Maritime University of Szczecin
Akademia Morska w Szczecinie
2010, 21(93) pp. 72–76 2010, 21(93) s. 72–76
The determination and analysis of the total measurement
uncertainty of roundness deviation
Wyznaczanie i analiza złożonej niepewności pomiarowej
odchyłki okrągłości
Krzysztof Nozdrzykowski
Maritime University of Szczecin, Marine Engineering Faculty Department of Machinery Construction and Operation Basics
Akademia Morska w Szczecinie, Wydział Mechaniczny, Zakład Podstaw Budowy i Eksploatacji Maszyn 70-205 Szczecin, ul. Podgórna 51/53, e-mail: k.nozdrzykowski@am.szczecin.pl
Key words: analysis, measurement uncertainty, roundness deviation Abstract
This author presents the results of an analysis of the effect of directly measured quantities and their measurement (standard) deviations on the value of the totalmeasurement uncertainty of roundness deviation. The methods described refer to the determination of roundness deviation based on the mean square element and basic mathematical transformations and relationships that provided a basis for the analysis. Its results are given in the form of diagrams followed by final conclusions.
Słowa kluczowe: analiza, niepewność pomiarowa, odchyłka okrągłości Abstrakt
W artykule zaprezentowane zostały wyniki analizy wpływu wielkości mierzonych bezpośrednio oraz ich nie-pewności pomiarowych (standardowych) na wartość złożonej nienie-pewności pomiarowej odchyłki okrągłości. Przedstawiona została metodyka wyznaczania odchyłki okrągłości według elementu średniokwadratowego oraz podstawowe przekształcenia i zależności matematyczne, w oparciu o które dokonano wspomnianej ana-lizy. Wyniki analizy zapisano w postaci wykresów i sformułowanych wniosków końcowych.
Introduction
As the result of unavoidable errors made in the technological machining process, the designed ideal geometrical shape of manufactured machine com-ponents is only approximately similar to the nomi-nal shape assumed by the designer. Deviations from the nominal shape profile are critical for the life and correct co-operation of moving and mating surfaces of the components. However, if these errors do not exceed the limits defined by relevant standards [1, 2, 3], then the product can be regarded as acceptable. The set of requirements specifying the geometry of an object (or an assembly of compo-nents) is known as the Geometrical Product Speci-fication (GPS). When GPS requirements are satis-fied, the product has essential characteristics, such
as functionality, safety, reliability and most impor-tantly, interchangeability.
According to GPS, a number of essential para-meters has to be stated in order to unequivocally describe the geometrical shape of machine compo-nents from the viewpoint of their use. One of these parameters is the allowable deviation from the ideal shape, referred to as the shape deviation. The defi-nition of shape deviation makes use of the so called fitting elements or reference elements. For the roundness deviation of cylindrical machine compo-nents this element can be a circle defined as fitting circle or reference circle (Fig. 1) [4]. The determi-nation of roundness deviation relative to a properly defined circle consists in identifying the greatest distance ΔR of points in the actual profile of the circle. Due to a number of advantages, the
commonly adopted reference circle is the least squares mean circle [4, 5].
Determination of the roundness deviation relative to the mean square circle
Holding objects in vee blocks is the most frequent method of supporting large cylindrical machine parts. However, the correct geometrical assessment of shape deviations depends on two factors: reference elements properly defined in the course of processing the results of measurements, and previous careful analysis of errors of the measurement method used. Vee block measure-ments belong to the so called reference methods of profile measurements, in which a vee block based element is rotated. The exact definition of reference feature in this case is rather difficult, and the least squares mean circle is the most proper for the evaluation of roundness deviation [6].
Practical measurement of roundness deviation ΔR requires that the greatest and smallest indica-tions of the measuring instrument are identified during the rotation of the object measured. As the examined object based in vee blocks is being ro-tated, the measured ΔF differs more or less from its real value ΔR. According to commonly accepted theory of mathematical description of machine component cylindrical profile roundness (using the harmonic analysis of shape profile), any profile can be developed into the trigonometric Fourier series [4]. Displacement lengths of the sensor pin measur-ing a given shape profile as the function of the rota-tion angle of the object supported by two vee blocks can be mathematically described as follows [7]:
3 3 3 3 1 3 3 cos k n Fn Fn n C F (1) where: 3 FnC – harmonic component amplitude;
3
Fn
– phase displacement of particular harmonic components; n3 – number of harmonic component.
