, >0, >0. t t (1) I. INTRODUCTION. Where

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1 Acceleraed Life Tesing Model for a Generalized Birnbaum-Saunders Disribuion Yao Cheng and E. A. Elsayed Deparmen of Indusrial and Sysems Engineering Rugers Universiy Piscaaway, NJ ygli0708@gmail.com, elsayed@rci.rugers.edu Absrac Faigue failures caused by cyclic sresses are commonly modeled by Birnbaum-Saunders (B-S) and Weibull disribuions. Someimes, maerials wih high cycle faigue exhibi bimodal failure raes which are difficul o model wih Weibull disribuion. The objecive of his paper is o invesigae a general Birnbaum-Saunders (GB-S) disribuion which covers diverse hazard raes such as increasing, upside down, mulimodal and ohers. This paper also uilizes GB-S disribuion o model faigue failure daa and validae is performance under differen condiions. GB-S-based-ALT model is also developed o provide accurae reliabiliy and lifeime predicion a operaing condiions using daa from acceleraed life esing experimens. Index Terms- Birnbaum-Saunders Disribuion, SB-S, GB-S, ALT, inverse power-law, general log-linear model, MLE I. INTRODUCTION Faigue, recognized as one of he main causes of failures of mechanical and elecrical componens, is a class of srucural damage ha occurs when maerial is exposed o cyclic applicaion of sress wih varying or consan ampliudes. Failure caused by faigue in meallic srucures is a pervasive phenomenon and failure of srucural maerials under cyclic applicaion of sress or srain is now a problem of increasing ineress of indusry because mos of he mechanical componens wor under cyclic sresses wih varying or consan ampliudes during heir lifeime of operaion. A single saic sress or srain, which is far below he hreshold of he srucure and causes no damage o he srucure if applied once, could induce faigue failure if applied repeaedly. Thus, failure daa is imporan, however, obaining failure daa under design sress level o predic componens reliabiliy and lifeime direcly is no always feasible. This paper models faigue failure observed in maerials which are subjec o single-ype-muliple-level mechanical sress wih a general Birnbaum-Saunders (GB-S) disribuion which is more flexible in modeling diverse failure raes. Addiionally, his paper invesigaes he applicaion of acceleraed life esing models o he GB-S disribuion in order o esimae reliabiliy a normal operaing condiions. A. Birnbaum-Saunders Life Model Birnbaum and Saunders [] propose he sandard Birnbaum- Saunders (SB-S) disribuion o model faigue failure ime of unis when a dominan crac, which is caused by cyclic or oher ypes of sresses, surpasses or reaches a predeermined crac lengh hreshold. If T denoes a specimen s lifeime, he cumulaive densiy funcion (cdf) of T is approximaely given by: Pr T = =,, >0, >0 is he shape parameer and is he scale parameer of he disribuion, is he failure hreshold level, is he sandard deviaion of he failure ime daa and is he cumulaive normal disribuion funcion. The expeced ime o failure, variance, sewness and urosis of his disribuion are invesigaed by Birnbaum and Saunders []. Desmond [] noes ha he SB-S disribuion applies even when he assumpion of SB-S is relaxed, ha is, crac incremen in a cerain cycle no only depends on he curren loading bu also is affeced by he oal crac size caused by previous cycles. The h momen of SB-S disribuion can be obained by he momen generaing funcion (MGF) as saed by Riec [3]. I is shown ha he SB-S hazard funcion is unimodally upsidedown and he funcional approximaion of he changing poin is given by Kundu [4]. Saisical analysis for he SB-S disribuion is also developed. Since exac disribuions of he MLEs are no available, Engelhard and Wrigh [5] presen asympoic ()

