Assessing the level of confidence for expressing extended uncertainty through control errors on the example of a model of a means of measuring ion activity


ACTA IMEKO 
ISSN: 2221-870X 
June 2021, Volume 10, Number 2, 199 - 203 

 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 199 

Assessing the level of confidence for expressing extended 
uncertainty: a model based on control errors in the 
measurement of ion activity  

Oleksandr Vasilevskyi1 

1 Vinnytsia National Technical University, 95 Khmelnitsky Shose str., 21021, Vinnytsia, Ukraine 

 

 

Section: RESEARCH PAPER  

Keywords: measurements; uncertainty; level of confidence; control errors; activity of ions; expanded uncertainty 

Citation: Oleksandr Vasilevskyi, Assessing the level of confidence for expressing extended uncertainty through control errors on the example of a model of a 
means of measuring ion activity, Acta IMEKO, vol. 10, no. 2, article 27, June 2021, identifier: IMEKO-ACTA-10 (2021)-02-27 

Section Editor: Maik Rosenberger, Ilmenau University of Technology, Germany 

Received March 31, 2020; In final form March 14, 2021; Published June 2021 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Oleksandr Vasilevskyi, e-mail: o.vasilevskyi@gmail.com  

 

1. INTRODUCTION 

There is a need to develop methods for estimating the 
uncertainty of dynamic measurements that meet international 
requirements for evaluating and expressing the quality of 
measurements, which is a topical scientific task in the field of 
metrology. International standards support the use of combined 
uncertainty uc(y) as a parameter for quantifying the uncertainty of 
the measurement result [1]-[4]. 

Although uc(y) is widely used to express uncertainty, in some 
cases, such as in trade, industry, regulatory acts or issues of health 
and safety, it is advisable to further specify a measure of 
uncertainty that determines the interval for the measurement 
result. This additional measure of uncertainty, which 
corresponds to interval estimation, is called expanded 
uncertainty. 

It is therefore relevant to determine a particular measurement 
technique to evaluate the trust level for calculating expanded 
measurement uncertainty.  

The purpose of the article is to develop a mathematical 
apparatus for estimating the confidence level in calculating the 
expanded uncertainty of measurements of ion activity, taking 
into account the manufacturer's (developer's) and consumer’s 
control errors, which will make it possible to establish the 
interval around the measurement result within which the 

majority of the distribution of values can be attributed to the 
measured value. 

2. MAIN MATERIALS OF THE RESEARCH 

In the literature [1]-[9], only partial consideration is given to 
methods of establishing a trust level for calculating the expanded 
uncertainty of measurement. A mathematical apparatus that 
would allow a reasonable confidence level in the uncertainty 
measurement is not described. It is therefore advisable to 
propose and describe a methodology for estimating trust level 
based on the control errors of the manufacturer and the 
consumer, which will make it possible to establish the value of 
the coefficient of coverage k for the calculation of the expanded 
uncertainty of measurement. As an example, the developed 
means was used to measure the activity of ions. 

The confidence level for calculating the expanded uncertainty 
is proposed based on the control errors of the manufacturer and 
the consumer using the formula 

𝐷 = 1 − 𝑃𝑛 = 1 − (𝛼 + 𝛽), (1) 

Where 𝛼 represents the control errors of the manufacturer, 𝛽 
represents the control errors of the consumer and 𝑃𝑛  represents 
the total value of control errors. 

The procedure for determining the confidence level is 
described on the basis of the combined uncertainty of the results 

ABSTRACT 
A method for estimating the level of confidence when determining the coverage factor based on control errors is proposed, using the 
example of measurements of ion activity. Using information on tolerances and uncertainty, it is possible to establish a reasonable interval 
around the measurement result, within which most of the values that can be justified are assigned to the measured value. 

mailto:o.vasilevskyi@gmail.com


 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 200 

of reusable measurements of ion activity using a measuring 
instrument built on the principle of converting voltage to 
frequency [10], [11], which is described by the following 
transformation equation: 

𝑁U/F =
𝑈pow 𝑓0 𝜏

(𝑈0 −
𝛼t(273.15 + 𝑡)

