1. Understanding the Accuracy of Assembly Variation Analysis Methods ADCATS 2000 Robert Cvetko June 2000

2. ADCATS 2000Slide 2 Problem Statement q There are several different analysis methods q An engineer will often use one method for all situations q The confidence level of the results is seldom estimated

3. June 2000ADCATS 2000Slide 3 Outline of Presentation q New metrics to help estimate accuracy q Estimating accuracy (one-way clutch) í Monte Carlo (MC) í RSS linear (RSS) q Method selection technique to match the error of input information with the analysis

4. Sample Problem One-way Clutch Assembly

5. June 2000ADCATS 2000Slide 5 Clutch Assembly Problem  e c b a c q Contact angle important for performance q Known to be quite non-quadratic q Easily represented in explicit and implicit form

6. June 2000ADCATS 2000Slide 6 Details for the Clutch Assembly í Cost of “bad” clutch is $20 í Optimum point is the nominal angle

7. June 2000ADCATS 2000Slide 7 Monte Carlo Benchmark (One Billion Samples)

8. June 2000ADCATS 2000Slide 8 10,000 Sample Monte Carlo There is significant variability even using Monte Carlo with 10,000 samples.

9. June 2000ADCATS 2000Slide 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 -3-20123 Estimate of the Mean Probability Density for the Estimate of the Mean 16 samples  = 0.25  1 sample  = 1  4 samples  = 0.5 

10. June 2000ADCATS 2000Slide 10 New Metric: Standard Moment Error q Dimensionless measure of error in a distribution moment q All moments scaled by the standard deviation Estimate True

11. June 2000ADCATS 2000Slide 11  SER1 for Monte Carlo

12. June 2000ADCATS 2000Slide 12  SER2 for Monte Carlo

13. June 2000ADCATS 2000Slide 13  SER3-4 for Monte Carlo

14. June 2000ADCATS 2000Slide 14 Standard Moment Errors

15. June 2000ADCATS 2000Slide 15 10,000 Sample Monte Carlo You don’t have to do multiple Monte Carlo Simulations to estimate the error!

16. June 2000ADCATS 2000Slide 16 Application: Quality Loss Function

17. June 2000ADCATS 2000Slide 17 Estimating Quality Loss with MC

18. RSS Linear Analysis Using First-Order Sensitivities

19. June 2000ADCATS 2000Slide 19 New Metric: Quadratic Ratio q Dimensionless ratio of quadratic to linear effect q Function of derivatives and standard deviation of one input variable

20. June 2000ADCATS 2000Slide 20 Calculating the QR q The variables that have the largest %contribution to variance or standard deviations q The hub radius a contributes over 80% of the variance and has the largest standard deviation

21. June 2000ADCATS 2000Slide 21 Linearization Error q First and second-order moments as function of one variable q Simplified SER estimates for normal input variables

22. June 2000ADCATS 2000Slide 22 Linearization of Clutch q The QR is effective at estimating the reduction in error that could be achieved by using a second- order method q 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 SER10.01410.01560.0157 SER2-0.0004 -0.0034 SER30.08440.09360.0944 SER4-0.0119-0.0144-0.0441 Error Estimates Obtained From:

23. Method Selection Matching Input and Analysis Error and Matching Method with Objective

24. June 2000ADCATS 2000Slide 24 Error Matching q “Things should be made as simple as possible, but not any simpler”-Albert Einstein q Method complexity increases with accuracy q Simplicity í Reduce computation error í Design iteration í Presenting results Input Error Analysis Error

25. June 2000ADCATS 2000Slide 25 Converting Input Errors to  SER2 q Incomplete assembly model q Input variable q Specification limits q Loss constant

26. June 2000ADCATS 2000Slide 26 Design Iteration Efficiency Accuracy MSM DOE RSS MC

27. June 2000ADCATS 2000Slide 27 Conclusions q Confidence of analysis method should be estimated q Confidence of model inputs should be estimated q New metrics - SER and QR help to estimate the error analysis method and input errors q Error matching can help keep analysis models simple and increase efficiency