Post on 19-Dec-2015
Understanding the Accuracy of Assembly Variation Analysis Methods
ADCATS 2000
Robert Cvetko
June 2000
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
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
Sample Problem
One-way Clutch Assembly
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
June 2000 ADCATS 2000 Slide 6
Details for the Clutch Assembly
Cost of “bad” clutch is $20
Optimum point is the nominal angle
Variable Mean Standard Deviation a - hub radius 27.645 mm 0.01666 mm c - roller radius 11.430 mm 0.00333 mm e - ring radius 50.800 mm 0.00416 mm
Contact Angle Value (degrees)Upper Limit 7.6184
Nominal Angle 7.0184Lower Limit 6.4184
June 2000 ADCATS 2000 Slide 7
Monte Carlo Benchmark
Value (mean)..................................... 7.014953 (Standard Deviation)........... 0.219668 (Skewness)............................ -0.094419 (Kurtosis)............................... 3.023816
2.6814,4062,1666,572
Quality Loss ($/part)..................
Contact Angle for the Clutch
Lower Rejects (ppm).................Upper Rejects (ppm).................Total Rejects (ppm)....................
(One Billion Samples)
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.
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
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
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
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
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
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
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!
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
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
2222
ecba ecba
RSS Linear Analysis
Using First-Order Sensitivities
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
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
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
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:
Method Selection
Matching Input and Analysis Errorand Matching Method with Objective
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
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
June 2000 ADCATS 2000 Slide 26
Design Iteration Efficiency
Design Iteration
Efficiency
Accuracy
MSMDO
E
RSS
MC
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