44 

AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume15|Number1|March2020|pp44–56 Journalhome:apem‐journal.org

https://doi.org/10.14743/apem2020.1.348 Originalscientificpaper

  

A comparison of the tolerance analysis methods in the open‐loop assembly 

Kosec, P.a, Škec, S.a,b,*, Miler, D.a aUniversity of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Zagreb, Croatia bTechnical University of Denmark, Kongens Lyngby, Denmark   

A B S T R A C T   A R T I C L E   I N F O

Dimensionalandgeometrictolerancesaffectboththecostandthefunctionali‐ty of a given product. Finding the acceptable trade‐off between the two isamong the commonengineering tasks.Thus,many tolerance analysismeth‐ods are developed to help engineers and assist in the decision‐making pro‐cess.Inthisarticle,theauthorshaveassessedfourtoleranceanalysismethodsby applying them to the open‐loop assembly. The results obtained by thetolerance chart (worst‐case) method, Monte‐Carlo simulation, vector‐loopanalysis,andtheUnifiedJacobian‐torsormodelwereanalysedandcompared.Additionally,theoverviewandapplicationguidelinesareincludedforeachofthemethods, aiming tohelpboth researchersandpractitioners.The resultshaveconfirmedthattherearesignificantvariationsintheoutputsacrosstheobservedmethods,implyingtheneedforinformedmethodselection.

©2020CPE,UniversityofMaribor.Allrightsreserved.

  Keywords:Assembly;Open‐loopassembly;Toleranceanalysis;Computeraidedtolerancing;Tolerancechartanalysis;UnifiedJacobian‐torsormodel;MonteCarlomethod;Vector‐loopanalysis

*Correspondingauthor:stanko.skec@fsb.hr(Škec,S.)

Articlehistory:Received6September2019Revised11March2020Accepted17March2020

  

1. Introduction 

During the design phase, tolerances are assigned to nominal dimensions, ensuring successfulassemblywhile retaining themanufacturing costs at anacceptable level.As the complexityofmechanical design increases, keeping trackof the tolerancesbecomesharder.Tomitigate theproblem, tolerance analysismethods of various complexity are available. Themethods rangefromsimple,1Dtolerancechartanalysis,toadvancedproceduresrequiringtheuseofadvancedmathematicalmodels.Examples includevector loop,UnifiedJacobian‐torsor,T‐maps,andSkinmodelShapes.Furthermore, toleranceanalysismethodscanbedividedby several criteria: anapproachtotheanalysis,identificationprocess,andthecalculationprocedureofdependentdi‐mensions[1]. Beforeanalysing,tolerancesareassignedtoassemblyfeaturesandareorganisedintostacks,easingthevariationanalysis.Stacksarethenusedtoanalysetheassemblybyreadingthedraw‐ings or by assigning tolerances on computer‐aided drawing (CAD). The tolerances are thenstacked into loops usingpoints, surfaces, vectors, or joints, amongothers –depending on themethod[1].Manualchartingisfrequentlyusedwhensolvingsimpleproblemsconsistingoffewdimensions.Asthenumberofdimensionsincreases,itsreliabilitydecreases–itiserror‐proneandtiresome.Additionally,manualanalysisishardtoperformin2Dand3Dtolerancingprob‐lems.

A comparison of the tolerance analysis methods in the open‐loop assembly 

Advances in Production Engineering & Management 15(1) 2020  45

The tolerancing problem complexity further increases when the geometric tolerances arenecessary [2].Geometric tolerances aredefinedby3D tolerance zones, renderingmost of thesimplermethods unusable. Thus, computer‐aided tools (CAT)were developed, increasing thecapabilities in terms of the number of available approaches andmathematical models. Manysuch toolsaredevelopedandsuccessfullyapplied(VisVSA,3DCS,CETOL,OpTol) [3] in the in‐dustrialenvironment.Unfortunately,variousproprietaryCATtoolsusedifferentmathematicalmodelstodefineandanalysetolerances,meaningthattheobtainedresultsmaydiffer[4]. State‐of‐the‐artCATtoolsallowuserstomodelassemblystackswithpoint‐to‐pointfeatures.Thecontributingtolerancesareidentifiedandarrangedintosuitablestacksorloops[5]aseachmethodiscompatiblewithaspecificstackingproceduretobuildthestackingequation.Inrecentpapers,manyresearchershavestudieddifferencesandsimilaritiesoftoleranceanalysismeth‐ods. Studies considered the contributing tolerances frommultiple directions [6], the angulardeviationoftheadjustableelement,oracriticalassemblyfeature(functionalrequirement)[6].Also,theform[1]andinteractionofthemultipletolerancesinthe3Dcontextisdefinedbythegeometricdrawingandtolerancing(GD&T)standards[1,10].DuetofrequentchangesinGD&Tstandards[7]suchasISO8015[2]andASMEY14.5[8,9],continuoussupportofthetoleranceanalysismethodsisneeded. Various assembly applications are described as a system of open‐loop or closed‐loop thatmustbesolvedtogether.Theopen‐loopdescribesadimensionstackterminatedwithagaporacriticalassemblyfeature.Theclosed‐loopdefinesaclosureconstraintfortheassembly,implyingthatadjustableelementsareintheassembly.Thus,thecriticaldifferencebetweentheopen‐loopandclosed‐loopassemblies is theexistenceofgap; in theopen‐loopassemblies,weanticipatethatgapdimensionmustbeproperlytolerancedtoallowustoformanengineeringfitwithan‐otherpart(fortheschemaoftheopen‐loopassembly,(pleaseseeFig.3).Thoseelements,gaporfunctionalrequirement,aretheresultofparttoleranceaccumulation.Iftherearenoadjustablecomponents,thereisnoneedforclosed‐loops–theassemblymodeliscomposedonlyofopen‐loops[2]. Inrecentstudies,methodsfortoleranceanalysiswerecomparedusingtheclosed‐loopexam‐ples.Theaimwastodeterminetheadvantagesandshortcomingsofeachmethod,alongwiththedifferences inoutput (e.g. [10‐15]).To thebestofourknowledge,mentionedresearchstudieshavenotconsideredtheopen‐loopassemblies.Hence,thecontributionofthearticleathandistheevaluationofthetoleranceanalysismethodsonopen‐loopproblems.Furthermore,besidesthescientificcontribution,thisarticleaimstoprovidethepractitionerswithasimplereviewandguidelinesfortheapplicationofeachmethod.Toachievethis,wehavecomparedfourdifferentmethods:tolerancechartmethod,MonteCarlomethod,vectorloopmodel,andUnifiedJacobian‐torsormodel.Eachmethodwasappliedtoanopen‐loopassembly,allowing forcomparison inperformancesandoutcomes.

