EVALUATION OF MANUFACTURING TOLERANCE USING A STATISTICAL METHOD AND EXPERIMENTATION

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1 ISSN Int j smul model 11 (2012) 1, 5-16 Orgnal scentfc paper EVALUATION OF MANUFACTURING TOLERANCE USING A STATISTICAL METHOD AND EXPERIMENTATION Barkallah, M.; Louat, J. & Haddar, M. Mechancs Modellng and Producton Research Unt (U2MP) Unversty of Sfax, Sfax Engneerng School (E.N.I.S.), B.P 1173, 3038 Sfax, Tunsa E-Mal: bark_maher@yahoo.fr Abstract Ths paper proposes a study of a mllng process plannng to determne the tolerancng for manufacturng usng statstcal tool. The devaton between the machned surface and the surface correspondng to the nomnal one due to combned errors s expressed n terms of the small dsplacement torsor parameters. These parameters are consdered as random varables, dsplacements are expresses as varances. Expermentaton s realsed on a set of manufactured parts whch conssts of measurng varous defects on machne-tool. The expermental results and the smulaton results are compared n ths paper. (Receved n March 2011, accepted n August Ths paper was wth the authors 1 month for 1 revson.) Key Words: Manufacturng Tolerancng, Statstcal Analyss, Smulaton, Three-Dmensonal Model, Process Plannng 1. INTRODUCTION Manufacturng tolerancng s ntended to determne the ntermedate geometrcal and dmensonal states of the part durng ts manufacturng process. These manufacturng dmensons also serve to satsfy not only the functonal requrements gven n the defnton drawng, but also the manufacturng constrants. Ths topc has drawn consderable attenton, and many papers have been publshed on one, two and three-dmensonal tolerancng. A very broad revew of the tolerancng method s presented n Wang et al. [1]. In ndustry, the most common approaches are smply undrectonal and often based on the Δl method [2-4]. Such approaches are not able to handle small angular devatons from one machnng phase to the next. The purpose of threedmensonal approaches s to manage the small angular devatons occurrng between the varous machnng phases. Prelmnary work motvatng the development of the 3D tolerance propagaton technques s regarded as the spatal dmensonal chan technque [5], the propagaton of poston errors s taken nto account n terms of a knematc chan, where the ndvdual error s represented as matrces wth three-dmensonal and three angular poston errors. For pars of functonal elements n a knematc chan model s assocated wth a set of sx vrtual jonts, three for small translatons, and three for small rotatons [6]. Three-dmensonal tolerance propagaton models based on the concept of a small dsplacement torsor (SDT) are used to smulate threedmensonal fxturng and machnng errors and ther mpacts on the geometry of the fnshed part. The SDT s a mathematcal object that represents the dsplacement of a rgd body usng three rotatons and three translatons. Ths approach models the nfluence of a process plan on functonal tolerances as a chan of torsors [7, 8]. The SDT-based and three-dmensonal tolerance propagaton overcomes such lmtatons. Based on the SDT method, a detaled model of mechancal parts, part-holders, and machnng operatons was developed [9, 10] and extended to tolerance synthess [11, 12]. Vectoral tolerancng can be appled to geometrcal DOI: /IJSIMM11(1)

