l berkley davis copyright 2009 mer301: engineering reliability lecture 16 1 mer301: engineering...

L Berkley DavisCopyright 2009

MER301: Engineering ReliabilityLecture 16

1

MER301: Engineering Reliability

LECTURE 17:

Measurement System Analysis and Uncertainty Analysis-Part 2



2

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



3

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



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



5

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


Engineering ReliabilityLecture 16

6

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



7

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


21

Gage Performance Characteristics


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



21



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

(High End)

Linearity



21



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

(High End)

Linearity



8

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)


2instrument

222, ityrepeatibilinstrumentSTtmeasuremen

22222ilityreproducibityrepeatabilinstrumentLTm



21



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

(High End)

Linearity



9


2222

222

0

0

0

ilityreproducibityrepeatabilinstrumenttmeasuremen

Y



biasactualobserved

tmeasuremenbiasactualobserved

tmeasuremen

YYYY

YYY

From pages119-120…



10

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



11

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




29

Emissions Sampling

NOxInstrument Yactual-

NOx fromGas turbine

Cal/ZeroGases

Yobs- NOx Reading

Heated Sampling L ine

Calibration Gas

Sample Conditioning


222tmeasuremenactobs

biasactobs



21



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

(High End)

Linearity



12

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?



21



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

(High End)

Linearity



13


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



21



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

(High End)

Linearity

2222222 0 ilityreproducibityrepeatabilinstrumentactmactobs



14


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




15


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




16


22222

2222

222

2222

222 0

0

ilityreproducibityrepeatabilinstrumentactualobserved

ilityreproducibSTtmeasuremenLT

ityrepeatabilinstrumentST



biasactualobserved




17

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)




18

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



19

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



20

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



21

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



22

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


Viscometer Example-Lecture 17

23

0.2=1.0%reproducible

Empirical Instrument Constant

Y=fn(K,densities, time)



24

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



25

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

l berkley davis copyright 2009 mer301: engineering reliability lecture 16 1 mer301: engineering...

Documents

bias of measurement

measurement system bias

measurement precision

measurement system variation

bias error precision

accuracy error

precision accuracy

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