𝑛𝑎
𝑝𝑋𝑖 ) 𝑘2

 
(2) 

where рХі is the activity of the ions; 𝛼t is the temperature 
coefficient of steepness, equal to 198.4 × 10–3 °C–1; t is the 
temperature of the medium under investigation, k2 is the gain 
scaling factor (DA1 and DA2 in Figure 1), Upow is the value of 
the reference voltage of the voltage converter at a frequency of 
10 V; τ = R C is the time constant of the voltage converter at a 
frequency that is used to set the full scale output frequency range 
(R = 1 kΩ, С = 47 µF), f0 is the frequency of the quartz resonator 
microcontroller (20 МHz), U0 is the standard potential of the 
reference electrode at the initial isopotential point and na is the 
charge of the ith ion [11]. 

The electrical functional circuit of the ion activity measuring 
device based on the conversion of voltage to frequency is shown 
in Figure 1. 

The general law of uncertainty when measuring ion activity 
depends on many factors, such as the activity of interfering ions 
due to the limited properties of ion-selective electrodes or the 
presence of measurement error due to temperature, zero drift, 
instability of the power supply, etc., and it can be difficult to 
identify which factor is dominant [10]. This allows us to adopt 
the law of distribution for the centred value of the error when 
measuring the normal activity of ions, which we describe with 
the expression 

𝑝(𝑝𝑋) =
1

𝑢𝑐 (𝑝𝑋)√2𝜋
e

−
(𝑝𝑋−𝑝�̄�)2

2 𝑢𝑐
2(𝑝𝑋)  (3) 

where 𝑝�̅� is the estimated value of ion activity and uc(pX) is the 
combined uncertainty of the measurement of ion activity. 

The value of the combined uncertainty of ion activity 
measurement when using a measuring device built on the basis 
of the conversion of voltage to frequency is calculated by the 
formula 

𝑢𝑐
2(pX) = 𝑢𝐴

2(𝑝𝑋) + 𝑢𝑐𝐵
2  , (4) 

𝑢𝑐𝐵
2 = ∑ 𝑐𝑖

2𝑢𝑖
2(𝑝𝑋)

𝑁

𝑖=1

+ 

+2 ∑ ∑ 𝑐𝑖 𝑐𝑗 𝑟(𝑝𝑋, 𝑡)𝑢𝑖 (𝑝𝑋)𝑢𝑗 (𝑡)

𝑁

𝑖=2

𝑁−1

𝑗=1

 , 

(5) 

where uA(pX) is the evaluation of type A standard uncertainty, 
ui(pX) represents components of type B uncertainty for 
measurements of ion activity with the respective sensitivity 
coefficients сi, uj(t) represents components of type B uncertainty 
for measurements of temperature with the corresponding 
sensitivity coefficients cj and r(pX, t) is the correlation coefficient 
between ion activity (pX) and temperature (t) [10]-[14]. 

Taking the transformation equation (2) into account, the type 
B combined uncertainty is determined by the formula 

𝑢𝑐𝐵1
2 = (

𝜕𝑁U/F

𝜕𝑝𝑋𝑖
)

2

[𝑢𝐵𝐸
2 + 𝑢𝐵𝑈/𝐹

2 ] + (6) 

+ (
𝜕𝑁U/F

𝜕𝑈0
)

2

𝑢𝐵𝛩𝑈0
2 + (

𝜕𝑁U/F

𝜕𝑡
)

2

𝑢𝐵𝑃
2 + 

+ (
𝜕𝑁U/F

𝜕𝑈𝑝𝑜𝑤
)

2

𝑢𝐵𝑈pow
2  , 

𝑢𝑐𝐵
2 = 𝑢𝑐𝐵1

2 + 2
𝜕𝑁U/F

𝜕𝑝𝑋𝑖
𝑢𝑐𝐵1

𝜕𝑁𝑡U/F

𝜕𝑡
𝑢𝑐𝐵𝑡 × 

×
∑ (𝑡𝑖𝑙 − 𝑡𝑖 )(𝑝𝑋𝑗𝑙 − 𝑝𝑋𝑗 )

