STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS ON THE MACHINE TOOL

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1 Metrol. Meas. Syst., Vol. XVIII (011), No. 1, pp METROLOGY AND MEASUREMENT SYSTEMS Idex , ISSN STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS ON THE MACHINE TOOL Staisław Adamczak, Adam Jausiewicz, Włodzimierz Makieła, Krzysztof Stępień Kielce Uiversity of Techology, Chair of Mechaical Techology ad Metrology, Al lecia P. P. 7, Kielce, Polad, , Abstract This paper deals with the experimetal validatio of the suitability of the method for measurig radial variatios of compoets o the process tool. The tests were coducted usig a computerized PSA6, which was compared to a Talyrod 73. The results of measuremet of roudess deviatios as well as roudess profiles were aalyzed for a sample of 70 shafts. The roudess deviatios were assessed by determiig the experimetal errors, while the profiles obtaied with the tested device were compared to those registered by the referece device usig three correlatio coefficiets. Keywords: roudess, experimetal method error, process tool. 011 Polish Academy of Scieces. All rights reserved 1. Itroductio The method for measurig roudess profiles o the process tool which bases o the variatios i the compoet radius belogs to a group of o-referece methods. It requires fixig a compoet i the cetres, rotatig it ad registerig the variatios i the radius i the rotatio agle fuctio by meas of a probe fitted perpedicular to the axis of rotatio [1,, 3]. The measuremet data may cotai a error, which is related to the type of probe applied ad the accuracy with which the ceter holes were made. The method error was determied theoretically [4]. It ca also be estimated experimetally by comparig the results obtaied with the tested istrumet to those from the referece istrumet. This paper discusses the results of statistical tests ad calculatios for the aalyzed ad referece istrumets, PSA6 ad Talyrod 73, respectively. The assessmet was made usig the experimetal error of roudess deviatio ad the correlatio-based compariso of the measured roudess profiles. I statistical testig, we used a sample of 70 groud shafts with ceter holes selected at radom from a batch. The measuremets were coducted uder laboratory coditios at the Kielce Uiversity of Techology, applyig equipmet suitable for measuremet of form profiles. The sequece ad rage of the calculatios were as follows: a. Determiatio of the experimetal error of the method for measurig roudess profiles, accordig to the priciples of statistical iferece, takig roudess deviatios ito cosideratio: the procedure for the estimatio ad test of sigificace for the mea value of the experimetal error [1, 5], the procedure for the estimatio ad test of sigificace for the variaces ad mea deviatios of the experimetal error [1, 5],, Article history: received o Ja. 5, 011; accepted o Feb. 8, 011; available olie o Mar. 1, 011.

2 S. Adamczak et al.: STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS the estimatio of the cofidece iterval of a sigle method error ad measuremet accuracy, the procedure for the test of cocordace betwee the distributio of the method error i a populatio with the theoretical distributio [1, 6] b. Statistical compariso of roudess profiles usig the correlatio calculus: compariso of roudess profiles by meas of cross-correlatio coefficiets, compariso of roudess profiles by meas of Pearso s liear correlatio coefficiets, compariso of roudess profiles by meas of Spearma s rak correlatio coefficiets. First, the referece istrumet, Talyrod 73, was used to measure the roudess profile of each compoet of a sample. I this way, a real roudess profile was obtaied. The measuremet results for a give roudess profile i the digital form ad the values of the roudess deviatio, Z, ad the amplitudes of the particular harmoics of this profile were trasmitted to the memory of the test-stad computer. Statistical iferece, i.e. drawig coclusios o the properties of the geeral populatio basig o the results obtaied for a sample or samples draw from this populatio, requires estimatig the values of the parameters of the distributio (poit estimatio), determiig the cofidece itervals (parameter estimatio) or testig statistical hypotheses. To reduce the probability of errors to a miimum, it was ecessary to: select a appropriate statistical method accordig to the data cocerig the aalyzed properties ad the tests to be coducted, use a represetative sample, strictly follow the procedure of each statistical method, apply the statistical methods selected for each test oly oce, maitai the assumed level of sigificace throughout the etire statistical hypothesis test, maitai the assumed level of cofidece whe determiig the cofidece iterval.. Experimetal error The basic method for idetifyig the accuracy of a measuremet istrumet is to compare its results with those obtaied by meas of a referece istrumet [1]. Oe of the most importat parameters used for this purpose is the experimetal error ( EBM) described by the followig relatioship: xi yi EBM =, (1) y i where: EBM experimetal error, x i measuremet result obtaied with the tested istrumet; y i measuremet result obtaied with the referece istrumet. The calculated experimetal errors were used to determie the accuracy of the aalyzed istrumet ad, accordigly, validate its suitability for certai applicatios. Table 1 shows rages of relative method errors established i our previous research ad the correspodig applicatios. 36

