Uncertainty of Measurement for Design Engineers

Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 43 (2016) 309 – 314 14th CIRP Conference on Computer Aided Tolerancing (CAT) Uncertainty of measurement for design engineers Władysław Jakubieca*, Wojciech Płowuchaa, Paweł Rosnera a University of Bielsko-Biała, Willowa 2, PL43-309 Bielsko-Biała * Corresponding author. Tel.: +48-33-8279-321; fax: +48-33-8279-300. E-mail address: [email protected] Abstract Paper presents simple software for evaluation of uncertainty for measurements carried out on CMMs. The input data are coefficients from MPEE formulae for CMM as well as metrological characteristic (type of dimension or geometrical deviation). The evaluated uncertainty enables the designer preliminary judgement if the available measuring equipment enables proper verification of the products (if the assumed tolerances are not too narrow comparing to possibilities in the range of product verification). Attention is brought to the fact that on up-to-date technical drawings, to achieve unambiguous interpretation, more often tolerances of position and tolerance of profile any surface are used. The operation of the EMU-CMMUncertaintyTM software is discussed. On a few examples of coordinate measurement models the software principle of work is explained and uncertainty budgets presented. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing. Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing Keywords: coordinate measuring machines, uncertainty evaluation, geometrical product specifications 1. Introduction Nomenclature Most of geometrical measurements in machine building industry is carried out by means of coordinate measuring systems, in particular coordinate measuring machines (CMM). The uncertainty of coordinate measurements depends on many factors, as in the case for any other measurements. The most important factors are accuracy of measuring machines, environmental conditions, measuring strategy (differing by designer of the measuring process and, to a certain extent, geometrical shape of the workpiece), as well as the workpiece itself (e.g. its stiffness, form deviations, surface roughness, CTE) [1,2]. It is often not duly realized that measurement uncertainty for different geometrical characteristics can differ significantly even if carried out on the same measuring machine. According to standards ISO 8015 [3] and ISO 14405-1 [4], it is important that designer defines the accuracy requirements in an unambiguous way because otherwise it can generate additional source of uncertainty. This means especially that the designer should avoid using toleranced dimensions and instead apply tolerance of position and/or profile any surface. Designer, when specifying the requirements for manufacturing accuracy (tolerances), takes into account the CMM coordinate measuring machine CTE coefficient of thermal expansion u standard uncertainty U expanded uncertainty MPEE maximum permissible error of indication x, y, z coordinates A, B, C, D, S characteristic points R radius l geometrical deviation ߲݈Ȁ߲‫ݎ‬ sensitivity coefficient function of the object. Moreover, he should know the measurement capability of the manufacturer i.e. achievable measurement uncertainty. He must also remember that it is commonly accepted that the ratio of the measurement uncertainty and the tolerance shall not be greater than 1:5. Two methods of uncertainty evaluation of coordinate measurement are known and standardized to certain extent. First is the experimental method based on the multiple measurement of the calibrated workpiece having similar form to the manufactured workpiece. The second is the simulation 2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 14th CIRP Conference on Computer Aided Tolerancing doi:10.1016/j.procir.2016.02.027 310 Władysław Jakubiec et al. / Procedia CIRP 43 (2016) 309 – 314 technique which in the online version enables evaluating the uncertainty after performing the measurement. However, the designer needs to know estimations of the measurement uncertainty already at the design stage, when setting tolerance values for particular characteristics. Authors recommend that designers use the software for offline evaluation of measurement uncertainty of coordinate measurements presented in this publication. Use of the software requires basic knowledge in coordinate measuring technique. In particular the knowledge about how to design the measuring process as well as about the CMM types and working principle. 