2 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... tolerance analyss, see [13] for example. A smulaton-based tolerance stack-up analyss was proposed [14], the Monte Carlo method s used to perform the smulaton. The dfferent possble errors are studed and t s assumed that Part/Fxture contact s perfect. Alternatvely, a graphcal representaton of part features, process plans, and functonal requrements defned wth an ISO standard can be employed to analyze three-dmensonal tolerance specfcatons and to generate manufacturng specfcatons compatble wth ISO standards [15]. Based on the TTRS approach [16], a set of algorthm allowng the development of manufacturng specfcaton was proposed [17, 18]. The dfferent revewed papers do not dscuss the estmated sources of error whether from a theoretcal model and/or expermental measurements. To perform realstc smulatons, t s now necessary to dspose of expermental data values. In ths paper a 3-D geometrcal manufacturng smulaton model s presented. It descrbes the geometrcal errors of part surfaces wth a statstcal approach. The am of realstc smulaton s to antcpate the behavour of the manufacturng process wth respect to ts capacty to obtan parts n conformty wth drawng offce specfcatons. Conformty of parts beng manufactured s verfed by means of tolerancng specfcatons for manufacturng [19-22]. Passng from one to another of these types of tolerancng specfcaton entals a specfcaton transfer operaton whch nvolves seekng the tolerancng specfcaton for manufacturng and a resoluton method. Smulaton s valdated by comparng smulated tolerancng values wth those obtaned from an expermental case study. 2. THE PROBLEM UNDER STUDY 2.1 Functonal defnton drawng The part to be studed s wrtten n accordance wth ISO standards by defnng the poston and sze of the tolerance zones wthn a reference system as presented n Fg. 1. In ths paper, a sngle functonal requrement C3 s studed. The reference B s a plane tangent to the real part surface. The nomnal plane s defned by a theoretcal dmenson of 5 mm from reference B. The tolerance zone 0.4 mm s the space between two planes centred on the nomnal plane. All ponts on the real specfed surface must le wthn the tolerance zone. C2 0.4 B 0.4 B C3 (59 ) C1 0.2 cz A 0.02 CZ 44 z A 5 39 ( 59 ) z y x C4 O.O5 B 20 (70) B Fgure 1: Dmensonng and tolerancng specfcatons on a desgn draft. 2.2 Tolerancng specfcaton for manufacturng The manufacturng process plan presented n Fg. 2 s performed on a numercal control (NC) machne tool. The machned surfaces have been denoted S and the raw surfaces BB. 6

3 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... The functonal requrement C3 s the localsaton of surface S 1 n relaton to surface S 6 (B). These surfaces (S 1 and S 6 ) are machned respectvely n phases 10 (S 1 ), and 20 (S 6 ). The surfaces are not generated n the same phase. The functonal tolerance s obtaned wth two phases. Phase number Drawng Phase Actve surface Phase 00: Recept of raw B2 y 00 z 00 4,5 B3 O 00 B1 1,2,3 Phase 10: Mllng on the NC machne: - prmary supports BB1 - secondary supports BB3 - tertary supports BB B2 y 10 z 10 O 10 O 1 z 1 y 1 2 X10 O 10 z 10 4,5 B3 SP 1 = B 1 SP 2 = B 3 SP 3 = B 2 S u 10 = S 1, S 2, S 3 Mllng S 1, S 2 and S 3 B1 1,2,3 Phase 20: Mllng on the NC machne: - Prmary supports S 2 - Secondary supports BB4 - Tertary supports BB2 6 B2 z 20 Y 20 4 y z 6 B4 4,5 SP 1 = S 2 SP 2 = B 4 SP 3 = B 2 S u 20 = S 4, S 5, S 6 Mllng S 4, S 5 and S 6 2 1,2,3 Fgure 2: Manufacturng process plan. The strategy for searchng for specfcatons to be ncluded n the phase contract s supported by the structure of specfcatons for desgn and process plannng. In ths paper, we suppose that the qualtatve generaton of manufacturng specfcatons s already dealt wth the ratonal method of 3D manufacturng tolerancng [17, 18]. The reducton of reference surfaces n a geometrc specfcaton of manufacturng s done n an automated manner by applyng dfferent equatons, orgnally developed by [23]. The manufacturng specfcatons (generated by a transfer) obtaned for the studed requrement C3 are represented on manufacturng drawngs presented n Fg. 3. We propose through a statstcal method based on small dsplacement torsor parameters, to treat the second pont whch s the determnaton of manufacturng tolerances n a quanttatve aspect. Ths work wll enable: to valdate a plannng process takng n to account the dfferent dspersons of manufacturng process (fxturng, machnng,...), to determne, for every manufacturng specfcaton, the adopted tolerances. 7

