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Correlation uncertaintyApplication to gear conformityJ.Y. Dantana,*, J.P. Vincenta, G. Goch (1)b, L. Mathieu (1)caLCFC, Arts et Me tiers ParisTech, 4 rue Augustin Fresnel, 57070 Metz Cedex 3, FrancebBIMAQ, Linzer Strabe 13, D-28359 Bremen, GermanycLURPA, ENS Cachan, Paris-Sud 11, Av. du Prsdt Wilson, 94235 Cachan Cedex, France1. IntroductionThe tolerancing process is defined through all the activitiesinvolved by geometric product variations management: tolerancedesign, manufacturing tolerance analysis, and tolerance verifica-tion. Tolerance verification permits closing the process loop, tocheck the product conformity, and to verify assumptions made bythe designer.Metrology always aims at providing reliable information as abasis for decisions. Measurement results are affected by measure-ment uncertainty, which leads to technical and economical risks inindustrial companies 1. By assessing the risks and the connectedconsequences of the decisions (conformity verification), thesignificance of the measurement result can be evaluated. Thesimulations of the functional chain of conformity assessmentsgenerate an estimation of the significance of measurements independence of measurement uncertainty and other types ofuncertainties.In fact, uncertainty is present in all areas of design, manu-facturing and metrology. The notion of uncertainty is by now wellentrenched in metrology. In the ISO TS17450-2 2, the notion ofuncertainty is generalized to specification and verification. Theuncertainties through the product life cycle span from the designintent to the inspection activity. The uncertainty is divided intocorrelation uncertainty, specification uncertainty and measure-ment uncertainty (Fig. 1 upper part):?Correlation uncertainty characterizes the fact that the IntendedFunctionality and the Specified Characteristics may not beperfectly correlated.?The specification uncertainty characterizes the ambiguity in thespecification expression.?Andthemeasurementuncertaintyisconsideredbythemetrologists and well described in GUM 3. The measurementuncertainty includes all the causes of variation of the quantity tobe measured to the result of inspection.Metrologists, standards and research activities focus more onthe measurement uncertainties. Srinivassan said: Correlationuncertainty, in particular, is an uncharted territory. Standardsdont tell us how to find this. 4. This paper proposes a formalismto express and evaluate the correlation uncertainty, and illustratesit on gear conformity assessment.2. Gear illustration of uncertaintiesThis illustration focuses on the pitch error which is defined inthe AGMA 2009-B01 5.To explain the measurement uncertainty, the definition of thepitch error can be taken as an example: the pitch of a bevel gear isdefinedasthearclengthbetweenallconsecutiveleftorrightflanksof one gear, measured at the pitch diameter d in a distance R fromthe apex of the reference cone. To estimate these deviations, somemetrology strategies are proposed that consist in probing onespecific point of all flanks, either by using a one probe device or atwo probes device. These strategies provide short quality inspec-tion times but are very sensitive to the measurement uncertainties(Fig. 2). Indeed, if there is a gap between the real inspection circleand the theoretical inspection circle, the pitch that has to bemeasured at the datum circle will differ from the measured pitch.To evaluate the pitch error, some mathematical models andcalculation steps are needed, based on the set of measured points6. To decrease these measurement uncertainties, Guenther 7proposes a new metrology strategy.CIRP Annals - Manufacturing Technology 59 (2010) 509512A R T I C L EI N F OKeywords:MetrologyDecision makingUncertaintyA B S T R A C TMetrology always aims at providing reliable information as a basis for decisions of the conformityassessment.Thesedecisionsareaffectedbythemeasurementuncertainty(TheGuidetotheExpressionofUncertainty in Measurement describes the measurement uncertainty evaluation) and the correlationuncertainty,whichcharacterizesthefactthattheIntendedFunctionalityandtheSpecifiedCharacteristics may not be perfectly correlated. To evaluate the risk of a wrong decision concerningthe conformity assessment due to the correlation uncertainty, this contribution proposes a model for theexpression and an evaluation method of the correlation uncertainty based on the Axiomatic Designmatrix and the Monte Carlo Simulation.? 2010 CIRP.* Corresponding author.Contents lists available at ScienceDirectCIRP Annals - Manufacturing Technologyjournal homepage: /cirp/default.asp0007-8506/$ see front matter ? 