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DeterministicDesignofMachinesusingErrorBudgetMethodsIntheprocessofdesigninganewmachine,thedesignertypicallystartswithideas,sketches,quickcalculations(atleastfirstorderguestimates)offeasibility,andthengetsreadytostartinvestigatingideasinmoredetail.However,beforeinvestingalotoftimeinsolidmodelsandfiniteelementanalysis,itcanbehelpfultomoredeeplyconsiderinadeterministicwaytheperformanceofdifferentconceptseitherwhollyorpartially(e.g.,aparticularaxis).Thetechniqueoffirstordererrorapportionmentamongaxesandcomponentsfollowedbymoredetailederrorbudgetingcanbehelpfultoinvestigatedesignideasandoptions.Firststep:ErrorapportionmentThedesignerneedstohaveanideaofthenumberofaxesneededtoaccomplishthetask,andtheoverallaccuracy(orrepeatability)ofthemachine.Assumingnoparticularaxisshould“workharder”thananothertodeliverperformance,thetotalallowableerrorcanbeapportionedamongstaxesandcomponentswithinaxes.ThiscanbedonewiththespreadsheetAxis_error_apportionment_estimator.xls.Thinkofthisasatreasuremap,orshoppinglist,orhuntinglicense….Itgivesinitialdirectiontothedesigneffort.Ithelpsthedesignerstartthecreativedesignprocesswithanideaofwhatcomponentsmightbemostfeasibletouse.Noneedtousesuperultraprecisionbearingsinadesktopprinterwhensimplertodesignwithslidingcontactwilldo…Remember,timeismoneyandyouarebornandthenonedaytransition…notimetowasteondeadenddesignpathswithnohopeofsuccess!

Thedesignengineerentersthetotalnumberofmovingaxesanticipatedforthemachine(e.g.,forasimple3Dprinter,threeaxes).Thetotalallowableerroristhattheusershouldrealizewhenusingthemachine.Therearefourprincipalerrorsthatoccurineachaxis:Geometricerrorsareerrorsintheshapeofthepartsfromwhichthemachineismadeand


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inaccuraciesinhowtheyareputtogether(e.g.,squarenessofaxes).Thermalerrorsareerrorsduetoexpansionofelements,suchasifthemachineisbuiltinacontrolledenvironmentandusedinanoldbuildingwithnotemperaturecontrol.Loadinducederrorsaredeflectionsofthemachinecausedbythepat,orthemachine’sownweightasitmoves,orcuttingforces.Processerrorsareerrorsintheprocessitself,suchassaggingprintmaterial,orthekerfonawaterjet.Foreacherrortypethedesignerhastorelyonexperience(oreducatedguessbecausedesignisiterative-justtryingtoreduceiterations!)inordertosettheapportionmentfactor.Thisfactorcanbefrom0-1or0-10oranyrange,justbeconsistentandthespreadsheetwillnormalize.Thefactorisameasureofhowdifficultitwillbeforthedesignertocreateamachinetonotexceedallowableerrorofthetype.Onascaleofzeroto1,a1meansthiswillbeachallengingerrorsogivemeallIcanhavetoworkwith.Thespreadsheetthiscalculateshowmucherrorofeachtypeisallottedperaxis.Itdoesthisbasedonlinearsumoftheerrors,rootsquaresum,andaverageoflinearsumandrootsquaresum.Rarelydoalltheerrorsalllinearlysumtobeasbadaspossible(itperhapsdependshowbadlyyourdesignancestorshavesinned?).Usuallysomeare+andsome–sosomecancellationoccurs.Butrarelyareyousoluckythattheycombinewithrootsquaresum.Hencetheexpectedvaluetodesignwithistheaverage.Nexteachaxishasthemajorelementsofbearings,structure,actuator(includingtheservo),sensors,andcables.Theerrorallottedforanaxismustnowbedividedupamongstthesecomponents,andonceagaintheapportionmentfactormustbeenteredbasednexperienceoraneducatedguess.Formostmachinesthecablesdonotpullontheaxisthatmuchandtheaxisisplentystiffsoasmallnumberamountoferrorisallottedforcables.Forawaferstepper,it’sadifferentstory…Wenowhaveanideaofthedimensionalperformancerequirements(e.g.,accuracyorrepeatability)fortheprimaryelementsinthesystem.Whenveryhighperformanceisrequiredwhereitseemslikewewillhavetoworkveryhardtomeettherequirements.Thenextstepistoassigncoordinatesystemsandthentoestimatetheerrormotionsintheelementsandstiffnessofconnections(e.g.,boltedjoints,bearingelements)betweencoordinatesystemsandthecomplianceofstructuralelements.Secondstep:ConceptDevelopmentwiththeaidofanErrorBudgetAnerrorbudgetcanbeusedtofirstassessjustgeometricerrors,andthedesignercanassumeifgeometricerrorscanbemanagedaccordingtolimitsallocatedintheerrorapportionmentphase,thentheload-inducederrorscouldalsobedealtwithbyappropriatecomponentsizing.Thedesignercanthenproceedusingsimplecalculationstosizeelementstosolidmodeltheconcept,andthenuseFEAtopredictdeflectionsandmakesuetheyarewithinthelimitssetbytheerrorapportionmentphase.Thisrequiresaniterativeprocessandsomehopethatwhentheentiremachineisassembledallwillbewell.


