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Predictive Force Modeling of Peripheral Milling

Permanent Link: http://ufdc.ufl.edu/UFE0022663/00001

Material Information

Title: Predictive Force Modeling of Peripheral Milling
Physical Description: 1 online resource (194 p.)
Language: english
Creator: Bhattacharyya, Abhijit
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: analytical, coefficients, constants, cutting, endmilling, force, helical, instantaneous, mechanistic, milling, model, runout, uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Milling is one of the most important subtractive manufacturing processes. Cutting force predictions in milling are useful for the structural design of machine tools, selection of optimum cutting parameters, design of workholding fixtures, tool stress analysis, spindle bearing design, and the real time monitoring of tool wear and breakage. Force models are also used in predicting the stability of the milling process, surface location error predictions, as well as surface finish predictions. In this dissertation, closed form analytical mechanistic cutting force models, using linearized lumped parameter cutting coefficients, are considered. Existing analytical mechanistic force models of helical peripheral milling, which use linearized cutting coefficients, either utilize four different sets of analytical expressions to describe the forces for one complete cutter rotation, or do not have closed form solutions. Numerical models are computationally-intensive, fail to provide the insights that analytical models do, and do not permit symbolic manipulation. In this work, two equivalent versions of closed form analytical expressions for chip thickness and chip width are developed using Heaviside unit step function and Fourier series approaches. The distinguishing feature is that single expressions describe the chip thickness and chip width during the entire cutter rotation. These expressions are then applied to develop single, analytical closed form expressions for each of the three orthogonal components of the cutting force, in a fixed coordinate frame, for helical peripheral milling. The model can be calibrated using partial radial immersion experiments. A procedure has been developed to calculate the variances in measured model input parameters. The propagation of uncertainties through the model is determined by Type A and Type B evaluations to develop an overall expanded uncertainty for placement of confidence intervals on cutting force predictions. The availability of single expressions for force components permits the derivation of compact expressions for sensitivity coefficients for use in the uncertainty analysis. Extensive experimental tests, as well as comparisons with established numerical models, verify the fidelity of the predictions. The model is extended to be able to predict cutting forces in cases of runout or differential pitch cutters. A refined formulation using instantaneous cutting coefficients, instead of the usual method of average coefficients, has also been developed, further increasing the accuracy of force predictions. Experiments show that the refined model is able to predict the force patterns and magnitudes very accurately for the entire range of radial immersions.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Abhijit Bhattacharyya.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Schueller, John K.
Local: Co-adviser: Mann, Brian P.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022663:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022663/00001

Material Information

Title: Predictive Force Modeling of Peripheral Milling
Physical Description: 1 online resource (194 p.)
Language: english
Creator: Bhattacharyya, Abhijit
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: analytical, coefficients, constants, cutting, endmilling, force, helical, instantaneous, mechanistic, milling, model, runout, uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Milling is one of the most important subtractive manufacturing processes. Cutting force predictions in milling are useful for the structural design of machine tools, selection of optimum cutting parameters, design of workholding fixtures, tool stress analysis, spindle bearing design, and the real time monitoring of tool wear and breakage. Force models are also used in predicting the stability of the milling process, surface location error predictions, as well as surface finish predictions. In this dissertation, closed form analytical mechanistic cutting force models, using linearized lumped parameter cutting coefficients, are considered. Existing analytical mechanistic force models of helical peripheral milling, which use linearized cutting coefficients, either utilize four different sets of analytical expressions to describe the forces for one complete cutter rotation, or do not have closed form solutions. Numerical models are computationally-intensive, fail to provide the insights that analytical models do, and do not permit symbolic manipulation. In this work, two equivalent versions of closed form analytical expressions for chip thickness and chip width are developed using Heaviside unit step function and Fourier series approaches. The distinguishing feature is that single expressions describe the chip thickness and chip width during the entire cutter rotation. These expressions are then applied to develop single, analytical closed form expressions for each of the three orthogonal components of the cutting force, in a fixed coordinate frame, for helical peripheral milling. The model can be calibrated using partial radial immersion experiments. A procedure has been developed to calculate the variances in measured model input parameters. The propagation of uncertainties through the model is determined by Type A and Type B evaluations to develop an overall expanded uncertainty for placement of confidence intervals on cutting force predictions. The availability of single expressions for force components permits the derivation of compact expressions for sensitivity coefficients for use in the uncertainty analysis. Extensive experimental tests, as well as comparisons with established numerical models, verify the fidelity of the predictions. The model is extended to be able to predict cutting forces in cases of runout or differential pitch cutters. A refined formulation using instantaneous cutting coefficients, instead of the usual method of average coefficients, has also been developed, further increasing the accuracy of force predictions. Experiments show that the refined model is able to predict the force patterns and magnitudes very accurately for the entire range of radial immersions.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Abhijit Bhattacharyya.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Schueller, John K.
Local: Co-adviser: Mann, Brian P.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022663:00001


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Dr.JohnK.Schueller,myreveredadvisor,madeitpossibleformetogobacktoschool.Hismethodoftrainingappearstobeunique.Hewouldaskonlythesimplestofquestionswhichsentmehuntingfordeeperanswers.Heletmesetmyownpace,providedakindofintellectualfreedomthatwastheenvyofmyfellowstudentsinthelaboratory,andencouragedmetokeepmyworkuncomplicated.Itisimpossibleformetorepaymydebttohim.Dr.BrianP.Mannhasbeenaninspirationaswellasacoach.Hisworkethicandself-disciplineputshisgraduatestudentstoshame.Dr.FredJ.Taylorhasbeenmostgeneroustomeforreasonsthatarequiteunfathomable.Hesucceededinteachingmeasemblanceofdigitalsignalprocessing,asubjectwhosenon-trivialnatureisnotlostuponme.Inadditiontothedepthofhisteachings,hecombinesabreadthofvisionwithasenseofhumorthatendearshimtothestudentcommunity.Dr.JohnC.Ziegertintroducedmetothefascinatingsubjectofuncertainty.Theimprintofhisteachingisevidentalloverthisdocument.Whileteachingmanufacturingtoseniorundergraduatestudents,itwasmyeorttoadopthismethodsofinstructionasbestonecouldhopetoimitate.Dr.TonyL.Schmitztaughtmethefundamentalsofthestructuraldynamicsofproductionmachinery.Hesuggestedtheideaoflettingmeteachtheseniorundergraduateclass.Hehasmorephysicalstaminathanmostpeople,allowinghimtoworklonghourstoproducealargevolumeofvaluablecontributionstotheliterature.Dr.NormanG.Fitz-Coyhasawayofinuencingthemindsofhisstudentsthatishardtoexplain.Hegavemeaperspectiveintothesubjectofdynamicsthatisnotpossibletoobtainfromthetextbooks.Hislecturestitillatethesenses.Itwasmygoodfortunetohavebeenhisstudent,watchinghisblackboardfromthebackbench,andgiving 3

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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 10 LISTOFFIGURES .................................... 11 LISTOFSYMBOLS .................................... 16 ABSTRACT ........................................ 21 CHAPTER 1OVERVIEWANDMOTIVATION ......................... 23 1.1MotivationforPredictiveForceModelingofPeripheralMilling ....... 25 1.2ReviewoftheOpenLiterature ......................... 26 1.3JusticationandScopeoftheWork ...................... 34 2DESCRIPTIONOFTHEPERIPHERALMILLINGPROCESS ......... 37 2.1TheMachineandtheCuttingTool ...................... 37 2.2ChipAreaattheToolWorkpieceInterface .................. 38 2.3ChapterSummary ............................... 40 3CHIPGEOMETRYINHELICALPERIPHERALMILLING ........... 43 3.1ToolChipContactCongurations ....................... 44 3.2AnalyticalExpressionsforL,T,andChipWidth,b 46 3.2.1HeavisideUnitStepFunctionFormulation .............. 46 3.2.2FourierTrigonometricSeriesFormulation ............... 47 3.3AnalyticalExpressionsforChipThickness .................. 49 3.4EectofProcessParametersonChipThicknessandChipWidth ..... 50 3.5ChapterSummary ............................... 50 4MECHANISTICFORCEMODELFORSTRAIGHTFLUTEDENDMILLS .. 57 4.1ForceModelforaSingleTooth ........................ 58 4.2AverageForceBasedEstimatesofKtandKn 59 4.3MultipleToothFormulation .......................... 60 4.4EectsofToothRunout ............................ 60 4.5CuttingCoecientModel ........................... 61 4.6ForcePredictionExample ........................... 62 4.7ChapterSummary ............................... 62 6

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.. 65 5.1ThePartialRadialImmersionExperiment .................. 65 5.2ExperimentalExtractionofCuttingCoecients ............... 66 5.3VariancesofModelInputParameters ..................... 68 5.4ChapterSummary ............................... 71 6PROPAGATIONOFUNCERTAINTIESTHROUGHTHEFORCEMODELFORSTRAIGHTFLUTEDENDMILLS ...................... 75 6.1PropagationofUncertaintiesThroughaMathematicalModel ....... 75 6.2VariancesofCuttingCoecientsandEectiveFeedRates ......... 77 6.2.1PropagationofAverageForceUncertaintytoCuttingCoecients 77 6.2.2PropagationofRadialRunoutUncertaintytoEectiveFeeds .... 78 6.3PropagationofUncertaintiesThroughtheCuttingForceModel ...... 78 6.3.1SensitivityCoecientsofComponentUncertainties ......... 78 6.3.2PropagationofTypeAUncertainties ................. 79 6.3.3PropagationofTypeB1Uncertainties ................. 80 6.3.4PropagationofTypeB2Uncertainties ................. 80 6.4ExpandedUncertainty ............................. 81 6.4.1ExpandedUncertaintyCoverageFactorTypeA ........... 81 6.4.2ExpandedUncertaintyCoverageFactorTypeB ........... 83 6.4.3OverallExpandedUncertainty .................... 83 6.5ForcePredictionwith95%CondenceInterval ................ 84 6.6ChapterSummaryandOutlookfortheForthcomingChapters ....... 84 7INSTANTANEOUSRIGIDFORCEMODELFORHELICALPERIPHERALMILLING ....................................... 91 7.1ForceModelforaSingleToothedCutter ................... 92 7.2ModelingforMultipleTeeth .......................... 94 7.3FormulationforCuttingCoecientIdentication .............. 95 7.4VericationoftheAnalyticalSolution ..................... 97 7.4.1TheDegenerateCaseofStraightFlutedCutters ........... 97 7.4.2ComparisonwithaNumericalSolutionforHelicalEndmills ..... 98 7.5ExperimentalDeterminationofModelInputParameters .......... 99 7.6VariancesofModelInputParameters ..................... 101 7.7PropagationofInputParameterUncertaintiesThroughtheForceModel 103 7.7.1PropagationofTypeAUncertainties ................. 103 7.7.2PropagationofTypeB1Uncertainties ................. 104 7.7.3ExpandedUncertainty ......................... 105 7.8ForcePredictionResults ............................ 106 7.9PropertiesoftheAnalyticalForceModel ................... 106 7.9.1Gibbs-WilbrahamDistortion ...................... 107 7.9.2ComputerImplementationIssues ................... 107 7.9.3RelativeMeritsoftheTwoVariants .................. 108 7

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............................... 109 8EFFECTSOFDIFFERENTIALTOOTHPITCHONTHEHELICALFORCEMODEL ........................................ 127 8.1FormulationforDierentialPitchEectsonForceComponents ...... 128 8.2ExperimentalModelCalibration ........................ 128 8.3ForcePredictionResults ............................ 130 8.4ChapterSummary ............................... 130 9EFFECTSOFRADIALRUNOUTONTHEHELICALFORCEMODEL ... 135 9.1FormulationfortheEectsofRunoutonForceComponents ........ 136 9.2VericationWithaNumericalSolution .................... 136 9.3ExperimentalModelCalibration ........................ 136 9.4ForcePredictionResults ............................ 138 9.5ChapterSummary ............................... 139 10FORCEMODELINGWITHINSTANTANEOUSCUTTINGCOEFFICIENTS 149 10.1InstantaneousCuttingCoecients ...................... 149 10.2VariancesofModelInputParameters ..................... 151 10.3PropagationofInputParameterUncertaintiesThroughtheForceModel 152 10.4ForcePredictionResults ............................ 152 10.5ChapterSummary ............................... 153 11DYNAMICINFLUENCESINFORCEMEASUREMENTS ........... 162 11.1DescriptionoftheForceMeasurementChain ................. 162 11.2FrequencyResponseoftheForceMeasurementChain ............ 162 11.3ChapterSummary ............................... 164 12FUTUREWORK ................................... 168 12.1StabilityandSurfaceLocationProblemFormulation ............ 168 12.2AugmentedForceModel ............................ 174 13CONCLUSION .................................... 177 APPENDIX AZELLNER'SREGRESSION ............................. 179 BDERIVATIONOFFOURIERCOEFFICIENTS .................. 182 B.1FourierCoecientsforL 182 B.2FourierCoecientsforT 183 B.3FourierCoecientsforbinTypeICutting .................. 184 B.4FourierCoecientsforbinTypeIICutting ................. 185 8

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....................................... 187 BIOGRAPHICALSKETCH ................................ 194 9

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Table page 2-1Computationofentryandexitangles(Ref.Fig. 2-5 ) .............. 40 3-1ConstructionofFouriercoecientsLk;Mk;Tk;Rk;BkandCk 51 5-1Experimentalconditions:straightutedendmill ................. 72 5-2Meancuttingconstantswiththestraightutedendmill ............. 72 5-3Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 5-1 ............................... 72 6-1Summaryofexperimentalconditionsusedforvericationofforcepredictions,holdingthecuttingconditionsofTable 5-1 xed ................. 85 7-1Parametersforcomparisonwiththenumericalsolution ............. 111 7-2Experimentalconditions:3-uted,45helixendmill ............... 111 7-3Meancuttingconstantswiththe3-uted,45helicalendmill .......... 111 7-4Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 7-2 ............................... 111 7-5Conditionsforexperimentalvericationofforcepredictionswiththe3-uted,45helixendmill .................................. 112 8-1Experimentalconditions:dierentialpitchcutter ................ 131 8-2Meancuttingconstantswithdierentialpitchcutter .............. 131 8-3Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 8-1 ............................... 131 8-4Conditionsforexperimentalvericationofforcepredictionswithdierentialpitchcutter ..................................... 131 9-1Experimentalconditions:cutterwithradialrunout ............... 140 9-2Meancuttingconstantswithcutterhavingradialrunout ............ 140 9-3Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 9-1 ............................... 140 9-4Conditionsforexperimentalvericationofforcepredictionswithradialrunout 140 10-1SummaryofexperimentalconditionsusedtoverifyforcepredictionsbasedoninstantaneousKtc;nc;ac,withtheconditionsofTable 9-1 heldxed ....... 154 10

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Figure page 1-1Facemillingandperipheralmilling(endmilling) ................. 36 1-2Straightutedandhelicalutedendmills ..................... 36 2-1Peripheralmillingonaverticalmillingmachine ................. 41 2-2Endmillterminology ................................ 41 2-3Instantaneouschiparea ............................... 42 2-4Circularpathapproximationforchipthickness .................. 42 2-5Up-millinganddown-millingcongurations .................... 42 3-1Developmentofahelicalcuttingedgeandthecorrespondinguncutchip .... 52 3-2Progressoftoolchipcontactzoneasthehelicalendmillrotates ........ 52 3-3EvolutionofL,andTasfunctionsofp 53 3-4Conceptofaveragedmeanchipthicknessinhelicalmilling ........... 54 3-5Evolutionofmeanchipthicknessfordierenthelixangles ............ 54 3-6Variationoftheaveragedmeanchipthickness, .................................. 55 3-7Variationoftheaveragedmeanchipthickness, ...... 55 3-8Chipwidthevolutionforvaryinghelixanglesat50%RI 56 3-9Chipwidthevolutionforvaryinghelixanglesat25%RI 56 4-1Uncutchiparea ................................... 63 4-2Transformationofforcesfromarotatingframe(t;n)toaxedframe(x;y). .. 63 4-3Idealizationofradialrunoutanditseectonthefeedpertooth ........ 64 4-4Forcepredictionexampleforastraightutedendmill .............. 64 5-1Experimentalset-up:straightutedendmill ................... 73 5-2Experimentalestimationofentryangleusingaphasorsignal .......... 74 5-3Linearregressionttingofcuttingcoecients:drymillingoflowcarbonsteelusingastraightutedendmill ........................... 74 6-1Variationofsensitivitycoecientswithp 86 11

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............. 87 6-3Variationofcomponentcombineduncertaintiesandoverallexpandeduncertainty 87 6-4Exampleof95%condenceintervalplacementonpredictedforces ....... 88 6-5Forcepredictionverication:fT=0:150mm/tooth,50%RI,up-milling,runout15m. .................................... 88 6-6Forcepredictionverication:fT=0:100mm/tooth,50%RI,up-milling,runout15m. .................................... 89 6-7Forcepredictionverication:fT=0:050mm/tooth,50%RI,up-milling,runout15m. .................................... 89 6-8Forcepredictionverication:fT=0:200mm/tooth,75%RI,cutstartswithh=0,runout15m. ................................ 90 6-9Forcepredictionverication:fT=0:150mm/tooth,75%RI,cutstartswithh=0,runout15m. ................................ 90 7-1Dierentialprojectedchipareasinhelicalperipheralmilling .......... 112 7-2Projecteddierentialfrontalchiparea ...................... 113 7-3Projecteddierentialaxialchiparea ....................... 113 7-4VariationoftheintegralsI2,I2,andI3 114 7-5Vericationoftheanalyticalmodel:thedegeneratecaseofzerohelix ..... 115 7-6Comparisonofanalyticalandnumericalsolutionsfor30helix,75%RI 116 7-7Residualsshowingthedierencebetweentheanalyticalandnumericalsolutionsfor30helix,75%RI 117 7-8Analyticalsolutionsforvariousconditionsinhelicalperipheralmilling ..... 118 7-9Experimentalset-up:diameter12.7mm,3-utedendmillwith45helix .... 119 7-10Linearregressionttingofcuttingcoecients:drymillingof6061-T6aluminumalloy,usinga3-uted,45helix,endmill ..................... 119 7-11Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,zerorunout,25%RI,down-milling ........... 120 7-12Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:200mm/tooth,zerorunout,25%RI,down-milling ........... 120 7-13Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,zerorunout,10%RI,down-milling ........... 121 12

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........... 121 7-15Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,zerorunout,5%RI,down-milling ............ 122 7-16Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:200mm/tooth,zerorunout,5%RI,down-milling ............ 122 7-17Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,zerorunout,75%RI,cutbeginswithh=0 ...... 123 7-18Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:200mm/tooth,zerorunout,75%RI,cutbeginswithh=0 ...... 123 7-19Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,zerorunout,25%RI,up-milling ............. 124 7-20Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,zerorunout,10%RI,up-milling ............. 124 7-21Gibbs-Wilbrahamdistortion ............................ 125 7-22ConvergenceandcomputationalburdenoftheAnalyticalFouriersolution ... 126 8-1Dierentialtoothpitchexample .......................... 132 8-2Linearregressionttingofcuttingcoecients:drymillingof6061-T6aluminumalloy,usingadierentialpitch,30helix,endmill ................ 132 8-3Dierentialeect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,30helix,fT=0:200mm/tooth,5%RI,down-milling ......... 133 8-4Dierentialeect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,30helix,fT=0:100mm/tooth,20%RI,down-milling ......... 133 8-5Dierentialeect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,30helix,fT=0:100mm/tooth,25%RI,up-milling .......... 134 8-6Dierentialeect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,30helix,fT=0:050mm/tooth,50%RI,up-milling .......... 134 9-1Comparisonwithnumericalsolutionwithrunoutincluded ........... 141 9-2Linearregressionttingofcuttingcoecients:drymillingof6061-T6aluminumalloy,usinganequispacedtooth,45helix,endmill,10mrunout ....... 142 9-3Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,5%RI,down-milling 142 13

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.. 143 9-5Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,10%RI,down-milling 143 9-6Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:200mm/tooth,runout10m,10%RI,down-milling 144 9-7Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,20%RI,down-milling 144 9-8Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,50%RI,down-milling 145 9-9Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,50%RI,down-milling 145 9-10Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:025mm/tooth,runout10m,50%RI,up-milling .. 146 9-11Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,75%RI,cutendswithh=0 ...................................... 146 9-12Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,75%RI,cutstartswithh=0 ...................................... 147 9-13Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,100%RIslotting ... 147 9-14Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,100%RIslotting ... 148 9-15Runouteect.Predictedvs.experimentalforces:Averagecuttingcoecientmodel,45helix,fT=0:200mm/tooth,runout10m,100%RIslotting ... 148 10-1Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,5%RI,down-milling ..... 155 10-2Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,5%RI,up-milling ....... 155 10-3Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,10%RI,down-milling .... 156 10-4Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:200mm/tooth,runout10m,10%RI,down-milling .... 156 14

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...... 157 10-6Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,50%RI,down-milling .... 157 10-7Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,50%RI,down-milling .... 158 10-8Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:025mm/tooth,runout10m,50%RI,up-milling ...... 158 10-9Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,75%RI,cutendswithh=0 159 10-10Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,75%RI,cutstartswithh=0 159 10-11Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:050mm/tooth,runout10m,100%RI,slotting ....... 160 10-12Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:100mm/tooth,runout10m,100%RI,slotting ....... 160 10-13Predictedvs.experimentalforces:Instantaneouscuttingcoecientmodel,45helix,fT=0:200mm/tooth,runout10m,100%RI,slotting ....... 161 11-1FRFsoftheforcemeasuringchain:RealandImaginaryparts ......... 165 11-2FRFsoftheforcemeasuringchain:MagnitudeandPhase ........... 166 11-3EectofforcemeasuringchainFRFsonmeasuredforcesignals ........ 167 12-1Dynamicchipthickness ............................... 175 12-2SchemeforgenerationofstabilitycontourusingTFEA ............. 176 15

