Transcript MER035 Lecture 1

MER301: Engineering
Reliability
LECTURE 17:
Measurement System Analysis and
Uncertainty Analysis-Part 2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
1
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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
2
Measurement System Analysis
Y observed  Y actual    bias   Y measuremen
t
 observed   actual    bias  0



2
observed

2
actual
0
2
measuremen t
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.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
3
Gage Performance relative to
required Tolerance Band
% GRR 
5 . 15  
 Me asure
 Proc ess
2
 Me asure
2
mea su rmen t
Tolerance
 100 %
 Observe d
2
 Process
 Measure
 Obse rv ed
 Proc ess
2
 Me asure
2
 20
 Process
 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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
4
Summarizing how it all fits together…..
When a set of measurements are made, the results are always
observed values, Y obs  Y act    bias   Y m

 obs   act    bias  0


2
obs

2
act
 0
% GRR 
2
m
mea su rmen t
Tolerance
 100 %
If the actual mean and standard deviation are known then the
measurement system bias and variance can be calculated
  bias   obs   act


5 . 15  
2
2
 m2   obs
  act
If the item being measured is a standard reference
2
 m2   obs
0
If the measurement system bias and variance are known then the
actual mean and actual variance can be calculated
2
2
 act
  obs
  m2
 act   obs    bias
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
5
Measurement System Errors
True
Accuracy
(Bias)
Time 2
Time 1
Repeatability
(precision)
Observed Average
True
Average
Operator A
Observed
Average
(Low End)
Accuracy
(Low End)
Stability
True
Average
Observed
Average
(High End)
Accuracy
(High End)
Operator B
Reproducibility
L Berkley Davis
Copyright 2009
Linearity
Engineering Reliability
Lecture 16
6
Elements that contribute to Accuracy
and Precision Errors
 Instrument Capability



Resolution
Gage Repeatability
Linearity
G a g e P e rfo rm a n c e
C h a ra
A c cc
u rate
c y ris tic s
T ru e
(B ia s )
T im e 2
T im e 1
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
S ta b ility
O b se rve d
A v e ra ge
T ru e
(L o w E n d )
A v e ra ge
T ru e
A v e ra ge
A ccu ra cy
(L o w E n d )
O p e ra to r A
O b se rve d
A v e ra ge
(H ig h E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
R e p ro d u c ib ility
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g
L in e a rity
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
21
 Measurement System - Short Term (ST)
 Instrument Capability
 Equipment Calibration(Bias)
 Test/Re-Test Study(Repeatability)
G a g e P e rfo rm a n c e
C h a ra
A c cc
u rate
c y ris tic s
T ru e
(B ia s )
T im e 2
T im e 1
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
S ta b ility
O b se rve d
A v e ra ge
T ru e
(L o w E n d )
A v e ra ge
T ru e
A v e ra ge
A ccu ra cy
(L o w E n d )
O p e ra to r A
O b se rve d
A v e ra ge
(H ig h E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
R e p ro d u c ib ility
L in e a rity
 Measurement System - Long Term (LT) Use
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g
 Measurement System - Short Term Use
 Reproducibility
 Stability
L Berkley Davis
Copyright 2009
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
21
G a g e P e rfo rm a n c e
C h a ra
A c cc
u rate
c y ris tic s
T ru e
(B ia s )
T im e 2
T im e 1
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
O b se rve d
A v e ra ge
T ru e
(L o w E n d )
A v e ra ge
O p e ra to r A
A ccu ra cy
(L o w E n d )
S ta b ility
T ru e
A v e ra ge
O b se rve d
A v e ra ge
(H ig h E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
R e p ro d u c ib ility
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g
Engineering Reliability
First Two are MER301:
Entitlement….Third
is Reality
Lecture 16
L in e a rity
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
21
7
Elements that contribute to Precision or
Variation Errors
 Instrument Capability