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Amoredeterministicapproachistodoasecondroundorerrorbudgetingwhereappliedloadsandthestiffnessofjointsandbearingsandcomplianceofstructuralelementsareaddedtothemodel.Thisdiscreteelementmodeleffectivelyconsiderstheneteffectofgeometricerrorsandloadinduceddeflectionstopredictoverallmachineperformance.Beingspreadsheetbased,thedesignercanrapidlyspottrendsandadjustmemberandbearingsizes(stiffness)accordinglytoconvergeonaworkabledesign.THENthedesigncanbesolidmodeledandcheckedwithFEAagainsttheandthemodelconsidersToillustratethelatterapproach,thisdocumentwillguideusersthroughusingerrorapportionmentanderrorbudgetspreadsheetsbyapplyingthemtoabridgestylemachineintendedforuseasawoodrouterinahomeshop.Wheretostart?Adesigneroftenstartsatthetoolandworksbackwardstothebase.Therationalisthatthedesignershouldhaveanideaofthestiffesttoolthatwillbeusedandthepowerandspeedofthespindle.Figure1showssomestickfigureconceptsfortheto-be-designedrouter.Keepingearlystagedesignsinthehand-drawnphaseenablesthedesignertospendmoretimewithyoungfamilymemberscreatingconceptsinCrayon,asshowninFigure2,aswellaspreventingbusinessoverlordsfromassumingthedesignisfinishedandsotheystarttakingorders…Italsohelpsthedesignertorelaxandmorefreelythinkofdesignoptions…Drawinganimalsandfunshapeswithinadesignhelpskeepyounghelpershappilyengagedandpromotesfree-spiritedcreativethoughtinthedesigner.Itislikeadithersignalthatissometimesusedtohelpaservoovercomefriction.


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Figure1:EarlyStickFigureconceptsforroutersasmotivatedbyconsiderationofFunctionalRequirements,DesignParameters,Analysis,References,Risks,&Countermeasures(FRDPARRC1)

Figure2:Crayons,bigsheetsofpaper,blocks…,andayounghelperdowondersforcreativityandhappinessofallinvolved.Giventhatafullsheetofplywoodisideallytobecut,abridgestylemachineisselectedastheapproachtofirstconsider.Eveninthestickfigurephase,itisagoodideatostartto

1BigAwesome(J)Designacronym(createdtomakefunofTMS-TLAs).SeeforexampleGraham,M,Slocum,A,MorenoSanchezR.,“TeachinghighschoolstudentsandcollegefreshmanproductdevelopmentbyDeterministicDesignwithPREP”,ASMEJournalofMechanicalDesign(SpecialIssueonDesignEngineeringEducation),July2007,Vol.129,pp


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thinkaboutcoordinatesystemsthatwillrepresentthemajorelementsofthemachine.Thebestwaytoassigncoordinatesystemsistothinkofthecentersofstiffnessofthesystem,particularlywherethebearingsattach,whicharesimplyrepresentedinthestickfigure.Startsimpleandthinkintermsofcantileverandsimplysupportedbeams,soyoumighthavemorecoordinatesystemsthanmovingdegreesoffreedom.Thebridgeconceptpaththathasbeendecideduponcanbeenvisionedasbeingmodeledwith7CoordinateSystems:

1. ToolinToolHolder2. Spindle3. Zaxis4. Yaxis5. VerticalRiser6. Xaxis7. Cuttingpointonworkpiece

Asdiscussedinmoredetailbelow,everyCoordinateSystemNhasanorigintowhereitwillattachtothenextCoordinateSystemN+1,andaPointOfInterest(POI)towhichthepreviousCoordinateSystem(N-1)isattached.Figure3showsafirstpassisometricview,stillhand-drawntopreventtooearlylock-intodesigndetailsascanbethecasewhensolidmodelsaretooearlycreatedinthedesignphase.


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Figure3:Whiteboardsketchedbridgemachineconceptsforarouter(createdatwhiteboardbasedoninspirationfromearlysketchescreatedasinFigure2).2AsthebridgestructureinFigure3wassketched,thedesignerstartedthinkingabouthowitmightbebuiltandthusalsomodeled:whereshouldcoordinatesystemsbeplaced?Indeed,manycorrectionsandadjustmentswererapidlymadealongthewayusinganeraserandmarkers.AbigDONOTERASEnoteensuredthatthedesigncouldremainforseveraldayssoitcouldmaturewiththehelpofpeerreviewbycolleagueswhowanderedintotheofficewereveryvaluable.Stickynotesweresometimesaddedandusedtocaptureideasthat

2Canyoufindallthesmileyfaces?Ideallytherewouldalsobesmallwildanimalshiddenwithin.Stayingalittlesillyhelpskeepthedesignerfreethinkingsoastonotgettooenamored,andhenceresistanttochange,withtheirowndesigntooearly.


