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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 2008 Factorial Fit: Color versus Solv/React, Cat/React, ... Estimated Effects and Coefficients for Color (coded units) Term Constant Solv/React Cat/React Temp React Purity React pH Solv/React*Cat/React Solv/React*Temp 1.4350 Effect Coef 2.7700 0.7175 -1.4650 -0.7325 -0.2725 -0.1363 4.5450 2.2725 -0.7025 -0.3513 1.1500 0.5750 -0.9125 -0.4562 Solv/React*React Purity -1.2300 -0.6150 Solv/React*React pH Cat/React*Temp Cat/React*React Purity Cat/React*React pH Temp*React Purity Temp*React pH React Purity*React pH 0.4275 0.2925 0.1200 0.1625 0.2138 0.1462 0.0600 0.0812 -0.8375 -0.4187 -0.3650 -0.1825 0.2125 0.1062 Normal Probability Plot of the Effects (response is Color, Alpha = .10) 99 D Effect Type Not Significant Significant F actor A B C D E N ame S olv /React C at/React Temp React P urity React pH 95 90 80 Percent 70 60 50 40 30 20 10 5 1 -2 -1 0 1 2 Effect 3 4 5 Lenth's PSE = 0.8475 Factorial Fit: Color versus React Purity Estimated Effects and Coefficients for Color (coded units) Term Constant React Purity Effect Coef SE Coef 2.770 4.545 2.272 T P 0.4147 6.68 0.000 0.4147 5.48 0.000 S = 1.65876 R-Sq = 68.20% R-Sq(adj) = 65.93% Analysis of Variance for Color (coded units) Source Main Effects DF Seq SS Adj SS Adj MS 1 82.63 38.52 38.52 F P 82.63 82.628 30.03 0.000 38.52 38.52 2.751 2.751 Residual Error 14 Pure Error Total 14 15 121.15 Nor mal P r obability P lot of the Residuals (response is Color) 99 90 Percent 50 10 1 -4 -3 -2 -1 0 Residual 1 2 3 4 Residuals V er sus the Fitted V alues (response is Color) 2 Residual 0 -2 0 1 2 Fitted Value 3 4 5 Residuals Ver sus React P ur ity (response is Color) 2 Residual 0 -2 -1.0 -0.5 0.0 React Purity 0.5 1.0 Cube Plot (data means) for Color 3.875 4.865 -2.385 1 1.795 Cat/React 6.005 5.425 1 React Purity 0.715 -1 -1 Solv/React 1.865 -1 1 Factorial Fit: Resist versus A, B, C, D Estimated Effects and Coefficients for Resist (coded units) Term Constant A B C D A*B A*C A*D ... Analysis of Variance for Resist (coded units) Source Main Effects 2-Way Interactions Residual Error Curvature Pure Error Total DF Seq SS Adj SS Adj MS F P Effect Coef SE Coef 60.433 47.700 23.850 -0.500 -0.250 80.600 40.300 -2.400 -1.200 1.100 0.550 T P 0.6223 97.12 0.000 0.7621 31.29 0.000 * 0.7621 -0.33 0.759 0.7621 52.88 0.000 * 0.7621 -1.57 0.190 0.7621 0.72 0.510 72.800 36.400 -2.000 -1.000 0.7621 47.76 0.000 * 0.7621 -1.31 0.260 4 17555.3 17555.3 4388.83 944.51 0.000 3 10610.1 10610.1 3536.70 761.13 0.000 4 1 3 18.6 5.6 13.0 18.6 5.6 13.0 4.65 5.61 4.33 1.30 0.338 11 28184.0 Normal Probability Plot of the Standardized Effects (response is Resist, Alpha = .10) 99 Effect Type Not Significant Significant C AC A F actor A B C D N ame A B C D 95 90 80 Percent 70 60 50 40 30 20 10 5 1 0 10 20 30 40 Standardized Effect 50 60 Factorial Fit: Resist versus A, C Estimated Effects and Coefficients for Resist (coded units) Term Constant A C A*C ... Analysis of Variance for Resist (coded units) Source Main Effects 2-Way Interactions Residual Error Curvature Pure Error Total DF