SELECTED PROBLEMS OF MEASUREMENT UNCERTAINTY - PART 2

Transcription

1 Technical Science Abbrev.: Techn. Sc., No 11, Y. 8 SELECTED PROBLEMS OF MEASUREMENT UNCERTAINTY - PART Sylwester Kłysz 1,, Janusz Lisiecki 1 1 Division for Reliability&Safety of Aeronautical Systems Air Force Institute of Technology in Warsaw Faculty of Technical Sciences University of Warmia and Mazury in Olsztyn Key words: measurement uncertainty, A-type and B-type uncertainty evaluation, normal (Gaussian) and rectangular distribution, uncertainty budget Abstract The rules for evaluation of measurement uncertainty are presented in this paper. It includes the list of definitions for basic terms that are associated with the issue and explains how to evaluate the measurement uncertainty for well-defined physical parameters that are considered as the measurands (part 1). Next, the methods for evaluation of measurement uncertainty are exhibited on the examples when measurement uncertainty is estimated for verification of a micrometer as well as evaluated for basic reliability parameters (part ). WYBRANE ZAGADNIENIA NIEPEWNOŚCI POMIARU CZ. Sylwester Kłysz 1,, Janusz Lisiecki 1 1 Zakład Niezawodności i Bezpieczeństwa Techniki Lotniczej Instytut Techniczny Wojsk Lotniczych w Warszawie Wydział Nauk Technicznych Uniwersytet Warmińsko-Mazurski w Olsztynie Słowa kluczowe: niepewność pomiaru, niepewność typu A i typu B, błąd pomiaru, rozkład normalny i prostokątny, budżet niepewności

2 Sylwester KŁYSZ, Janusz LISIECKI Streszczenie W pracy podano zasady postępowania przy wyznaczaniu niepewności pomiaru. Przedstawiono podstawowe pojęcia dotyczące tego zagadnienia oraz sposób wyznaczania niepewności pomiaru dobrze określonej wielkości fizycznej stanowiącej wielkość mierzoną (cz.1). Przedstawiono sposób szacowania niepewności pomiaru na przykładzie szacowania niepewności przy sprawdzaniu mikrometru i szacowania niepewności podstawowych parametrów wytrzymałościowych (cz.). 1. Estimation of uncertainty for verification of a micrometer The error of micrometer readout can be calculated with use of the formula [Piotrowski, Kostyrko ]: E x = (L W + P t ) ± U(E x ), (1) where: L the maximum readout from the micrometer for three measurements of dimensions, W rated length of the standardized plate (W n ) with account for the correction factor deviations of the plate length, P t correction factor for temperature conditions, U(E x ) the expanded uncertainty at the confidence level of 1-α =,95. The value of the temperature correction factor is neglected. Uncertainty equation Due to the fact that input parameters are uncorrelated, the standard uncertainty connected with the already determined absolute deviation for the micrometer readout can be expressed by the formula: ( ) = ( u E c u ), () c x where: c i sensitivity coefficients, i.e. the partial derivatives of the measurement function for the function components. In this case c i = 1 or c i = -1. The standard uncertainty for the limit error can be then expressed in the following form: c x i i u ( E ) = u ( L) + u ( W ) + u ( P ). () t

3 Selected problems of measurement uncertainty - part Determining of component standard uncertainties The standard uncertainty for the micrometer readout is determined on the basis of the instrument resolution r, that for this case is, (readout with use a magnifying glass), by means of the B-type method and with the assumption of the rectangular distribution of the uncertainty: u ( L) = r =, 58 mm. (4) The standard uncertainty for the standardized plate length is determined on the basis of the expanded uncertainty U for determining of the deviation of the plate length from its rated length as provided by the calibration certificate. The B-type method is applied with assumption of the Gaussian distribution. U u ( W ) =. (5) The standard uncertainty for the temperature correction factor u(p t ) is determined under the assumption that coefficients of thermal expansion for the standardized plate and for the micrometer are the same: α W = α L = α t = 11,5 1-6 o C -1 and Δt = t W - t L (where: t W = o C, t L ambient temperature in the laboratory room T I = ± 1 C). The uncertainty u(δt), as determined with use of the B-type method (rectangular distribution) amounts to t. Thus: u(p t ) = W α t t. (6) Expanded uncertainty If all values of component uncertainties u(e x ) are comparable, the expanded uncertainty at the confidence level of 1 α =,95 can be calculated with the formula: U(E x ) = u c (E x ). (7) If the uncertainty u c (E x ) has one dominating component (e.g. u(l)) and the rectangular distribution is assumed, then the distribution can be considered as the distribution for readout errors and then the expanded uncer-

