L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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MER301: Engineering Reliability
LECTURE 17:
Measurement System Analysis and Uncertainty Analysis-Part 2
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis Total Error in a measurement is defined as the difference
between the Actual Value and Observed Value of Y Two general categories of error – Accuracy or Bias and
Precision Accuracy or Bias of Measurement System is defined as the difference
between a Standard Reference and the Average Observed Measurement Precision of a Measurement System is defined as the standard deviation of
Observed Measurements of a Standard Reference Total Error = Bias Error + Precision Error for independent random variables
Measurement System Error is described by Average Bias Error (Mean Shift)and a statistical estimate of the Precision Error (Variance)
Measurement System Analysis is a Fundamental Part of Every Experiment
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MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Bias or Accuracy error is a constant value and is dealt with by calibrating the measurement system
Variation or Precision error is a random variable which depends on the measurement equipment(the instruments used) and on the measurement system repeatability and reproducibility. Instrument Capability Analysis, Test/retest (repeatability)and Gage R&R studies are used to quantify the size of these errors.
222 0
0
tmeasuremenactualobserved
biasactualobserved
tmeasuremenbiasactualobserved YYY
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MER301: Engineering ReliabilityLecture 16
Gage Performance relative to required Tolerance Band
R&R less than 10% - Measurement system is acceptable. R&R 10% to 30% - Maybe acceptable - make decision
based on classification of characteristic, hardware application, customer input, etc.
R&R over 30% - Not typically acceptable. Find the problem using root cause analysis(fishbone), remove root causes
GRR is a measure of “noise” in the data GRR is a measure of “noise” in the data
%100% 15.5 Tolerance
measurmentGRR
4
Process2
2 2Measure
Process2
2 20Measure
Measure
Measure
Process
Process
Observed
Observed
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Summarizing how it all fits together…..
When a set of measurements are made, the results are always observed values,
If the actual mean and standard deviation are known then the measurement system bias and variance can be calculated
If the item being measured is a standard reference
If the measurement system bias and variance are known then the actual mean and actual variance can be calculated
mbiasactobs YYY
222 0 mactobs
actobsbias 222actobsm
222mobsact
022 obsm
0 biasactobs
biasobsact
%100% 15.5 Tolerance
measurmentGRR
L Berkley DavisCopyright 2009
Engineering ReliabilityLecture 16
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Measurement System Errors
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average
(Low End)
Observed Average
(High End)
Linearity
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MER301: Engineering ReliabilityLecture 16
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Elements that contribute to Accuracy and Precision Errors
Instrument Capability Resolution Gage Repeatability Linearity
Measurement System - Short Term (ST) Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)
Measurement System - Long Term (LT) Use Measurement System - Short Term Use Reproducibility Stability
First Two are Entitlement….Third is Reality
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Elements that contribute to Precision or Variation Errors
Instrument Capability Resolution Gage Repeatability Linearity
Measurement System- Short Term (ST) Use Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)
Measurement System - Long Term (LT) Use Measurement System - Short Term Use (ST) Reproducibility(Gage R&R) Stability(Gage R&R)
First Two are Entitlement….Third is Reality
2instrument
222, ityrepeatibilinstrumentSTtmeasuremen
22222ilityreproducibityrepeatabilinstrumentLTm
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
2222
222
0
0
0
ilityreproducibityrepeatabilinstrumenttmeasuremen
Y
ilityreproducibityrepeatabilinstrumenttmeasuremen
tmeasuremenactualobserved
biasactualobserved
tmeasuremenbiasactualobserved
tmeasuremen
YYYY
YYY
From pages119-120…
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Updating how variances all fit together
When a set of measurements are made, the results are always observed values,
If the actual mean and standard deviation are known then the measurement system bias and variance can be calculated
If the item being measured is a standard reference
If the measurement system bias and variance are known then the actual mean and actual variance can be calculated
mbiasactobs YYY 2222222 0 ilityreproducibityrepeatabilinstrumentactmactobs
222mobsact
22222 0 ilityreproducibityrepeatabilinstrumentobsm
222222ilityreproducibityrepeatabilinstrumentactobsm
)( 22222ilityreproducibityrepeatabilinstrumentobsact
biasactualobserved
biasobsact
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Elements that contribute to Accuracy and Precision Errors
Instrument CapabilityResolutionGage RepeatabilityLinearityMeasurement System- Short Term(ST) UseInstrument CapabilityEquipment CalibrationSystem Repeatability
Measurement System- Long term (LT) UseMeasurement System -Short Term(ST) Use ReproducibilityStability 2222
ilityreproducibityrepeatabilinstrumenttmeasuremen
Union CollegeMechanical Engineering
MER301: Engineering ReliabilityLecture 16
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Emissions Sampling
NOxInstrument Yactual-
NOx fromGas turbine
Cal/ZeroGases
Yobs- NOx Reading
Heated Sampling L ine
Calibration Gas
Sample Conditioning
tmeasuremenbiasactualobserved YYY
222tmeasuremenactobs
biasactobs
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Establish magnitude and sources of
measurement system error due to bias and precision errors
Tools Instrument Capability Analysis Test/Re-test – system precision/repeatability Calibration - bias “Continuous Variable” Gage R&R (Gage
Reproducibility and Repeatability) Attribute Variable Gage R&R Destructive Gage R&R
How Can we Address Accuracy and Precision Errors?