4 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... C3 t1 B1 B3 B C2 3 t2 CZ B1 Phase10 Mllng of surfaces S 1,S 2 and S 3 (S 1 : toleranced surface) 6 B ,5 B3 B1 1,2,3 C3 tf t6 2 B4 B t4 6 C4 20 Phase20 Mllng of surfaces S 4,S 5 and S 6 (S 6 : datum reference B ) 6 B ,5 B4 2 1,2,3 t5 2 C1 tf Fgure 3: Manufacturng drawngs. 3. THREE-DIMENSIONAL TOLERANCING MODEL The proposed work conssts on developng a statstcal method for the smulaton of 3D manufacturng tolerancng whch allows the evaluaton of the manufacturng tolerances zone. Ths method based on the components of the small dsplacements torsors whch are consdered as random varables. The small dsplacement torsors (SDT) assocated wth every surface of the chan s transported on the target surface and act as generators of dsturbances. They are appled n several ponts defnng the nomnal surface and so generate ponts of the resultant tolerance zone. 3.1 Statstcal aspect of devaton torsor The small dsplacement torsor (SDT) s used to defne the geometrcal defects of machned surfaces. The SDT concept s based upon the movements of a rgd body [6]. It s defned by two vectors representng the values of three small rotatons Ω ( α, βγ, ) and three small translatons DO ( u, v, w) of a pont O of the body. In ths case, the dsplacement of any pont M of the body s obtaned by: 8

5 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... D = D + Ω OM (1) M O The small dsplacement of the consdered surface s characterzed by a torsor defned n the coordnate frame, R as: Ω T [ T] = [ ] ( OR = αβγu v w (2), ) ( OR, ) D O ( OR, ) In ths study, the sx components of the SDT are consdered as random varables. It should be noted that a torsor s defned by a pont and a bass. The change of a pont and the base of torsor s acheved by a base pont and a 6 6 transformaton matrx. For any change of torsor reference of a wrench, the followng matrx relaton s then used: 1 Ω ( x [ ] 2, y2, z2) P 12 0 Ω ( x1, y1, z1) = (3) D O P D 2( x2, y2, z2) 12 κ 1 P 12 O 1( x 1, y 1, z1) where represents the matrx of pre-vector product of the vectoroo. 2 κ Components of the devaton torsor expressed at the base pont O 1( x 1, y 1, z 1 ) were assumed to be ndependent. The covarance matrx of the random vector defnng ths torsor s dagonal. In general, the lnear change of varable Y = K X, where X and Y are random vectors and K s a constant matrx, the covarance matrx of the random vector Y s obtaned by: cov(y) = K T. cov(x). K (4) The covarance matrx assocated wth the devaton torsor of the surface S expressed at x, y, z s therefore deduced from relatonshps (3) and (4). It s then pont O 2 n the base ( ) necessary to work n a unque reference, whch wll be called part reference. Eqs. (3) and (4) allow deducng the statstcal characterstcs of the resultng devaton torsor. The resultng devaton s the sum of random devatons torsors ntervenng n the equaton of the chan. For each devaton torsor nvolved n ths relatonshp, there s a passage matrx P k whch allows changng the local coordnate system (centre of the surface studed) to the global coordnate system (reference part). The statstcal characterstcs of the resultng devaton (T r ) are deducted from the followng equaton: cov( T ) P cov( T) P P cov( T ) P r T T = The varance n the dsplacement of a pont M n the normal drecton of the surface s obtaned by projectng followng a untary normal vector (shown as n S n Fg. 4). The projecton s expressed by the relaton: T V = σ 2 = M, DM n ns Σ ns ns s (5) (6) 3.2 Study of a machnng step The objectve of ths study s to determne the devaton between the real surface and the correspondng nomnal surface, dependng on the dfferent nfluental defects n the S manufacturng process. Ths devaton s defned by dsplacng pont M of real surface wth respect to the correspondng pont M on the nomnal part model as presented n Fg. 4. N 9