Seq SS Adj SS Adj MS F P Effect Coef SE Coef 60.43 47.70 23.85 80.60 40.30 72.80 36.40 T P 0.6537 92.44 0.000 0.8007 29.79 0.000 * 0.8007 50.33 0.000 * 0.8007 45.46 0.000 * 2 17543.3 17543.3 8771.6 1710.43 0.000 1 10599.7 10599.7 10599.7 2066.89 0.000 8 1 7 41.0 5.6 35.4 41.0 5.6 35.4 5.1 5.6 5.1 1.11 0.327 11 28184.0 Residuals Versus the Fitted Values (response is Resist) 3 2 1 Residual 0 -1 -2 -3 0 20 40 60 80 100 Fitted Value 120 140 160 180 Normal Probability Plot of the Residuals (response is Resist) 99 95 90 80 Percent 70 60 50 40 30 20 10 5 1 -5.0 -2.5 0.0 Residual 2.5 5.0 Addendum to solution to Problem 2: manual test of curvature (courtesy R. Schwenke) <--I 1 "oV +* I vet'$*-q c-c'^IL 'iJ;^ * 1,u^c/'*' k.t^lr- : ( 1, 1 If L r 1- t/r< E- T( * : 5"dD6T ^L ( 53.3 -f 6l q) t4rt : ( q-6l.tt),*(ez.et o (. \ ac / I I rl I + ( f ( ,? - 6l ([o;_Q.i ,l)"t -.*--*_%* 3 r" 3 7(7 -D F '= 6 s . 60 6 1 l,KZe + ,= $ F = =. l,J 3{I E{ ( "It;s) =:0, 3a 7/ tr){\ t , =l Jr=3 ou,(. ( c--* \.q L*"r{. t t r' o . / rt a t c , I JJ")v7 tP *D y'1.t nn{^ t {* t .A; . Problem 3: example solution (courtesy X. Su) a. Run 1 2 3 4 (1) a b ab I 1 1 1 1 Design Factors A B AB -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 Sum of squares 0.02983 0.015826 0.005171 0.000856 0.00095 0.052634 0.1963 0.0914 0.1107 0.065 Degrees of Freedom 1 1 1 1 20 24 Replicate Results 0.2185 0.1914 0.1814 0.0891 0.0925 0.0855 0.1071 0.1109 0.1115 0.065 0.0667 0.0662 0.2092 0.0913 0.1145 0.0664 Totals 0.9968 0.4498 0.5547 0.3293 Effects estimate -0.07724 -0.05626 0.03216 Source of variation A B AB Curvature Residual Error Total Mean Square 0.02983 0.015826 0.005171 0.000856 4.75234E-05 F0 628 333.1789 108.8632 18.02105 P-value 1.40194E-16 6.13524E-14 1.54196E-09 0.000396599 Since A, B, AB and curvature are significant (P<0.05), they have to be included in the regression model. There is also evidence of pure quadratic curvature. Using Minitab: Response Surface Regression: Replicates versus A, B The following terms cannot be estimated, and were removed. B*B The analysis was done using coded units. Estimated Regression Coefficients for Replicates Term Constant A B A*A A*B Coef 0.10190 -0.03862 -0.02813 0.01463 0.01608 SE Coef 0.003083 0.001541 0.001541 0.003447 0.001541 T 33.053 -25.054 -18.249 4.244 10.432 P 0.000 0.000 0.000 0.000 0.000 S = 0.00689372 R-Sq = 98.19% PRESS = 0.00148511 R-Sq(pred) = 97.18% R-Sq(adj) = 97.83% Analysis of Variance for Replicates Source Regression Linear Square Interaction Residual Error Pure Error Total DF 4 2 1 1 20 20 24 Seq SS 0.051684 0.045656 0.000856 0.005171 0.000950 0.000950 0.052634 Adj SS 0.051684 0.045656 0.000856 0.005171 0.000950 0.000950 Adj MS 0.012921 0.022828 0.000856 0.005171 0.000048 0.000048 F 271.88 480.35 18.02 108.82 P 0.000 0.000 0.000 0.000 Unusual Observations for Replicates Obs 6 16 StdOrder 6 16 Replicates 0.219 0.181 Fit 0.199 0.199 SE Fit 0.003 0.003 Residual 0.019 -0.018 St Resid 3.10 R -2.91 R R denotes an observation with a large standardized residual. Estimated Regression Coefficients for Replicates using data in uncoded units Term Constant A B A*A A*B Coef 