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Ck Heavisideunitstepfunction 16

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T0 ) overallexpandeduncertainty symmetricmomentmatrixoftwo-stageAitkenestimators ) 17

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) ) standarduncertainty covariance 18

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) combineduncertainty i ^i h x y t;n;a 19

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angularspindlespeed t;n;a 20

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Millingisoneofthemostimportantsubtractivemanufacturingprocesses.Cuttingforcepredictionsinmillingareusefulforthestructuraldesignofmachinetools,selectionofoptimumcuttingparameters,designofworkholdingxtures,toolstressanalysis,spindlebearingdesign,andtherealtimemonitoringoftoolwearandbreakage.Forcemodelsarealsousedinpredictingthestabilityofthemillingprocess,surfacelocationerrorpredictions,aswellassurfacenishpredictions. Inthisdissertation,closedformanalyticalmechanisticcuttingforcemodels,usinglinearizedlumpedparametercuttingcoecients,areconsidered.Existinganalyticalmechanisticforcemodelsofhelicalperipheralmilling,whichuselinearizedcuttingcoecients,eitherutilizefourdierentsetsofanalyticalexpressionstodescribetheforcesforonecompletecutterrotation,ordonothaveclosedformsolutions.Numericalmodelsarecomputationally-intensive,failtoprovidetheinsightsthatanalyticalmodelsdo,anddonotpermitsymbolicmanipulation.Inthiswork,twoequivalentversionsofclosedformanalyticalexpressionsforchipthicknessandchipwidtharedevelopedusingHeavisideunitstepfunctionandFourierseriesapproaches.Thedistinguishingfeatureisthatsingleexpressionsdescribethechipthicknessandchipwidthduringtheentirecutterrotation.Theseexpressionsarethenappliedtodevelopsingle,analyticalclosedformexpressionsforeachofthethreeorthogonalcomponentsofthecuttingforce,inaxedcoordinateframe, 21

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Aprocedurehasbeendevelopedtocalculatethevariancesinmeasuredmodelinputparameters.ThepropagationofuncertaintiesthroughthemodelisdeterminedbyTypeAandTypeBevaluationstodevelopanoverallexpandeduncertaintyforplacementofcondenceintervalsoncuttingforcepredictions.Theavailabilityofsingleexpressionsforforcecomponentspermitsthederivationofcompactexpressionsforsensitivitycoecientsforuseintheuncertaintyanalysis. Extensiveexperimentaltests,aswellascomparisonswithestablishednumericalmodels,verifythedelityofthepredictions.Themodelisextendedtobeabletopredictcuttingforcesincasesofrunoutordierentialpitchcutters.Arenedformulationusinginstantaneouscuttingcoecients,insteadoftheusualmethodofaveragecoecients,hasalsobeendeveloped,furtherincreasingtheaccuracyofforcepredictions.Experimentsshowthattherenedmodelisabletopredicttheforcepatternsandmagnitudesveryaccuratelyfortheentirerangeofradialimmersions. 22

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Peripheralmillingisasubtractiveprocessforthemanufactureofdiscreteprismaticparts.Alargevarietyofcomponentswhichareusedinaircraft,automobiles,ships,railroads,medicalequipment,spacevehicles,powergenerationandprocessequipment,aswellastoolsfortheproductionofelectroniccomponentsandplasticparts,couldnotbeeconomicallymanufacturedwithoutemployingsomevariantoftheperipheralmillingprocess.Materialisremovedfromarawworkpiecebyinterruptedmachining,usingatoolhavingwelldenedcuttingedges,intheformofchips(swarf),toproduceanishedparthavingadesiredshapeandsize.Thisresearchfallswithinthesubjectareaofmachiningofmetallicalloys,butthemachiningofnon-metallicmaterialssuchascomposites,wood,andplasticsisalsoimportant. TheStatisticalAbstract ( 2008a )oftheU.S.CensusBureaushowsthatmetalworkingmachineryshipmentsintheUnitedStatesintheyear2005totaled$2.80billion,outofwhich$2.08billionweremetalcuttingtypeofmachines,whiletherestwereofmetalformingtype. TheStatisticalAbstract ( 2008b )oftheU.S.CensusBureaushowsthatgrossvalueofnewordersandexportsforU.S.machinetoolsintheyear2006totaled$4.38billion(27,288units),outofwhich$3.70billion(23,670units)weremetalcuttingtypeofmachines,whiletherestwereofmetalformingandothermanufacturingtechnologytype.Thoughthesenumbersappeartobemodest,theimpactoftheseequipmentontheeconomyisallencompassing,aectingvirtuallyeveryproductmanufacturedinthemodernworld. Approximately33%oftheabovementioned$2.08billionworthofmetalworkingmachines,shippedin2005intheU.S.,wereofthetypewhichusemillingasthemainmachiningprocess(millingmachinesandmachiningcenters).Machineslistedintheabovereportundertheheadings`boringanddrillingmachines',`othermetalcuttingmachinetools',`remanufacturedmetalcuttingmachinetools',and`stationtypemachines'also 23

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Inmilling,theworkpieceisfedacrossarotatingmillingcuttertoproduceprismaticparts.Twocommoncategoriesofmillingarefacemillingandperipheralmilling(orendmilling).Figure1-1showsthedierencebetweenthesetwotypes.Facemillinggeneratesasurfacethatisperpendiculartotheaxisofcutterrotation.Peripheralmillingproducessidewallsgeneratedbytheperipheryofthecutterparalleltotheaxis.Usually,asecondarysurfaceisalsogeneratedwhichisakintofacemilling,asshowninthegure.Inpureperipheralmilling,onlysidecuttingoccurs.Arbormountedslabmillsoftenoperateinthismode.Variantsoftheseprocessesincludeballendmilling,threadmilling,andplungefacemilling.Millingcuttersmaybestraightutedorhelical(Fig. 1-2 ). Thedemandforcomplexprismaticpartsiseverincreasingbecausesuchpartsoerthedesigneralotofexibility.Hence,millinghasassumedacentralroleamongmachiningprocesses.Modernmillingprocessescombinecapabilitiesofhighmaterialremovalrateswithaccuratesurfaceplacement,nesurfacenishandclosetolerancesonsurfaceatness. Inthecommercialaerospaceindustry,monolithicmachiningofverylargealuminumalloypartshasgainedpopularity.Monolithicpartdesignconferseconomicadvantagesbyreducingthenumberofparts,eliminatingexpensiveassemblyandimprovingpartaccuracy.Thetradeoinvolveshighspeedmachiningofthinribsusingslenderendmills.Themachinistfacesaproblemhavingacombinationofaexibletoolandaexibleworkpiece.Theattendantlossofstabilitylimitsmaterialremovalratesandstabilityboundarieshavetobecorrectlydeterminedtondzonesofstablemachiningwherehighmaterialremovalratesmaybepossible( Tlusty 2000 ). Theboomintheplasticconsumerdurablesmarkethasfosteredamushroomingmoldanddieindustry.High-speedfree-formmillingofferrousalloys,usingslenderendmills, 24

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Nelsonetal. 1998 )resultinhighspeciccuttingpressuressothatthecuttersexperienceveryhighcuttingforces.Otherimportantapplicationsofmillingincludethemachiningoftitaniumalloysfordefenseandaerospaceapplications( Leighetal. 2000 ),andhighvelocitymachiningofaluminumalloysforautomobilecomponents.Manyoftheexperimentsreportedinthisdocumentwereconductedonaluminumalloys.Miniaturemachiningisincreasinginimportance.Smalldiameterendmillsarelimitedbylateralforce,makingitimportanttoestimateforcecomponentswhileselectingmachiningparameters( Schuelleretal. 2007 ). Ananalysisofthemillingprocessinvolvesthepredictionofcuttingforces,stabilitylimits,surfaceplacementaccuracy,andsurfacenish. Kurdi ( 2005 )haspresentedamethodofoptimizationofthesurfacelocationandmaterialremovalrateinmillingwhileconsideringtheuncertaintyinthemillingmodel.Samplingmethodswereused(LatinHypercubeandMonteCarlo)toplacecondenceintervalsonthestabilitylimits.Condenceintervalswerenotplaceddirectlyonthepredictedcuttingforces.Thereare 25

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Sawin ( 1926 )presentedoneoftherstmodelsthatappearedintheliteratureaddressingboth,straightutedcutters,aswellashelicalcutters.Thiswasamechanistic 26

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Salomon ( 1926 )developedamodelinwhichtheworkdonebyastraightutedcutterwasconsideredtobeafunctionofthegeometry.Apowerlawwasusedtorelatethespeciccuttingpressuretotheinstantaneouschipthickness.Thecircularpathapproximationwasusedtodeterminetheinstantaneouschipthickness. Klien ( 1937 )developedtangentialforceequationsforstraightutedandhelicalcuttersusingtheFourierseries.Thecuttingcoecientwasconsideredtobeaconstantforagivenworkpiecematerial. Martellotti( 1941 ; 1945 )studiedthekinematicsofmilling,statedthecirculartoolpathapproximation,anddevelopedexpressionsforthetruechipthicknessconsideringthetruetrochoidaltoolpath.Thechipthickness,andpowerconsumedindownmillingwasshowntobegreaterthaninupmilling.However,downmillingwasshowntohaveotheradvantagessuchasimprovedtoollife. Kienzle ( 1952 )denedaparameterwhichistheforcerequiredtomachineachipofsize1mm2.HedenedthecuttingcoecientasaproductofthisparameterandapowerofthechipthicknesswhichissimilartoSolomon'smodel. Sabberwal ( 1961 )arguedthatforagivensetofmachiningparameterstheworkdoneinmaterialremovalshouldbeindependentofthehelixangle.Onthisbasisheexpectedtheinstantaneousspeciccuttingpressuretobeindependentofthehelixangleofthecutter.Heveriedthisobservationformachiningofhightensilesteel(EN28)withdierentcuttershavinghelixanglesrangingfrom0to30.Healsofoundtheobservationtobetrueforcuttingmildsteel(0and50helix)andaluminum(0and45helix).Basedonthisobservation,itiscommonpracticetoextractcuttingcoecients 27

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Ku ( 2006 )hasprovidesomeevidencethatcuttingcoecientscoulddependonhelixangle.Suchcomparisonsarecontingentuponthefactthatallotherfactorsremainthesame,especiallytheothergeometricalparametersofthecutters. Sabberwal ( 1961 )modeledthetangentialcuttingforcesinhelicalmillingbymultiplyingthefrontalchipareawiththespeciccuttingpressure.However,thenormalcuttingforcewasnotmodeled,norwastheaxialforce.Therefore,thismodelwassuitableforpowercalculationsonly.Thefeedforceandthelateralforcecouldnotbedetermined.Animportantcontributionmadeby Sabberwal ( 1961 )wastohighlighttheexistenceofdierenttoolengagementconditionsforhelicalmilling.Hedeterminedtwodierenttypesofcuttingdependingonthewhetherornotthetrailingedgeofthetoothenteredthecutbeforetheleadingedgeleftthecut.Eachofthesetwotypesofcuttingwasidentiedashavingthreephasesasthecutprogressed.Duringtherstphase,thechipwidthincreased,stayedconstantinthesecondphase,anddecreasedinthethirdphase.Inalaterwork, TlustyandMacNeil ( 1975 )namedthetwotypesofcutasTypeIandTypeII.ThethreephasesweretermedA,B,andC.Thisistheterminologyusedinthisdocumentinthefollowingchapters. TlustyandMacNeil ( 1975 )alsopresentedtherstanalyticalsolutionforthetwocomponentsofcuttingforceintheplaneofcutterrotationforhelicalperipheralmilling.Thetangentialcuttingforcewasconsideredproportionaltothechipareawiththespeciccuttingforce,K,beingafunctionofworkpiecematerial,toolgeometry,andtheaveragechipthickness.Theradialforce(whichisgenerallycalledthenormalforceinthisdocument)wasassumedtobe30%ofthetangentialforce.Thiswasatwodimensionalmodel. Thetangentialandnormal(radial)forcecomponentsaredenedinarotatingframeofreferenceattachedtotherotatingcutter.Inthe TlustyandMacNeil ( 1975 )model,theseweretransformedintocomponentsinanon-rotatingframethrougharotation 28

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Theaforementionedmodelswereoneortwodimensional,describingcomponentsofthecuttingforceintheplane.Thisisanadequatedescriptionofforcesinpureperipheralmillingwithstraightutedtools,whichisessentiallyorthogonalmachining.However,forhelicalmilling,theaxialcomponentoftheforcemustalsobemodeledforacompletedescription.Thisisaninstanceofobliquemachining. Usuietal. ( 1978a b )presentedamethodinwhichthisobliquecuttingtoolwasmodeledasasinglepointtoolbeingacollectionoforthogonalcuttingconditionsalongthecuttingedge.Theyusedanenergymethodtopredictforcecomponents.Theinputparametersforthemodelaretheshearangle,thefrictionangle,andtheshearstress.Theoutputsaretheshearenergyandthefrictionenergybasedonwhichthecuttingforcesareobtained. Tsai ( 2007 )hasdevelopedaversionofthismodelsuitableforrotatingcutterswhichcanbeappliedtoahelicalmillingsituation.Thedicultywiththisapproachisthattheinputparameters,namelytheshearangleandthefrictionangle,arediculttoidentifyinagivenmachiningsituation.Forthisreason,themechanistictypeofmodel,wherethecuttingcoecientscanbeexperimentallydetermined,isgenerallypreferred.However,thisapproachcouldbeusefulinpredictingforcesduringcutterdesign,becauseatthatstage,thetoolhasnotbeenmanufactured,andexperimentaldeterminationofcuttingcoecientsisimpossible. Piispanen ( 1937 1948 )and Merchant ( 1945 )haddevelopedtheconceptsofshearangleandfrictionanglefororthogonalmachining. AltintasandLee ( 1996 )presentedageneral 29

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Kline ( 1982 )developedacomprehensivemechanisticforcemodelmovingaheadoftherigidforcemodelbyconsideringcutterdeections,cutterrunout,andmodelingthetruetrochoidaltoolpath.Thisresultedinaseriesofpapers.In DeVorandKline ( 1980 )thecuttingcoecientswereexperimentallyestimatedusingaverageforces.Thecoecientswerecalledempiricalconstants,butitwasshownthattheydependedonmachiningparameters.Theconstantassociatedwithtangentialforcecomponentwascalledtheunithorsepowerorthespeciccuttingenergy.Inttingtheconstantsinthemodeltoexperimentalaverageforcedata,whilemachining4340steelat320BHN,themultiplecorrelationcoecients(R2values)were0.984forKT(thetangentialcoecient)and0.936forKR(thefractionofKTwhichyieldsthenormalcoecient).Inadditiontotheforces,theforcecenter,forcedistribution,andcutterdeectionwerepredictedusingnumericalmethods.Anumericalmethodwasalsousedtotrackthetruetrochoidaltoolpath.Bydoingsoitwaspossibletomakeadistinctionbetweenupmillinganddownmillingforces,whichwasnotpossiblewiththecircularpathapproximation.Forcepredictionswerefoundtoagreewithexperimentwithin5to10%.Thecorrespondencewasbetteratsmallradialdepthsofcut,butdeterioratedastheradialdepthofcutincreased.Animportantndingconcernedcutterdeections.Itwasshownthatforawidevarietyofcuttingconditionswhichproducedthesameresultantforce,themagnitudeanddirectionofdeectionsvariedwidely. Klineetal. ( 1982 )developedafurtherversionofthismodelcapableofpredictingforcecharacteristicsincorneringcutsduringwhichtheradialengagementkeepschanging.Thiswasalsoanumericalmodel.Theymentionedthatsuchcorneringcutsarecommonlyencounteredinaerospacemachiningoperations. KlineandDeVor ( 1983 )showedthat 30

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SutherlandandDeVor ( 1986 )extendedKline'sworkandsuggestedaexiblemodelincludingrunout.Thisisanumericalmodelinwhichaneectivefeedratewascomputedwhichbalancesthecuttingforcesandtheresultantsystemdeections.Theydiscoveredthatsystemdeectionstempertheeectsofrunouttoreducethepeakforceandmaximumsurfaceerror. Yellowley ( 1988 )notedthatmillingtorqueandforcescouldberepresentedintermsofFourierseries.YellowlydevelopedtheFourierrepresentationofforcesintheplaneforstraightutedcutters.Healsodevelopedasymbolicrepresentationfortheresultanttorquebyconsideringthetoothtobemadeupofelementalstraightcuttingedgeshavinganangularseparation.However,hedidnotcomputetheFouriercoecients,nordidhecomputecuttingforcesforthehelicalmill. MontgomeryandAltintas ( 1991 )presentedanumericaldynamicmodelforhelicalmillingusingwhichforcescouldbepredictedunderstaticordynamicconditions.Thetrochoidaltoolpathwasusedtodeterminetheuncutchipthickness.Thegeometryoftoolandworkpiecemotionwascapturedusingstructuralmodeling.Thiswasalsoatwodimensionalmodel.Anothertwodimensionalmodelforhelicalmillingwasproposedby AltintasandSpence ( 1991 )foracomputationallyecientschemeofforceprediction.Thiswasaclosedformanalyticalsolutioninwhichthreedierentexpressionshadtobeusedasthecutprogressedthroughthethreephasesofcutting.Toderiveaverageforcesforcalculatingthecuttingcoecients,theexpressionsweresimpliedundertheassumptionofazerohelixbasedonSabberwal's( 1961 )observationthatthehelixanglehadnoinuenceoncuttingcoecients.Ageneralizedmechanisticmodelforhelicalmillingforceswas 31

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EnginandAltintas ( 2001 )whichtookintoconsiderationthetrochoidaltoolpath.Thenalexpressionsfortheforcesareintheformofintegralswhichhavetobeevaluatedbytheuserforaspecicsetofcuttingconditions. AbrariandElbestawi ( 1997 )publishedathreedimensionalclosedformsolutionwheretheforceswereexpressedasalinearcombinationofasetofbasisfunctions.However,thesimplicityofusinglinearizedcuttingcoecientswaslostasthesewerereplacedwithamatrixwhichincorporatedthehelixangle. Ehmanetal. ( 1997 )haveprovidedacomprehensivereviewofvariousforcemodelsthatappearedintheliteraturebefore1997.Theirreviewalsoencompassesmanyaspectsofdynamicforcemodeling.Inthisdissertation,theemphasisisonstaticforcemodeling.Atwodimensional,Fourierseriesbasedsolutionpresentedby Schmitz ( 2005 )and SchmitzandMann ( 2006 )includestheeectofthehelixanglebydividingthecutterintothinaxialslicesandsummingtheeectstoobtainthetotalforce.Recently,athreedimensionalmodelwasdevelopedby Mannetal. ( 2008 )usinganequivalentcomplexFourierseriesrepresentationofforcestofacilitatesymbolicmanipulation.Thismodelwasappliedtostabilityandsurfacelocationanalysis.Inthepaper,theintegrationshavenotbeensymbolicallysolved.Ifthatisdone,thiscouldpossiblyprovideaclosedformsolutionfortheforcemodelitself. Severalauthorshavereportedontheuseofdierentialpitchcutters,especiallyformitigatingtheeectsofregenerativechatteraswellasreducingdimensionalsurfaceerrors(e.g. Slavicek 1965 ; Vanherck 1967 ; Tlustyetal. 1983 ; ShiraseandAltintas 1996 ).However,aforcemodelforadierentialpitchhelicalendmill,whichhasasingleclosedformanalyticalexpressioncoveringtheentiredomainofonecutterrotation,hasnotbeenreportedintheliterature. KlineandDeVor ( 1983 )showedthatrunoutincreasestheaveragechipthicknessforthoseteethwhichhavehigherrunout,increasestheratioofthemaximumtoaverageforce,andshiftsthefrequencycontentoftheforcesignalawayfromthetoothpassingfrequencytothespindlerotationalfrequency.Theratiooftherunouttothefeedratewasidentied 32

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ArmaregoandDeshpande ( 1991 )presentedanumericalmodelconsideringcuttereccentricity.Theirmodelcouldpredictforceswellexceptinthosecaseswherecutterdeectionswerelarge,forwhichcasetheyhadadeectionmodel. Schmitzetal. ( 2007 )investigatedtheeectsofrunoutonsurfacenish,surfacelocation,andstabilityusingaforcemodelwhichisatime-domainsimulationdescription. LiandLi ( 2004 )proposedatheoreticalforcemodelwithcutterrunoutandtrochoidaltooltrajectory.Theydiscretizedthecutterintoslicesandsumtheeectsofforceduetoeachslicetoobtainthecompleteforce. JunzWangandZheng ( 2003 ),and Koetal. ( 2002 )identiedcuttingforcecoecientsinthepresenceofrunout. Forrealisticdynamicmodelingofmillingprocesses,theconsiderationofrunoutisanimportantfactor.Recognizingthisfact, AltintasandChan ( 1992 )presentedadigitalsimulationmodelfordynamicmillinginwhichtheyincludedtheeectsofrunout. Weinertetal. ( 2007 )haveshownthatrunoutaectssurfacequalityeveninstablemachining. Wangetal. ( 1993 )presentedanin-processcontrolmethodologytocompensateforsurfacenishimperfectionsduetocutterrunoutinmillingprocesses.Therunoutgeometrywasidentiedfromcuttingforcemeasurementsandthemanipulationofradialdepthofcuttofollowtheinversetrajectoryofrunout.Thustheeectsofrunoutwerecounteractedforsurfacenishimprovement. Thesubjectofcuttingcoecientshasalsoreceivedmuchattentionintheliterature. Gradiseketal. ( 2004 )haspresentedaverycomprehensivestudyofcuttingcoecientsinwhichthedependenceofthesecoecientsonvariousprocessparametershasbeenexperimentallystudiedindetail.Oneoftheimportantconclusionswasthatthecoecientsdonotdierinupmillinganddownmilling.Whereasmostofthestudieslistedhereuseanaveragedvalueofcuttingcoecients,thisisonlyanapproximatewaytohandletheproblem.Sincethecoecientschangewithprocessparameters,theyevolve 33