2. Methods and materials 

Inthisresearchstudy, fourtolerancingmethodswerecompared: tolerancechartmethod[16],MonteCarlomethod,vector‐loopmodel[15],andUnifiedJacobian‐torsor(seeSection2.1).Eachmethodisdescribed,alongwiththestepsnecessarytoapplyit.Thoseincludetolerancingprob‐lemidentification,mathematicalmodelling,andcalculationprocedures.

2.1 Used tolerance analysis methods 

Tolerancechartmethod isthemost frequentlyusedtoleranceanalysismethodintheindustry[16],mostlyduetoitssimplicity.Itiswidelyusedforsolvingproblemsconcernedwithdimen‐sionaltolerances,althoughtherecentimprovementsenableditsapplicationtogeometrictoler‐ances [15].Themethod isone‐dimensional; inorder toapply it to themulti‐dimensionalgeo‐metrictolerances,theymustbeconvertedto1Dspace[15]. Tolerancechartmethodcanbeperformedonboththepartandassemblylevel.Forassemblylevel,partsincludedinthetolerancechainrepresentoneofthetoleranceend‐points(maximumorminimum).Eachpartisplacedagainstitsmatingpartinoneofitstoleranceend‐points.Asa

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result,theworst‐casetolerancechartmethodillustratestheminimalandmaximalvariationofafunctionalrequirementbasedonthevaluesinthetolerancechain[9]. Whenperformingthetolerancechartmethodanalysis,thefirststepistosetagoalbylabel‐lingthechainstartingandendingpoints[16].Thestartingpointisselectedononeedgeandtheending point on the opposite edge of the analysed feature (see Fig. 1). The chain indicator isplacedtodeterminethedirectionofthedimensionvectorandiseitherpositiveornegative[17].Thevectorpointingtowardthechainend‐pointismarked“⊕”,andthevectorpointingoppositeoftheend‐pointismarked“⊖“.Theindicatorshowswhethertoaddorsubtractdimensionsandtolerancesduringthestackcalculation.Additionally,itsimplifiestheinterpretationoftolerancechart results [16]. The resultingdimension chain is the shortest possible and consists only ofknowndimensions‐dimensionssetbydesigners. Tolerancechartmethodwasusedinrecentstudies[5,6,9,15‐17],mostlyasareferenceforthecomparisonofadvanced toleranceanalysismethods. Itsmost importantadvantage issim‐plicity;nocomputationaltoolsareneededasitcanbecarriedoutbyhand.Thedownsideisthattheuserhastokeepinmindallthestandardrules[2,8]forcreatingthestacks,makingthepro‐cesserror‐prone.Besides,thetolerancechartmethodcreatesstacksinonedirectionandignoresthecontributionsofothers,possiblyprovidingunsatisfactoryresults.

Description � � TOL

Dimension1 50 ±1.5Dimension2 12 ±0.5Dimension3 14 ±0.7Dimension4 9 ±0.2Sum 50 35 ±2.9

⊕ ⊖

50 35 15

2.9

Finaldimension:

15 2.9Fig.1Tolerancechartmethod

Aplethoraofstatisticalapproacheswasintroducedtoconductnon‐linearstatisticaltoleranceanalysis.A typicalexample is theMonteCarlosimulation(MCS)basedon thealgorithmof thesamename.Itutilisesrandomsamplinginputvaluestocalculatetheoutputresults.Foragiveninputvector ,thenumberofsamplingvalues isdetermined , , … , .Byusingthemath‐ematical model (transfer function) new output vector of same length is found

, , … , . Finally, the output results are analysedby calculating statistical data such asmean,standarddeviation,orrange. MonteCarlosimulation(MCS)isabeneficialtoolfortoleranceanalysisofmechanicalassem‐blies.Itsmainadvantageisflexibilityandabilitytousevariousnon‐normalinputoroutputdis‐tributions[18].Alargesetofsamplepartsiscreatedbyrandomlyassigningatolerancevaluetoeachnominaldimension.Valuesareselectedwithinthetoleranceintervaltosimulatethemanu‐facturingvariation[18].Theprocessisrepeateduntilenoughoutputdataisacquiredtoenabletheuseofstatisticaltechniques.Itallowsthecalculationofthemeanvalue,standarddeviation,range,upperandlowerspecificationlimit,andshareofrejectedsamples[19].

Definetheproblem

Assigntheexpectedtolerancedistributiontoeachdimension

Estimatetherequirednumberofruns

Analysethedata

Applytransferfunction

Randomlygeneratetolerances(atinput)

Fig.2MonteCarlosimulationforimplicitassemblyconstraints

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Advances in Production Engineering & Management 15(1) 2020  47

Inthisarticle,MCSisappliedasanextensiontotheTolerancechartmethod.AmodifiedformofMCS(McCATS)accountingfortheimplicitassemblyvariationswasused,assuggestedin[20].Inthemodifiedsimulation,therandompartsaresenttotheassemblyfunction,whichiterativelysolvesthetolerancechartequationsforthedependentassemblyvariations[20].Theprocessisrepeated until a sample of a suitable size to produce the assembly histogram is created. ThestepsnecessarytocarryoutthetoleranceanalysisusingtheMCSareshowninFig.2.