3 Metrol. Meas. Syst., Vol. XVIII (011), No. 1, pp Table 1. Rages of relative method errors i surface texture measuremet ad the correspodig applicatios [1]. Measuremet accuracy Type of applicatio rage [%] % 5% Measuremet of stadards: roughess, waviess, form profiles 5% 15% Scietific research 10% 5% Measuremet of surface texture uder idustrial coditios 3. Statistical determiatio of the method error usig roudess deviatios A sample cosistig of 70 groud shafts was used to establish the experimetal error through statistical testig. The aalysis icluded: the estimatio ad test of sigificace for the mea value, the estimatio ad test of sigificace for the variaces ad mea deviatios, the estimatio of the cofidece iterval of a sigle method error for the assumed cofidece, the determiatio of the cocordace of the distributio of the method error i a populatio with the theoretical distributio Procedures for the estimatio ad test of sigificace for the mea value of the experimetal error I order to compare the mea values of the experimetal error, it was ecessary to measure roudess deviatios i the same cross-sectios usig three differet istrumets. First, the value of the roudess deviatio was established by meas of the referece istrumet, i.e. the Talyrod 73, with higher spidle rotatio accuracy (spidle ruout of 0 m). The, the experimetal errors were calculated. Their mea values were determied usig the followig relatioship: 1 EBM = EBM i, () i = 1 where: umber of samples, EBM i relative method error of the roudess deviatio for each elemet of the sample. The mea value of the method error i the aalyzed populatio was estimated usig the followig procedure: a. Determie the method error icludig the roudess deviatio of the profile measured at a computerized test stad, b. Estimate the mea value of the method error, c. Estimate the iterval of cofidece for the mea value of the method error with ormal distributio ad the ukow mea deviatio usig the followig formula: where: s s E B M u p ; E B M + u p, EBM the mea value of the method error, s the calculated mea deviatio, u p the quatile of the ormal distributio read from the tables i Refs.[1, 6, 7, 8]. (3) 37

4 S. Adamczak et al.: STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS To explai whether or ot the divergece betwee the mea values is radom, it was ecessary to coduct a test of the meas for a case whe the values of the mea square errors are ukow. The mea values could be thus compared by calculatig the value of t s from: t s = s z x x , (4) where: x 1, x - the mea values beig compared, 1, umber of measuremets, s z = ( 1) ( 1) ( 1) + ( 1) s + s 1 1 1, (5) where: s 1, s mea square errors of the measured values of x 1, x The calculated value of t s was compared with its critical value, t(p, k) read from the tables i Ref. [1]. The critical value is determied whe P = 0.95 ad the umber of degrees of freedom is calculated from the followig relatioship. k = 1 +. (6) If t s exceeds the critical value, it ca be assumed that the divergece is sigificat; if ot, the divergece is radom (isigificat). 3.. Procedures for the estimatio ad test of sigificace for the variaces ad mea deviatios of the experimetal errors Whe establishig the relatioships betwee the variaces, oe eeds to estimate ad test the sigificace for the variaces. This is particularly importat whe the accuracy of istrumets is checked. The procedure to estimate the populatio variace for a case whe the mea value of the method error is ukow ivolves: a. calculatig the variaces value accordig to the formula: ( ) i 1, (7) s = 1 i= 1 EBM EBM where: umber of samples, EBM i the method error for the measuremet of a elemet i the sample, EBM the estimated mea method error for the sample. b. calculatig the value of the coefficiet F usig the followig relatioship: s F 1. s = > (8) 1 The calculated value of the coefficiet F was compared with the critical value read from the tables i Ref. [1]. If the value of F is greater tha the critical value, the divergece betwee the aalyzed variaces is sigificat. However, whe F is less tha the critical value, the divergece is assumed to be radom (isigificat). 38