4 4 Sylwester KŁYSZ, Janusz LISIECKI tainty at the level of confidence of 1 α =,95 can be calculated with the formula: U(E x ) = 1,65 u c (E x ). (8) If the uncertainty u c (E x ) has two dominating components (e.g. u(l) and u(w)) and the rectangular distribution is assumed for the both components with the respective spans R 1 = a 1 and R = a, the components can be superposed to make up the trapezoidal even-armed distribution with the span of R = a = (a 1 + a ) and upper base of b = aβ, where β = (a a 1 )/(a + a 1 ). Then the composed uncertainty amounts to u c (E x ) = x ) u ( x ) ( 1 u + = 1 a a +. (9) The expanded uncertainty at the confidence level of P = 1 α =,95 can be calculated by the formula: U(E x ) = 1. (1) (1 P)(1 β ) u c( E x ) 1+ β 6 Finally, the indication error value shall be: E x =max {d - U(E x ), d + U(E x )}, (11) where: d the maximum deviation from the W dimension, d = L W. Note: uncertainty components u c (E x ) can be considered as insignificant and then neglected if u c (E x ) * u c (E x ),5 u c (E x ), * (where u c (E x ) the composed uncertainty with one or two components ignored), i.e. when omission of one or two components in the formula for the uncertainty calculation results in alteration of such uncertainty by not more than 5%. Uncertainty budget All the informations for analysis of uncertainty are brought together in the table 1. Table 1

5 Selected problems of measurement uncertainty - part 5 Uncertainty budget for calculation of the composed uncertainty for verification of a micrometer Parameter symbol X i Parameter estimation x i Standard uncertainty Contribution of into the standard uncertainty for each individual standardized plate (of n) L Eq. (4) c i u 1 (L 1 ) c i u (L ) c i u (L ) c i u n (L n ) W Eq. (5) c i u 1 (W 1 ) c i u (W ) c i u (W ) c i u n (W n ) P t Eq. (6) c i u 1 (P t1 ) c i u (P t ) c i u (P t ) c i u n (P tn ) E x L - W Eq. () Eq. () Eq. () Eq. () 1.. Estimation of uncertainties for measurements of strength parameters obtained from the static tensile test Estimation of uncertainty for determination of the tensile strength R m Formula for the measurement result (for a single specimen) Tensile strength R m = f (F m, d ) = F m = S where: F m maximum force recorded during the tensile test, S initial cross-section of the specimen, d initial average diameter of the specimen. 4 π F m d, (1) Uncertainty equation Due to the fact that input parameters are uncorrelated, the standard uncertainty connected with the determined tensile strength of the specimen is defined by the following formula: ( R ) ( c u ) u, (1) = m i i where: c i sensitivity coefficients, i.e. the partial derivatives of the measurement function for the i th function components, u i standard uncertainties for individual components.

6 6 Sylwester KŁYSZ, Janusz LISIECKI In this case c Fm = 4 and c do = 8 π F m d, thus: u 4 8F ( R ) m u ( F ) + u ( d ) m m =. (14) Determination of component standard uncertainties The uncertainty for measurement of the specimen diameter u ( d ) is calculated by means of [Rozporządzenie Ministra... 4]: a) on the basic of the arithmetical means for the series of six measurements (the A-type method), with the t-student distribution assigned (for p α = 68,7%): u ( dk d ) k = 1 ( d s ) = 1,11, (15) n n( n 1) where: n number of measurements, or b) on the basic of the micrometer resolution, with use of the formula: where u(d m ) amounts to,89 mm.,1 u ( d m ) =, (16) The value which is higher is adopted for further calculations. Major factors that affect total uncertainty of measurement of the F force include: 1) uncertainty of the measurement of the force UF F m m uw ( Fm ) =, (17) where: U Fm the uncertainty of measurement (in percents), attributable to the dynamometer of the machine, as read from the calibration certificate for the force that is the closest to the force value F m measured during the tensile test and for the selected range of the applied measuring head, ) zero adjustment in the force-measuring path, ) possible misalignment of the force applied,