2 disribuions o consruc confidence inervals of he parameers. Two modified momen esimaors ha boh improve momen esimaion (MME) and maximum lielihood esimaion (MMLE) are proposed. MME and MMLE exhibi heir own pros and cons under differen condiions as sudied by Kundu and Balarishnan [6]. Dupuis and Mills [7] provide robus esimaion of he parameers and quaniles of SB-S disribuion since in pracice he colleced daa do no always follow he SB-S model. The inference procedure for he SB-S disribuion wih symmerically incomplee daa is derived by Desmond [] since in pracice i is common o end a life esing before all unis under es fail. Desmond [8] develops a log-linear model based on he SB-S disribuion which considers random effecs and sudies he performance of various esimaion and predicion mehods. SB-S disribuion is considered as one of he normal disribuion family and is relaionship wih similar disribuions is discussed and invesigaed by Desmond [8], Kundu [9], ec. Desmond [] saes ha SB-S disribuion is more flexible han Inverse Gaussian (IG) disribuion, whereas he IG disribuion seems o have applicaions for incomplee daa bu SB-S disribuion has difficuly in incorporaing such daa. To generalize he SB-S model, wo GB-S disribuions are proposed by inroducing a second shape parameer. One of he GB-S disribuions proposed by Owen [0] considers he effec of sequence of loading and he crac exension hus he crac exension is modeled as a memory process. Owen [] also proposes anoher GB-S disribuion which is discussed in deails in his paper. This GB-S disribuion builds relaionship beween he lifeime T and sandard normal variable Z as: T Z T and are he shape parameers and is he scale parameer of his GB-S disribuion. Univariae and mulivariae exensions of he SB-S disribuion are given in Díaz-García and Dominguez-Molina []. Also, Díaz-García and Leiva-Sánchez [3] use a biological model o provide a more general derivaion of SB-S disribuion which yields a B- S disribuion family incorporaing lognormal and oher disribuions. B. Acceleraion Model Saisics-based models are usually used o he failure imes of faigue failure daa due o is abiliy in capuring he random variaion of failure imes due o differen paerns or levels of cyclic forces. To model faigue failure daa, researchers usually uilize he normal disribuion family o provide an accurae descripion of he failure ime disribuion. The SB-S disribuion is derived from he normal disribuion family. Compared wih Weibull, lognormal and oher disribuions which fi failure daa well especially wihin he cenral region () of he disribuion, SB-S disribuion has been shown o provide an accurae descripion of failure daa especially under low sress levels. Neverheless, predicing reliabiliy using faigue daa a normal operaing condiions migh no be feasible due o he exensive ime and resource needed. Therefore, esing unis a acceleraed or oher condiions and uilizing he failure imes observed a differen levels of sresses o predic he lifeime a operaing levels of sress is an appropriae alernaive approach. This ype of es is ermed an acceleraed life es (ALT). A model which relaes reliabiliy and lifeime under severe condiions o normal environmens is called an ALT model. The sress is no only referred o he mechanical force bu also includes oher ypes of sresses, such as humidiy, volage, emperaure, ec. Elsayed [4] classifies ALT models mainly ino several groups. The mos widely used are he parameric models (saisics-based) and he physics-saisics-based models. The parameric models assume ha failure ime a differen sress levels are relaed o each oher by a common failure ime disribuion wih differen parameers (usually a mean or scale parameers). The life-sress funcion, which is he funcion of applied sress, subsiues he scale parameers