 N 2 ss 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ~a 11 ~a 12 ~a 13 . . . ~a 1 s ~a 21 ~a 22 . . . ~a 2 s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 ::: 000 ::: 0b P 1 11 P 1 12 P 1 13 ::: P 1 1s )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 1 1s 0 ::: 00 P 1 21 P 1 22 P 1 22 ::: P 1 2s )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 1 2s 0 ::: 00 . . . . . . . . . . . . . . . . . . 0 ::: 00 P 1 s1 P 1 s2 P 1 s3 ::: P 1 ss )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 1 ss 0 ::: 00 000 ::: P 2 11 P 2 12 P 2 13 ::: P 2 1s )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 2 1s 000 ::: P 2 21 P 2 22 P 2 22 ::: P 2 2s )Tj/T1_4 7.97 Tf6.586 0 Td(1 P 2 2s . . . . . . . . . ::: . . . . . . . . . . . . . . . . . . 000 ::: P 2 s1 P 2 s2 P 2 s3 ::: P 2 ss )Tj/T1_4 7.97 Tf6.58701 0 Td(1 P 2 ss 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ~a 11 ~a 12 ~a 13 . . . ~a 1 s ~a 21 ~a 22 . . . ~a 2 s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 n )Tj/T1_6 7.97 Tf6.58701 0 Td(1 + 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ~ 0 ~ C 1 1 ~ C 1 2 . . . ~ C 1 s ~ C 2 1 . . . ~ C 2 s 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; (3.47) where N j ii = Z t j 0 h M q i ( j ( t ))+ C q _ i ( j ( t ))+[ K q )Tj/T1_3 11.955 Tf11.955 0 Td(b K c ( j )] i ( j ( t )) i i ( j ( t )) d j ( t ) ; (3.48) P j ii = )Tj/T1_3 11.955 Tf9.299 0 Td(b Z t j 0 K c ( j ( t )) i ( j ( t )) i ( j ( t )) d j ( t ) : (3.49)

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49 C j i = b Z t j 0 ~ f o ( j ( t )) i ( j ( t )) d j ( t ) : (3.50) Equation(3.47)takestheformofadiscretelinearmapthatcanbewrittenas A ~a n = B ~a n )Tj/T1_6 7.97 Tf6.586 0 Td(1 + ~ C or ~a n = Q ~a n )Tj/T1_6 7.97 Tf6.586 0 Td(1 + ~ D (3.51) where Q isthemonodromyoperator.Thecoecientsoftheassumedsolution ~a n representthevelocityanddisplacementatdiscretepointsintime.Thisprovidesa dynamicmapoverasingletimedelay. 3.3.3 Stability Thestabilityorthedegreeofstabilityinmillingdependsuponanumberof factorssuchas,cuttingconditions,specifccuttingenergyofthematerial,feedrate, cuttingspeed,toolgeometry,andtoolwear.Themostinruentialparameterischip width,whichcanberegardedasthemeasureofstabilityprovidedthatallother cuttingconditionsarestandardized.Foragivenvibration,andvariationinchip thickness,thefeedbackprovidedbythegeneratedvariableforceisproportionalto chipwidth.Thechipwidthrepresentsagainintheclosed-loopofselfexcitation. Withsucientlysmallchipwidth,cuttingisalwaysstable,aschipwidthincreases alimitingvalueisreachedforagivenspindlespeedbeyondwhichchatteroccurs. Thecharacteristicmultipliersofthedynamicmap Q ,determinethestability ofthegoverningequation,baseduponwhethertheyresidewithintheunitcircle [6{8,51,54,56].Thesystemisstableforagivensetofparameters,( b; n),ifall thecharacteristicmultipliershaveamoduluslessthanone.Dependinguponthe mannerinwhichthecharacteristicmultiplier, ,leavestheunitcircle,thetwo distinctroutestoinstabilityare;1)Acharacteristicmultiplierleavestheunitcircle through-1(realaxis),resultingina period-doubling bifurcation;and2)Apairof complexconjugatecharacteristicmultipliersexitstheunitcircleawayfromreal axis,resultingina Hopf or Neimark bifurcation.

