MAE430 Reliability Engineering in ME Term Project II Jae Hyung Cho 20101103 Andreas Beckmann 20156476.

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Contents Project I results summary Results using theoretical probability distribution Results using the graphical procedure Conclusion 3

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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

2

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.

4

Contents

• Project I results summary• Results using theoretical probability distribution• Results using the graphical procedure• Conclusion

5

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

7

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

21

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• ,

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