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Wanetal. 2007 ). Thedimensionsofthecuttingcoecientsarethoseofpressure[N/mm2].Forthisreason,theyarevariouslyreferredtoasspeciccuttingpressures( Jayarametal. 2001 ; Perezetal. 2007 ),specicforcecoecients( Gradiseketal. 2004 ),speciccuttingforce( KoandCho 2005 ),orspeciccuttingenergy( Kline 1982 )intheliterature.Occasionally,theyarealsocalledcuttingconstants( AltintasandSpence 1991 ).However,inthisdocumenttheterm`cuttingconstants'isusedforadierentpurposeinthesensethatcuttingconstantsaretrueinvariants,whereastheothertermssuchascuttingcoecients,specicforce,etc.,areprocessparameterdependent.Inanycase,themeaningofterminologyisusuallyclearfromthecontext. AltintasandBudak 1995 ; Mannetal. 2005 ; Insperger 2003 ).However,experimentalvericationisusuallydoneusinghelicalendmills( Mannetal. 2003 ).Helicalendmillsarepreferredbecausetheydistributetheforceoveralongercuttingedgelength,reducingthelocalpressure.Forthisreason,thisresearchisdirectedtowardsdevelopingafullyanalyticalclosedformforcemodelwhichwouldhaveasingleexpressionforeachforcecomponentthatisvalidfortheentirecutterrotation. 34

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35

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Facemillingandperipheralmilling(endmilling)processes Figure1-2. Straightutedandhelicalutedendmills.Twoutedandfourutedcuttersareshownforeachofthetwovarieties. 36

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Thischapterprovidesabriefqualitativedescriptionoftheperipheralmillingprocessaccompaniedwithanexplanationoftheterminology.Therawworkpieceandthecuttingtoolareheldinamachinewhichprovidestherelativemotionandpowertoremovematerialintheformofchips.Thegeometryofthechipholdsthekeytoanunderstandingoftheprocess.Thechapterconcludesbymathematicallycharacterizingthechipforthesimplestpossibletoolgeometry. 2-1 showsaschematicsketchofaperipheralmillingset-upona3-axisverticalmillingmachine.Theendmill,alsocalledthetoolorcutter,isheldinatoolholder,calledthechuck,whichismountedonapoweredmachinespindle.Theworkpieceisfedacrossthetoolsuchthatthereisalinearrelativemotionbetweentheendmillandthepart,inadditiontotherotationofthetool.Thecutterrotatesatthespindlespeed,.MovementsoftheX-Y-Zaxesprovidetherelativemotionbetweenworkpieceandtool.Machineswithadditionalrotaryaxesareverycommon.Materialisremovedintheformofchips(swarf)viasuccessivepassesofcuttingedges(teeth)acrosstheworkpiece.Inperipheralmilling,theaimistoproducesidewallsgeneratedbythesidesofthemillingcutter,thoughfacemillingactionalsotakesplace,i.e.,asurfaceparalleltothefaceoftheendmillisalsogenerated.Occasionally,pureperipheralmillingmaybeperformedwithonlysidecutting. Theendmillsarerotary,multipointcuttingtoolshavingeitherhelicalorstraightutes(zerohelix).ThegeometryofhelicalperipheralmillingisillustratedinFig. 2-2 whichshowsthehelixangle,,andthediameter,D.Thecommanded(nominal)axialdepthofcut,a,theradialdepthofcut,ar,thefeedpertooth,fT,andthespindlespeed,,arexedparameterssetbythemachinist.Thecuttingedgesarecalledteethwiththechipspacesbeingcalledgulletsorutes.Thetermuteisalsolooselyusedtoreferto 37

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Radialimmersion,RI,ar 2-3 illustratesthedenitionsoftheinstantaneouschiparea,instantaneouschipthickness,andinstantaneouschipwidthforastraightutedendmillandahelicalendmill. Thechipareaisnotconstantandchangesasthecutterrotates.Tocalculatetheinstantaneouschiparea,thechipthickness,h,andchipwidth,b,havetobecalculated.Forastraightutedendmill,thechipwidthisaxedconstantandequalsthecommanded(nominal)axialdepthofcut,a.Forahelicalendmill,thechipwidthvariesasthecutterrotates.Inmilling,thechipthickness,h,variesasthecutprogresses.Thechipthicknessisdenedbyprojectingthechipontheplaneofrotation.Forstraightutedcutters,theprojectionisastraightlineallowingthespecicationofaninstantaneouschipthickness.Forhelicalcutters,theprojectionisanareaofnonuniformthickness.Itisconvenienttospecifyameaninstantaneouschipthickness,hm. Therelativemotionofthecuttingedgewithrespecttotheworkpieceisacombinationofthecutterrotationandthetranslatingfeed.Thetruepathofthemillingtoothintheplane,withrespecttotheworkpiece,istrochoidal.Martellotti's( 1941 )simpliedcirculartoolpathapproximation(Fig. 2-4 )isusedinthisdocumentbecauseitiscommonpracticeandyieldsgoodresults.Thecircularpathapproximationhadbeenusedbyearlierresearchers,suchas Salomon ( 1926 )and Sawin ( 1926 ).Thecutterrotationmaybetrackedusingarotationangle,,denedwithrespecttoanarbitraryreferenceline.Atany 38

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Cutter-workpieceengagementcongurationsareshowninFig. 2-5 .Inup-milling,thecutbeginswithzerochipthickness,whereasindownmillingthecutendswithazerochipthickness.RIvaluesexceeding50%resultinmixed-modecongurationsoftwopossibletypes.Thecutmayeitherbeginwithzerochipthickness,orendwithzerochipthickness.Aradialimmersionof100%resultsinmachiningofdoublewalledchannelsandiscalledslotting.Theradialimmersioncanalsoberelatedtotheanglethetooltipstartsandexitsthecut,designatedstandex,respectively,throughageometricalcalculation.Thedomainofstandexis[0;].ExpressionsforcalculationsofentryandexitanglesaregiveninTable 2-1 Toothengagementextent,[st;ex],forastraightutedcutterhavingbeendened,thechipthicknessEq. 2{2 isrewrittenformallyas Forastraightutedcutter,theinstantaneousuncutchipareais Martellotti ( 1941 )proposedthattheaverageundeformedchipthicknesscouldberelatedtothecomponentsofthecuttingforce.Theconceptofaveragechipthicknessisinvokedtofacilitatetheexperimentaldeterminationofcuttingcoecients,aswellasaveragecuttingforces.UsingEq. 2{2 theaveragechipthicknessforcuttingwithstraight 39

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Table2-1. Computationofentryandexitangles(Ref.Fig. 2-5 ) RadialUp/DownEntryangleExitangle immersionmillingstex RI0:5Down-milling0arccos(12RI) 0 RI=1:0Fullslotting 0

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Schematicsketchofaperipheralmillingset-uponaverticalmillingmachine. Figure2-2. Endmillterminology:Helixangle,,anddiameter,D,denethetoolgeometry.Thecommanded(nominal)axialdepthofcut,a,theradialdepthofcut,ar,thefeedpertooth,fT,andthespindlespeed,,arexedmachiningparameters.Atwouted,righthandedhelicalendmillisshown. 41

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Instantaneouschiparea(boldoutlines),instantaneouschipthickness,h,andinstantaneouschipwidth,b,inperipheralmillingusing(a)straightutedendmillsand(b)helicalutedendmills.Thehelicalcutteryieldsavariablechipwidthasitrotates,andanonuniforminstantaneouschipthickness.So,ameaninstantaneouschipthickness,hm,isdenedforthehelicalcutter. Figure2-4. Circularpathapproximationforchipthicknesscalculation. Figure2-5. Examplesofcutter-workpieceengagementshowingup-milling(conventionalmilling)anddown-milling(climbmilling)congurationsforRI<50%.Theangularorientationofanypointonthecuttingedge,,isafunctionoftimeduetocutterrotation.Theleadingpointofthetoothentersthecutatstandexitsthecutatangleexwhicharexedangularpositionsinspaceintheplaneofcutterrotation. 42

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Theaimofthischapteristoanalyticallycapturetheshapeandsizeofthechipasthecutterrotates.Thisknowledgewillbeappliedinlaterchaptersinthedeterminationofclosedformanalyticalexpressionsforthecuttingforcecomponents,whichisoneofthemaingoalsofthisresearch.Theexpressionsforchipthicknesswillalsobeusedincharacterizinglumpedparametercuttingcoecientsinthecuttingforcemodel.Forstraightutedcutters,theexpressionsarewellknown( Tlusty 2000 )andarestatedinthepreviouschapter.Thischapterdemonstratesthecomplexitiesinanalyticallydescribingthechipinhelicalperipheralmilling.Thedistinguishingfeatureofthedevelopmentisthatsingleclosedformanalyticalexpressionsdescribethechipovertheentirecutterrotation. Twodierent,butcompletelyequivalent,versionsoftheanalyticalexpressionsarepresented.OneisbasedontheHeavisideunitstepfunctionapproach,andtheotherisaFouriertrigonometricseriesmodel.Whentheexpressionsareusedtodescribecuttingforces,theyresultintwodierent,butequivalentversionsofcuttingforcemodels.Thephysicsoftheprocessisdescribedinexactlythesamemanner,butthesetwoversionshaveadierentmathematicalstructurewhichhavecertainimplicationsfortheuser. IntheHeavisideunitstepfunctionbasedmodelanexactrepresentationoftheforceisobtained,andthismodelisusefulformostapplications.TheFourierseriesmodelrequiresaninnitesumforexactresults,buttruncatedseriessumsconvergeveryquickly.Theconvergencepropertiesarediscussedinmoredetailinchapter 7 .Whensymbolicmanipulationsoftheanalyticalexpressionsarerequiredforanyapplication,themathematicalstructureofthetwodierentmodelsmaydictatethechoicefortheuser.ThederivativesoftheHeavisidestepfunctionbasedexpressions,withrespecttothecutterrotationangle,arediscontinuous.Ifcontinuousderivativesarerequired,theFourierseriesbasedformulationmustbeused. 43

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60t+2 Np(3{1) where =thespindlespeedinrevolutions/minute(rpm),and Insteadystate,underconditionsofstablemachining,thetimeevolutionofdependentvariablessuchaschipthickness,chipwidth,andcuttingforces,iscyclicintime.Itsucestostudytheevolutionofalldependentvariablesoftheproblemoveraperiodofonecutterrevolution.Inusingthevariousmodelsdevelopedinthisdocument,onewouldmerelysubstitute toobtaintimeevolutionofthedependentvariables.Theproblemissolvedfortheprincipalvaluesofp2[0;2) andanyfurtherreferencetotimeissuppressed.Theargumentofallsubsequenttrigonometricfunctionsisp,shiftedbyappropriateconstants,dependingonthesituation.Theterm\instantaneous"willbetakentorefertothecurrentangularpositionofthepthtooth,p. Figure 3-1 showsthedevelopmentofahelicalcuttingedgeandthecorrespondinguncutchip.Thissketchisalsousefulforeaseofvisualizationofsomesubsequentgures.Thetoolchipcontactzoneisshownasagraylineinthedevelopedsketch.isthehelixangleofthecutter.Theangularorientation,,ofanypointonthetoolchipcontactzoneoranypointonthecuttingedge,istakenfromanarbitraryreference. 44

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3-1 showstheangularorientationsoftheleadingandtrailingpointsofthetoolchipcontactzonewhichareatLandT,respectively.Thesetwoangularorientationsarefunctionsofp.Inthisdocument,thesefunctionaldependenciesareanalyticallyexpressedinsingleequationsinclosedformandvalidfortheentirecutterrotation.Practicallyusefulexpressionsforchipthickness,chipwidth,cuttingforces,andcuttingcoecientsarederivedintheformofsingleanalyticalexpressionsinclosedformasfunctionsofLandT,andhenceasfunctionsofp.Closedformanalyticalexpressionsareusefulinprovidinginsightintothephysicalprocess,canbeeasilyprogrammedintocalculatorsforquickcalculations,anddonotsuerfromnumericaldivergence.Inaddition,symbolicmanipulationsarepossible. ArepresentativetoolchipcontactcongurationisshowninFig. 3-1 inwhichthecylindricalsurfaceoftheendmillandtheuncutchiphavebeendevelopedontoaplane. Sabberwal ( 1961 )and TlustyandMacNeil ( 1975 )identieddierentcongurationsoftoolchipengagementinhelicalmillingaccordingtowhichtwopossibletypesofcuttingthatmayoccur,eachhavingthreedistinctphases. Figure 3-2 showsthesepossibletoolchipcontactcongurations.InTypeIcutting,thechipwidthattainsavalueequaltothecommandedaxialdeptofcutatsomepointduringthecut,i.e.,b=a,forsomeportionofthecut.InTypeIIcutting,bD (3{4) 45

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TlustyandMacNeil ( 1975 )whodevelopedseparateanalyticalexpressionsforcuttingforcesforthedierentphases.Theirresulthasfourseparatesetsofequations,oneeachforthephasesAandC,andtwoseparateequationsforphaseBcorrespondingtoTypeIandTypeIIcutting. Thesolutionpresentedheretreatsthethreephasesasasingleone.AlgebratakescareofthetwotypesofcuttinggovernedbyEqs. 3{3 and 3{4 .AsingleexpressioneachforLandTisdevelopedwhichcoversallthedierentpossibletoochipcontactzonecongurations. ByinspectionofthegeometryshowninFigure 3-2 ,theevolutionoftheintermediatevariablesL,andTcanbestudiedasfunctionsofpinTypeIandTypeIIcutting.TherelationshipsaresketchedinFig. 3-3 inwhichtherelationbetweenbandpisalsosketchedsinceitisobtaineddirectlybyinspection.Now,itisjustamatterofwritingsingleanalyticalexpressionstocharacterizethesesequencesofstraightlinesegmentswhichareperiodicoveronecutterrotation.UponinspectingFig. 3-3 ,itcanbeseenthatthevariouslinesegmentsinLandTstartandendatthesamevaluesoftheargumentp.So,itturnsoutthattheanalyticalexpressionsarethesameforTypeIandTypeIIcutting,andtheuserdoesnothavetoworryaboutthisdistinction. 3-3 revealsthatthefunctionsL,T,andbconsistofnitelinesegmentsinthedomainp2[0;2).Theequationsoftheselinesegmentscanbewrittenbymultiplyingtheequationofthecorrespondingstraightline(whichhasinniteextents)withafunctionwhichhasunitvalueinthedomainofthelinesegment,andvanishes 46

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where&issomexedvalueofP.Thefunctionassinglevaluedeverywhereinthedomain. TheunitstepfunctionbasedexpressionsforLandTarewrittenbyinspectionofFig. 3-3 .TheequationsfortheindividualstraightlinesegmentsforLandTaregiveninAppendix B .Theequationsofthestraightlinesaremultipliedbyasuitabledierenceofthestepfunctions.TheresultingfunctionalrelationshipsareL=p[H(pst)H(pex)]+ex[H(pex)H(pex2atan=D)] (3{6)T=2664(p2atan=D)[H(pst2atan=D)H(pex2atan=D)]+stH(pst)H(pst2atan=D)3775 Thechipwidth,b,isobtainedbyinspectionofgeometryinFig. 3-2 ,andmaybeexpressedasafunctionofpthroughtheintermediatevariablesLandT 3-3 ),yieldingafundamentalperiodof2,andmakingitpossibletowritethesefunctionsintrigonometricserieswithappropriateFouriercoecients( Kreyszig 2006 )asfollows whereL0;Lk;MkandT0;Tk;RkaretherelevantFouriercoecients. 47

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3{8 fromtheHeavisideformulationstillholds.Therefore,forthepurposeofcomputation,aseparateexpressionforbneednotbederived.However,insomeapplications,especiallyiftheintentistostudythevariationofbalone whereB0;Bk;CkaretherelevantFouriercoecients. BasedonthevariationsgiveninFig. 3-3 ,theFouriercoecientsarederivedbypiecewiseintegrationsovertheperiodp2[0;2)usingstandardprocedures(detailsinAppendix A ).AverageFouriercoecientsinEqs. 3{9 3{11 are: 2(2ex2st) 2+2atan Dex(3{12) 2(2ex2st) 2+2atan Dst(3{13) 2(3{14) ResultsofthecalculationsforLk;Mk;Tk;Rk;BkandCkarelistedinTable 3-1 .Forinstance,thecoecient,Lk,maybeformedbyinspectionofcolumn2ofTable 3-1 whichshowsthatLkhasfourtermsinthesummation ( 2005 )and Xuetal. ( 1998 )haveoeredinterestingstudiesofdepth-of-cutvariationsinendmillingandballendmilling,respectively. 48

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2-3 whichshowsasketchofachipinhelicalmilling.Themeaninstantaneouschipthickness,hm,maybecomputedbytakingthemeanovertheentireangularextentspannedbythechip Uponsimplication, wherethesymbolhisshorthandnotationandthesamplingfunction(sinecardinal)isdenedas sinc(&),sin(&) Themeaninstantaneouschipthickness,hm,variesasthetoolmovesacrossthechip.Figure 3-4 showstheevolutionofhmoverthecompletetoothpassageacrosstheuncutchip.Theaveragedmeanchipthickness, DZsthm(p)dp=fT DZsthdp(3{19) 49

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Dst(3{20) Inthismanner,themeaninstantaneouschipthicknessandtheaveragedmeaninstantaneouschipthicknesshavebeenexpressedanalytically. 3-5 showsanexampleoftheevolutionofthemeanchipthickness,hm,asthecutterrotates.Asthehelixangleincreases,thespanofangulartool-chipcontactincreases.Thisresultsinaloweringof Figure 3-6 illustratesthevariationof 3-7 illustratesthevariationof Theevolutionofchipwidthisstudiedusingtwoexamples.TheresultsofcalculationfortheseexamplesaregiveninFigs. 3-8 and 3-9 .Asthehelixangleincreases,theangularspanoftoothchipcontactincreases,andatsomehelixangle,thecuttransitionsfromTypeItoTypeII. TheidenticationofthevariablesLandTmakesitpossibletowritesingleexpressionsforthechipthicknessandchipwidthinhelicalperipheralmillingwhich 50

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Thereasonforcalculatingthechipthicknessandchipwidthinhelicalmillingistobeabletousetheseexpressionsinderivingamechanisticmodelforcuttingforces.Thenextchapterisdevotedtointroducingthereadertothemathematicalstructureofsuchmodels. Table3-1. ConstructionofFouriercoecientsLk;Mk;Tk;Rk;BkandCk D)]0001 D)]0ex D)]ex D)]001 51

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Developmentofahelicalcuttingedgeandthecorrespondinguncutchip.Theangularorientation,,ofanypointonthetoolchipcontactzone,oronthecuttingedge,isreckonedfromanarbitraryreference.LeadingandtrailingpointsofthecontactzoneareatLandT,respectively.Thetooltipisatp.Theentryandexitangles(standex)arexedorientations,aisthecommandedaxialdepthofcut,andisthehelixangle.Arepresentativetoolchipcontactcongurationisshown. Figure3-2. Progressoftoolchipcontactzoneasthehelicalendmillrotates.InTypeIcutting,b=a,inPhaseBofthecut.InTypeIIcutting,b
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EvolutionoftheintermediatevariablesL,andTasfunctionsofpinTypeIandTypeIIcutting.Evolutionofthechipwidth,b,isalsoshown.ThefunctionalrelationshipsareobtainedbyinspectionofFig. 3-2 53

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Conceptoftheaveragedmeanchipthickness, Figure3-5. Evolutionofthemeanchipthickness,hm,asthecutterrotates,fordierenthelixanglesattwodierentradialimmersions,andthecorrespondingaveragedmeanchipthickness, 54

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Variationoftheaveragedmeanchipthickness, Figure3-7. Variationoftheaveragedmeanchipthickness, 55

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Chipwidthevolutionforvaryinghelixanglesoveronecutterrotationforasingleutedendmill,basedontheanalyticalsolution(RI=50%;a=4mm,D=12mm,up-milling). Figure3-9. Chipwidthevolutionforvaryinghelixanglesoveronecutterrotationforasingleutedendmill,basedontheanalyticalsolution(RI=25%;a=4mm,D=12mm,up-milling). 56

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Thepurposeofthischapteristoprovidethereaderwithabackgroundonthemethodofmechanisticforcemodelingandtheassociatedterminology.Themechanisticmodelchosenforthisdemonstrationisatwodimensionalforcemodelforperipheralmillingwithstraightutedendmillswhichiswidelyused( Tlusty 2000 ).Itcapturesthecomponentsofthecuttingforceintheplaneofcutterrotation.Formillingwithstraightutes,withonlyperipheralcutting,theaxialforcecomponentvanishes.Soatwodimensionalmodelsuces. Inamechanisticforcemodel,thegeometryofthechipareaisdetermined.Thechipareaismultipliedbyappropriatecuttingcoecientstoobtaintherespectiveforcecomponents.Thesecuttingcoecientsarelumpedparameters.Theycapturetheeectsofthematerialpropertiesoftheworkpiecematerialsbeingmachined,tribologicalaspectsoftoolworkinterfacefrictionincludingtheapplicationofcuttinguids,theimpactofthespecictoolgeometrysuchastherakeandclearanceangles,andthedependenceoncuttingconditionssuchasthemachiningparametersused.Suchmodelsneedtobeexperimentallycalibratedforanyspecicapplicationonagivenmachiningset-up,whichisthesubjectofdiscussioninthenextchapter. Themechanismofmachiningofductilemetallicalloysinvolvescutting(shearingatinteratomicplanes)andplowingactions.Shearingactionisthedominantmodeofpowerconsumptioningeneralapplications.Formicromachiningapplications,especiallywhentheorderofmagnitudeofthefeedisthesameastheedgeroundingradiusofthesharpcuttingedges,theplowingactioncanconsumecomparableamountsofpower.Thelumpedparametercuttingcoecientswhicharerelatedtothechipareacapturethecutting(shearing)actionwellandmaybecalled(shear)cuttingcoecients.Plowingisbettermodeledasbeingproportionaltothelengthofthecuttingedgeincontactwiththechip, 57