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werenotquitereadytoreplaceexistinglines.Theprocesswascatalyzedandlubricatedbylunchesandcoffeebreaks.Throughout,theprocessenabledthinkingaboutcoordinatesystemsthatrepresentmotionaxesandconnectionsbetweenmajorstructuralelements.Theentireprocessshowsthatitisgoodtouseawhiteboardforearlydevelopment.This“fuzziness”isanimportantpartofthecreativephase.Analysis(asinequationsandalsocriticalthinkingaboutobtainingcomponents,manufacturing…)canbethoughtofasalensthathelpsbringadesignintofocus.Anoteaboutassigningcoordinatesystems,whichwillbediscussedinmoredetailbelow.EveryCoordinateSystem(CS)hasaPointOfInterest(POI)andaPointOfAttachment(POA),whichbyconventionhereisattheoriginoftheCS.Accordingly,oneCS’sorigin(POA)islocatedatanother’sPOI.Meanwhile,errorsinmachinesaredominatedbysystematicerrors(e.g.,axesarenotboltedtogetherexactlysquare,orrepeatablestraightnesserrorinanaxis),randomerrors(e.g.,bearingnoise),anddeflections.Thelatterareduetolimitedstiffnessatjointsandbearingsandcomplianceinstructures.Itisthuslogicaltofollowthestructuralloopofthemachine,whichtracesthepathofforcesthroughthemachinefromtoolpointbackthroughthemachineandupintotheworkpiecewherethetoolisacting,andplaceaCS’sPOAataconnectionbetweenmajorelements.HenceeachCS’sPOIwillbecollocatedwiththenextCS’soriginwhichisataconnectionbetweenmajorelements…ThestiffnessoftheconnectionbetweencoordinatesystemscanthusberepresentedbyspringsbetweenoneCS’sPOAanditsadjoiningCS’sPOI.Withinacoordinatesystem,thecomplianceofthestructurethatconnectsthePOAtothePOIcanthenberepresentedbyafull(6x6)compliancematrixasdiscussedbelow.FigurativelythesequenceisshowninFigure4.

Figure4:Modelingamachineasasequenceofcompliantbeamsanddiscretespringconnections.


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Oncetheoverallconcepthasmostlycongealed,differentviews,closeupsandcrosssectionsofregionscanbesketchedtohelpthedesignerbetterunderstandasshownforexampleinFigures5and6.Onemethodofdesignsynthesisistostartatthetooltip,becausethat’swheretheactionis,andkeepinginmindtheforcesandaccuracydesired,createelementsalongthestructurallooptoenablethedesiredmotionstooccur,andconceptuallysizetheminaccordancewiththeirdistancefromthetoolpoint.ThisisthephasewhereonemovesfromastickfiguretoatleastquasirealityJ.Followingthislogic,considerthemachineinFigure3andthecrosssectioninFigure5.CS_1’sorigin(POA)islocatedatthecenterofthecolletthatholdsthetool.ThePOIisthetooltipandthecomplianceofthetool(1/stiffness)canbeeasilydetermined.ThestiffnessofthetoolheldinthecolletandthecolletinthetaperedreceiverofthespindleshaftwouldbemodeledasasetoflinearandrotaryspringswhereresultantforcesandmomentsontheCS’sorigincauselinearandangulardisplacementsrespectively.Thismightrequireexperimentationoraliteraturesearchassurelythishasbeentestedbefore.

Figure5:Whiteboardsketchedcrosssectionofthetool,collet,andspindleontheendoftheZ-axisstructure.NoteYaxisispositiveintothepagesoapositiveXdirectionforcewouldcauseanegativerotationabouttheYaxis.CS_2needstobeabletomodelthestiffnessofthespindlebearingsaswellasthespindleshaft.Ifitwerelocatedatonebearingortheother,theangularerrorcreatedbytheresultantmomentfromtheforceatthetoolpointmaynotbecompletelycaptured.Yetthespindleshaftmayalsobeasignificantsourceofcompliance.ThejointbetweenthespindlehousingandendoftheZ-Axisstructureislikelytobefarmorerigidthanthespindlebearings,andhencewewillfocusoureffortsonthespindlebearings.PlacingCS_2atthecenterofthespindleshaft,andmodelingitsattachmentstiffnesstothePOIofCS_3(theendoftheZ-Axis)asthelinearandangular(pitchandyaw)stiffnessofthespindlebearingsandhousingwillaccuratelyreflecttheeffectspindlebearingdeflectionshaveonthemotionofthetooltip.


