Uncertainty of coordinate measurement of geometrical deviations

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Procedia CIRP 00 (2017) 000–000 Procedia CIRP 75 (2018) 361–366 www.elsevier.com/locate/procedia

15th 15th CIRP CIRP Conference Conference on on Computer Computer Aided Aided Tolerancing Tolerancing –– CIRP CIRP CAT CAT 2018 2018

CIRP Design Conference, May 2018, France deviations Uncertainty of measurement of geometrical Uncertainty 28th of coordinate coordinate measurement of Nantes, geometrical deviations a Wojciech Płowucha A new methodology to analyze the functional Wojciech Płowuchaa** and physical architecture of of 2, Bielsko-Biała, existing products forUniversity an assembly oriented family identification University of Bielsko-Biała Bielsko-Biała ,, Willowa Willowa 2, PL PL 43-300 43-300product Bielsko-Biała, Poland Poland * Corresponding author. E-mail address: [email protected] a a

* Corresponding author. E-mail address: [email protected]

Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France Abstract Abstract

University of simplified for measurement * At Corresponding Tel.: +33 3 87 37 54 30; models E-mail address: [email protected] At University author. of Bielsko-Biala, Bielsko-Biala, simplified models for the the coordinate coordinate measurement of of different different geometrical geometrical characteristics characteristics (size, (size, distance, distance, angle angle

and and geometrical geometrical deviations) deviations) have have been been developed. developed. The The most most important important assumption assumption for for measurement measurement model model is is taking taking into into account account the the obvious obvious fact that coordinate measurement is indirect measurement. To develop the models of different geometrical characteristics the mathematical fact that coordinate measurement is indirect measurement. To develop the models of different geometrical characteristics the mathematical minimum minimum number number of of required required points points was was assumed. assumed. Models Models use use information information on on CMM CMM accuracy accuracy only only in in the the form form of of formula formula for for maximum maximum Abstract permissible error of length measurement E L,MPE verified by actual acceptance or reverification test results. The presented uncertainty budgets permissible error of length measurement EL,MPE verified by actual acceptance or reverification test results. The presented uncertainty budgets enable enable analysis analysis of of influence influence of of measurement measurement uncertainty uncertainty of of particular particular coordinates’ coordinates’ differences differences on on the the total total measurement measurement uncertainty. uncertainty. In© business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of ©today’s 2018 The Authors. Published by Elsevier B.V. 2018 The Authors. Published by Elsevier B.V. © 2018 Authors. Published byofElsevier B.V. agile andThe reconfigurable production systems emerged to cope of with products and product families. To design and optimize production Peer-review under responsibility responsibility the Scientific Scientific Committee of the various 15th CIRP CIRP Conference on Aided Tolerancing CIRP CAT CAT 2018. Peer-review under of Committee the 15th Conference on Computer Computer Tolerancing --CAT CIRP 2018. Peer-review under thethe Scientific Committee the 15th CIRP Conference AidedAided Tolerancing 2018. aim systems as well as responsibility to choose theofoptimal product matches,ofproduct analysis methods on areComputer needed. Indeed, most of the- CIRP known methods to Keywords: coordinate measurement; uncertainty evaluation; type B evaluation; task-specific uncertainty analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and Keywords: coordinate measurement; uncertainty evaluation; type B evaluation; task-specific uncertainty nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable 1. assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and 1. Introduction Introduction a functional analysis is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the Nomenclature similarity betweenon families byevaluation providing design to both, Nomenclature production system planners and product designers. An illustrative Research uncertainty of coordinate Research onproduct uncertainty evaluation of support coordinate example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of measurements last for many years [1] and recently become measurements last for many years [1] and recently become CMM coordinate CMMof––the coordinate measuring machine thyssenkrupp Presta France not is then carried out to give a first industrial evaluation proposed measuring approach. machine particularly particularly important important not only only for for metrologists metrologists [2, [2, 3]. 3]. The The u – standard uncertainty; index denotes ©research 2017 Thecarried Authors. Published by Elsevier B.V. u – standard uncertainty; index denotes deviation deviation e.g. e.g. ST ST –– out at University of Bielsko-Biała in this field research carried out at University of Bielsko-Biała in this field straightness Peer-review under in responsibility of the scientific committee of the 28th CIRP Design Conference 2018. straightness was presented some publications [4, 5]. The publications

