mae430 reliability engineering in me term project ii jae hyung cho 20101103 andreas beckmann...
DESCRIPTION
Contents Project I results summary Results using theoretical probability distribution Results using the graphical procedure Conclusion 3TRANSCRIPT
MAE430 Reliability Engineering in METerm Project II
Jae Hyung Cho 20101103Andreas Beckmann 20156476
Contents
• Project I results summary• Results using theoretical probability distribution• Results using the graphical procedure• Conclusion
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Contents
• Project I results summary• Results using theoretical probability distribution• Results using the graphical procedure• Conclusion
3
Project I results summary
• Jae Hyung’s Data Set (n = 63)– Best fitting distribution: Biexponential Distribution– Best CDF estimation method: Median Rank
• Andreas’s Data Set (n = 59)– Best fitting distribution: Weibull Distribution– Best CDF estimation method: Symmetric S. C. D.
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Contents
• Project I results summary• Results using theoretical probability distribution• Results using the graphical procedure• Conclusion
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Strength and Stress
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Data Set 1 42 207 371 440
60 225 380 449
96 232 386 449
96 233 390 461
114 235 393 468
130 239 393 473
132 239 399 484
134 259 400 490
150 262 405 499
150 268 410 506
159 299 412 514
186 305 413 527
187 306 414 544
188 315 415 546
194 340 422 606
205 345 435
Mean Strength
Data Set 2 16 139 281 428
27 140 292 430
30 151 313 441
30 152 314 446
40 154 323 450
49 157 325 460
53 168 336 463
81 175 360 488
87 189 364 513
89 207 384 513
93 209 398 547
99 236 408 561
108 238 421 573
123 240 423 601
134 242 424
Mean Stress
Calculation of PDF Using Wolfram Alpha
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Strength: BiexponentialCDF calculation: Median Rankξ = 118.9061X0 = 391.3413
Strength: WeibullCDF calculation: Symmetric S.C.D.m = 1.41297ξ = 308.2052
Theoretical probability distribution
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Strength
Stress
Numerical Integration Using Matlab
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>> f = @(x)(1-exp(-exp((x-391.3413)/118.9061))).*(0.000430008.*exp(-0.000304329.*(x.^1.41297)).*(x.^0.41297));
>> P_f = integral(f, 0, Inf)
P_f = 0.3792
>> g = @(x)(1-exp(-(x/308.2052).^(1.41297))).*(0.000312936*exp(-0.0372099.*exp(0.00841*x)+0.00841*x)); >> R = integral(g, 0, Inf)
R = 0.6208
Stress-based Strength-based
The two formulas yield the same result !
Integration from to : Same Result
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% f_stress_smaller0 = 0; % F_stress_smaller0 = 0;% f_stress_larger0 = 0.000430008.*exp(-0.000304329.*(x.^1.41297))*(x.^0.41297);% F_stress_larger0 = 1-exp(-(x./308.2052).^(1.41297));% f_strength = 0.000312936.*exp(-0.0372099.*exp(0.00841.*x)+0.00841.*x);% F_strength = 1-exp(-exp((x-391.3413)/118.9061)); % term_for_R_smaller0 = (f_strength * F_stress_smaller0);% term_for_R_larger0 = (f_strength * F_stress_larger0);
integrand_R_smaller0 = @(x) 0.000312936.*exp(-0.0372099.*exp(0.00841.*x)+0.00841.*x) .* 0;integrand_R_larger0 = @(x) (0.000312936.*exp(-0.0372099.*exp(0.00841.*x)+0.00841.*x)) .* (1-exp(-(x./308.2052).^(1.41297)));integrand_Pf_smaller0 = @(x) 0 .* ( 1-exp(-exp((x-391.3413)./118.9061)));integrand_Pf_larger0 = @(x) (0.000430008.*exp(-0.000304329.*(x.^1.41297)).*(x.^0.41297)) .* (1-exp(-exp((x-391.3413)./118.9061)));
R = integral(integrand_R_smaller0, -inf , 0) + integral(integrand_R_larger0 , 0, inf)Pf = integral(integrand_Pf_smaller0, -inf , 0) + integral(integrand_Pf_larger0 , 0, inf)
R = 0.620773031855217
Pf = 0.3792290345758780.6208 + 0.3792 = 1
Contents
• Project I results summary• Results using theoretical probability distribution• Results using the graphical procedure• Conclusion
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𝑷 𝒇
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Lower Limit
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Upper Limit
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Triangle Method
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𝑅
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Lower Limit
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Upper Limit
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Triangle Method
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Comparison of and
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TheoreticalGraphical ( calculation)
Lower Upper Triangle
(=1-)
TheoreticalGraphical ( calculation)
Lower Upper Triangle
(=1-)
Most conservative values
Graphical method calculation check:
Contents
• Project I results summary• Results using theoretical probability distribution• Results using the graphical procedure• Conclusion
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Conclusion
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Theoretical Calculation vs. Graphical Calculation• Data sets do not exactly represent biexponential and Weibull
distributions Difference between theoretical and graphical method results
Lower vs. Upper vs. Triangle (for )• The upper limit method is the most conservative• Conservative design is preferred in mechanical engineering
Selected Reliability and Probability of Failure• ,