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Butwhatofthedeflectionsofthespindleshaft,thecompliantstructureconnectingCS_2’sPOAtoitsPOIwhichistheoriginofCS_1?Thespindleshaftconservativelylookslikeasimplysupportedbeamunlessitisaslowspeedspindlewithasignificantsetofbacktobackmountedrollingelementbearingsatitsfront.Ineithercase,lineardeflectionofthesimplysupportedbeamisnotsignificantbecauseCS_1isconsideredlocatedatasupportpoint(thefrontspindlebearing)butangulardeflectionsareimportant:TheslopedeflectionofthespindleshaftatthePOIdirectlyaffectsangulardeflectionandhenceAbbeerrorofthetooltip.SincetherelativepositionofCS_2andCS_1doesnotchangewecanlocateCS_2atthecenterofthespindleshaft,andmodelthecompliantbeamthatconnectsCS_2’sPOAtoitsPOIasasimplysupportedbeamwhoselengthisthatoftheentirespindleshaft.Movingontotherestofthesystem,considerFigure3andFigure6whichshowsasideviewofthemachine,whereitshouldbeclearthatCS_3’soriginisattheCenterofStiffness(COS)oftheZbearings(e.g.,twolinearguide(profilerail)cartridges(genericallycalledbearingblocks)oneachoftwoprofilerails).ThestiffnessofthesebearingswiththelocalstructurewilldominatetheattachmentstiffnessoftheCS_3’sPOAtoCS_4’sPOI.ThecompliancebetweenCS_3’sPOAandPOIwillbethatoftheZ-Axisstructure,sometimescalledtheram,whichcanbemodeledasacantileverbeam,whoselengthchangesinaccordancewiththepositionofthespindle,whichiscontrolledbytheZ-Axisactuator.ThatZ-AxisactuatorwillthusbetheZ-Axiscompliancevalueinthe{6x6}compliancematrix,wheretheotherentriesareduetothoseofacantileverbeam.NoteinthisdesigntheZ-AxisbearingblocksareattachedtoachunkystructuretowhichtheY-Axisbearingblocksarealsoattached.InthiscaseCS_3’sPOA(origin)isattachedtoCS_4’sPOIandtherereallyisnocompliantstructurebetweenthetwo.HencethecompliancematrixforCS_4’s“structure”wouldjustbethenullmatrixofzeros3.

3Inaspreadsheetthisiseasiertodo,becauseoldvaluescanbedeletedandthedefaultvalueis“0”insubsequentcalculationsthatreferencethecells.Elseiftheanalysiswasbasedonusingastiffnessmatrix(insteadofthecompliancematrix)theuserwouldhavetodeletevaluesandenter1E12orsomesuchsuperhugevalue


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Figure6:Whiteboardsketchedsideviewofthemachine.Thebridgeofthemachinerepresentsaclosedstructuralloop,whereasthematrixmethodsthataretobeusedtoanalyzeerrors(HomogeneousTransferMatrices,HTMs)isaserialbasedapproach.ThesolutiontothisquandaryistovisualizehowthemachinedeformsandconsiderthephilosophyofPOAspringsandthecompliancematrixthatrepresentsthestructurebetweenthePOAandthePOI.Withthisinmind,placetheoriginofCS_5atthetopofoneoftherisersthatconnecttheX-Axisbearingstothebridgebeam.Thebridgebeamisconsideredasimplysupportedbeam,whentheY,ZaxisassemblyisallthewaytotherightandthusfarawayfromtheoriginofCS_5onemightthinkthisrepresentsaHUGEmomentthatwillcauseaninordinately(andinaccurate)Abbeerror.However,noteinFigure3howtheaxesdirectionshavebeendonatedasCS_5’andCS_5’’(e.g.,Z’5andZ’’5).Thisindicatesthattheattachmentpointsarefarapart.Giventhattheyareeachverystiff,themomentstiffnessoftwolinearspringsrepresentingtheattachmentpoints,whicharespacedfarapart,isalsoHUGE(seethelinearspringsspacedapartyieldingrotationalspringanalysismethodontheleftsideofFigure4)