7 Selected problems of measurement uncertainty - part 7 4) ambient temperature during the test and velocity of the load application, 5) sampling (recording frequency). The error that results from the aforementioned factors was evaluated to ±1%. Therefore, the uncertainty for measurement of the maximum force shall be calculated with the formula Uncertainty budget u,1 F ) =. (18) m ( Fm The information for further analysis of uncertainty is presented in the Table. Table Uncertainty budget for calculation of the composed uncertainty for tensile strength Parameter symbol X i Parameter estimation x i Standard uncertainty Sensitivity coefficient c i F m Eq. (18) d 4F m R m Eq. (15) or (16) 4 8F m Contribution into the composed standard uncertainty c i u(f m ) c i u( d ) Eq. (14) Determination of the expanded uncertainty The relative expanded uncertainty with the expansion coefficient k α = at the confidence level of 1 α =,95 is calculated with the following formula: u( Rm ) U ( R ) = 1%. (19) m 1... Estimation of the uncertainty for determination of the proof stress R p, (at non-proportional elongation) Formula for the measurement result (for a single specimen) R m

8 8 Sylwester KŁYSZ, Janusz LISIECKI Proof stress R p, = f (F,, d ) = F, S 4F, =, () where: F, tension force that is applied during the test and then brings to permanent elongation of the specimen equal to,% of the measurement length that corresponds to the extensometer base, S initial cross-section of the specimen, d initial average diameter of the specimen Uncertainty equation Due to the fact that input parameters are uncorrelated, the standard uncertainty connected with the determined proof stress is defined by the following formula: ( ) = ( u R c u ), (1) p, i i where: c i sensitivity coefficients, i.e. the partial derivatives of the measurement function for the i th function component, u i standard uncertainties for individual components. In this case c F, = 4 and c do = - 8F,, thus: u 4, ( R ) u ( F ) + u ( d ) p, c, 8F =. () Determination of component standard uncertainties The uncertainty for measurement of the specimen diameter ( ) d , equations (15 17). Overall uncertainty for measurement of the force F, : u par. ( F ) = u ( F ) + u ( F ) u ( F ), () uc,,, +, E where subsequent components result from:

9 Selected problems of measurement uncertainty - part 9 uncertainty of the force measurement (see formula (18)) u ( F ),,1 F, =, recording frequency during automatic measurement F ( 1) F, ( ) u( F ),, =, where: F,(1) the closest value of force that was higher than F,, F,() the closest value of force that was lower than F, ; inclination of the straight line that is parallel to the linear section of the σ-ε curve that is described by the formulas: σ,e = E( ε,) or F F, E = ( ε,), ε u where:,e ) ε, ε F( ε,) ( ε) F ε ( F ) = u ( F ) + u ( ε + u ( ε),,1 Fmax where: u( F ) = and u( F ) max Kε u ( F ) = u ( F ) + u ( F ), min max,1 Fmin =, min u ( ε) = u ( ε ) + u ( ε ), max Kε εmax εmin where: u ( εmax ) = and u ( εmin ) =, K ε u ( ε) = ε, where: K ε accuracy class of the applied extensometer Uncertainty budget The informations for further analysis of uncertainty are presented in the Table. Table Uncertainty budget for calculation of the composed uncertainty for proof stress Parameter Parameter Standard Sensitivity Contribution into the min

10 1 Sylwester KŁYSZ, Janusz LISIECKI symbol X i estimation x i uncertainty coefficient c i F, Eq. () 4 composed standard uncertainty c i u c (F, ) d 4F, R p, Eq. (15) 8F, c i u( or (16) d ) Eq. () Determination of the expanded uncertainty The relative expanded uncertainty with the expansion coefficient k α = at the confidence level of 1 α =,95 is calculated with the following formula: u( Rp, ) U ( R ) = 1%. (4) p, Rp, 1... Estimation of the uncertainty for determining of the elongation at rupture A Formula for the measurement result (for a single specimen) For measurement with use of an extensometer the elongation at rupture is: A = f (ε t, ε,1 ) = ε t - ε,1, (5) where: ε t deformation of the specimen at rupture; ε,1 deformation at the limit of elasticity R s Uncertainty equation Due to the fact that the measurement is taken directly and the composed uncertainty u(a) is the resultant of two components, the composed standard uncertainty can be calculated with the formula: ( A) u ( ε ) u ( ε ) u = +. (6) t c,1

11 Selected problems of measurement uncertainty - part 11 Determination of component standard uncertainties The uncertainty for measurement of the deformation ε t is determined on the basis of the accuracy class attributable to the longitudinal extensometer K ε, with use of the B-type method and the rectangular distribution is adopted for this uncertainty: Kε εt u( εt ) =. (7) The uncertainty of measurement of the deformation ε,1 : ( ε ) u ( ε ) + u ( ε ) + u ( ) uc,1,1,1 ε, 1FE =, (8) where subsequent components result from: the accuracy class of the longitudinal extensometer ε,1 ( ε,1) = K ε u, recording frequency during automatic measurements ε,1(1) ε,1() u ( ε,1) =, where: ε,1(1) the closest value of deformation that was higher than ε,1 ; ε,1() the closest value of deformation that was lower than ε,1 ; inclination of the straight line that is parallel to the linear section of the σ-ε curve that is described by the formula: 4F,1 σ,1e = E( ε,1,1), hence ε,1 = +, 1, thus: E F F u( ε 4 8,1 4,1,1FE ) = u ( F,1) u ( d ) u ( E) d E + + d E d E, π π π,1 F, 1,1 = (see equation (8); u ( d ) see equation (5) and (6); u ( E) see equation (). where: u( F )

