estimation of measurement uncertainty - Sciendo

RESEARCH WORKS OF AFIT Issue 22, pp. 81 ÷ 114, 2007

Janusz LISIECKI Sylwester KŁYSZ Instytut Techniczny Wojsk Lotniczych

DOI 10.2478/v10041-008-0004-4

ESTIMATION OF MEASUREMENT UNCERTAINTY

The paper presents rules for evaluation of measurement uncertainty. 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 parameter that is considered as the measured. Next, the method for evaluation of measurement uncertainty is exhibited on the examples when measurement uncertainty is estimated for verification of a micrometer as well as evaluated for basic reliability parameters. Keywords: measurement uncertainty, A-type uncertainty, B-type uncertainty, measuring error standard deviation, normal (Gaussian) distribution, rectangular distribution, uncertainty budget, effective number of degrees of freedom (ENDF).

1. Introduction In accordance to requirements of the standard PN-EN ISO 10012 [1]: – measurement uncertainty should be evaluated for every measurement process that is covered by any measurement management system, – measurement uncertainty should be recorded and all the known reasons for measurement variation must be documented, – evaluation of the uncertainty should take account for not only uncertainty of the measurement equipment calibration, but also for all the other components of uncertainty that are essential for the specific measurement process. Relevant methods for the analysis should be applied, – if any uncertainty components are so insignificant as compared to other components that evaluation of such insignificant components is unjustified in technical and economical aspects, calculation thereof should be given up and the decision should be recorded, – efforts devoted to evaluation and recording of measurement uncertainty should be commeasurable to the significance of the measurement results to the product quality.

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Janusz LISIECKI, Sylwester KŁYSZ

In accordance to requirements of the standard PN-EN ISO/IEC 17025, the accredited laboratory: – when it performs its own calibration and verification procedures it should have in place the working procedure (instruction) for evaluation of measurement uncertainty for every calibration or verification, – it should have in place the working procedures on evaluation of measurement uncertainty, – when the character of the applied research method disables strict, metrological and statistically justifiable calculation of measurement uncertainty the efforts should be made to identify all the uncertainty components and estimate them in reasonable manner based on available knowledge on capabilities of the method and gathered past experiences, such a way of uncertainty evaluation should de described, – in case when examinations are carried out by means of a well-known method where boundary limits for main sources of measurement uncertainty and the methods for result reporting are truthfully defined, such results should be reported in accordance with the relevant statements of the research procedure or instruction. – sources of uncertainties include the applied standards and reference materials, used methods and equipment, environmental conditions, properties and status of the objects that are subject to examination or calibration as well as the performing staff.

2. Metrological definitions [3÷7] Measuring error – represents the difference between the measurement result and the exact value of the measured quantity. As the exact value is actually unknown, the error cannot be accurately determined. Systematic error – represents the difference between the averaged value for infinite number of measurement results for the same quantity measured that are carried out under repeatable condition and the exact value of the quantity measured. It is the error that under repeatable conditions remains constant or varies following the defined rule. Correction factor – represents the value that is algebraically added to the rough measurement result in order to compensate the systematic error. The correction factor equals to the value of the estimated systematic error but with the reverse sign.


Estimation of measurement uncertainty

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Random error – represents the difference between the specific measurement result and the averaged value for infinite number of measurement results obtained for repeatable measurement conditions. It is the error that under repeatable conditions varies in random way. Gross (excessive) error – results from incorrect execution of the measurement procedure, e.g. negligent reading of instrument indication, use of a defective instrument, inappropriate use of an instrument. When a measurement result with a gross error appears in a series of measurements, such a results should be discarded. Deviation – represents the difference between the specific value and the referenced benchmark, Permissible limiting error (of a measuring instrument) – represent the boundary values of measuring error that are permitted by technical conditions or requirements related to the specific measuring instrument, True (exact) value – represents the value that conforms to the definition of the specific value. In other words, it is the value that would be obtained as a result of an error-free measurement. Apparent true (exact) value – represents the value that is assigned to the specific value and considered, sometimes provisionally, as the determined value with some uncertainty that is acceptable for the certain application. It means that if under some circumstances uncertainty of the benchmark can be considered as negligible insignificant, the value assigned to that benchmark can be provisionally accepted as exact (true). Measurement result – represents the value that is assigned to the quantity measured and achieved by taking measurements, Rough result – the result of a measurement before correction of the systematic error, Corrected result – the result of a measurement after correction of the systematic error, Measurement accuracy – degree of coincidence between the measurement result and the exact (true) value of a quantity measured. Measurement accuracy is of quality nature and no figures can be assigned thereto. Resolution (of an indicating instrument) – the smallest difference of indication attributable to a measuring instrument that can be clearly distinguished. Resolution of instrument is one of the reasons for uncertainty of measurement instrument. The value for instrument resolution should be defined in the following way: – in case of analogue indicating devices the resolution is expressed as a ratio of the indicating arrow width on the instrument scale over the scale interval size multiplied by the scale interval value,


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– in case of digital indicating devices the resolution is expressed as a half of distinguishable variations plus a single scale interval for unstable indications or just a single scale interval when indications vary by not more than one scale interval. The following values of instrument resolution are adopted for analogue devices: – 1/2 of the scale interval, when the scale interval size is less than 1.25 mm, – 1/5 of the scale interval, when the scale interval size is between 1.25 and 2.5 mm, – 1/10 of the scale interval, when the scale interval size is more than 2.5 mm. Calibration – a sequence of operation aimed at establishing, under specific conditions, the relationships between the values measured by a measuring instrument or values represented by the measurement standard or by a reference material and the corresponding values of parameter established by standards of the measurement units. Result of the calibration process make it possible to assign appropriate values of the measured parameter to observable readouts of a measuring instrument or to find out correction factors for observable readouts. Verification – a sequence of operation aimed at establishing conformity of the measuring instrument to requirements of the verification regulations, recommendations of technical standards or technical specifications. Most frequently the requirements are expressed as permissible limiting errors ±Eg. Standard uncertainty (u) – uncertainty of a measurement result expressed as the standard deviation, A-type uncertainty (uA) – the uncertainty calculated by the method of statistical analysis of a series of single observations (mostly with use of the normal (Gaussian) distribution of measurement results). B-type uncertainty (uA) – the uncertainty calculated by methods other than the ones for the A-type case (mostly with use of the rectangular distribution that describes systematic errors caused by unrecognisable systematic influence). Combined standard uncertainty (uc) – the uncertainty that is defined when a number of uncertainty component occurs, for direct measurements is calculated as the square roof of the sum of squares of component uncertainties, for indirect measurements the summation of squares of component uncertainties are added with relevant weight coefficients in accordance with the rule of uncertainty propagation, Expanded uncertainty (U) – is obtained as a product of the combined standard uncertainty by the expansion factor k, U = k · uc