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andthustheresultinginaccuracyofanAbbeerrorcausedbyangulardeflectionofthecoordinatesystemisverysmall,andintheworstcasejustaddstotheconservativenatureofthemodel.ThesimplysupportedbeammodelofthebridgeisusedtodeterminethecompliancematrixforthestructurebetweenCS_5’sPOAanditsPOI(whichisalsoCS_4’sPOA).AstheY-AxisservomovestheY,Zaxisassemblyfromlefttoright,theslopewillchangefromnegativetozero(whenitsinthemiddle)andbacktopositive.HencetheAbbeerroratthetooltipwillbeaccuratelyaccountedfor.ThesamephilosophyisusedforCS_6whichmodelstheX–Axis.NotethatthecompliancematrixforthestructurebetweenCS_6’sPOAandPOIisalittlemoreinterestingbecauseofthetworisers.Thisjustmeansthatthebendingstiffnessesarethecombinedstiffnessesofthetworiserelements,butthetorsionalstiffnessabouttheZaxisisbasedonthebendingstiffnessoftherisersspacedapart(analogoustothemethodshownontheleftsideofFigure4).NotethattheeffectivecenterofCS_6isthusatthecenterofthetable.However,intermsofthespreadsheetmodel(andcoordinatesentered)itwillbekeptatthecenterofstiffnessoftheleftsideXbearings.Thefinalcoordinatesystemisthenplacedontheworkpieceonthetableatthepointwherethetoolwillcontactthework.TomodeltheeffectofspringsandcompliancesassociatedwithCS_7takessomethoughtexperiments.Thisisessentiallythesameconditionaswiththebridge,whereCS_4(analogousheretoCS_7)movedalongthebridgeandCS_5’andCS_5’’(analogoustoCS_6’andCS6’’here)werelocatedattheendsofthebridge.Reciprocity!ThePOAforCS_7isthusmodeledasthestiffnessoftheworkpiecefromwherethetoolcontactsittowhereitisattachedtothebed(inmostcasesthisshouldideallybeinfinitewithrespecttothestiffness(1/compliance)ofthetool).ThecomplianceofCS_7fromitsPOAtoitsPOIisthecomplianceofthemachineframeandbedandismodeledasasimplysupportedstructure.However,somefinesseisrequiredtoapplytheresultantforcesandmomentsfromCS_6tothecompliancematrix.TheresultantforcesandmomentsontheoriginofCS_7areappliedtothecompliancematrixtoobtaintheapparentdeflectionsofthePOIofCS_7.Thestructuralloopisthuscompleteandmodeled.Figure7summarizescoordinatesystems1through7andtheirorigins(pointsofattachment)andpointsofinterest.Suchatablecanbehelpfulforthedesignerwhenbuildingthedetailederrorbudget.Indeed,recordingsketchesmadeandfreeformwritingtorecordthethinkingassociatedwithplacementofcoordinatesystemsandmodelingofstiffnessesandcompliancescanbeinvaluabletothedesignerandassociates.Theactofwriting,knowingotherswillread,helpstohonethoughtsandresolvequandaries.Italsomakesiteasierforpeerreviewtohelpidentifyandcorrectissues.


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Figure7:Summaryofcoordinatesystems1through7andtheirorigins(pointsofattachment)andpointsofinterest.“Stiffness”(Compliance)ofelementsbetweenaCS’sPOAandPOIInordertoassignreasonablevaluestothedistancesbetweencoordinatesystemsoriginsfortheconceptshowninFigure3sotheerrorspreadsheetcanbeutilized,asenseofproportionofthesizeofthestructuralelementsisneeded.Firststartwithanestimateoftheloadsonthetool,andgiventheallowableerrorofstructuralelements(Figure1),thestiffnessrequiredofeachaxiscanbeestimated.Giventhatsomeoftheparametersareessentiallya“guess”basedonwhatmayseemreasonable,theresultingstiffnessguestimatesshouldbetreatedasfuzzyestimatessubjecttochangeandbarterasoneideaorelementistradedoffforanother.Thetooltipisthusthepointofinterest.Aspreadsheetisagoodwayforadesignertoquantifyinitialguestimates,asshownfortherouterforexampleFigure8.


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Figure8:Initialassessmentofforcesonthecuttingtool.NextfromthefirstguesserrorapportionmentshowninFigure1,weseeforthethree-axisrouterwearedesigninghere,eachaxis’structureisallocated10micronsoferror(deflection).Thismeansthateachaxismusthaveastructuralstiffnessofabout85.5N/0.010mm=8550N/mm.Assumingfornowthejointsaremuchstifferthananyonebeam,somethingwewillhavetopaycarefulattentiontoduringthedesignphase.GiventhebridgetypestructureofFigure3,for1.3mtravel(soastocuta48”widesheetofplywood)thebridgewouldbeatleast1.4mbetweentheuprights.Remember,thestiffness(compliance)goeslikeL^3sobefrugalwithaddingspaceforwhatseemslikeeaseofdesignorpackaging.Inorderfora1.4mlongsimplysupportedbeamtoyieldtheastiffnessof8550N/mminthemiddleofthebeam,theequationfordeflectionofasimplysupportedbeamloadedbyaforceinthemiddleisFL^3/48EI,asquarealuminumcrosssectionbeammusthaveamomentofinertiaofabout7E6mm^4.Thisseemslarge,butcanbeobtainedwithastandardstockextrudedaluminumsquaretube6inx6in,x1/8”wall,whichwouldbeaveryefficientbeam(highstiffnesstoweight).Tohelpbuilddesignintuition,itsinterestingtoexplorepossibilities.Assumewewerewantingtomachinemetalwithhigherprecisionthanasimplerouterwouldprovideandwewantedastiffnessinoursimplysupportedbeamofabout64kN/mm(8X).Thisrequiresasquarebeamofabout200mmx200mmx12mmwall(stockUSsizeisabout8inx8inx½”wall),orequivalentlya250mmx250mmx6mmwallthickness.Consideringthebeamis1400mmlong,theratiooflengthtoheightofthelatterisabout5.6.A300mmx300x5mmthickbeamwouldhavethesamecrosssectionalareaandbe40%stiffer!ThinkofSaintVenantforbeamsthathavethesamestiffness,thelength/crosssectiondimensionproportionsare7,5.6,and4.7,andtherelativeareas(weights)are1.00,0.65,and0.39respectively.Howtochoose?Thebiggercrosssection,smallerwallthicknessbeamwillbemuchlighterandhencehavehighernaturalfrequency,butcanitbeobtained?Indeedtherearelimitedsizesofstocksquarealuminumtubing,with8inx8inx1/2inwallthelargest.Onehastogotosteeltogetlargertube.A1400mmlong8inx8inx¼”wallsquaresteeltubewould


