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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 1ˆ 2 2 Chi - Square Distributi on : mean n 1, variance 2n 1 n 1ˆ 2 2n 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 n2 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 n6 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 Kˆ 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