The function described by the relationship (1) is periodic and has minimum and maximum values, their difference being the measured roundness dev-iation ΔF3:
max 3
min 3 3 F F F (2)Developing the function (1), which is a mathe-matical description of the measured roundness dev-iation, we can write:
3 3 3 3 1 3 1 3 3 cos sin k n Fn k n Fn n B n A ΔF (3)where An3, Bn3 are components of the amplitudes
Cn3 of particular harmonics of the measured
round-ness profile and are expressed by these relation-ships: 3 3 3 n cos 3 n n F F F C n A (4) 3 3 3 n sin 3 n n F F F C n B (5)
Then the phase shift of the subsequent harmonic is defined by this equation:
3 3 3 3 tg n n n F F F A B n (6)
In numerical calculations the relationships (4) and (5) correspond to:
p n p p p F n r n A p n 3 1 cos 2 3
(7) p n p p p F n r n B p n 3 1 sin 2 3
(8) or these equations: p n p p p F n p n y n A p n π 2 cos 2 3 1 3
(9) p n p p p F n p n y n B p n π 2 sin 2 3 1 3
(10)where: yp – discretized values of the function
ΔF3(φ), np – number of intervals adopted for
discre-tization, n3 – number of the subsequent harmonic. The above equations are used for discretized yp
values of analogue signals, obtained during mea-surements with continuous radius rp recording (yp –
deviation of p-th point of the roundness profile from the adopted reference circle). The coordinates of the least squares mean reference circle centre have the values xs = AFn3, ys = BFn3 calculated from
the relationships(7) and (8) for the harmonic n3 = 1. To calculate the value of the least squares mean circle radius this relationship can be used [8]:
p p n p p s n r r
1 (11) Relationships (1–11) allow to do necessary cal-culations in order to determine the measured roundness deviation ΔF3, to present the measured profile in the form of a trigonometric Fourier series (so called finite cosinusoidal transform) and to de-pict a profile in the graphical form in the polar orCartesian coordinate system. The reference element for the polar system will be a least squares mean circle with a radius rs, while for the Cartesian
sys-tem a mean line m [4].
Determination and analysis of the total measurement uncertainty of roundness deviation
According to the principles of determining com-plex measurement uncertainty [9, 10, 11], the mea-surement uncertainty of the measured deviation ΔF3 can be written in this general form:
p p n p n p p p p p F F r r F 1 1 2 2 3 2 2 3( ) ( ) 3 (12) By substitutions (equations (2–8)) and transfor-mations, we can formulate the final formula defin-ing the measurement uncertainty of the measured roundness deviation expressed with directly mea-sured quantities rp, φp and their standarduncertain-ties Δrp and Δφp:
(13) sin sin cos 1 cos cos sin 1 sin sin sin cos cos cos 2 2 1 2 2 2 min 3 max 3 3 1 3 3 max 3 min 2 2 2 3 3 max 3 min 1 2 3 3 max 3 min 3 3 3 3 3 3 3 3 3 p k n p p n p k n p p p k n p n p k n p p F n n n n n n n n r n n n n n n n p p
where: p p n p π 2 For measurements where the radius rp is
conti-nuously recorded, the measurement uncertainty of roundness deviation ΔΔF3 can be written as:
np p p p F r r F 1 2 2 3( ) 3 (14)and taking account of (9) and (10):
2 1 2 2 2 3 3 max 3 min 1 2 3 3 max 3 min 3 3 3 3 3 sin sin π 2 sin cos cos π 2 cos 2 p k n p n p k n p p F r n n n p n n n n p n n p
(15) On the basis of the relationship (13) the direct influence of standard uncertainties Δrp, Δφp ofmeasured quantities on the total complex uncertain-ty value ΔΔF3 was examined. Such analysis allows to specifically affect the final value of the calcu-lated total measurement uncertainty (taking into account the economic factor resulting from increas-ing or decreasincreas-ing standard deviation of direct mea-surements). Examples of simulated calculations results of the total uncertainty value of the mea-sured roundness deviation based on the relationship (13) for a specific set of input data are shown in figures 1 and 2. This set of input data consisted of discretized values yp of the function described by
the relationship (1) (written down with a set of
n3 = 15 harmonics obtained from measurements of a real object), determined at points φp
correspond-ing to boundary points np of equal intervals as
re-sulting from the function given in (13). The values of angles φmax and φmin and roundness deviation ΔF3 were also determined for these data (angles φmax and φmin are angles at which the function (1) as-sumes the maximum ΔF3(φmax) and minimum ΔF3(φmin) values). In figures 1–2 the total mea-surement uncertainty of roundness deviation is ex-pressed in the form relative to the roundness devia-tion ΔF3 as: 3 3 3 F w F F (16)
Changes of ∆ΔF3 were examined in intervals of standard uncertainty changes of directly measured quantities: for Δrp 0÷0,1, for Δφp 0÷0,2,
respec-tively. This analysis took also account of the impact of the number of intervals taken for discretization on the value w∆ΔF3.