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haveastiffnessof6392N/mm.Whatisthe“opposite”ofarectilineartube?ARoundtube!Therearefarmoreoptionsinroundtubing,butwhataretheimplicationsforbearingmountingandmountingthetubetotherestfthemachine?A10indiameterroundtubewith½”wallthicknesshasthesamecrosssectionalareaasthe8inx½”wallsquaretube,butis20%stifferinbendingand60%stifferintorsion.Whatwouldyouchoose?Backtotherouter,considerFigure4andFigure8below.ThecompliancematrixrelatesthestiffnessofaCoordinateSystem’sstructurefromthePOItothecenterofstiffness(CoordinateSystemN’sorigin).Thismatrixcanbecreatedbytheuserinthecoordinatesystems’worksheets.A6x6compliancematrixisneededsotheangulardeflectionsofastructureinacoordinatesystemNareaddedtothenextcoordinatesystem’s(N-1)origindeflections.Thenwhentheerrorgains4forCoordinatesystemN-1areapplied,theywilleffectivelyincludetheAbbeerrorscausedbyastructuralmember’sangulardeflectioncausingotherelementsattachedtoittotilt.Ifacoordinatesystemisplacedatthecenterofstiffnessofbearingsthatallowmotion,thecomplianceinthedirectionofmotionisconsideredtobethestiffnessoftheservo(e.g.,ballscrew,coupling,bearingsetc.).AttheoriginofthecoordinatesystemtherewillbesomesortofattachmenttothenextCoordinatesystem’sPOI.AteachCoordinatesystemorigintherearethreeresultantforcesandmomentscomputedfromtheforcesandmomentsinitiallyactingonCS_1’sPOIplusanyadditionalforcesandmomentsaddedalongthewaytoaCoordinateSystem’sPOIororigin.Ifthereareforces,suchasfromcenterofmass,in-between,theycanalwaysberesolvedintoforcesandmomentsappliedtotheorigin.Henceonlythreelinearandthreeangularstiffnessvaluesareneededtodescribetheattachmentofacoordinatesystem(justavector).

Figure9:ImageoftheattachmentpointstiffnessandcompliancematrixforacoordinatesystemSinceuserswilltendtoforgettoentervalues,andthedefaultinExcelforanemptycellis“0”,thecompliancematrixisusedintheerrorbudgetspreadsheet.Italsomakesmore4Somepositiveangularerrorsresultinnegativedisplacements,thussincerandomerrorshavenosign,forallthecoordinatesystems,themagnitudeoftheamplificationofangularerrorsmustbeindividuallycomputedforeachpotentialangularerror.Atableoftheseamplificationfactors,orerrorgains,canthenbeusedtocomputetheAbbeerrorsattributedtorandomangularerrorsabouteachoftheaxesofeachofthecoordinatesystems.Theerrorgainsforthesystematicerrors,ontheotherhand,dohaveasign.


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sensegiventhatwehaveforcesandwanttodeterminedeflections,soX=C*F,whereC=1/k.Notethematrixwillalwaysbesymmetricinthisanalysis.Theelementsforwhich6x6compliancematricesareneededareshowninFigure7.Itisdesirabletohaveaworksheetwithintheerrorbudgetspreadsheetforenteringthedimensionsofmembersandtocalculatethestiffnessmatricessincetheformulaswillbesimilar.NotethatwhenequatingthelengthofabeambyreferringtoacellintheworksheetAxes'CSandLoadsthatsometimesthecoordinatesmaybenegative,butlengthsofbeamsarealwayspositive,sousetheABS()function.Alsonotewhenenteringformulasforcompliancesremembertoconsiderthesign.Forexample,amomentmaycauseadeflectioninanegativedirection5.ItishelpfultohavetheformulasfordeflectionandslopeofcantileverandsimplysupportedbeamssubjecttoforcesandmomentsasshowninFigure10.Itisverydifficulttohaveabeamwithatruezeroslopeconditionatbothends,anditismoreconservativetoassumeasimplysupportedcondition.Noteitisimportanttobecarefulofthesignoftheappliedforceormomentandtheresultingdeflectionsandslopes.Theseformulastakethisintoaccount,suchthatifahorizontalbeamwerealsoattachedtothepointwheretheforceormomentwasapplied,theresultingAbbeerrorwouldhavethepropersign.

5NOTEExcelhasastrange“feature”whereenteringaformulasuchas“-L/(E*I)”returnsapositivevalue!Entering“-1*L/(E*I)”returnsanegativevalue.Thesameistruewhenareferenceisraisedtoapower:-1*B130^2yields–B1302;however,-B130^2yieldsB1302.