Thevector loopmodel isastack‐uptechniqueusedtoextendthestackanalysis to twoandthree‐dimensional assemblies [1]. The ideaof the vector loopmethod is touse vectors tode‐scribethedimensionsandassociatedtolerances.Vectorsarearrangedinloopstodeterminetheassemblydeviations.Toleranceanalysisproblemsaresolvedusingthekinematicconcept;con‐tactpointsaresetaskinematicjoints.Anumberofpossiblemotionsisdefinedforeachjoint(i.e.degreesof freedom),alongwiththe localdatumreferenceplane.Threetypesofvariationsaredescribed in vector loopmodel: dimensional variations (lengths and angles), kinematic varia‐tions(smalladjustmentsbetweenmatingpoints,joints)andgeometric/featurevariations(posi‐tion,roundness,angularity)[1]. Dimensionalandgeometrictolerancesaredescribedasadditionaldegreesoffreedomonthekinematic joints [1]. Kinematic simplification is required to represent geometric tolerances insuch way. Thus, in the vector loop model, geometric tolerances are included only at matingpoints, in thedirectiondefinedby the typeofkinematic joint [1].Theyaredescribedasaddi‐tional translational and rotational transformations (displacementvectors, rotationmatrices)–asgapswithzero‐lengthnominaldimensionvectors. Theassemblygraphisadiagramthatrepresentstheanalysedassembly,includingitsparts,dimensions,matingconditions, functionalelements,andfunctionalrequirements.Thegraphisusedtorepresentanylineardimensionintheassemblyasavector(seeFig.3).Vectorsarecon‐nectedandformchainsorloops,reflectinghowassemblypartsstack‐uptogether.Theassociat‐edtoleranceisincludedasasmallkinematicadjustmentofsuchavector(gap)[1,12].Suchrep‐resentationallowsustodeterminethefunctionalrequirementsofanassembly.Stack‐upfunc‐tionsarebuiltbyincludingthevectorvariationsinvolvedineachchainintoimplicitkinematicequations.Assuch,theycanthenbesolvedusingvariousmathematicalapproaches[1,6]. Foreachpartinthetolerancechain,alocaldatumreferenceframe(DRF)isaddedtoidentifytherelevantfeaturesofapartfortoleranceanalysis.DRFsarethenconnectedusingdatumpathsrepresentinggeometric layouts,whichdefine thedirectionandorientationof vectors formingthe loop[1].Theyarecreatedbystackingandchainingthedimensionsthat locate thecontactpointbetweentwoparts.Aftercreatingdatumpaths,thevectorloopscanbecreatedbyconnect‐ingdatums.Loopscanbeopenorclosed,dependingonthefunctionalrequirementofthetoler‐anceanalysis.Thenumberofclosedloopsiscalculatedas 1,where isthenumberofthematingpoints,and thenumberofparts.

A

B

C

D

Box

Circle

Square

L1L2

L3

gg

Fig.3Assemblygraphandtheexampleofvectorloops

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48  Advances in Production Engineering & Management 15(1) 2020

Afterdefining thevector loops, the calculation is carriedout [1,11].Whenconsidering theclosed‐loop problem, the equations are often non‐linear; theymust be linearized using directlinearizationmethod[1,11],producingapproximateresults.Thus,vector loop,whenusingdi‐rect linearization, is unable to generate true worst‐case results [4, 11]. When the open‐loopproblemisconsidered,deviationsarecalculateddirectlyusingexplicitequations[11].

UnifiedJacobian‐torsor(JT)method[21]isa3Dtoleranceanalysismethod.ItusestheJaco‐bianmatrix to relate the functional requirement (FR) and virtual joints displacements. JT ad‐vancesthepunctualsmall‐displacementvariablesof theJacobianformulationtorepresenttol‐erancezonesusingthetorsormodelandintervalarithmetic.ItoffersmoreoutputinformationontheFR,reducingthesizeoftheanalysedmodelsinceitisnolongerpoint‐based[21]. Torsormodelusessmalldisplacementscrewstoestablishtolerancezonesofpoints,curves,andsurfaces[21,22].Eachrealsurfaceismodelledbyasubstitutionsurfacedefinedbyasetofscrewparametersthataremodellingthedeviationsfromnominalgeometry[23].Screwparam‐etersarearrangedintorsorscontainingtranslationalcomponentsofapoint , , and , , asrotationalcomponentswithrespecttothenominalgeometry:

, (1)

where isDRFusedtoevaluatethescrewcomponents.Torsormodelcanfullydefinethetoler‐ancezonesduetoitsabilitytoshapespatialvolumeswithinwhichthesurfacesaredeviating[10]. TheprocedureofUnifiedJacobian‐torsormethodconsistsof4steps[24].Thefirststepistoidentifyallfunctionalelements(FEs)affectingtheFRbydistinguishingkinematicchainsinvolv‐ingthefunctionalconditionorpartunderstudy.Functionalelementcanbeanypoint,curveorsurfaceofapartandcreatesinternalorkinematicpairs[21].Thesecondstepistoassociateatorsoror screwparameter to each element (surface, axis) of thekinematic chain.Torsors ex‐pressthedegreesoffreedomandtheallowableelementdisplacementsandtheirbounds.Smalldisplacementsareapplied toparts’ geometrical featuresaffecting theFR [21], afterwhich theJacobianmatrix is used to determine relative positions and orientations of torsorswithin thechosen kinematic chain (step three) [22]. The final step is to combine torsor and Jacobian toprovideamatrixequation.Solvingaresultingmatrixusing intervalalgebraprovides the func‐tionalconditionbounds.