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Amechanisticmodelpresentedby Sabberwal ( 1961 )relatedthetangentialforcetothechipareawiththecuttingcoecientbeingtheconstantofproportionality.Thecoecientwascalledthespecicpressure,andwasexperimentallydemonstratedtobeindependentofthehelixangleforafewworkmaterials.Thisideaisusedasthebasisofthemodelpresentedinthischapter. Tlusty 2000 ).Theaxialchipareaiszero.Hence,theaxialcuttingforcecomponentvanishes.Theproblemisrendered2-dimensionalforstraightutedcutters.Thecomponentsoftheforceintheplaneneedonlybeconsidered.Figure 4-1 showstheinstantaneousorientationsofthesetwoforcecomponents.Theorientationsoftheseforceschangewiththerotationofthecutter,i.e.,thesecomponentsareexpressedinarotatingcoordinateframeattachedtotheendmill.Eq. 2{4 yieldstheuncutchipareaasAc=bh(=afTsin).Forcecomponentsonthepthtooth,asafunctionofitsangularorientation,p,maybewrittenas 58

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whereKtandKnarelumpedparametercuttingcoecientswhichdependonthetool-workpiece-cuttinguidcombination,aswellasonmachiningparameterssuchascuttingspeedandchipthickness.Theymaybeobtainedbyexperimentduringtheprocessofmodelcalibrationforanyspecicsituation.KtandKnaredenedinarotatingframeofreference,andarecalledthetangentialandnormalcuttingcoecients,respectively. Inaxedframeofreference,thecomponentsofthecuttingforceexperiencedbyasingletoothareobtainedusingthetransformationsshowninFig. 4-2 ,resultinginthefollowingfamiliar( Tlusty 2000 )relationshipsfortheforcecomponentswhenthetoothisinthecut andFx;y0whenp=2[st;ex](toothoutofthecut). ( 1961 )showedthatthemeancuttingcoecientcouldberelatedtotheaveragevaluesofthecuttingforce.Forasingletooth,theaveragedcomponentsofthecuttingforce, Uponsimplication 59

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Inverting wheretheaverageforcecomponentsareobtainedfromexperiments,andallothertermsontherighthandsideoftheaboveequationareknown. where andpisarbitrary(1pN),beingjustareference. ( 1983 )identiedtheratiooftherunouttothefeedrateasanimportantparameterwhichdeterminestheeectofrunoutonthecuttingforcesystem. 60

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4-3 inwhichatwoutedcutterisconsideredforthesakeofsimplicity. LetfT1andfT2betheeectivefeedspertoothexperiencedbythetwoteeth,andfTbethenominal(commanded)feedpertooth.ThefollowingrelationsholdfT1+fT2=2fT withthenotationconventionsuchthatfT1>fT2.SolvingEqs. 4{9 and 4{10 simultaneously and Foracutterwithmorethantwoteeth,theexpressionsforeectivefeedcanbederivedusingsimilararguments. 61

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Sabberwal ( 1961 )isadopted,leadingto wheret;nandt;narecuttingconstantswhosevaluesdependonthecombinationoftoolmaterialandworkmaterial,thespeciccuttinggeometry,aswellascuttingconditions,suchasthetypeofcuttinguidbeingused.Thesecuttingconstantsaretrueconstants,asopposedtothecuttingcoecientswhichhavejustbeenmodeledasbeingfunctionsofchipthickness.Ingeneral,thecuttingcoecientsarefunctionsofmachiningparameters,i.e.,thespeed,feed,andtheaxialaswellasradialdepthsofcut( Gradiseketal. 2004 ).Inthisdocument,thecoecientsareevaluatedexperimentallyforxedvaluesofcuttingspeedandaxialdepthofcut.Hence,theydependonlyonthefeedandradialdepthofcut,which,together,determinethechipthickness. 4-4 .Themethodofcalibratingthemodelthroughexperimentalinputparameterdeterminationisdiscussedinthenextchapter. 62

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Figure4-1. Theuncutchiparea,Ac=bh,isrelatedtotheinstantaneoustangentialforcecomponent,Ft,andtheinstantaneousnormalforcecomponent,Fn.Arighthanded,straightutedendmillhavingtwoteethisillustratedindownmillingconguration. Figure4-2. Transformationofforcesfromarotatingframe(t;n)toaxedframe(x;y). 63

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Idealizationofradialrunoutanditseectonthefeedpertooth.Thenominal(commanded)feedpertoothisfT.Thetotalindicatedreading(T.I.R.)oversuccessiveteethisameasureoftherelativerunoutbetweenthetwoteeth.Therunoutisexaggeratedforillustrativepurposes.Acaseofup-millingwithatwoutedcutter,havingequispacedteeth,isshown. Figure4-4. Forcepredictionexample:2-utedendmillhavingdiameter12:7mm,feed0:150mm/tooth,50%radialimmersion,axialdepth-of-cut0:5mm,cuttingspeed72m/min,up-milling,nominalrunout0:015m.Workmaterial:lowcarbonsteel(HV170).Toolmaterial:solidcarbide.Cuttingconstants:t=7:179;t=0:4145;n=7:006;n=0:5203. 64

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Theaimofthischapteristofamiliarizethereaderwiththemethodofcalibrationofamechanisticforcemodel.Thetwodimensionalcuttingforcemodeldescribedinthepreviouschapterisexperimentallycalibrated.Themodelinputparameterswhicharerequiredtomakeforcepredictionsarethevaluesofthecuttingcoecients(orthecuttingconstants)andthevaluesofrelativeradialrunoutbetweensuccessiveteeth. Experimentaldeterminationintroducesuncertaintiesinthemodelparameters.Theseuncertaintiesintheinputparameterspropagatethroughtheforcemodel.Theanalysisofuncertaintypropagationisthesubjectofthenextchapter.However,thevariancesofthemodelinputparametersaredeterminedinthischapter.Thesevariancesarearesultofrandomandsystematiceectsinexperimentalmeasurements.Thoseeectsareidentied,andtheuncertaintiesattachedtothemodelinputparametersarequantiedinthischapter. 5-1 showstheexperimentalset-upsforcuttingforcemeasurementandradialrunoutmeasurement.Cuttingcoecientswereextractedfordrymachininglowcarbonsteel(hardnessHV170)usinga25%radialimmersionexperiment.Astraightuted,uncoated,solidcarbideendmillofSGS 65

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5-2 showsasketchoftheorientationofthephasorsignalwithrespecttothefeeddirection.Thisprovidesareferenceangularpositionintherotationsothatthetheoreticalandexperimentalforcesignalscanbealigned. Cuttingcoecientswereextractedbasedonasetofexperimentswith25%radialimmersion.ThexedconditionsoftheexperimentsaregiveninTable 5-1 .TheaveragechipthicknesscanbevariedbychangingthefeedbasedonEq. 2{5 .Valuesofcuttingcoecientswererecordedforeachlevelofaveragechipthickness,andlinearregressionwasusedtoestablishrelationshipsbetweenthem.Usingthesecuttingcoecients,forcepredictionsweremadeforupmilling(50%RI),andmixedmode(75%RI),andexperimentallyveriedfordierentvaluesoffeed. 4{2 and 4{7 indicatethatthecuttingcoecientsarepossiblycorrelated.So,thecovariancesassociatedwithKtandKnneedtobeconsidered.Experimentswereconductedatsevendierentfeeds,intherange0:0500:250mm/tooth,thetimetracesofforcecomponentswererecorded,andaverageforcevalues,(fx;fy),werecomputed.Theexperimentalestimatesoftheaveragecuttingforcecomponentsarerepresentedusingthecorrespondinglowercaseletterstodistinguishsamplemeasuresfrompopulationmeasures.Ateachfeedrate,theexperimentwasrepeatedvetimes.Forceswereaveragedoverone 66

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4{7 .Sevensetsofdata,withvereplicationsineachset,werecollectedforttingalinearmodelaccordingtoEq. 4{13 .Twoparameters,theslopeandtheintercept,werettedto35pointsineachregressionyielding352=33degreesoffreedom( TaylorandKuyatt 1994 ,pp.9)forthestandarduncertaintiesassociatedwitheachofthefourcuttingconstantst;nandt;n. Thecuttingconstantst;nant;nofEq. 4{13 maybeobtainedusingmultiresponselinearregression. Kurdi ( 2005 )hassolvedamultiresponseregressionproblemhavingasimilarmathematicalstructure,usingthetheorypresentedby Zellner ( 1962 ).Themethodtsalinearregressionmodelandenablestheevaluationofthevariance-covariancematrixbetweentheresponses,andthevariance-covariancematrixoftherandomerrorintheregressionmodel,whichpermitstheextractionofthevariance-covariancematrixofthecuttingconstants(detailsinAppendix A ).Thisinformationisrequiredfortheevaluationofuncertaintyfromrandomeectsinthemeasuringprocessforthecuttingconstants,whenmakingpredictionsusingtheforcemodel. Zellner's( 1962 )methodofestimatingthemultipleresponseparameters,andthevariancecovariancematrixoftheseestimators,wasusedincalculations.ThettedregressionlinesaredisplayedinFig. 5-3 .Basedontheregression,theestimatedcuttingconstants,t;nandt;n,arecalculatedandgiveninTable 5-2 Thenominalvaluesofcuttingcoecients,obtainedfromtheexperiment,maybecomputedbyusingthenominalvaluesofthecuttingconstantsintheEq. 4{13 withtheaveragechipthickness, 67

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0:0239[mm] Hence,theexpressionsshouldstrictlybeusedonlyforthisrangeof ISO 1995 ).TheNISTTechnicalNote1297,1994Edition( TaylorandKuyatt 1994 ),isamorecompactguidelineauthoredbytwooftheprimaryauthors( Kackeretal. 2007 )oftheGUM.Thisdocumentisopentothegeneralpublic.So,mostreferencesaremadetotheNISTdocument.Thesedocumentsclassifyuncertaintiesintotwodierentcomponents.UncertaintycomponentswhichareevaluatedbystatisticalmethodsareclassiedasTypeA,whereasthosewhichareevaluatedbyothermeansareclassiedasTypeB. Themeasuredvaluesofthelumpedparametercuttingcoecients,andcuttingconstants,aresubjecttorandomeectsowingtorandomvariationsinmaterialproperties,toolchipinterfacefriction,etc.Thisrandomeectcanbequantiedusingstatisticalevaluation.ThevariancesofthecuttingconstantsareestimatedusingaTypeAevaluationwhichisbasedonastatisticalanalysisofthemeasurementdata. Thevariance-covariancematrixofthecuttingconstants,t;nandt;n,asobtainedusingZellner's( 1962 )method,isgiveninTable 5-3 .Thediagonalelementsarethe 68

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Measuringinstrumentscancausesystematicerrors.Inthiscasetherearetwomeasurementdevices.Thedynamometer,amplier,anddataacquisitioncardsystemformsachainformeasurementofthecuttingforcecomponents.Adialindicator,havinganiteresolution,hasbeenusedtomeasuretheradialrunoutofsuccessiveteeth.TheeectsofthesetwomeasuringdevicesonthevariancesofmodelinputparametersarequantiedusingtheTypeBevaluation.Todistinguishbetweenthetwomeasurements,thevariancesofmeasuredforcesaredesignatedasbeingevaluatedusingaTypeB1analysis,andthevariancesofmeasuredradialrunoutsofsuccessiveteetharedesignatedasbeingevaluatedusingaTypeB2analysis. Thesevariancesinthemodelinputparameters(cuttingconstants)duetorandomeectsareobtaineddirectlybyusingtheTypeAevaluation.However,thevariancesinthemodelinputparameters(cuttingcoecientsandeectivefeedpertooth)duetothesystematiceectsarenotobtaineddirectlyusingtheTypeBevaluation.Themeasurementvalueswhichareaectedbysystematicerrorsaretheaverageforcecomponents,andtheradialrunouts.TheTypeB1andB2evaluationsyieldthevariancesofthesemeasurementsaswillbeshowninwhatfollows.Thesevariancesneedtobepropagatedtothecuttingestimatedvaluesofcuttingcoecientsandeectivefeedspertooth.Thetaskofpropagatingthevariancesisaccomplishedinthenextchapter.Inthischapter,onlythevariancesoftheaverageforcesandradialrunoutaredeterminedusingtheTypeB1andB2evaluations,respectively. Dieck 1997 ).TheSchwarzinequality,jcovar(g1;g2)j2var(g1)var(g2),isusedasacrosscheckforthenumericalcalculations( Taylor 1997 ). 69

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Thevariancesinmeasuredvaluesofaverageforcecomponents,u2(fx;y;z),maybecalculatedbasedonestimatesofstandarduncertaintiesinforcemeasurementsprovidedbytheinstrumentmanufacturer( Cadille 2008 )whoiscertiedtoISO9001andISO17025(forcalibration).Accordingtothemanufacturer'scerticate,thetotaluncertaintyofforcemeasurementusingmulticomponentdynamometershavingpiezoelectricchargedevices,iscalculatedasp TheuncertaintyassociatedwithrunoutmeasurementiscapturedusingaTypeB2evaluationasdiscussedin TaylorandKuyatt ( 1994 ).Thedialindicatorusedinrunoutmeasurementresolvesto0:0025mm.Thecosineerrorofthelevertypeindicatorwasneglectedinthisanalysis.Basedonarectangular(uniform)distributionofthehalfinterval,thevarianceoftheradialrunoutmeasurement( TaylorandKuyatt 1994 )is 70

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1962 )method.Thustherandomeectsarecaptured. Thevariancesofcuttingcoecientsandeectivefeedspertooth,duetosystematiceects,arecapturedindirectly.Thevariancesofthemeasuredaveragecuttingforceswereestimatedbasedontheinstrumentmanufacturer'sestimateswhichassumeanunderlyingnormaldistributionforthevariances.Thevarianceofrunoutmeasurementwasestimatedassuminganunderlyingrectangulardistribution.Thesevariancesneedtobepropagatedtothevariancesinestimatedvaluesofthemodelinputparameters(Kt,Kn,andfTi). Theideaofquantifyinguncertaintiesincuttingcoecients,fortheirpropagationthroughmodelsforstabilityandsurfacelocationanalysis,hasappearedintheliterature( Kurdi 2005 ; Duncanetal. 2006 ).However,theideaofquantifyingvariancesinmodelinputparametersforplacingcondenceintervalsonpredictedcuttingforces,andthedeterminationofthevariance-covariancematrixofcuttingconstantstocapturetherandomeectsforthelogarithmiccuttingcoecientmodelchosenhere,arecontributionsofthisresearch.Thepropagationofthemodelinputparameteruncertaintiesthroughthecuttingforcemodelisthesubjectofthenextchapter. 71

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Experimentalcuttingconditions:drymachiningoflowcarbonsteel(HV170)withastraightuted,uncoated,solidcarbideendmill EndmilldiameterCuttingspeedNo.ofteethAxialdepthofcut 12:7mm72m/min2(equispaced)0:5mm Table5-2. MeanvaluesofestimatedcuttingconstantsforcuttingconditionsofTable 5-1 ttnn Goodnessoft:Adj.R2=0:943Adj.R2=0:903 Table5-3. Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 5-1 ttnn t0.0003120.0007020.000249 n0.0071610.002456 n0.000873 72

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Experimentalset-upshowingthestraightutedendmillheldinacolletchuckandmountedontheverticalspindle,thelasertachometer,andtheworkpiecemountedonthedynamometerwhichisheldonthemachinetable.Theset-upforradialrunoutmeasurementusingthedialindicatorisalsoshown. 73

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Experimentalestimationoftheangularorientationofatoothattheentryofacutusingasynchronizingphasorsignal.Acaseofup-millingwith50%radialimmersionisillustrated. Figure5-3. Linearregressionttingofcuttingcoecients,asafunctionoftheaveragechipthickness,fordrymillingoflowcarbonsteel(HV170),usingastraightutedsolidcarbideendmill,having2equispacedteeth,andanominalrunoutof15m.Experimentalpointsaredenotedbysmallcircles. 74

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Experimentalcalibrationoftheforcemodelsintroducesuncertaintiesinthemodelinputparameters.Theanalysisofthepropagationoftheseuncertaintiesthroughtheforcemodelisoneoftheimportantcontributionsofthisresearch.UncertaintypropagationcanbestudiedbyanalyticalmethodsorbysamplingmethodssuchastheMonteCarlo.Inthisdocument,theanalyticalmethodofpropagationhasbeenadoptedwhichisfacilitatedbytheclosedformsolutionsoftheforcemodels.Inthischapter,modelparameteruncertaintiesarepropagatedthroughthetwodimensionalforcemodelforstraightutedendmills.Thisservestodemonstratethetechnique.Thesameprocedurewillbeusedinsubsequentchapterswhereclosedformforcemodelsaredevelopedforhelicalperipheralmilling. Thoughtheredonotappeartobeanystudiesintheliteraturethathavequantiedtheuncertaintiesassociatedwithpredictedinstantaneouscuttingforcesforperipheralmilling,examplesofapplicationsinwhichsuchanuncertaintyanalysisisimportantincludetherealtimemonitoringoftoolwearandtheautomatedsensingoftoolbreakage,bothofwhicharebasedonforcesensingprinciples.Inthemetalcuttingcommunity,thereisinterestinquantifyinguncertaintiesassociatedwithmodelbasedpredictionsofdierentaspectsofthemachiningprocess.Forinstance,inhisinvestigationoftheoptimizationofthemillingprocessesunderuncertainty, Kurdi ( 2005 )hasstudiedthevariancesofcuttingcoecients,andplacedcondenceintervalsonstabilityboundaries. Inthischapter,individualstandarduncertaintiesarepropagatedthroughthemodeltoobtainanestimateofthecombineduncertainties.Thecombineduncertaintyof 75

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TaylorandKuyatt ( 1994 ),whenameasurandYisnotmeasureddirectly,butisfoundfromNotherquantitiesX1;X2;:::XNusingafunctionalrelation thenanestimateofthemeasurand,oroutputquantityY,denotedbythelowercasey,isfoundusinginputestimatesx1;x2;:::;xNfortheNinputquantitiesX1;X2;:::XNusingthefunctionalrelationship Thecombinedstandarduncertaintyofthemeasurementresulty,denotedbyuc(y)istakentorepresenttheestimatedstandarddeviationofy,andisthepositivesquarerootoftheestimatedvarianceu2c(y)givenby @xi2u2(xi)+2N1Xi=1NXj=i+1@g @xi@g @xju(xi;xj)(6{3) whereu2( )arevariancesandu( TaylorandKuyatt ( 1994 )refertoEq. 6{3 asthelawofpropagationofuncertainty.Thecombinedstandarduncertaintiesofthepredictedforcesarederivedusingthislawforwhichthefunctionalrelation`g'isgivenbyEq. 4{2 ,whichdenesthefunctiongoverningtheforcecomponents.Thesensitivitycoecientsarethepartialderivativesoftheforcecomponentswithrespecttothemodelinputparameterswhichhaveuncertaintiesattachedtothem,namely,thecuttingconstants,thecuttingcoecients,andtheeectivefeedrates. 76

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5.3 ofchapter 5 ,itwasobservedthatthevariancesofmodelinputparametersduetorandomeectsweredirectlyavailablefromtheTypeAevaluationintheformofavariance-covariancematrixforthecuttingconstantst;nandt;n.However,thevariancesofmodelinputparametersduetosystematiceectswerenotavailabledirectlyfromtheTypeB1andB2evaluations.TheTypeB1evaluationprovidedanestimateofthevariancesofthemeasuredaverageforcesfEq. 5{4 5{5 4{2 and 4{8 4{7 yieldsthefollowingsensitivitycoecientsforuseinpropagationoftheuncertaintyinaverageforcemeasurementstotheuncertaintiesincuttingcoecients Fx@Kt=@ Fy@Kn=@ Fx@Kn=@ Fy9>>>>>>>=>>>>>>>;=2(exst) UsingtheabovesensitivitiesinEq. 6{3 thevariancesofcuttingcoecients,duetothesystematiceects,canbeobtained Thevaluesofu(fx;y)intheEq. 6{5 abovearesetat1:207%ofthenominalvaluesoftheaverageforcecomponentsbasedontheinstrumentmanufacturer'sestimatesasexplainedindetailinchapter 5 .Forsimplicity,anypossiblecorrelationbetweenKtandKnfrommeasurementchannelcrosstalkisneglected,i.e.,u(kt;kn)issettozero. 77

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4{11 and 4{12 yieldthefollowingsensitivitycoecientsforuseinpropagationoftheuncertaintyinrelativerunouttotheuncertaintiesineectivefeeds Equations 5{5 and 6{6 maybeusedinEq. 6{3 toestimatethevariancesoftheeectivefeedforeachindividualtooth 4{2 .Hence,theirsensitivitiesmustbecomputed. whereKtandKnareparameterizedint;nandt;n,accordingtoEq. 4{13 78

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4{2 SensitivitycoecientswithrespecttotheeectivefeedsareusedinthedeterminationofTypeB2componentuncertainties.ThesearefoundusingEqs. 4{2 2{5 ,and 4{13 ThesensitivitiesinEqs. 6{9 6{11 arefunctionsofp.Forillustrativepurposes,theyareplottedinFig. 6-1 foraparticularcombinationofmachiningparameters. 5-2 Theestimatedvariancesofthecuttingconstantsfdenotedbyu2( )gandtheestimatedcovariancesofthecuttingconstantsfdenotedbyu( 5-3 .Theseuncertaintieshavetobepropagatedtothepredictedforcesusingthesensitivities,sij,givenbyEq. 6{9 Thesesensitivitiesarethemselvesafunctionofthecuttingconstants.hence,theyareevaluatedattheestimatedparameterst;nandt;n.Thecombinedcomponentuncertaintiesofpredictedforcesforthepthtooth,ucA(fx;y),duetotheuncertaintiesinthecuttingconstants,areobtainedusingthesensitivities,sij,inthepropagationlawEq. 6{3 79