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50 3.3.4 Surface Location Error Thesteadystatedierenceinthefnalcutsurfaceandpredictedsurfaceis knownasthesurfacelocationerror.Itiscausedduetorelativevibrationsbetween thetoolandtheworkpiece.Duetothephenomenonofsurfacelocationerror,the machinedsurfacemaybeplacedaboveorbelowthedesiredsurface.Theschematic representationofthestateofunder-cutorover-cutisshowninFigure(3{6). Assumingallotherfactorsareabsent,thissectionpresentsananalyticalmethodto predictthesurfacelocationerrorduetorelativevibrationbetweenthetoolandthe workpiece[4,47,54,57{59].The x -displacementand y -displacementcoordinatesof thetoothwhenitentersthecut(up-milling)andexiststhecut(down-milling),are givenbythecoecientvector ~a n .Consideringthatinthestablecasethemotion isperiodic,andthereforetheperiodiccoecientscanbefoundfromthedynamic mapas ~a n = ~a n )Tj/T1_4 7.97 Tf6.586 0 Td(1 = ~a n : (3.52) Substitutingequation(3.52)intotheequation(3.41)orequation(3.51)gives ~a n =( I )Tj/T1_5 11.955 Tf11.955 0 Td(Q ) )Tj/T1_4 7.97 Tf6.58701 0 Td(1 ~ D : (3.53) Thevector ~a n canbeusedtotothespecifysurfacelocationerrorforeachand everyspindlespeed(n),andthedepthofcut( b ). 3.4 Convergence Afniteelementsolutioncanbemodifedbyusingthreedierentapproaches: 1)Theh-refnementinwhichthesameclassofelementscontinuetobeusedbut arechangedinsize.Themeshisrefneduntiltheconvergenceisachieved;2)The p-refnementinwhichthesizeoftheelementiskeptconstantbuttheorderof theinterpolatingpolynomialisincreasedtoattaintheconvergedsolution.The rateofconvergenceforagivensetofvariablesismorerapidwithp-refnementas

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51 Figure3{6:Schematicrepresentingthestateofunder-cutandover-cutinthecuttingplane.

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52 comparedtoh-refnement;and3)Thehp-refnementinwhichboththenumberof elementsandthepolynomialorderisincreasedtogetconvergedsolution. Erroranalysisaccountsfordiscretizationerrorandassumesallotherforms oferrorareabsent.Discretizationerroriscausedbyusingafnitenumberoftrial functions, ( ( t )),andadiscretesetofcoecients, a i ,inassumedsolutionto approximatetheexactsolutionto x ( ( t )).Completenessandcontinuityaretwo minimumconditionsrequiredtoapproximatethesecondordersystemwithalinear combinationoftrialfunctions.Itcanbeshownthattheeigenvaluesconvergeatthe samerateastheglobalenergy.Therateofconvergenceofglobalenergyisofthe orderof O ( E 2( p )Tj/T1_2 7.97 Tf6.586 0 Td(m +1) ),where2 m istheorderofgoverningdierentialequation, p is thepolynomialorder,and E isthenumberoftimeelements.Inphysicalsystems theeigenvalueiseithertheglobalenergyortheratioofglobalpotentialenergyto globalkineticenergy[44]. Errorintheenergynormcanbewrittenas k e k 2 = k ~x ( t ) k 2 )-222(k ~ x 0 ( t ) k 2 ; (3.54) where k ~x ( t ) k 2 representstheenergyformof ~x ( t ).Consideringeigenvalues( )have thesamerateofconvergenceasglobalenergy,equation(3.54)canbewrittenas k e k 2 = 2 )Tj/T1_1 11.955 Tf11.955 0 Td( 0 2 ; (3.55) where isthetrueeigenvalueand 0 istheeigenvaluefromanapproximated solution.Theconvergenceisconsideredtohavebeenobtainedwhenpercentage errorislessthanfvepercent. Figure(3{7)representstheeectofmeshrefnement,(h-refnement),and polynomialrefnement,(p-refnement),ontheconvergenceofafourdegreeof freedommillingsystemthemodalparametersofwhicharegiveninTable(3{1). Thepercentageerrorrerectstheclosenessoftheapproximatedeigenvalue,tothe