6 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... Fgure 4: Devaton between the toleranced surface and the nomnal part. S N S The devaton between pont M and nomnal surface M M s obtaned by summng two devatons: N S N M M S M M n = M M n + M M n (7) where: N M - M M n : Devaton between nomnal part surface and nomnal machne surface. Ths N devaton s the dsplacement of pont M, whch can be calculated wth the small dsplacement torsor. M S - M M n : Devaton of machned surface S wth respect to the nomnal machne surface (programmed surface). Ths devaton can n theory be drectly measured at each pont M usng a Renshaw probe that assumes machne-tool defects are smallest as ether tool defects or part deformatons. In our approach, we do not treat the case of a part, but the mage of a seres of parts. Present when manufacturng a surface, the followng dspersons n the postonng system can by recovered when manufacturng a surface: - Dspersons about the poston of the toleranced element. - Dspersons about the poston of the referenced elements. These dspersons depend on the manufacturng process such as machnng and postonng of the workpece. Based on the hypothess of small dsplacement around the nomnal poston, due to varaton of postonng and dmensonally stable solds, t can be shown that the varance n the dsplacement of a pont of a surface n a gven drecton can be calculated by: T 2 V = σ = M,n D ns Σ s pos,m M n ns (8) s where: - [ n S (a, b, c )] s the sngle column matrx of the unt vector coordnates defnng the calculaton drecton, - Σ pos, M s the 3 3 matrx to calculate the varance at each pont M of the termnal surface. Its components depend on the coordnates of M (x, y, z), the coordnates of the part-holder support ponts A (X A, Y A, Z A ), the normal at contact n A (n x, n y, n z ) and the varaton of the poston at the ponts of contact δ. 10

7 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... To get the total local varance, addng the varance of machnng defects obtaned by drect measurement on the machne at each pont M usng a Renshaw probe can be suffcent: 2 T σ = n n 2 Total / s Σ pos,m s + σ (9) ns machnng These results are true f the ponts of the parts are centred on the average poston or are offset n relaton to ths poston. 4. NUMERICAL VALIDATION OF DEVELOPPED MODEL 4.1 Influence of dspersons n phase 10 In phase 10, the part studed s placed on the sostatc part-holder by a prmary support (A 1, A 2 and A 3 ) on the surface BB1, a secondary support (A 4 and A 5 ) on the surface B 2B and tertary support (A 6 ) on the surface BB3. Fg. 5 llustrates the poston of A 1 to A 6 and M 1 to M 4 n phase 10. A 6 A 4 A 5 A3 A 1 A 2 Fgure 5: Part-holder n phase 10. In a machnng phase, the defects generated by a manufacturng process are caused by defects n postonng of the workpece and machnng defects. Defects resultng from the postonng of the pece depend on two types of defects: - Defects of perpendcularty between the three reference surfaces of the part. - The qualty of support, and clampng surfaces. These random defects were evaluated from measurements made on a seres of 50 peces. Table I gves the characterstc nformaton of the postonng for determnng the expresson of matrx components pos, M. Usng dfferent dspersons due to work holdng errors, and takng nto account both the measurement and the coordnates of ponts of the target surface, we thus obtaned the varous possble varatons n the normal drecton normal to the plane of the pont M. Table II contans the nformaton needed to conduct the calculaton of the varance of the poston at dfferent ponts n a gven drecton. By addng ths varance to the varance of machnng defects (V machnng ) n phase 10, we obtan the varance of the dsplacement of ponts M of the toleranced surface wth respect to the system reference. 11

