1、11FTM08AGMA Technical PaperA ComprehensiveSystem for PredictingAssembly Variation withPotential Application toTransmission DesignBy K. Chase and C. Sorensen,Brigham Young UniversityA Comprehensive System for Predicting Assembly Variationwith Potential Application to Transmission DesignKenneth W. Cha
2、se and Carl D. Sorensen, Brigham Young UniversityThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractRecent advances in tolerance analysis of assemblies allow designers
3、 to:S Predict tolerance stack-up due to process variations.S Examine variation in clearances and fits critical to performance.S Use actual production variation data or estimates from prior experience.S Use engineering design limits to predict the percent rejects in production runs.Acomprehensivesyst
4、emhasbeendevelopedformodeling1D,2D,and3Dassemblies,whichincludesthreesources of variation: dimensional (lengths and angles), geometric (GD theclosedlooplocates theArm as it slides inor out toaccom-modate dimensional variation. In this case, to properly grip the Reel, the Gap must be negative over th
5、e fullrangeof dimensional variations. As is often thecase, theopen loopdepends onelements of the closedloopfor its solution. In this case, RLis the result of the vector chain in the closed loop, so the loops are coupled.Figure 4. General 2-D vector loop, showing relative angles between adjacent vect
6、ors6 11FTM08Figure 5. Vector assembly model showing an open and a closed loop, representing a lockinghub assemblyThe vectors in a vector loop are not simply pin jointed together. To accurately represent solid bodies, thevectorsmustbefixedtothepartstheyrepresent. Thus,therelativeanglebetweentwovector
7、smayrepresenta machined angle between two surfaces on the same part, in which case the nominal angle and tolerancewould be specified. Alternately, if two adjacent vectors are fixed to two mating parts, their angles or lengthsmay vary kinematically, describingthedegreesof freedombetweentheparts, inwh
8、ichcaseonly thenominallengths andangles of thekinematic variables would beknown. Their variations could only be determinedbyan assembly tolerance analysis.Kinematic degrees of freedomThe kinematic degrees of freedom, which describe the small displacements between mating parts, may beadded to a vecto
9、r assembly model by inserting kinematic joints into the vector loops. Figure 6 shows 12common kinematic joints, which may be used to represent mating surfaces in assemblies. The arrows andnumbers indicate the degrees of freedom in each case.Figure 6. 3-D kinematic joints representing mating surfaces
10、 and degrees of freedom inassemblies7 11FTM08Vector models havebeenwidely usedtorepresent therigidbodykinematics ofmechanisms. They mayalsobeusedtomodelstaticassemblies. Themajordifferencesbetweenakinematicmodelofamechanismandakinematicmodelofastaticassemblyaretheinputsandoutputs. Formechanismanalys
11、is,theinputsarelargemotions of one or more of the members; the outputs are the rigid body displacements, velocities, etc. of themembers. For static assemblies, theinputs aresmall variations inthedimensions andgeometric form ofthemembers; the outputs are the small rigid body displacements that occur
12、due to production variations. For amechanismmodel,thesolutiondescribesthemotionofasinglemechanismwithtime,fromonepositiontothenext. For a static assembly, a statistical solution predicts the variation of all assemblies compared to thenominal assembly, or the change from one assembly to the next.Geom
13、etric variationsGeometricvariationsofform,orientationandlocationarethefinalsourceofvariationtobeincludedinavectorassemblymodel. Suchvariationscanonlyintroducevariationintoanassemblywheretwopartsmakematingcontact. Themannerinwhichgeometricvariationpropagatesacrossmatingsurfacesdependsonthenatureof th