2010 CIRP.doi:10.1016/j.cirp.2010.03.040To illustrate the specification uncertainties, the followingexample can be used: the AGMA 2009-B01 5 states thatMeasurements for determining single pitch variation, cumulativepitch variation, and index variation are made at the tolerancediameter, relative to the gear datum axis of rotation, tangent to thetolerancediameterintheplaneofrotation.Thetolerancediameter,dt, is the diameter where the mean cone distance Rm and themidpoint of the working depth intersect. Furthermore, Rm is thedistance from the apex of the pitch cone to the middle of the facewidth. But it is not explained how to obtain the middle point ofthe face width: is it taken on a tooth randomly chosen, is it a meanof all teeth, how is the middle of the face width calculated, and sowhich the metrology strategy is suitable to obtain it? In this case,the specification is ambiguous. It characterizes the specificationuncertainties.CorrelationuncertaintiescharacterizethefactthattheIntendedFunctionality and the Specified Characteristics may not beperfectly correlated. In the case of gears, the deviations of thegeometry impact the transmission error (difference between thetheoretical and real instantaneous angle of rotation of the wheel).For example, a pitch error will lead to a step of the transmissionerror (Fig. 3).In practice, the designer limits the pitch error in order to limitthe transmission error. Indeed, even if the relative variations of areal gear ratio are minor, accelerations induced are not negligibleand a jump of the angular velocity must be avoided in order toreduce noise level and vibrations. Unfortunately, there is noexplicit relation between the transmission error and the geome-trical Specified Characteristics (like pitch errors, runouts, formdeviations, etc.) 8. In this case there exist correlation uncertain-ties between the deviations and the transmission error or thekinematic characteristics.3. Formalisation of correlation uncertaintyIn the GUM 3, the measurement uncertainty is a calculablenumber, taken to be the probability distribution that characterizesknowledgeofmeasurand.Intheproposedformalism,thecorrelation uncertainty is a calculable number: CU, taken to bethe confidence interval or the probability distribution thatcharacterizes knowledge of the relationship between IntendedFunctionalities and Specified Characteristics.To formalise the correlation uncertainty, it is necessary toidentify the correlations or the relationships between IntendedFunctionalities and Specified Characteristics of the product. Well-known techniques to deal with these correlations are the SuhsAxiomatic Design matrix 9.In fact, one can formalise these correlations by using Suhsdesign matrix to express the relationship between the functionrequirements (FRs) and design parameters (DPs).The correlations can be expressed mathematically in terms ofthe matrix equation:fFRgm? A?m?pfDPgp(1)where FRmis a vector of independent Functional Requirements,DPpis the vector of design parameter, Aij=FRi/DPjare thesensitivity coefficients.Functional Requirements (FRs) are defined, in axiomatic design,as a minimum set of independent requirements that completelycharacterize the functional needs of the product in the functionaldomain. An FR is specified in terms of its nominal value withallowable variations or desired accuracy (design range). Theentirety of possible values (or the probability density function ofvalues) of the chosen system to satisfy FR is called the systemrange. The FR is satisfied only if the design range and the systemrange have a common area; common range. When the systemrange is not completely included in the design range, there is afinite uncertainty that the FR may not be satisfied.In the context of product conformity assessment and correla-tion uncertainty, the Functional Requirements (FRs) are theIntended Functionalities; the design parameters (DPs) are thespecified or measured characteristics. The first proposal is toformalisethecorrelationuncertaintybyintervalsofeachcomponent of the matrix: CUijis the confidence interval of theFig. 1. Global view of GPS uncertainty.Fig. 2. Influence of the inspection circle deviations on the pitch error estimation.J.Y. Dantan et al./CIRP Annals - Manufacturing Technology 59 (2010) 509512510coefficient Bijwhich models the correlation uncertainty of Bij.fIFgm? B?m?pfSCgp(2)where IFmis a vector of independent Intended Functionalities,SCpis the vector of Specified Characteristics, Bij=IFi/SCjare thesensitivity coefficients. The matrix B mathematically representsthelinearinterrelationbetweenIFsandSCs. Thelinearizationofthis interrelation introduces an error.The second proposal is to formalise the correlation betweenIntendedFunctionalitiesandMeasurandsofeachIntendedFunctionality. A measurand of one Intended Functionality is notmeasured directly, but is an estimation of the Intended Function-ality. It is determined from N Specified Characteristics SC1,SC2, ., SCNthrough a functional relation f:fMIFg ffSCg(3)where MIF is a vector of independent Measurands of IntendedFunctionalities, SC is the vector of Specified Characteristics, thefunction f should express not simply a physical law but thefunctional chain.Without correlation uncertainty, the conformity assessment ofa product is trivial. The conformance is proofed, if the value of theMeasurand associated with an Intended Functionality is locatedwithin the Functional Requirement. Then, the correlation betweenIntendedFunctionalitiesandMeasurandsofeachIntendedFunctionality can be expressed mathematically simplify by theidentity matrix:fIFg Id? ? fMIFg(4)In reality, Measurands of Intended Functionalities are alwaysafflicted by the correlation uncertainty. Thus, they are not theperfect image of the Intended Functionalities. The correlationbetweenIntendedFunctionalitiesandMeasurandsofeachIntended Functionality can be expressed mathematically in termsof the matrix equation:fIFgm? C?m?p? fMIFgp(5)The correlation uncertainty is modeled by an interval or by aprobabilitydistributionofeachcomponentofthematrix.CUijistheconfidence interval of the