𝑛𝑖𝑗
𝑙=1

√∑ (𝑡𝑖𝑙 − 𝑡𝑖 )
2𝑛𝑖𝑗

𝑙=1
∑ (𝑝𝑋𝑗𝑙 − 𝑝𝑋𝑗 )

2𝑛𝑖𝑗
𝑙=1

 

(7) 

where 𝑁 𝑡𝑈/𝐹 =
4 𝑈pow 𝑓0 𝜏

𝑘1 𝐸𝑠 𝛼𝑡 𝑡
 is the equation of transformation for 

measurements of temperature (Figure 1) [10], Es = І R0 is the 
sensor supply voltage (R0 is the sensor resistance at a temperature 
of 0 °С) and k1 is the gain scaling factor (DA3 in Figure 1),  

𝜕𝑁U/F

𝜕𝑝𝑋𝑖
=

𝛼𝑡 (273.15 + 𝑡) 𝑈pow  𝑓0 𝜏

[𝑈0 −
𝛼𝑡 (273.15 + 𝑡)

𝑛𝑎
𝑝𝑋𝑖 ]

2

𝑘2𝑛𝑎

= 12.98 𝑝𝑋 −1 

is the sensitivity factor for ion activity at a temperature of 25 °С, 

𝜕𝑁U/F

𝜕𝑈0
= −

𝑈pow 𝑓0 𝜏

[𝑈0 −
𝛼𝑡 (273.15 + 𝑡)

𝑛𝑎
𝑝𝑋𝑖 ]

2

𝑘2

= −0.22 V−1 
 

is the coefficient of the voltage sensitivity of the standard 
potential of the reference electrode, 

𝜕𝑁U/F

𝜕𝑡
=

𝛼𝑡  𝑝𝑋𝑖  𝑈pow 𝑓0 𝜏

[𝑈0 −
𝛼𝑡 (273.15 + 𝑡)

𝑛𝑎
𝑝𝑋𝑖 ]

2

𝑘2𝑛𝑎

= 0.3 °C−1 

is the coefficient of sensitivity at an additional measured 
temperature, 

𝜕𝑁U/F

𝜕𝑈pow
=

𝑓0𝜏

(𝑈0 −
𝛼𝑡 (273.15 + 𝑡)

𝑛𝑎
𝑝𝑋𝑖 ) 𝑘2

= −9.08 V−1 

is the coefficient of the voltage sensitivity of the stable power 
supply source of the voltage-to-frequency converter (VFC), nij is 
the number of pairwise measurements of temperature and ion 

activity, 𝑢𝐵𝐸 = 𝛾
𝑝𝑋max

100%√3
−3 pX is the type B uncertainty caused 

by the presence of a primary electrode with a low accuracy class 

(  = 0.7 %), 𝑢𝐵𝛩𝑈0 =
𝛩𝑈0

√3
= 0.95 mV is the type B uncertainty 

caused by the instability (𝛩𝑈0 = 1.65 mV) of the standard 

potential of the reference electrode, 𝑢В𝑝 =
𝛥t Ubiasmax

𝑈Pout√3
≈ 2.89 ⋅

10−3 °С is the type B uncertainty caused by the existence of a 
bias voltage (Ubiasmax 𝑈Pout⁄ = 0.2·10

–3) of the operational 

amplifier when the temperature deviates (𝛥𝑡 = 5 °С) from the 

nominal value, 𝑢𝐵𝑈POW
=

𝛩𝑈𝑃

√3
≈ 1.16 mV is the type B 

uncertainty caused by instability in the reference voltage power 

supply source (𝛩𝑈𝑃  = 2 mV) and 𝑢𝐵VFC =
𝛿nonl𝑝𝑋max

100%√3
−6  pX is the 

type B uncertainty caused by the nonlinearity (𝛿nonl) of the 
voltage converter (AD650) [10]-[14]. 