2. Geometrical characteristics of machine parts The geometrical characteristics of machine parts are dimensions and geometrical deviations, i.e. deviations of form, orientation, location and run-out. Up-to-date approach to geometrical product specification (GPS) assumes that toleranced dimensions should only be applied to so called features of size [4,5]. In other cases one should use geometrical tolerances [6,7,8,9], where a special role has the tolerance of position and the tolerance of profile any surface. Tolerances of dimensions for elements which are not features of size shall only be applied in case of low requirements (e.g. in case of using general tolerances [10]), i.e. in case where the lack of unambiguous specification of geometry does not interfere with the proper acceptance or rejection of the product [11]. 3. Coordinate measuring systems Now-a-days the term coordinate measuring systems is applied to [12]: classical CMM, measuring arms, optical and multisensor machines, laser and structured light scanners, CTs, laser-trackers, and others in which the measurement information are coordinates of points in certain coordinate system. This information is next processed by special software installed on a computer being integral part of the measurement system, in order to evaluate necessary geometrical characteristics. The information processing is usually two-stage. On the first stage, from sets of points the definition parameters of geometrical features (planes, circles, cylinders etc.) are calculated. The parameters are vectors determining location and orientation of the features and one or two scalars describing linear or angular dimensions of the features. On this stage various association criteria are used (e.g. Gaussian, minimax). Some parameters are directly geometrical characteristics to be measured (e.g. diameter of a cylinder o circle, cone angle). On the second stage, using geometrical links (constructions) further characteristics are evaluated which describe the relations between two or more geometrical features (e.g. position deviation of hole’s axis in regards to datum system). The software of coordinate measuring machines includes functions and options which enable evaluation of necessary characteristics. 4. Accuracy of coordinate measuring systems To describe the accuracy of any measuring device one or more metrological characteristics can be defined [13]. For example, for well-known device which is a micrometre, following characteristics are defined [14]: x full surface contact error (limited by MPEJ), x repeatability (limited by MPER), x partial surface contact error (limited by MPEE). Basic metrological characteristics for coordinate measuring machines are: x error of indication for size measurements E (limited by MPEE) [15,16], x probing error P (limited by MPEP) [17]. In 2009, the standard concerning performance tests for classical CMMs [16] introduced new characteristics: x length measurement error with zero ram axis stylus tip offset, E0 (limited by E0,MPE), x length measurement error with ram axis stylus tip offset of L, E150 (limited by EL,MPE), x repeatability range of the length measurement error, R0 (limited by R0,MPL). Other parts of the standard ISO 10360 dealing with acceptance and reverification tests for coordinate measuring systems introduce many other metrological characteristics, specific for different types of these systems. E.g. for CMMs with optical distance sensors [18] among others one can find: x PS:X:Opt – probing size error (limited by PS:X:Opt,MPE), x EBi:X:Opt – bi-directional length measurement error (limited by EBi:X:Opt,MPE). A few new parts of the standard are under development. 5. Uncertainty of coordinate measurement In the literature concerned with uncertainty of geometrical measurements [1] many sources of errors (uncertainty components) are mentioned. Usually as the most important following influence groups are mentioned: measuring equipment, environmental conditions, operator and the measured workpiece. As mentioned in Introduction, the uncertainty of coordinate measurements depends mainly on accuracy of the measuring machine, environmental conditions, measuring strategy and characteristics of the workpiece (shape, dimensions, CTE). The measurement strategy applied by the CMM operator is the element which is highly determined by the designer considering the up-to-date geometrical product specification. The difficulty in the evaluation of uncertainty of coordinate measurements comes, in great extent, from the fact that in general case the uncertainty of measurement of different geometrical characteristics even measured on the same measuring machine is different and for each characteristic must be evaluated separately. Two method of uncertainty evaluation of coordinate measurement are known. First method is experimental consisting on multiple measurements of calibrated workpiece od the shape and dimensions similar to the manufactured workpieces [19,20]. Second is simulation technique [21,22]. The on-line version of the simulation software is part of the 311 Władysław Jakubiec et al. / Procedia CIRP 43 (2016) 309 – 314 CMM control and evaluation software and calculates the uncertainty for particular characteristics after the measurement is finished. None of these methods is suitable for use by the designer at the stage of tolerancing of the designed parts. 