(1)


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The expanded uncertainty determines limits for the uncertainty interval that can be associated with a defined confidence level. Confidence level (p) – represents the probability that the true (exact) value belongs to the interval of the measurement uncertainty (within the confidence interval), which can be written as: P = P{x0 ∈ (x-U; x+U)}

(2)

3. Expression of uncertainty in measurement Every measurement is aimed at determination of the measured value, i.e. value of the specific quantity that is the objective of measurements. Yet measurement result is only approximated or evaluated (estimated) value of the measured quantity and therefore the result is complete only when it is quoted with the estimated uncertainty. Following the Manual Expression of Measurement Uncertainty [4]: “Uncertainty (of measurement) is the parameter that is associated with measurement result and exhibits dispersion of values that can be assigned to the measured value in justified manner.” Expression of uncertainty in measurement refers to a well-defined physical quantity that is considered as the measured value and can be characterized by z single value obtained from measurement. The term “value” (measurable quantity) stands for the property of the phenomenon, material or substance that can be qualitatively described and quantitatively evaluated. Uncertainty of measurement is a result of random errors that can occur during measurement. It is important to distinguish the term “error” and the term “uncertainty of measurement”. Every error represents a random variable, while uncertainty of measurement is a parameter for distribution of error probabilities. In general [6], every result of measurement Y is composed of the following: – rough result Ys, – totalised correction coefficient PΣ to compensate the determinable errors, – expanded uncertainty U(Y) of the measured quantity Y so that Y = (Ys + PΣ) ± U(Y)

(3)

The components Ys and PΣ are burdened by partial uncertainties U(Ys) and U(PΣ) that make up together the overall uncertainty U(Y). After having taken account for correction factors the measurement result can be expressed in the following form:


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Y = Ypop ± U(Y)

(4)

where: Ypop = Ys + PΣ is the correct result. Methods for calculation of uncertainty deal with corrected results, i.e. after making compensation of the component for the systematic error caused by recognizable systematic impact. The compensation is carried out by adding correction factor to rough measurement or by multiplying the same by correction coefficient. It is assumed that corrected result of measurement represents random variable with its respective mathematical expectation that are equal to corresponding true (exact) value.

4. Analysis of sources for uncertainty Major sources of errors that take place during the measurement process and make up the overall uncertainty of measurement are the following [4, 6, 8]: • Inaccurate determination of a measured quantity and /or misinterpretation of the definition of the measured quantity. Accurate definition for every measured quantity is of high importance. Every imprecision in that definition may be a reason for errors. If the measurements are dedicated to the velocity of sound propagation in dry air one has to accurately define composition of the air, its temperature and pressure. • Inaccuracy of the used measuring equipment Measuring equipment is the primary source of errors. Due to a number of reasons, such as calibration errors, errors in manufacturing of the instrument scale (graduation errors), non-linearity errors, changes of dimensions and other properties of equipment components in pace with the equipment ageing or due to other external or environmental factors the equipment may exhibit inaccuracy of indication, dispersion of indications or hysteresis error, etc. • Qualifications and psychical /physical condition of the measuring staff with account for resolution of the instrument Imperfectness of human senses as well as inappropriate layout of instrument on the measurement bench may serve as reasons for misreading of instrument indications. • Errors of the measurement method (approximation and simplifying assumptions inherent to the method and procedure of measurement)



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Such errors occur when the applied prevents from accurate measurement of the desired parameter. For example, measurement of length with use of thrust instrument that brings about to elastic deformation of measured objects and therefore to alteration of the measured dimensions, a contact thermometer introduced into the measured material changes temperature of the same, a voltmeter with long unscreened wires measures voltage of the source affected by interferences. • Non-representative sampling This is the error component caused by the fact that the measured sample is not a representative specimen for the defined measured quantity. • Ambient conditions Ambient conditions, such as temperature, pressure, humidity, etc. represent the factors that affect the measuring system and therefore eventual result of measurement. External factors, e.g. electric and magnetic fields, vibration, dust, shocks, nuclear radiation may significantly influent measurement result. Ambient factors affect both the measuring system and the measured quantity. • Calculation errors that results e.g. from rounding of the calculation results. • Inaccurate values for the predefined and known constants and other parameters provided by external sources and then used for data processing procedures • Dispersion of values for the measured quantity obtained from observations. Is caused by the fact that observations are repeated under only apparently identical conditions.

5. Procedure for estimation of the full result of measurement The procedure for estimation of the full result of measurement is outlined on the drawing below:


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Janusz LISIECKI, Sylwester KŁYSZ The mean value x = rough result of measurements

Results of individual measurements

P – correction factor

µ - exact (true) value

corrected result correctness Standard uncertainties type B type A accuracy Composed uncertainty

Expanded uncertainty

Fig. 1. Graphic representation of the procedure for estimation of measurement uncertainty [9]

6. Measurement equation The dependence of measurement result, considered as the random variable Y, on a number of random variables X1, …, Xn, associated with the measurement process, i.e. directly measurable quantity, correction factors, physical constants and errors thereof is described by the formula: Y = f(X1, X2, X3, ..., Xn-1, Xn)

(5)

The function f of the above equation may express no physical law and merely describe the measurement process. It should cover all the parameters that can affect the modelling of the measurement result. Initial parameters (arguments for the function f) should be defined as accurately as practically feasible in order to determine values thereof in unambiguous manner.



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The estimation of the output parameter Y, that is adopted as the measurement result, shall be the value of y that is determined by the same equation as above, but with the X1, …, Xn values substituted with their estimators x1, …, xn, namely: y = f(x1, x2, x3, ..., xn-1, xn)

(6)

7. Determining of standard uncertainties for all the components [4, 6, 9] At the outset, it is necessary to estimate standard uncertainties u(xi) for all the initial parameters that influent the research process. For the research (measurements) where series of observations are carried out the A-type method of statistical analysis series of observations is applicable. If uncertainty of the input value cannot be determined on the basis of a series of measurements, the B-type method is applied, where the standard uncertainty is evaluated on the basis of information about possible range of variations for the measured quantity in question. The A-type method (for the measured value or errors of its measurements described by Gaussian distribution) achieves the best evaluation (estimator) for the mathematical expectation µ associated with the random variable x as the arithmetical mean x provided that n independent observations were made under repeatable measurement conditions.

x=

1 n

n

∑x

(7)

i

1

The standard uncertainty of the A-type is equivalent to standard deviation of the mean value over a series of experiments. The standard deviation is calculated by the following formula:

()

uA x =

u(x )

(8)

n

where: n

u( x ) =

∑(

1 xi − x n − 1 i =1

)2

(9)

The foregoing formulas are true only for sufficiently long series of observations (n > 30 is usually assumed), when the mean value is a trustworthy estimation for the mathematical expectation.


