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```Understanding the Accuracy of
Assembly Variation Analysis Methods
Robert Cvetko
June 2000
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
Slide 2
Outline of Presentation
New metrics to help estimate accuracy
 Estimating accuracy (one-way clutch)

 Monte
Carlo (MC)

Method selection technique to match the
error of input information with the analysis
June 2000
Slide 3
Sample Problem
One-way Clutch Assembly
Clutch Assembly Problem

c


c

b
a
June 2000
e
Contact angle
important for
performance
Known to be quite
Easily represented in
explicit and implicit
form
Slide 5
Details for the Clutch Assembly
Contact Angle
Upper Limit
Nominal Angle
Lower Limit
Value (degrees)
7.6184
7.0184
6.4184
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
June 2000
Slide 6
Monte Carlo Benchmark
(One Billion Samples)
Contact Angle for the Clutch
 (mean).....................................
 (Standard Deviation)...........
 (Skewness)............................
 (Kurtosis)...............................
Quality Loss (\$/part)..................
Lower Rejects (ppm).................
Upper Rejects (ppm).................
Total Rejects (ppm)....................
June 2000
Value
7.014953
0.219668
-0.094419
3.023816
2.681
4,406
2,166
6,572
Slide 7
10,000 Sample Monte Carlo

Monte Carlo - 1,000 Runs
Run #1
Max/Min
Std Dev
(10,000 Samples)
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
There is significant variability even using
Monte Carlo with 10,000 samples.
June 2000
Slide 8
One-Sigma Bound on the Mean
Estimate of the Mean versus Sample Size
Probability Density for the
Estimate of the Mean
1.6
1.4
16 samples
* = 0.25 
1.2
1.0
4 samples
* = 0.5 
0.8
0.6
1 sample
*= 1 
0.4
0.2
0.0
-3
-2
-1
0
1
2
3
Estimate of the Mean
June 2000
Slide 9
New Metric: Standard Moment Error


Dimensionless
measure of error in
a distribution
moment
All moments scaled
by the standard
deviation
June 2000
Estimate
SERi 
True
ˆ i   i 

i
Slide 10
SER1 for Monte Carlo
ˆ 1  1
 Standard Normal Distributi on

n
 SER1
 ˆ 1  1  1
 Variance 

   n
1

n
SER1 versus One-Simga Bound
1.E+01
1.E+00
SER (log scale)

2
SER1
1.E-01
1Simga
SER1a
SER1b
1.E-02
1.E-03
1.E-04
1.E-05
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
Sample Size (log scale)
June 2000
Slide 11
SER2 for Monte Carlo
n  1ˆ 2
2
 Chi - Square Distributi on :
mean  n  1, variance  2n  1
 n  1ˆ 2 
  2n  1
Variance 

2


 ˆ 
2
Variance  2  
  2  n  1
 SER2 
2
n  1
1.E+01
1.E+00
SER (log scale)
 ˆ
 
2
Variance  2  2  
  2  2  n  1
SER2 versus One-Simga Bound
1.E-01
1Simga
SER2a
SER2b
1.E-02
1.E-03
1.E-04
1.E-05
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
Sample Size (log scale)
June 2000
Slide 12
SER3-4 for Monte Carlo
 SER3
4

n2
 SER4
SER4 versus One-Simga Bound
1.E+01
1.E+01
1.E+00
1.E+00
1.E-01
1Simga
SER3a
SER3b
1.E-02
1.E-03
SER (log scale)
SER (log scale)
SER3 versus One-Simga Bound
100

n6
1.E-01
1Simga
SER4a
SER4b
1.E-02
1.E-03
1.E-04
1.E-04
1.E-05
1.E-05
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
1.E+00
Sample Size (log scale)
June 2000
1.E+02
1.E+04
1.E+06
1.E+08
Sample Size (log scale)
Slide 13
Standard Moment Errors
One-Sigma Bound for SER1-4 versus Sample Size
SER (log scale) -
1.E+00
1.E-01
SER4
SER3
SER2
SER1
1.E-02
1.E-03
1.E-04
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)
June 2000
Slide 14
10,000 Sample Monte Carlo

Monte Carlo - 1,000 Runs
Run #1
Max/Min
Std Dev
(10,000 Samples)
7.01111
7.02288/ 7.00846
.002203
Est 68%
Conf Int
± .002212
2
0.04893
0.05036/0.04598
.000717
± .000692

-.00086
-.00011/-.00184
.000263
± .000212

0.00732
0.00788/ 0.00628
.000251
± .000233
You don’t have to do multiple Monte Carlo
Simulations to estimate the error!
June 2000
Slide 15
Application: Quality Loss Function
L()
L   L( ) f ( )d
  K    min  f ( )d
 K 2  K 1   min 
f()
2
m
2
1
 L 

 L 




 SER1 
 SER2  SER2 

SER
1






2
 L ,Total 2  
2


 1 

2 
  Kˆ 2 

  2 K ( ˆ 1  m) ˆ 2 



n
n

1









June 2000

2
Slide 16
Estimating Quality Loss with MC
%Error in Quality Loss for Monte Carlo
100.000%
%Error (log scale)
10.000%
1.000%
%Error
1Sigma
0.100%
0.010%
0.001%
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
1.E+10
Sample Size (log scale)
June 2000
Slide 17
Using First-Order Sensitivities
           
    a     b     c     e 
 a   b   c   e 
2
2
2
2


Dimensionless
linear effect
Function of
derivatives and
standard deviation
of one input
variable
June 2000
1 f
a
2
baa
2

a
QR a 

f
ba
a
2
Slide 19
Calculating the QR
1st Derivative
2nd Derivative
Standard Deviation


Input Variable
a
c
e
-11.91
-23.73
11.82
-20.11
-81.05
-20.41
0.01666 0.00333 0.00416
-0.0141 -0.0057 -0.0036
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
June 2000
Slide 20
Linearization Error


First and second-order
moments as function
of one variable
Simplified SER
estimates for normal
input variables
June 2000
SER1a  QR
SER2a  2QR 2
SER3a  6QR  8QR 3
SER4a  60QR 2  60QR 4
Slide 21
Linearization of Clutch
Error Estimates Obtained From:
Method of
Benchmark
Ratio of a
System Moments
SER1
SER2
SER3
SER4


0.0141
-0.0004
0.0844
-0.0119
0.0156
-0.0004
0.0936
-0.0144
0.0157
-0.0034
0.0944
-0.0441
The QR is effective at estimating the reduction in
error that could be achieved by using a secondorder method
If the accuracy of the linear method is not enough,
a more complex model could be used
June 2000
Slide 22
Method Selection
Matching Input and Analysis Error
and Matching Method with Objective
Error Matching


Input
Error
Analysis
Error

simple as possible, but not
any simpler”-Albert Einstein
Method complexity increases
with accuracy
Simplicity

Reduce computation error
 Design iteration
 Presenting results
June 2000
Slide 24
Converting Input Errors to SER2




Incomplete assembly
n
2
model
 SER2   %Contribut ion i SER2,i 
i 1
Input variable
2 X
Specification limits  SER2 
XL  X
Loss constant
1
June 2000
 SER2 
K
K
Slide 25
Design Iteration Efficiency
Design
Iteration
Efficiency
MSM
MC
Accuracy
June 2000
Slide 26
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

June 2000