5 Metrol. Meas. Syst., Vol. XVIII (011), No. 1, pp Estimatio of the cofidece iterval of a sigle method error ad measuremet accuracy The cofidece iterval of a sigle method error for the assumed level of sigificace was read from the ormal distributio tables. The cofidece iterval was calculated usig the followig relatioships: ( EBM u s; EBM + u s), (9) p where: EBM the mea value of the method error, u p the quatile of the ormal distributio, s the mea square deviatio of the method error. The measuremet accuracy of the aalyzed methods was determied usig the followig relatioship: max. p DP = EBM ± u p s (10) Formula (10) was used to qualitatively assess the measuremet accuracy of each method. It icluded roudess deviatios of the compoets from a give statistical sample, which were measured twice: first with the referece istrumet, ad the with the tested istrumet. Accuracy defied i this way icludes experimetal errors related to the systematic ad radom errors of the measuremet of this deviatio Procedure for the test of the cocordace betwee the distributio of the method error i a populatio with the theoretical distributio All the cosideratios were based o the assumptio that the results of the experimet are distributed ormally. The doubts cocerig the ormality of distributio will be fially dissolved if the procedure for the tests of the cocordace betwee the method error distributio ad the theoretical distributio is applied. I this aalysis, we used oe of the most popular tests determiig the level of cocordace with the ormal distributio test χ [1, 3] Assessig the results of statistical tests for the method error Table icludes results of the statistical tests of the method error established for the measuremet of radius variatios o a process tool for a sample of 70, estimatio of the tests, estimatio of the cofidece itervals ad tests of cocordace betwee the method error distributio ad the theoretical distributio. The mea values of the relative experimetal error of the method ad the cofidece itervals of a sigle value of the error idicate that the accuracy of measuremet of roudess deviatios for the aalyzed samples rages from 10% to 5% [1]. Table shows results of statistical testig of the experimetal method error calculated with respect to the Talyrod 73. The sigificace test for the mea values ad the variaces idicates that the divergece betwee the values calculated o the basis of the radomized test ad the critical values is radom. The test of cocordace with the theoretical distributio cofirms that the results of the experimetal error are i agreemet. The statistical testig of the experimetal error made it possible to calculate the accuracy of the method for measurig roudess profiles o the process tool based o the variatios i compoet radius. The accuracy was approximately 18%, which cofirms the istrumet suitability for aalyzig surface structure uder idustrial coditios. 39

6 S. Adamczak et al.: STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS Table. Results of the statistical determiatio of the experimetal error for the PSA 6 based o the results obtaied by meas of the Talyrod 73. Sample type Groud shafts Number of samples 70 Observed value of EBM mi the method error EBM max 0.53 Mea value Cofidece iterval for P= ±0.013 (u p =1.96) Test of sigificace for the mea radom divergece value Test of sigificace for the variaces radom divergece Mea deviatio s Test of cocordace with the The error distributio i cocordace theoretical distributio with the theoretical distributio Method accuracy MA [%] 18 % 4. Method for comparig roudess profiles by meas of the correlatio calculus 4.1. Compariso of roudess profiles usig cross-correlatio coefficiets The aalysis ad estimatio of the experimetal method error were used to assess the suitability of a istrumet for accurate measuremet. It is extremely importat to determie how similar the measured profiles are, because, theoretically, a istrumet may provide us with approximate results despite the fact that a completely differet roudess profile is registered. It is possible to visually compare two results obtaied for the same cross sectio with differet measuremet istrumets. The compariso, however, is oly of qualitative character. To compare the measured profiles quatitatively, oe eeds to use the correlatio calculus [1], by determiig the cross-correlatio fuctio. It ca be represeted usig the relatioship for the so-called coefficiet of cocordace: π Z ( ϕ) Z ( ϕ + ϕ ) dϕ r( φ ) =, (11) π π Z ( ϕ) dϕ + Z ( ϕ) dϕ T W T W 0 0 where: Z T (ϕ) profile measured by applyig the tested method, Z W (ϕ) profile measured usig the referece method, ϕ 0 shift betwee the measured profiles. The coefficiets ca be used to compare eve a sigle measuremet result. They may rage from 1 to 1. Whe the cross-correlatio coefficiet is egative, there is defiitely o agreemet betwee the measured profiles. Positive coefficiets ca be estimated basig o the rules provided by I. P. Guilford [1]. For the etire sample ( = 70), we determied: the mea value ad cofidece iterval, the cofidece iterval for a sigle cross-correlatio coefficiet. Usig the calculated mea values of the coefficiets of cocordace betwee the compared roudess profiles ad the estimated cofidece itervals for a sigle value of the coefficiet 40