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
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Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Instrument Capability Analysis….. Resolution-smallest increment that the gage can resolve in the
measurement process. Gage should be able to resolve tolerance band into ten or more parts. Resolution Uncertainty =
Instrument Precision- measure of instrument repeatability or instrument “noise”.. Found by repeated measurements of the same test item. Uncertainty =
Linearity- consistency of the measurement system across the entire range of the measurement system. Linearity Uncertainty =
The variations are combined as follows
00 4 u
rru 4
llu 4
2222
2222
lroinstrument
lroinstrument uuuu
Union CollegeMechanical Engineering
Engineering ReliabilityLecture 16
21
Gage Performance Characteristics
Repeatability (precision)
Reproducibility
Operator B
Operator A
Stability
Time 1
Time 2
Observed Average
Accuracy (Bias)
True
True Average
True Average
Accuracy(Low End)
Accuracy(High End)
Observed Average(Low End)
Observed Average
(High End)
Linearity
2222222 0 ilityreproducibityrepeatabilinstrumentactmactobs
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Measurement System Short Term Use Includes Instrument Capability Repeatability - variation when one operator repeatedly
makes the same measurement with the same measuring equipment Test/Re-test Study
Calibration/Bias
Measurement System-Long Term Use Includes Measurement System –Short Term Use Reproducibility- variation when two or more operators make
same measurement with the same measuring equipment Stability-variation when the same operator makes the same
measurement with the same equipment over an extended period of time
2222222 0 ilityreproducibityrepeatabilinstrumentactmactobs
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
Measurement System-Short Term Use Repeatability-variation when one operator repeatedly
makes the same measurement with the same measuring equipment Test/Re-test Study
Measurement System - Long Term Use Reproducibility- variation when two or more operators
make same measurement with the same measuring equipment
Stability-variation when the same operator makes the same measurement with the same equipment over an extended period of time
2222222 0 ilityreproducibityrepeatabilinstrumentactmactobs
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 16
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Measurement System Analysis
22222
2222
222
2222
222 0
0
ilityreproducibityrepeatabilinstrumentactualobserved
ilityreproducibSTtmeasuremenLT
ityrepeatabilinstrumentST
ilityreproducibityrepeatabilinstrumenttmeasuremen
tmeasuremenactualobserved
biasactualobserved
tmeasuremenbiasactualobserved YYY
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MER301: Engineering ReliabilityLecture 17
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Measurement System Analysis Instrument Capability
Resolution Gage Repeatability Linearity
Measurement System - Short Term (ST) Use Instrument Capability Equipment Calibration Test/Re-Test Study
Measurement System - Long Term (LT) Use Measurement System (Short Term Use) Reproducibility(Gage R&R) Stability(Gage R&R)
First Two are Entitlement….Third is Reality
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MER301: Engineering ReliabilityLecture 17
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Mathematics of Measurement System Analysis
The Partial Derivative(Propagation of Errors) Method can be used to estimate variation when some X’s are related to actual product variation and other X’s are related to the measurement system (some may relate to both)