The combined uncertainty values are calculated in [10] and 
[14], and the maximum uncertainty value for measuring the 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 201 

activity of phosphate ions is 17.21 × 10–3 pX in the range of 
measurements from 6 to 0.3 pX. 

A compatible two-dimensional confidence level density 
when measuring the activity of ions, taking into account the 
allowable deviation of the measurement error ε, which is 
established by the consumer, is described by the expression 

𝑝(𝑝𝑋, ) =
1

2 π ⋅ 𝑢𝑐 (𝑝𝑋) ⋅ 𝑢
e

− 
(𝑝𝑋−𝑝�̄�)2

2 𝑢𝑐
2(𝑝𝑋)

 − 
(Δ )2

2 𝑢𝜀
2

. (8) 

In most practical cases, as noted in [9] and [15], the 
admissible deviation ε of the parameter controlled by the user, 
depending on the combined uncertainty of the measurement 
results uc(y), is determined by the formula 

[%] =
6 𝑢𝑐 (𝑦)

𝑑
100 =

6 𝑢𝑐 (𝑦)

𝑋max − 𝑋min 
100 , (9) 

where d is the width of the tolerance field determined by the 
values of the upper Xmax and lower Xmin range of measurement 
for the physical quantity.  

By substituting the values of the maximum combined 
uncertainty when measuring ion activity uc(pX) = 17.21·10–3 pX 
and the values of the upper (Xmax = 6 pX) and the lower (Xmin = 
0.3 pX) ranges of the measurement limit in equation (9), we 
obtain the value of the permissible error set by the consumer, 
which is ε = 1.81 %. 

The uncertainty uε of the tolerance field of the controlled 
parameter at which the result can be considered reliable in 

practice is recommended to be equal to 6uc(y) [9], [13], [15]-[18]. 
Thus, uε = 6 × uc(y) = 6 × 17.21·10–3 = 103.26·10–3 pX. 

Taking into account expression (8), the manufacturer’s 
control errors α are estimated by the formula 

𝛼 = ∫ [∫
e

− 
(Δ𝑝𝑋)2

2 𝑢𝑐
2(𝑝𝑋)

 − 
(Δ )2

2 𝑢𝜀
2

2 π ⋅ 𝑢𝑐 (𝑝𝑋) ⋅ 𝑢

𝑋𝑚𝑖𝑛

−∞

+
𝑋𝑚𝑎𝑥

𝑋𝑚𝑖𝑛

 

+ ∫
e

− 
(Δ𝑝𝑋)2

2 𝑢𝑐
2(𝑝𝑋)

 − 
(Δ )2

2 𝑢𝜀
2

2 π ⋅ 𝑢𝑐 (𝑝𝑋) ⋅ 𝑢
𝑑𝛥

∞

𝑋𝑚𝑎𝑥

] 

(10) 

and the consumer’s control errors 𝛽 are estimated by the formula 

𝛽 = ∫ ∫
e

− 
(Δ𝑝𝑋)2

2 𝑢𝑐
2(𝑝𝑋)

 − 
(Δ )2

2 𝑢2

2 π ⋅ 𝑢𝑐 (𝑝𝑋) ⋅ 𝑢
𝑑Δ𝑝𝑋𝑑Δ +

𝑋max−∆𝑝𝑋

𝑋min+∆𝑝𝑋

𝑋min

−∞

 

+ ∫ ∫
e

− 
(Δ𝑝𝑋)2

2 𝑢𝑐
2(𝑝𝑋)

 − 
(Δ )2

2 𝑢2

2 π ⋅ 𝑢𝑐 (𝑝𝑋) ⋅ 𝑢
𝑑Δ𝑝𝑋𝑑Δ

𝑋max−∆𝑝𝑋

𝑋min+∆𝑝𝑋

∞

𝑋max

 . 