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Fig.2 depicts how the coefficient

1 n

depends on the length of the meas-

urement series.

Fig. 2. The diagram for the

1

n

The correction factor

coefficient as a function of n [10]

1

for the experimental standard uncertainty drops n rapidly for small n value (for n = 4 it is halved) but for large n (above a dozen) tends to decrease rather slowly. It is why excessive expanding of measurement series seems to be unjustified. Needless to say that it is really difficult to make sure that measurement conditions shall be repeatable for long series of measurements. However, number of measurements should be high enough to guarantee that the arithmetical mean is a trustworthy estimation for the mathematical expectation. Long series of measurements (n > 30) should be launched when the combined standard uncertainty up(x) is to be found out. Such a type of uncertainty is used later on when similar measurements are taken under the same conditions. Then, after completion of the series of n1 new measurements, the kind of measurement (observation) distribution becomes known and the uncertainty can be determined by the following formula: uA ( x ) =

up ( x ) n1

(10)

If kind of measurement (observation) distribution remains unknown, the required number of measurements in a series depends on the desired confidence interval for the expanded uncertainty.


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Fig. 3. Dependence of the confidence level p on the expansion coefficient k for the Gaussian distribution [10]

Table 1 Expansion coefficient k calculated from the t-Student distribution for selected confidence levels p and various number of degrees of freedom ν p

68,27

70

90

95

95,45

99

99,73

1

1,84

1,96

6,31

12,71

13,97

63,66

235,80

2

1,32

1,39

2,92

4,30

4,53

9,92

19,21

3

1,20

1,25

2,35

3,18

3,31

5,84

9,22

4

1,14

1,19

2,13

2,78

2,87

4,60

6,62

5

1,11

1,16

2,02

2,57

2,65

4,03

5,51

6

1,09

1,13

1,94

2,45

2,52

3,71

4,90

7

1,08

1,12

1,89

2,36

2,43

3,50

4,53

8

1,07

1,11

1,86

2,31

2,37

3,36

4,28

ν

9

1,06

1,10

1,83

2,26

2,32

3,25

4,09

10

1,05

1,09

1,81

2,23

2,28

3,17

3,96

∞

1,00

1,03

1,65

1,96

2,00

2,58

3,00

The foregoing data allow making a conclusion with respect to reasonable selection of the length for series of measurements. For ν = ∞ the expansion coefficients k calculated for the t-Student distribution as well as for Gaussian distribution coincide. On the other hand, substantial differences are observed for short series of measurements. The differences even increase in pace with growth of the desired confidence level. The reason for this phenomenon lies in large uncertainty when the standard deviation of mean is calculated for relatively small number of samples. For not very high confidence levels e.g. p ≤ 70%, differences between results calculated for the both distribution schemas are significant for only short series of measurements, when n ≤ 5 and become negligible for longer series (n > 5).


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Hence the conclusion is to be made that for n ≤ 5 the expansion coefficient should be calculated from the t-Student distribution whilst for n > 5 they can be derived from Gaussian distribution. For high levels of confidence (px ≥ 95%) differences between data calculated from Gaussian and t-Student distributions are pretty large, even for a sample size (more than 10 measurements). Therefore for the desired high confidence levels p (e.g. 99%) long series of measurements should be preferred (e.g. n > 10). As one can easily see, the confidence level for the standard uncertainty u( x ) = σ x equals to 68.2% whereas for the 2σx expanded uncertainty U = 2σx is as high as 95.4%. These confidence levels are sufficient for laboratory applications. However, in some very specific cases, in particular when human health and safety is in question, even higher confidence levels are required. Then the desired confidence level pα is presumed and then the appropriate expansion coefficient is to be found in the reference tables. The confidence level for the expansion coefficient of 3 (U = 3σx) equal to 99.7% is considered as very high and near the certainty. In case of the B-type method, where we have to deal with a single measurement result or with quality acquired from documents or literature references, the boundary limits a+ i a- can be evaluated for the input variable Xi and then the standard uncertainty is calculated by the formula: uB ( x i ) =

a k

(11)

a+ + a− 2 The value uB(xi) as calculated in the foregoing manner is referred to as the standard uncertainty of B-type. It is the method for analysis of conditions where underlying reason for error may occur and is recommended for analysis and estimation of instrumental (equipment) errors. If the partial standard uncertainty u(xi) cannot be determined on the basis of repeatable observations (a series of measurements), the standard uncertainty can be found out based on information about possible range of variations for the parameter in question with the attention paid to such factors as: – early acquired measurements, – experience and general knowledge about the applied materials and instruments properties, – specifications provided by the equipment manufacturer, – data obtained as a result of calibration or acquired from other certificates,

where: a =



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– data from tables of physical and technical parameters tables. If in any available documents, tables of physical and technical parameters or in other sources the variation range for the analyzed parameter is defined, e.g.: – the expanded uncertainty U is defined as the multiple of its standard deviation - the standard uncertainty is calculated by dividing of the first parameter by the relevant expansion coefficient k, – the expanded uncertainty U is defined as the interval with the specified confidence interval – the Gaussian distribution is assumed and the adequate coefficient is adopted for the specified confidence level, – neither k nor confidence level is specified – the “3σ” rule should be applied by adopting k = 3. The following features attributable to any source of errors should be taken into account: 1) availability of information on the probability distribution applicable to the specific parameter (e.g. Gaussian, rectangular, triangular), 2) availability of boundaries of the variability interval for the specific parameter value, xi belongs to (a-, a+), 3) probability of the fact that the specific value of the xi parameter falls into the predefined interval. Examples of formulas for estimation of the standard uncertainty of B-type are gathered in Table 2. Table 2 Rules for estimation of the B-type standard uncertainties for selected parameters [9] Characteristics of parameters that are covered by the B-type estimation

Standard uncertainty u(xi)

Examples for application of the B-type estimation

– distribution of all the possible values for the Xi parameter is of the Gaussian type – values of the Xi parameter fall within a the estimated boundaries from a- to for k = 3; u(xi) = ; p = 0.99 3 a+, – the best approximation of the xi a – when a represents the uncervalue coincides with the middle of for k = 2; u(xi) = ; p = 0.95 tainty that is specified in the 2 (a+ + a- ) calibration certificate, for k = 1; u(xi) = a; p = 0.68 , the interval – when a represents the uncer2 tainty of the reference data – value of the kp coefficient that corre- u(xi) = 1.48a (please see the and parameter values of k are spond with the confidence level is note) known for such data. already known, Note: as well as in case when the (a + a- ) value of the k coefficient that correwhere a = + 2 spond with the confidence level is unknown but there is 50% of probability that the value for the input parameter Xi falls into the interval from a- to a+