14、e contact.Figure 7 illustrates this concept. Consider a cylinder on a plane, both of which are subject to surfacewaviness,representedbyatolerancezone. Asthetwopartsarebroughttogethertobeassembled,thecylin-der could be on the top of a hill or down in a valley of a surface wave. Thus, for this case, t
15、he center of thecylinder will exhibit translational variation from assembly-to-assembly in a direction normal to the surface.Similarly, the cylinder could be lobed, as shown in the figure, resulting in an additional vertical translation,depending on whether the part rests on a lobe or in between.Inc
16、ontrast tothecylinder/planejoint, theblock onaplane, showninFigure 7, exhibits rotational variation. Intheextremecase,onecorneroftheblockcouldrestonawavinesspeak,whiletheoppositecornercouldbeatthebottomofthevalley. Themagnitudeofrotationwouldvaryfromassembly-to-assembly. Wavinessonthesurface of the
17、block would have a similar effect.In general, for two mating surfaces, we have two independent surface variations, which introduce variationinto the assembly. How they propagate depends on the nature of the contact, that is, the type of kinematicjoint. Figure 8shows two3-D joints, subject tosurfacev
18、ariation. An unconstrainedbody has six degrees offreedom (arrows). Contact between mating surfaces creates constraints. Each arrow is either a K or an F.ThearrowsmarkedwithaKindicatethekinematicdegreesoffreedominthejoint. ThearrowsmarkedbyanFindicateaconstraineddirectionforformvariationpropagation.K
19、inematic displacementspropagatealongthekinematic axes. Geometric form variations can only propagate through the constrained axes of the joint.Asanestimateofthemagnitudeofassemblyvariationproducedbysurfacevariation,wecanusethegeomet-ric tolerancezonespecifiedas designlimits andthelengthof contactbetw
20、eenthematingparts,as definedinequation1aand1b. For translational variation, theextrememagnitudedis assumed tobe equal to half thetolerancezone. For rotational variation, the extremeangle dis formed by the contact length extendedoverthe peak-to-valley height.Figure 7. Propagation of 2-D translational
21、 and rotational variation due to surface waviness8 11FTM08Figure 8. Simultaneous propagation of translational and rotational variation due to surfacewaviness in 3-DTranslational variation(1a)d 12tol zoneRotational variation(1b)d= tan1tol zonecontact lengthSincetheextremevalueisprobablyarareoccurrenc
22、e,settingthetolerancezoneequal tothe 3 limits ofanormal distribution will make an extreme less likely to occur in the assembly model. A catalog of models forgeometricvariationshasbeendefinedforeachofthe12jointsshowninFigure 6,correspondingtoeachoftheANSI Y14.5 geometric tolerance specifications 2.Th
23、e models for geometric variation are only approximations to permit the effects to be included in toleranceanalysis. More study is needed to develop improved models. In particular, the propagation of surface vari-ation in assemblies needs to be characterized andverified. The interactionof geometric v
24、ariations withsizevariations and the consequences of the Envelope Rule are other issues which need to be resolved.Assembly tolerance specificationsAmanufacturedproductmustperformproperlyinspiteofdimensionalvariation. Toachievethis,engineeringdesignrequirements mustincludeassemblytolerancelimits. The
25、designer mustassignupperandlowerlim-its to those gaps, clearances and overall dimensions of an assembly which are critical to performance.Assembly tolerancelimitsareappliedtothestatistical distributionof theassembly variations,as predictedbytolerance analysis, to estimate the number of assemblies wh
