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David Rumpf, StatisticianGE Aircraft Engines Page 1 Computer Experiments to Predict Propagation of Variation An Aircraft Engine Blade Assembly Case Study Presented at the Fall Technical Conference King of Prussia, PA October 2002 By Special thanks to Robert Shankland, GEAE Engineering for his patience and expertise in running the analytic computer stress model.

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David Rumpf, StatisticianGE Aircraft Engines Page 2 Blade Assembly Stress Study Goals: 1)A DFSS (design for six sigma) design which meets LCF life requirements 2)Tolerance requirements for X1, X2, X3 What was available: A computer model which evaluates stress for any specific set of X1, X2 and X3 values. Run time ~ 1 hour Two stress points, Y1 and Y2 as shown. Two outcomes for each location, mean stress and alternating stress. Alternating is bigger driver for part low cycle fatigue life. What was needed: Non-linear transfer functions for Y1a,b and Y2a,b versus X1, X2 and X3 which could be used for multiple Monte Carlo models, ~1000 iterations each, for stress versus tolerances on X1, X2 and X3. X1=interference fit Y1a,b= notch mean and alternating stress Y2a,b = rabbet fillet mean and alternating stress X2=CP rabbet interference fit X3=CP drop a function of three drops

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David Rumpf, StatisticianGE Aircraft Engines Page 3 Six Sigma, Producibility and Robust Design Quality plan relies on inspection. Expect to have rework, scrap and MRB activity. 6 xbar +/- 3 +/- 6 will fit within tol Combined Engineering, Manufacturing 6 Goal Improved Manufacturing Process “sigma level” Robust Design allows Wider Tolerance to meet Customer Need Typical Historical Situation Process“sigma level” 2 xbar +/- 3 only +/- 2 fit within tol Tolerance meets Customer Need Quality plan focus on parameter control and process monitoring. First time yield 100%. Reaching six sigma goal requires combined Manufacturing/Design Effort

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David Rumpf, StatisticianGE Aircraft Engines Page 4 Statistical/Design of Experiment Opportunities Manufacturing Process Improvement: Screening designs Factorial designs Leveraging EVOP Quality Improvement metrics Review by Vice-president Engineering, meeting Customer Needs and improving producibility: Quality Function Deployment Voice of the Customer Robust Design: Screening Factorial Response Surface Designs Producibility Scorecards Review by Vice-president Focus of this presentation

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David Rumpf, StatisticianGE Aircraft Engines Page 5 Robust Design Y = Stress or Useful Life X’s are parameters which impact Y which could include: Part Key Characteristic values Environment Mating part Key Characteristic values Customer usage pattern X’s are typically a combination of controllable and noise (uncontrollable) factors Y = f(X C, X N ) Goals: Target Y Minimize Y, that is, variation in Y

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David Rumpf, StatisticianGE Aircraft Engines Page 6 Why Robust Design Statistical/Engineering method for product/process improvement (Taguchi’s idea) Two types of factor, control (Xc) and hard to control (Xn or noise) Control factor levels can change target Hard to control factors have variation during normal process or usage Robust design: Set Xc to take advantage of non-linearity in Y = f(Xn) Design space is typically non-linear Non-linear Response 0 2 4 6 8 10 12 14 1234567 Response X n

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David Rumpf, StatisticianGE Aircraft Engines Page 7 Wu and Hamada recommend a two step process Experiments: Planning, Analysis and Parameter Design Optimization, CFJ Wu and M Hamada, Wiley 2000 Obtain Transfer Functions: Ybar = f 1 (X C ) Y = f 2 (X C ) Typically one finds different sets of X’s in the two transfer functions If Target is goal: Minimize variation in Y, the harder objective Minimize Ybar distance from target If maximum or minimum is the goal: Optimize Ybar Minimize variation in Y An alternative approach is non-linear optimization of Z where Z = |Target – Ybar|/ Y

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David Rumpf, StatisticianGE Aircraft Engines Page 8 Statistical Issues with Analytic Models Designing a new part: Typically done analytically Often a complex, time consuming process to obtain a result for a single set of parameter values Examples include finite element analysis models, system models, etc Leads to serious optimization and simulation issues Recommended approach: Run designed experiment, typically Response Surface, to capture non-linear effects Use RSM transfer function for Optimization Simulation to estimate effect of variation in X’s on Y Statistical Problem: Analytic models have no random variation, always the same answer for a set of X values RSM assumes normally distributed error in residuals from model fit Residuals from analytic model are entirely lack of fit.