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– distribution of all the possible values for the Xi parameter is of the rectangular (uniform) type – the best approximation of the xi value coincides with the middle of (a + a- ) , the interval + 2 – there is 100% probability that value of the input parameter Xi falls within the interval from a- to a+ – distribution of all the possible values for the Xi parameter is of the triangular type – the best approximation of the xi value coincides with the middle of (a + a- ) , the interval + 2 – there is 100% probability that value of the input parameter Xi falls within the interval from a- to a+ – distribution of all the possible values for the Xi parameter is represented by the arc sin function, – the best approximation of the xi value coincides with the middle of (a + a- ) , the interval + 2 – there is 100% probability that value of the input parameter Xi falls within the interval from a- to a+ – distribution of all the possible values for the Xi parameter still remains partly unknown, – there is 100% probability that the value of the input parameter Xi falls within the boundaries of the interval from a- = xi - b- do a+ = xi + b+ that can be asymmetrical with respect to the best approximation of the measured value, – it is adopted, by substitution, that distribution of all the possible values of the parameter Xi is of the rectangular type, and its value falls within the boundaries of the interval from b- to b+.

u(xi) =

where a =

u(xi) = u(xi) =

a 6

(a+ + a- ) 2

u(xi) =

where a =

3

(a+ + a- ) 2

u(xi) =

where a =

a

a 2

(a+ + a- ) 2

a+ − b− 12 a+ − a−

12 for the symmetric interval (a+ - a-) = 2a 2

u (xi) =

a

2

3

– when a represents the limit error for the measuring instrument, – when a represents the limit error for the reference value for which the coefficient value of k remains unknown, – uncertainty of the resolution (r = 2a) of the indicating instrument (e.g. slide caliper, micrometer), – when probability of errors occurrence is higher in the vicinity of the interval boundaries is less than in the middle of the field of errors, – when dispersion of the instrument indications it insufficient as compared to its resolution and there is no chance to carry out measurements with an instrument with higher resolution.

– a measurement of the diameter of an oval-shaped object with random cross-section (deviations of the outline from the average circle exhibit the arc sin distribution),

– when a represents the resolution of an digital indicating device, – when a is the width of the variation range of readouts cause by the hysteresis (indication are different depending on the fact whether readouts had been performed for increasing or decreasing values of the Xi parameter), – when a is the limit error that is a result of rounding or truncation of figures during calculations.

Calculation results, obtained in the above manner do not have to be worse than those that are based on repeatable observations. Table 3 presents the values of “uncertainties of uncertainties” that origin in the statistic way, depending on the number of observations.


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Table 3 Ratios of experimental standard deviations of the mean value for n independent observations with Gaussian distribution over standard deviation of the mean value [4] Number of observations

σ[s (q )] / σ(q )

n

%

2

76

3

52

4

42

5

36

10

24

20

16

30

13

50

10

The data in the table exhibit that even for n=10 observations the “uncertainty of uncertainties” is still as high as 24%. This allows to conclude that calculation of standard uncertainty with use of the A-type method do not have to be more dependable than in case when the standard uncertainty is calculated with use of the B-type method and in many cases of experiments when number of measurements must be limited, the components obtained from calculations with use of the B method may be even better defined than respective components that are obtained from calculations with the A method.

8. Determining of the composed standard uncertainty 8.1. Direct measurements For the quantity that are measured in direct way, when the standard uncertainties of both A and B types are taken into account, the composed standard uncertainty uC is the square root of the sum of squares for the both uncertainty values. uC = u A2 + uB2

(12)

8.2. Indirect measurements In most cases the evaluate value y in not measured in direct way but determined on the basis of measurements for other quantities xi that remain in strict relation by a specific function (see measurement equation (6)).


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y = f(x1, x2, x3, ..., xn-1, xn)

(13)

Based on the total differential the law for propagation of uncertainties is formulated in the following way: n

uC ( y ) =

∑ 1

2

⎛ ∂f ⎞ 2 ⎜ ⎟ u ( xi ) ⎝ ∂xi ⎠

ui ( y ) = ci ⋅ u ( xi )

(14)

(15)

where: u(xi) – standard uncertainties for measurement of input values, calculated with use of either A or B method; the composed standard uncertainty uC(y) is then the estimate of the standard deviation σy and characterizes dispersion of values that by justified reasons can be associated with the measured quantity y, ∂f = c i – partial derivatives that are referred to as sensitivity coefficients. ∂x i The law for propagation of uncertainties is true if the input variables are uncorrelated, which is the most common practice. However, if the provision on mutual independence of input values is not met, the formula with account for covariances must be applied: uC2

2

m

m −1 m

⎛ ∂f ⎞ (y ) = ⎜ ⎟ u 2 ( xi ) + 2 ∂x i ⎠ i =1 ⎝ i =1

∑

⎛ ∂f ⎞⎛ ∂f ⎞ ⎟⎜ ⎟u (x i , x j ) i ⎠⎝ ∂x j ⎠ j = i +1

∑ ∑ ⎜⎝ ∂x

(16)

The final result for measurement of the quantity Y is calculated on the basis of the defined function (measurement equation) where arithmetical means of directly measurable quintiles are substituted: y = f ( x1, x2,....., xn )

(17)

8.3. Uncertainty budget If individual input values are mutually independent, the information that is essential for analysis of the measurement uncertainty can be brought together into a table (see Table 4). Such a table assures clarity of the analysis and is referred to as the uncertainty budget.


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Tabela 4 Uncertainty budget for calculation of a composed uncertainty for uncorrelated input values [8] Value symbol Xi

Value estimation xi

X1

x1

X2

x2

. . .

. . .

XN

xN

Y

y

Standard uncertainty

Probability distribution

Sensitivity coefficient ci

Contribution to the standard composed uncertainty ui(y)

Gaussian

c1

u1(y) = c1·uA(x1)

rectangular

c2

u2(y) = c2·uB(x2)

. . .

. . .

. . .

triangular

cN

uN(y) = cN·uB(xN)

u(xi) u A (x1) = uB (x 2 ) =

. . . uB (xN ) =

U 2 x2

2 3

xN 2 6

uc(y)

9. Determining of the expanded uncertainty In case of direct measurements the expanded uncertainty U is the product of the expansion coefficient and the composed standard uncertainty: U = k · uC(y)

(18)

where uC(y) is calculated with use of the formula for direct measurements The expansion coefficient for the guaranteed confidence level pα should be calculated on the basis of the distribution for a standardized random value, where the distribution is a convolution of both the Gaussian and uniform distributions, when the samples size is large. Otherwise, when the sample size is small, the t-Student and uniform distributions should be applied. For indirect measurements, the expanded uncertainty, denoted as U, is obtained by multiplication of the standard composed uncertainty uC(y) by the expansion coefficient k: U = k · uC(y)

(19)

where uC(y) is calculated with use of the formula for indirect measurements (14) and the result is often give in the conventional form:


98


Y=y±U

(20)

Exact calculation of expansion coefficient for the desired confidence level is a sophisticated job for indirect measurements as it needs to know the function of the probability density distribution for the random value that is used for modelling of measurement results y. Such a function is a convolution of component distribution for random variables that are used for modelling of input parameters. Calculation of such convolutions is difficult, except for some specific cases that include convolution of any number of Gaussian distributions, which is the Gaussian distribution itself and its parameters can be easily calculated. It is why approximated methods are practically used for calculation of the expansion coefficient.