was presented in some publications [4, 5]. The publications E MPE – maximum permissible error of length measurement EL, L, MPE – maximum permissible error of length measurement bring a lot of attention to possibility of simplifying the the x, y, x, y, zz –– coordinates coordinates procedure procedure of of uncertainty uncertainty evaluation evaluation in in the the way way it’s it’s possible possible to to A, B ... –– characteristic A, B ... characteristic points points apply apply also also in in the the industrial industrial conditions. conditions. Moreover, Moreover, special special AB, AS ... –– vector of coordinates’ differences AB, AS ... vector of coordinates’ differences for for points points A A and and attention attention is is paid paid to to the the fact fact that that the the models models are are unambiguously unambiguously B, A and S etc. B, A and S etc. with 1.compatible Introduction product– range and coordinates characteristics manufactured and/or compatible with the the principle principle of of the the coordinate coordinate measuring measuring ofabthe (x, y, of vector 3 – respective ab11,, ab ab22,, ab ab respective coordinates (x, the y, z) z)main of the the vector in 3 this technique. assembled in system. In this context, challenge technique. AB, differences for points A B AB, i.e. i.e. coordinates’ coordinates’ differences points A and and B single Comparing publications, Due to theto fastprevious development in the theexamples domainin of modelling analysisofisgeometrical now not for only to cope with Comparing to the the previous publications, the examples in this this ll –– general generaland indication deviation as function of indication of geometrical deviation as function of paper uncertainty which communication and present an ongoing trend budgets of digitization and products, a limited product range or existing product families, paper additionally additionally present uncertainty budgets which enable enable respective coordinates’ differences of points x i respective coordinates’ differences of points x i detailed of components influencing the digitalization, manufacturing enterprises are facing important also to be able to analyze and to compare products to define detailed analysis analysis of the the main main components influencing the but sensitivity coefficients 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙/𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝑖𝑖𝑖𝑖 – 𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙/𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 coefficients 𝑖𝑖𝑖𝑖 – sensitivity calculated uncertainty. The examples can be easily adopted for challenges in today’s market environments: a continuing new product families. It can be observed input that classical existing calculated uncertainty. The examples can be easily adopted for uncertainties of particular quantities uuxixi –– standard standard uncertainties of in particular input quantities calculations for different data. tendency towards reduction of product development times and product families are regrouped function of clients or features. calculations for different data. The models of the shortened product lifecycles. In addition, there is expressing an increasing assembly oriented product families are hardly to find. The developed developed models consist consist of the the formula formula expressing the However, Thus uncertainty of measurement can be evaluated by Thus uncertainty of level, measurement can be mainly evaluated by geometric deviation l as a function of differences of coordinates demand of customization, being at the same time in a global On the product family products differ in two geometric deviation l as a function of differences of coordinates means of the formula for uncertainty of indirect measurement means of the formula for uncertainty of indirect measurement of characteristic characteristic points competition with points competitors allworkpiece. over the world. This trend, main characteristics: (i) the number of components and (ii) the of xxii of of the the workpiece. [6]. Due to of of characteristic [6]. of Due to use use of of differences differences of coordinates coordinates ofelectronical). characteristic which is inducing the development from macro to micro type components (e.g. mechanical, electrical, points one can of between input points one methodologies can assume assume lack lack of correlation correlation between input )) 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = (1) markets, in diminished lot sizes due to augmenting Classical considering mainly single products = 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖results (1) . quantities x i quantities x i. product varieties (high-volume to low-volume production) [1]. or solitary, already existing product families analyze the To cope with this augmenting variety as well as to be able to product structure on a physical level (components level) which identify possible optimization potentials in the existing causes difficulties regarding an efficient definition and 2212-8271 2212-8271 © © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. production system, it is important to have a precise knowledge comparison of different product-- CIRP families. Addressing this Peer-review under responsibility of the Scientific Committee of the 15th CIRP Conference on Aided CAT Peer-review under responsibility of the Scientific Committee of the 15th CIRP Conference on Computer Computer Aided Tolerancing Tolerancing CIRP CAT 2018. 2018. bring a Assembly; lot of attention to Family possibility of simplifying Keywords: Design method; identification