0.101900 -0.0386200 -0.0281300 0.0146300 0.0160800 y = 0 + 1x1 + 2x2 + 11x12 + 12x1x2 = 0.101900 - 0.0386200x1 - 0.0281300x2 + 0.0146300x12 + 0.0160800 x1x2 Normal Probability Plot (response is Replicates) 99 95 90 80 Percent 70 60 50 40 30 20 10 5 1 -0.02 -0.01 0.00 Residual 0.01 0.02 The normal probability plot looks skewed, with very little data lying on the blue line. The residuals do not seem to be following a normal distribution. Versus Fits (response is Replicates) 0.02 0.01 Residual 0.00 -0.01 -0.02 0.050 0.075 0.100 0.125 0.150 Fitted Value 0.175 0.200 Variances of the residuals is shown to grow with increasing fitted values. However, the residuals are equally distributed above and below the center line. b. Using transformed data sets exp(y): From Minitab: Response Surface Regression: Replicates versus A, B The following terms cannot be estimated, and were removed. B*B The analysis was done using coded units. Estimated Regression Coefficients for Replicates Term Constant A B A*A A*B Coef 1.10728 -0.04396 -0.03236 0.01779 0.01933 SE Coef 0.003733 0.001867 0.001867 0.004174 0.001867 T 296.581 -23.550 -17.337 4.262 10.357 P 0.000 0.000 0.000 0.000 0.000 S = 0.00834830 R-Sq = 98.00% PRESS = 0.00217794 R-Sq(pred) = 96.88% R-Sq(adj) = 97.60% Analysis of Variance for Replicates Source Regression Linear DF 4 2 Seq SS 0.068342 0.059599 Adj SS 0.068342 0.059599 Adj MS 0.017086 0.029800 F 245.15 427.58 P 0.000 0.000 Square Interaction Residual Error Pure Error Total 1 1 20 20 24 0.001266 0.007477 0.001394 0.001394 0.069736 0.001266 0.007477 0.001394 0.001394 0.001266 0.007477 0.000070 0.000070 18.17 107.28 0.000 0.000 Unusual Observations for Replicates Obs 6 16 StdOrder 6 16 Replicates 1.244 1.199 Fit 1.221 1.221 SE Fit 0.004 0.004 Residual 0.023 -0.022 St Resid 3.14 R -2.92 R R denotes an observation with a large standardized residual. Estimated Regression Coefficients for Replicates using data in uncoded units Term Constant A B A*A A*B Coef 1.10728 -0.0439614 -0.0323629 0.0177910 0.0193347 Regression model: y = 0 + 1x1 + 2x2 + 11x12 + 11x12 + 12x1x2 = 1.10728 -0.0439614x1 -0.0323629x2 + 0.0177910x12 + 0.0193347x1x2 Normal Probability Plot (response is Replicates) 99 95 90 80 Percent 70 60 50 40 30 20 10 5 1 -0.02 -0.01 0.00 Residual 0.01 0.02 0.03 Versus Fits (response is Replicates) 0.03 0.02 Residual 0.01 0.00 -0.01 -0.02 1.050 1.075 1.100 1.125 1.150 Fitted Value 1.175 1.200 1.225 Both plots do not show much improvement from the previous plots. Using transformation 1/y: From Minitab: Response Surface Regression: Replicates versus A, B The following terms cannot be estimated, and were removed. B*B The analysis was done using coded units. Estimated Regression Coefficients for Replicates Term Constant A B A*A A*B Coef 9.81930 3.06367 2.01027 0.27219 0.02012 SE Coef 0.12827 0.06414 0.06414 0.14341 0.06414 T 76.551 47.769 31.344 1.898 0.314 P 0.000 0.000 0.000 0.072 0.757 S = 0.286823 R-Sq = 99.39% PRESS = 2.57085 R-Sq(pred) = 99.05% R-Sq(adj) = 99.27% Analysis of Variance for Replicates Source Regression Linear Square Interaction Residual Error DF 4 2 1 1 20 Seq SS 268.849 268.545 0.296 