understanding the accuracy of assembly variation analysis methods adcats 2000 robert cvetko june...
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Understanding the Accuracy of Assembly Variation Analysis Methods
ADCATS 2000
Robert Cvetko
June 2000
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June 2000 ADCATS 2000 Slide 2
Problem Statement
There are several different analysis methods An engineer will often use one method for
all situations The confidence level of the results is
seldom estimated
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June 2000 ADCATS 2000 Slide 3
Outline of Presentation
New metrics to help estimate accuracy Estimating accuracy (one-way clutch)
Monte Carlo (MC)RSS linear (RSS)
Method selection technique to match the error of input information with the analysis
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Sample Problem
One-way Clutch Assembly
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June 2000 ADCATS 2000 Slide 5
Clutch Assembly Problem
e
c
ba
c
Contact angle important for performance
Known to be quite non-quadratic
Easily represented in explicit and implicit form
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June 2000 ADCATS 2000 Slide 6
Details for the Clutch Assembly
Cost of “bad” clutch is $20
Optimum point is the nominal angle
Variable Mean Standard Deviation a - hub radius 27.645 mm 0.01666 mm c - roller radius 11.430 mm 0.00333 mm e - ring radius 50.800 mm 0.00416 mm
Contact Angle Value (degrees)Upper Limit 7.6184
Nominal Angle 7.0184Lower Limit 6.4184
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June 2000 ADCATS 2000 Slide 7
Monte Carlo Benchmark
Value (mean)..................................... 7.014953 (Standard Deviation)........... 0.219668 (Skewness)............................ -0.094419 (Kurtosis)............................... 3.023816
2.6814,4062,1666,572
Quality Loss ($/part)..................
Contact Angle for the Clutch
Lower Rejects (ppm).................Upper Rejects (ppm).................Total Rejects (ppm)....................
(One Billion Samples)
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June 2000 ADCATS 2000 Slide 8
Monte Carlo - 1,000 RunsRun #1(10,000 Samples) Max/Min Std Dev
7.01111 7.02288/ 7.00846 .002203
2 0.04893 0.05036/0.04598 .000717
-.00086 -.00011/-.00184 .000263
0.00732 0.00788/ 0.00628 .000251
10,000 Sample Monte Carlo
There is significant variability even using Monte Carlo with 10,000 samples.
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June 2000 ADCATS 2000 Slide 9
One-Sigma Bound on the Mean
Estimate of the Mean versus Sample Size
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
-3 -2 -1 0 1 2 3
Estimate of the Mean
Pro
bab
ility
De
nsi
ty f
or
the
E
sti
ma
te o
f th
e M
ean 16 samples
= 0.25
1 sample = 1
4 samples = 0.5
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June 2000 ADCATS 2000 Slide 10
New Metric: Standard Moment Error
Dimensionless measure of error in a distribution moment
All moments scaled by the standard deviation
i
iiSERi
ˆ
Estimate True
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June 2000 ADCATS 2000 Slide 11
SER1 for Monte Carlo
SER1 versus One-Simga Bound
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08
Sample Size (log scale)
SE
R (
log
sc
ale
)1Simga
SER1a
SER1b
n
n
n
SER
SER
1
1ˆVariance
onDistributi Normal Standardˆ
1
1121
11
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June 2000 ADCATS 2000 Slide 12
SER2 for Monte Carlo
SER2 versus One-Simga Bound
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08
Sample Size (log scale)
SE
R (
log
sc
ale
)
1Simga
SER2a
SER2b
1
2
1
2ˆVariance
1
2ˆVariance
12ˆ1
Variance
12 variance,1mean
:onDistributi Square-Chiˆ1
2
2
2
2
2
2
2
2
2
2
2
n
n
n
nn
nn
n
SER
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June 2000 ADCATS 2000 Slide 13
SER3-4 for Monte Carlo
2
43
nSER6
1004
nSER
SER3 versus One-Simga Bound
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08
Sample Size (log scale)
SE
R (
log
sc
ale
)
1Simga
SER3a
SER3b
SER4 versus One-Simga Bound
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08
Sample Size (log scale)
SE
R (
log
sca
le)
1Simga
SER4a
SER4b
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June 2000 ADCATS 2000 Slide 14
Standard Moment Errors
One-Sigma Bound for SER1-4 versus Sample Size
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08
Sample Size (log scale)
SE
R (
log
sca
le)
-
SER4
SER3
SER2
SER1
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June 2000 ADCATS 2000 Slide 15
Monte Carlo - 1,000 Runs Est 68%Run #1(10,000 Samples) Max/Min Std Dev Conf Int
7.01111 7.02288/ 7.00846 .002203 ± .002212
2 0.04893 0.05036/0.04598 .000717 ± .000692
-.00086 -.00011/-.00184 .000263 ± .000212
0.00732 0.00788/ 0.00628 .000251 ± .000233
10,000 Sample Monte Carlo
You don’t have to do multiple Monte CarloSimulations to estimate the error!