2212-8271©©2017 2018The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-review under responsibility of scientific the Scientific Committee of the 15th CIRPConference Conference on Computer Aided Tolerancing - CIRP CAT 2018. Peer-review under responsibility of the committee of the 28th CIRP Design 2018. 10.1016/j.procir.2018.04.071

Wojciech Płowucha / Procedia CIRP 75 (2018) 361–366 Wojciech Płowucha / Procedia CIRP 00 (2018) 000–000

362 2

𝑁𝑁𝑁𝑁 𝑢𝑢𝑢𝑢(𝑙𝑙𝑙𝑙) = �∑𝑖𝑖𝑖𝑖=1 �

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

2

2 � 𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

(2)

Standard uncertainties uxi 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 length measurement of the CMM and that normal distribution applies with k = 3.

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 = 𝐸𝐸𝐸𝐸𝐿𝐿𝐿𝐿,𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 /3

(3)

In all examples the uncertainty budget is composed for measurement on CMM with EL,MPE = 2 + 4L/1000) µm, L - in mm. 2. Reasons for assuming k = 3 Normal distribution with k = 3 corresponds to the probability P= 0,9973, that all indication errors of CMM are less than EL, MPE. Here we assume that considered CMM conforms the requirements. This means that evaluated indication errors during verification procedure are smaller than EL, MPE for all 105 measurements of the calibrated lengths. Thus the probability that indication errors are less or equal to the permissible value is greater than (1 – 1/105). If necessary different value of k may be assumed or it can be calculated from the verification results. Fig. 1 depicts example (quite typical) of CMM verification results for CMM with EL, MPE = 2 + 4L/1000 µm.

Fig. 2 presents histogram of normalized indication errors Est for the range (-1, 1) according to formula. 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝐸𝐸𝐸𝐸 ⁄𝐸𝐸𝐸𝐸𝐿𝐿𝐿𝐿,𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

(4)

The standard deviation (in regard to 0) calculated for the normalized errors s = 0,268 which gives the coefficient k = 1/s = 3,74. 3. Measurement uncertainty of straightness Fig. 3 depicts an example of tolerance specification and modelling of straightness deviation measurement. The model refers to a simplified method of classical measurement of straightness. The point S is to lay near the line connecting points A and B and usually is close to the middle of the line segment AB, where the sag arrow is the largest.

Fig. 3. Measurement model of deviation of axis straightness

The tolerance zone is the cylinder of the diameter equal to the tolerance value [7]. The straightness deviation l in this case is modelled as the distance between point S and line AB. 𝑙𝑙𝑙𝑙 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ×

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 |𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴|

(5)

Two vectors occur in the formula and each consists of differences of three coordinates: AB(ab1, ab2, ab3) and AS(as1, as2, as3), therefore l is function of 6 variables denoted generally as xi, i = 1..6.

Fig. 1. Example of actual CMM verification results (errors of indication for different measured calibrated lengths in mm)

𝑙𝑙𝑙𝑙 =

�𝑒𝑒𝑒𝑒 2 + 𝑓𝑓𝑓𝑓 2 + 𝑔𝑔𝑔𝑔2 𝑚𝑚𝑚𝑚

where: 𝑒𝑒𝑒𝑒 = 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠2 ∙ 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏3 − 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠3 ∙ 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏2 , 𝑓𝑓𝑓𝑓 = 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠3 ∙ 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏1 − 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠1 ∙ 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏3 , 𝑔𝑔𝑔𝑔 = 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠3 ∙ 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏1 − 𝑎𝑎𝑎𝑎𝑠𝑠𝑠𝑠1 ∙ 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏3 , 𝑚𝑚𝑚𝑚 = 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏12 + 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏22 + 𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏32 . The uncertainty of measurement of straightness equals the uncertainty of measurement of distance between point S and straight-line AB and is function of coordinate differences for pairs of points AS and AB [8]. 𝑢𝑢𝑢𝑢𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑢𝑢𝑢𝑢(𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

Fig. 2. Histogram of normalized error values Est for EL,MPE = 2 + 4L

(6)

(7)