Figure 1 presents the change in w∆ΔF3 depending on Δrp and Δφp for np = 60, while figure 2 shows
the change in w∆ΔF3 depending on Δrp and Δφp for
np = 360.
The analysis results have shown that the value of the total measurement uncertainty ΔΔF3 is much
more affected by measurement uncertainty of Δrp
than that of Δφp. In the assumed boundaries of Δrp
and Δφp changes, the relative change in the value
w∆ΔF3 is, respectively, for the graph w∆ΔF3 = f(Δrp,
Δφp) with np = 60 (Fig. 1) 0÷6.7310–3, while for
the graph w∆ΔF3 = f(Δrp, Δφp) with np = 360 (Fig. 2)
0÷3.2310–3. By increasing the number of points describing a circular profile that make up a basis for the assessment of roundness deviation ΔF3 we ob-tain a more favourable result, i.e. reduced error
w∆ΔF3.
Fig. 1. Change in the relative value of complex measurement uncertainty of roundness deviation w∆ΔF3 depending on
stan-dard uncertainties of the radius Δrp and angle of rotation Δφp
for np = 60
Rys. 1. Zmiana względnej wartości złożonej niepewności pomiarowej odchyłki okrągłości w∆ΔF3 w zależności od
nie-pewności standardowych promienia Δrp i kąta rotacji Δφp dla
np = 60
Fig. 2. Change in the relative value of complex measurement uncertainty of roundness deviation w∆ΔF3 depending on
stan-dard uncertainties of the radius Δrp and angle of rotation Δφp
for np = 60
Rys. 2. Zmiana względnej wartości złożonej niepewności pomiarowej odchyłki okrągłości w∆ΔF3 w zależności od
nie-pewności standardowych promienia Δrp i kąta rotacji Δφp dla
np = 60
The character of changes in measurement uncer-tainty wΔΔF3 dependent on measurement uncer-tainty Δrp and the number of points assumed for
the description of roundness profile based on the relation (15) is shown in figure 3. Therefore, the measurement methods used should assure an accu-rate division of object rotation angle as well as the capability of dividing the rotation into possibly large number np. As it was emphasized before, for
a given measuring task and a desired measurement uncertainty of roundness deviation ΔF3 we can select measurement systems that will guarantee appropriate measurement uncertainties of directly measured quantities Δrp, Δφp and to define a proper
number of points describing a circular profile for assessment.
Fig. 3. Change in the relative value of total measurement uncertainty of roundness deviation w∆ΔF3 depending on
stan-dard uncertainties of the radius Δrp and the number of points np
assumed for the description of roundness profile
Rys. 3. Zmiana względnej wartości całkowitej niepewności pomiarowej odchyłki okrągłości w∆ΔF3 w zależności od
nie-pewności standardowych promienia Δrp i liczby punktów np
przyjętych do opisu profilu okrągłości
Summary
The presented results of an analysis of the im-pact of directly measured quantities and their stan-dard measurement uncertainties have shown une-quivocally that from the viewpoint of the consi-dered complex measurement uncertainty of the roundness deviation, the quantities affecting the measurement uncertainty are the measurement un-certainty of the radius Δrp, and of the rotation angle
interval Δφp and the number np of points adopted
for description of a circular profile. Comparing separately the influence of each single parameter we can state that the effect of uncertainty Δrp and np
is more significant than that of uncertainty Δφp.
Therefore, the selection of these quantities should
wF3 10–3 p rp wF3 10 –4 rp np rp p wF3 10–3
be adjusted to the desired measurement uncertainty ΔΔF3 and characteristic dimensions of the object being measured that result from a specific mea-surement task.
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Recenzent: dr hab. inż. Tadeusz Iglantowicz, prof. ZUT Zachodniopomorski Uniwersytet
