ADT

2015 12 41 12 December 2015 Journal of Beijing University of Aeronautics and Astronautics Vol 41 No 12 http bhxb buaa edu cn jbuaa@ buaa edu cn DOI 10 13700 /j bh 1001-5965 2014 0790 1 2 * 1 2 1 1 100191 2 100191 AD AD B114 3 A 1001-5965 2015 12-2225-07 1 4 AD 2-3 5 1 2 3 6 7 2014-12-15 2015-02-12 2015-03-18 14 37 www cnki net /kcms /detail /11 2625 V 20150318 1437 004 html 61104182 YWF-14-KKX-004 1989 liule89@ gmail com * 1980 leexy@ buaa edu cn J 2015 41 12 2225-2231 Liu L Li X Y Jiang M Evaluation method for accelerated degradation testing with interval analysis J Journal of Beijing University of Aeronautics and Astronautics 2015 41 12 2225-2231 in Chinese

2226 2015 8 b 2 f u b 1 b 2 = 槡 2πu 3 - b 2 2 u - b 1 [ ] u > 0 2b 2 1u 4 9-10 R t C σ μ = Φ( C - μλ t σ 槡 Λ t ) - exp ( 2μC 2 ) Φ σ ( - C + μλ t ) 5 σ 槡 Λ t t r = Λ -1 R -1 μ C σ R = r 6-1 r X - X - μ μ - μ - 1 2 1 1 1 - - - Wiener 15 Z Y = A Y = A 0 + A 1 y 1 + + A p y p 7 Y = 1 y X t 1 y p p X t = μλ t + σb Λ t 1 A = A 0 A 1 A p X 0 0 X t = X t - A i = a i c i a i c i X 0 σ B A i 7 Λ t t μ Z= a 0 c 0 + a 1 c 1 y j1 + + a p c p y jp = a y j c y j 8 Arrhenius a = a 0 a 1 a p c = c 0 c 1 c p Eyring ln μ = β 0 + β 1 φ s 2 y j = 1 y j1 y jp φ s β 0 β 1 ΔX i ~ N μδt i σ 2 Δt i 3 ΔX i = X t i - X t i - 1 Δt i = Λ t i - Z * Y = A * Y = A * 0 + A * 1 y 1 + + A * p y p 9 Λ t i - 1 Z * Y = A * Y = A * 0 + A * 1 y 1 + + A * p y p C 10 Λ ~ IG C /μ C 2 /σ 2 IG b 1 b 2 A * i = a i d i A * i = a i b 1 > 0 b 2 > 0 c i

12 min a d d ( n 2227 y j y j ) d + ξa a a y j + d y j Z - subject a y j - d y j Z - to d i 0 i = 0 1 p max a c 2 n c ( n y j y j ) c + ξc c a y i + c y i Z - subject a y j - c y j Z - to c i 0 i = 0 1 p 2 n 11 12 Z Y Z * Y Z Y Z * Y 13 13 A * i d i c i + d i d i 0 7 min a c d d ( n y j y j ) d + ξ a a + c c a y j + c y j + d y j Z - a y j - c y j - d y j Z - subject a y i + c y i Z - to a y j - c y j Z - c i 0 d i 0 i = 0 1 p 2 n 14 1 2 m j t ijk 1 E X t = μλ t Λ t - X - X - 1 μ - ij μ μ - ij σ 2 13 - - 1 1 Fig 1 Flowchart of interval accelerated degradation testing analysis CSAD x - ijk x - ijk i i = 1 2 K j 2 n i k k = t 2 1 ijk x - ijk x - ijk 7 p = 1 11 - A 1 μ - ij μ - ij σ

2228 2015 18 19 3 σ K σ 2 i = 1 = n i m j k = 2 e ijk - e ij k -1 2 15 K n i m j Δt ijk i = 1 k = 2 e ijk e ijk = 3 1 x ijk - μ ijk Λ t ijk e * ijk = x - * ijk - x - 18 ijk 2 * 2 + x - ijk - - x 槡 ijk 16 16 15 σ 2 μ - 0 μ - 0 1 2 1 2 1 able 1 Information on accelerated wear degradation i μ - ij μ - testing for metal alloy ij μ - i μ - i n { i μ - i = min { μ - ij μ - } μ - n i i = max { μ - ij μ - 17 } s i μ - i μ - i 7 p 14 a c d s 0 μ - 0 μ - 0 2 2 2 3 Fig 2 Degradation data for accelerated wear testing under 5 6 three applied weights 2 1 μ - 0 μ - 0 σ 1 Δ i Δ i ~ N m ζm m ζ R R - t - R t C σ μ 0 μ - 0 μ - 0 = C σ μ - 0 R t t - - r t r = Λ -1 R -1 μ - 0 Λ -1 R -1 μ - 0 C σ - μ 0 C σ R = r C σ R = r 18 19 3 /g /h 1 10 4 2 5 10 20 50 100 200 500 2 50 4 2 5 10 20 50 100 200 500 3 100 4 2 5 10 20 50 100 200 500 ζ = 0 02 ζ = 0 2 i ± Δ i m m