coefficient Cij, which models thecorrelation uncertainty of Cij.If the result of a measurement is close to the specification limitsand the coefficient Cijis smaller than 1, there exists risk to reject agood product; respectively, if the coefficient Cijis greater than 1,there exists risk to accept a bad product. Moreover, the correlationuncertainty increases the zone in which neither conformance nornon-conformance can be detected.This second proposal allows to model the correlation uncer-tainties of non-linearizable interrelations between the IntendedFunctionalities and the Specified Characteristics.Monte Carlo methods (MCM) are a perfect tool for evaluatinguncertainties without any restrictions, neither on the form of themodel nor on the number of output quantities. MCM and softwarefor modeling and simulating the functional chain are exemplarilyused for computing the correlation uncertainties.Therefore in conformity assessments, based on this proposedapproach and classical approaches for measurement uncertaintypropagation, it is possible to evaluate the probability of correct orwrong decisions (Fig. 1, lower part).4. Gear illustration of correlation uncertainty evaluationTo illustrate the impact of correlation uncertainty on theconformityassessment,fourspecificationmodelsarecompared:Inthe case of bevel gears, the pitch angular error could be definedbetween two points (one of each flank) or between two fittingfeatures (one of each flank), which are defined using a criterion(least square, MinMax, Tchebycheff) 10. Two Intended Function-alities and two Specified Characteristics were considered:?IFFi0: the maximum range of the transmission error,?IFfi0: the tooth-to-tooth maximum range of the transmissionerror,?SCcpe: the cumulative angular pitch error,?SCpe: the angular pitch error.Fig. 4 shows the global approach for the evaluation of thecorrelation uncertainties:Fig. 3. Influence of a pitch error on the transmission error 6.Fig. 4. Global approach for the correlation uncertainty evaluation.J.Y. Dantan et al./CIRP Annals - Manufacturing Technology 59 (2010) 5095125111. In the first step, a substituted geometry with randomdeviations is generated. The substituted model is an image of areal gear. This geometry is a set of analytic polynomial surfaces:smooth Bezier surfaces.2. A virtual acquisition of the substituted model is performed.Each Bezier surface is discreted into a set of equidistant points.3. The virtual metrologic process of the set of points obtained instep 2 is performed. The pitch errors are estimated for eachfitting criterion. A mathematical model based on the metrologyis created.20and 4. The meshing is simulated using the Tooth ContactAnalysis (TCA) program and leads to an estimation of thetransmission error 8.30and 5. Kinematic characteristics are evaluated.A Monte Carlo (step 6) simulation reiterates all the steps fromone to five, and for each studied criterion a set of characteristics isobtained. In the presented study three quality class gears havebeen tested: 10, 7 and 4. For each class, 100 geometries have beengenerated and simulated: for each criterion and each class, allstudied kinematic characteristics are calculated.A set of values for the Intended Functionalities and theMeasurands associated with the Intended Functionalities is thusobtained. For each criterion and each Intended Functionality, thefollowing relationships and quantities are calculated (step 7):?the linear regression between Intended Functionalities andMeasurands of each Intended Functionality: the equation of thelinear regression shows the trend of the correlation,?the confidence intervals of the estimated slope of the linearregression: it allows to affirm that the exposed values arecontained in an interval by allowing a risk of 5% in the caseinvestigatedhere.Itenablestoestimatethecorrelationuncertainty,?the linear correlation coefficient R provides information aboutthe quality of the correlation.The following equations show the results obtained for eachcriterion performed on ISO class 10, 7 and 4 gears.?1 point,IFFi0IFfi0?0:99?0:12?0:70?0:07?MIF;1PtMIF;1Pt?;R2Fi0R2fi0#0:720:30?Least square,IFFi0IFfi0?1:02?0:12?0:72?0:07?MIF;LSMIF;LS?;R2Fi0R2fi0#0:750:30?MinMax,IFFi0IFfi0?1:05?0:1?0:74?0:06?MIF;MMMIF;MM?;R2Fi0R2fi0#0:800:33?Tchebycheff,IFFi0IFfi0?1:03?0:08?0:83?0:05?MIF;TcMIF;Tc?;R2Fi0R2fi0#0:870:48?The Tchebycheff criterion offers the best correlation regardingthe functional aspect. The worst correlation is given by the onepoint fit. The least square criterion is close to the one pointcriterionregardingthecorrelation,andtheMinMaxcriterionoffersthe second best correlation after Tchebycheff criterion.The correlation coefficients of IFfiare small. In fact, this intendedfunctionalitydependsonthepitcherrorandontheformdeviationofthe gear flank. But the evaluated correlation uncertainty does nottake into account the form deviation as Specified Characteristic.5. ConclusionThe approach proposed here is based on a thorough and anintegrated view of functionality, specification and verification.The decision making during product conformity assessment isnot affected with only the measurement uncertainty, but alsowith the correlation uncertainty and the specification uncer-tainty. Based on the Axiomatic Design matrix, the proposedformalism of the correlation uncertainty allows to model alinear or a non-linear interrelation between the In