9.1. Approximated methods for calculation of the expanded uncertainty [10] Method I – with use of predefined values for expansion coefficient It consists in application of the coefficient k = 2 for the confidence level p ≈ 95% and k = 3 for the confidence level p ≈ 99%. It is the method that is used for measurement experiments that involve independent input parameters Xi and: 1) distributions for all the components of the standard composed uncertainty uC(y) are of Gaussian type, 2) distributions for the components of the standard composed uncertainty uC(y) are of rectangular type, but with the same width and not less than 3 distributions are available, 3) there is quite many input parameters Xi (practically not less than 4), the composed standard uncertainty uC(y) is free of domination by the component of the standard uncertainty that is calculated by means of the A-type method for the series of only few observations or by the component of the standard uncertainty that is calculated by means of the B-type method from the presumed rectangular distribution (i.e. the uncertainties ciuA(xi) and ciuB(xi) contribute to the uC(y) in comparable degree) and uC(y) is much higher than a single component ciuB(xi). Method II – geometrical sum It consists in calculation of the expanded uncertainty Ui for every single input parameter (for every component of the error) and then the expanded uncertainty for the output value Uy is calculated as the square root of the sum of squares for all the component uncertainties, in accordance with the formula:



2 U y = U x21 + U x22 + ... + U xn

99

(21)

Expansion coefficients for component uncertainties must be calculated for the same confidence level. Method III – ordinary (algebraic) sum Adding in the uncertainty domain:

U = UA + UB or Uy = Ux1 + Ux2 + … + Uxn

(22)

The method assumes the worst case for accumulation of errors, i.e. the situation when all the component errors are of the maximum value and are of the same sign. It is very little probably, which means that the method exaggerates uncertainty of measurements and is the most pessimistic one. However, the method is used for workshop measurements due to easy calculation as well as for specific cases or for instances, when tolerances are to be fund out or when components of machinery must be manufactured with defined play. Method IV – dominating component It is recommended for the cases when one of the component uncertainties of either A or B types is the dominating factor. If uA >> uB, the substitutions k = kA must be made in the formula (18), Similarly, if uB >> uA the substitution in the formula (18) should be k = kB. For the Gaussian probability distribution and for the confidence level of 95,45% the expansion factor should be k = 2, whereas for the confidence level of 99,73% - k = 3 For the rectangular probability distribution and for the confidence level of 95% the expansion factor should be k = 1,65, whereas for the confidence level of 99% - k = 1,71. Method V – effective degrees of freedom For this method the expanded uncertainty is denoted as Up and calculated by the following formula:

Up = kp · uC(y) = tp(νeff) · uC(y)

(23)

where tp stands for the t coefficient attributable to the t-Student distribution that can be read from relevant reference tables in accordance with the selected confidence level p and for the effective number of the degrees of freedom νeff. The figure νeff is to be calculated by the Welch-Satterthwaite formula


100


ν eff =

uC4 (y ) 4

⎛ ∂f ⎞ ui4 ( xi ) ⎟ ⎜ νi ∂xi ⎠ i =1 ⎝ N

∑

(24)

where: uC(y) – composed standard uncertainty for the output value, u(xi) – standard uncertainties for the input parameters, i = 1, 2, ..., N, νi – number of the degrees of freedom for u(xi). Practical calculation of the number of degrees of freedom for u(xi): – if distribution for the u(xi) is of the Gaussian type, then

νi = n – 1, – if distribution for the u(xi) is of the rectangular type with the presumed width a is adopted with no uncertainty as the limits for such of a, then u(xi) = 2 3 decomposition are known and then νi →∞, hence 1/ νi →0, – if the source of error is rated to the B group and available information of that error source indicate that the standard uncertainty u(xi) can be merely ∆u( xi ) , then estimated with the specific relative error δui = u( xi ) νi ≈

1 1 ⋅ 2 2 δui

For instance, under the assumption that u(xi) is estimated with the error of 1 1 ≈8. 25%, number of the degrees of freedom shall amount to: νi ≈ ⋅ 2 0,0625

10. Examples for estimation of measurement uncertainty 10.1. Estimation of uncertainty for verification of a micrometer [6] 10.1.1. Measurement equation The error of micrometer readout can be describe with use of the formula: Ex = (L – W + Pt) ± U(Ex) [mm]


(25)


101

where: L – the maximum readout from the micrometer for three measurements of dimensions; W – rated length of the standardized plate (Wn) with account for the correction factor – deviations of the plate length, Pt – correction factor for temperature conditions, U(Ex) – the expanded uncertainty at the confidence level of 1 – α = 0,95. The value of the temperature correction factor is neglected (0).

10.1.2. 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:

∑ (c ⋅ u )

uc (E x ) =

i

i

2

(26)

where: ci – sensitivity coefficients, i.e. the partial derivatives of the measurement function for the function components, In this case ci = 1 or ci = -1. The standard uncertainty for the limit error can be then expressed in the following form: uc (E x ) = u 2 (L ) + u 2 (W ) + u 2 (Pt ) [mm]

(27)

10.1.2.1. 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 0.002 (readout with use a magnifying glass), by means of the B method and with the assumption of the rectangular distribution of the uncertainty. u (L ) =

r

2 3

= 0,00058 mm

(28)

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 method is applied with assumption of the Gaussian distribution. u (W ) =

U [mm] 2


(29)

102


The standard uncertainty for the temperature correction factor u(Pt) is determined under the assumption that coefficients o thermal expansion for the standardized plate and for the micrometer are the same: αW = αL = αt = 11,5 ·10-6 oC-1 and ∆t = tW - tL (where: tW = 20oC, tL – ambient temperature in the laboratory room TI = 20 ± 10°C), The uncertainty u(∆t), as determined with use of the B method ∆t . Thus: (rectangular distribution) amounts to 2 3 u(Pt) = W · αt ·

∆t

2 3

[mm]

(30)

10.1.3. Uncertainty budget All the information for analysis of uncertainty are brought together in the table 5. Table 5 Uncertainty budget for calculation of the composed uncertainty for verification of a micrometer Parameter symbol Xi

