Project I 11.27.2014 Group 8 Saeb Moosavi, Shakil Bin Zaman MAE 430 Introduction To Reliability In Mechanical Engineering Design Professor Ji Ho Song MAE 430 Project I 1
Contents • Data set 1 (Saeb) Linearity Test • K-S test Data set 2 (Shakil) Distribution analysis • Data set 3 (1+2: Shakil + Saeb) • Conclusion MAE 430 Project I 2
600 38 128 270 376 66 120 47 134 382 DATA SET 1 – Saeb 187 656 266 26 data 661 337 57 681 384 565 53 632 373 141 7 378 253 MAE 430 Project I 3
600 38 128 270 376 Target: select the best distribution of given data 66 120 47 Step1 Cumulative distribution function estimation 134 382 187 Step2 Probability estimation using probability paper 656 266 661 337 Step3 Goodness of fit test (Kolomogorov-Smirnov test) 57 681 384 565 53 632 373 141 7 378 26 Data 253 MAE 430 Project I 4
C.D.F. estimation method • C.D.F. estimation method F ( x j ) = j / n • Simple cumulative distribution(X) F ( x j ) = ( j - 0.5) / n • Symmetric simple cumulative distribution F ( x j ) = j /( n +1) • Mean rank F ( x j ) = ( j - 0.3) /( n + 0.4) • Median rank F ( x j ) = ( j -1) /( n -1) • (Mode rank) (X) F ( x j ) = ( j - 0.375) /( n + 0.25) • The rest method Simple C.D. & Mode rank can’t be use because F(x) = 0 or 1 MAE 430 Project I 5
Probability distribution MAE 430 Project I 6
Symmetric simple cumulative distribution Equation y = a + b* 2 2 Adj. R-Squar 0.91477 Value Standard Err Equation y = a + b* Norm Intercept -1.2884 0.09701 Adj. R-Squar 0.88426 Norm Slope 0.0043 2.61958E-4 1 Value Standard Erro 1 B Intercept -4.3861 0.32344 B Slope 0.83085 0.05996 0 Norm 0 B -1 -1 LogNormal Normal -2 -2 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A 2 2 Equation y = a + b* Adj. R-Squar 0.9666 1 Value Standard Erro 1 C Intercept -6.3174 0.21831 C Slope 1.0894 0.04047 0 0 -1 -1 C C Equation y = a + b -2 -2 Adj. R-Squ 0.8072 Value Standard Er C Intercept -2.091 0.18333 -3 Weibull -3 C Slope 0.0050 4.95067E-4 Biexponential -4 -4 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A MAE 430 Project I 7
Mean Rank 2.0 Equation y = a + b* 2.0 Adj. R-Squa 0.92916 1.5 Equation y = a + b Value Standard Err 1.5 C Intercept -1.1853 0.08076 Adj. R-Squa 0.88285 C Slope 0.0039 2.18083E-4 Value Standard Err 1.0 1.0 C Intercept -4.0022 0.29714 C Slope 0.7581 0.05509 0.5 0.5 0.0 C 0.0 C -0.5 -0.5 -1.0 -1.0 LogNormal -1.5 Normal -1.5 -2.0 -2.0 0 100 200 300 400 500 600 700 2 3 4 5 6 7 A A 2 2 Equation y = a + b Adj. R-Squ 0.84288 Value Standard Er Equation y = a + b 1 1 Adj. R-Squa 0.95881 C Intercept -1.923 0.14794 Value Standard Err C Slope 0.0046 3.9949E-4 C Intercept -5.6530 0.21671 0 C Slope 0.97004 0.04018 0 -1 -1 C C -2 -2 Weibull Biexponential -3 -3 -4 -4 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A MAE 430 Project I 8
Median Rank Equation y = a + b* Adj. R-Squa 0.92212 2.0 2.0 Value Standard Err C Intercept -1.2423 0.08907 Equation y = a + b 1.5 1.5 C Slope 0.0041 2.40535E-4 Adj. R-Squ 0.88386 Value Standard Er 1.0 1.0 C Intercept -4.212 0.31123 C Slope 0.7979 0.0577 0.5 0.5 0.0 0.0 C C -0.5 -0.5 -1.0 -1.0 LogNormal -1.5 Normal -1.5 -2.0 -2.0 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A 2 Equation y = a + b*x 2 Adj. R-Squar 0.84288 Value Standard Error Equation y = a + C Intercept -1.9237 0.14794 1 Adj. R-Squ 0.9635 1 C Slope 0.00464 3.9949E-4 Value Standard Er C Intercep -6.008 0.21666 0 0 C Slope 1.0339 0.04017 -1 -1 C C -2 -2 -3 Weibull Biexponential -3 -4 -4 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A MAE 430 Project I 9
The Rest Method Equation y = a + b* 2 Equation y = a + 2 Adj. R-Squar 0.91972 Adj. R-Squ 0.88405 Value Standard Err Value Standard Er C Intercept -1.2586 0.09174 C Slope 0.0042 2.4773E-4 C Intercep -4.273 0.31543 1 1 C Slope 0.8094 0.05848 0 0 C C -1 -1 LogNormal Normal -2 -2 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A 2 Equation y = a + b* 2 Adj. R-Squar 0.81911 Value Standard Err Equation y = a + C Intercept -2.041 0.17181 1 1 Adj. R-Squ 0.96477 C Slope 0.0049 4.63975E-4 Value Standard Er C Intercep -6.115 0.21693 0 0 C Slope 1.0530 0.04022 -1 -1 C C -2 -2 Weibull -3 -3 Biexponential -4 -4 2 3 4 5 6 7 0 100 200 300 400 500 600 700 A A MAE 430 Project I 10
R-correlation coefficient Symm.S.C Mean Rank Median Rank Rest R 2 R 2 R 2 R 2 R R R R Normal 0.9181 0.95822 0.9319 0.9654 0.9252 0.96189 0.9229 0.96069 Log-Normal 0.8888 0.94281 0.7667 0.87565 0.8884 0.9426 0.974 0.98694 Weibull 0.9679 0.98384 0.756 0.8695 0.8954 0.9463 0.787 0.88717 Bi-exponential 0.8149 0.90272 0.8491 0.9215 0.8491 0.9215 0.8263 0.90903 MAE 430 Project I 11
Kolmogorov-Smirnov Test (K-S test) Normal & LogNormal α = 0.198 D n α = 0.01 Weibull & Extreme Value α = 0.204 D n n = 26 Normal & LogNormal α = 0.170 D n α = 0.05 Weibull & Extreme Value α = 0.175 D n MAE 430 Project I 12
K-S Test Symmetric simple cumulative distribution σ = 232.56 σ = 1.203 μ = 299.53 μ = 5.279 1.0 1.0 0.8 0.8 0.6 0.6 B B 0.4 0.4 Normal 0.2 a = 0.01, D = 0.198 0.2 LogNormal a = 0.05, D = 0.170 a = 0.01, D = 0.198 a = 0.05, D = 0.170 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A m = 1.0894 ξ = 200 1.0 1.0 ξ = 329.96 X 0 = 418.2 0.8 0.8 0.6 0.6 B B 0.4 0.4 Weibull Biexponential 0.2 0.2 a = 0.01, D = 0.204 a = 0.01, D = 0.204 a = 0.05, D = 0.175 a = 0.05, D = 0.175 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A MAE 430 Project I 13
K-S Test Mean Rank σ = 256.41 σ = 1.319 1.0 1.0 μ = 303.92 μ = 5.279 0.8 0.8 0.6 0.6 B B 0.4 0.4 Normal LogNormal 0.2 0.2 a = 0.01, D = 0.198 a = 0.01, D = 0.198 a = 0.05, D = 0.170 a = 0.05, D = 0.170 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A m = 0.97 ξ = 217.39 1.0 1.0 ξ = 339.54 X 0 = 418.04 0.8 0.8 0.6 0.6 B B 0.4 0.4 Weibull 0.2 Biexponential 0.2 a = 0.01, D = 0.204 a = 0.01, D = 0.204 a = 0.05, D = 0.175 0.0 a = 0.05, D = 0.175 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A MAE 430 Project I 14
K-S Test Median Rank σ = 243.9 σ = 1.253 1.0 1.0 μ = 303 μ = 5.279 0.8 0.8 0.6 0.6 B 0.4 B 0.4 LogNormal Normal 0.2 a = 0.01, D = 0.198 0.2 a = 0.01, D = 0.198 a = 0.05, D = 0.170 a = 0.05, D = 0.170 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A m = 1.0339 ξ = 215.52 1.0 1.0 ξ = 333.95 X 0 = 414.59 0.8 0.8 0.6 0.6 B B 0.4 0.4 Weibull Biexponential a = 0.01, D = 0.204 0.2 0.2 a = 0.01, D = 0.204 a = 0.05, D = 0.175 a = 0.05, D = 0.175 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A MAE 430 Project I 15
K-S Test The Rest Method σ = 238.09 σ = 1.235 1.0 1.0 μ = 299.66 μ = 5.279 0.8 0.8 0.6 0.6 B 0.4 B 0.4 Normal LogNormal 0.2 a = 0.01, D = 0.198 0.2 a = 0.01, D = 0.198 a = 0.05, D = 0.170 a = 0.05, D = 0.170 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A m = 1.053 ξ = 204.08 1.0 1.0 ξ = 332.69 X 0 = 416.53 0.8 0.8 0.6 0.6 B B 0.4 0.4 Weibull a = 0.01, D = 0.204 Biexponential 0.2 0.2 a = 0.05, D = 0.175 a = 0.01, D = 0.204 a = 0.05, D = 0.175 0.0 0.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 A A MAE 430 Project I 16
Discussion According to K-S test results the best distribution function for this data set is Weibull distribution because data are more concentrated in the center of limit line in this distribution. MAE 430 Project I 17
262 500 328 91 608 119 15 164 DATA SET 2 – Shakil 211 89 668 21 data 278 319 116 330 449 74 128 622 223 98 MAE 430 Project I 18
Symmetrical Simple Cumulative Distribution C Equation y = a + b*x D Adj. R-Square 0.85896 Linear Fit of C Linear Fit of D 2.0 Value Standard Error 2.0 C Intercept -1.15535 0.14318 C Slope 0.00452 4.41996E-4 1.5 1.5 Equation y = a + b*x 1.0 1.0 Adj. R-Square 0.82697 Value Standard Error D Intercept -3.10377 0.35302 0.5 D Slope 3.90427 0.4305 0.5 0.0 0.0 C D -0.5 -0.5 LogNormal -1.0 Normal -1.0 -1.5 -1.5 -2.0 -2.0 0 100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0 1.2 A A E Linear Fit of E 1 K Equation y = a + b*x Linear Fit of K 2 Adj. R-Square 0.71158 Value Standard Error Equation y = a + b*x Adj. R-Square 0.89182 K Intercept -1.88455 0.25577 Value Standard Error 0 K Slope 0.00517 7.89547E-4 1 E Intercept -4.44447 0.34076 E Slope 4.9369 0.41554 0 -1 E -1 K -2 -2 Biexponential Weibull -3 -3 -4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 100 200 300 400 500 600 700 A A MAE 430 Project I 19
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