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53 Figure3{7:Percentageerrorreductionwiththeincreaseinnumberofelementsfor variousinterpolatingpolynomialorders.Solidlinerepresentsconvergenceofcubic Hermiteswithlinearweightingfunctions.Dottedline,dashedline,anddashedline withdotsrepresentconvergenceofthird,ffth,andseventhorderpolynomialsusing Galerkinapproachrespectively. trueeigenvalue,obtainedusingtimemarchingmillingsimulation.Itisworthy tonotethereductioniserrorfromapproximately35percentto10percentusing thesameinterpolatingpolynomialsbutbyweightingthepolynomialsusinga Galerkinapproachasopposedtothelinearweightingfunctions.Asisevident fromthegraph,thetheconvergenceisobtainedforafourdegreeoffreedom millingsysteminlessthan5elementswhileusing5 th and7 th orderpolynomials. Thefollowingparameterswereusedtocomputethestabilityboundariesand surfacelocationerrorforthesystemgiveninTable(3{1): K t =6 10 8 [ N=m 2 ],

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54 Table3{1:ModalparametersforafourDOFmillingsystem-1. M [kg] C [Ns/m] K [N/m] 0 : 027000 00 : 03000 000 : 0270 0000 : 030 7000 0200 0070 0002 1 10 6 000 01 : 6 10 6 00 001 10 6 0 0001 : 6 10 6 K n =1 : 8 10 8 [ N=m 2 ],feed h = : 127[ mm=tooth ],atworuteendmillwith diameter D =12 : 7[ mm ],andaradialimmersionof5percent.Stabilityand surfacelocationpredictionsforthesystemareshowninFigure(3{8)andhasbe spotcheckedagainstEulersimulationinFigure(3{9).Itrequires10elementsto produceconvergedsolutionwiththe h-version approach,whichgreatlyincreases thecomputationaltime,whereas,theconvergenceisobtainedbyusingonlytwo elementscoupledwith5 th orderpolynomialswhenthesystemisanalyzedusing hp-version approach.Therealadvantageofusing hp-version timefniteelement analysisisthatitmakestheprocesslesscomputationallytaxingbyincreasingthe internaldegreesoffreedomoftheinterpolatingpolynomialsandthereforereduces thesizeofthemonodromyoperatormatrix. Consideranother4-DOFmillingsystemthemodalparametersofwhich aregiveninTable3{2.Thefollowingparameterswereusedtocomputethe Table3{2:ModalparametersforafourDOFmillingsystem-2. M [kg] C [Ns/m] K [N/m] 10 6 4 : 05000 05 : 4200 004 : 050 0005 : 42 636 : 5000 0987 : 800 00636 : 50 000987 : 8 10000 01800 00100 00018 stabilityboundariesandsurfacelocationerrorforthesystemgiveninTable3{2: K t =7 10 8 [ N=m 2 ], K n =2 : 1 10 8 [ N=m 2 ],feed h = : 2[ mm=tooth ],atworuteend millwithdiameter D =19 : 07[ mm ],andaradialimmersionof100percent.

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55 Figure3{8:StabilityandsurfacelocationErrorpredictionforafourdegreeoffreedommillingsystem.TheparametersareshowninTable3{1.PlotErepresentsthe surfacelocationerroratadepthofcut b =2[ mm ]andplotErepresentssurface locationerrorat b =4[ mm ]. Theparameterdomainwasexploredfrom5000[rpm]to40000[rpm]instepsof 50[rpm],andfrom0 : 01[mm]to7[mm]depthofcutinincrementsof0 : 25[mm], usingEulersimulation, h-version ,and hp-version TFEA.AsisevidentfromFigure 3{10,thestabilityresultsobtainedusing hp-version TFEAareinperfectagreement withthoseobtainedfromEulersimulation.TheadvantageofusingTFEAover Eulersimulationisthetimerequiredtogettheresults.Whileitrequiredover1400 minutestogettheresultsfromEulersimulation,theresultsfrom hp-version TFEA wereobtainedinlessthan105minutes.Figure3{11representstheconvergenceof TFEAresultsinlargesurfacelocationerrorregions.