12 1 Sylwester KŁYSZ, Janusz LISIECKI 1... Uncertainty budget The informations for further analysis of uncertainty are presented in the Table 4. Table 4 Uncertainty budget for calculation of the composed uncertainty for elongation at rupture Parameter symbol X i Parameter estimation x i Standard uncertainty ε t Eq. (6) c i u(ε t ) Contribution into the composed standard uncertainty ε,1 Eq. (7) c i u c (ε,1 ) A ε t - ε,1 Eq. (5) Determination of the expanded uncertainty The relative expanded uncertainty with the expansion coefficient k α = at the confidence level of 1 α =,95 is calculated with the following formula: u( A) U ( A) = 1%. (9) A Estimation of the uncertainty for determination of the Young s (elastic) modulus E Formula for the measurement result (for a single specimen) The Young s (elastic) modulus: E = f (ΔF, d, Δε) = S F ε = 4 ε F, () where: ΔF difference between the upper (F max ) and lower (F min ) level of forces for the range of elastic deformations, S initial cross-section of the specimen, d initial average diameter of the specimen,

13 Selected problems of measurement uncertainty - part 1 Δε difference between the upper (ε max ) and lower (ε min ) level of deformations for the range of elastic deformations (measurement with an extensometer) Uncertainty equation Due to the fact that input parameters are uncorrelated, the standard uncertainty connected with the determined Young s (elastic) modulus is defined by the following formula: ( ) = ( u E c i u i ), (1) where: c i sensitivity coefficients, i.e. the partial derivatives of the measurement function for the i th function component, u i standard uncertainties for individual components. In this case c ΔF = u 4 ε d π 4 ε ; c d = F ε F 4 F and cδε =, thus: π d ε ε 4 F ε ( E) = u ( F ) + u ( d ) + u ( ε) o Determination of component standard uncertainties The uncertainty for measurement of the specimen diameter ( ) d o , equations (15 17). The uncertainty for measurement of the force ΔF: o o o. () u - par. u ( F ) = u ( F ) + u ( F ), () max where: the uncertainty values for measurement of the upper (F max ) and lower (F min ) level of the force (see formula (18)): min,1 Fmax,1 F u( Fmax ) = and min u( Fmin ) =. The uncertainties for measurement of the upper (ε max ) and lower (ε min ) level of deformation as determined on the basis of the accuracy class of the longitudinal extensometer K t, with use of the B-type method (the rectangular distribution):

14 14 Sylwester KŁYSZ, Janusz LISIECKI u ( ε) = u ( ε ) + u ( ε ), (4) max K where: ε εmax K u ( εmax ) = and ε εmin u ( εmin ) = Uncertainty budget The informations for further analysis of uncertainty are presented in the Table 5. Table 5 Uncertainty budget for calculation of the composed uncertainty for the Young s (elastic) modulus Parameter symbol X i Parameter estimation x i Standard uncertainty Sensitivity coefficient c i ΔF Eq. () 4 min Contribution into the composed standard uncertainty c i u c (ΔF) d Eq.(15) 8F, or (6) c i u( d ) 4 F Δε Eq. (4) ε c i u(δε) E 4 F ε Eq. () Determination of the expanded uncertainty The relative expanded uncertainty with the expansion coefficient k α = at the confidence level of 1 α =,95 is calculated with the following formula: u( E) U ( E) = 1%. (5) E