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David Rumpf, StatisticianGE Aircraft Engines Page 9 Case Study was a Learning Process for the GEAE Author Initial approach: (Note: All results coded for proprietary reasons) Full Factorial with center-point, 9 computer runs, ~ 9 hours run time Y’s = life required log transformation Choose to use Y’s = stress Interactions and curvature were significant, see Y1a graph below Results led team to an RSM design in 3 factors. Largest interaction and curvature effects

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David Rumpf, StatisticianGE Aircraft Engines Page 10 RSM Design Face centered central composite design, illustrated for two factors below We ran 15 runs, no repeated center-points since computer model has no random variation Factor A Factor B RSM analysis requires 9+ runs for a 2 factor design, 15+ runs for a 3 factor design.

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David Rumpf, StatisticianGE Aircraft Engines Page 11 RSM Results Analysis plan and results: Chose most parsimonious model via backwards selection based on p values Concerns: 1)High residuals, especially for Rabbet stress, cause concern. 2)R-sq not as high as desired Questions: 1)Does this transfer function fit well enough for engineering need? 2)Is there a better way to fit analytic/computer model results

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David Rumpf, StatisticianGE Aircraft Engines Page 12 Enter Professor William Welch and a Space Filling Design GEAE author looked for help. Jeff Wu suggested Professor W. Welch of the University of Waterloo New Experimental Plan: Space filling design instead of DOE or RSM design Recommended for computer/analytic experiments Multiple levels to provide better estimate for non-linear and interaction effects 33 runs for 3 factors Doubled the number of runs Spaces levels for each factor at 1/32 nd of the range Plots below show experimental grid, 2 factors at a time Computer experiment was run for the 33 sets of conditions (~30 hours of run time)

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David Rumpf, StatisticianGE Aircraft Engines Page 13 Approximating Random Variable Function Model Treat Deterministic output Y(x) as a realization of a random function (stochastic process) Y(x) = Ybar(x) + Z(x) Sacks et al, Statistical Science, 1989 Intuition: Model correlation between Z(x) and Z(x’) for any two input vectors x and x’ x close to x’ – correlation large x far from x’ – correlation small Leads to a distribution of Y(x) at any x given the Y’s at the design points Perform diagnostic tests on model Accuracy of prediction? Standard error of prediction?

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David Rumpf, StatisticianGE Aircraft Engines Page 14 Accuracy comparison (e.g., Notch-Alt or Y1b) -------------------------------------------------- Approximating Cross-Validated Model RMSE --------------------------------------------------- Polynomial – 2 nd degree 0.71 Polynomial – 3 rd degree 1.04 Random function 0.48 3 rd degree polynomial fits even worse than 2 nd degree! Error = Y – fitted Y Fitted Y is leave-one-out cross validated (take observation out and predict it) RMSE = 1/n sum (Y – fitted Y) 2

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David Rumpf, StatisticianGE Aircraft Engines Page 15 Diagnostic Checking of Random-Function Model Accuracy assessment: Plot Y versus fitted Y Standard error (se) assessment: Plot (Y – fitted Y) / se(fitted Y) versus fitted Y (Fitted Y is leave-one-out cross validation) Actual

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David Rumpf, StatisticianGE Aircraft Engines Page 16 Visualization of Input-Output Relationships e.g. Y1b as a function of RetArm Other two inputs (CPRabbet and CPDrop) averaged out

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David Rumpf, StatisticianGE Aircraft Engines Page 17 Propagating Variation Through the Random Function Model CPRabbet, CPDrop, RetArm have independent N(mu, sigma) distributions e.g., set mu = center of range Sample CPRabbet, CPDrop, RetArm and pass through model to get a distribution of e.g., Y1b values

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David Rumpf, StatisticianGE Aircraft Engines Page 18 Conclusions RSM approach: Good starting point Will work fine for Ybar and simple underlying Physics Space Filling Design: Allows us to model responses with very nonlinear underlying Physics Random-function model: Provides valid standard errors of prediction Can adapt to nonlinearities in data Fast, so can quickly propagate variation inputs => outputs via Monte Carlo

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