Parameter estimation xi

L W Pt

0

Ex

L-W


Contribution of u(xi) into the standard uncertainty for each individual standardized plate (of n)

wzór (28)

ci u1(L1)

ci u2(L2)

ci u3(L3)

ci un(Ln)

wzór (29)

ci u1(W1)

ci u2(W2)

ci u3(W3)

ci un(Wn)

wzór (30)

ci u1(Pt1)

ci u2(Pt2)

ci u3(Pt3)

ci un(Ptn)

wzór (27)

wzór (27)

wzór (27)

wzór (27)

10.1.4. Expanded uncertainty If all values of component uncertainties u(Ex) are comparable, the expanded uncertainty at the confidence level of 1 – α = 0.95 can be calculated with the formula: U(Ex) = 2 · uc(Ex) [mm]

(31)

If the uncertainty uc(Ex) 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 uncertainty at the level of confidence of 1 – α = 0.95 can be calculated with the formula: U(Ex) = 1,65 · uc(Ex) [mm]

(32)

If the uncertainty uc(Ex) has two dominating components (e.g. u(L) i u(W)) and the rectangular distribution is assumed for the both components with the re-



103

spective spans R1 = 2a1 and R2 = 2a2, the components can be superposed to make up the trapezoidal even-armed distribution with the span of R = 2a = 2(a1 + a2) and upper base of b = 2aβ, where β = (a2 – a1)/(a2 + a1). Then the composed uncertainty amounts to uc(Ex) =

a12 + a22

u 2 ( x1) + u 2 ( x2 ) =

3

(33)

The expanded uncertainty at the confidence level of P = 1 – α = 0.95 can be calculated by the formula: U(Ex) =

1 − (1 − P )(1 − β2 ) 1 + β2 6

⋅ uc (E x )

(34)

Finally, the indication error shall be: Ex =max [d - U(Ex),d + U(Ex)]

[mm]

(35)

where: d – the maximum deviation from the W dimension, d = L – W. Note: uncertainty components uc(Ex) can be considered as insignificant and then neglected, if uc(Ex) – uc* (Ex) ≤ 0,05 uc(Ex) (where uc* (Ex) – 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%.

10.2. Estimation of uncertainty for measurements of strength parameters obtained from the static tensile test [11] 10.2.1. Estimation of uncertainty for determination of the tensile strength Rm 10.2.1.1. Formula for the measurement result (for a single specimen)

Tensile strength Rm = f (Fm, d0 ) =

Fm S0

=

4Fm πd 02


(36)

104


where: Fm – maximum force recorded during the tensile test for the specimen; S0 – initial cross-section of the specimen; d0 – initial average diameter of the specimen. 10.2.1.2. 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: u (Rm ) =

∑ (c

i

⋅ ui )

2

(37)

where: ci – sensitivity coefficients, i.e. the partial derivatives of the measurement function for the ith function components, ui – standard uncertainties for individual components. In this case cFm =

4 πd02

and cdo = −

8Fm πd 03

, thus:

2

2

⎛ 4 ⎞ ⎛ 8F ⎞ u (Rm ) = ⎜⎜ 2 ⎟⎟ u 2 (Fm ) + ⎜⎜ − m3 ⎟⎟ u 2 (d0 ) ⎝ πd0 ⎠ ⎝ πd0 ⎠

(38)

Determination of component standard uncertainties

The uncertainty for measurement of the specimen diameter u (d 0 ) is calculated by means of: a) on the basic of the arithmetical means for the series of six measurements (the A method), with the t-Student distribution assigned (for p = 68,27%): n

u (d 0s ) = 1,11⋅

∑ (d k =1

− d0 )

2

0k

n (n − 1)

(39)

where: n – number of measurements or b) on the basic of the micrometer resolution, with use of the formula: u (d0 m ) =

0,01 2 3

where u(d0m) amounts to 0.00289 mm.


(40)

105


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 uw (Fm ) =

UFm ⋅ Fm

(41)

200

where: UFm – 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 Fm measured during the tensile test and for the selected range of the applied measuring head, 2) zero adjustment in the force-measuring path; 3) possible misalignment of the force applied; 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 u(Fm ) =

0,01 ⋅ Fm 3

(42)

10.2.1.3. Uncertainty budget

The information for further analysis of uncertainty is presented in the Table 6. Table 6 Uncertainty budget for calculation of the composed uncertainty for tensile strength Parameter symbol Xi



Fm

formula (42)

d2

formula (39) or (40)

Rm


4 2

πd 0

−

8Fm 3

πd 0

4Fm

Contribution into the composed standard uncertainty u(xi)

ci · u(Fm)

ci · u( d 0 )

formula (38)

2

πd 0


106


10.2.1.4. Determination of the expanded uncertainty

The relative expanded uncertainty with the expansion coefficient k = 2 at the confidence level of 1 – α = 0.95 is calculated with the following formula: U (Rm ) =

2 ⋅ u (Rm ) ⋅ 100% Rm

(43)

10.2.2. Estimation of the uncertainty for determination of the proof stress Rp0,2 (at non-proportional elongation) 10.2.2.1. Formula for the measurement result (for a single specimen)

Proof stress Rp0,2 = f (F0,2, d0 ) =

F0,2 S0

4F0,2

=

πd02

(44)

where: F0,2 – tension force that is applied during the test and then brings to permanent elongation of the specimen equal to 0.2% of the measurement length that corresponds to the extensometer base, S0 – initial cross-section of the specimen; d 0 – initial average diameter of the specimen. 10.2.2.2. 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 p0,2 ) =

∑ (c ⋅ u ) i

i

2

(45)

where: ci – sensitivity coefficients, i.e. the partial derivatives of the measurement function for the ith function component, ui – standard uncertainties for individual components. In this case cF0,2 =

4 πd02

and cdo = -

8F0,2 πd 03

, thus:

2

2

⎛ 8F ⎞ ⎛ 4 ⎞ 2 ⎟ u (F0,2 ) + ⎜ − 0,2 ⎟ u 2 (d 0 ) u (R p0,2 ) = ⎜⎜ ⎜ πd 3 ⎟ 2 ⎟ ⎝ ⎝ πd 0 ⎠ 0 ⎠


(46)


107

Determination of component standard uncertainties The uncertainty for measurement of the specimen diameter u (d0 ) – par.