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Figure10:Deflectionandslopeformulasforcommonbeamsusedtomodelmachines.Positionoftheforceonasimplysupportedbeamisfromlefttoright.Arighthandcoordinatesystemisassumed.Itisveryimportanttocreatesketchesofbeamsbeingmodeledwiththeirassignedcoordinatesystems,andcomparetheorientationstotherighthandsystemsusedforshown.AcheckoftheequationsshowninFigure10correspondstotheshapesshowninthesketches,whichcanbeverifiedbysupportingandloadinganobjectandobservinghowitdeforms.AsshowninFigure11,thesevaluesrepresentthecompliances,anditisgoodtoseethattheyaresymmetric:theslopecausedbyaforcehasthesamesignandvalueasadeflectioncausedbythemoment.

Figure11:NormalizedresultsofdeflectionandslopeformulasofFigure11appliedtocommonbeamsusedtomodelmachines.Foraservocontrolledaxes,whichcontrolsthepositionofaCSorigin(POA)wrtanothercoordinatesystem’sPOICSorigin(POA@CenterofStiffness),theaxialstiffness(compliance)ofthesystematthePointOfAttachment(POA)willbedominatedbytheactuator,suchasaballscreworbelt.Thecomplianceoftheshaft(orbelt)canbecomputedandthenafactor(typically3fortheattachment(e.g.,nut)tothemovingmemberandthesupportbearings)accountsforthecomplianceofotherelementsusedtoattachit.Figure12showsanimagefromtheworksheetCS_3Z-Axis.


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Figure12:ResultsofdeflectionandslopeformulasappliedtocommonbeamsusedtoWearenowreadytoapplytheabovetoeachofthecoordinatesystemsasshowninFigure7usingtheworksheetbearings,servo,structure.CS_1ToolHolderStructurecompliance(fromacoordinatesystem’sPOAtoitsPOI)FortheToolHolder,seeFigure5,thecomplianceisthatofthetoolwhichismodeledasafractionofthatofacylindertoaccountfortheflutes.Figure13showstheToolHoldersystemcompliancecalculationsdoneintheworksheetCS_1Toolholder.Notehowfiguresarepastedtohelpthedesignercheckforthesignofthecompliancesinaccordancewithcoordinatesystemplacementonthebeamelement.Thisshouldbedoneforeachcoordinatesystem.Notethe“Note:”onthelastcell.Thedesignershouldleavenotesonhowtheymodeledthesystem.Notescanalsobeattachedtoindividualcalculationcells.Notethattheworksheetismadegeneric,soifcertaincellswiththeirentiresarenotneeded,thosecellsshouldjustbecoloredblue.Noneedtodeleteorclearentriesastheymaybeusedinthenextroundofuseofthespreadsheet.

Figure13:CompliancecalculationsfortheToolHoldercoordinatesystem.Attachmentcompliance(POAtonextCSPOI)


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Theattachmentcomplianceforthetoolholder,ataperedcolletforexample,willbeanorderofmagnitudegreaterthanthestiffnessofthetoolitself,andevenofthespindleshaft.Itishelpfulthatwhenevercopyingasetofcells,pasting,andthenupdatingtheformulas,tochangethecellbackgroundcolorasaremindertomakesuretochangeallthecells!Thecompliancematrixreferencesthecellstotheleft,soifadditionalcoordinatesystemsaretobeaddedforanewmachine(orCS),theentireblockcanjustbecopiedandpasted,andthencheckedfortheorientationofthestructurewithrespecttothecoordinatesystemsandthecompliancematrix.Ascellsarechanged,changetheirbackgroundcolorandthuspreventmissingone.CS_2SpindleStructurecompliance(fromacoordinatesystem’sPOAtoitsPOI)Thespindlecoordinatesystemisplacedatthecenterofthespindleshaftbetweenthespindlebearings,asdiscussedindetailinconjunctionwithFigure5above,andtheentriesfordeterminingthecompliancematrixfromtheworksheetCS_2SpindleareshowninFigure14.

Figure14:CompliancecalculationsfortheToolHoldercoordinatesystem.Attachmentcompliance(POAtonextCSPOI)Notethattheforcesappliedtothesimplysupportedbeamareconsideredtobeatthefrontspindlebearingandthustheycannotcreateanangulardeflectionandthecorrespondingcompliancetermscomputetozero,asdotheoffaxismomenteffectsondisplacement.Attachmentcompliance(POAtonextCSPOI)ThestiffnessofthePOAofCS_2tothePOIontheZ-Axisisassumedtobedominatedbythespindlebearings.TheattachmentpointsareassumedtobeattheequatorpointsofthefourbearingsCS_3Z-Axis