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6{11 basedontheTypeB2evaluation.TheseuncertaintiesmaybepropagatedtothepredictedforcesusingEq. 4{2 andthesensitivitiesexpressedinEq. 6{10 toyieldthecombinedcomponentuncertaintiesofpredictedforcesforthepthtooth,ucB1(fx;y),byapplyingEq. 6{3 6{7 and 6{8 basedontheTypeB2evaluation.TheseuncertaintiesmaybepropagatedtothepredictedforcesusingEq. 4{2 andthesensitivitiesexpressedinEq. 6{5 toyieldthecombinedcomponentuncertaintiesofpredictedforcesfor 80

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6{3 wherethesubscriptonfTpisdroppedforconvenience. Turzeniecka 2000 ; Fotowicz 2006 ). TaylorandKuyatt 1994 ,pp.4),anda2ucA(fx;y)widthaboutthenominalvaluedenesanintervalinwhichthemeasurementresultisbelievedtoliewithalevelofcondenceofapproximately95%,i.e.,thecoveragefactorfortheexpandeduncertaintyisA=2. IfucA(fx;y)themselveshavenon-negligibleuncertainty,aconventionalproceduremaybeusedtoproduceacoveragefactorthatproducesanintervalhavingtheapproximatelevelofcondencedesired.TheeectivedegreesoffreedomforthecombineduncertaintiesareestimatedbasedontheWelch-Satterthwaite(W-S)expression( TaylorandKuyatt 81

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,pp.8-9,Eqs.B-1andB-2)asfollows and wheredfeffjucA(fx)anddfeffjucA(fy)aretheeectivedegreesoffreedomforthecombinedTypeAcomponentuncertaintiesoffxandfyrespectively.Thesuperscript?,usedinthetermsu?cA(fx;y)inEqs. 6{15 and 6{16 ,indicatesthatnocorrelationisconsideredamongtheinputsincalculatingthesecombineduncertainties,i.e.thecovariancetermsinEq. 6{12 arenotconsidered,assuggestedby Willink ( 2007 )inhisgeneralizationoftheW-Sexpressionforusewithcorrelateduncertaintycomponents. Todeterminethecuttingconstantsinchapter 5 ,experimentswereconductedatsevendierentfeeds.Ateachfeed,vereplicationsweretaken.Twoparameters(slopeandintercept)werettedto75=35pointsineachregression,yielding352=33degreesoffreedomforthestandarduncertaintiesassociatedwitheachofthefourcuttingconstantst;n;andt;n( TaylorandKuyatt 1994 ,pp.9). ThecoveragefactorfortheTypeAcomponentexpandeduncertaintyisdeterminedbasedontheStudent'st-distributionforanydesiredcondenceinterval,andmaybereadotheTableB.1in TaylorandKuyatt ( 1994 )knowingthevalueofdfeffjucA(fx;y).Theeectivedegreesoffreedomarefunctionsoftheangularpositionofthepthtooth,p.Therefore,thecoveragefactorisalsoafunctionofp.ThefunctionaldependenceisillustratedforanexamplecaseinFig.( 6-2 ).Forthisparticularcase,itmaybenotedthattheeectivedegreesoffreedomarequitelarge(31),sothatthecoveragefactorapproaches2.Byinspectionofthegure,A'2:04maybetakenasagoodconservative 82

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FortheTypeB2uncertaintiesduetosystematiceectsoftherunoutmeasuringdevice,therectangulardistributionisassumedwithinnitedegreesoffreedom.BasedonthisthecoveragefactorfortheTypeB2componentexpandeduncertaintyisB2'1:65fora95%condenceinterval. Turzeniecka ( 2000 )hassuggestedvariousapproximatemethodsofcalculatingtheexpandeduncertaintyinsuchsituations.ThebestchoicedependsontherelativemagnitudesoftheTypeAandTypeBuncertainties.Intheproblemathand,thisratioisnotxed.ThisfactisclariedwithanexampleshowninFig. 6-3 .Moreover,theratioisafunctionofp,andprocessparameterssuchasthefeedandradialimmersion.Forsuchasituation, Turzeniecka ( 2000 )hassuggestedtherootsumofsquares(RSS)methodasagoodsolution.TheexpandeduncertaintyistheRSS ( 2000 )hasnameditthevectorsummethod 83

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whereFx;yarecalculatedbasedonEq. 4{2 Theaboveanalysisconsiderstheforcesanduncertaintiesassociatedwiththeengagementofasingletooth.Formultipletoothengagement,theforcesduetoeachtootharemerelysummed.However,theuncertaintiesmustbeobtainedbytherootsumofsquaresmethodbyconsideringthesumofthesquaresofeachofthecomponentcombineduncertaintiesforeverytoothindividually,andtakingthesquarerootofthisoverallsum. Alltheuncertainties,combinedaswellasexpanded,arefunctionsoftheangularpositionofthepthtooth,anexampleofwhichisgiveninFig. 6-3 .ForthissameexamplecondenceintervalsareplacedonpredictedforcesandshowninFig. 6-4 6-5 6-9 84

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Thoughtheuncertaintyanalysiswasperformedforacuttingforcemodelapplicableforstraightutedendmills,theprocedureisquitegeneral.Forhelicalendmills,thebasicforcemodelwilldier,buttheuncertaintyanalysismethodwillremainexactlythesame.Thistechniqueofuncertaintypropagationthroughamillingforcemodelisoneofthecontributionsofthisresearch.Theprocedurefordeterminingtheuncertainties,andthecorrespondingcondenceintervals,isreadilyimplemented.Theabilitytoprovideadefensibleuncertaintystatementtoaccompanycuttingforcepredictionshasapracticalbenet.Itenablestheprocessplannertodecidetheusefulnessofmodelbasedforcepredictionsinanyspecicapplication.Thesuccessoftheanalyticuncertaintyanalysisprocedurehingedonthefactthatclosedformanalyticalexpressionswereavailableforthecuttingforcesaswellascuttingcoecients.Evenso,forthetwodimensionalforcemodelforstraightutedcutters,theuncertaintyanalysiswasquitealgebraicallyinvolved. Intheforthcomingchaptersclosedformanalyticalsolutionswillbedevelopedforhelicalperipheralmilling.Themodelswillbemechanistichavingthesamestructureastheforcemodeldescribedintheselastthreechapters.Theclosedformsolutionswillalsopermittheuseofthesameuncertaintypropagationprocedureaswasadoptedinthischapter. Table6-1. Summaryofexperimentalconditionsusedforvericationofforcepredictions,holdingthecuttingconditionsofTable 5-1 xed ResultsdisplayedinFeed(mm/tooth)Radialimmersion(%)Up/downmillingormixedmode(>50%RI) Figure 6-5 0:15050Upmilling Figure 6-6 0:10050Upmilling Figure 6-7 0:05050Upmilling Figure 6-8 0:20075mixedmode,h=0attheentry Figure 6-9 0:15075mixedmode,h=0attheentry 85

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Variationofsensitivitycoecientswithangularpositionofthepthtooth:feed0:150mm/tooth,50%radialimmersion,up-milling,nominalrunout15m.OtherconditionsasinTable5-1. 86

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VariationoftheeectivedegreesoffreedomofcombineduncertaintiesofTypeA,andthecorrespondingcoveragefactorfora95%condenceinterval,withangularpositionofthepthtooth:feed0:150mm/tooth,50%radialimmersion,up-milling,nominalrunout15m.OtherconditionsasinTable5-1. Figure6-3. Variationofcomponentcombineduncertainties,andoverallexpandeduncertaintyfora95%condenceinterval,withangularpositionofthepthtooth:feed0:150mm/tooth,50%radialimmersion,up-milling,nominalrunout15m.OtherconditionsasinTable5-1. 87

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Exampleillustratingtheplacementof95%condenceintervalboundsonpredictedcuttingforces:feed0:150mm/tooth,50%radialimmersion,up-milling,nominalrunout15m.OtherconditionsasinTable5-1. Figure6-5. Predictedvs.experimentalforcesignals:feed0.150mm/tooth,50%radialimmersion,up-milling.Nominalrunout15m.OtherconditionsasinTable 5-1 88

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Predictedvs.experimentalforcesignals:feed0.100mm/tooth,50%radialimmersion,up-milling.Nominalrunout15m.OtherconditionsasinTable 5-1 Figure6-7. Predictedvs.experimentalforcesignals:feed0.050mm/tooth,50%radialimmersion,up-milling.Nominalrunout15m.OtherconditionsasinTable 5-1 89

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Predictedvs.experimentalforcesignals:feed0.200mm/tooth,75%radialimmersion,cutstartswithh=0.Nominalrunout15m.OtherconditionsasinTable 5-1 Figure6-9. Predictedvs.experimentalforcesignals:feed0.150mm/tooth,75%radialimmersion,cutstartsasup-milling.Nominalrunout15m.OtherconditionsasinTable 5-1 90

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Inthischapteranewmodelforcuttingforcesinhelicalperipheralmillingisdeveloped.Themodelhasdistinguishingfeaturesthatimbueitwithcertainadvantageswhencomparedwithothermodelsavailableintheliterature.Sincethisisamechanisticmodel,thechipareahastoberelatedtothecuttingcoecients.Themodelisthreedimensionalbecausetheaxialforcedoesnotvanishinhelicalmilling.Itisarigidmodelsincetheeectsoftoolorpartdeectionsonthechipareaareignored.Forsimplicity,theeectofradialrunoutofsuccessiveteethisignored.Thateectwillbeincorporatedinthenextchapter. Figure7-1showstheschemewhichischosentorelatethethreedierentialcuttingforcecomponentstotherespectiveareas.Thedierentialprojectedfrontalchiparea,dAf,isrelatedtothedierentialtangentialforcecomponent,dFt,andthedierentialnormalforcecomponent,dFn.Thedierentialprojectedaxialchiparea,dAa,isrelatedtothedierentialaxialforcecomponent,dFa.Incontrasttothetypicalapproach(e.g. EnginandAltintas 2001 )therakefacechiparea(thegrayregioninFig.7-1)isnotusedinthemodel.Aparticularfeature ofthemodelisthattheaxialprojectedchipareaisrelatedtotheaxialforcecomponent,unlikethegeneralpracticeofrelatingtherakefacechiparea.Thisensuresthattheaxialforcecomponentautomaticallyvanishesasthehelixanglegoestozeroinastraightutedcutter.Itisnotnecessarytoforcetheaxialcuttingcoecienttogotozeroforstraightutedcutters. Thethreedimensionalforcemodelforhelicalperipheralmillingpresentedinthischapterdiersfrompriorworkinseveralways.Eachforcecomponenthasasingle,closedformexpressionwhichisvalidfortheentirecutterrotation,andthestructureoflinearizedcuttingcoecientsisretained.Anyarbitraryvalueofradialimmersion,andhenceevensmallimmersions,canbeusedtoexperimentallydeterminethecuttingcoecients. 91

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whereKtc,KncandKacarelinearizedcuttingcoecientswhichmaybeobtainedexperimentallyduringtheprocessofmodelcalibrationforanyspecicsituation. Therotating(tangentialandnormal)componentsarerelatedtocomponentsinaxedcoordinateframeviaarotationmatrix(Fig. 4-2 ),whiletheaxialcomponentremainsdecoupled Thereaderwillnotethatthevalueofthedeterminantofthe33matrixintheaboveequationequals1.The(x;y;z)coordinatesformalefthandedsystem.Thisisaresultofthewaythe(x;y)coordinatesarelaidoutinFig. 4-2 .Thesenseofthecutterrotationisclockwise.Thepositivezaxisisalignedwiththesenseofrotation.Thischoiceisarbitrary,andhasnophysicalsignicance. SubstitutingfromEq. 7{1 andintegratingyieldsthetotalforces wheretheintegrationsarecarriedoutovertheappropriatelimits.Theselimitsareexplicitlyshowninasubsequentstep,afterapplyingcertaintransformations. 92

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7{3 ,alocalcoordinate,,hasbeeninvokedinFig. 7-2 .Byinspectionofthegeometry TLd(7{4) Again,fromFig. 7-2 ,thedierentialelementoffrontalchiparea,dAf,is wherethelocalchipthicknessatisobtainedusingthecircularpathapproximation,h=fTsinfEq. 2{2 EliminatingdusingEq. 7{4 yieldsdAfasafunctionof dAf=bfT BasedonFig. 7-3 ,thedierentialelementoftheprojectedaxialchiparea,dAa,is ThethreeintegralsRsindAf,RcosdAfandRdAamaynowbecomputed:ZsindAf=bfT ThesethreeresultsmaybesubstitutedintoEq. 7{3 toobtain,uponsimplication 93

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2[1cos(T+L)sinc(TL)](7{13) and wherethesamplingfunction(sinc)hasbeendenedearlierinEq. 3{18 Intheaboveequations,thevariablesL,Tandbarefunctionsofp.ThefunctionalexpressionsforL,Tandbhavebeenderivedinchapter 3 .TheEq. 7{11 representstheforcecomponentsonasingletooth,asafunctionofp,inclosedform. 4.3 thecomponentsofthetotalforceareasummationoftheforcecomponentsoftheindividualteeth where andpisarbitrary(1pN),beingjustareference. Foruniformlyspacedteeth,ihasaconstantvalue.Fordierentialtoothspacingthetoothpitchangles,i,aredirectlyobtainedfromthespecicationsoftheendmill.The 94

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Martellotti ( 1941 )proposedthattheaverageundeformedchipthicknesscouldberelatedtothecomponentsofthecuttingforce. Sawin ( 1926 ), Salomon ( 1926 ),and Sabberwal ( 1961 )showedthatthecuttingcoecientvarieswiththechipthickness. Thecoecientsalsodependonotherprocessparameterssuchascuttingspeed( ShinandWaters 1997 )andtoolgeometry( Jayarametal. 2001 ).Forsimplicity,thecuttingspeed,axialdepthofcut,andtoolgeometryarekeptxedintheexperimentsreportedinthispaper.Hence,theresultsreportedhereholdonlyforthespecictypeoftoolgeometryused,thestatedcuttingspeed,andtheaxialdepthofcutusedintheexperiments. Basedon Sabberwal ( 1961 ),thecoecientsmaybeexpressedasexponentialfunctionsoftheaveragedmeanchipthickness, wheretc;nc;acandtc;nc;acarecuttingconstants.Thevaluesoftheseconstantsdependonthecombinationoftoolmaterialandworkmaterial,thespeciccuttinggeometry,aswellascuttingconditions,suchasthetypeofcuttinguidbeingused. Thecuttingcoecientsdescribedabovecanbeexperimentallyextractedbasedonasmallsetofcuttingtests.ThecoecientsKtc;nc;accorrespondingtoagivenfeedper 95

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3{19 .TheexerciseisrepeatedforasetofdierentvaluesoffT.Thus,amappingisestablished,generatingafunctionaldependenceofKtc;nc;acon 7{16 Forasinglehelicaltooth,thecomponentsofthecuttingforcemaybeaveragedoveronecutterrevolution.Theaveragedcomponentsofthecuttingforce, 22Z00BBBB@b1b20b2b1000D31CCCCAdp3777758>>>><>>>>:KtcKncKac9>>>>=>>>>;(7{17) Usingshorthandnotation where Solvingforthecoecientsyields fT0BBBB@I1 where 7{19 haveclosedformexpressionsfusingEqs. 7{12 7{14 andEqs. 3{6 3{8 or 3{9 3{11 96

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7-4 forthepurposeofillustration. Inthissection,aprocedurehasbeendevelopedfortheidenticationofcuttingcoecients.Theformulationisindierenttowhetherthecuttingcoecientsareafunctionofhelixangle.Theexpressionsarereadilycomputedforanyvalueofradialimmersion.Hence,lowimmersiontestscanbeusedtodeterminethecuttingcoecients.Foranygivencombinationofworkpiecematerial,toolmaterial,toolgeometry,andcuttingconditionssuchasuseofcuttinguid,thecuttingcoecientsmaybeexperimentallyderived.Thecoecientsarevariableastheyareafunctionofchipthickness.So,cuttingconstantshavebeeninvokedwhicharetrueinvariants.Intheforthcomingsectionstheconstantsareexperimentallydeterminedandtheforcemodelisveried. 4 .Next,theresultsoftheanalyticalsolutionarecomparedwithanumericalsolutionavailableintheliteraturefrom Tlusty ( 1985 2000 ). (7{21)L+T=8><>:2p;p2(st;ex)0;elsewhere (7{22)b=8><>:a(aconstant);p2(st;ex)0;elsewhere (7{23) 97

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7{11 resultsinthefamiliarrelationshipsfEq. 4{2 ThisveriesthattheexpressionsinEq. 7{11 arecorrectforthedegeneratecase,=0.Incomputerimplementation,theseequationsarenotwellbehaved,duetothecottermwhichgrowswithoutboundas!0.Theformulationisrenderednominallyuselessfor=0.However,theequationsworkaccuratelyforextremelysmallvaluesofhelixanglesothatthehelixangleiszeroforallpracticalpurposes.Figure 7-5 demonstratestheresultsofcomputationforastraightutedendmillusing=0:0000001.Byinspectionoftheforcesignalsitcanbeconcludedthattheresultsofthisanalyticalformulationareaccurateforstraightutedcutters. ( 1985 2000 )haspresentedanumericalmodel,basedonthemethodproposedby DeVorandKline ( 1980 )and Klineetal. ( 1982 ),inwhichthecutterisbrokenupintoinnitesimalslabsaxially.Theeectsofeachslabarenumericallyintegratedtoobtainthecompleteforce.Tlusty's( 2000 )algorithmwasusedtogeneratethenumericalresultsforhelicalmillsdisplayedinthissection. Figure 7-6 showsacomparisonofthenumericalsolutionandtheproposedanalyticalsolutionfora30helixcutterfor75%radialimmersion,withthecutstartingwithh=0.Theresultsagreeveryclosely. TheresidualsareplottedinFig. 7-7 whichshowsthatthedierencebetweenthenumericalandanalyticalsolutionsisnegligible.Thisisarepresentativeresult. Tlusty ( 1985 2000 )haspublishedsuchplotsfordierentcombinationsofradialimmersionandup/down-milling.ThosenumericalresultsarereproducedinFig. 7-8 using 98

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Whenslottingwithfourteeth,thetotaltoothengagementangleis180andhalfthecutterisalwaysengaged.Thisimpliesthattwoutesarealwaysimmersedinthecut.Iftheyareequispaced,thelagofcorrespondingpointsonsuccessiveteethis90.Therefore,thetotalcontributionofthetwoteethtothechipthicknessisconstant.Thetotalaxialengagementoftheteeththatareincontactwiththechipalsoremainsconstant.Itisexpectedthatthetotalforceintheplanehq 7-8 showtheresultsofslotmillingwithacutterhaving30helixangleand4equispacedteeth.Theanalyticalsolutionsshowtheexpectedconstantforcemagnitude. 5 exceptforthedetails. Figure 7-9 showstheexperimentalset-up.Cuttingcoecientswereextractedfordrymachiningthealuminumalloy6061-T6usinga50%radialimmersionexperiment.AKennametalsolidcarbideendmill,styleHPF37A,withtitaniumdiboridecoating,having45helix,12.7mmdiameter,and3utes,wasused.ASchunkTRIBOSrpolygonalclampingtoolholderwasemployedtominimizerunout.Spindleadaptionwashollowshank,taperandfacecontactHSK-63A.Uponclampingthetoolinthespindle,norunoutwasmeasurablewhentheendmillwasindicatedusingadialgagehavingaleastcountof2:5m(0.0001"),whichmeansthatthetheoreticalformulationthatassumeszerorunoutcanbeexperimentallyveried.Otherexperimentalconditionswerethesameasthoseusedfortheexperimentdescribedinchapter 5 .Figure 7-9 alsoshowsasnapshotofasetofrecordedforcesignals. 99

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7-2 .Areasonablylargespectrumof Gradiseketal. ( 2004 )haveshownthatthecuttingcoecientsareinsensitivetothemillingconguration,beingthesameforupmilling,downmilling,andmixedmodemilling.Usingcoecientsextractedbasedondownmillingexperiments,forcepredictionshavebeenmadefordownmilling,upmilling,andmixedmode(>50%radialimmersion),andexperimentallyveriedinaforthcomingsection. Equations 7{11 and 7{20 indicatethatthecuttingcoecientsintheplane(KtcandKnc)arepossiblycorrelated.Theaxialforceandtheaxialcuttingcoecientareindependent.InconsideringtheuncertaintiesassociatedwiththepredictionsofFxandFy,thevariances,aswellasthecovariancesassociatedwithKtcandKncneedtobeconsidered,i.e.,itisamultiresponse,multivariateproblem.Inpredictingtheaxialforcecomponent,thevarianceassociatedwithKacsuces,i.e.,itisaunivariateproblem. The50%radialimmersionexperimentswereconductedateightdierentfeedrates,equispacedwithintherange0.025-0.200mm/tooth.Timetracesofforcecomponentswererecorded,andaverageforcevalues,(fx;fy;fz),werecomputed.Ateachfeedrate,theexperimentwasrepeatedvetimes.Forceswereaveragedoveronerotation,onapertoothbasis.Tocontrolthevariability,theaveragewastakenover100successiverotations.Valuesofcuttingcoecients,correspondingtoeachfeedrate,werederivedusingEq. 7{20 .Thus,eightsetsofvedatapointseachwerecollectedtotalinearmodelfEq. 7{16 100