10.2.1.2., equations (39–42). Overall uncertainty for measurement of the force F0,2: u (F0,2 ) = uF2 (F0,2 ) + u 2 (∆F0,2 ) + u 2 (F0,2E )

(47)

where subsequent components result from: – uncertainty of the force measurement (see formula (42)) 0,01 ⋅ F0,2 uF (F0,2 ) = 3 – recording frequency during automatic measurement F0,2(1) − F0,2(2 ) u (∆F0,2 ) = 2 3 where: F0,2(1) – the closest value of force that was higher than F0,2, F0,2(2) – the closest value of force that was lower than F0,2; – inclination of the straight line that is parallel to the linear section of the σ−ε curve that is described by the formulas: σ02E = E (ε − 0,002 ) or F0,2E =

∆F (ε − 0,002) ∆ε 2

2 2 ⎛ ∆F (ε − 0,002) ⎞ ε − 0,002 ⎞ ⎛ ∆F ⎞ ⋅ u 2 (ε ) 2 2 ⎟ u (F0,2E ) = ⎛⎜ ⋅ u ( ∆ ε ) + ⎟ ⋅ u (∆F ) + ⎜⎜ − ⎜ ⎟ ⎟ ∆ε ⎝ ⎠ ⎝ ∆ε ⎠ ( ∆ε )2 ⎝ ⎠

u (∆F ) = u 2 (Fmax ) + u 2 (Fmin )

where: where: u (Fmax ) =

0,01 ⋅ Fmax 3

and u (Fmin ) =

0,01 ⋅ Fmin 3

u (∆ε ) = u 2 (εmax ) + u 2 (εmin )

where: u (εmax ) =

K ε ⋅ εmax

3

i u (εmin ) =

K ε ⋅ εmin

u (ε ) =

3 Kε ⋅ ε

3

where: Kε – accuracy class of the applied extensometer.


108


10.2.2.3. Uncertainty budget

The informations for further analysis of uncertainty are presented in the Table 7. Table 7 Uncertainty budget for calculation of the composed uncertainty for proof stress Parameter symbol Xi



F0,2

formula (47)

d2

formula (39) or (40)

Rp0,2



4 2

πd 0 −

8F0,2 3

πd 0

4F0,2

ci · u(F0,2)

ci · u( d 2 )

formula (46)

2

πd 0


The relative expanded uncertainty with the expansion coefficient k = 2 at the confidence level of 1 – α = 0.95 is calculated with the following formula: U (Rp 0,2 ) =

2 ⋅ u (R p 0,2 ) R p 0,2

⋅ 100%

(48)

10.2.3. Estimation of the uncertainty for determining of the elongation at rupture A 10.2.3.1. Formula for the measurement result (for a single specimen)

For measurement with use of an extensometer the elongation at rupture is: A = f (εt, ε0,001) = εt - ε0,001

(49)

where: εt – deformation of the specimen at rupture; ε0,001 – deformation at the limit of elasticity Rs. 10.2.3.2. 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:



109

u ( A ) = u 2 (ε t ) + uc2 (ε 0,001 )

(50)

Determination of component standard uncertainties The uncertainty for measurement of the deformation εt is determined on the basis of the accuracy class attributable to the longitudinal extensometer Kε, with use of the B method and the rectangular distribution is adopted for this uncertainty. u (εt ) =

K ε ⋅ εt

(51)

3

The uncertainty of measurement of the deformation ε0,001: uc (ε 0,001 ) = u 2 (ε 0,001 ) + u 2 (∆ε 0,001 ) + u 2 (ε 0,001FE )

(52)

where subsequent components result from: – the accuracy class of the longitudinal extensometer: u (ε 0,001 ) =

K ε ⋅ ε 0,001

3

– recording frequency during automatic measurements ε 0,001(1) − ε 0,001( 2 ) u (∆ε 0,001 ) = 2 3 where: ε0,001(1) – the closest value of deformation that was higher than ε0,001; ε0,001(2) – the closest value of deformation that was lower than ε0,001; – inclination of the straight line that is parallel to the linear section of the elongation curve that is described by the formula: 4F0,001 σ 0,001E = E (ε 0,001 − 0,001) , hence ε 0,001 = + 0,001 , thus: πd 02 ⋅ E 2

2

2

⎛ 4 ⎞ ⎛ 8F ⎞ ⎛ 4F0,001 ⎞ 2 ⎟ ⋅ u 2 (F0,001 ) + ⎜ − 0,001 ⎟ ⋅ u 2 (d 0 ) + ⎜ − ⎟ u (ε 0,001FE ) = ⎜⎜ ⎟ ⎜ ⎟ ⎜ πd 2 ⋅ E 2 ⎟ ⋅ u (E ) 2 3 π ⋅ π d E d E ⎝ 0 ⎠ ⎝ ⎠ ⎝ ⎠ 0 0

where: u (F0,001 ) =

0,01 ⋅ F0,001 3

(see equation (42);

u(d 0 ) – see equation (39) i (40); u (E ) – see equation (56).

10.2.3.3. Uncertainty budget The informations for further analysis of uncertainty are presented in the Table 8.


110


Table 8 Uncertainty budget for calculation of the composed uncertainty for elongation at rupture Parameter symbol Xi




εt

formula (50)

ci · u(εt)

ε0,001

formula (51)

ci · uc(ε0,001)

A

εt - ε0001

formula (49)


The relative expanded uncertainty with the expansion coefficient k = 2 at the confidence level of 1 – α = 0.95 is calculated with the following formula: U (A) =

2 ⋅ u(A) ⋅ 100% A

(53)

10.2.4. Estimation of the uncertainty for determination of the Young’s (elastic) modulus E 10.2.4.1. Formula for the measurement result (for a single specimen)

The Young’s (elastic) modulus: E = f (∆F,

d 0 , ∆ε) =

∆F S 0 ⋅ ∆ε

=

4 ∆F πd02 ⋅ ∆ε

(54)

where: ∆F – difference between the upper (Fmax) and lower (Fmin) level of forces for the range of elastic deformations; S0 – initial cross-section of the specimen; d 0 – initial average diameter of the specimen ∆ε – difference between the upper (εmax) and lower (εmin) level of deformations for the range of elastic deformations (measurement with an extensometer). 10.2.4.2. 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:


111


u (E ) =

∑ (c ⋅ u ) i

i

2

(55)

where: ci – sensitivity coefficients, i.e. the partial derivatives of the measurement function for the ith function components, ui – standard uncertainties for individual components. In this case c∆F =

4 πd02 ⋅ ∆ε

; cd = 0

8 ⋅ ∆F πd03 ⋅ ∆ε

2

and c∆ε = −

2

4 ⋅ ∆F πd02 ⋅ ∆ε2

, thus:

2

⎛ ⎞ 2 ⎛ 8 ⋅ ∆F ⎞ 2 ⎛ 4 ⋅ ∆F ⎞ 2 4 ⎟⎟ u (∆F ) + ⎜⎜ − ⎟⎟ u (do ) + ⎜⎜ 2 ⎟ u (∆ε ) u (E ) = ⎜⎜ 2 3 2⎟ ⎝ πdo ⋅ ∆ε ⎠ ⎝ πdo ⋅ ∆ε ⎠ ⎝ πdo ⋅ ∆ε ⎠

(56)

Determination of component standard uncertainties The uncertainty for measurement of the specimen diameter u (do ) - par.