26、ich will be within the specifications.Designers need to control more than just gaps and clearances in assemblies. Orientation and position offeatures may also be important to performance. To be a comprehensive design tool, a tolerance analysissystemmustprovideasetofassemblytolerancespecificationsabl
27、etocoverawiderangeofcommondesignrequirements.A system of assembly tolerance specifications patterned after ANSI Y14.5 has been proposed 3. ThoseANSIY14.5featurecontrolswhichrequireadatumappeartobeusefulasassemblycontrols. However,thereis a distinct difference between component tolerance and assembly
28、 tolerance specifications, as seen inFigure 9. Inthecomponenttolerancespecificationshown,theparallelism tolerancezoneisdefinedasparal-leltodatumA,areferencesurfaceonthesamepart. By contrast,theassemblyparallelism tolerancedefinesa tolerance zone on one part in the assembly, which is parallel to a da
29、tum on another part. In order todistinguish an assembly tolerance specification from a component specification, new symbols have beenproposed. The feature control block and the assembly datum have been enclosed in double boxes.9 11FTM08Figure 9. Comparison of component and assembly tolerance specifi
30、cationsAnexampleof aparallelism assembly specificationis anautomobiledoor assembly. Afully assembleddoor,with hinge plates attached, is mounted in place on theframe. When thedoor is closed, thegap betweentheouteredgeofthedoorandthedoorframemustbeuniformwidthallthewayaroundthedoor. Notonlyisthisarequ
31、irementforproperoperationofthedoor,italsocontrolsthefitwiththegasket,assuringitwontleak. Thisis also a measure of the craftsmanship and precision of the body assembly process.Atoleranceanalysisvectormodelwouldbeginthevectorloopononesideofthegap,atthedoorpost,proceedthrough the frame members to the h
32、inge, through it and through the interior frame of the door, to the edgeoppositethedoorpost. Withtwohingesandadoorstop,therewouldlikelybeoneortwoclosedloopsinvolvedinpositioningthedoor. Thegapnominalandvariationcouldbecalculatedatthispoint,aswellasseveralothercheck points. In addition, the nominal a
33、ngle andvariation betweenthe twoedges couldbe predicted, basedonthedimensional andGDxiis the specified tolerance for dimension xi.Ifallthetolerancesare 3 ,theresultantassemblyvariationdURSSwillalsobe 3 . Theequationsmaybemodified to obtain higher quality (sigma) levels.Useful toolsThere are several
34、useful graphical tools for evaluating the results of your tolerance analysis, as illustrated inFigure 19. These are discussed in 10.a) The Percent Contribution by each variation source is calculated as shown in equations 3a and 3b. Itrepresents thechangeinthegapduetoonevariable, dividedby thesumof a
35、llthechanges(times 100toconvert to percent). A typical plot is shown in Figure 19.Worst case(3a)%Cont = Gapxjxj GapxixiRSS(3b)%Cont = Gapxjxj2 Gapxixi2b) TheTornado plot is thesensitivities of theGaptoeachvariable, independent of thetolerance, sortedindescending order of magnitude, and retaining neg
36、ative values.c) TheDistributionplotshows theprobabledistributionof theGap, basedonthespecifiedtolerancesandcalculatedsensitivitiesofthevariationsources. Thesemakeiteasy totry differenttolerances ordifferentprocessestoseetheeffectsonyourtolerancestackandthepredictedqualitylevel (%of assembliesthatwil
37、l meet the engineering specs).15 11FTM08a) Percent contribution chart b) Tolerance sensitivity chartc) Normal distribution plot - for predicting rejectsFigure 19. Useful tools for evaluating tolerance analysis resultsCAD implementationFigure 20 shows the structure of a Computer-Aided Tolerancing Sys