for differen levels of sresses while he shape parameer remains he same. Acceleraed failure ime (AFT) model is one of ypical parameric models. The physics-saisics-based models explain he relaionship beween applied sress and failure rae by uilizing he parameers of he physics of he device in conjuncion wih he saisical parameers o obain realisic models. The general log-linear relaionship is a general life-sress relaionship which incorporaes oher models, for example, he Arrhenius model, he inverse power law model and he Eyring model. Such lifesress relaionships can be applied in a specified underlying disribuion and has he effec of changing he mean or scale of he failure disribuion, bu he shape parameers remain he same. Ofen, applying life-sress relaionships o a disribuion increases he number of he unnown parameers. The inverse power law model is applied o he SB-S disribuion and he corresponding inference procedures are invesigaed by Owen [5]. II. GENERALIZED BIRNBAUM-SAUNDERS MODEL The hazard funcion of SB-S disribuion is resriced o be unimodal which fails o cover a wide range of failure rae ypes as menioned above. This paper invesigaes a generalized B-S model ha overcomes he limiaion of SB-S disribuion in modeling diverse failure raes. As saed in Eq., wih he second shape parameer inroduced, he cdf and hazard funcion of he GB-S model are: F;,, = (3) The hazard rae funcion is

3 f(;,, ) h (;,, ) R (;,, ) a e (4) Noe ha SB-S is a special case of he GB-S disribuion wih =0.5. A. Characerisics and Properies Wih Eq. and he general binomial heorem, he ransformaion beween T and Z is achieved as: For Z or, r r T Z, 0 For - Z, r r s T s Z, 0s 0 s For Z - or Z, r r r s r T s Z 0s 0 s Thus, he rh momen of he GB-S model is: r r r E r T I I (5) 0s 0 s s s s I, a s I r s r r s s, r s I is worh noing ha when Z, he momens of T do no exis. The expecaion, variance, sewness and urosis are obained as he special case of momens. The sewness and urosis are no affeced by he scale parameer. Figure reveals ha when becomes smaller, he urosis increases sharply. The pdf of GB-S disribuions remains unimodal for differen values of. Anoher imporan observaion abou he inverse of T is ha: if T belongs o GB-S disribuion wih parameers, and, hen T - also belongs o GB-S - disribuion wih he corresponding parameers, and, respecively. The sewness and urosis of T are he same as he sewness and urosis of T. B. GB-S Hazard Rae Figure shows GB-S hazard raes wih consan and varying since, he scale parameer of he disribuion has no effec on he shape so i is fixed a uniy for all cases. Also for all value of, he hazard rae funcion is always upsidedown when =0.5. In general, he hazard funcion of GB-S disribuion covers hree ypes of failure condiions: he hazard funcion can be eiher increasing or mulimodal when 0.5 ; he hazard funcion can be eiher upside down or mulimodal when 0.5 and he hazard funcion is always an upside down funcion of when 0.5. III. ACCELERATED MODELS Alhough he GB-S disribuion is more flxible in covering differen ypes of failure raes, is performance in modeling failure daa, especially acceleraed failure daa, needs o be invesigaed. This secions begins wih he developmen of GB- S-based acceleraed models, as well as Weibull and SB-S accelraed models, eiher in specific or general forms. In secion I, hese models are applied o he experimenal daa and heir performances is compared. A. Inverse Power Law-Based-Acceleraed Model Once a baseline lifeime disribuion wih scale parameer or mean is adoped and an approporiae acceleraion form is seleced according o he applied sress ype, he unnown parameers can be esimaed by observing failure imes a elevaed sress levels which are hen used o predic reliabiliy a normal operaing condiions.