Resolution
2
Gage
 Repeatability
instrument
Linearity
G a g e P e rfo rm a n c e
C h a ra
A c cc
u rate
c y ris tic s
T ru e
(B ia s )
T im e 2
T im e 1
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
T ru e
A v e ra ge
O b se rve d
A v e ra ge
(L o w E n d )
A ccu ra cy
(L o w E n d )
O p e ra to r A
S ta b ility
T ru e
A v e ra ge
O b se rve d
A v e ra ge
(H ig h E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
R e p ro d u c ib ility
L in e a rity
 Measurement System- Short Term (ST) Use
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g
 Instrument Capability
2
2
2
 
Equipment
Calibration(Bias)




measuremen t , ST
instrument
repeatibil
 Test/Re-Test Study(Repeatability)
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
21
ity
 Measurement System - Long Term (LT) Use
 Measurement System - Short Term Use (ST)
2
2
2
2
2








 Reproducibility(Gage
R&R)
m
LT
instrument
repeatabil ity
reproducib
 Stability(Gage R&R)
L Berkley Davis
Copyright 2009
First Two are Entitlement….Third
is Reality
MER301: Engineering Reliability
Lecture 16
ility
8
Measurement System Analysis
Y observed  Y actual    bias   Y measuremen
t
 observed   actual    bias  0
 observed   actual  0   measuremen
2
2
 Y measuremen
t
 Y
0
mea suremen t
 measuremen
2
L Berkley Davis
Copyright 2009
2
  Y instrument   Y repeatabil
ity
  Y reproducib
ility
From pages
119-120…
  instrument   repeatabil
2
t
t
2
MER301: Engineering Reliability
Lecture 16
  reproducib
2
ity
ility
9
Updating how variances all fit together

When a set of measurements are made, the results are always
observed values, Y obs  Y act    bias   Y m
2
2
2
2
2
 obs
  act
 0   m2   act
  instrument
  repeatabil


2
reproducib ility
If the actual mean and standard deviation are known then the
measurement system bias and variance can be calculated
m
2

 observed   actual    bias
2
2
2
2
  obs   act   instrument   repeatabil
  reproducib
2
ity
If the item being measured is a standard reference
2
2
2
 m2   obs
 0   instrument
  repeatabil

ity
ity

ility
2
reproducib ility
If the measurement system bias and variance are known then the
actual mean and actual variance can be calculated
2
2
 act
  obs
  m2
2
2
2
2
 act
  obs
 ( instrument
  repeatabil
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
ity
 act   obs    bias
2
  reproducib
)
ility
10
Elements that contribute to Accuracy
and Precision Errors
G a g e P e rfo rm a n c e
C h a ra
A c cc
u rate
c y ris tic s
T ru e
(B ia s )
T im e 2
T im e 1
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
T ru e
A v e ra ge
S ta b ility
O b se rve d
A v e ra ge
(L o w E n d )
T ru e
A v e ra ge
A ccu ra cy
(L o w E n d )
O p e ra to r A
O b se rve d
A v e ra ge
(H ig h E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
 Instrument Capability
R e p ro d u c ib ility
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g
Resolution
Gage Repeatability
Linearity
Heated Sampling Line
Cal/Zero
Gases
Instrument Capability
Equipment Calibration
System Repeatability
NOx
Instrument
Calibration Gas
Sample
Conditioning
Yactual-
NOx from
Gas turbine
 Measurement SystemLong term (LT) Use
 obs   act    bias
2
2
2
 obs
  act
  measuremen
t
Yobs- NOx Reading
Y observed  Y actual    bias   Y measuremen
Measurement System -Short
Term(ST) Use
Reproducibility
Stability
2
 2
L Berkley Davis
Copyright 2009
21
Emissions Sampling
Measurement SystemShort Term(ST) Use
measuremen t
L in e a rity
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
Union College
Mechanical Engineering
instrument

MER301: Engineering Reliability
Lecture 16
t
MER301: Engineering Reliability
Lecture 16
2
repeatabil ity
29