The uncertainty of measurement of distance between a point and a straight-line is analysed depending on the location of the point S relative to the points A and B as well as the distance AB. In presented example the characteristic points are centre points and not directly probed points. Such simplification leads to insignificant overestimation of the uncertainty. In the example, the element is oriented parallel to the x axis and its length is 100 mm – the analyses is performed for the



Wojciech Płowucha / Procedia CIRP 75 (2018) 361–366 Wojciech Płowucha / Procedia CIRP 00 (2018) 000–000

following coordinates of characteristic points of the tolerated element: A(100, 100, 100) and B(200, 100, 100). For the position of the point S symmetrically with respect to points A and B, the standard uncertainty is the highest and amounts to 0.75 μm. The uncertainty budget for this position is presented in the table 1. Table 1. Uncertainty budget for measurement of point – straight line distance (Fig. 3) for the case when point S is lying over the line AB – S(150, 100, 100.01) Component

xi, mm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

as1

50

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

0.73

as2

0

0

0.67

0

as3

0.01

1

0.67

0.67

ab1

100

0

0.80

0

ab2

0

0

0.67

0

ab3

0

-0.50

0.67

0.33

u=

0.75

0

It should be noted that one of the uncertainty components occurs with a weight of 1 and one with a weight of 0.5, and both concern a small measured value. The components related to large measured values and related to the length of the measured object occur in the budget with a weight equal zero, which means that the length of the measured object does not affect the measurement uncertainty of straightness. 4. Measurement uncertainty of flatness Fig. 4 depicts an example of tolerance specification and modelling of flatness deviation measurement. The model refers to a simplified method of classical measurement of flatness. The point S is to lay near the gravity centre of points A, B and C.

363 3

The result do not differ significantly over the plane - the largest value of the measurement uncertainty is 0.74 µm. Table 2. Uncertainty budget of the flatness measurement for S(200,150,0) Component

xi, mm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

as1

150

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

0.87

as2

100

0

0.80

0

as3

0.01

1

0.67

0.67

ab1

300

0

1.07

0

ab2

0

0

0.67

0

ab3

0

-0.33

0.67

0.22

ac1

150

0

0.87

0

ac2

300

0

1.07

0

ac3

0

-0.33

0.67

0.22

u=

0.74

0

It can be seen that one of the uncertainty components occurs with a weight of 1 and two with a weight of 0.33, and all concern a small measured value. The components related to large measured values and related to the size of the measured object occur in the budget with a weight equal 0. 5. Measurement uncertainty of parallelism The parallelism tolerance may be applied to straight line or plane. 5.1. Parallelism of a line related to a datum plane Fig. 5 depicts an example of tolerance specification and measurement modelling of parallelism of axis related to a datum plane. The point S is to lay near the end of toleranced axis.

Fig. 4 Measurement model of deviation of flatness

The tolerance zone is limited by pair of parallel planes apart by the tolerance value [7]. The uncertainty of measurement of flatness equals the uncertainty of measurement of distance between point S and plane ABC and is function of coordinate differences for pairs of points AS, AB and AC [9]. 𝑢𝑢𝑢𝑢𝐹𝐹𝐹𝐹𝐿𝐿𝐿𝐿 = 𝑢𝑢𝑢𝑢(𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

(8)

Table 2 presents the uncertainty budget of flatness measurement. The measured surface is a square with side length 400 mm. The points A(5, 5), B(395, 5) and C(200, 395) define the reference plane. Point S can be located in any place near the plane but usually around the central point of the plane.

Fig. 5. Measurement model of deviation of parallelism of axis and plane

The tolerance zone is limited by two parallel planes the distance equal tolerance value apart and parallel to the datum [7]. The uncertainty of measurement of parallelism equals the uncertainty of measurement of distance between point S and plane containing point K and parallel to plane ABC [9]. 𝑢𝑢𝑢𝑢𝑀𝑀𝑀𝑀𝑃𝑃𝑃𝑃𝐿𝐿𝐿𝐿 = 𝑢𝑢𝑢𝑢(𝐾𝐾𝐾𝐾𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

(9)

Table 3 presents the uncertainty budget of parallelism measurement. The points A(50, 200, 10), B(350, 350, 10) and

Wojciech Płowucha / Procedia CIRP 75 (2018) 361–366 Wojciech Płowucha / Procedia CIRP 00 (2018) 000–000

364 4

C(350, 50, 10) define the datum plane and point K(50, 200, 200) the toleranced axis. The largest value of the uncertainty is 0.82 µm for the point S(350, 200, 200).