15 Selected problems of measurement uncertainty - part 15. Recapitulation On the basic of presented above methods for determination of measurement error and uncertainties the analysis of micrometer verification results and test proficiency results (in with accredited testing laboratory of Air Force Institute of Technology have been participated) were done. The research covered verification of the analogue micrometer with its measurement range 5 mm, serial No , manufactured by MITUTOYO company. For reference the set of gauge plates was used, No 7149, with its calibration certificate No M /4. The verification procedure was carried out at the ambient temperature of ±, o C, in accordance with the instruction from the metrological surveillance IW-1-11-L5, for the full measurement range of the micrometer when the gauge plates with rated sizes of W n = 1; 1,5; 1,5; ; 5; 8; 1; 15; ; 5 mm were subsequently used. Verification results are collected in Table 6. Table 6 Results of the micrometer verification, [mm] Length of the standardized plate W n readout I L 1 readout II L Readout / established value readout III L maximum deviation from the W dimension d measurement uncertainty U(E x ) Indication error d+u(e x ) d-u(e x ) 1 1, 1, 1,1,154,9,,1 1,5 1,51 1,5 1,51,91,9 1,9, 1,5 1,5 1,5 1,5,8,9,9 -,9,,1,,16,9,,1 5 5, 5,1 5,1,148,9,,1 8 8, 8,1 8,1,16,9 1,9,1 1 1,1 1,1 1,1,948,9 1,9, ,1 15,1 15,1,9,9 1,8,,,1,1,1911,9,8 1, 5 5, 5, 5,1,1798,9,7,9 Verification result was accepted as passing one, because the indication error equal,8 μm was less than the permissible limit error for micrometric instruments E g = ±4 μm (PN-8/M-5).

16 16 Sylwester KŁYSZ, Janusz LISIECKI In 5 the Laboratory for Material Strength Testing (LMST) participated in the survey of competence the proficiency test (PT) (scope of strength tests for round steel bars under room temperature). The survey was organized by the Institut für Eignungsprüfung (Germany). The survey brought together research laboratories from 9 countries and 78 of the participants had the accreditation in accordance with the standard EN ISO/IEC 175. The participants were assigned to provide uncertainties for measurement of each parameter. The examination results along with uncertainties values, calculated in accordance with the foregoing formulas, are presented in Table 7. Based on comparison of data covered by the report [Wyrażanie niepewności 1999] submitted to the Institut für Eignungsprüfung with the information presented in Table 7 the conclusions about general matching of the results can be made. The LMST laboratory has fulfilled the requirements related to the competence survey and was granted with the certificate. Table 7 Measurement results obtained by the LMST laboratory during the competence survey in 5 No of specimens R p, [MPa] U(R po, ) [%] R m [MPa] U(R m ) [%] A [%] U(A) [%] E [MPa] U(E) [%] ±,6 796 ±1,4 1,9 ±1, ±,1 71 ±,1 795 ±1,1 14, ±1, ±,1 718 ±, 79 ±1,8 1,5 ±1, ±, ±,1 784 ±1,18 16,1 ±1, ±, ±,1 781 ±1,4 14,8 ±1, ±, ±,1 79 ±1, 14,1 ±1, ±,11 Average values 715 ±,4 79 ±1, 14,6 ±1, ±,1 (LMST) Values (participants) 71 ±, 79 ±1, 16, ±, ±4, Values (survey organizers) 691 ±, 786 ±1,6 15,7 ±1, n/a n/a

17 Selected problems of measurement uncertainty - part 17 References: AGARD Conference Proceedings No.186, Impact damage Tolerance of structure. 41 st Meeting of the Structures and Materials Panel, Ankara. Analiza błędów i niepewności pomiarów, 7. Arendarski J.. Niepewność pomiarów. Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa. Atluri S.N Computational methods in the mechanics of fracture. Computational methods in mechanics, Vol., North Holland. CWA 1561-:5. Measurement uncertainties in mechanical tests on metallic materials-part : The evaluation of uncertainties in tensile testing. Dokument EA-4/ Wyrażanie niepewności pomiaru przy wzorcowaniu. Międzynarodowy słownik podstawowych i ogólnych terminów metrologii GUM. -Piotrowski J., Kostyrko K.. Wzorcowanie aparatury pomiarowej. Wydawnictwo Naukowe PWN, Warszawa. PN-EN ISO 11:4. Systemy zarządzania pomiarami. Wymagania dotyczące procesów pomiarowych i wyposażenia pomiarowego. PN-EN ISO 9:1 Systemy zarządzania jakością. Podstawy i terminologia. PN-EN ISO/IEC 175:5 Ogólne wymagania dotyczące kompetencji laboratoriów badawczych i wzorcujących. Proficiency Test-Tensile Test Steel-Round bar at room temperature (TTSRR 5) - Final Raport. 6. Institut für Eignungsprüfung. Rozporządzenie Ministra Gospodarki, Pracy i Polityki Społecznej w sprawie w sprawie wymagań metrologicznych, którym powinny odpowiadać maszyny wytrzymałościowe do prób statycznych. 4. Wyrażanie niepewności pomiaru Przewodnik GUM.