0.008 1.645 Adj SS 268.849 268.545 0.296 0.008 1.645 Adj MS 67.212 134.272 0.296 0.008 0.082 F 817.00 1632.15 3.60 0.10 P 0.000 0.000 0.072 0.757 Pure Error Total 20 24 1.645 270.495 1.645 0.082 Unusual Observations for Replicates Obs 17 StdOrder 17 Replicates 11.696 Fit 11.125 SE Fit 0.128 Residual 0.571 St Resid 2.23 R R denotes an observation with a large standardized residual. Estimated Regression Coefficients for Replicates using data in uncoded units Term Constant A B A*A A*B Coef 9.81930 3.06367 2.01027 0.272187 0.0201165 Regression model: y = 0 + 1x1 + 2x2 + 11x12 + 11x12 + 12x1x2 = 9.81930 + 3.063674x1 + 2.01027x2 + 0.272187x12 + 0.0201165x1x2 Normal Probability Plot (response is Replicates) 99 95 90 80 Percent 70 60 50 40 30 20 10 5 1 -0.75 -0.50 -0.25 0.00 Residual 0.25 0.50 Versus Fits (response is Replicates) 0.50 0.25 Residual 0.00 -0.25 -0.50 5.0 7.5 10.0 Fitted Value 12.5 15.0 Comparing these two new plots, there is much improvements in the sense that the residual VS fitted value plots do not show a growth in variance. Also, the normal probability plot shows a more well fitted data to line, hence randomly distributed data. Thus, the last transformation 1/y seems more appropriate for fitting the current regression model. Response Surface Regression: y versus x1, x2, z The analysis was done using coded units. Estimated Regression Coefficients for y Term Constant x1 x2 z x1*x1 x2*x2 z*z x1*x2 x1*z x2*z ... Analysis of Variance for y Source Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total ... Estimated Regression Coefficients for y using data in uncoded units Term Coef DF Seq SS Adj SS Adj MS F P Coef SE Coef 87.3333 9.8013 2.2894 -6.1250 -13.8333 -21.8333 0.1517 8.1317 -4.4147 -7.7783 T P 1.681 51.968 0.000 1.873 1.873 5.232 0.001 1.222 0.256 1.455 -4.209 0.003 3.361 -4.116 0.003 3.361 -6.496 0.000 2.116 4.116 0.072 0.945 1.975 0.084 2.448 -1.804 0.109 2.448 -3.178 0.013 9 2034.94 2034.94 226.105 13.34 0.001 3 3 3 8 3 5 789.28 953.29 292.38 135.56 90.22 45.33 789.28 263.092 15.53 0.001 953.29 317.764 18.75 0.001 292.38 135.56 90.22 45.33 97.458 16.945 30.074 9.067 3.32 0.115 5.75 0.021 17 2170.50 Constant 87.3333 x1 x2 z x1*x1 5.8279 1.3613 -6.1250 -4.8908 x2*x2 z*z x1*x2 x1*z x2*z -7.7192 0.1517 2.8750 -2.6250 -4.6250 Response Surface Regression: y versus x1, x2, z The analysis was done using coded units. Estimated Regression Coefficients for y Term Constant x1 x2 z x1*x1 x2*x2 x1*x2 x1*z x2*z ... Analysis of Variance for y Source Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total ... DF Seq SS Adj SS Adj MS F P Coef SE Coef 87.361 9.801 2.289 -6.125 -13.760 -21.760 8.132 -4.415 -7.778 T P 1.541 56.675 0.000 1.767 1.767 5.548 0.000 1.296 0.227 1.373 -4.462 0.002 3.019 -4.558 0.001 3.019 -7.208 0.000 3.882 2.095 0.066 2.308 -1.912 0.088 2.308 -3.370 0.008 8 2034.86 2034.86 254.357 16.88 0.000 3 2 3 9 4 5 789.28 953.20 292.38 135.64 90.31 45.33 789.28 263.092 17.46 0.000 953.20 476.602 31.62 0.000 292.38 135.64 90.31 45.33 97.458 15.072 22.578 9.067 2.49 0.172 6.47 0.013 17 2170.50 Residual Plots for y Normal Probability Plot of the Residuals 99 Residual