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June 2000 ADCATS 2000 Slide 16
Application: Quality Loss Function
2
2
2
21
2
2
2
12
,
1
2ˆ
1ˆ)ˆ(2
21
nK
nmK
SER
L
SER
LSERSERTotalL
2
min12
min )(
)()(
KK
dfK
dfLL
m
f()
L()
1
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June 2000 ADCATS 2000 Slide 17
Estimating Quality Loss with MC%Error in Quality Loss for Monte Carlo
0.001%
0.010%
0.100%
1.000%
10.000%
100.000%
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10
Sample Size (log scale)
%E
rro
r (l
og
sca
le)
%Error
1Sigma
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2222
ecba ecba
RSS Linear Analysis
Using First-Order Sensitivities
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June 2000 ADCATS 2000 Slide 19
New Metric: Quadratic Ratio
Dimensionless ratio of quadratic to linear effect
Function of derivatives and standard deviation of one input variable
a
aaa
a b
b
a
fa
f
QR
2
2
2
1
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June 2000 ADCATS 2000 Slide 20
Calculating the QR
The variables that have the largest %contribution to variance or standard deviations
The hub radius a contributes over 80% of the variance and has the largest standard deviation
a c e1st Derivative -11.91 -23.73 11.822nd Derivative -20.11 -81.05 -20.41Standard Deviation 0.01666 0.00333 0.00416QR (quadratic ratio) -0.0141 -0.0057 -0.0036
Input Variable
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June 2000 ADCATS 2000 Slide 21
Linearization Error
42
3
2
60604
863
22
1
QRQRaSER
QRQRaSER
QRaSER
QRaSER
First and second-order moments as function of one variable
Simplified SER estimates for normal input variables
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June 2000 ADCATS 2000 Slide 22
Linearization of Clutch
The QR is effective at estimating the reduction in error that could be achieved by using a second-order method
If the accuracy of the linear method is not enough, a more complex model could be used
Quadratic Ratio of a
RSS vs. Method of
System Moments
RSS vs. Benchmark
SER1 0.0141 0.0156 0.0157SER2 -0.0004 -0.0004 -0.0034SER3 0.0844 0.0936 0.0944SER4 -0.0119 -0.0144 -0.0441
Error Estimates Obtained From:
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Method Selection
Matching Input and Analysis Errorand Matching Method with Objective
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June 2000 ADCATS 2000 Slide 24
Error Matching
“Things should be made as simple as possible, but not any simpler”-Albert Einstein
Method complexity increases with accuracy
Simplicity Reduce computation error Design iteration Presenting results
Input Error
Analysis Error
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June 2000 ADCATS 2000 Slide 25
Converting Input Errors to SER2
Incomplete assembly model
Input variable Specification limits Loss constant
n
iiSERiSER
1
2,22 ion%Contribut
XX L
XSER
2
2
KSER K 1
2
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June 2000 ADCATS 2000 Slide 26
Design Iteration Efficiency
Design Iteration
Efficiency
Accuracy
MSMDO
E
RSS
MC
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June 2000 ADCATS 2000 Slide 27
Conclusions Confidence of analysis method should be
estimated Confidence of model inputs should be
estimated New metrics - SER and QR help to estimate
the error analysis method and input errors Error matching can help keep analysis
models simple and increase efficiency