2.2 Assembly model for case study 

Theabove‐describedmethodswerecomparedbyanalysinga3Dtolerancingproblem.Theas‐semblyconsistingofthecantileverandtherotatinghandle(openloop)wasusedasanexample.Thus,boththedimensionalandgeometrictoleranceswereconsidered.Thefunctionalrequire‐mentdeviationsareassessedusingeachofthemethods,whiletheresultsarecomparedinSec‐tion3.5.Thenominaldimension(DIM),upperdeviation limit(UDL),and lowerdeviation limit(LDL)were calculated.Thecomparison is focusedon the similaritiesanddifferencesbetweenresultsobtainedbyeachmethod.Differencesinproceduresandcalculationapproachesarealsoobserved.

Asimplerotatinghandleassemblyconsistingoffourpartswasusedtocarryoutthecompari‐sonbetween themethods (seeFig.4).Thepole (2) is fixed to thebottomplate (1),while thelever (3) ismountedonto the journal locatedon thepole.Thehandle (4) is installed into theborelocatedonthelever.Toleranceswereassignedtoallthedimensionsapartfromadistancebetweenthehandle(4)andthebaseplate(1),whichisselectedasafunctionalrequirement).

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Advances in Production Engineering & Management 15(1) 2020  49

Fig.4Casestudymodel

Thepositionaltolerancebetweenthetopsurfaceofthebaseanditscut‐outwasinclud‐ed.Thecontactbetween thebaseand thecylindricalbaseof thepole is considered ideal.Theparallelismtolerancebetweentheaxisofthecylindricalpinlocatedonthepoleandthebottomsurface of the polewas also included. The lever ismounted onto the pole pin (see Fig. 4) byclearancefit⌀45H8/g7.Ontheoppositesideofthelever,thehandleismountedintotheborewithaclearancefit⌀45G6/h7.Regardingthegeometrictolerances,theparallelismbetweentwolever bores and perpendicularity between the handle and themounting sleevewallwere re‐quired. Eachmethodwasthenappliedtotheabove‐describedopen‐loopassembly.Theresultswerecomparedaccordingtothreecriteria:

identificationofthecontributingtolerances, calculationofthedependentdimension(functionalrequirement), analysisofcalculationdifferencescomparedtotheassemblieswithclosedloops.

AnassemblygraphwascreatedforeachofthemethodsexceptfortheMonteCarlosimula‐tion,asitisbasedonTolerancechartmethod.

3. Results and discussion 

3.1 Tolerance chart results 

Tolerancechartmethodismostlyusedfordimensionaltolerances,eventhoughtherecentmodi‐ficationshaveenabledtheanalysisofgeometrictolerancesaswell[15].Thegeometrictoleranc‐esaretobetransformedintotheirdimensionalcounterparts.Yet,suchtransformationdoesnotaccount fortheangularsurfacedeviation. Inthisarticle, thetolerancechartmethodisappliedonlytodimensionaltolerances.

Tolerancechain consistingofbaseplateboredepth (a), lengthbetween thepolebasisandpole journal axis (b), tolerance fitbetweenpole journal and leverbore (c andd),distancebe‐tweenleverboreaxes(e),andtolerancefitbetweenlowerleverboreandhandle(fandg).Asmentioned, thedistancebetweenthetopsurfaceof thebaseplate(part1)andhandle(4) isafunctionalrequirement.IndicesUandLwereincludedtodenotetheupperandlowerdeviationlimit,respectively.

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50  Advances in Production Engineering & Management 15(1) 2020

 

Dimension TOL.a 15 0b 317 0c 22.5 cut=–0.009/2;clt=–0.034/2d 22.5 dut=0.039/2;dlt= 0e 150 0f 22.5 fut=0.025/2;dlt=0.006/2g 22.5 gut=0/2;glt=‐0.025/2

Sum 362 210

Nominalfunctionalrequirementdimension:152mm

Stackequationfortheupperdeviationlimit:0.008mm

Stackequationforthelowerdeviationlimit:0.062 mm

Fig.5Applicationoftolerancechartmethod

The tolerance stack coordinate system is definednext; the startingpoint is set at the baseplatesurface(1).Theupwarddimension isshowninFig.5 isselectedaspositiveandmarkedwiththeindicator“⊕”,whilethedownwardisnegativeandmarkedwith“⊖”.Thedirectionofthe tolerance chain is chosen arbitrarily, but it is important to respect the specifieddirectionalong the chain. Finally, the results are calculatedby adding and subtracting values along thetolerancechainandshowninFig.5.

3.2 Monte Carlo simulation results 

MonteCarlosimulationwasappliedfollowingtheprocedureexplainedinSection2.1.Determin‐ingtheappropriatedistributiontoeachof the toleranceswas thecrucialstep,as itaffects theresults.Thedistributionofgeometrictolerancesalongwiththeintervalbetweentheupperandlowerdeviationlimitmostfrequentlyfollowsthenormaldistribution. Tolerance fits are asymmetrical, requiring the use of the skewed distribution according to[19].Distributionof inputvalues isdescribedusing±3 processrange(6 ).Sincedataaboutthemanufacturingprocesswasnotavailable,3DCSCATSsoftwarewasused todetermine thedistributionmodelsoftolerancefits.Accordingto3DCS,tolerancefitshaveunimodalcontinuousprobabilitydistributioncalledPearson1.Sincetolerancelimitsassignedtodimensioncareneg‐ative,thedistributionmodelisskewedleftfromthenominaldimension.Thesamecanbecon‐cludedforthetoleranceassignedtodimensiong.Ontheotherhand,tolerancelimitsassignedtodimensionsdandfarepositive,anddistributionisright‐skewed.