7 Metrol. Meas. Syst., Vol. XVIII (011), No. 1, pp oe ca assume that the correlatio betwee the compared roudess profiles, i.e. oes measured with the radial method ad oes obtaied with the referece istrumet is very high. This is cofirmed also through qualitative (visual) compariso of these profiles (Figs.1 ad ) Table 3. Results of statistical tests of cross-correlatio coefficiets for the pair of istrumets: the PSA 6 ad the Talyrod 73 Sample type Groud shafts Number of samples 70 Observed mi value max Mea value Cofidece iterval for P=0.95 (u p =1.96) 0.995±0.00 Mea deviatio s Cofidece iterval for the variaces 0.009± Figures 1 ad show the visual compariso of roudess profiles measured usig the referece istrumet (Talyrod 73) ad the aalyzed istrumet (PSA6). The results illustrated below were obtaied for three compoets with differet values of the coefficiets of cocordace. Fig. 1 Visual compariso of two roudess profiles i rectagular coordiates measured with two differet istrumets (the Talyrod 73 solid lie, the PSA 6 dotted lie: a) compariso for compoet No 16 (crosscorrelatio coefficiet ), b) compariso for compoet No 6 (cross-correlatio coefficiet ), c) compariso for compoet No 68 (cross-correlatio coefficiet ) 41

8 S. Adamczak et al.: STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS Fig. Compariso of the amplitude spectra of the roudess profiles measured with two istrumets: the Talyrod 73 shaded bars, ad the PSA 6 white bars: a) compariso for compoet No 16, b) compariso for compoet No 6, c) compariso for compoet No Compariso of roudess profiles usig Pearso s liear correlatio coefficiets A alterative solutio for a cross-correlatio fuctio is to apply the Pearso s liear correlatio fuctio to assess the cocordace of profiles. The compariso required chagig the coordiates of the measuremet poits ito amplitudes ad phase shifts for harmoics from to 15. That was possible by applyig a Fast Fourier Trasform. The estimatio was carried out usig the obtaied values of amplitudes of the particular harmoics. Establishig the cocordace betwee profiles by meas of phase shifts was less importat as this testifies to the repeatability of positioig of a compoet i a istrumet [1]. Oce the values of the amplitudes of the harmoics from the two istrumets (the referece measuremet istrumet ad the PSA 6) were grouped ito 14 sets for each 4

9 Metrol. Meas. Syst., Vol. XVIII (011), No. 1, pp harmoic umber, they were statistically aalyzed, which ivolved establishig the coefficiet of correlatio betwee the amplitudes of the harmoics obtaied by usig the referece method ad the tested method. The, the matrix of the correlatio coefficiets was calculated. Pearso s correlatio coefficiets were calculated usig the followig relatioship: r = sxy s s where: s XY - covariace of the X i Y i sets, s X - mea deviatio for the values of the X i set, s Y - mea deviatio for the values of the Y i set. where: X Y XY X X i Yi i = 1, 1 s = ( C C )( C CY ), (13) C X i - values of the amplitudes of the particular harmoics for profile Z p (ϕ) C X - mea value of the amplitudes of the particular harmoics for profile Z p (ϕ) C - values of the amplitudes of the particular harmoics for profile Z a (ϕ) Y i C Y - mea value of the amplitudes of the particular harmoics for profile Z a (ϕ) The SADCOM program was used to calculate the correlatio matrices. The results are tabularized i Table 4. Usig the I.P. Guilford table, oe ca determie the relatioships betwee the obtaied correlatio coefficiet ad the degree of correlatio betwee the obtaied roudess profiles. Table 4. Pearso s correlatio matrix for measuremets by applyig the tested method ad the referece measuremet istrumet Talyrod 73 (1) I additio, the hypotheses were verified for each correlatio coefficiet by calculatig the statistic accordig to: r t =, (14) 1 r where: r estimated correlatio coefficiet, umber of samples. After assumig the probability (P = 0.95) ad the umber of degrees of freedom ( ), we read the critical values, t kr, ad compared them to the statistic, t. It was the possible to 43