Component ks1

xi, mm 300

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

uxi = EL, MPE/3, µm 1.07

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

Component

xi, mm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

ks1

300

0

1.07

ks2

300

0

1.07

0

ks3

0

1

0.67

0.67

0

ab1

300

0

1.07

0

150

0

0.87

0

0

1

0.67

0.67

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

Table 3. Uncertainty budget of the parallelism measurement for S(350, 200, 200) 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

Table 4. Uncertainty budget of the parallelism measurement for S(350, 350, 200)

0

ks2

0

0

0.67

0

ab2

ks3

0

-1

0.67

0.67

ab3

ab1

300

0

1.07

0

bc1

0

0

0.67

0

-300

0

1.07

0

0

0.50

0.67

0.33

u=

1.00

ab2

150

0

0.87

0

bc2

ab3

0

0.50

0.67

0.33

bc3

ac1

300

0

1.07

ac2 ac3

0

-150

0

0.87

0

0

0.50

0.67

0.33

u=

0.82

One can see that one of the uncertainty components occurs with a weight of 1 and two with a weight of 0.5, and all concern a small measured value. The components related to large measured values and related to the size of the measured object occur in the budget with a weight equal to zero, which means that the size of the measured object nor the distance of the toleranced element from the datum do not affect the uncertainty of the parallelism measurement. 5.2. Parallelism of planes Fig. 6 depicts an example of tolerance specification and measurement modelling of parallelism of planes. The point S is to lay anywhere on the toleranced plane but usually near the opposite corner to location of point K.

The budget analysis shows that two of the uncertainty components occurs with a weight of 1 and one with a weight of 0.5, and all concern a small measured value. The components related to large measured values and related to the size of the measured object occur in the budget with a weight equal to zero, which means that the size of the measured object nor the distance of the toleranced element from the datum do not affect the uncertainty of the parallelism measurement. 5.3. Parallelism of axes Fig. 7 depicts an example of tolerance specification and measurement modelling of parallelism of axes. The datum axis is represented by two points A and B. The toleranced axis is given by points K and S. The uncertainty of measurement of parallelism equals the measurement uncertainty of distance of a point S from the straight line parallel to the axis AB and going though point K. The point S is to lay anywhere near the toleranced axis but usually near the end of it.

Fig. 6. Measurement model of parallelism of planes

The tolerance zone is limited by two parallel planes the distance equal tolerance value apart and parallel to the datum [7]. The uncertainty of measurement of parallelism equals the uncertainty of measurement of distance between point S and plane containing point K and parallel to plane ABC [8]. 𝑢𝑢𝑢𝑢𝑀𝑀𝑀𝑀𝑃𝑃𝑃𝑃𝐿𝐿𝐿𝐿 = 𝑢𝑢𝑢𝑢(𝐾𝐾𝐾𝐾𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

(10)

Table 4 presents the uncertainty budget of parallelism measurement. The points A(50, 200, 10), B(350, 350, 10) and C(350, 50, 10) define the datum plane and point K(50, 200, 200) the toleranced axis. The largest value of the uncertainty is 1.0 µm for the point S(350, 350, 200).

Fig. 7. Measurement model of parallelism of axes

The tolerance zone is limited by a cylinder of diameter equal tolerance value and parallel to the datum axis [7]. The uncertainty of measurement of parallelism equals the uncertainty of measurement of distance between point S and straight-line passing through point K and parallel to line AB [8]. 𝑢𝑢𝑢𝑢𝑀𝑀𝑀𝑀𝑃𝑃𝑃𝑃𝐿𝐿𝐿𝐿 = 𝑢𝑢𝑢𝑢(𝐾𝐾𝐾𝐾𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

(11)

Table 5 presents the uncertainty budget of parallelism measurement. The points A(50, 50, 10) and B(250, 50, 10) define the datum axis and point K(50, 350, 10) defines the



Wojciech Płowucha / Procedia CIRP 75 (2018) 361–366 Wojciech Płowucha / Procedia CIRP 00 (2018) 000–000