2
reproducib ility
11
How Can we Address Accuracy
and Precision Errors?
 Establish magnitude and sources of
measurement system error due to bias
and precision errors
 Tools
G a g e P e rfo rm a n c e
C h a ra
A c cc
u rate
c y ris tic s
T ru e
(B ia s )
T im e 2
T im e 1
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
T ru e
A v e ra ge
O b se rve d
A v e ra ge
(L o w E n d )
S ta b ility
T ru e
A v e ra ge
O b se rve d
A v e ra ge
(H ig h E n d )
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




L Berkley Davis
Copyright 2009
O p e ra to r A
A ccu ra cy
(L o w E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
R e p ro d u c ib ility
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g
MER301: Engineering Reliability
Lecture 16
L in e a rity
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
21
12
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 = u   4  
0
0
Instrument Precision- measure of instrument repeatability or
instrument “noise”.. Found by repeated measurements of the
same test item. Uncertainty = u r  4   r
Linearity- consistency of the measurement system across the
G a g e P e rfo rm a n c e
entire range of the measurement system.
C h a ra c te ris tic s
Linearity Uncertainty = u l  4   l
The variations are combined as follows
T ru e

A c c u ra c y
(B ia s )
R e p e a ta b ility
(p re c is io n )
O b se rve d A v e rag e
T ru e
A v e ra ge
u
2
instrument

2
instrument
 u o  u r  u l
2

2
o
2

2
r

2
2
2
2
2
 obs
  act
 0   m2   act
  instrument
  repeatabil
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
2
2
l
ity
T im e 2
T im e 1
O p e ra to r A
O b se rve d
A v e ra ge
(L o w E n d )
A ccu ra cy
(L o w E n d )
S ta b ility
T ru e
A v e ra ge
O b se rve d
A v e ra ge
(H ig h E n d )
A ccu ra cy
(H ig h E n d )
O p e ra to r B
R e p ro d u c ib ility
U n io n C o lle g e
M e c h a n ic a l E n g in e e rin g

L in e a rity
E n g in e e rin g R e lia b ility
L e c tu r e 1 6
2
reproducib ility
21
13
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

2
obs
 Includes Measurement System –Short Term Use
2
2
2
2
2
2



0










 Reproducibilityvariation
when tworepeatabil
or more
act
m
act
instrument
ity operators
reproducib ility
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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
14
Measurement System Analysis
 Measurement System-Short Term Use
 Repeatability-variation when one operator repeatedly
2
2
2
2
2
2
 obs
  act
 0 same
  m2 measurement
 act
  instrument
  repeatabil


makes
the
with
the same
ity
reproducib ility
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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 16
15
Measurement System Analysis
Y observed  Y actual    bias   Y measuremen
t
 observed   actual    bias  0
2

 observed
0
2
actual


2
measuremen t

2
ST

2
instrument

2
LT

2
measuremen t
2

 observed
L Berkley Davis
Copyright 2009

2
instrument
2
actual

2
measuremen t
2
repeatabil ity

2
reproducib ility
2
repeatabil ity


2
ST

2
instrument
2
reproducib ility

2
repeatabil ity
MER301: Engineering Reliability
Lecture 16

2
reproducib ility
16
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)
L Berkley Davis
Copyright 2009
First Two areMER301:
Entitlement….Third
is Reality
Engineering Reliability
Lecture 17
17
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)
Y  fn  X 1 , X 2 , X 3 ,........, X n 
 Y 
2
 
 Y2  


X
1 


2
X1
 Y
 
 X 2
2

 


2
X2
 Y
         
 X N
2

 


2
Xn
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
2
 observed

L Berkley Davis
Copyright 2009
2
X  actual

MER301: Engineering Reliability
Lecture 17
2
X  measure
18
Example 17.1-Variation Equations
Y  fn ( x 1 , x 2 , x 3 ,.... x n )  x 1  x 2k  x 3l    x nm
j
Y
V (Y )   Y2  (
 Y2
Y
2
x n2
Y
2
 Y2
Y