(11) 

The admission field is the value of the measured value ΔpX. 
In this case, this is the activity of the pX ions, which is defined 
by the formula 

RS 232 PC

RD

TD

MCU

MSP430F123

CA1

CA0
19

20

Vcc

Vss

2

4

XOUT

XIN

5

6

TA0

TA1

TA2

22

23

24
UTxD

URxD

15

16

RST
7

+ 5 V

P2.0

P2.1

8

9

20

МГц

Vss

Vdd
+ 5 V

2

9

5
H

CLK

DI

H
T

1
6

1
1

4

3

P3.0

P3.1

11

12

P3.3

P3.1

P3.2

P3.3

13

14C6

C7

DD4

1

DD5

pX

U

3

2

4

+ 5 V

7
6

100M

LMC 6001

- 5 V

3

2

4

+ 5 V

7

AD820

6

- 5 V

18k

6k

0

1

2

3

А0

А1 АGND

100k

100n

3

4

7

6

U/F

AD650

2

1

8

5

+ V in

R

Ct

Ct

F out

LC

Vcc

Vss

561КП1

+ 5 V

Y

DA2

DA4

DD2

R1

R8

R9

R3

R6

R12

R 10

C1

C3

T

UT

3

2

4

+ 5 V

7
AD820

6

R2

3

4

7

6

U/F

AD650

2

1

8

5

+ V in

R

Ct

Ct

F out

LC

Vcc

Vss

+ 5 V

DD1

R7

R5

R11

C2
C4

R4

R5

IN

GND

1171

CП42
2

1

3

DD2

R13

C8

R6

DD3

DA1

DA3

C1

C5

 

Figure 1. The functional electrical circuit of the device for measuring ion activity. 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 202 

Δ𝑝𝑋 =
𝑝�̄�

100
𝛿max , (12) 

where 𝑝�̅� is the estimation of the ion activity in the upper 
measurement range, 𝛿max is the maximum relative measurement 
error (for measurements of ion activity, this value is 0.7 % in the 
range of measurements from 6 to 0.3 pX). 

Consequently, when calculating the control errors of the 
manufacturer and the consumer, the field of admission ∆pX can 
be calculated by (12), which, with the corresponding numerical 
values, is equal to 2.1·10–3 pX. 

By substituting calculated tolerances in the formula for 
estimating the control errors of the manufacturer (10) and the 
consumer (11) and solving them using the Maple 12 package, we 

obtain the following numerical values: 𝛼 = 0.24·10–67 (𝛼 ≈ 0), 
and 𝛽 = 0.00196. The total value of the control errors is Pn = 
0.00196, and the confidence level for calculating the expanded 
uncertainty of the measurement, according to formula (1) will be 
equal to D = 1 − Pn = 1 − 0.00196 = 0.998. 

Figure 2 illustrates the change in the manufacturer’s and 
consumer’s control errors depending on the parameter μ, which 
is equal to uε/uc(pX). This parameter establishes the relationship 
between the permissible uncertainty of uε, which is specified by 
the consumer (via normative documents), and the combined 
uncertainty of uc(pX), which is set by the manufacturer. The 
figure shows uε < uc(pX), uε ≈ uc(pX) and uε > uc(pX).  

Thus, based on the constructed characteristic of the change 
in reliability (Figure 2), which is obtained by calculating the 
consumer and manufacturer’s control errors, a range can be 
calculated within which the majority of the distribution of values 
that can be attributed to the measured value is likely to be located, 
depending on the value of the accepted tolerance for the 
monitored parameter. 

As can be seen from Figure 2, with a tolerance of six 
combined uncertainties, the confidence level is 99.8 %. 

Given the level of confidence based on control errors, the 
expanded uncertainty can be calculated from the formula  

𝑈 = 𝑘𝑝 ⋅ 𝑢𝑐 (𝑦) = 𝑘99.8 ⋅ 17.21 ⋅ 10
−3 , (13) 

where kp is the coverage ratio for the established confidence level 
(p = D), which is taken from the Student's table. 

Thus, with 30 degrees of freedom and a confidence 
probability of 99.8 %, the coverage factor kp is 3.385, and with 
10 degrees of freedom and a probability of 99.8 %, it is 4.14. 
Accordingly, if these coverage factor values are entered into (13), 
the expanded uncertainty will be U99.8 = 71.25·10-3 pX for 10 
degrees of freedom and U99.8 = 58.26·10–3 pX for 30 degrees of 
freedom. 