6. Off-line software for evaluation of uncertainty of coordinate measurements – version of the design engineers The software EMU-CMMUncertaintyTM, developed by the authors of this publication, in basic version is used for evaluation of coordinate measurements carried out on particular measuring machine (it considers dimensional parameters of the CMM and its geometrical errors as well as probing errors estimated by experiment) working in specific environmental conditions (certain range of temperature) and with applied measuring strategy. As measuring strategy is understood here the position and orientation of the workpiece in the measuring volume, styli used for probing and the probing strategy [23,24,25]. The only information on the measuring machine required by the version of the software offered to the designers is the maximum permissible error MPEE of the machine. The measurement models use minimal number of characteristic points of the workpiece and the measured characteristic is expressed as function of differences of coordinates of these points. The operation of the software requires choosing a characteristic (Fig. 1 and 2) and specify the information on the dimensions of the workpiece (Fig. 3). Fig. 2. Dialog window for choosing specific case of geometrical tolerance Information on the dimensions of the measured workpiece is specified in the dialog window including example drawing of the measured characteristic in the workpiece coordinate system (Fig. 3). Fig. 3. Example dialog window for specifying the dimensions of the workpiece. 7. Example measurement models Fig. 1. Main window of the software – the toolbar for choosing the characteristic groups is marked with red frame. The characteristics are grouped according to the classification of the dimensions (first 2 symbols from left side in red frame) and geometrical deviations (remaining symbols). Each symbol is assigned to combo box menu which enables choosing particular case (Fig. 2). The models used in this software are fully consistent with the applicable rules [26]. Coordinate measurement is treated as indirect measurement. The models consist of the formula expressing the characteristic as a function of differences of coordinates of characteristic points of the workpiece. Standard uncertainties of measurements of particular differences of coordinates are evaluated by type B method assuming that biggest possible error is equal to maximum permissible error of indication of the CMM (a = MPEE). Assuming normal distribution (k = 0,5) this gives ‫ ݑ‬ൌ Ͳǡͷ ή ‫ܧܲܯ‬ா (1) 312 Władysław Jakubiec et al. / Procedia CIRP 43 (2016) 309 – 314 Assuming that particular differences of coordinates are not correlated the uncertainty of measurement can be evaluated by means of the formula [26]: ‫ݑ‬௖ ሺ݈ሻ ൌ డ௟ ටσே ௜ୀଵ ቀడ௥ ೔ ଶ ቁ ‫ݑ‬ଶ ሺ‫ݎ‬௜ ሻ (2) where: u(ri) – standard uncertainties of particular measurands, ߲݈Ȁ߲‫ – ݎ‬sensitivity coefficient. Three models will be presented: first applies to size (diameter), second to orientation deviation (perpendicularity) and third to location tolerance (coaxiality). In all examples the uncertainty budget is composed for measurement on CMM with MPEE = (1,5 + L/300) Pm, L - in mm. 7.1. Model of circle diameter measurement The model of measurement of circle diameter is presented on Fig. 4. This model applies to any probing strategy, for which probing points are at least approximately equally distributed on the circle circumference. which, when converted using the abbreviated notation (e.g. y31 = y3 – y1), gives following formula for diameter D ‫ ܦ‬ൌ ʹ ή ඥሺͲǡͷ ή ‫ݔ‬ଷଵ ൅ ‫ ݐ‬ή ‫ݕ‬ଷଵ ሻଶ ൅ ሺͲǡͷ ή ‫ݕ‬ଷଵ െ ‫ ݐ‬ή ‫ݔ‬ଷଵ ሻଶ The diameter D is function of 8 differences of coordinates x31, y31, x21, y21, x42, y42, x43, y43. The uncertainty budget is presented in Table 1. Analysis of the budget shows that there are four rows with the sensitivity coefficients are not zero and are equal to 0,35. Finally, standard uncertainty is ca. uc = 0,6 μm. The expanded uncertainty of radius measurement is than 1,2 μm (for the diameter Uc = 2,4 μm). for comparison, the