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56 Figure3{9:Poincaresection,1/tooth,andsimulationdataforcasesA,B,C, andDshowninFigure(3{8).StablecuttingprocessesareshownincasesA(n= 11400 rpm;b =2 mm ),B(n=17300 rpm;b =2 mm ),C(n=34800 rpm;b =2 mm ), andD(n=28500 rpm;b =4 mm ). 3.5 Interpolated Polynomials TheformulationofinterpolatingpolynomialshasbeendiscussedinChapter 2,butispresentedagainforconvenience.Forconvergenceofasecondorder dierentialsystem,thetrialfunctionsorinterpolatedpolynomials,mustsatisfyat leasttwoconditionsC 0 continuityandcompleteness[44].

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57 Figure3{10:Stabilitylobesforafourdegreeoffreedommillingsystem.TheparametersareshowninTable3{2.SolidlinerepresentsTFEApredictionsandbroken linerepresentsthestabilitypredictionfromEulersimulation. Completenesswasensuredbyincreasingthedegreeofthepolynomialswhilepreserving C 0 continuity.Toobtainabetterconvergencetheinterpolatingpolynomials arealsoconstrainedfor C 1 continuity.Todriveapolynomialwith C 1 continuity, thepolynomialanditsderivativewereinterpolatedateachinter-elementnode. Thisrequirestwoboundaryconditionsateachinter-elementnode.

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58 Figure3{11:TFEAconvergesinlargeSLEregionsasnumberofelementsisincreased. ,dashedline,diamonds,andlinewithdotsrepresentSLEprediction using8,12,25,and50elementsrespectivelywith h-version TFEA.SolidlinerepresentsSLEpredictionsfromEulersimulation.DotsrepresentSLEpredictionusing hp-version TFEAwith5elementsand7 th orderpolynomials. Thesolutiontakestheform s X i =1 ~ i ( ( t ))=1 ; (3.56a) s X i =1 _ ~ i ( ( t ))=1 ; (3.56b) where ~ isaninterpolatedpolynomial.Polynomialsareoforder p whichisrelated tothetotalnumberofpolynomialsas p = s )Tj/T1_0 11.955 Tf11.955 0 Td(1 : (3.57)

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59 Expressionforasetof5 th orderinterpolatingpolynomialstakestheform, 1 ( )=1 )Tj/T1_0 11.955 Tf11.955 0 Td(23 ( t ) t j 2 +66 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(68 ( t ) t j 4 +24 ( t ) t j 5 ; (3.58a) 2 ( )= t j " ( t ) t j )Tj/T1_0 11.955 Tf11.955 0 Td(6 ( t ) t j 2 +13 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(12 ( t ) t j 4 +4 ( t ) t j 5 # ; (3.58b) 3 ( )=16 ( t ) t j 2 )Tj/T1_0 11.955 Tf11.955 0 Td(32 ( t ) t j 3 +16 ( t ) t j 4 ; (3.58c) 4 ( )= t j " )Tj/T1_0 11.955 Tf9.298 0 Td(8 ( t ) t j 2 +32 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(40 ( t ) t j 4 +16 ( t ) t j 5 # ; (3.58d) 5 ( )=7 ( t ) t j 2 )Tj/T1_0 11.955 Tf11.955 0 Td(34 ( t ) t j 3 +52 ( t ) t j 4 )Tj/T1_0 11.955 Tf11.955 0 Td(24 ( t ) t j 5 ; (3.58e) 6 ( )= t j " )Tj/T1_5 11.955 Tf11.291 16.85699 Td( ( t ) t j 2 +5 ( t ) t j 3 )Tj/T1_0 11.955 Tf11.955 0 Td(8 ( t ) t j 4 +4 ( t ) t j 5 # : (3.58f) Figure(3{12A)andFigure(3{12B)representthedisplacementandvelocity constraintsinlocaltimerespectivelyforffthorderinterpolatedpolynomials ( p =5 ;s =6 ).Localtimeisdividedintothreenodeswhichincludetwoboundary nodesandoneintermittentnode.Thesumofdisplacementandvelocityateach nodeisone.Constraininginterpolatedpolynomialsinthisfashionhelpsmaintain completeness, C 1 continuity,andorthogonality. 3.6 Stability Experimental Verifcation Stabilitytestswereperformed[11]andcomparedwiththeanalyticalpredictionsforacomplianttoolandrigidworkpiece.Thecuttingpressuresvaluesfor (7050-T7451)aluminumareshowninTable3{4. Table3{3:Cuttingpleasures K t [ N=m 2 ] K n [ N=m 2 ] 5 : 36 10 8 1 : 87 10 8