Those X’s that represent actual characteristics of the quantity Y contribute to the product variation while those associated with measurements of Y will contribute to measurement variation
nXXXXfnY ,........,,, 321
2
2
2
2
2
2
2
1
221 nX
NXXY X
Y
X
Y
X
Y
222measureXactualXobserved
L Berkley DavisCopyright 2009
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Example 17.1-Variation Equations
2
2
223
2
222
2
221
2
22
2
2
2
22
2
213212
321
2
2
2
22
2
2
2
22
2
22
2
2
22
22
21
2
2
12
21
2
2
2222
3
22
2
22
1
2
321321
321
21
321
)()(
)(
)()()(
)()()()()(
),....,,(
n
xxxxY
n
x
n
xmn
lkj
mn
lkjn
n
x
n
n
n
x
n
nxxY
xn
xxxY
mn
lkjn
xm
xl
xk
xj
Y
xm
xxxxxm
xxxx
x
xx
Y
Y
x
xx
Y
Y
x
xx
Y
Y
x
xx
Y
Y
x
Y
x
Y
x
Y
x
Y
x
YYV
xxxxxxxxfnY
n
nnn
n
n
Fuel ConsumptionExample
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MER301: Engineering ReliabilityLecture 17
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The Uncertainty Variables and
The quantity is a measure of the uncertainty in the value of the variable .It is a band wide that is a 95% confidence interval on the value of Define an Uncertainty Variable for any variable as
so that
A dimensionless Relative Uncertainty is defined as
i
X
i
X
i
XX XXX
uu iii
i
24
iii XXXu 24
iXuii XX 24iX
iX
u u
iX
iXiXY 2
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 17
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Measurement System Uncertainty
The quantity is a measure of relative uncertainty in the measurement R and is an uncertainty band wide arising from variation in the x’s. It represents a 95% CI on the size of the variation expected in the reading
The equation for relative uncertainty for a measurement system can be written as
The individual x terms can be written as a relative uncertainty uX
RuRuu RR /
R4
i
X
i
X
i
XX XXX
uu iii
i
24
2
22
2
22
22
22
12
22 ........21
R
u
X
R
R
u
X
R
R
u
X
R
R
uu nX
N
XXRR
tmeasuremenR RRY 22
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 17
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Example 17.2: Uncertainty Equations….
222222222
2
22
23
22
22
22
21
22
2
2
2
22
23
22
22
22
21
22
2
2
2
2
223
2
222
2
221
2
22
2
2
2
22
2
22
2
2
22
22
21
2
2
12
21
2
2
321321
321
321
321
321
21
)()()()()(
)4()4()4()4()4(
)()()(
),....,,(
n
n
n
n
n
xxxxY
n
xxxxY
n
xxxxY
n
xxxxY
n
x
n
nxxY
mn
lkjn
umulukuju
x
um
x
ul
x
uk
x
uj
Y
u
xm
xl
xk
xj
Y
xm
xl
xk
xj
Y
xx
Y
Y
x
xx
Y
Y
x
xx
Y
Y
x
Y
xxxxxxxxfnY
Viscometer and Triangle Examples
L Berkley DavisCopyright 2009
Viscometer Example-Lecture 17
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0.2=1.0%reproducible
Empirical Instrument Constant
Y=fn(K,densities, time)
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 17
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Example 17.3 Uncertainty in Liquid Mass Flow Rate
The mass flow rate of water through a tube is to be determined by collecting water in a beaker. The mass flow rate is calculated from the net mass of water collected divided by the time interval.
Where
Error Estimates are: Mass of full beaker, Mass of empty beaker, Collection time interval,
t
m
dt
dmm
gmgmum ff42,400
gmgmum ee 42,200 sec4.0sec2.0,10 tut
ef mmm
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 17
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Lecture 17 Summary Review of Measurement System Analysis from Lecture
16 Instrument Capability Measurement System in Short Term (ST) Use…
Instrument capability Repeatability Calibration/Bias
Measurement System in Long Term (LT) Use… Measurement System in Short term Use… Reproducibility(Gage R&R) Stability(Gage R&R)
Mathematics of Measurement System Analysis and Uncertainty Analysis
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