12 2229 0 07 0 1 0 3 0 5 0 7 1 μm m C = 50 μm 1 5 g 2 φ s = W W 2 Λ t = ln t m = 0 5 μm 3 1 2 1 ζ = 0 02 0 m μ - ij μ - ij σ = 0 677 2 2 W i μ - i μ - i 14 3 μ - 0 μ - 0 = 0 784 4 0 938 6 2 0 887 18 19 4 0 9 exp 46 72 55 19 h Fig 3 3 3 μ Draft coefficient μ under three applied weights 5 ~ 7 μ 0 σ 5 a m 1 0 86 0 91 5 b μ 0 m μ 0 σ 6 4 Fig 4 m = 0 5 μm Interval reliability curves when m = 0 5 μm 3 2 5 m m 0 01 0 03 0 05 Fig 5 Center and radius of draft coefficients under different m values in two cases

2230 7 2015 6 m Fig 6 Diffusion coefficients under different m values in two cases 3 Fig 7 m R = 0 9 R = 0 9 under different m values in two cases m σ 5 6 7 a 7 b 1 m 2 m 4 11 Ye Z S μ 0 1 2 μ 0 σ References 1 Nelson W B Accelerated testing Statistical models test plans and data analysis M New York John Wiley & Sons 2009 493-544 2 Meeker W Q Escobar L A Lu C J Accelerated degradation tests Modeling and analysis J echnometrics 1998 40 2 89-99 3 J 2007 28 8 1002-1007 Deng A M Chen X Zhang C H et al A comprehensive review of accelerated degradation testing J Acta Armamentarii 2007 28 8 1002-1007 in Chinese 4 Wang Z Huang H Z Du L Reliability analysis on competitive failure processes under fuzzy degradation data J Applied Soft Computing 2011 11 3 2964-2973 5 Gonzalez-Gonzalez D S Alejo R J P Cantu-Sifuentes M et al A non-linear fuzzy regression for estimating reliability in a degradation process J Applied Soft Computing 2014 16 137-147 6 Alefeld G Mayer G Interval analysis heory and applications Center and radius of reliable lifetimes when J Journal of Computational and Applied Mathematics 2000 121 1-2 421-464 7 anaka H Lee H Interval regression analysis by quadratic programming approach J IEEE ransactions on Fuzzy Systems 1998 6 4 473-481 8 J 2008 29 3 611-615 Wang J Qiu Z P Wang X J Interval analysis for stress intensity factors J Acta Aeronautica et Astronautica Sinica 2008 29 3 611-615 in Chinese 9 Yu I Chang C L Applying Bayesian model averaging for quantile estimation in accelerated life tests J IEEE ransactions on Reliability 2012 61 1 74-83 10 Chateauneuf A Accelerated life testing and degradation modeling J Reliability Engineering & System Safety 2014 131 228 Xie M Stochastic modelling and analysis of degradation

12 for highly reliable products J Applied Stochastic Models in Business and Industry 2014 31 1 13-32 12 Park C Padgett W J Stochastic degradation models with several accelerating variables J IEEE ransactions on Reliability 2006 55 2 379-390 13 Pan Z Q Balakrishnan N Multiple-steps step-stress accelerated degradation modeling based on Wiener and Gamma processes J Communications in Statistics-Simulation and Computation 2010 39 7 1384-1402 14 Li X Y Jiang M Sun F Q et al Constant stress AD for superluminescent diode and parameter sensitivity analysis J Eksploatacja I Niezawodnosc-Maintenance and Reliability 2010 2231 2 21-26 15 Wang X Jiang P Guo B et al Real-time reliability evaluation with a general Wiener process-based degradation model J Quality and Reliability Engineering International 2014 30 2 205-220 16 Escobar L A Meeker W Q A review of accelerated test models J Statistical Science 2006 21 4 552-577 17 Chhikara R S Folks J L he inverse Gaussian distribution heory methodology and applications M New York CRC Press 1988 23-29 18 Meeker W Q Escobar L A Statistical methods for reliability data M New York John Wiley & Sons 1998 631 Evaluation method for accelerated degradation testing with interval analysis LIU Le 1 2 LI Xiaoyang * 1 2 JIANG ongmin 1 1 School of Reliability and Systems Engineering Beijing University of Aeronautics and Astronautics Beijing 100191 China 2 Science and echnology on Reliability and Environmental Engineering Laboratory Beijing 100191 China Abstract raditional evaluation methods of accelerated degradation testing AD are based on precise degradation data to conduct reliability and lifetime assessment However with interfere of the uncertainties from human factors the test data can be imprecise represented by interval rather than precise data Under this consideration an interval analysis method for AD evaluation was proposed based on Wiener process which included possibility and necessity models Interval regression method was firstly used to transfer the problems of modeling interval degradation data under different accelerated stress levels into quadratic programming problems he interval drift coefficients under different stress levels with possibility model and diffusion coefficient were obtained hen the interval drift coefficients were extrapolated to normal stress condition with accelerated model under necessity model and further to analyze the relationship between measurement uncertainty and reliability and lifetime evaluation results Finally the numerical study was used to present and verify the proposed methodology and conduct uncertainty sensitivity analysis he results show that both reliability and lifetime evaluation results are effected by epistemic uncertainty of measurement and their correctness can be ensured with decreasing epistemic uncertainty Key words accelerated degradation testing AD interval analysis reliability life evaluation epistemic uncertainty sensitivity analysis