8 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... Table I: Dsperson on each part-holder support n phase 10. Poston and orentaton of the support (phase 10) Ponts A 1 A 2 A 3 A 4 A 5 A 6 X A 2,851 52,769 27, ,321 Y A -59,259-59,259-9,463-21,021-46,021 0 Z A ,927 11,927 28,483 n x n y n z Varance δ 2,74E-05 2,65E-05 1,79E-05 6,29E-05 6,53E-05 2,03E-04 Table II: Dspersons on the target surface n phase 10. Ponts of the surface1 (Phase 10) Ponts M 1 M 2 M 3 M 4 X Y 50,5-50,5-50,5-50,5 Z a b c V postonnng E E E E-04 V machnng 9,4567E-06 1,78E-05 1,4649E-05 1,4124E-05 total varance 0, , , , Tolerance nterval (1/0) 0, , , , , Influence of dspersons n phase 20 In phase 20, the studed part s placed on the sostatc part-holder by a prmary support (A 1, A 2 and A 3 ) on the surface 2, a secondary support (A 4 and A 5 ) on the surface BB4 and tertary support (A 6 ) on the surface B 3B. Fg. 6 llustrates the poston of A 1 to A 6 and M 1 to M 4 n phase 20. A 6 A 4 A 5 A 3 A 1 Fgure 6: Part-holder n phase

9 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... Table III gves the characterstc nformaton of the postonng for determnng the expresson of matrx components pos, M n phase 20. Table III: Dspersons on each part-holder support n phase 20. Poston and orentaton of the support (phase 20) Ponts A 1 A 2 A 3 A 4 A 5 A 6 X A 3 52,77 27, Y A -59,26-59,26-9, Z A ,5 n x n y n z Varance δ 1,5176E-05 9,816E-06 1,291E-05 1,2374E-05 1,35E-05 1,6327E-05 To obtan the varance, the same procedure used n phase 10 s appled. Table IV present the dsplacement varance of the toleranced surface (6) summts ponts (M ) n ts normal drecton wth respect to the reference system of the manufacturng specfcaton n phase 20. Table IV: Dspersons on the target surface (6) n phase 20. Ponts of the surface 6 (Phase 20) Ponts M 1 M 2 M 3 M 4 X Y Z a b c V postonnng E E E E-04 V machnng 1,46465E-05 1,4816E-05 1,5098E-05 1,47E-05 total varance 0, , , , Tolerance 0, , , , nterval (6/0) 0, Results and comparsons Consderng a normal dstrbuton followng the dstrbutons shown n Fgs. 8, 9 and 10, the dsperson s evaluated n ts maxmum result as follows: - Phase 10: for a varance Phase 20: for a varance Localsaton (C3): for a varance To check the consstency of smulaton results wth respect to expermental results, a comparson s made wth measurements on a seres of 50 peces shown n Fg. 7. The results of the computed accumulaton of the manufacturng defects generated by two phases (partholder defects, workpece locatng dsperson and machnng defects) s then compared wth the values determned after fnal control of the functonal condton C3 over the coordnate 13

10 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal... measurng machne (CMM). Table V shows ths comparson. It s observed that the dfference between the analytcal model and expermental measurement s reasonably small. Fgure 7: Fnal nspecton on CMM. Fgs. 8, 9 and 10 dsplay the normal probablty plots of the sample data and from these fgures the sample data seems to be normal. The hstograms of the 50 observatons wth a densty lne and specfcaton lmt (LSL, T, USL) and the results of the process-capablty analyss are dsplayed. Probablty plot of observed values for δ 1/ 0 Hstograms and process-capablty analyss for δ 1/ 0 Percent Normal 95% CI de δ 1/ 0 Mean 0,07294 StDev 0,01941 N 50 AD 0,420 P-Value 0,314 cble cble USL Potental (Wthn) Capablty Cp 1,13 CPL 1,25 CPU 1,00 Cpk 1,00 CCpk 0,00 Process Data LSL 0,00000 Target 0,00000 USL 0,13118 Sample Mean 0,07294 Sample N 50 StDev(Wthn) 0, ,00 0,02 0,04 0,06 0,08 δ 1/ 0 0,10 0,12 0,14 0,00 0,02 0,04 0,06 0,08 0,10 0,12 Fgure 8: Dstrbuton of measured values (phase 10). Probablty plot of observed values for δ 6/ 0 Hstograms and process-capablty analyss for δ 6/ 0 Normal 95% CI de δ 6/ 0 Percent Mean 0,05874 StDev 0,02123 N 50 AD 0,247 P-Value 0,742 cble USL Potental (Wthn) Capablty Cp 0,96 CPL 0,92 CPU 1,00 Cpk 0,92 CCpk 0,00 Process Data LSL 0,00000 Target 0,00000 USL 0,12242 Sample Mean 0,05874 Sample N 50 StDev(Wthn) 0, ,00 0,02 0,04 0,06 0,08 0,10 0,12 δ 6/ 0 0,00 0,02 0,04 0,06 0,08 0,10 0,12 Fgure 9: Dstrbuton of measured values (phase 20). 14