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toleranced axis. The largest value of the uncertainty is 1.0 µm for the point S(250, 350, 10.01). Table 5. Uncertainty budget of the parallelism measurement for S(250, 350.01, 10.01) 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

200

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

0.93

as2

0

0

0.67

0

as3

0.01

1

0.67

0.67

ab1

200

0

0.93

0

ab2

0

0

0.67

0

ab3

0

-1

0.67

0.67

u=

0.94

Component as1

xi, mm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

Table 6 presents the uncertainty budget of circular runout measurement. The points A(100, 100, 100) and B(130, 100, 100) define the datum axis. The diameter of the toleranced element is 50 mm. The largest value of the uncertainty is 1.31 µm for the point S(180, 100, 125). Table 6. Uncertainty budget of the circular runout measurement for S(180, 100, 125)

It should be noted that two non-zero uncertainty components occur with a weight of 1 and both concern a small measured value. The components related to large measured values and related to the size of the measured object occur in the budget with a weight equal to zero, which means that the size of the measured object nor the distance of the toleranced element from the datum do not affect the uncertainty of the parallelism measurement. 6. Measurement uncertainty of runout In the following the examples of circular and total runout are discussed (Fig. 8 and 9). The tolerance zone of total radial runout is limited by two coaxial cylinders with a difference in radii equal to the tolerance value, the axes of which coincide with the datum [7]. The runout deviation is the smallest zone between radii of the coaxial circles or cylinders coinciding with the datum and surrounding all points of actual cross-section or surface (toleranced element), in this case these are the points S1 and S2. The uncertainty of runout measurement equals the geometric sum of uncertainties of two deviations of point-line distance. It turns out that neglecting the insignificant differences in uncertainty associated with different angular positions and assuming for further calculations only the largest uncertainty of point-line distance measurement one can find that the uncertainty of the runout is √2 times greater than the uncertainty of the distance between the point and the line. 𝑢𝑢𝑢𝑢𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = √2 ∙ 𝑢𝑢𝑢𝑢(𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

Fig. 8. Measurement model of circular radial runout

0

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

50

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

0.73

bs2

0

0

0.67

0

bs3

25

1

0.70

0.70

ab1

30

0

0.71

0

ab2

0

0

0.67

0

ab3

0

-1.67

0.67

1.11

u=

1.31

Component

xi, mm

bs1

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm 0

Analysis of the budget shoes that the uncertainty component connected with the diameter of the toleranced element occurs in the budget with the weight of 1 and the weight factor of the second non-zero uncertainty component arises from the ratio of distance between the toleranced element to the datum and width of the datum. In this example 50 mm / 30 mm = 1.67. 6.2. Total radial runout Fig. 9 depicts an example of tolerance specification and measurement modelling of total radial runout. The datum axis is represented by two points A and B. The toleranced feature is given by points S1 and S2 representing the extreme radii and lying anywhere between the points A and B.

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This applies to both runout examples described below.

6.1. Circular radial runout Fig. 8 depicts an example of tolerance specification and measurement modelling of circular radial runout. The datum axis is represented by two points A and B. The toleranced cylindrical surface is given by points S1 and S2 representing the extreme radii.

Fig. 9. Measurement model of total radial runout

Table 7 presents the uncertainty budget of total runout measurement for following data: A(100, 100, 100), B(200, 100, 100), S(150, 100, 125)). The largest value of the uncertainty is 0.78 µm for the point S(180, 100, 125). The budget shows that the uncertainty component connected with the diameter of the toleranced element occurs in the budget with the weight of 1 and the weight factor of the second

Wojciech Płowucha / Procedia CIRP 75 (2018) 361–366 Wojciech Płowucha / Procedia CIRP 00 (2018) 000–000