Runs 2000Nominal 152.000mmMean 151.999mmSTD 0.018mm6STD 0.110mmLSL 151.000mmUSL 153.000mmEST.TYPE PearsonIEST.LOW 151.958mmEST.HIGH 151.995mmEST.RANGE 0.085mm

Fig.6MonteCarloassemblyresultsandhistogram

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Advances in Production Engineering & Management 15(1) 2020  51

Next step is to define the variationmodel function. SinceMonte Carlo is applied to tolerancechartmethod,tolerancestackequationsareusedtodefineit.Afterrunningthesimulationfor =2000timeswithrandomizedinputvariables,anoutputmodelforFRwascreated.Avariationanalysisprovidesdescriptivestatistics,inferentialstatisticsdata,andahistogram(showninFig.6).Byaddingandsubtractingtheinputvariablesusingthevariationmodelfunctions,distribu‐tionofFRtoleranceswasfound.TheresultingfunctionalrequirementdistributionisalsoPear‐son1.However,itisnotasprofoundlyleft‐orright‐skewedasaretheinputvariables.

3.3 Vector‐loop results 

Anassemblygraphdescribingtheopen‐loopoftheassemblyanditsvectorlooptolerancechain(ordatumpath)wascreated. Itwasusedto identifyof thenumberofvectorchainsand loopsinvolvedintheassembly(Fig.4).Sinceeachpart is incontactwithitstwoneighbouringpartsonlyonce,thisassemblycontainsoneopenloop.Samecanbeseenontheassemblygraph(Fig.7)where each arrow is representing the contact between parts. Vector loop is open at the gap(notedg)betweentheHandle(Part4)andBase(Part1).

The datumpath (Fig. 7, right) connects the point, surface, axis orDRF of a partwith nextpart’spoint,surfaceaxisorDRF.DRFshavebeenassignedtoeachpartwithrespecttotheorigincoordinatesystematthetopoftheBase(Part1).AlltheDRFshaveahorizontal ‐axisandverti‐cal ‐axis.Origincoordinatesystemissetinsuchawaythatpositivedirectionof axiscorre‐spondswiththepositivedirectionofatolerancechaininTolerancechartmethod.Thiseasesthetrackingandmethodcomparison.

Thegeometrictoleranceswerealsoaccountedfor.Eachtolerancewasrepresentedasanad‐ditionalvectorofmagnitudeequalto± /2,where isthewidthofcorrespondingtolerancefield(seeFig.4;0.2forthepositional,and0.1forparallelismandperpendicularitytolerances).Theadditionalvectorsrepresentgapsbetweenpartscontactingpointsandweredenotedbasedonthecorrespondingnominaldimension.ThepositiontoleranceontheBasecut‐outwithrespecttothedatumA(apos)isrepresentedasatranslationvectorofthesurfaceinthe ‐direction(Fig.7).TheparallelismsappliedtothePole’spin( ),andLeverholes( )withrespecttothedatumBwerealsorepresentedastranslationvectoralongthe ‐axis[1].Perpendicularityap‐pliedtothehorizontalaxisoftheHandle( )withrespecttothedatumDcanbedescribedasatranslationalong ‐axis[1].Accordingtotheassemblygraphthereare 3contactingpointsand 4parts,resultingin0closedloops(Eq.1wasused).Thereisalsooneopen‐loopfunc‐tionalrequirement.

 

Var. Tol.description Gapvectorlengthapos Position(Basetoptocutout) ±0.1bpar Parallelism(Pole’stobottom) ±0.05cd Dimensionaltol. (Pole’spin) cdu=–0.009;cdl=–0.034dd Dimensionaltol. (Lever’supperhole) ddu=0.039;ddl=0epar Parallelism(Leverholeaxes) ±0.05fd Dimensionaltol.(Lever’slowerhole) fdu=0.025;fdl=0.009gd Dimensionaltol. (Handle) fdu=0.0;fdl=–0.025hpar Perpendicularity(Handleaxis) ±0.05

Uppervalueoffunctionalrequirement:

2 2

2 2152.241mm

Lowervalueoffunctionalrequirement:

2 2

2 2151.680mm

Fig.7Vectorloopassemblygraphandresults

 

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52  Advances in Production Engineering & Management 15(1) 2020

3.4 Unified Jacobian‐torsor results 

Beforecreatingtheassemblygraph(Fig.8),itwasnecessarytoidentifythefunctionalelements(FE)andfunctionalrequirements(FR).Also,itwasrequiredtodifferentiatebetweentheinter‐nal and kinematic pairs. For the assembly at hand, there are four internal and one kinematicpair.Firstinternalpair(FE0‐1)islocatedontheBase(Part1),asthepositionaltolerancedefinedbetweenitstopandcut‐outsurfacecorrespondstofunctionalsurfaces0and1ontheassembly.TheparallelismtolerancesdefineFE2‐3andFE4‐5.InternalpairFE6‐7isdefinedbytheperpendicu‐laritytolerancesetontheHandle(Part4). Onlykinematicpair(FE1‐2)issetbetweentheBasecut‐outandPole’sbottom.However,thecontactisassumedtobeidealsothatitwillnotimpacttheanalysis.Twomorecontactsdefinedbytolerancefits(betweenPoleandLever,andbetweenLeverandHandle)werenotsetaskin‐ematicpairseventhoughtheyareinphysicalcontact.ThismeanstheyaredefinedasimportantconditionstobesatisfiedbetweentwoFEs.So,accordingto[13],theyarethendefinedasfunc‐tionalrequirementsthatwillbetakenintoaccountintheanalysisaskinematicpairs.

