Download Thesis Defense

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
Understanding the Accuracy of
Assembly Variation Analysis Methods
ADCATS 2000
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
ADCATS 2000
Slide 2
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
June 2000
ADCATS 2000
Slide 3
Sample Problem
One-way Clutch Assembly
Clutch Assembly Problem

c


c

b
a
June 2000
e
ADCATS 2000
Contact angle
important for
performance
Known to be quite
non-quadratic
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
Cost of “bad” clutch
is $20
 Optimum point is
the nominal angle

Variable
Mean
Standard Deviation
a - hub radius 27.645 mm
0.01666 mm
c - roller radius 11.430 mm
0.00333 mm
e - ring radius 50.800 mm
0.00416 mm
June 2000
ADCATS 2000
Slide 6
Monte Carlo Benchmark
(One Billion Samples)
Contact Angle for the Clutch
 (mean).....................................
 (Standard Deviation)...........
 (Skewness)............................
 (Kurtosis)...............................
Quality Loss ($/part)..................
Lower Rejects (ppm).................
Upper Rejects (ppm).................
Total Rejects (ppm)....................
June 2000
ADCATS 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
ADCATS 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
ADCATS 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 
ADCATS 2000
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
ADCATS 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
ADCATS 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)
ADCATS 2000
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
ADCATS 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
ADCATS 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

ADCATS 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
ADCATS 2000
Slide 17
RSS Linear Analysis
Using First-Order Sensitivities
           
    a     b     c     e 
 a   b   c   e 
2
2
2
2
New Metric: Quadratic Ratio


Dimensionless
ratio of quadratic to
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
ADCATS 2000
2
Slide 19
Calculating the QR
1st Derivative
2nd Derivative
Standard Deviation
QR (quadratic ratio)


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
ADCATS 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
ADCATS 2000
Slide 21
Linearization of Clutch
Error Estimates Obtained From:
RSS vs.
RSS vs.
Quadratic
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
ADCATS 2000
Slide 22
Method Selection
Matching Input and Analysis Error
and Matching Method with Objective
Error Matching


Input
Error
Analysis
Error

“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
June 2000
ADCATS 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 
ADCATS 2000
K
K
Slide 25
Design Iteration Efficiency
RSS
Design
Iteration
Efficiency
MSM
MC
Accuracy
June 2000
ADCATS 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
ADCATS 2000
Slide 27
Related documents