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60 Figure3{12:Fifthorderinterpolatedpolynomialsplottedasafunctionofthe normalizedlocaltime. Themodalparametersobtainedfora12.75[mm]diameter,106[mm]overhang, carbideendmillareshowninTable3{4.Thefollowingcuttingparameterswere Table3{4:Modalparametersfor12.75[mm]endmill. Diameter[mm] M [kg] C [Ns/m] K [N/m] 12.75 0 : 04360 00 : 0478 4 : 2680 04 : 355 9 : 14 10 5 0 01 : 00 10 6 usedtoverifytheanalyticalstabilitypredictions:radialimmersionof0.635[mm] andafeedrateof0.127[mm/tooth]werekeptconstantwhilethespindlespeed,( n ), anddepthofcut,( b ),werechangedforeachcuttingtest.Areferencesurfacewas createdbeforeandaftereachcuttingtest.

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61 Figure3{13:ExperimentalvalidationofTFEAstabilitypredictionsforthe 12 : 75[ mm ]carbideendmilldescribedinTable(3{4).Thesymbol representsa stablecase, representsanunstablecaseand . representsborderlinestability. Cuttingspeedanddepthofcutforeachreferencesurfacewaschosenbasedonstabilityandminimumsurfacelocationerror.Thesignalfromthecapacitanceprobes wassampledat 25 [KHz],tomeasurethetooldisplacementsatapproximately 19[mm]fromthetooltip.Abarcodewaspaintedontheshankofthetooltorecord theonceperrevolutionsignalusingalasertachometer.TimeseriesdatafromcapacitanceprobesandalasertachometerwasrecordedusinganNIdataacquisition board(SC-2345),interfacedwithLabview7.0,usinga16bitNIDAQCard-6036E. Thetimeseriesdatawassampledfor1/toothsignal.

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62 Figure3{14:Poincaresection,phasespace,and1/toothdataforcasesA,B, C,andDshowninFigure(3{13).Casesshowninthisfguredepictaninstabilityalsoknownasripbifurcation.Unstablecuttingprocessesareshownin casesA(n=15000 rpm;b =0 : 5 mm ),B(n=15000 rpm;b =0 : 76 mm ), C(n=15000 rpm;b =1 mm ),andD(n=15300 rpm;b =1 : 5 mm ). Thecutwasconsideredtobestableifthe1/toothsignalhadasteadyconstant value.TheTFEAstabilitypredictionsandexperimentalverifcationisshownin Figure(3{13).Cuttingtestsmarked(A-D)inFigure(3{13)depictaninstabilityalso knownasripBifurcation.Thiskindofinstabilityoccurswhenthecharacteristic multiplieroftheeigenvalueleavestheunitcirclethrough )Tj/T1_3 9.963 Tf7.74901 0 Td(1 .Thiscanbephysicallyunderstoodasaphenomenonwherethetoothofthecuttersuccessivelyrips betweentwopointsduringthecourseofarevolution.

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63 Figure3{15:ExperimentalvalidationofTFEAstabilitypredictionsforthe 19 : 05[ mm ]carbideendmilldescribedinTable(3{5).Thesymbol representsa stablecase,symbol representsaclearcutunstablecaseand . representsborderlinestability. ThistypeofunstablebehaviorisshowninFigure(3{14)byusing1/toothsamples data,aPoincareplot,andaPhasespaceplot.Theexistenceofaquasiperiodic solutionisindicatedbyanovalshapedphaseplotandtwofxedpointsbetween whichthefrstreturnalternatesasisevidentfromPoincaresection.Pseudo velocity, x 2 ,forphasespaceplotswascomputedusingtheautocorrelationfunction. Forthisapproachthelongtimeaverageover t oftheproductof x ( t ) and x ( t + ) in atimeseriesiscomputed.Foraperiodicresponse,theautocorrelationfunctionwill detectitsperiodicity.