Uppervalueoffunctionalrequirement:152.28mm

Lowervalueoffunctionalrequirement:151.143mm

Fig.8Jacobian‐torsormethodandresults

Jacobianmatrixandsmall‐displacementtorsorvectorwerecalculatedforeachinternalandkinematic pair. A small‐displacement torsor vectorwas also calculated for eachFE (based ontorsorrepresentingthetolerancezone[21]):

, (2)

Foreachtolerance, translationalandrotationalcomponents insidethe tolerancezoneweredetermined[21].Sincethedirectionofafunctionalrequirementisalongthe ‐axisandrotationsthatwould influence functional requirement are around and ‐axis, , and componentmustbecalculated. Contactbetweenfunctionalsurfaces0and1intheFE0‐1isaplanarcontactwithnormalcon‐tainingonetranslationcomponent( )andtworotationalcomponents( , )[12,21].Compo‐nentsarecalculatedusingtheequationsforplanarsurfaceaccordingto[21].InternalpairsFE2‐3,FE4‐5, andFE6‐7, and functional requirementsFR3‐4 andFR5‐6 are defined by the tolerance fits.Theyusetranslationalcomponentsvandwandrotationalcomponents and ofaslippingpiv‐otwith the axis [21]. Below is a table containing displacements torsors for each internal andkinematicpair.

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Table1DisplacementtorsorsforeachinternalandkinematicpairFE0‐1 FE2‐3 FR3‐4 FE4‐5 FR5‐6 FE6‐7

0 0

0 0

0.1 0.1

0.0013 0.0013

0.0013 0.0013

0 0

0 0

0.05 0.05

0.05 0.05

0 0

0.004 0.004

0.004 0.004

0 0

0.00365 0.0045

0.00365 0.0045

0 0

0.0003 0.0004

0.0003 0.0004

0 0

0.05 0.05

0.05 0.05

0 0

0.004 0.004

0.004 0.004

0 0

0.025 0.045

0.025 0.045

0 0

0.002 0.0004

0.002 0.0004

0 0

0.05 0.05

0.05 0.05

0 0

0.001 0.001

0.001 0.001

Jacobianmatrixiscalculatedaccordingtotheprocedurepresentedin[21].Itspurposeistocalculatetheeffectofthetraditionaltorsorsetforeachfunctionalelement(FE)onthefunctionalrequirement(FR)oftheassembly[21].Finally,aftercalculatingsmall‐displacementtorsorvec‐torsandJacobianmatricesforeachFE,thesamecanbedoneforFR:

∙ 0.058 0.239 .0.434 0.184 .

0.001 0.005 0.0040.001 0.011 0.010

(3)

3.5 Comparison of the results obtained by different methods 

Afteranalysingtheassemblypresented inSection3.1usingthe fourmethods,resultsarepre‐sented inTable 2. Abbreviations are used to ease the result disambiguation; TCwasused fortolerancechainmethod,MCforMonteCarlosimulation,VLforthevector loopmethod,andJTfortheUnifiedJacobian‐torsormethod.SinceitisnotpossibletoincludegeometrictolerancesinTCandMCanalysis,tworesultswereprovidedforVLandJTmethods.Thefirstbatchofresultsincludeddimensionaltolerances,whilethesecondincludesboth.Besidesthequantitativeanaly‐sis results, qualitativeproperties such as the scope andperceived complexity of eachmethodwereassessed. ProprietaryCATtools thatareused inday‐to‐dayworkareoftenperceivedasblackboxes.Thatmeansthattheusersarefrequentlynotfamiliarwiththeunderlyingprocessesandmathe‐maticalmodels.Besides,thetoleranceanalysismethodsusedinCATtoolsareoftennotcomply‐ingtothetechnicalstandards.Forthisreason,wehaveanalysedunderlyingtoleranceanalysismethods,aimingtodeterminetheiradvantagesandshortcomings.

Whenconsideringthedimensionaltolerances(DT)exclusively,TC,MC,andVLprovidesimi‐lar results,withFRu being practically equal. Contrary to the upper value of the functional re‐quirement, the deviations in lower (FRl ) are greater – TC andVL providemore conservativeresultswhencomparedtoMC.AsignificantdeviationinFRuwasfoundwhencalculatingitusingJT.Unlikeotheranalysedmethods, in Jacobian‐torsormethod toleranceanalysis iscarriedoutusingatolerancezoneasabasis(insteadofpoints),causingtheafore‐mentionedvariationsinresults.Byusing zones and surfaces insteadof points, it is possible to create amore crediblerepresentationofarealisticcase.

Table2Methodcomparison Scope FRu FRl Δ Applicability Complexity

Tolerancechainmethod DT 151.992 151.938 0.054 Forsimple1DtasksSimple,carriedoutbyhand

MonteCarlosimulationmethod

DT 151.995 151.958 0.037Simpletasks,statisticalanalysis

Statisticaltoolsrequired

VectorLoopmethod

DT 151.991 151.930 0.061Multi‐dimensional

Carriedbyhand/morecomplex

D&GT 152.243 151.143 1.100UnifiedJacobian‐torsormethod

DT 152.369 151.946 0.423 Multi‐dimensional,automation

Requiresmathemati‐caltools,enablesautomation D&GT 152.665 151.053 1.612

 

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54  Advances in Production Engineering & Management 15(1) 2020

TheMCmethodistheleastconservativeduetoitsstatisticalapproach–whencarryingoutthetoleranceanalysisusingtheMCmethod,mostextremecasesareexcludedfromtheanalysisandcountedaswrite‐offs.Theadvantageofsuchanapproachisthatitreducesthecostofmanu‐facturingequipment; it is lessexpensivetowrite‐offaportionofparts, thentopurchasemoreaccuratemanufacturingtools.Thus,theanalysismethodshouldbeselectedinaccordancewiththe manufacturing process. Statistical approaches are suitable when analysed products aremanufactured in large series,whileprototypesandone‐of‐a‐kindproductswarrant theuseofmorecomplexandconservativemethods,suchasJT.