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TaylorandKuyatt 1994 ,pp.9)forthestandarduncertaintiesassociatedwitheachofthefourcuttingconstantstc;ncandtc;nc. Thecuttingconstantstc;ncandtc;ncofEq. 7{16 maybeobtainedusingmultiresponselinearregression,whereasthecuttingconstantsacandaccanbeobtainedusingsimplelinearregression.Forthemultiresponselinearregression,Zellner's( 1962 )methodofestimatingthemultipleresponseparameters(thetwo-stageAitkenestimators),andthevariancecovariancematrix(themomentmatrix)oftheseestimatorswasused,followingthesamemethodasoutlinedinchapter 5 .ThelinearregressionlinesareplottedinFig. 7-10 .Theestimatedcuttingconstantstc;ncandtc;ncaregiveninTable 7-3 .Alineartwasnotfoundsuitablefortheaxialcuttingconstants(R2=0:144inlinearregression).ThemeanvalueofKac=395:7N/mm2wastted,resultinginthevaluesac=ln(395:7)=5:981andac=0. Thus,foranyparticularvalueoftheaveragedmeanchipthickness, (7{25)Knc=enc( (7{26)Kac=395:7[N/mm2] (7{27) 5 ,applieshere.Inthiscasetherearetwosourcesofuncertainty.Therstistherandomeectsofthevariationofworkpiecematerialproperties,toolchipinterfacefrictioncoecients,etc.,onthecuttingconstants.ThiseectisevaluatedusingaTypeAanalysis.ThesecondsourceisthesystematiceectoftheforcemeasuringinstrumentationchainwhichisevaluatedusingaTypeB1analysis.Therunoutisnotconsideredinthisversionofthemodel,thoughitwasmeasured(andfoundtobenominallyzeroontheexperimentalset-up). 101

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7-4 .Thediagonalelementsarethevariances,andtheo-diagonalelementsaretherespectivecovariances.Additionally,thevarianceoftheindependentcuttingconstant,ac,duetorandomeects,wasfoundtobe0.00247562. SystematiceectsinthemeasurementprocessareassessedusingaTypeB1analysisinthesamemannerasinchapter 5 ,exceptfordierencesinthedetails.Equation 7{20 yieldsthefollowingsensitivitycoecientsforuseinpropagationoftheuncertaintyinaverageforcemeasurementstotheuncertaintiesincuttingcoecients Fx@Ktc=@ Fy@Knc=@ Fx@Knc=@ Fy@Kac=@ Fz9>>>>>>>>>>=>>>>>>>>>>;=2 fT8>>>>>>>>>><>>>>>>>>>>:I1=(I21+I22)I2=(I21+I22)I2=(I21+I22)I1=(I21+I22)1=(DI3)9>>>>>>>>>>=>>>>>>>>>>;(7{28) Usingtheabovesensitivitiesthevariancesofcuttingcoecients,duetothesystematiceects,canbeobtained Thevaluesofu(fx;y;z)intheEq. 7{29 abovearesetat1:207%ofthenominalvaluesoftheaverageforcecomponentsbasedontheinstrumentmanufacturer'sestimatesasexplainedindetailinchapter 5 .Forsimplicity,anypossiblecorrelationbetweenKtcandKncfrommeasurementchannelcrosstalkisneglected,i.e.,u(ktc;knc)issettozero. 102

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7{11 togetherwithEq. 7{16 wherethecuttingcoecientsareparameterizedinthecuttingconstantsaccordingtoEq. 7{16 Thesesensitivitiesareexpressedintermsofthecuttingcoecientsintheaboveequationbecausetheexpressionsremaincompactwhenwritteninthismanner.Theyarefunctionsoftheangularpositionofthepthtooth,p. Thevariancesofcuttingconstantsduetorandomeects,availablefromtheTypeAevaluation,maybepropagatedtothepredictedforcesusingEq. 7{11 andthesensitivitiesexpressedinEq. 7{30 toyieldthecombineduncertaintiesucA(fx;y;z)ofpredictedforcesfor 103

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6{3 {z }estimatedvariances+20BBBB@s11s12s11s13s11s14s12s13s12s14s13s140s21s22s21s23s21s24s22s23s22s24s23s240000000s35s361CCCCA8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:u(tc;tc)u(tc;nc)u(tc;nc)u(tc;nc)u(tc;nc)u(nc;tc)u(ac;ac)9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;| {z }estimatedcovariances(7{31)wherelowercasetc;nc;acandtc;nc;acareinputestimatesforthevaluesofinputquantitiestc;nc;acandtc;nc;ac,respectively,andsijarethesensitivitycoecientsevaluatedattheestimatedparameterstc;nc;acandtc;nc;ac. 7{11 104

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7{29 basedontheTypeB1evaluation.TheseuncertaintiesmaybepropagatedtothepredictedforcesusingEq. 7{11 andthesensitivitiesexpressedinEq. 7{32 toyieldthecombineduncertaintiesucB1(fx;y;z)ofpredictedforcesforasingletoothbyapplyingEq. 6{3 TaylorandKuyatt 1994 ),i.e.,A=2. TheTypeB1evaluationofthestandarduncertaintiesofdynamometerforcemeasurementswascarriedoutbasedonthemanufacturer'scerticateundertheassumptionofanormaldistributionforthemeasurementsandtheircombineduncertainties.Itisassumedthatthemanufacturer'sestimateisbasedonasucientlylargenumberofobservationstojustifythechoiceoftheusualcoveragefactorof2fortheexpandeduncertaintyfora95%condenceintervalbasedontheStudent'st-distribution,i.e.,B1=2.Forboththesourcesofuncertainty,theunderlyingdistributionsarenormal.Theexpandeduncertaintymaybecomputedastherootsumofsquaresofthecomponentexpandeduncertainties.Thetotalexpandeduncertaintyis 105

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whereFx;y;zarecalculatedbasedonEq. 7{11 7-5 .Threedierentcuttingcongurationsweretestednamely,down-milling,up-milling,andmixedmodemilling(>50%radialimmersion).ThecuttingspeedandaxialdepthofcutwerekeptxedfTable 7-2 Figures( 7-11 7-20 )showtheresultsofforcepredictionsinwhichupperandlowerboundsof95%condenceintervalofpredictedforcesaredisplayedgraphically,alongwiththeexperimentallyobtainedsignals,whichverifythepredictions.Reliablepredictionswereobtainedforvariouscombinationsoffeed,radialimmersion,andupordownmillingcongurations.Thehighdelityoftheaxialforcecomponentpredictionindicatesthatconsideringtheaxialforcetobeproportionaltotheprojectedaxialareaisjustied.Thecuttingcoecients,derivedonthebasisofdownmilling,sucedforupmillingandmixedmodepredictions,asanticipatedin Gradiseketal. ( 2004 ). Ingeneral,thepredictionswerefoundtobequitereliable.However,certaindecienciesarenoticeable.Thepeakforcesarenotpredictedverywellinquiteafewofthecases.Themodeloverpredictsthepeakforces.Forhighimmersioncase(suchas75%RI),theexperimentalsignalsaresomewhatskewedwithrespecttothepredictedforcepatterns.Thesedecienciesofthemodelareaddressedinchapter 10 whereamodicationisproposedwhichcorrectstheseshortcomings. 106

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3-3 Gibbs 1898 1899 )ofthesefunctions.ThepartialFouriersumsdonotconvergetothefunctionvalues.Atthepointofdiscontinuity,theconvergenceistothemidpointofthejump,butthereconstructedfunctionoscillateswildlyintheneighborhoodofthejumpdiscontinuity.TakingalargenumberoftermstoformthepartialFouriersumdrivestheoscillationsveryclosetothejumpdiscontinuities.Theoscillationsdonotvanish,nomatterhowmanytermsaresummed.Intheneighborhoodofthejumpdiscontinuity,theyovershootthetargetbyabout9%( BrandoliniandColzani 1999 ),independentofthenumberoftermsinthepartialFouriersum.Thus,functionvaluesofreconstructedLandTareuncertaintowithin9%intheneighborhoodofthejumpdiscontinuities.Theproblemdoesnotaectthereconstructionofbasithasnojumpdiscontinuities.Forthisreason,thecomputedvaluesofFxandFyareinaccurateinasmallneighborhoodofthejumpdiscontinuitiesofLandT. Figure 7-21 showstheeectsofincreasingthenumberoftermsinthepartialFouriersums.EventhoughLandToscillate,theimpactonFx,Fy,andFtotalisseentobeverysmallbecausethechipwidthgoestozeroatthepointswhereLandToscillate.Inallthecalculationsofforcethathavebeenmadebytheauthors,thesmoothnessofFx,Fy,andFtotalwasneverobservedtobesignicantlyaectedbytheGibbs-WilbrahamphenomenonaectingLandT.SincethechipwidthalwaysgoestozeroatthepointofthesejumpdiscontinuitiesofLandT,thisfactordoesnotplayasignicantroleinpracticalimplementationofthecode. 107

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Figure 7-22 displaystheconvergencecharacteristicsoftheanalyticalFouriersolution,includingtheprocessingtime.Convergenceisachievedwithlessthan400termsinthepartialFouriersum.ThecomputationalburdenfortheHeavisidemethodiscomparativelylessduetotheabsenceofseriessummation. TheHeavisidestepfunctionisimplementedinavarietyofwaysinsoftware.Somepackagesdonotallotafunctionvalueatthediscontinuity.Othersprovideanaveragevalueofhalfatthispoint.Theprogrammermustbeawarehowthisaectsthecomputation. 108

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7-5 Forapplicationsrequiringsymbolicmanipulations,theHeavisidemethodisconvenientandcompact.Ifaccuracyrequirementsarehigh,theFouriermethodbecomesimpracticalbecausethenumberoftermstobemanipulatedisverylarge. Theresultshavebeenshowntobevalidfortheentireparameterspacecoveringhelicalaswellasstraightutedperipheralmilling,partialorfullimmersioncutting,multipleteethinthecut,aswellasforup-milling,down-millingandslottingbycomparisonwithestablishednumericalmethods. Experimentalvalidationwasperformedbycalibratingthemodelforthealuminumalloy6061-T6usinga45helixendmill.Thedelityofpredictionswasshowntobehigh.Inderivingtheaxialcuttingforcecomponent,theprojectedaxialchipareahasbeenconsidered,whichleadstogoodpredictionsoftheaxialcuttingforce. 109

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Incomputercalculations,forthedegeneratecaseofstraightutecutters,averysmallhelixanglemustbespecied,astheexpressionsarenotwellbehavedforzerohelixangle.However,analytically,ithasbeenshownthattheexpressionsreducetothefamiliarrelationshipsforzerohelix,showingthattheformulationremainsvalid.Therefore,theexpressionsmaybeusedinapplicationsinvolvinganalyticmanipulations,suchasndingderivativesorintegrals. Thisisarigidmodelwhichdoesnotaccountforradialrunoutortheeectsoftoolorworkpiecedeectionsonthecuttingforce.Theexperimentalvericationwasdonewithanendmillhavingteethwithequalpitch(angularspacing),i.e.,thethreeteethwerespacedat120toeachother.Itwasnotedthatthemodeloverpredictspeakforces,anddoesnotpredicttheskewnessoftheforcesignalswhichareexperimentallyobservedespeciallyinhighimmersionexperiments.Thesedecienciesareaddressedinchapter 10 wherethemodelismodied.However,beforemakingthemodication,proceduresforincorporatingtheeectsofdierentialtoothspacingandradialrunoutofsuccessiveteethareconsidered.Extensionsofthismodeltoincludethesetwoeectsareincludedinthenexttwochapters,respectively. 110

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Parametersinthenumericalsolutionof Tlusty ( 1985 2000 ) EndmilldiameterHelixangleFeedpertoothNo.ofteethCuttingcoecientsKtcKnc Experimentalcuttingconditions:drymachiningofthealuminumalloy6061-T6withaTiB2coatedsolidcarbideendmill EndmilldiameterHelixangleCuttingspeedNo.ofteethAxialdepthofcut 12:7mm45240m/min34mm Table7-3. MeanvaluesofestimatedcuttingconstantsforcuttingconditionsofTable 7-2 tctcncnc Goodnessoft:Adj.R2=0:981Adj.R2=0:992 Table7-4. Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 7-2 tctcncnc tc0.00011722-0.00000330-0.00000115 nc0.000686260.00022796 nc0.00007974 111

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Summaryofexperimentalconditionsusedforvericationofforcepredictions,withthecuttingconditionsofTable 7-2 heldconstant ResultsdisplayedinFeed(mm/tooth)Radialimmersion(%)Up/downmillingormixedmode(>50%RI) Figure 7-11 0:05025Downmilling Figure 7-12 0:20025Downmilling Figure 7-13 0:10010Downmilling Figure 7-14 0:20010Downmilling Figure 7-15 0:0505Downmilling Figure 7-16 0:2005Downmilling Figure 7-17 0:10075mixedmode,cutstartswithh=0 Figure 7-18 0:20075mixedmode,cutstartswithh=0 Figure 7-19 0:10025Upmilling Figure 7-20 0:10010Upmilling Figure7-1. Dierentialprojectedchipareasinhelicalperipheralmilling:Thedierentialprojectedfrontalchiparea,dAf,isrelatedtothedierentialtangentialforcecomponent,dFt,andthedierentialnormalforcecomponent,dFn.Thedierentialprojectedaxialchiparea,dAa,isrelatedtothedierentialaxialforcecomponent,dFa.Therakefacechiparea(thegrayregionwhichislabeledwithbandhm)isnotusedinthemodel.Atwouted,righthandedhelicalendmillisillustrated. 112

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Dierentialelementoftheprojectedfrontalchiparea,dAf. Figure7-3. Dierentialelementoftheprojectedaxialchiparea,dAa. 113

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VariationoftheintegralsdescribedinEq. 7{19 fordownmillingusinga45helixangle.ThesignofI1changesinupmilling,withI2andI3remainingthesame.Theintegralsareindependentofendmilldiameter. 114

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Vericationoftheanalyticalmodelforthedegeneratecaseofstraightutedcutterusing=0:00000001tosimulatethezerohelixcaseforasingle-toothedendmill.Theconventionalsolutionplots(dashedlines)wereobtainedbyusingEq. 7{24 whereasEq. 7{11 wasusedforthecurrentanalyticalsolutionplots(solidlines).Parametersused:D=30mm,fT=0:10mm/tooth,a=10mm,RI=25%,Kt=2000N/mm2,Kn=0:30Kt. 115

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Comparisonoftheclosedformanalyticalsolutionwithanumericalsolutionpublishedby Tlusty ( 1985 2000 )foracaseof0:75%RI,withh=0attheentryofthecut.ParametersusedaregiveninTable 7-1 .ThetotalforceintheplaneisdisplayedFtotal=p 116

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ResidualsshowingthepercentagedierencebetweentheanalyticalandnumericalsolutionsdisplayedinFig. 7-6 117

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Analyticalsolutionsforvariousconditionsinhelicalperipheralmilling.ParametersaregiveninTable 7-1 .Theresultsmatchcloselywiththenumericalsolutionspublishedby Tlusty ( 1985 2000 ),withdierencesbeingwithin0:5%.ThetotalforceintheplaneisshownFtotal=p 118

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Experimentalset-upshowingthediameter12.7mm,3-utedendmillwith45helixheldinapolygonalchuckwiththeblackandwhitereectingbandwrappedaroundit,thelasertachometer,andtheworkpiecemountedonthedynamometer.AscreenshotoftheLabViewrvirtualinstrumentusedtorecordforcedataisalsoshown. Figure7-10. Linearregressionttingofcuttingcoecients,asafunctionoftheaveragedmeanchipthickness,fortheexperimentalconditionsgiveninTable 7-2 using50%radialimmersion.Experimentalpointsaredisplayedas. 119

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Predictedvs.experimentalforcesignals:feed0:050mm/tooth,25%radialimmersion,down-milling.OtherconditionsasinTable 7-2 Figure7-12. Predictedvs.experimentalforcesignals:feed0.200mm/tooth,25%radialimmersion,down-milling.OtherconditionsasinTable 7-2 120

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Predictedvs.experimentalforcesignals:feed0.100mm/tooth,10%radialimmersion,down-milling.OtherconditionsasinTable 7-2 Figure7-14. Predictedvs.experimentalforcesignals:feed0.200mm/tooth,10%radialimmersion,down-milling.OtherconditionsasinTable 7-2 121

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Predictedvs.experimentalforcesignals:feed0.050mm/tooth,5%radialimmersion,down-milling.OtherconditionsasinTable 7-2 Figure7-16. Predictedvs.experimentalforcesignals:feed0.200mm/tooth,5%radialimmersion,down-milling.OtherconditionsasinTable 7-2 122

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Predictedvs.experimentalforcesignals:feed0.100mm/tooth,75%radialimmersion,cutbeginswithh=0.OtherconditionsasinTable 7-2 Figure7-18. Predictedvs.experimentalforcesignals:feed0.200mm/tooth,75%radialimmersion,cutbeginswithh=0.OtherconditionsasinTable 7-2 123

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Predictedvs.experimentalforcesignals:feed0.100mm/tooth,25%radialimmersion,up-milling.OtherconditionsasinTable 7-2 Figure7-20. Predictedvs.experimentalforcesignals:feed0.100mm/tooth,10%radialimmersion,up-milling.OtherconditionsasinTable 7-2 124

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Gibbs-Wilbrahamdistortioneects.kisthenumberoftermsinthepartialFouriersumforb,L,andT.DistortioneectsonLandT,closetothejumpdiscontinuities,areobservedtobefairlysignicantfork=100.Propagationoftheeecttothecuttingforceisnegligibleforallpracticalpurposesbecausethechipwidth,b,vanishesatthepointsofthejumpdiscontinuitiesofLandT.ThetotalforceintheplaneisshownFtotal=p 125

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ConvergenceandcomputationalburdenoftheAnalyticalFouriersolution(dashedlines).Theindex`k'referstothenumberoftermsinthepartialFouriersummation.Computationtimeincludescomputationofthenumericalsolution(solidlines).APCwitha1.6MHzprocessorwasused.Thecomputationsrefertothecaseofa=10,RI=0:75,h=0attheentryofthecut,withotherparameterslistedinTable 7-1 .ThetotalforceintheplaneisshownFtotal=p 126

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Inthischapter,theinstantaneousrigidforcemodelforhelicalendmillingisextendedtoincludetheeectsofdierentialtoothpitch.Itwasnotedinchapter 1 thatseveralauthorshavereportedontheuseofdierentialpitchcuttersformitigatingchatterandreducingdimensionalsurfaceerrors(e.g. Slavicek 1965 ; Vanherck 1967 ; Tlustyetal. 1983 ; ShiraseandAltintas 1996 ).Theextensionofthehelicalforcemodelofchapter 7 tocoverdierentialpitcheectsisverystraightforward,merelyrequiringthecomputationoftheeectivefeedforeachindividualtooth,andthephaseangleatwhicheachtoothentersthecut.ThisinformationisappliedinEq. 7{15 toobtaintheresults.Thecalibrationofthemodelfollowsexactlythesameprocedureasinchapter 7 .Intheexperimentalset-upusedformodelcalibration,thedialindicatorhavingaresolutionof2:5m(0:0001")couldnotresolvetherunout.Hence,theeectsofrunoutwerenotrequiredtobeconsidered.Runouteects,ifany,couldbetreatedusingthemethodtobedescribedinthenextchapter,whichissimilartotheprocedureoutlinedinchapter 4 Anoteonthecommercialavailabilityofdierentialpitchcuttersisinorder.Facemillswithindexableinsertsarecommonlymanufacturedwithunequalpitch(e.g.,theseries2J6B,2J4B,and2J2BofIngersollmake,series8000VA19ofStellrammake,andseries245,and490ofSandvikCoromantmake,etc.).Ontheotherhand,solidendmillswithdierentialpitcharelesscommon,thoughtheiravailabilityisimproving(e.g.theHPHVandHPHVTseriesofKennametalmake,thePluraseriesofSandvikCoromantmake;afourutedendmillofFuturaCarbideSRL,Italymake,etc.).Fortheexperimentalresultsreportedinthischapter,adierentialpitchendmillofKennametalmakewasused(detailsinTable 8-1 ).Thesewerethesametoolsusingwhich Powell ( 2008 )reportedtheresultsofatimedomainsimulationofstabilityboundariesusingvariablepitchendmills. 127

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8-1 showsanexampleofdierentialtoothpitch. 7{15 ,theeectivefeedpertooth,fTi,andthephaseshift,foreachtoothentry,i,arerequired.Letfrdenotethelinearfeedperrevolutionofthecutter.Forthe4-utedcutterusedintheexperiments,thetoothpitchspacingwas97-83-97-83,resultingintheeectivefeedsforthefourteethbeingfT1=(97=360)fr,fT2=(83=360)fr,fT3=(97=360)fr,andfT4=(83=360)fr,respectively.Therespectivepitchanglesfollowtheangularseparationpattern.ThisinformationisusedinEq. 7{15 ,alongwiththedetailsofthecuttergeometryandcuttingparameterslistedinTable 8-1 ,toyieldthepredictedcuttingforcesafterexperimentalmodelcalibration. 7 ,exceptthattheendmillandtoolholderweredierent(Table 8-1 ).A10%partialradialimmersiontestwasusedtoextractcuttingcoecients. Experimentswereconductedatninedierentfeedrates,withintherange0.025-0.250mm/tooth,thetimetracesofforcecomponentswererecorded,andaverageforcevalues,( fy; fz),werecomputed.Ateachfeedrate,theexperimentwasrepeated5times.Forceswereaveragedoveronerotationonapertoothbasis.Toreducethevariability,theaveragewastakenfor50successiverotations.ValuesofKtc,Knc,andKac,correspondingtoeachfeedrate,werederivedusingEq. 7{20 .Thus,ninesetsofvedatapointseachwerecollectedforttingalinearmodelaccordingtoEq. 7{16 .Twoparameters(slopeandintercept)werettedto45points,yielding43degreesoffreedomforeachregression.Thecuttingconstantstc;ncandtc;ncofEq. 7{16 wereobtainedusingZellner's( 1962 )multiresponselinearregression,whereasthecuttingconstantsacand 128