11 Barkallah, Louat, Haddar: Evaluaton of Manufacturng Tolerance Usng a Statstcal Probablty plot of observed values for δ 1/ 6 Normal 95% CI de δ 1/ 6 Mean 0,1306 StDev 0,02590 N 50 AD 0,492 P-Value 0,209 cble Hstograms and process-capablty analyss for δ 1/ 6 USL Potental (Wthn) Capablty Cp 1,34 CPL 1,68 CPU 1,00 Cpk 1,00 CCpk 0,00 Percent Process Data LSL 0,00000 Target 0,00000 USL 0,20833 Sample Mean 0,13062 Sample N StDev(Wthn) 0, ,050 0,075 0,100 0,125 0,150 0,175 0,200 0,225 δ 1/ 6 0,00 0,03 0,06 0,09 0,12 0,15 0,18 0,21 Fgure 10: Dstrbuton of measured values δ 1/ 6 (C3). Table V: Comparson between smulaton and expermental results. Tolerance nterval (1/0) - phase 10 t1 B1 B3 B2 Smulaton Expermental 0,082 0,116 Tolerance nterval (6/0) - phase 20 0,079 0,127 t6 2 B4 B2 Tolerance nterval (1/6) 0.4 B 0,281 0,31 5. CONCLUSION A three-dmensonal statstcal approach for determnng the manufacturng tolerances has been presented, based on the components of the small dsplacements torsors whch are consdered as random varables. Passng from a part to a seres of manufactured parts s obtaned not by consderng small dsplacements (representatve of geometrc defects) but by the varance on these dsplacements. Through a three-dmensonal smulaton method, we determne the geometrc tolerance values of dfferent manufacturng specfcatons. The nput data for smulaton are estmated varance and the results be expressed, too, for a varance. An expermental campagn was set up whch conssts of measurng varous defects on machnetool and then ensurng that the defects obtaned on the fnal part are n good agreement wth those obtaned from smulaton. It was found out that the dfference was reasonably small. Followng these studes an automatc tool based on software can be developed to respond to the ncreasng needs of ndustrals n ths feld. REFERENCES [1] Ru, W.; Thmm, G. L.; Yongsheng, M. (2010). Revew: geometrc and dmensonal tolerance modellng for sheet metal formng and ntegraton wth CAPP, Internatonal Journal of Advanced Manufacturng Technology, Vol. 51, No. 9-12, , do: /s [2] Anselmett, B.; Bourdet, P. (1993). Optmsaton of a workpece consderng producton requrements, Computers n Industry, Vol. 21, 23-34, do: / (93)90042-y [3] Duret, D. (1995). Détermnaton des ntervalles de tolérance en fabrcaton par une approche graphque, Actes sémnare tolérancement et chaînes de cotes, ENS Cachan [4] Bouazz, Z.; Masmoud, F. (2006). Manufacturng smulaton: Computer aded tolerancng for process plannng, Internatonal Journal of Smulaton Modellng, Vol. 5, No. 1, 5-15, do: /ijsimm05(1)

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