Theapplicabilityofmethodsregardingthetolerancingproblemdimensionalityshouldbead‐dressednext.Asappliedinthisstudy,byusingTCandMConly1Dproblemcontainingdimen‐sionaltolerancescanbesolved.Thisdrawbackcanbepartiallymitigatedbyconvertingthege‐ometrictolerancesintotheirdimensionalcounterparts;however,methodsremainlimitedto1Dproblems. In comparison, VL and JT were developed with having the 2D and 3D tolerancingproblemsinmind.Besidesthedimensionaltolerances,bothmethodscanbeusedtoanalysethegeometric tolerances aswell. However, there is a significant difference betweenVL and JT intermsoftolerancerepresentation.Theformer,vectorloop,observesasetoftolerancessimulta‐neously, forgoingthepossible interactionsamongthem.The latter,UnifiedJacobian‐torsor, in‐cludesboththetranslationalandrotationalcomponents, thusincludingdifferenttolerancesascomplementary. Theprocedurecomplexityofeachmethodshouldalsobeconsidered.TC isby far themoststraightforwardmethodandcanbecarriedbyhand. It issuitable forsimpletolerancingprob‐lems that engineers solvedaily. Second is theVL,which requires an additional schemaof thevectorloop.Byprocedurecomplexity,MCcomesnext.Itisastatisticalmethod,meaningthatitrequires a large sample inorder toprovide significant results–experiencewith similarpartsandtheirtolerancesisnecessary.ThelastmethodisJT,whichisfoundtobethemostcomplex.To carry it out, it requires the detection of functional elements, functional requirements, andkinematicpairs. When comparing the method performance on open and closed‐loop tolerancing, changesweredetectedonlyinVL.Thevector‐loopmethodisaffectedbytheproceduresystemofopenandclosedloops.Incaseswhereonlyopen‐looptolerancesareused,thevectorloopmethodisreducedtoexplicitequations.Thisallowsforadirectcalculationofthefunctionalrequirementvalues.Inotherwords,theVLmethodlosesitsadvantagetoTC. The limitations of the study should also be considered. Each of themethods is carried outstrictlyaccordingtotheliterature,withoutadditionaldatamanipulation(forexample,geometrictolerances were not converted to dimensional ones). Additionally, when carrying out MC, itshouldbestressedthatpreviousknowledgeaboutthemanufacturingprocessandmanufactur‐ingtoolpropertiesisnecessarytoenablesatisfyingapproximationoftolerancedistribution. Lastly, during theplanningof theproductdesignprocess, in addition to tolerance analysismethods,engineersshouldalsoconsiderapplyingthetoleranceoptimisationmethods.Severalstudieshavebeencarriedoutonthesubject,suchas[25,26].Usingoptimisationalgorithmstotolerancingproblemsallowsus to findtheoptimal trade‐offbetweenthetolerances,manufac‐turingcosts,andqualityloss[25].Suchanapproachwouldsurelyincreasethedesigneffective‐ness,increasingitsmarketsuccess.

4. Conclusion 

Simple tolerancing problemwas used to assess the similarities and differences between fourtolerance analysis methods: Tolerance chart (“Worst‐case analysis”), Monte Carlo Simulationmethod,Vector‐loopmethod,andUnifiedJacobian‐torsor.Open‐loopassemblywasusedtoillus‐tratetheproblem‐solvingprocessusingeachofthemethods.Basedontheresults,theauthorshaveconcludedthefollowing:

A comparison of the tolerance analysis methods in the open-loop assembly

• Tolerance chart and Monte Carlo Simulation methods do not account for the geometrical tolerances. This results in overly optimistic results; however, both methods are only suita-ble for solving simple, 1D tolerancing problems.

• The unified Jacobian-torsor method was found to be most conservative (i.e. provided the most substantial deviations in functional requirement), followed by Vector-loop, Toler-ance chart, and Monte Carlo Simulation, respectively.

• Tolerance chart is the simplest and thus suitable for solving many day-to-day tolerancing problems. Monte Carlo Simulation and Unified Jacobian-torsor require more detailed anal-ysis and know-how and are suitable for more pressing problems. Vector loop can be con-sidered the middle ground – it offers good results at the moderate complexity.

• When comparing the method performance in open-loop assemblies to closed-loop ones, differences are detected only in Vector-loop method.

The field of tolerance analysis is fruitful, and there is more work to be done. Following this study, the authors aim to analyse the performance of tolerancing methods by carrying out an industrial case study. The part deviations measured during the quality assurance are to be com-pared to the values provided by analysis methods, providing additional insight.

Acknowledgement This paper reports on work funded by the Croatian Science Foundation project IP-2018-01-7269: Team Adaptability for Innovation-Oriented Product Development - TAIDE.

References [1] Polini, W. (2011). Geometric tolerance analysis, In: Colosimo, B., Senin, N. (eds.), Geometric tolerances: Impact on

product design, quality inspection and statistical process monitoring, Vol. 2, Springer, London, United Kingdom, 39-68, doi: 10.1007/978-1-84996-311-4_2.

[2] International organisation for standardisation (2011). ISO 8015-2011 – Geometrical product specifications (GPS) – Fundamentals – Concepts, principles and rules, ISO, Geneva, Switzerland.

[3] Sigurdarson, N., Eifler, T., Ebro, M. (2018). The applicability of CAT tools in industry – Boundaries and challenges in tolerance engineering practice observed in a medical device company, Procedia CIRP, Vol. 75, 261-266, doi: 10.1016/j.procir.2018.04.066.