38、tem (CATS) integrated with a commer-cial3-DCADsystem. TheCA TS Modelercreatesanengineeringmodelofanassemblyasagraphicalandsymbolic overlay, linked associatively to the CAD model. Pop-up menus present lists of joints, datums,GD afullspectrumofassemblymodelingelements;combinedwithasystematicmodelingpr
39、ocedure,withestablishedrules. The system is understandable by virtue of the use of a few model elements, common to engineeringand manufacturing, and the similarity to current tolerancing practices.TheAnalyzer includes built-instatistical evaluationtools for toleranceanalysis, graphical output,andane
40、ffi-cient solver, which make it suitable for design revision. Tolerance synthesis tools permit allocation of toler-ances to the contributing component dimensions and sizing them to assure the set of assigned tolerancesmeettheassemblyspecs. CADintegrationhasresultedinacompletelygraphicalCAD-basedappl
41、icationforcreating vector assembly models and evaluating assembly variation in manufactured products.AllthepiecesareinplaceforafullyfunctionalCAD-basedtoleranceanalysisanddesigntool,althoughfuturerefinements and enhancements are sure to be added. The efforts to make the system understandable andeasy
42、touse,aswellasintegratingitwiththedesignersownCADsystem,willhelptowinacceptanceandusein the engineering/manufacturing community.References1 ANSI Y14.5-2009, Dimensioning and Tolerancing.2 Chase, K.W., Gao, J., and Magleby, S.P., Tolerance Analysis of 2-D and 3-D Mechanical Assemblieswith Small Kinem
43、atic Adjustments, Advanced Tolerancing Techniques, John Wiley,1997.3 Carr,C.D.,AComprehensiveMethodforSpecifyingToleranceRequirementsforAssemblies, BrighamYoung University, April 1993.4 Drake,Jr.,P.,Dimensioning and Tolerancing Handbook, McGraw-Hill, 1999.- Chase, K.W., Multi-Dimensional Tolerance A
44、nalysis, Chapter 13, pp. 1-27.- Chase, K.W., Minimum-Cost Tolerance Allocation, Chapter 14, pp. 1-23.5 Chase, K.W., Gao, J., and Magleby, S.P., General 2-D Tolerance Analysis of Mechanical Assemblieswith Small Kinematic Adjustments, Journal of Design and Manufacturing, vol. 5 no. 4, 1995.6 Chase, K.
45、W., Gao, J., Magleby, S.P., and Sorenson, C.D., Including Geometric Feature Variations inTolerance Analysis of Mechanical Assemblies, IIE Transactions, vol. 28, pp. 795-807, 1996.18 11FTM087 Gao,J.,Chase,K.W.ChaseandMagleby,S.P.,General3-DToleranceAnalysisofMechanicalAssem-blies with Small Kinematic
46、 Adjustments, ,IIE Transactions, vol. 30, pp. 367-377, 1998.8 Gao, J., ”Nonlinear Tolerance Analysis of Mechanical Assemblies”, Brigham Young University, 1993.9 Gao, J., Chase, K. W., and S. P. Magleby, ”Comparison of Assembly Tolerance Analysis by the DirectLinearization and Modified Monte Carlo Si
47、mulation Methods,” Proc. of the ASME Design EngineeringTechnical Conferences, Boston, MA, 1995, pp. 353-360.10 Geng, H., Manufacturing Engineering Handbook, McG-Hill, 2004.- Chase,K.W.,BasicToolsforToleranceAnalysisofMechanicalAssemblies,Chapters2-5,pp.1-13.11 Merkley,K.G.,Chase,K.W.,Perry,E., AnInt
48、roductiontoToleranceAnalysisofFlexibleSystems,MSCWorld Users Conference, Newport Beach, CA, June 199612 Tonks,M.R.,Chase,K.W.,Smith,C.C.,PredictingDeformationofCompliantAssembliesUsingCovari-ant Statistical Tolerance Analysis,9thCIRP International Seminar on Computer-Aided Tolerancing,Tempe, AZ, Apr
49、il 2005.13 Glancy,C.G.,Chase,K.W.,ASecond-OrderMethodforAssemblyToleranceAnalysis, Proceedingsofthe ASME Design Engineering Technical Conferences, Las Vegas, NV, Sept. 1999.14 Wittwer,J.W.,Chase,K.W.,andHowell,L.L., TheDirect LinearizationMethodAppliedtoPositionErrorin Kinematic Linkages, Mechanism and Machine Theory, Vol 39, No 7, pp 681-693, 2004.15 Smith, D.K., and Parkinson, A.R., Constraint Analysis of Assemblies Using Screw Theory andTolerance Sensitivities, Brigham