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8-2 Basedontheregression,theestimatedcuttingconstantstc;ncandtc;ncarecalculatedandgiveninTable 8-2 .Alineartwasnotfoundsuitablefortheaxialcuttingconstants(smallvaluesofR2inlinearregression).ThemeanvalueofKacwastted.Thus,foranyparticularvalueoftheaveragedmeanchipthickness, (8{1)Knc=289:7( (8{2)Kac=379:5[N/mm2] (8{3) Thevariance-covariancematrixforthecuttingconstants,tc;ncandtc;nc,obtainedusingZellner's( 1962 )methodaregiveninTable 8-3 .Thevarianceoftheindependentcuttingconstantacwasfoundtobe0.03940389.Thus,thevariancesofmodelinputparameters,duetorandomeects,areavailablefortheTypeAuncertaintyevaluation. FortheTypeB1evaluationofuncertainty,forthevariancesofcuttingcoecientsduetosystematiceects,theprocedureoutlinedinchapter 7 holdsgood.Sincethereisnorunout,theTypeB2evaluationforvariancesofeectivefeedisnotrequired. Thepropagationofuncertaintiesinmodelinputparametersfollowsonthesamelinesasinchapter 7 .AsucientlylargenumberofexperimentsallowsthechoiceofA1=2astheexpandeduncertaintycoveragefactorforthecombinedTypeAcomponentuncertaintiesfora95%condenceinterval.TheunderlyingapproximatenormalassumptionfortheforcemeasurementuncertaintiesallowsthechoiceofcoveragefactorB1=2astheexpandeduncertaintycoveragefactorforthecombinedTypeAcomponentuncertaintiesforthe95%condenceinterval.Therootsumofsquaresmethodisusedtocalculatetheoverallexpandeduncertainty. 129

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8-3 )-( 8-6 )showtheresultsofforcepredictionsforthedierentialpitchendmill.Upperandlowerboundsof95%condenceintervalofpredictedforcesaredisplayedgraphically,alongwiththeexperimentallyobtainedsignals,whichverifythepredictions. Reliablepredictionswereobtainedforvariouscombinationsoffeed,radialimmersion,andupordownmillingcongurationsgiveninTable 8-4 .Thedecienciesofthemodelwhichwerehighlightedinthepreviouschapter,stillhold,i.e.,insomecases,thepeakforcesareoverpredicted.Thecuttingcoecients,derivedonthebasisofdownmilling,sucedforupmillingandmixedmodepredictions.Thedierentialpitchgivesrisetoaphasingoftheforcesaccordingtotheangularpositionofthetooth.Theresultingvariationsinfeedpertoothgivesrisetovariationsintheamplitudeoftheforcesfromtoothtotooth. 7 .Theeectsofdierentialtoothspacingwereincorporatedinastraightforwardmannerusingtheappropriatevaluesoftheeectivefeed,andthephaseangleoftoothentry,foreachindividualtooth.Theprocedureforpropagationofuncertaintiesinmodelinputparametersthroughtheforcemodelisthesameasinchapter 7 .Theforcepredictionswereexperimentallyveried.Thehelicalforcemodelisfurtherextendedtoincludetheeectsofradialrunoutofsuccessiveteethinthenextchapter. 130

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Experimentalconditions:drycuttingofaluminumalloy6061-T6;Kennametalmakeuncoatedsolidcarbideendmillhavingdierentialpitch(97-83-97-83);Schunkmakehydrogripchuck EndmilldiameterHelixangleCuttingspeedNo.ofteethAxialdepthofcut 19:05mm30250m/min44mm Table8-2. MeanvaluesofestimatedcuttingconstantsforcuttingconditionsofTable 8-1 tctcncncacac Table8-3. Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 8-1 tctcncnc tc0.000032390.000255640.00007451 nc0.003400830.00095239 nc0.00027760 Table8-4. Summaryofexperimentalconditionsusedforvericationofforcepredictionswithradialrunout,holdingthecuttingconditionsofTable 8-1 xed ResultsdisplayedinFeed(mm/tooth)Radialimmersion(%)Up/downmillingormixedmode(>50%RI) Figure 8-3 0:2005Downmilling Figure 8-4 0:10020Downmilling Figure 8-5 0:10025Upmilling Figure 8-6 0:05050Upmilling 131

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Exampleofdierentialtoothpitch,alsocalledunequaltoothspacing,orvariablepitch.Thespecictoothspacingshown(97-83-97-83)belongstothecutterusedintheexperimentswhoseresultsarereportedinthischapter. Figure8-2. Linearregressionttingofcuttingcoecients,asafunctionoftheaveragedmeanchipthickness,fordrymillingof6061-T6aluminumalloy,usingadierentialpitch,4-uted,30helix,solidcarbideendmillhavingzeronominalrunout.Experimentalpointsaredenotedby\M". 132

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Dierentialpitcheects.Predictedvs.experimentalforcesignals:4-uted,30helixcutterwith97-83-97-83toothspacingresultinginamplitudeandphasevariationsofforcesfromtoothtotooth;feed0.200mm/tooth,5%radialimmersion,down-milling.OtherconditionsasinTable 8-1 Figure8-4. Dierentialpitcheects.Predictedvs.experimentalforcesignals:4-uted,30helixcutterwith97-83-97-83toothspacingresultinginamplitudeandphasevariationsofforcesfromtoothtotooth;feed0.100mm/tooth,20%radialimmersion,down-milling.OtherconditionsasinTable 8-1 133

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Dierentialpitcheects:Predictedvs.experimentalforcesignals:4-uted,30helixcutterwith97-83-97-83toothspacingresultinginamplitudeandphasevariationsofforcesfromtoothtotooth;feed0.100mm/tooth,25%radialimmersion,up-milling.OtherconditionsasinTable 8-1 Figure8-6. Dierentialpitcheects:Predictedvs.experimentalforcesignals:4-uted,30helixcutterwith97-83-97-83toothspacingresultinginamplitudeandphasevariationsofforcesfromtoothtotooth;feed0.050mm/tooth,50%radialimmersion,up-milling.OtherconditionsasinTable 8-1 134

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Inthischapter,theinstantaneousrigidforcemodelisextendedtoincludetheeectsofradialrunoutofsuccessiveteeth.Theaccuracyoftoolholdingandspindlemountinginmodernmachinetoolshasimprovedtremendouslyinrecentyears,especiallywiththeadventofadvancedtoolholdingmethodssuchaspolygonalchucksandhydrogripchucks,coupledwithprecisespindleadaption,suchastheHSKinterfacewhichprovidessimultaneoustaperandfacebuttingforthetoolholderwhenclampedinthespindle.Theprecisiongroundsolidendmillsaremanufacturedtoextremelyclosetolerances.Asaresult,itispossibletoholdthetoolpointrunouttoaverylowvalueuponassemblyoftheendmillintheholderandmountingonthemachinespindle.Ongoodsetups,itiseasytokeeptheradialrunoutdowntojustafewmicrons.Underpracticalconditionsontheshopoor,itisdiculttoreducetherunouttozero.Suchaneortwouldnotbeeconomicallyjustiableformostsituations.Therefore,itisnecessarytoincludetheeectsoftoolrunoutfortheforcemodeltohavepracticalvalue. Inchapter 1 itwasmentionedthattherearemanyforcemodelsavailableintheliteraturewhichhavesuccessfullypredictedforceswithrunouttakenintoconsideration.Here,thepurposeistoprovideaclosedformanalyticalforcemodel,whichhasasingleequationcoveringtheentirecutterrotation,whichwillpredictforcesunderconditionsofnon-zerorunout.Astraightforwardextensionofthemodelproposedinchapter 7 achievesthisobjectivebysuitablycalculatingtheeectivefeedforeachindividualtoothtobeappliedtoEq. 7{15 .Anadditionalmodelinputparameterisintroduced,i.e.,theeectivefeedofeachtooth,fTi,whichisassessedbyanactualmeasurementoftherunoutupontoolassemblyandmountingonthespindle.Thisinputisaectedbyuncertaintyofthemeasuredvalueofrunoutwhichisachievedusingameasuringdevicesuchasadialindicator.ThisintroducestheTypeB2evaluationofuncertaintyinthisinputparameter.Thetreatmentisanalogoustothecaseofthestraightutedmodelofchapter 4 135

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4.4 ofchapter 4 applies.Forsimplicity,atwoutedcutterisconsidered.AllthefourEqs. 4{9 4{12 apply,withthetermshavingthesamemeanings.TheEqs. 4{11 and 4{12 yieldtheeectivefeedsofthetwoteethgiventhemeasuredvalueofradialrunout,,betweenthesetwoteeth.Foracutterwithmorethantwoteeth,theexpressionsforeectivefeedcanbederivedusingsimilararguments.TheseeectivefeedsareusedinEq. 7{15 toyieldtheforcecomponentsforacutterwithradialrunout. Tlusty ( 1985 2000 )fora2-utedendmillwith10mrunout.Foraparticularsetofparameters,thecomparisonisshowninFig. 9-1 .Thedierencebetweenthetwosolutionsislessthan0:5%. 7 ,exceptthattheendmillandtoolholderweredierent(Table 9-1 ).Uponclampingthetoolinthespindle,thestaticradialrunoutwasmeasuredbyindicatingtheendmillswithadialgagehavingaleastcountof2.5m(0.0001").Thenominalvalueofthemeasuredrunoutwas10m.A25%partialradialimmersiontestwasusedtoextractcuttingcoecients. Experimentswereconductedatninedierentfeedrates,withintherange0.025-0.250mm/tooth,thetimetracesofforcecomponentswererecorded,andaverageforcevalues,( fy; fz),werecomputed.Ateachfeedrate,theexperimentwasrepeated5times.Forceswereaveragedoveronerotationonapertoothbasis.Toreducethevariability,theaveragewastakenfor50successiverotations.ValuesofKtc,Knc,andKac,correspondingtoeachfeedrate,werederivedusingEq. 7{20 .Thus,ninesetsofvedatapointseachwerecollectedforttingalinearmodelaccordingtoEq. 7{16 .Twoparameters(slopeandintercept)werettedto45points,yielding43degreesoffreedom 136

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7{16 wereobtainedusingZellner's( 1962 )multiresponselinearregression,whereasthecuttingconstantsacandacwereobtainedusingsimplelinearregression.ThettedregressionlinesaredisplayedinFig. 9-2 Basedontheregression,theestimatedcuttingconstantstc;ncandtc;ncarecalculatedandgiveninTable 9-2 .Alineartwasnotfoundsuitablefortheaxialcuttingconstants(smallvaluesofR2inlinearregression).ThemeanvalueofKacwastted.Thus,foranyparticularvalueoftheaveragedmeanchipthickness, (9{1)Knc=121:3( (9{2)Kac=428:5[N/mm2] (9{3) Thevariance-covariancematrixforthecuttingconstants,tc;ncandtc;nc,obtainedusingZellner's( 1962 )methodaregiveninTable 9-3 .Thevarianceoftheindependentcuttingconstantacwasfoundtobe0.00678924.Thus,thevariancesofmodelinputparameters,duetorandomeects,areavailablefortheTypeAuncertaintyevaluation. FortheTypeB1evaluationofuncertainty,forthevariancesofcuttingcoecientsduetosystematiceects,theprocedureoutlinedinchapter 7 holdsgood.Inaddition,theTypeB2evaluationhastobeperformedtoextractthevariancesoftheeectivefeedduetouncertaintyinthemeasuredvalueoftheradialrunout.Thisanalysishasbeenperformedinchapters 5 and 6 basedonwhichthevariancesintheeectivefeedsofthetwoteetharegivenbyEqs. 6{7 and 6{8 ,andarenotedbelowforreadyreferenceu2(fT1)=u2(fT2)=@fT1;2 7 .Theadditionaltaskhereisthepropagation 137

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6 .Thesensitivitiesofthecuttingforcestotheeectivefeedratesaredetermined.Thesesensitivitiesareusedtopropagatetheuncertaintiesintheeectivefeedratestoyieldthecombinedcomponentuncertainties(TypeB2)ofthepredictedcuttingforcecomponents. ThesensitivitiesofcuttingforcecomponentswithrespecttotheeectivefeedsarefoundusingEq. 7{11 Theuncertaintiesintheeectivefeedratesoftheithtooth,asexpressedinEq. 6{7 and 6{8 ,maybepropagatedtothepredictedforcesusingEq. 7{11 andthesensitivitiesexpressedinEq. 9{4 toyieldthecombineduncertaintiesucB2(fx;y;z) wherethesubscriptonfTiisdroppedforconvenience. 9-3 )-( 9-15 )showtheresultsofforcepredictionsforthe2-uted,45helix,cutterhavinganominalradialrunoutof10m.Theforcesarepredictedreasonablywellforvariouscombinationsoffeed,radialimmersion,andupordownmillingcongurationsgiveninTable 9-4 .However,thedeciencyinthemodelshowsupinseveralcaseswherethepeakforcesareoverpredicted,andtheexperimentalforcepatternsareskewedwithrespecttothepredictedpatterns.Theexpectedeectoftherunouttofeedratioisclearlyobserved.Thelargerthisratio,thelargeristhevariationinforcesfromtoothtotooth. 138

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139

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Experimentalcuttingconditions:drycuttingofaluminumalloy6061-T6;TiB2coatedsolidcarbideendmill(KennametalcatalogNo.HPF45A750S2150)havingtwoequispacedteeth;Polygonalchuck(SchunkcatalogNo.203794) EndmilldiameterHelixangleCuttingspeedNo.ofteethAxialdepthofcut 19:05mm45250m/min24mm Table9-2. MeanvaluesofestimatedcuttingconstantsforcuttingconditionsofTable 9-1 tctcncncacac Table9-3. Symmetricvariance-covariancematrixofcuttingconstantsforexperimentalconditionsofTable 9-1 tctcncnc tc0.000038850.000054110.00001795 nc0.001845000.00058139 nc0.00019285 Table9-4. Summaryofexperimentalconditionsusedforvericationofforcepredictionswithradialrunout,holdingthecuttingconditionsofTable 9-1 xed ResultsdisplayedinFeed(mm/tooth)Radialimmersion(%)Up/downmillingormixedmode(>50%RI) Figure 9-3 0:1005Downmilling Figure 9-4 0:1005Upmilling Figure 9-5 0:10010Downmilling Figure 9-6 0:20010Downmilling Figure 9-7 0:05020Upmilling Figure 9-8 0:05050Downmilling Figure 9-9 0:10050Downmilling Figure 9-10 0:02550Upmilling Figure 9-11 0:05075mixedmode,cutendswithh=0 Figure 9-12 0:05075mixedmode,cutstartswithh=0 Figure 9-13 0:050100Slotting Figure 9-14 0:100100Slotting Figure 9-15 0:200100Slotting 140

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Comparisonoftheclosedformanalyticalsolutionwithanumericalsolutionpublishedby Tlusty ( 1985 2000 )fora2-utedendmillwith10mrunout,0:50%RI,down-milling,a=4mm,fT=0:100mm/tooth,D=19:05mm,=45,Ktc=2000N/m2,Knc=0:3Ktc.ThetotalforceintheplaneisdisplayedFtotal=p 141

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Linearregressionttingofcuttingcoecients,asafunctionoftheaveragedmeanchipthickness,fordrymillingof6061-T6aluminumalloy,usinganequispacedtooth,2-uted,45helix,TiB2coated,solidcarbideendmillhavinganominalradialrunoutof10m.Experimentalpointsaredenotedby\#". Figure9-3. Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,5%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. 142

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Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,5%radialimmersion,upmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. Figure9-5. Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,10%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. 143

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Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.200mm/tooth,10%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Smallrunouttofeedratio(0.05)resultsinsmallvariationinforcesfromtoothtotooth. Figure9-7. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,20%radialimmersion,upmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. 144

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Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,50%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. Figure9-9. Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,50%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. 145

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Runouteects:Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.025mm/tooth,50%radialimmersion,upmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.40)resultsinlargevariationinforcesfromtoothtotooth. Figure9-11. Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,75%radialimmersion,cutendswithh=0.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. 146

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Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,75%radialimmersion,cutstartswithh=0.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. Figure9-13. Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,100%radialimmersionslotting.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. 147

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Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,100%radialimmersionslotting.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. Figure9-15. Radialrunouteect.Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.200mm/tooth,100%radialimmersionslotting.Nominalrunout10m.OtherconditionsasinTable 9-1 .Smallrunouttofeedratio(0.05)resultsinsmallvariationinforcesfromtoothtotooth. 148

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Sofar,allforcepredictionshavebeenmadeusingxedvaluesofthecuttingcoecients,Ktc;nc;ac,whichwerecalculatedbasedonanaveragedmeanchipthickness, 7 ,whichwasbasedonaveragecuttingcoecients,areaddressedinthischapter.Instantaneouscuttingcoecientsareusedtorenetheforcemodeltoyieldaccurateforcepredictions. Sincethecuttingconstantsareinvariants,itisplausiblethattherelationshipbetweentheinstantaneouscuttingcoecientsandtheinstantaneoushmhastheformK=ehm,forxedvaluesofcuttingspeed,axialdepthofcut,andotherxedcuttingconditionssuchasthetypeofcuttinguidbeingused.Thevaluesofthecuttingconstants,tc;nc;acandtc;nc;ac,maybefoundusingtheaverageforcesasoutlinedinchapter 7 .Theunderlyingassumptionisthatthevaluesofthecuttingconstantsidentiedinthismannercanbeusedtocalculatetheinstantaneouscuttingcoecients.Thevalidityofthisassumptionwillbetestedbycomparisonofexperimentalforcesignalswithpredictedforcesbasedontheinstantaneouscuttingcoecients.TheeectofconsideringthedependenceofKtc;nc;aconhmisstudiedinthischapter.Theeectsofradialrunoutare 149

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Theinstantaneouscuttingcoecientsarefunctionsofp,andmaybeexpressedthroughtheintermediatevariablesLandTusingEq. 3{17 wherethesamplingfunction(sinc)hasbeendenedinEq. 3{18 TherelationbetweeninstantaneouscuttingcoecientsandinstantaneouscuttingforcecomponentsmaybeobtainedbyinvertingthecuttingforceEq. 7{11 toyield Theaboveequationraisesaninterestingpossibility.Iftheinstantaneousforcescouldbemeasuredusingasingleexperimentalcut,theinstantaneousvaluesofthecuttingcoecientscouldbeobtained.Thesevaluescanbedirectlyrelatedtotheinstantaneoushmtoobtaintherelationbetweenthecuttingcoecientsandhm.Therearecertaindrawbacksofthisscheme.First,theinstantaneousforcescannotbemeasuredasaccuratelyasaverageforcesbecausetheexperimentalforcesignalshavehighfrequencyoscillations(wiggles)ofsmallmagnitudesuperimposedonthemduetomeasuringinstrumentdynamics.Thiseectisdiscussedinchapter 11 .Thewigglesdonotaectaverageforcevaluesappreciably.Second,theinvertedmatrixinEq. 10{4 becomesnumericallyillconditionedforsmallvaluesofhm. 150

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7 and 9 respectively.However,thevariancesofcuttingcoecients(obtainedusingaTypeB1evaluation)arenolongerconstantnumbers,butarefunctionsofpbecausetheinstantaneousforces,andthesensitivitiesofcuttingcoecientstotheinstantaneousforces,arefunctionsofp. Equation 10{4 yieldsthefollowingsensitivitycoecientsforuseinpropagationoftheuncertaintyininstantaneousforcemeasurementstotheuncertaintiesininstantaneouscuttingcoecients Usingtheabovesensitivitiesthevariancesofcuttingcoecients,duetothesystematiceects,canbeobtained whichisanalogoustoEq. 7{29 wherethevariancesofaverageforcecomponentsarereplacedbyvariancesofinstantaneousforcecomponents,u2(fx;y;z).Thevaluesofu(fx;y;z)intheEq. 10{6 abovearesetat1:207%ofthenominalvaluesoftheinstantaneousforcecomponentsbasedontheinstrumentmanufacturer'sestimates,asexplainedindetailinchapter 5 .Forthesakeofsimplicity,anypossiblecorrelationbetweenKtcandKnc

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Thesensitivitycoecientshavingbeenobtained,theuncertaintiesinmodelinputparametersarepropagatedusingEqs. 7{31 7{33 ,and 9{5 toobtainthecombinedcomponentuncertaintiesucA(fx;y;z),ucB1(fx;y;z),anducB2(fx;y;z),respectively,wherethevaluesofu2(ktc;nc;ac)obtainedfromEq. 10{6 aretobeusedinEq. 7{33 ,andinstantaneouscuttingcoecientsaretobeusedinallrelevantexpressionsinsteadofxedvaluesofKtc;nc;ac. 10-1 .TheresultsaredisplayedinFigs.( 10-1 )-( 10-13 ).Comparedwiththepredictionsmadeunderthesameconditionsinchapter 9 ,wherethecuttingcoecientshadxedvaluesbasedontheaveragedmeanchipthickness,thematchwithexperimentalresultsisobservedtobemuchcloserusingtheinstantaneouscuttingcoecientmodel. Thedecienciesnotedintheearliermodelofchapters 7 8 ,and 9 arerectiedbytheuseofinstantaneouscuttingcoecients.Noteworthyisthefactthatmagnitudesoftheforces,aswellasthepatternsarepredictedaccurately.Theskewnessoftheexperimentalforcesinhighimmersionexperimentsisalsosuccessfullycaptured.Aglaringdemonstrationofthisisprovidedbyacomparisonoftheguresinthischapterwiththecorrespondingguresintheearlierchapter 9 .Furthermore,ingeneral,theexperimentalforcepatternsliewithinthe95%condenceintervalboundswhentheinstantaneouscuttingcoecientsareused.Allthesepiecesofevidencestrengthentheclaimofthecorrectnessoftheclosedformanalyticalcuttingforcemodel. 152