10 S. Adamczak et al.: STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS defie whether or ot there exists ay correlatio. I the correlatio matrix, the highlighted umbers idicate that the correlatio occurs, while those i a white backgroud represet ull correlatio Compariso of roudess profiles usig Spearma s rak correlatio coefficiets Aother fuctio used to compare measured roudess profiles is the Spearma correlatio. Sice the method has previously proved to be effective, it was used for this aalysis. The basic defiitio of Spearma s correlatio fuctio is as follows: if the compared properties ca be ordered icreasigly, the it is possible to apply a rak correlatio fuctio. This was particularly sigificat i the experimetal tests coducted as part of this aalysis. Determiig Spearma s rak correlatio coefficiets for measuremet of differet compoets will imply that there is a correlatio betwee the obtaied roudess profiles. The procedure to derive Spearma s coefficiets of the rak correlatio betwee the amplitudes of the particular compared roudess profiles was as follows: determiig the amplitudes of each harmoic i the X ad Y sets, from the highest to the lowest accordig to variable Z T, ad the accordig to aother variable, i.e. Z W, for each pair of raks Z T ad Z W, determiig the differece D by subtractig the lower rak from the higher oe, assessig a rak correlatio coefficiet usig the formula 6D rs = 1, ( 1) where: D sum of squares of differeces betwee the raks determied by meas of relatioship (16) umber of samples. = ( xi yi ), i= 1 (15) D r r (16) where: r xi rak of a elemet of the X i set, r yi rak of elemets of the Y i set. As show i Table 5, the calculated coefficiets of the rak correlatio betwee the values of amplitudes of the particular harmoics for each sample were give i the form of the matrix of Spearma s correlatio coefficiets. I the matrix, the ull correlatio results were highlighted. Table 5. Spearma s correlatio matrix 44

11 Metrol. Meas. Syst., Vol. XVIII (011), No. 1, pp Estimatig the results of statistical testig for the compared roudess profiles usig the correlatio calculus Pearso s correlatio coefficiets were calculated for each set of amplitudes of the compared roudess profiles usig a special computer program, SADCOM [9]. The profiles were measured by meas of: o-referece istrumets equipped with ROFORM ad CYFORM software, referece istrumets equipped with the SAJD software, ZEISS coordiate machies with the CALYPSO software. Some of the calculatio results obtaied by applyig Pearso s ad Spearma s correlatio coefficiets for the amplitudes of each harmoic are provided i Table 6. Table 6. Pearso s ad Spearma s coefficiets of correlatio betwee the two istrumets, the PSA6 ad the Talyrod 73, for the amplitudes of each harmoic Harmoic umber Pearso s coefficiet Spearma s coefficiet From Table 6 it is clear that all the aalyzed relatioships are correlated, which was deoted by the plus sig (+) at each correlatio coefficiet. The values of the correlatio coefficiets show that there is a very high or high correlatio betwee the amplitudes of the harmoics that are domiat for the compared roudess profiles ad the amplitudes of the harmoics that have a sigificat effect o the profile form ( = 10). For the other amplitudes of each harmoic, the correlatio is i most cases very high, high ad moderate. 5. Coclusio The statistical tests show that there exists a high correlatio betwee the roudess deviatios as well as the roudess profiles measured with two differet methods. The 18 % accuracy cofirms that the aalyzed method is suitable for measurig surface texture uder idustrial coditios. Visual compariso of roudess profiles also shows that the tested istrumet ca be used for accurate measuremets. The correlatio betwee measured profiles was very high whe the profiles were compared usig a fuctio of correlatio, i.e. crosscorrelatio, Pearso s correlatio ad Spearma s correlatio. The tests cofirm that the aalyzed method ca be effectively used for measurig roudess profiles uder idustrial coditios. 45

12 S. Adamczak et al.: STATISTICAL VALIDATION OF THE METHOD FOR MEASURING RADIUS VARIATIONS OF COMPONENTS Refereces [1] Adamczak, S. (008). Measuremet of surface texture. Form profiles, waviess ad roughess. Warsaw. WNT. (i Polish). [] Jakubiec, W., Maliowski, J. (004). Metrology of Geometrical Quatities. Warsaw: WNT. (i Polish). [3] Pada, A., Jurko, J. (008). Bearigs for pumps ad compressors. Proceedigs of the 1th Iteratioal scietific ad egieerig coferece Hervico 008. Kielce Uiversity of Techology. Polad, [4] Adamczak, S., Jausiewicz, A. (004). Determiig the theoretical method error for measuremet of roudess profiles o productio machies. CEEPUS Sciece Report, Project PL-17, Measurig Techology i Advaced Maufacturig Systems. Kielce Uiversity of Techology, [5] Adamczak, S., Michalski, D. (004). Statistical Methods For Compariso of Measurig Istrumets Accuracy. Metrology ad Measuremets Systems, 11 (3), [6] Tiku, M.L. (1980). Goodess of fit statistics based o the spacigs of complete or cesored samples. Australia Joural of Statistics,, [7] Whitehouse, D.J. (1994). Hadbook of Surface Metrology. Istitute of Physics. Bristol&Philadelphia. [8] Whitehouse, D. J. (00). Surfaces ad their measuremet. Lodo. HPS. [9] Adamczak, S., Michalski, D. (00). SADCOM user maual. Kielce Uiversity of Techology. (i Polish) 46