3. CONCLUSIONS 

The work describes the characteristics of the change in the 
manufacturer’s and consumer’s control errors, on the basis of 
which a general characteristic of the change in the control errors 
that occurs during the measurement of ion activity is obtained. 
Using this characterisation, it is possible to determine graphically 
the confidence level within which most of the distribution of 
values obtained by measuring the activity of ions is probably 
located. 

The obtained nomograph makes it possible to determine 
with a high probability the confidence level at a given tolerance 
for the controlled parameter to determine the coverage factor in 
calculating the extended uncertainty of ion activity measurement. 

The described approach for determining confidence 
probability on the basis of control errors can be applied to any 
type of measurements, provided a separate calculation is 
performed to characterise the control errors of the manufacturer 
of the measuring instrument and the consumer (i.e. errors of the 
first and second kind). 

 

Figure 2. Characteristics of the change of control errors depending on the parameter μ. 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 203 

REFERENCES 

[1] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Evaluation of 
measurement data – guide to the expression of uncertainty in 
measurement, Joint Committee for Guides in Metrology, Bureau 
International des Poids et Mesures JCGM 100:2008 (2008).  

[2] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Evaluation of 
measurement data – supplement 2 to the 'Guide to the expression 
of uncertainty in measurement' – extension to any number of 
output quantities, Joint Committee for Guides in Metrology, 
Bureau International des Poids et Mesures JCGM 102:2011 
(2011). 

[3] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Evaluation of 
measurement data – an introduction to the ‘Guide to the 
expression of uncertainty in measurement’ and related documents, 
Joint Committee for Guides in Metrology, Bureau International 
des Poids et Mesures JCGM 104:2009 (2009). 

[4] ISO/IEC, Uncertainty of measurement – part 1: introduction to 
the expression of uncertainty in measurement, ISO/IEC Guide 
98-1:2009 (2009). 

[5] O. M. Vasilevskyi, Calibration method to assess the accuracy of 
measurement devices using the theory of uncertainty, Int. J. 
Metrol. Qual. Eng. 5(4) (2014) 403, 9 pp.  
DOI: 10.1051/ijmqe/2014017  

[6] IEC, Application of uncertainty of measurement to conformity 
assessment activities in the electrotechnical sector, IEC GUIDE 
115-2007 (2007). 

[7] ISO/IEC, General requirements for the competence of testing 
and calibration laboratories, ISO/IEC 17025:2005 (2005). 

[8] O. M. Vasilevskyi, Methods of determining the recalibration 
interval measurement tools based on the concept of uncertainty, 
Technical Electrodynamics 6 (2014) pp. 81-88. 

[9] S. F. Beckert, W. S. Paim, Critical analysis of the acceptance 
criteria used in measurement systems evaluation, Int. J. Metrol. 
Qual. Eng. 8 (2017) art. 23.   
DOI: 10.1051/ijmqe/2017016 

[10] O. M. Vasilevskyi, V. M. Didych, Elements of the Theory of 
Constructing Potentiometric Means of Measuring Control of Ion 
Activity with an Increased Probability [Monograph], Vinnytsia: 
VNTU, 2013. Online [Accessed 2 June 2021]  
http://vasilevskiy.vk.vntu.edu.ua/file/d23a8f23c4fdc081c30ebb0
9bc7182e6.pdf  

[11] V. M. Didych, O. M. Vasilevskyi, V. O. Podzharenko, 
Potentiometer facilities of ions activity measurement of humus 
elements in soil, Visnyk of Vinnytsia Politechnical Institute 5 
(2008) pp. 5-10. Online [Accessed 2 June 2021]  
https://visnyk.vntu.edu.ua/index.php/visnyk/article/view/639/
638 

[12] O. M. Vasilevskyi, Metrological characteristics of the torque 
measurement of electric motors, Int. J. Metrol. Qual. Eng. 8 (2017) 
7, 9 pp.  
DOI: 10.1051/ijmqe/2017005  