4 f (; ) a exp a (7) Fig.. GB-S urosis for differen and =0.3 = =.5 The esimaion of he unnown model parameers in Eq. 7 can be obained by maxmizing he lielihood funcion for he observed acceleraed failure daa. Assuming wo sress levels and are applied and he corresponding wo failure ime daa ses are obained The lielihood funcion is obained as: h() n i L(,,, ;, ) f,,, ;, (8) ij i ij i i j Fig.. GB-S hazard raes wih consan and varied The inverse power law model is commonly used for nonhermal acceleraed sresses and is given as: h, - = >0, >0 h() is a quanifiable life measure, such as mean life orcharacerisic life represens he sress level, are model parameers. By subsiuing he scale parameer wih he acceleraed life model h, he acceleraed inverse power law GB-S model can hen be wrien as: F ; (6) i is he ih sress level j is he jh failure daa in he corresponding daa se n i is he number of observaions in ih daa se ij represens he jh failure observaion in he daa se obained under ih sress level i is he ih sress level In his paper, daa are obained from he Insrumen Developmen Uni of he Physical Research Saff, Boeing Aircraf Company, by subjecing meal-coupons o repeaed alernaing sresses and srains. The hree daa ses obained under hree sress levels are lised below, 4 Sample (Sress/cycle:. 0 psi): 3.70, 7.06, 7.06, 7.46, 7.85, 7.97, 8.44, 8.55, 8.58, 8.86, 8.86, 9.30, 9.60, 9.88, 9.90, 0.00, 0.0, 0.6, 0.8, 0.0, 0.55, 0.85,.0,.0,.08,.5,.0,.34,.40,.99,.00,.00,.03,.,.35,.38,.5,.58,.6,.69,.70,.90,.93, 3.00, 3.0,3.3, 3.5, 3.30, 3.55, 3.90, 4.6, 4.9, 4.0, 4.0, 4.50, 4.5, 4.75, 4.78, 4.8, 4.85, 5.0, 5.05, 5.3, 5., 5., 5.30, 5.40, 5.60, 5.67, 5.78, 5.94, 6.0, 6.04, 6.08, 6.30, 6.4, 6.74, 7.30, 7.50, 7.50, 7.63, 7.68, 7.8, 7.8, 7.9, 8.0, 8.68, 8.8, 8.90, 8.93, 8.95, 9.0, 9.3, 9.40, 9.45, 0.3,.00,.30,.5,.68, 4.40

5 4 Sample (Sress/cycle:.6 0 psi):.33,.58,.68,.76,.90,3.0,3.,3.5,3.8,3.,3.,3.9, 3.35,3.36,3.38,3.38,3.4,3.4,3.4,3.44,3.49,3.50,3.50,3.5, 3.5,3.5,3.5,3.56,3.58,3.58,3.60,3.6,3.63,3.66,3.67,3.70, 3.70,3.7,3.7,3.74,3.75,3.76,3.79,3.79,3.80,3.8,3.89,3.89, 3.95,3.96,4.00,4.00,4.00,4.03,4.04,4.06,4.08,4.08,4.0,4., 4.4,4.6,4.6,4.6,4.0,4.,4.3,4.6,4.8,4.3,4.3,4.33, 4.33,4.37,4.38,4.39,4.39,4.43,4.45,4.45,4.5,4.56,4.56,4.60, 4.64,4.66,4.68,4.70,4.70,4.73,4.74,4.76,4.76,4.86,4.88,4.89, 4.90,4.9,5.03,5.7,5.40, Sample 3 (Sress/cycle: 3. 0 psi): 0.7,0.9,0.96,0.97,0.99,.00,.03,.04,.04,.05,.07,.08,.08,.08,.09,.09,.,..3,.4,.4,.4,.6,.9,.0,.0,.0,.,.,.3,.4,.4,.4,.4,.4,.8,.8,.9,.9,.30,.30,.30,.3,.3,.3,.3,.3,.3,.3,.3,.33,.34,.34,.34,.34,.34,.36,.36,.37,.38,.38,.38,.39,.39,.4,.4,.4,.4,.4,.4,.4,.4,.44,.44,.45,.46,.48,.48,.49,.5,.5,.5,.55,.56,.57,.57,.57,.57,.58,.59,.6,.63,.63,.64,.66,.66,.68,.70,.74,.96,. To examine he performance of inverse power law GB-S model, failure daa from any wo of he hree samples can be uilized o esimae he unnown parameers of he model and hese esimaed parameers can be used o predic he reliabiliy under design sress. The esimaed reliabiliy is hen compared wih he heoreical reliabiliy (observed daa se). We use daa ses and o esimae he unnown parameers. The log-lielihood funcion of he inverse power law GB-S model can be wrien as: 0 j j j j j j j l 03log 03log + log log j j j j 0 j Taing parial derivaives of he log-lielihood funcion wih respec o,, and yields he following four equaions: (9) l 03 j j 3 j 0 j j 3 j j j l 03 l j log j j j j j 0 j log j j j j j j log j j j 0 j log j j j j j j j j 0 j j j j j j 0 j j j