value of maximum permissible error of length measurement for the investigated CMM for measuring the length equal to the circle radius (50 mm) is MPEE = 1,7 μm, and for the length equal to diameter (100 mm) MPEE = 1,8 μm). Table 1. The uncertainty budget for circle diameter measurement Differences of coordinates, mm MPEE, Pm x31 = 100 y31 = 0 x42 = 0 y42 = 100 x21 = 50 y21 = 50 x43 = 50 y43 = 50 1,83 1,5 1,5 1,83 1,67 1,67 1,67 1,67 Standard Sensitivity uncertainty ui, coefficient Pm wl wri 0,917 0,75 0,75 0,917 0,833 0,833 0,833 0,833 wl u i , Pm wri 0,7 0 0,7 0 0 0,7 0 0,7 uc = To calculate diameter on the basis of coordinates of 4 points (roughly uniformly distributed on a circle) the geometrical property was used, that centre point S(x0, y0) is an intersection point of bisectors of the chords AC and BD, and the radius R is arithmetic mean of the distance of points A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) from the centre S. Equations of the bisectors are: ଶ ൝௬భା௬ య ଶ ൅ ‫ ݐ‬ή ሺ‫ݕ‬ଷ െ ‫ݕ‬ଵ ሻ ൌ െ ‫ ݐ‬ή ሺ‫ݔ‬ଷ െ ‫ݔ‬ଵ ሻ ൌ ௫మ ା௫ర ଶ ௬మ ା௬ర ଶ ൅ ‫ ݏ‬ή ሺ‫ݕ‬ସ െ ‫ݕ‬ଶ ሻ െ ‫ ݏ‬ή ሺ‫ݔ‬ସ െ ‫ݔ‬ଶ ሻ 0,64 0 0,53 0 0 0,58 0 0,58 1,37 7.2. Model of perpendicularity of axes measurement Fig. 4. The model of measurement of circle diameter using 4 points. ௫భ ା௫య (8) The model of measurement of perpendicularity of axes is presented on Fig. 5. (3) Coordinates of the centre point ‫ݔ‬଴ ൌ ‫ݕ‬଴ ൌ ௫భ ା௫య ௬భ ା௬య where: ଵ ‫ݐ‬ൌ ή ଶ ଶ ଶ ൅ ‫ ݐ‬ή ሺ‫ݕ‬ଷ െ ‫ݕ‬ଵ ሻ െ ‫ ݐ‬ή ሺ‫ݔ‬ଷ െ ‫ݔ‬ଵ ሻ ௫రమ ήሺ௫మభ ା௫రయ ሻା௬రమ ήሺ௬మభ ା௬రయ ሻ ௫రమ ή௬యభ ି௫యభ ή௬రమ (4) (5) To calculate the deviation of perpendicularity of axes on the basis of coordinates of 4 so called characteristic points the following formula was used (see [24]): (6) The radius of the circle can be calculated as the distance of point A from the calculated centre point S(x0,y0) equals ܴ ൌ ඥሺ‫ݔ‬଴ െ ‫ݔ‬ଵ ሻଶ ൅ ሺ‫ݕ‬଴ െ ‫ݕ‬ଵ ሻଶ Fig. 5. The model of measurement of axes perpendicularity using 4 points; (a) specification; (b) model. (7) ݈ൌ ௫యర ௫మభ ା௬యర ௬మభ ା௭యర ௭మభ మ ା௬ మ ା௭ మ ට௫మభ మభ మభ (9) Deviation of perpendicularity of axes l is function of 6 differences of coordinates x34, y34, z34, x21, y21, z21. Władysław Jakubiec et al. / Procedia CIRP 43 (2016) 309 – 314 The uncertainty budget is presented in Table 2. In this case the budget includes two sensitivity coefficients equal 1 (remaining are equal 0). The expanded uncertainty Uc = 2,1 μm. In this case the measured characteristic (perpendicularity deviation) has very small value for which MPEE = 1,5 μm. 313 version of the EMU-CMMUncertaintyTM software can be used for this purpose. The presented examples of uncertainty budgets for different characteristics confirm that uncertainty of different characteristics measured on the same CMM can be significantly different. Table 2. The uncertainty budget for axes perpendicularity measurement Differences of coordinates, mm MPEE, Pm x21 = 100 y21 = 0 z21 = 0 x34 = 0 y34 = 0 z34 = 100 1,83 1,5 1,5 1,5 1,5 1,83 Standard Sensitivity uncertainty ui, coefficient Pm wl wri 0,917 0,75 0,75 0,75 0,75 0,917 wl u i , Pm wri 0 0 1 1 0 0 This publication is prepared within the EU founded project entitled “Geometrical Product Specification and Verification as a toolbox to meet up-to-date technical requirements (GPSVToolbox)” no. 2015-1-PL01-KA202-016875 carried out within the scope of Erasmus+ programme Key Action 2 – Strategic Partnership [27]. 0 0 0,75 0,75 0 0 1,06 uc = References 7.3. Model of coaxiality measurement The model of measurement of coaxiality is presented on Fig. 6. Fig. 6. The model of measurement of coaxiality using 3 points; (a) specification; (b) model. To calculate the coaxiality deviation on the basis of coordinates of 3 points following formula was used (see [25]): ሺ௬భయ ௭భమ ି௭భయ ௬భమ ሻమ ାሺ௭భయ ௫భమ ି௫భయ ௭భమ ሻమ ାሺ௫భయ ௬భమ ି௬భయ ௫భమ ሻమ ݈ൌට మ ା௬ మ ା௭ మ ௫భమ భమ భమ (10) Deviation of coaxiality l is function of 6 differences of coordinates x12, y12, z12, x13, y13, z13. The uncertainty budget is presented in Table 3. In this case the sensitivity coefficient equal 5 needs special attention. It causes the expanded uncertainty rise to Uc = 10 μm. Table 3. The uncertainty budget for coaxiality measurement. Differences of coordinates, mm MPEE, Pm x12 = 20 y12 = 0 z12 = 0 x13 = 100 y13 = 0 z13 → 0 1,57 1,5 1,5 1,83 1,5 1,5 Standard Sensitivity uncertainty ui, coefficient Pm wl wri 0,783 0,75 0,75 0,917 0,75 0,75 Acknowledgements 0 0 5 0 0 1 uc = wl u i , Pm wri 0 0 3,75 0 0 0,75 5,099 8. Conclusions The knowledge on estimate of measurement uncertainty at the design stage can help the designer in selecting dimensional and geometrical tolerances. 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