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64 Anothersetofexperiments[10]wereperformedusingatworute,19.05[mm] diameter,106[mm],overhang,carbideendmill.Themodalparametersobtained aftermodalanalysisonthetoolareshowninTable3{5. Table3{5:Modalparametersfor19.05[mm]endmill. Diameter[mm] M [kg] C [Ns/m] K [N/m] 19.05 0 : 0610 00 : 061 4 : 3260 03 : 858 1 : 667 10 6 0 01 : 669 10 6 Experimentswereperformedon(7050-T7451)aluminumwheretheworkpiecewasdown-milledatafvepercentradialimmersionandafeedrateof 0 : 178 [mm/tooth].Figure(3{15)showsthesuperimposedexperimentalresultsandanalyticalpredictionsfromTFEA.Casesmarked(E-F)onthestabilitydiagramdepictan unstablebehaviorknowhasHopfBifurcation.Thiskindofinstabilityresultswhen twocomplexconjugateeigenvaluesexittheunitcircleawayfromtherealaxis. Figure(3{16)depictsadensecollectionofpointsontwoclosedloops.Suchdense collectionofpointsonclosedloopsofaPoincaresectionarecharacteristicofatwoperiodicquasiperiodicsolutions.Thecasesmarked(G-H)depictripbifurcation whichareverifedinFigure(3{16). 3.7 Summary and Conclusions Inthischapter,weanalyzethestabilityandsurfacelocationoftimedelayed millingsystemsgovernedbyordinarylineardierentialequations.Thedelayed dierentialequationisanalyzedforstabilityusingtwodierenttechniques: 1)asingleelement(p-version)timefniteelementmethod;and2)amultiple element(hp-version)timefniteelementmethod.Inbothcasesthetruesolution isapproximatedusingasetofinterpolatedpolynomials.Stabilityofthesystemis determinedbytheeigenvaluesofthediscretelinearmap.Theformerapproachuses onlyonetimefniteelementbutasetofhigherorderorthogonalpolynomialsto obtainconvergence.Ontheotherhand,thehp-versionuseshigherorderorthogonal

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65 Figure3{16:Poincaresection,phasespace,and1/toothdataforcasesE,F, G,andHshowninFigure(3{15).Unstablecuttingprocessesareshownin casesA(n=16200 rpm;b =2 : 54 mm ),B(n=16200 rpm;b =3 : 66 mm ), C(n=17300 rpm;b =2 : 54 mm ),andD(n=17300 rpm;b =3 : 66 mm ).CasesE andFresultfromanunstableHopfbifurcationandcasesGandHarearesultof ripbifurcation.

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66 polynomialsandanincreasingnumberofelementstoensureconvergence.Therate ofconvergenceofthep-versionisveryhigh.Afasterconvergencerateisobtained when p isincreasedascomparedtoonlyincreasingthenumberofelementsin h-version[5{7,51,53,54].Howeverduetosymbolicmanipulation,thep-version becomesmorecomplicatedandcomputationallyexpensiveasthepolynomialorder isincreasedbeyondacertainvalue.Toovercomethislimitationthe hp-version of timefniteelementstakeadvantageofboth h-version and p-version toobtainfaster convergencebyincreasingforpolynomialorderandthenumberofelements. Infuturework,amoreversatilehp-versionoftimefniteelementmethodcan beupdatedbyincorporatingtheeectofhelixangleandtoolgeometriesonthe stabilityandsurfacelocationerror.Resultsobtainedfrombothapproacheswere verifedbyactualcuttingtests.Thedevelopedtemporalfniteelementapproachis showntobeapowerfulandrexibleapproachtothesolutionofordinarydierential equationswithtimedelaysuchasmilling.