6 l log log j j j j j 0 j j log j j j j 0 j log log j j Newon s ieraive mehod is applied o solve he parial derivaives of he log lielihood funcions. The ieraion ends when he esimae converges. Usually for a nonlinear equaion, here exiss more han one local opimal soluion. These soluions are reurned o he lielihood funcion and he global opimal value is obained accordingly. To compare he performance of GB-S disribuion wih oher models, he inverse power law Weibull acceleraed model and he inverse power law SB-S acceleraed model are developed as follow. The Weibull acceleraed model: - - F ; -e (0) f ; e () Taing he logarihm of he lielihood funcion and parial derivaes wih respec o he unnown parameers of Weibull acceleraed model we obain as: l 03 log 03log + log n ij i j j j j 0 log 03 log log 0 log 0 () j n l 03 ij i j 03log log 0 loglog log log j j l j j j j l log 0log 0 j j j j The parameers esimaion procedure is similar o ha of GB-S. The deails of he power law SB-S acceleraed model are given in Owen [5]. B. A General Log-linear Acceleraion Form The inverse power law acceleraed model is limied o modeling he relaionship beween lifeime and mechanical sress. To examine he GB-S acceleraed model in a more general case, an exponenial form of life-sress relaionship, incorporaing he inverse power law model, is considered: h Z exp a a z 0 Z is he sress vecor (varied ypes of sress can be used) a and a are model parameers. 0 When expa 0, expaz, he exponenial model yields he inverse-power acceleraed model. Subsiuing he scale parameer, we obain he general acceleraion models for Weibull, SB-S and GB-S disribuions respecively, as expa0az FWeibull ; e (3)

7 Exponenial-form ALT models heoreical SB-S Weibull GB-S F() Fig.3. cdf of hree inverse power law models and heoreical cdf heoreical SB-S Weibull GB-S Inverse power law ALT models 0.6 F() Fig.4. cdf of hree general models and heoreical cdf F ; SBS (4) expa a z expa a z 0 0 F ; GBS expa a z expa a z 0 0 (5) The lielihood funcions and parial derivaes wih respec o each model s unnown parameers can be found respecively. I. COMPARISON As discussed earlier, an ALT model can be used o esimae reliabiliy performance under he desired sress level by uilizing failure daa obained a differen sress levels o obain he parameers of he model. In his secion, we consider all he scenarios where any wo of he daa ses are uilized o esimae he parameers of each proposed model and he prediced reliabiliies of each model are compared wih he hird daa se. Applying he daa ses and o each model and comparing wih he hird daa se, we obain he following esimaed parameers and he sum of squared errors (SSE) beween he observed and esimaed reliabiliies for each model are obained as summarized in Table I: Table I. Parameers and SSEs of each acceleraed model Inverse power law Weibull acceleraed model ˆ 0.347, ˆ 775.6, ˆ =.756 SSE=9.98 Inverse power law SB-S acceleraed model ˆ 0.49, ˆ 8.86, ˆ =5.548 SSE=5.089 Inverse power law GB-S acceleraed model ˆ 0.64, ˆ =0.333, ˆ , ˆ =5.855 SSE=.003 General Weibull acceleraed model ˆ.68, a ˆ 3.85, a ˆ.868 SSE= General SB-S acceleraed model ˆ 0.48, aˆ 6.674, aˆ SSE= General GB-S acceleraed model ˆ 0.58, ˆ =0.5, aˆ 6.758, aˆ SSE=.896 The Weibull acceleraed model resuls in he larges SSEs for all scenarios implying ha is predicion as an Acceleraed Failure Time (AFT) model for hese faigue daa is inaccurae. The GB-S acceleraed model has he smalles SSEs for boh inverse power law and he general cases. For he general acceleraed models, he esimaes of he SB- S and GB-S s shape parameers are almos idenical (esimaes of are close o 0.5). For he inverse power law acceleraed