10.2.1.2., equations (39÷42). The uncertainty for measurement of the force ∆F: u (∆F ) = u 2 (Fmax ) + u 2 (Fmin )

(57)

where: the uncertainty values for measurement of the upper (Fmax) and lower (Fmin) level of the force (see formula (42)): u (Fmax ) =

0,01 ⋅ Fmax 3

i u (Fmin ) =

0,01 ⋅ Fmin 3

The uncertainties for measurement of the upper (εmax) and lower (εmin) levels of deformation as determined on the basis of the accuracy class of the longitudinal extensometer Kt, with use of the B method (the rectangular distribution): u (∆ε ) = u 2 (εmax ) + u 2 (εmin )

where: u (εmax ) =

K ε ⋅ εmax 3

i u (εmin ) =

K ε ⋅ εmin 3

.


(58)

112


10.2.4.3. Uncertainty budget The informations for further analysis of uncertainty are presented in the Table 9. Table 9 Uncertainty budget for calculation of the composed uncertainty for the Young’s (elastic) modulus Parameter symbol Xi



∆F

formula (57)

d0

formula(39) or (40)

∆ε

formula (58)

E



4

ci · u(∆F)

2

πd 0 −

−

8F0,2

ci · u( d 0 )

3

πd 0

4 ⋅ ∆F 2

πd 0 ⋅ ∆ε

2

4 ∆F

ci · u(∆ε)

formula (56)

2

πd 0 ⋅ ∆ε


The relative expanded uncertainty with the expansion coefficient k = 2 at the confidence level of 1 – α = 0.95 is calculated with the following formula: U (E ) =

11.

2 ⋅ u (E ) ⋅ 100% E

(59)

Recapitulation

On the basic of presented above analytic 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 0÷25 mm, serial No 102-217-9157246, manufactured by MITUTOYO Company. For reference the set of gauge plates was used, No 714390, with its calibration certificate No M11-419-578.3/2004. The verification procedure was carried out at the ambient temperature of 20 ± 0,2oC, in accordance with the instruction from the metrological surveillance IW-


113


31-11-L5, for the full measurement range of the micrometer when the gauge plates with rated sizes of Wn = 1; 1,05; 1,5; 2; 5; 8; 10; 15; 20; 25 mm were subsequently used. Verification results are collected in Table 10. Table 10 Results of the micrometer verification, [mm] Length of the standardized plate Wn

Readout / established value

readout I L1

readout II L2

readout III L3

maximum deviation from the W dimension, d

Indication error measurement uncertainty U(Ex)

d-U(Ex)

d+U(Ex)

1

1.000

1.000

1.001

0.001054

0.0009

1.9

0.1

1.05

1.051

1.050

1.051

0.000921

0.0009

1.8

0.0

1.5

1.500

1.500

1.500

0.000008

0.0009

0.9

-0.9

2

2.000

2.001

2.000

0.001060

0.0009

2.0

0.2

5

5.000

5.001

5.001

0.001048

0.0009

1.9

0.1

8

8.000

8.001

8.001

0.001006

0.0009

1.9

0.1

10

10.001

10.001

10.001

0.000948

0.0009

1.8

0.0

15

15.001

15.001

15.001

0.000920

0.0009

1.8

0.0

20

20.002

20.001

20.001

0.001911

0.0009

2.8

1.0

25

25.002

25.002

25.001

0.001798

0.0009

2.7

0.9

Verification result was accepted as passing one, because the indication error Ex equal 2,8 µm was less than he permissible limit error for micrometric instruments Eg = ±4 µm (PN-82/M-53200). In 2005 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 29 countries and 78 of the participants had the accreditation in accordance with the standard EN ISO/IEC 17025. 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 11. Based on comparison of data covered by the report [12] submitted to the Institut für Eignungsprüfung with the information presented in Table 11 the conclusion 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.


114


Table 11 Measurement results obtained by the LMST laboratory during the competence survey in 2005 No of specimens

Rp0,2 [MPa]

U(Rpo,2) [%]

Rm [MPa]

U(Rm) [%]

A [%]

U(A) [%]

E [MPa]

U(E) [%]

1

719

±3,06

796

±1,24

13,9

±1,18

185 000

±2,13

2

721

±3,01

795

±1,21

14,3

±1,17

186 400

±2,13

3

718

±3,03

792

±1,28

13,5

±1,17

187 200

±2,14

4

714

±3,01

784

±1,18

16,1

±1,17

185 300

±2,11

5

704

±3,12

781

±1,24

14,8

±1,17

186 200

±2,18

6

712

±3,01

792

±1,20

14,1

±1,18

188 700

±2,11

Average values (LBWM)

714,7

±3,04

790,0

±1,23

14,6

±1,17

186 500

±2,13

Values declared by the survey participants (the median)

712,4

± 2,0

790,0

± 1,3

16,2

± 2,0

186 500

± 4,0

Values declared by the survey organizers

691,1

± 2,3

786,2

± 1,36

15,7

± 1,2

n/a

n/a

References 1. Analiza błędów i niepewności pomiarów, www.eti.pg.gda.pl/katedry/kose/dydaktyka 2. Arendarski J.: Niepewność pomiarów. Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa 2003. 3. CWA 15261-2:2005 Measurement uncertainties in mechanical tests on metallic materials-Part 2: The evaluation of uncertainties in tensile testing. 4. Dokument EA-4/02: Wyrażanie niepewności pomiaru przy wzorcowaniu. 1999. 5. Międzynarodowy słownik podstawowych i ogólnych terminów metrologii. GUM. 1996. 6. Piotrowski J., Kostyrko K.: Wzorcowanie aparatury pomiarowej. Wydawnictwo Naukowe PWN, Warszawa 2000. 7. PN-EN ISO 10012:2004 Systemy zarządzania pomiarami. Wymagania dotyczące procesów pomiarowych i wyposażenia pomiarowego. 8. PN-EN ISO 9000:2001 Systemy zarządzania jakością. Podstawy i terminologia. 9. PN-EN ISO/IEC 17025:2005 Ogólne wymagania dotyczące kompetencji laboratoriów badawczych i wzorcujących. 10. Proficiency Test-Tensile Test Steel- Round bar at room temperature (TTSRR 2005) Final Raport. Institut für Eignungsprüfung, 2006. 11. Rozporządzenie Ministra Gospodarki, Pracy i Polityki Społecznej w sprawie w sprawie wymagań metrologicznych, którym powinny odpowiadać maszyny wytrzymałościowe do prób statycznych. 2004. 12. Wyrażanie niepewności pomiaru, Przewodnik GUM. 1999.