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CHAPTER4 SUMMARY 4.1 Completed Work Thisthesisbuildsupontheprevious h-version [6{8,50{52]timefniteelement analysistocreateamoreversatile hp-version .Themotivationforthisresearchwas toreducethecomputationaltimerequiredtoachievetheconvergedsolution.Inthe initialworkdoneontimefniteelementanalysis,theconvergencewasobtainedby increasingthenumberofelements.Convergenceobtainedinthisfashionisknown as h-convergence .Thisledtoadramaticincreaseinthesizeofglobalmatriceswith theincreaseinnumberofelements,therebyincreasingthecomputationaltime. Alternatively,amoreeectiveapproachtoachievefasterconvergenceistoincrease theorderoftheinterpolatingpolynomials.Theconvergenceobtainedbyincreasing thepolynomialorderwhilekeepingthenumberofelementsconstantisknownas p-convergence .Although p-version providesamorerapidconvergenceascompared to h-version ,thechoiceofachievingconvergencebyonlyincreasingthepolynomial orderisclearlyinecientbeyondacertainpoint.Theoptimumsolutiontothe problemiscombiningtheconvergencecapabilitiesofboth h-version and p-version toformamorepowerful hp-version .Inthisapproachconvergenceisachievedby increasingboththepolynomialorderandthenumberofelements. Chapter2presentsananalysisonarepresentativeproblemintheformof Mathieu'sequation.AlthoughMathieu'sequationdoesnotrepresentaspecifc physicalsystem,itcloselymodelssomeofthephysicalsystemswithtimedelay (i.e.,derectionofellipticalmembranes,milling,motionofroboticarms,rotating shafts,motionofthehelicopterblades,etc).AnalysispresentedinChapter2looks atthestableandunstabledomainsofMathieu'sequationintheparameterspace. 67

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68 The p-version timefniteelementanalysisapproachisappliedtothisproblem topredictstableregionsanddierentkindsofunstablebehaviorsuchasrip bifurcationandHopfbifurcation.Theconvergenceisobtainedbyincreasingthe polynomialorderwhilekeepingthenumberofelementstoone.Thischapteralso presentsauniquewaytoextractFloquettransitionmatrixforafreelyvibrating systemwithtimefniteelementanalysis.Particularemphasisislaidonerror analysisandconvergencecriterion. Chapter3discussestheapplicationof hp-version oftimefniteelementmethod toanalyzethestabilityandsurfacelocationerrorforamultipledegreeoffreedom millingsystem.A4DOFcasestudydiscussedinChapter3clearlydemonstrates theabilityof hp-version TFEAtoreducethecomputationaltimebyproviding fasterconvergencecapability.Thepredictedresultsarevalidatedwiththehelpof experimentalresultsperformedonvariousmachinetools.Timeseriesdatafortool derectionandonceperrevolutiontimingsignalwasacquiredusingnon-contact sensorssuchascapacitanceprobesandlasertachometerrespectively.Astrong agreementwasfoundbetweenpredictedandexperimentalresults. 4.2 Future Work TFEAisaversatilesemi-analyticaltechniquethatcanbeappliedtoanalyzea varietyofsystemswithtimedelayandparametricexcitation.Incontexttomilling, furtherresearchisrequiredtoincludetheeectsofmultiplecuttingteeth,and cutterhelixangleandcuttergeometryonstabilityandsurfacelocationerror.By incorporatingtheabilitytopredictsurfacequalityforarexibletoolandrexible workpiece,TFEA hp-version canprovideanidealtoolforon-lineapplications.

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BIOGRAPHICALSKETCH IwasborninIndiaonFeb21 st 1979.Mymajorismechanicalengineering.I receivedmyundergradfromThaparInstituteofEngineeringandTechnology,Patiala,India.AftergraduatingwithhonorsIdidmyMSinmechanicalengineering attheUniversityofFlorida.MygraduateGPAis3.77ona4pointscale.Myfeld ofresearchisrelatedtotimefniteelementanalysisofthestabilityofthesystems governedbytime-periodicdelayeddierentialequations.Aftergraduatingfroma prestigiousinstitutelikeUFandunderthewatchfuleyesofmyadvisorIamready totakemyskillsettothenextlevelofperformance. 74