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hm,theactualvaluesofKtc;nc;acdropdown.Hence,thepeakforcesinthecyclearelowerthanthosepredictedusingtheaveragedvaluesofKtc;nc;ac,andviceversa.Thevariationofhmalsoexplainstheskewnessinthecuttingforcepatterns. TheincorporationoftheinstantaneousKtc;nc;acintheforcemodelwasseentobestraightforwardduetothestructureofthemodel,andsowastheuncertaintyanalysis.Thechoiceofthemodeldependsontheuser.Ifveryhighaccuracyofmagnitudeandformarerequiredoftheforcepredictions,theinstantaneouscoecientformulationmaybeused.Itwasalsoseenthatforhighradialimmersions,theinstantaneouscoecientmodelmaybepreferable.Formostpurposes,theaveragedcoecientmodelofchapter 7 wouldsuce. 153

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SummaryofexperimentalconditionsusedtoverifyforcepredictionsbasedoninstantaneousKtc;nc;ac,withtheconditionsofTable 9-1 heldxed ResultsdisplayedinFeed(mm/tooth)Radialimmersion(%)Up/downmillingormixedmode(>50%RI) Figure 10-1 0:1005Downmilling Figure 10-2 0:1005Upmilling Figure 10-3 0:10010Downmilling Figure 10-4 0:20010Downmilling Figure 10-5 0:05020Upmilling Figure 10-6 0:05050Downmilling Figure 10-7 0:10050Downmilling Figure 10-8 0:02550Upmilling Figure 10-9 0:05075mixedmode,cutendswithh=0 Figure 10-10 0:05075mixedmode,cutstartswithh=0 Figure 10-11 0:050100Slotting Figure 10-12 0:100100Slotting Figure 10-13 0:200100Slotting 154

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,5%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. Figure10-2. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,5%radialimmersion,upmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. 155

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,10%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. Figure10-4. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.200mm/tooth,10%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Smallrunouttofeedratio(0.05)resultsinsmallvariationinforcesfromtoothtotooth. 156

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,20%radialimmersion,upmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. Figure10-6. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,50%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. 157

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,50%radialimmersion,downmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. Figure10-8. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.025mm/tooth,50%radialimmersion,upmilling.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.40)resultsinlargevariationinforcesfromtoothtotooth. 158

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,75%radialimmersion,cutendswithh=0.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. Figure10-10. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,75%radialimmersion,cutstartswithh=0.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. 159

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.050mm/tooth,100%radialimmersionslotting.Nominalrunout10m.OtherconditionsasinTable 9-1 .Largerunouttofeedratio(0.20)resultsinlargevariationinforcesfromtoothtotooth. Figure10-12. Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.100mm/tooth,100%radialimmersionslotting.Nominalrunout10m.OtherconditionsasinTable 9-1 .Moderaterunouttofeedratio(0.10)resultsinmoderatevariationinforcesfromtoothtotooth. 160

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Predictedvs.experimentalforcesignals:equispaced,2-uted,45helixcutter;feed0.200mm/tooth,100%radialimmersionslotting.Nominalrunout10m.OtherconditionsasinTable 9-1 .Smallrunouttofeedratio(0.05)resultsinsmallvariationinforcesfromtoothtotooth. 161

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Measurementofcuttingforcesusingthemeasurementchainformedbythedynamometer,cable,amplier,anddataacquisitioncard,wasoneofthemostimportantaspectsofthisresearch.Thepurposeofthischapteristoaddcertainqualitativecommentsonforcemeasurementsusingthismeasuringchain.Thisallowsthereadertoholdtheexperimentalforcesignalsinproperperspective. Theidealforcemeasuringinstrument(completelyin-phase,unitmagnituderesponse,withinnitebandwidth)isnotavailable.Thedynamicsoftheforcemeasuringchainmakesithardtoobtaincleanforcesignals.Thecompletecharacterizationofthisprocessdemandsalongandcarefulprocedure(e.g. Castroetal. ( 2006 ); TounsiandOtho ( 2000 )).Inthischapter,thefrequencyresponsefunction(FRF)oftheforcemeasurementchainisconsideredtoexaminetheinuencingdynamics.ExperimentalFRFsoftheforcemeasurementchainareexaminedtomakequalitativeremarksonthedelityofexperimentalforcemeasurements. 11-1 .Aninstrumentedpiezoelectricimpacthammermodel086C05,ofPCBPiezotronicsmake,wasusedtoexcitethedynamometerwithanimpactsignalineachofthex,y,andzdirections,andtheoutputforce,obtainedthroughtheforcemeasurementchain,wasrecorded. 162

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11-1 intheformofrealandimaginarycomponents.ThisdatawasconvertedtomagnitudeandphasecomponentsandisshowninFig. 11-2 .Whentheparttobemachinedismountedonthedynamometer,themassloadingeectchangesthefrequencyresponsefunction.Twoartifactswereusedwhichwererepresentativeofthepartsmachinedwhilecalibratingandverifyingtheforcemodels.Alightweightaluminumartifactwasclampedonthedynamometertomimicthepartsusedtocalibratethehelicalforcemodel,andaheavysteelpartwasmountedwhichhadbeenusedtocalibratethestraightutedforcemodel.AvisualinspectionshowsthattheFRFsareaected(Figs. 11-1 and 11-2 ). Tostudytheinuenceoftheforcemeasurementchainontheinputforces,itisconvenienttoconsiderthemagnitudeandphaseoftheFRFsinFig. 11-2 .ItcanbeseenthatthemagnitudetheofFRFsisclosetounityforfrequenciesbelow1000Hz,buttheFRFisnotexactlyatinthisrange.Similarly,thephaseisclosetozerointhisrangeoffrequencies,buttheFRFisnotat.Thefrequencycontentoftheinputforcesinthisrangewillbeampliedaccordingtothemagnituderesponse,andwillsueralagaccordingtothephaseresponse. Togetaqualitativesenseoftheseinuences,thefrequencycontentofahypotheticalforcesignalswithtoothpassingfrequencyof60HzisexaminedinFig. 11-3 .Attheintegralmultiplesofthetoothpassingfrequency,theinputsignalhasfrequencycontent.TheeectofthenonunitFRFmagnitudeatthesefrequencieswillamplifythelowpowercontentanddistorttheoutputsignal.Thisalsoexplainstheappearanceofhighfrequencywigglesintheexperimentalforcesignals. Theeectofthenonzerophaseatthetoothpassingfrequenciesandtheirharmonicsistointroducealagintheresponseofthesefrequencies.Thiswillcreateanadditionaldistortioninthesignal. 163

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Measurementsintroduceuncertainties.TheuncertaintyintheexperimentalFRFcreatesafurtherdicultybecauseitisabivariateproblem.TheFRFhasarealpartandanimaginarypart.So,themethodofquantifyingitsuncertaintyisspecialized.ThisquanticationofuncertaintiesinexperimentalFRFshasbeenstudiedby KimandSchmitz ( 2007 ).Thesethreefactors,themagnituderesponse,phaseresponse,anduncertaintiesinmeasuredvaluesoftheexperimentalFRFsarefurthercomplicatedbytherealizationthattheFRFitselfisnotaconstant.Asthemillingcutterremovesmaterial,thepartlosesmass,andthischangestheFRFconstantly. Fortheexperimentsreportedinthisdocument,themaximumfundamentaltoothpassingfrequencywas300Hz.Atthisfrequency,theresponseofthechainisreasonablyat,both,inmagnitudeandphase.However,theharmonicsaredistortedandgiverisetowigglesontheforcesignals.Qualitatively,thecalibrationoftheforcemodelisnotaected 164

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Figure11-1. DirectFrequencyResponseFunctions(FRFs)oftheforcemeasuringchainwiththedynamometermountedonanges,xedonamountingplate,andclampedtothemachinetable:InadditiontotheFRFofthebaredynamometer,twoothersetsofFRFwererecordedtoshowtheinuenceofthemassloadingduetothepartbeingclampedontothedynamometerwhenitisbeingmachined.Thelightweight(0.4kg)aluminumartifactisrepresentativeofthemachinedpartsusedtoverifythehelicalforcemodel,whereastheheavy(1.8kg)steelartifactwasthepartusedtoverifythestraightutedforcemodel. 165

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DirectFRFsoftheforcemeasuringchainshowingmagnitudeandphaseobtainedfromtheexperimentallydeterminedrealandimaginarypartsoftheFRFsshowninFig. 11-1 166

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DemonstrationoftheeectofnonunitmagnitudeoftheFRFatthepointswheretheinputforcesignalhasfrequencycontent. 167

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Inthisdissertation,theanalyticalmodelisdevelopedbyconsideringthetool,workpiece,andmachinestructuretoberigid.Aexiblemodelneedstobedevelopedtoincludetheeectsofcutterandworkpiecedeections.Theuseofthisforcemodelinotherapplicationssuchasstabilityanalysis,surfacelocationerror,andsurfaceproleerrorneedstobeexplored.Themodelalsoneedstobeappliedtosurfacenishpredictions. Baylyetal. 2003 ; Mann 2003 ; Mannetal. 2005 ).Recently,theeectsofhelicalgeometryhavebeenincludedintheTFEAmodel.Theproblemwassolvedforacompliantworkpiecerigidtoolmodelin Pateletal. ( 2008 ),andarigidworkpiececomplianttoolmodelin Mannetal. ( 2008 ).TheTFEAsolutionoftheproblemforasingledegreeoffreedom(SDOF)compliantstructurecanbeobtainedusingstandardprocedure( Mann 2003 ; Mannetal. 2005 ).Usingthehelicalforcemodelofchapter 7 ,theproblemisformulatedcompletelybelowforafutureresearchertosolveitbydevelopingsomeecientcomputationaltechnique. Figure 12-1 illustratestheconceptofdynamicchipthicknessinperipheralmilling.Intheidealcase,the(static)chipthicknessishs,intheabsenceofregenerationofwavinessofthemachinedsurface.Basedonthecircularpathapproximationonemaywrite Regenerationcausesadynamiccomponenthtobeimposedonhs.Thus,theactualchipthicknessis 168

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Thebreakupofthedynamicchipthickness,h,intoitscomponentsxandyisillustratedinFig. 12-1 Ifisthetoothpassingperiod,then x=x(t)x(t)(12{3) y=y(t)y(t)(12{4) wherexandyarefunctionsoftime,t. AnexpressionforhmaybewrittenbasedonthegeometrydisplayedinFig. 12-1 h=xsin+ycos(12{5) Oncethedynamicchipthicknessischaracterized,theforcemodelisdevelopedtoincludeitseects.Theinnitesimalprojected(frontal)areaoftheuncutchip,dAf,atanarbitraryangularorientation,,isdAf=(fTsin+xsin+ycos)d whereLandTaretheangularorientationsoftheleadingandtrailingedgesofthechipinthetool-chipcontactzone,andbistheinstantaneousprojected(frontal)axialdepthofcut. 169

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Thetangentialandnormalcomponentsofthecuttingforce,intheplaneperpendiculartothecutteraxis,areconvertedtocomponentsdFxanddFy,wherethenon-rotating(x;y)coordinatesystemhasitsoriginonthecutteraxis.Thecomponentsarerelatedviaarotationmatrix Integrationyieldsthetotalforcesonasingletooth wheretheintegrationsarecarriedoutovertheappropriatelimits.Theselimitsareexplicitlyshowninasubsequentstep,afterapplyingcertaintransformations. SubstitutingfromEq. 12{7 intotheabovetwoequationsrecaststheforcerelationshipsintermsoftheelementaldierentialfrontalchiparea,dA Ashorthandnotationisintroducedforconvenience 170

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12{6 ,thetwointegralsRsindA=RSdAfandRcosdA=RCdAfmaynowberewritten SubstitutingfromEqs. 12{13 and( 12{14 )intoEq. 12{11 yields {z }static+8><>:FxDFyD9>=>;| {z }dynamic(12{15) Thestaticcomponentoftheforceis {z }static=bfT whereFS11=TZLSCd 171

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{z }dynamic=b where Considertheproblemofmodelingaperipheralmillingprocessasasingledegreeoffreedom,linear-spring,mass,andviscous-dampersystem.Eitherthestructureorthetoolmaybeconsideredcompliant( Mann 2003 ).Letthesystembecompliantinasingledirectionandthexaxisbealignedinthisdirection.Foracutterwithasingletooth,Eq. 12{16 yieldsthestaticforcecomponentinthexdirection Incomputingthedynamicchipthickness,y0.ThedynamiccomponentoftheforcefEq. 12{21 Forsuchasystem,asummationofforcesyieldstheequationofmotion 172

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Basedon( Mann 2003 ,pp.17),thefollowingobservationsmaybemadeontheforcingfunctioninEq. 12{28 Equation 12{28 representsatimedelayeddierentialsystem.Itdoesnothaveaclosedformsolution( Mann 2003 ).Hence,temporalniteelementanalysis(TFEA)isappliedtoobtainstabilityandsurfaceplacementinformation.Thegeneralprocedureforsolutionisgivenin Mann ( 2003 )and Mannetal. ( 2005 ),andthereaderisreferredtothesepublicationsforclaricationofthenotationsthatfollow.ThesolutionprocedureisalsoshownintheintheformofaowchartinFig. 12-2 .UsingthisproceduretheTFEAsolutionisformulatedasfollows [T(j)L(j)]i(j)p(j)dj(12{30) ThedynamicforcingtermappearingwithintheparenthesisinEq. 12{29 ,andwithinthebracketsinEq. 12{30 ,isevaluatedusingEq. 12{22 .Theperiodic(static)forcingterm 173

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12{31 ,isevaluatedusingEq. 12{17 .Thiscompletestheformulationoftheproblem. 7{1 becomes wheredSistheelementaledgelengthinthetool-chipcontactzone.TheexpressionfordSisobtainedbyreferencetoFig. 7-2 Followingthesamederivationprocedureasinchapter 7 ,aresultanalogoustoEq. 7{11 canbeobtained.Now,sixcoecientshavetobeextractedexperimentallyusingthreeequations.Experimentscanbeconductedattwodierentvaluesofradialimmersiontoobtainasetofsixlinearlyindependentequations.Thesetofequationscanbesolvedforsixcoecients.Inthismanner,theaugmentedforcemodelcanbedeveloped. 174

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Dynamicchipthicknessh.Thecomponentsofthedynamicchipthickness,xandy,andtheirrelationtoh,areshowninthegeometricalconstruction.hsistheideal(static)chipthicknessintheabsenceofregenerationofwavinessofthemachinedsurface.Acaseofup(conventional)millingisillustrated. 175

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SchemeforgenerationofstabilitycontourusingTFEAforastructurewithSDOFcompliance.Thereadermayreferto Mann ( 2003 )or Mannetal. ( 2005 )forclaricationoftheterminology. 176

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Asingle,analyticalclosedformexpressionhasbeenderivedforeachofthethreecomponentsofthecuttingforce,inaxedcoordinateframe,forhelicalperipheralmilling.Akeyadvantageofthemethodisthatthealgebratakescareofallthedierentgeometricalpossibilities(TypeI,TypeIIcutting,andPhasesA,B,C)automatically.Separateexpressionsarenotrequired,andthereisnoneedtokeeptrackofthecuttingtypeorphaseincomputations.Thesesingleexpressionswillallowanalyststoconvenientlyusetheminanalyticalorsemi-analyticalapplicationswheresymbolicmanipulationsarerequiredtobeperformed.Anexampleofthisisthederivationofthesensitivitycoecientsintheuncertaintyanalysis. Theresultshavebeenshowntobevalidfortheentireparameterspacecoveringhelicalaswellasstraightutedperipheralmilling,partialorfullimmersioncutting,multipleteethinthecut,aswellasforup-milling,down-millingandslottingbycomparisonwithestablishednumericalmethods.Experimentalvalidationwasperformedbycalibratingthemodelforthealuminumalloy6061-T6usinga45helixendmill.Thedelityofpredictionswasshowntobehigh.Inderivingtheaxialcuttingforcecomponent,theprojectedaxialchipareahasbeenconsidered.Itwasshownthatthismethodleadstogoodpredictionsoftheaxialcuttingforce. Oneofthefeaturesofthemodelisthatthezerohelixassumptionisnotrequiredtobeinvokedwhileextractingthecuttingcoecients.Anotherfeatureisthatafullslottingcutisnotnecessarytoextractthecuttingcoecients.Anyvalueofpartialimmersionsuces.Thiscanleadtosavingsintestmaterialwhencuttingexpensiveworkpiecematerial,orusingmachineswithlimitedspindlepower. Incomputercalculations,forthedegeneratecaseofstraightutedcutters,averysmallhelixanglemustbespecied,astheexpressionsarenotwellbehavedforzerohelixangle.However,analytically,ithasbeenshownthattheexpressionsreduceto 177

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Thoughsomeofthepredictionsmadebythehelicalforcemodelofchapter 7 ,usingaveragecuttingcoecients,wereonlyapproximate,withthemodeloverpredictingpeakforcesforcertainsituationssuchashighimmersionmilling,orfailingtoaccuratelypredicttheskewnessoftheforcecomponents,theinstantaneouscuttingcoecientmodelofchapter 10 wasabletosolvealltheseissuessuccessfully.Inmostapplications,themodelusingaveragecuttingcoecientswouldbeadequate,butwhenveryaccuratepredictionisrequired,theinstantaneouscuttingcoecientmodelisavailable. Remarkshavebeenmaderegardingthefuturedirectionsofwork.Twoproblemshavebeenformulatedforfutureresearch.OneisastabilityandsurfacelocationproblemusingTFEA.Theotherisanaugmentedforcemodelusingedgecoecientsinadditiontocuttingcoecients. 178

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Followingthemethodemployedby Zellner ( 1962 ),themultiresponsemodelforthesimultaneousestimationofthecuttingconstantst;nandt;n,ofEq. 4{13 ,maybewrittenas whereisthenumberofexperimentaldatasets,Kiisa1vectorofthe(logarithmsofthe)cuttingcoecientresponses,iisa21vectorofunknownconstantparameters(thecuttingconstants),"iisarandomerrorvectorassociatedwiththeithresponse,and whereln Inmatrixnotation,theEq. A{1 mayberewrittenwithHappearinginblockdiagonalform: {z }K=0BB@Ht200Hn21CCA| {z }H8>>>>>>><>>>>>>>:ttnn9>>>>>>>=>>>>>>>;| {z }+8>><>>:"t1"n19>>=>>;| {z }"(A{3) Zellner ( 1962 )hasworkedoutanexampleinwhichhehasshownhowtoestimatethemultipleresponseparametersfthetwo-stageAitkenestimatorsgandobtainthevariancecovariancematrixfthemomentmatrixgoftheseestimators.Thatexampleissymbolicallyreproducedherewithsuitablechangesinnotation.RewritingEq. A{3 inasimpliedmanner,thesystemtobeestimatedmayberepresentedas 179

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^UT^U=(KHB)T(KHB)=KTKBTHTHB=0B@KTtKtKTtKnKTnKtKTnKn1CA0B@^TtHTtHt^t^TtHTtHn^n^TnHTnHt^t^TnHTnHn^n1CA=(2)fs0g whosediagonalelementsaretheestimatedcoecientestimatorvariances,ando-diagonalelementsareestimatedcovariances. 180

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Fortheexperimentsanalyzedinthispaper,Ht=Hn,makingthetwo-stageAitkenestimatorsthesameasthesingleequationleastsquaresestimators,butthevariance-covariancematrix,V(),isnotdiagonal,i.e.,thecovariancesarenon-zero. 181

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FouriercoecientsL0;Lk;Mk;T0;Tk;Rk;B0;Bk;andCk,arederivedbyinspectionofFig. 3-3 ,writingthefunctionalrelationshipsvalidforthefundamentalperiod,p2[0;2),andintegratingappropriatelyusingstandardFourierseriesprocedures( Kreyszig 2006 ). 3-3 .ThefunctionalrelationshipbetweenLand FouriercoecientL0 2(B{2) AstraightforwardcalculationyieldsEq. 3{12 FouriercoecientLk DZexexcosd375 182

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DZexexsind375 3-3 showsthatthevariationofTisthesameinTypeIandTypeIIcutting.ThefunctionalrelationshipbetweenTand,withinthefundamentalperiod,[0;2),is FouriercoecientT0 2(B{6) AstraightforwardcalculationyieldsEq. 3{13 FouriercoecientTk DZststcosd+ex+2atan DZst+2atan D(2atan D)cosd375 183

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DZststsind+ex+2atan DZst+2atan D(2atan D)sind375 3-3 .Thefunctionalrelationbetweenband,withinthefundamentalperiod,[0;2),is FouriercoecientB0jTypeI 2(B{10) AstraightforwardcalculationyieldsEq. 3{14 FouriercoecientBkjTypeI DZstDcot Dacosd+ex+2atan DZexDcot

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FouriercoecientCkjTypeI DZstDcot Dasind+ex+2atan DZexDcot 3-3 .Thefunctionalrelationbetweenband,withinthefundamentalperiod,[0;2),is FouriercoecientB0jTypeII 2(B{14) AstraightforwardcalculationyieldsEq. 3{14 185

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DZexDcot DZst+2atan DDcot FouriercoecientCkjTypeII DZexDcot DZst+2atan DDcot Theobservationisthatthecoecientshavethesamealgebraicformforboth,TypeIandTypeIIcutting.TheyneednotbelistedseparatelyinTable 3-1 186

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TheauthorwasborninBhilwara,India,onMarch8,1962.HewasraisedinIndiawhereheobtainedaBachelorofTechnologydegreeinmechanicalengineeringfromtheIndianInstituteofTechnology,Kanpur,in1984.Hethenworkedinsalesandapplicationsofcuttingtools,formingtools,andwear-resistantparts.In2003,hedecidedtogobacktoschooltoaugmenthiseducation.HeiscurrentlyadoctoralcandidatetheUniversityofFlorida. 194