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Structurecompliance(fromacoordinatesystem’sPOAtoitsPOI)ThestructurebetweentheZaxisPOAanditsPOI(theoriginoftheSpindle)isdominatedbythetubethatistheZ-Axisstructureitself:TheZ-axisbearingblocksarefixedtoaplatestructure,whichalsoboltstotheYaxisbearingblocks.TheZ-Axisprofilebearingrailsareboltedtothestructure,whichismodeledasanaluminumsquaretube.AsquaretubeisconsideredbecausethecuttingforcescanbeequalintheXandYdirectionsandsoequalstiffnessisneeded.NotethelengthofthestructureistiedtoAttachmentcompliance(POAtonextCSPOI)ThestiffnessofthePOAofCS_3tothePOIontheY-AxisisassumedtobedominatedbytheZ-Axis’bearingcarriages(blocks).CS_4Y-AxisStructurecompliance(fromacoordinatesystem’sPOAtoitsPOI)TheY-AxisPOIistheZ-Axisoriginandthetwoareveryclosetogethersothereisnoappreciablestructuralcompliancebetweenthetwo:Itsjustachunkyplate,sothecompliancesareallsetto“0”.Attachmentcompliance(POAtonextCSPOI)ThestiffnessofthePOAofCS_4tothePOIontheBridgeisassumedtobedominatedbytheY-Axis’bearingcarriages(blocks).CS_5BridgeStructurecompliance(fromacoordinatesystem’sPOAtoitsPOI)Thebridgespanstherisers,andtothebridgeareattachedbearingrailsonwhichtheYaxisbearingcarriagesmove.Thestructuralcomplianceofthebridgestructurethusmustbecalculated.Asdiscussedabove,foraclosedstructuralloopmachine,anopenkinematicmodelcanbeusedbyconsideringtheattachmentpointsforCS_5toCS_6tobewidelyspacedapart(atthetopofeachriser)andthencalculatetheangularstiffnessoftheattachmentaccordingly.Thisgivesahugerotationalstiffnesssotheangulardeflectionwillbeverysmall.Thestructuralcomplianceisthatofasimplysupportedbeam,sowhentheYaxisispositionedinthemiddleofthebridge,theangulardeflectionofthebridgewillactuallybezero.Neartheendsofthebridge,thecompliancebecomeshuge.Attachmentcompliance(POAtonextCSPOI)


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ThestiffnessofthePOAofCS_5tothePOIontheX-Axis(thetopoftherisers)isaboltedconnectionwhichifdonecorrectlyshouldbeasstiffasifwelded.Itismodeledas4pointsspacedatthecornersofthejoint.CS_6X-AxisStructurecompliance(fromacoordinatesystem’sPOAtoitsPOI)FromtheplatestowhichtheX-Axisbearingcarriagesareattacheduptothebridgearetherisers.Thecomplianceofthesestructuresmustbetakentogether,soforexampleintheXdirection,thecomplianceofthetworisersisone-halfofthatofasingleriser.ThetorsionalcomplianceabouttheZ-AxisistheeffectofthelateralcompliancesKoftherisersactingasaforcecoupleoverthelengthLofthebridgeandisthus2/KL2.TherollcomplianceabouttheX-Axisisafunctionofthecompressionstiffnessoftherisersadthebridgelengthandthusisquitehuge.Justbeforewecometotheendofthestructuralloopbackatthetoolpoint,itisimportanttonotethatthereareessentiallytwooptionsforplacementofthebed’sCSorigin.Thefirstistosimplylocateitinthecenterofthebedandcallthatthereferenceframeorigin.Thentheentiremachine’sperformanceiswrtthereferenceframe(centerofbed).ThesecondoptionistoassumethatBedcoordinateaxisoriginisalsoatthetoolpoint,andthusthepositionoftheX-AxiscoordinatesystemwithrespecttothebedshouldbesuchthattheoriginoftheBedcoordinatesystemislocatedatthetoolpoint.Thiscanbeachievedwiththeproductof-1andthesumofthecoordinatesalongthestructuralloopchain.Thisiswhatwillbeassumedforthisdesign.Attachmentcompliance(POAtonextCSPOI)ThestiffnessofthePOAofCS_6tothePOIontheBedisassumedtobedominatedbytheX-Axis’bearingcarriages(blocks).CS_7BedStructurecompliance(fromacoordinatesystem’sPOAtoitsPOI)ThestructureofthebedisassumedtobeasandwichpanelsuchasplywoodbondedtoeithersideofStyrofoaminsulationforahomemachineor1/8”aluminumsheetsbondedtoeithersideofhoneycomb.Thebedreallythusismoreofaplatestructure,butforsimplicitysakehereitisassumedtobeabeamoflengththesameasthebridge,andwidthtentimesitsthickness.Itendsupbeingasstiffasthebridge.TheX-Axisservopositionthusendsupnotbeingasensitiveparameter,whatismoreimportantisthepositionofthespindleascontrolledbytheYaxis.Attachmentcompliance(POAtonextCSPOI)


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ThestiffnessofthePOAofCS_7tothePOIodthe“next”referenceframe,giventhatthenextframeistobeatthetoolpointwouldactuallybetheattachmentofthepartbeingcuttothebed.Ifproperlyattached,itshouldbeverymuchorestiffthanthestiffnessofthebeditselfforexample.DiscussiononthelayoutofthespreadsheettoperformtherequiredanalysisThissectionisunderconstruction…itisnotneededforoperationbyauserofthespreadsheet,butitisgoodtodiscusstheanalysisprocess.Itwillbewrittenincomingweeks….ErrorgainsThefollowingfigureillustrateswhyerrorgainsareneeded,andforsystematicerrors,someofthegainsarenegative


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