[4] Corrado, A., Polini, W. (2017). Manufacturing signature in jacobian and torsor models for tolerance analysis of rigid parts, Robotics and Computer-Integrated Manufacturing, Vol. 46, 15-24, doi: 10.1016/j.rcim.2016.11.004.

[5] Ramnath, S., Haghighi, P., Chitale, A., Davidson, J.K., Shah, J.J. (2018). Comparative study of tolerance analysis methods applied to a complex assembly, Procedia CIRP, Vol. 75, 208-213, doi: 10.1016/j.procir.2018.04.073.

[6] Chen, H., Jin, S., Li, Z., Lai, X. (2014). A comprehensive study of three dimensional tolerance analysis methods, Computer-Aided Design, Vol. 53, 1-13, doi: 10.1016/j.cad.2014.02.014.

[7] Morse, E.P., Shakarji, C.M., Srinivasan, V. (2018). A brief analysis of recent ISO tolerancing standards and their potential impact on digitalization of manufacturing, Procedia CIRP, Vol. 75, 11-18, doi: 10.1016/j.procir.2018. 04.080.

[8] American Society of Mechanical Engineers (2004). ASME Y14.5 – Mathematical definition of dimensioning and tolerancing principles, ASME, New York, USA, 1-15.

[9] Shen, Z., Ameta, G., Shah, J.J., Davidson, J.K. (2005). A comparative study of tolerance analysis methods, Journal of Computing and Information Science in Engineering, Vol. 5, No. 3, 247-256, doi: 10.1115/1.1979509.

[10] Marziale, M., Polini, W. (2011). A review of two models for tolerance analysis of an assembly: Jacobian and tor-sor, International Journal of Computing Integrated Manufacturing, Vol. 24, No. 1, 74-86, doi: 10.1080/0951192X. 2010.531286.

[11] Chase, K.W., Magleby, S.P., Gao, J. (2004). Tolerance analysis of 2-D and 3-D mechanical assemblies with small kinematic adjustment, Advanced Tolerancing Techniques, Vol. 218, 1869-1873.

[12] Ghie, W. (2009). Statistical analysis tolerance using jacobian torsor model based on uncertainty propagation method, The International Journal of Multyphysics, Vol. 3, No. 1, 11-30, doi: 10.1260/175095409787924472.

[13] Ghie, W., Laperrière, L., Desrochers, A. (2010). Statistical tolerance analysis using the unified Jacobian-Torsor model, International Journal of Production Research, Vol. 48, No. 15, 4609-4630, doi: 10.1080/002075409028 24982.

[14] Wang, Y. (2008). Closed-loop analysis in semantic tolerance modeling, Journal of Mechanical Design, Vol. 130, No. 6, Article No. 061701, doi: 10.1115/1.2900715.

[15] Schleich, B., Wartzack, S. (2016). A quantitative comparison of tolerance analysis approaches for rigid mechani-cal assemblies, Procedia CIRP, Vol. 43, 172-177, doi: 10.1016/j.procir.2016.02.013.

Advances in Production Engineering & Management 15(1) 2020 55

Kosec, Škec, Miler

[16] Magadum, R., Allurkar, B.S. (2015). A comparative study of fasteners tolerance analysis methods, International Journal of Scientific & Engineering Research, Vol. 6, No. 6, 700-705.

[17] Fischer, B.R. (2015). Mechanical Tolerance Stackup and Analysis, CRC Press, Boca Raton, USA. [18] Yan, H., Wu, X., Yang, J. (2015). Application of Monte Carlo method in tolerance analysis, Procedia CIRP, Vol. 27,

281-285, doi: 10.1016/j.procir.2015.04.079. [19] Schenkelberg, F. (2016). Statistical tolerance analysis – Basic introduction, FMS Reliability Publishing, Los Gatos,

USA. [20] Chase, K.W., Gao, J., Magleby, S.P. (1995). General 2-D tolerance analysis of mechanical assemblies with small

kinematic adjustments, Journal of Design and Manufacturing Automation, Vol. 5, 263-274. [21] Desrochers, A., Ghie, W., Laperrière, L. (2003). Application of unified jacobian-torsor model for tolerance analy-

sis, Journal of Computing and Information Science in Engineering, Vol. 3, No. 1, 2-14, doi: 10.1115/1.1573235. [22] Ghie, W. (2012). Tolerance analysis using Jacobian-Torsor model: Statistical and deterministic applications, In:

Cakaj, S. (ed.), Modelling simulation and optimization – Tolerance and optimal control, InTech, Rijeka, Croatia, 147-160, doi: 10.5772/9043.

[23] Desrochers, A., Delbart, O. (1998). Determination of part position uncertainty within mechanical assembly using screw parameters, In: ElMaraghy H.A. (ed.), Geometric design tolerancing: Theories, standards and applications, Springer, Boston, USA, 185-196, doi: 10.1007/978-1-4615-5797-5_14.

[24] Peng, H., Lu, W. (2017). Three-dimensional assembly tolerance analysis based on the Jacobian-Torsor statistical model, In: Proceedings of 3rd International Conference on Mechatronics and Mechanical Engineering (ICMME 2016), Les Ulis, France, doi: 10.1051/matecconf/20179507007.

[25] Siva Kumar, M., Kannan, S.M., Jayabalan, V. (2009). A new algorithm for optimum tolerance allocation of complex assemblies with alternative processes selection, The International Journal of Advanced Manufacturing Technolo-gy, Vol. 40, No. 7-8, 819-836, doi: 10.1007/s00170-008-1389-5.

[26] Ramesh Kumar, L., Padmanaban, K.P., Balamurugan, C. (2016). Optimal tolerance allocation in a complex assem-bly using evolutionary algorithms, International Journal of Simulation Modelling, Vol. 15, No. 1, 121-132, doi: 10.2507/IJSIMM15(1)10.331.

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