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non-zero uncertainty component arises from the ratio of distance between the toleranced element to the datum and width of the datum. In this example 50 mm / 100 mm = 0.5. Table 7. Uncertainty budget of the total runout measurement for S(150, 100, 125) Component

xi, mm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

bs1

-50

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

0.73

bs2

0

0

0.67

0

bs3

25

1

0.70

0.70

ab1

100

0

0.71

0

ab2

0

0

0.67

0

ab3

0

0.50

0.67

0.33

u=

0.78

0

Component

xi, mm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

uxi = EL, MPE/3, µm

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖 , µm

ks1

-330

𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖

0

1.11

ks2

0

0

0.67

0

ks3

0.01

1

0.67

0.67

ac1

0

0.92

0.67

0.61

ac2

150

0

0.87

0

ac3

180

0

0.91

0

bc1

0

0.92

0.67

0.61

bc2

-150

0

0.87

0

bc3

180

0

0.91

0

u=

1.09

0

8. Conclusions

7. Measurement uncertainty of perpendicularity Fig. 10 depicts an example of tolerance specification and measurement modelling of perpendicularity of axis to the plane. The datum plane is represented by points A, B and C. The toleranced axis is given by points K and S which lay on the opposite ends of the toleranced element.

Fig. 10. Measurement model of perpendicularity

The tolerance zone is limited by a cylinder of diameter equal tolerance value and perpendicular to the datum plane [7, 10]. The uncertainty of measurement of perpendicularity equals the uncertainty of measurement of distance between point S and straight-line passing through point K and perpendicular to plane ABC. 𝑢𝑢𝑢𝑢𝑀𝑀𝑀𝑀𝑃𝑃𝑃𝑃𝑀𝑀𝑀𝑀 = 𝑢𝑢𝑢𝑢(𝐾𝐾𝐾𝐾𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴)

Table 8. Uncertainty budget of the perpendicularity measurement for S(20, 200, 60.01)

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Table 8 presents the uncertainty budget of perpendicularity measurement. The points A(10, 50, 10), B(10, 350, 10) and C(10, 200, 190) define the datum plane and point K(350, 200, 60) defines the toleranced axis. The largest value of the uncertainty is 1.09 µm for the point S(20, 200, 60.01). It can be seen that one of the non-zero uncertainty components occurs with a weight of 1 and two additional with weight of 0.92 and all of them concern a small (zero or almost zero) measured value. The components related to large measured values and related to the size of the measured object occur in the budget with a weight equal 0, which means that the size of the measured object does not affect the uncertainty of the parallelism measurement.

The presented measurement models and respective uncertainty budgets enable analysis of influence of measurement uncertainty of particular coordinates’ differences on the total measurement uncertainty. It can be seen that in case of proper orientation of the workpiece in relation to the CMM coordinate system the analysis of the resulting uncertainty budget can help to draw some conclusions on the factors influencing measurement uncertainty. The presented examples concern a few example cases of deviations of form, orientation and run out. It is worth noting that in the case of deviations of form and orientation, the components of uncertainty related to the dimensions of the workpiece do not have an impact on the uncertainty budget and have sensitivity coefficients equal zero. Authors started some experimental validation and plan comparison with other uncertainty evaluation software. References [1] Wilhelm R.G., Hocken R., Schwenke H. Task specific uncertainty in coordinate measurement. CIRP Annals 2001;50 (2):553-63. [2] Stryczek R. Alternative methods for estimating plane parameters based on a point cloud. Meas. Sci. Rev. 2017;17: 282-9. [3] Humienny Z., Turek P. Animated visualization of the maximum material requirement, Measurement 2012;45:2283-7. [4] Jakubiec W., Płowucha W., Rosner P. Uncertainty of measurement for design engineers. Procedia CIRP 2016;3:309-14. [5] Jakubiec W., Płowucha W., Starczak M. Analytical estimation of coordinate measurement uncertainty. Measurement 2012;45:2299-308. [6] JCGM 100:2008. Evaluation of measurement data. Guide to the expression of uncertainty in measurement. [7] ISO 1101:2017. Geometrical product specifications (GPS). Geometrical tolerancing. Tolerance of form, orintation, location and run-out. [8] Jakubiec W. Analytical estimation of uncertainty of coordinate measurements of geometric deviations. Models based on distance between point and straight line. Adv. Manuf. Sci. Technol. 2009;33:31-8. [9] Jakubiec W., Płowucha W. Analytical evaluation of the uncertainty of coordinate measurements of geometrical deviations. models based on the distance between point and plane. Adv. Manuf. Sci. Technol. 2013;37:516. [10] Humienny Z., Berta M. A Digital application for geometrical tolerancing concepts understanding. Procedia CIRP 2015;27:264-9.