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PREPRINTSERIESISSUENO200919STUTTGARTRESEARCHCENTREFORSIMULATIONTECHNOLOGYSRCSIMTECHSIMTECHCLUSTEROFEXCELLENCEPFAFFENWALDRING7A70569STUTTGARTPUBLICATIONSSIMTECHUNISTUTTGARTDEWWWSIMTECHUNISTUTTGARTDESFREYAADADALAUAAVERLAEXPEDIENTMODELINGOFBALLSCREWFEEDDRIVESSTUTTGART,FEBRUARY2011AINSTITUTEFORCONTROLENGINEERINGOFMACHINETOOLSANDMANUFACTURINGUNITSISW,UNIVERSITYOFSTUTTGART,SEIDENSTRAE36,70174STUTTGART/GERMANYALEXANDRUDADALAUISWUNISTUTTGARTDEWWWISWUNISTUTTGARTDEABSTRACTBALLSCREWFEEDDRIVESARETHEMOSTCOMMONLYUSEDMECHANISMTOPROVIDELINEARMOTIONINHIGHSPEEDMACHINETOOLSPOSITIONACCURACYANDTHEACHIEVABLECLOSEDLOOPBANDWIDTHOFSUCHDRIVESYSTEMSAREUSUALLYLIMITEDBYTHESTRUCTURALVIBRATIONMODESOFTHEMECHANICALCOMPONENTSHIGHERORDERPLANTMODELSALLOWFORABETTERUNDERSTANDINGOFTHESYSTEMDYNAMICS,IMPROVETHEDESIGNPROCESSOFFEEDDRIVESANDAREESSENTIALFORTHEDEVELOPMENTOFSOPHISTICATEDCONTROLSTRATEGIESTHISWORKPRESENTSACLEARLYARRANGEDANDVERSATILEMODEL,CAPABLEOFEXPRESSINGTHERELEVANTDYNAMICSOFBALLSCREWFEEDDRIVESKEYWORDSMACHINETOOLSIMULATIONBALLSCREWPLASTICITYINTRODUCTIONDYNAMICALFEEDDRIVESCOMMONLYUSEDINMACHINETOOLSANDPRODUCTIONUNITSREPRESENTCOMPLEXMECHANICALSTRUCTURESWITHMULTIPLEDEGREESOFFREEDOMDOFANDVARIOUSEIGENMODESNUMEROUSSCIENTIFICWORKSHAVEDEALTWITHTHESIMULATIONOFSUCHDRIVESYSTEMS,USINGFINITEELEMENTMETHODSFEMANDCOMPLEXMODELINGTECHNIQUESINORDERREPRESENTTHEDYNAMICALBEHAVIORASDETAILEDASPOSSIBLE13INCONTRARYTHEAPPROACHPRESENTEDINTHISPAPERAIMSATFINDINGANINTUITIVEANDEASYTOUSEMODEL,CAPABLEOFEXPRESSINGTHEMOSTRELEVANTCHARACTERISTICSOFABALLSCREWFEEDDRIVEWITHRESPECTTOTHEIRAPPLICATIONINMACHINETOOLSGENERALLYSPOKENABALLSCREWDRIVECANBECHARACTERIZEDASASYSTEMWITHTHREEDIFFERENTTYPESOFEIGENMODESAXIAL,ROTATIONALANDFLEXURALMODES4TYPICALLYTHEFIRSTAXIALANDROTATIONALMODEOFTHEBALLSCREWSHOWADOMINANTINFLUENCEONTHEOVERALLDYNAMICSWHILETHERELEVANCEOFHIGHERORDERMODESFORMOSTTECHNICALAPPLICATIONSISRATHERSMALLTHEEXCITATIONOFTHEFLEXURALMODESANDTHEIREFFECTONTHEFEEDMOTIONGREATLYDEPENDONFABRICATIONANDMOUNTINGTOLERANCESASWELLASONTHEOPERATINGCONDITIONSOFTHEDRIVESYSTEMPRESUMINGPROPERMOUNTINGANDOPERATINGCONDITIONS,THEEXCITATIONOFTHEFLEXURALMODESARECONSIDERABLYSMALLANDTHEREFOREWILLBENEGLECTEDINTHECOURSEOFTHISPAPERTHETWODOMINANTVIBRATIONALDEGREESOFFREEDOM,NAMELYTHEAXIALANDROTATIONALMODE,AREPRIMARILYINFLUENCEDBYTHEGEOMETRICALDIMENSIONSANDTHEPITCHOFTHEBALLSCREWSHAFTDEPENDINGONTHETRANSMISSIONRATIOASWELLASONTHEOPERATINGCONDITIONSTHESYSTEMCANBEHAVEQUALITATIVELYANDQUANTITATIVELYVERYDIFFERENTINMOSTCASESHOWEVER,THECONFIGURATIONOFAFEEDDRIVEALLOWSFORASIMPLEAPPROXIMATIONOFTHESYSTEMSCHARACTERISTICSINTHEFOLLOWINGAPRACTICALANDAPPLICATIONORIENTEDMODELINGTECHNIQUEFORBALLSCREWFEEDDRIVESWILLBEDERIVEDANDVERIFIEDBYMEASUREMENTRESULTSBALLSCREWFEEDDRIVESFORMACHINETOOLSFIGURE1SHOWSATYPICALBALLSCREWASSEMBLYWITHITSRELEVANTCOMPONENTSINTHISCASESERVOMOTORANDBEARINGSAREFIXEDTOTHEMACHINEBASEANDTHETORQUEOFTHESERVOMOTORISTRANSMITTEDTHROUGHANELASTICCOUPLINGONTOTHEBALLSCREWSHAFTNOTDEPICTEDINFIGURE1ARETHELINEARGUIDEWAYSCONSTRAININGTHEMOVEMENTOFTHEMACHINETABLEINAXIALDIRECTIONTHETRANSFORMATIONFROMTHEROTATIONALINTOTHELINEARMOTIONISREALIZEDBYTHEBALLSCREWSYSTEMWITHITSTRANSMISSIONRATIOI,DEFINEDASTHEDISTANCEOFTRAVELHDURINGONEREVOLUTIONOFTHESHAFT1MANUFACTURERSPROVIDEAGREATVARIETYOFBALLSCREWSYSTEMSWITHDIFFERENTCONSTRUCTIVEDESIGNS,DIMENSIONSANDTRANSMISSIONRATIOSTHEAVAILABLEPITCHESNOWADAYSLIEBETWEEN005AND3TIMESTHEBALLSCREWDIAMETER,WHILEFORMOSTAPPLICATIONSINMACHINETOOLSVALUESBETWEEN025AND1AREBEINGUSEDFIG1SCHEMATICCONFIGURATIONOFABALLSCREWFEEDDRIVETHEDYNAMICALBEHAVIOROFSUCHADRIVESYSTEMISGREATLYINFLUENCEDBYTHEPHYSICALDIMENSIONSOFTHEBALLSCREWSHAFTDIAMETERANDLENGTHOFTHESHAFTHAVEADECISIVEIMPACTONTHEOVERALLRIGIDITYANDTHEREFOREONTHEDYNAMICALBEHAVIOROFTHEDRIVESYSTEMFIGURE2SHOWSAMEASUREDFREQUENCYRESPONSEOFABALLSCREWDRIVE,DERIVEDFROMTHECOMPARISONOFMOTORANDMACHINETABLEVELOCITYFIG2MEASUREDFREQUENCYRESPONSEOFABALLSCREWFEEDDRIVECLEARLYNOTICEABLEARETHETWODOMINANTEIGENFREQUENCIESF1ANDF2,CHARACTERISTICALFORBALLSCREWFEEDDRIVESTHEFIRSTEIGENFREQUENCYINFIGURE2REPRESENTSANAXIALOSCILLATIONOFTHEMACHINETABLEWHILETHESECONDEIGENFREQUENCYCANBEPRIMARILYASCRIBEDTOANANGULARDEFLECTIONOFTHEROTATIONALSYSTEMALTHOUGHEACHMODEISARESULTOFAXIALANDANGULARDEFORMATIONS,ITISCONVENIENTTOCLASSIFYANAXIALANDAROTATIONALMODEDEPENDINGONTHEPREDOMINANTDEFORMATIONINBOTHCASESTHEFLEXIBILITYOFTHEBALLSCREWSHAFTHASAMAJORINFLUENCEONTHEABSOLUTEVALUEOFTHEEIGENFREQUENCIESTHISISDUETOTHEFACTTHATTHESHAFTREPRESENTSTHECOMPONENTWITHTHELEASTRIGIDITYWITHINTHEDRIVESYSTEM5FURTHERMORETHEPROPERTIESOFTHEBALLSCREWSHAFTARENOTCONSTANTBUTCHANGEOVERTHETRAVELRANGEOFTHEMACHINETABLEATTHISPOINTITSHOULDBENOTEDTHATFORMOSTTECHNICALAPPLICATIONSTHEEIGENFREQUENCYOFTHEAXIALMODEISARTICULATELYSMALLERTHANTHATOFTHEROTATIONALMODEONTHISACCOUNTTHEAXIALMODEISOFTENREFERREDTOASTHEFIRSTEIGENFREQUENCYOFABALLSCREWFEEDDRIVETHESTRUCTURALVIBRATIONMODESOFTHEMECHANICALSYSTEMAREOFPARTICULARIMPORTANCEWHENCONSIDERINGTHEDYNAMICALBEHAVIOROFAPOSITIONCONTROLLEDFEEDDRIVEESPECIALLYINHIGHSPEEDAPPLICATIONS,HIGHACCELERATIONANDJERKVALUESCANEXCITETHEMECHANICALSTRU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HEFIRSTEIGENFREQUENCYINCOMPARISONTOTHATTHESECONDEIGENFREQUENCYSEEMSTOCHANGERATHERINSIGNIFICANTLYWITHDIFFERENTOPERATINGCONDITIONSBOTHMODELSSHOWTHATTHEROTATIONALMODEREMAINSALMOSTCONSTANTFORAGREATRANGEOFPARAMETERSTHEDIFFERENCEBETWEENHYBRIDANDDISCRETEMODELEXPRESSEDINFIGURE5CANBEASCRIBEDTOTHESIMPLIFIEDREPRESENTATIONOFTHESHAFTINTHEDISCRETEMODELFIG5PERCENTALDIFFERENCEBETWEENHYBRIDANDDISCRETEMODELOFTHEBALLSCREWDRIVEFROMFIGURE5ITCANBESEENTHATTHEABSOLUTEDIFFERENCEASWELLASTHEPARAMETERSENSITIVITYOFTHEEIGENFREQUENCYGREATLYDEPENDSONTHEINTERACTIONBETWEENAXIALANDROTATIONALMODE,DETERMINEDBYTHETRANSMISSIONRATIOOFTHEBALLSCREWWITHINCREASINGTRANSMISSIONRATIOTHEINFLUENCEOFTHEROTATIONALRIGIDITYOFTHESHAFTONTHEOVERALLAXIALRIGIDITYINCREASESTHEPOSITIONDEPENDENT,EFFECTIVEAXIALRIGIDITYCANBEEXPRESSEDASFOLLOWS7INMANYCASESHOWEVER,THEINFLUENCEOFTHEROTATIONALRIGIDITYONTHEAXIALMODECANBENEGLECTEDDUETOTHETRANSMISSIONRATIOOFTHEBALLSCREWWITHTHEEFFECTIVEAXIALRIGIDITYANDTHEKNOWLEDGEOFTHECONSISTENCYOFTHEROTATIONALMODE,THETWOCHARACTERISTICEIGENFREQUENCIESOFABALLDRIVECANBEAPPROXIMATEDUSINGEQUATIONS8AND989DESPITETHECHOSENSIMPLIFICATIONSTHEDISCRETEMODELSHOWSGOODRESULTSFORTHEFIRSTTWOEIGENFREQUENCIESOFABALLSCREWSYSTEMTHISALLOWSFORANINTUITIVEANDCLEARLYARRANGEDSIMULATIONOFTHEDYNAMICALBEHAVIOR,VALIDFORAGREATRANGEOFPARAMETERSANDWELLSUITABLEFORMOSTPRACTICALAPPLICATIONSINTHEFOLLOWINGCHAPTERTHESIMULATIONRESULTSOFTHEDISCRETEMODELDEPICTEDINFIGURE4WILLBECOMPAREDTOMEASUREMENTSONANACTUALMACHINETOOLFEEDDRIVEEXPERIMENTALVERIFICATIONOFTHEMODELATTHEINSTITUTEFORCONTROLENGINEERINGOFMACHINETOOLSANDMANUFACTURINGUNITSISWATTHEUNIVERSITYOFSTUTTGARTEXISTSAGREATVARIETYOFTESTBENCHESWITHDIFFERENTSIZEBALLSCREWSANDNUMEROUSTRANSMISSIONRATIOSWHILETHERESEARCHONTHISTOPICISBASEDONEXPERIMENTSWITHDIFFERENTBALLSCREWTYPES,THEFOLLOWINGMEASUREMENTSFOCUSONATYPICALBALLSCREWSETUPFORMACHINETOOLSACCORDINGTOTHESPECIFICATIONSINTABLE3THEPARAMETERSUSEDFORTHESIMULATIONCOMPLYWITHTHOSEOFTHEEXPERIMENTALSETUPTHEVALUESFORRIGIDITYANDMASSAREBASEDONMANUFACTURERSDATAWHILETHEDAMPINGFACTORSCORRESPONDTOEMPIRICALVALUESTABLE3SPECIFICATIONSOFTHETESTBENCHWITHTHEGIVENFEEDDRIVEVARIOUSMEASUREMENTSHAVEBEENCONDUCTEDANDCOMPAREDTOTHESIMULATIONRESULTSFROMTHEDISCRETEMODELFIGURES6AND7EXEMPLARILYSHOWSIMULATEDANDMEASUREDFREQUENCYRESPONSESFORTHEGIVENFEEDDRIVEATDIFFERENTOPERATINGCONDITIONSFIG6SIMULATEDANDMEASUREDFREQUENCYRESPONSEFORTHEFEEDDRIVEATDIFFERENTMACHINETABLEPOSITIONSNOMINALDIAMETERBALLSCREWD040MMBALLSCREWLEADPH20MMDYNAMICALLOADVALUECN54600NNOMINALBALLDIAMETERDW6MMNUMBEROFBALLTRACKTURNS5TRAVELRANGE700MMMASSMACHINETABLE300600KGFIG7SIMULATEDANDMEASUREDFREQUENCYRESPONSEFORDIFFERENTMACHINETABLEMASSESTHEDISCRETEMODELUSEDTODESCRIBETHEMECHANICALSYSTEMSHOWSGOODCONCORDANCEWITHTHEBEHAVIOROFTHEACTUALFEEDDRIVEDESPITEOFTHECHOSENSIMPLIFICATIONSANDTHEIMPRECISENESSOFTHEMANUFACTURERSDATA,AGOODPREDICTABILITYCOULDBEACHIEVEDFURTHERMORETHEPREDICTEDPARAMETERDEPENDENCYOFTHEEIGENFREQUENCIESISCONFIRMEDBYTHEMEASUREMENTRESULTSTHEFREQUENCYOFTHEAXIALMODECHANGESDEPENDINGONMACHINETABLEMASSANDPOSITIONWHILETHEROTATIONALMODEOFTHEBALLSCREWDRIVEREMAINSALMOSTCONSTANTTHEINVARIANCEOFTHEROTATIONALMODETOWARDSOPERATINGCONDITIONS,ASALREADYDETECTEDINTHEFOREGOINGSIMULATIONS,CANBEEXPLAINEDBYTHELOWPASSCHARACTERISTICOFTHEMECHANICALSYSTEMAFTERTHEFIRSTEIGENFREQUENCYTHESYSTEMEXPERIENCESANAMPLITUDEDROPOFFOF20DB/DECADEHENCETHEENERGYIMPOSEDBYTHESERVODRIVEATHIGHERFREQUENCIESISMERELYTRANSFORMEDONTOTHEMACHINETABLETHEAXIALMOTIONOFTHETABLEATTHESECONDEIGENFREQUENCYISTHEREFOREQUITESMALLDUETOTHEKINEMATICCOUPLING,THEANGULARDEFLECTIONOFTHESHAFTWILLSTILLLEADTOANAXIALDISPLACEMENTWHICHINTHISCASEISPREDOMINANTLYEXECUTEDBYTHEDEFLECTIONOFTHESHAFT,BEARINGANDMOUNTINGCONSIDERINGTHEMINORCHANGESINTABLEPOSITIONATTHEROTATIONALMODEOFTHEBALLSCREWSYSTEM,ITISINTELLIGIBLETHATTHENEITHERTHEMASSNORTHEEFFECTIVEPOSITIONOFTHETABLEHAVEARELEVANTINFLUENCEONTHESECONDEIGENFREQUENCYOFTHEOSCILLATINGSYSTEMTHEPREDICTABILITYANDSIMULATIONOFTHEDOMINANTEIGENFREQUENCIESOFABALLSCREWDRIVEANDTHEIRDEPENDENCYTODIFFERENTOPERATINGCONDITIONSPROVIDESABETTERUNDERSTANDINGOFTHESYSTEMDYNAMICSANDISBENEFICIALFORTHEDESIGNPROCESSOFTHEMECHANICALCOMPONENTSASWELLASFORTHEIMPLEMENTATIONOFADEQUATEFEEDBACKCONTROLLERSINPARTICULARTHECONSISTENCYOFTHEROTATIONALEIGENMODEISQUITEFAVORABLEFORTHEDESIGNOFVIBRATIONREJECTIONANDCONTROLALGORITHMSSUCHASNOTCHFILTERSCONCLUSIONINTHISPAPERTWOWELLSTRUCTUREDANDCLEARLYARRANGEDMODELSFORBALLSCREWFEEDDRIVESHAVEBEENPRESENTEDADISCRETEANDAHYBRIDDISCRETECONTINUUMMODELBYSEPARATINGTHEPHYSICALCHARACTERISTICSOFTHESHAFTINTOANAXIALANDAROTATIONALSYSTEM,THEBALLSCREWDRIVECOULDBEMODELLEDEXCLUSIVELYUSINGDISCRETEBLOCKSINMATLAB/SIMULINKTHROUGHCOMPARISONWITHTHEMORECOMPLEXHYBRIDMODELITCOULDBESHOWNTHATTHESIMPLIFIEDDISCRETEMODELISWELLSUITABLEFORPREDICTINGTHEMOSTRELEVANTEIGENFREQUENCIESOFAFEEDDRIVEOVERAGREATRANGEOFPARAMETERSSIMULATIONRESULTSSHOWTHATTHEBALLSCREWSHAFTHASADECISIVEIMPACTONTHEOVERALLDYNAMICSOFSUCHADRIVESYSTEMFURTHERMORETHESECONDRELEVANTEIGENFREQUENCYPROVESTOBELESSSENSITIVEWITHRESPEC
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