[13] O. M. Vasilevskyi, V. M. Didych, A. Kravchenko, M. Yakovlev, I. 
Andrikevych, D. Kompanets, Y. Danylyuk, W. Wójcik, A. 
Nurmakhambetov, Method of evaluating the level of confidence 
based on metrological risks for determining the coverage factor in 
the concept of uncertainty, Proceedings Volume 10808, Photonics 
Applications in Astronomy, Communications, Industry, and 
High-Energy Physics Experiments, WUT Wilga Resort, Poland, 2 
– 10 June 2018. 
DOI: 10.1117/12.2501576 

[14] O. M. Vasilevskyi, V. Y. Kucheruk, V. V. Bogachuk, K. 
Gromaszek, W. Wójcik, S. Smailova, N. Askarova, The method of 
translation additive and multiplicative error in the instrumental 
component of the measurement uncertainty, Proc. SPIE 10031, 
Photonics Applications in Astronomy, Communications, 
Industry, and High-Energy Physics Experiments, WUT Wilga 
Resort, Poland, 29 May – 6 June 2016. 
DOI: 10.1117/12.2249195  

[15] D. J. Wheeler, An honest gauge R&R study, 2006 ASQ/ASA Fall 
Technical Conference, Columbus, US, 12 – 13 October 2006. 
Online [Accessed 2 June 2021]  
http://www.spcpress.com/pdf/DJW189.pdf  

[16] O. M. Vasilevskyi, P. I. Kulakov, I. A. Dudatiev, V. M. Didych, 
A. Kotyra, B. Suleimenov, A. Assembay, A. Kozbekova, 
Vibration diagnostic system for evaluation of state interconnected 
electrical motors mechanical parameters, Proc. SPIE 10445, 
Photonics Applications in Astronomy, Communications, 
Industry, and High Energy Physics Experiments, WUT Wilga 
Resort, Poland, 29 May – 6 June 2016.   
DOI: 10.1117/12.2280993 

[17] O. M. Vasilevskyi, V. Y. Kucheruk, V. V. Bogachuk, 
K. Gromaszek, W. Wójcik, S. Smailova, N. Askarova, The method 
of translation additive and multiplicative error in the instrumental 
component of the measurement uncertainty, Proc. SPIE 10031, 
Photonics Applications in Astronomy, Communications, 
Industry, and High-Energy Physics Experiments, WUT Wilga 
Resort, Poland, 29 May – 6 June 2016.   
DOI: 10.1117/12.2249195 

[18] V. O. Podzharenko, O. M. Vasilevskyi, Diagnostics of technical 
condition of electromechanical systems for the logarithmic 
decrement, Proceedings of Donetsk National Technical 
University (2005). 

 

 

https://doi.org/10.1051/ijmqe/2014017
https://doi.org/10.1051/ijmqe/2017016
http://vasilevskiy.vk.vntu.edu.ua/file/d23a8f23c4fdc081c30ebb09bc7182e6.pdf
http://vasilevskiy.vk.vntu.edu.ua/file/d23a8f23c4fdc081c30ebb09bc7182e6.pdf
https://visnyk.vntu.edu.ua/index.php/visnyk/article/view/639/638
https://visnyk.vntu.edu.ua/index.php/visnyk/article/view/639/638
https://doi.org/10.1051/ijmqe/2017005
https://doi.org/10.1117/12.2501576
http://vasilevskiy.vk.vntu.edu.ua/file/ref/bf7f9857715c0be9f72e906112b07474.pdf
http://vasilevskiy.vk.vntu.edu.ua/file/ref/bf7f9857715c0be9f72e906112b07474.pdf
https://doi.org/10.1117/12.2249195
http://www.spcpress.com/pdf/DJW189.pdf
https://doi.org/10.1117/12.2280993
http://vasilevskiy.vk.vntu.edu.ua/file/ref/bf7f9857715c0be9f72e906112b07474.pdf
http://vasilevskiy.vk.vntu.edu.ua/file/ref/bf7f9857715c0be9f72e906112b07474.pdf
https://doi.org/10.1117/12.2249195