Percent 90 50 10 1 -5 0 Residual 5 5 Residuals Versus the Fitted Values 5.0 Residuals Versus x1 (response is y) 0 2.5 -5 Residual 60 80 Fitted Value 100 0.0 Histogram of the Residuals 3.0 Frequency Residual Residuals Versus the Order of the Data 5 -2.5 0 -5.0 -5 2 4 6 8 10 12 14 16 Observation Order 18 -2 -1 0 x1 1 2 1.5 0.0 -4 -2 0 2 Residual 4 Residuals Versus x2 (response is y) 5.0 5.0 Residuals Versus z (response is y) 2.5 2.5 Residual 0.0 Residual -2 -1 0 x2 1 2 0.0 -2.5 -2.5 -5.0 -5.0 -1.0 -0.5 0.0 z 0.5 1.0 ^ 2 Contour Plot of y vs x2, x1 y < 50.0 - 60.0 - 70.0 - 80.0 - 90.0 > 90.0 Contour Plot of y vs x2, x1 y < > 40.0 50.0 60.0 70.0 80.0 80.0 1 x1 = -0.393500 x2 = 0.293782 y = 90.1692 50.0 60.0 70.0 80.0 1 40.0 50.0 60.0 70.0 x2 0 x2 Hold Values z -1 Hold Values z 0 0 -1 x1 = -0.109756 x2 = -0.308367 y = 90.0198 x1 = 0.708407 x2 = -0.555312 y = 90.2054 -1 -1 0 x1 1 -1 0 x1 1 Contour Plot of sqrt{Vz(y(x,z)]} vs x2, x1 sqrt{Vz(y(x,z)]} < 6 6 8 8 - 10 10 - 12 > 12 Contour Plot of y vs x2, x1 y < 30.0 40.0 50.0 60.0 70.0 > 30.0 40.0 50.0 60.0 70.0 80.0 80.0 1 1 0 x2 x2 0 Hold Values z 1 -1 -1 -1 0 x1 1 -1 0 x1 1 Note: the question was unclear as to whether the noise input z was controllable. If so, selecting z = -1 may give minimal sensitivity of the output to variation in z. If, however, we assume that z cannot be controlled, we must assume it to have zero mean and constant variance. The alternative solution following (courtesy H. Hu) shows a solution based on the assumption that z cannot be controlled. Problem 4 Montgomery 13-12 Reconsider the crystal growth experiment from Exercise 13-10. Suppose that x z is now a noise variable, and that the modified experimental design shown here has been conducted. The experimenters want the growth rate to be as large as possible but they also want the variability transmitted from z to be small. Under what set of conditions is growth greater than 90 with minimum variability achieved? x -1 -1 -1 -1 1 1 1 1 -1.682 1.682 0 0 0 x -1 -1 1 1 -1 -1 1 1 0 0 -1.682 1.682 0 z -1 1 -1 1 -1 1 -1 1 0 0 0 0 0 y 66 70 78 60 80 70 100 75 100 80 68 63 113 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 118 88 100 85 We use Minitab to do the robustness study. The experimental design is a "modified" central composite design in which the axial runs in the z direction have been eliminated. Response Surface Regression: response versus x1, x2, x3 The analysis was done using coded units. Estimated Regression Coefficients for response Term Constant x1 x2 x3 x1*x1 x2*x2 x1*x2 x1*x3 x2*x3 Coef 98.896 1.271 1.361 -6.125 -5.412 -14.074 2.875 -2.625 -4.625 SE Coef 5.607 3.821 3.821 4.992 3.882 3.882 4.992 4.992 4.992 T 17.639 0.333 0.356 -1.227 -1.394 -3.625 0.576 -0.526 -0.926 P 0.000 0.747 0.730 0.251 0.197 0.006 0.579 0.612 0.378 S = 14.1209 R-Sq = 65.33% PRESS = 9196.84 R-Sq(pred) = 0.00% R-Sq(adj) = 34.51% Analysis of Variance for response Source Regression Linear Square Interaction DF 8 3 2 3 Seq SS 3381.2 347.5 2741.3 292.4 Adj SS 3381.2 347.5 2741.3 292.4 Adj MS 422.65 115.84 1370.65 97.46 F 2.12 0.58 6.87 0.49 P 0.142 0.642 0.015 0.699 Residual Error