8 models, here exis significan differences among he hree models in erms of SSE and esimaes of parameers. Clearly, GB-S model provides he mos accurae predicion among all models. Similarly, daa ses and 3 are used o obain he parameers of he models which are hen used for reliabiliy predicion a he sress level of daa se. Liewise, daa ses and 3 are used o obain he parameers of he models which are hen used for reliabiliy predicion a he sress level of daa se. The GB-S shows slighly beer performance han he SB-S in erms of SSE. The esimaes of he parameers of he GB-S and SB-S acceleraed models are almos idenical for he general (exponenial) case. The raios of he esimaed shape parameers ( / ) of he SB-S and GB-S acceleraed models are close o he inverse-power-law case.. CONCLUSIONS SB-S models are widely acceped o model faigue failure daa. However, i is limied in modeling differen failure ypes of failure raes. In his paper, we generalize BS disribuion and develop wo GB-S-based-ALT models. Their performances are compared wih SB-S-based-ALT models and Weibull-based-ALT models using several muliple ses of experimen daa. The resuls show ha he GB-S-based-ALT model ouperforms he hree acceleraed. Exensive analysis of differen daa ses show ha developed GB-S ALT model can be used o provide accurae reliabiliy predicion for faigue and wear ou daa. REFERENCES [] Birnbaum, Z.W. and Saunders, S.C., 968. A new family of life disribuion. Applied Probabiliy, 5 (6), [7] Dupuis, D.J. and Mills, J.E., 998. Robus esimaion of Birnbaum-Saunders disribuion. IEEE Transacion on Reliabiliy, 47 (), [8] Desmond, A.F., 0. A mixed effecs log-linear model based on he Birnbaum-Saunders disribuion. Compuaional Saisics & Daa Analysis. 56 (), [9] Kundu, D, 00. Birnbaum-Saunders Disribuion. Available from: hp://home.ii.ac.in/~undu/pala-bs-.pdf [0] Owen, W.J., 006. A new hree-parameer exension o he Birnbaum-Saunders disribuion. IEEE Transacions on Reliabiliy, 55 (3), [] Owen, W.J., 004. Anoher loo a he Birnbaum-Saunders disribuion. Available from: hp:// Ownn.pdf. [] Díaz-García, J.A. and Dominguez-Molina, J.R., 005. Generalized Birnbaum-Saunders and sinh-normal disribuions. Comunicación Técnica, 5(), -9. [3] Díaz-García, J.A. and Leiva-Sánchez,., 00. A new family of life disribuion based on Birnbaum-Saunders disribuion. Comunicación Técnica, (), -6. [4] Elsayed, E.A., 0. Reliabiliy Engineering. nd ed. New Jersey: John Wiley & Sons. [5] Owen, O.J. and Padge, W.J., 000. A Birnbaum Saunders Acceleraed Life Model. IEEE Transacion on Reliabiliy, 49 (), 4-9. [] Desmond, A., 985. Sochasic models of failure in random environmens. The Canadian Journal of Saisics, 3 (), [3] Riec, J.R., 999. A Momen-generaing Funcion wih Applicaion o he Birnbaum-Saunders Disribuion. Communicaions in Saisics Theory and Mehod, 8 (9), 3-. [4] Kundu, D., Kannan, N. and Balarishnan, D., 008. On he hazard funcion of Birnbaum-Saunders disribuion and associaed inference. Compuaional Saisics & Daa Analysis, 5 (5), [5] Engelhard, M., Bain, L.J. and Wrigh, F.T, 98. Inferences on he Parameers of he Birnbaum-Saunders Faigue Life Disribuion Based on Maximum Lielihood Esimaion. American Saisical Associaion, 3 (3), [6] Ng, H.K.T, Kundu, D. and Balarishnan, D., 003. Modified momen esimaion for he wo-parameer Birnbaum-Saunders disribuion. Compuaional Saisics & Daa Analysis, 43 (003),