2
x 12
Y
(
(
2
Y
x n
Y
 x1
)2 
 j2 
 x1

)2 
)2 
2
xn

x
2
x1
2
1
L Berkley Davis
Copyright 2009

x 12

x n2

2
x1
 k2 
2
x1
(
x 22
Y
Y
x 2
(
2
)2 
Y
x 2
(
2
x2
)2 

x 22
x n2
( x 1  x 2k  x 3l    x nm ) 2
j

x
2
x2
2
2
 l2 

x
2
x3
2
3
2
x2
Y
x3
)2 
  
2
x3
   (
x n2
Y
2
(
Y
x n
Y
x n
)2 
)2 

   m 2 

x
2
xn
2
n
MER301: Engineering Reliability
Lecture 17
2
xn
x n2
 ( m  x  x 2k  x 3l    x nm 1 ) 2 
j
1
2
xn

2
xn
 m2 

x n2
2
xn
x n2
Fuel Consumption
Example
19
The Uncertainty Variables u  and
u
 The quantity u X i is a measure of the uncertainty in
the value of the variable X i .It is a band 4   X i   2  
wide that is a 95% confidence interval on the value
of X i Define an Uncertainty Variable for any
variable X i as
u X i  4  
so that
Xi
 2 
Y  X i  2 
Xi
Xi
 A dimensionless Relative Uncertainty is defined as
uXi 
L Berkley Davis
Copyright 2009
u X i
Xi

4 
Xi

Xi
MER301: Engineering Reliability
Lecture 17
 2 
Xi
Xi
20
Xi
Measurement System Uncertainty
u R  u R / R is a measure of relative
 The quantity
uncertainty in the measurement R and u R is an
uncertainty band 4   R wide arising from variation in the
x’s. It represents a 95% CI on the size of the variation
expected in the reading Y  R  2   R  R  2   measuremen t
 The equation for relative uncertainty for a measurement
system can be written as
u R2 
u R2
R2
 R
 
 X 1
2
 u X 1   R


 R 2   X
2


2
2
 R
 u X 2



........

 R2
 X
N


2
2
 u X n

 R2

2
 The individual x terms can be written as a relative
uncertainty uX
u X
4 X
 2  X
uX 


Xi
Xi
Xi
i
i
i
i
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 17
21
Example 17.2: Uncertainty Equations….
Y  fn ( x 1 , x 2 , x 3 ,.... x n )  x 1  x 2k  x 3l    x nm
j
 Y2
Y

2
Y
 Y2
Y
x 12
(
2
 j2 

2
x
(4   Y ) 2
Y
 x1
2
x1
2
1
)2 
2
2
x1
x 12

x
 j2 
 j2 

 k2 
(4  
2
( u Y ) 2
Y
Y
( u x1 ) 2
x 12
x1
x 22

Y
2
x2
2
2
(
2
 l2 
)2
x 12
 k2 
Y
x 2

2
x3
x
 k2 
( u x 2 ) 2
x 22
2
3
)2 

2
x2
  
x 22
Y
   m 2 

Y
(
x n
2
)2 

2
xn
x n2
2
xn
x n2
(4  
x2
x 22
 l2 
)2
 l2 
( u x 3 ) 2
x 32
(4  
x3
x 32
)2
   m 2 
MER301: Engineering Reliability
Lecture 17
(4  
xn
x n2
   m 2 
u Y2  j 2  u x21  k 2  u x22  l 2  u x23      m 2  u x2n
L Berkley Davis
Copyright 2009
x n2
( u x n ) 2
x n2
Viscometer and
Triangle Examples
22
)2
Viscometer
ExampleLecture 17
0.2=1.0%
reproducible
Y=fn(K,densities, time)
L Berkley Davis
Copyright 2009
Empirical Instrument Constant
23
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.

m 

Where
dm
dt

m
t
m  m f  m e

L Berkley Davis
Copyright 2009
Error Estimates are:
m f  400 , u f   2 gm  4 gm
 Mass of full beaker,
 Mass of empty beaker, m e  200 , u e   2 gm  4 gm
 Collection time interval,  t  10 , u    0 . 2 sec  0 . 4 sec
t
MER301: Engineering Reliability
Lecture 17
24
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 17
25