Lack-of-Fit Pure Error Total 9 4 5 17 1794.6 935.3 859.3 5175.8 1794.6 935.3 859.3 199.40 233.81 171.87 1.36 0.365 Unusual Observations for response Obs 8 StdOrder 9 response 100.000 Fit 81.449 SE Fit 10.836 Residual 18.551 St Resid 2.05 R R denotes an observation with a large standardized residual. Estimated Regression Coefficients for response using data in uncoded units Term Constant x1 x2 x3 x1*x1 x2*x2 x1*x2 x1*x3 x2*x3 Coef 98.8959 1.27146 1.36130 -6.12500 -5.41231 -14.0744 2.87500 -2.62500 -4.62500 Resi dual Plots for respo nse Normal Probability Plot 99 90 50 10 1 -20 -10 0 Resi dual 10 20 60 70 80 Fi tted Va lue 90 100 Residual Percent 20 10 0 -10 Versus Fits Histogram 6.0 Frequency Residual 4.5 3.0 1.5 0.0 -15 -10 -5 0 5 Resi dual 10 15 20 2 4 20 10 0 -10 Versus Order 6 8 10 12 14 16 Obser vation Order 18 20 The response model for the process robustness study is : y(x,z)=f(x)+h(x,z)+ = x x x x x x z x z x z y(x,z)=98.8959+1.27146x +1.36130x -5.41231x -14.0744x +2.875x x -6.125z-2.625x z-4.625x z Therefore the mean model is Ez[y(x,z)]=f(x)= 98.8959+1.27146x +1.36130x -5.41231x -14.0744x +2.875x x The variance model is Vz[y(x,z)]= , = (-6.125-2.625x1-4.625x2)2+ Now we assume that the low and high levels of the noise variable z have been run at one standard deviation either side of its typical or average value, so that =1 and since the residual mean square from fitting the response model is MSE=199.40will use =MSE=199.40 Therefore the variance model Vz[y(x,z)]= (-6.125-2.625x1-4.625x2)2+199.40 Following figures show response surface contour plots and three-dimensional surface plots of the mean model and the standard deviation respectively. C ont our P lot of M ean vs x 2, x1 1.5 60 50 70 Hold Values z 0 1.0 0.5 x2 0.0 -0.5 -1.0 -1.5 -1.5 -1.0 -0.5 80 90 70 80 50 60 0.0 x1 0.5 1.0 1.5 Co nt our Pl ot of St de v v s x 2, x1 1.5 1.0 0.5 18.0 19.5 16.5 21.0 22.5 Hold Values z 0 x2 0.0 -0.5 -1.0 -1.5 -1.5 -1.0 -0.5 0.0 x1 0.5 1.0 1.5 15.0 Surface Plot of Mean vs x2, x1 Hold Values z 0 100 80 Mean 60 2 40 0 -2 0 x1 2 -2 x2 S u r f ac e P l ot o f S t d e v v s x 2, x 1 Hold Values z 0 22.5 S t d e v 20.0 17.5 15.0 0 -2 -1 x1 -1 0 1 -2 1 x2 The objective of the robustness study is to find a set of operating conditions that would result in a mean response greater than 90 from the mean model with the minimum contour of standard deviation. The unshaded region of the following plot indicates operating conditions on x1 and x2, where the requirements for the mean response larger than 90 are satisfied and the response standard deviation do not exceed 14.5. C o n t o u r P lo t o f S t d e v , M e an 1.5 1.0 0.5 x2 0.0 -0.5 -1.0 -1.5 -1.5 -1.0 -0.5 0.0 x1 0.5 1.0 1.5 Stdev 14 14.5 Mean 90 100 Hold Values z 0 Actually, if we use Excel Solver, we can get a optimal solution for minimizing the standard deviation with the constraint that the mean value is greater than 90. The optimal solution is : mean=90 Stdv=14.18 X1=-0.83 x2=-0.58 This solution conforms to the analysis we did above. ...
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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