Reliability Assessment and Residual Life Prediction Method based on Wiener Process and Current Degradation Quantity

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1 Engineering Leers, 4:, EL_4 08 Reliabiliy Assessmen Residual Life Predicion Mehod based on Wiener Process Curren Degradaion Quaniy Huibing Hao, Chunping Li Absrac In his aricle, he populaion reliabiliy modeling individual residual life predicion are discussed. Firsly, hree inds of differen Wiener process models are used o characerize he degradaion daa, he unnown parameers are esimaed by using he Marov Chain Mone Carlo (MCMC mehod. Secondly, under hose degradaion models, he individual residual life predicion mehod is also obained on he basis of curren degradaion quaniy. Finally, a faigue cracs daa example is given o illusrae he usefulness validiy of he proposed model mehod. Numerical resuls show ha he rom effec model is well fied wih he acual degradaion daa, his model has smalles predicion error. Meanwhile he predicion accuracy is accepable, his predicion mehod provides a foundaion for mainenance decision. Index Terms Wiener process, residual life, curren degradaion quaniy, Marov chain Mone Carlo D I. INTRODUCTION UE o he advances in maerial science manufacure echnology, he lifeime of modern produc has been becoming longer longer, usually i is difficul o obain enough failure daa during research developmen period []. In his siuaion, degradaion daa can be used as an alernae resource for reliabiliy analysis. In he las decades, degradaion daa have played a more imporan role in reliabiliy assessmen han ever before []. As we now ha degradaion (e.g. wear, erosion faigue is a common phenomenon for elecro-mechanical sysem is componens. Degradaion can be mahemaically described wih a coninuous process in erms of ime. In Ref. [3], hree inds of mehod for degradaion daa analysis are proposed, i.e. linear regression mehod, degradaion pah mehod, sochasic process mehod. One highligh of sochasic process model is ha he lifeime can be defined as he firs hiing ime when he degradaion process reaches a failure hreshold. And sochasic process can flexibly describe he failure mechanism characerisics of operaing environmen, i has been widely Manuscrip received Ocober 8, 05; revised March 5, 06. This wor was suppored by he Humaniy Social Science Youh Foundaion of Minisry of Educaion of China (No. 5YJCZH055, he Naional Naural Science Foundaion of China (No H. B. Hao is wih he Deparmen of Mahemaics, Hubei Engineering Universiy, Hubei, 43000, China. haohuibing979@63.com. C. P. Li is he corresponding auhor wih he Deparmen of Mahemaics, Hubei Engineering Universiy, Hubei, 43000, China. lichunping35@63.com. used o model he degradaion pah, including Gamma process [4], Wiener process [5,7,8,9,0], Marov process [6], e al. Among hose sochasic processes, Wiener process is mos widely used. For example, LEE e al [5] TANG e al [8] used Wiener process model o describe he degradaion of ligh emiing diode; SU ZHANG [7] used i o deal wih degradaion daa of laser device. In reliabiliy sudy, beyond evaluaing producs reliabiliy, how o obain he residual lifeime of a produc is also of grea ineres. In Ref. [9], REN e al used fixed effec Wiener process o esimae residual life of an aircraf engine, he predicion accuracy is given. However, he above sudies consider only he fixed effec Wiener process. In his paper, he mixed effec rom effec Wiener process models are proposed, hen he reliabiliy assessmen individual residual life can be obained. Considering Marov chain Mone Carlo (MCMC mehod is convenien efficien o sample from complex disribuion, MCMC mehod is used o esimae he unnown parameers [5]. The res of he paper is organized as follows. In Secion, he differen degradaion models are inroduced. Then, he residual life predicion mehods are presened in Secions 3. In Secion 4, he parameers esimaion mehod based on he MCMC is presened. A numerical example wih faigue cracs daa is given in Secion 5. Finally, some conclusions are made in Secion 6. II. POPULATION DEGRADATION MODEL BASED ON WIENER PROCESS Due o he good mahemaical properies physical inerpreaions of Wiener process, i has been aen o describe he performance degradaion of producs. A well adoped form of Wiener process X (, 0 can be expressed as M X ( B ( ( where μ σ are drif diffusion parameers, respecively; B( is a sard Brownian moion which is used o describe ime-correlaed srucure. Le be he hreshold value of he produc. I is assumed ha he degradaion pah is described by he model M. Given he hreshold value ξ, he produc s lifeime T is defined as T inf{ X( } ( i is nown ha T follows inverse Gaussian disribuion wih probabiliy disribuion funcion (PDF (Advance online publicaion: 8 May 06

2 Engineering Leers, 4:, EL_4 08 ( ft (, exp 3 ( exp 3 (3 Then, basis on he PDF of lifeime T, we can obain he reliabiliy a ime as follow R( Pr( T ft ( x dx exp (4 where Ф( is he cumulaive disribuion funcion (CDF of sard normal disribuion. In mos cases, each sample uni usually experiences differen sources of variaions during heir operaion. Thus, i is more appropriae o incorporae uni o uni variabiliy in he degradaion process, he mixed effec model can describe he uni variabiliy. A convenional mixed effec Wiener process model can be expressed as M Y ( B ( (5 ~ N(, Assume ha he degradaion pah of produc is described by he model M. Considering ha he drif parameer μ is rom variable, by using he oal law of probabiliy, he PDF of he lifeime T can be reconsruced in model M as ft( ft(, d ( exp 3 (6 ( ( where ( is disribuion funcion of he sard normal disribuion. Then, he reliabiliy a ime can be expressed as R ( exp ( (7 Up o now, some paper have considered mixed effec Wiener process model heir applicaions (see in [4], [0], [] []. Bu in hose sudies, only he drif parameer μ is considered o be rom variable. In his paper, a rom effec Wiener process is used o characerize he degradaion daa, where μ σ of his model are regarded as rom variables. A rom effec Wiener process model can be expressed as M 3 Z( B( ~ G(, (8 ~ N(, / where β, α, θ λ are unnown parameers; G( N(, are gamma disribuion normal disribuion, respecively. I is noed ha model M 3 can be used o describe boh he variaion from uni o uni ime correlaed srucure. Similarly o he above, when μ σ are rom variables, by using he oal law of probabiliy, he PDF of lifeime T in model M 3 is given by ( ft ( exp 3 0 ( ( ( exp d d ( ( ( 3 ( [ ( ] ( he reliabiliy a ime can be expressed as ( R ( 0 3 x [ ( x] ( ( x ( x x dx III. INDIVIDUAL RESIDUAL LIFE PREDICTION BASED ON CURRENT PERFORMANCE DEGRADATION (9 (0 As we nown, Equaions (4 (7 (0 provide a basis mehod for produc reliabiliy evaluaion, hey can characerize he average survival rae of he populaion. Insead of he average populaion s characerisics, he residual life predicion of individual produc has imporan pracical applicaions, such as planning of mainenance aciviies, supply chain managemen, replenishmen of invenory sysem, e al. In his secion, he residual life predicion mehod based on curren performance degradaion is given by using he differen Wiener process model. Firsly, we focus on he residual life predicion mehod under he degradaion model M. According o he independen incremen propery of he Wiener process, given μ σ, we can ge he following X (, ~ N, ( X( s, ~ N( s, ( s, s. ( where X ( s X ( Xs (. Supposing ha a produc has operaed unil ime wihou failure, X( is he corresponding degradaion quaniy a ime, he condiional reliabiliy can be formulaed as R X ( x Pr T X ( x Pr X ( X ( x Pr X (, X ( x Pr X ( x Pr X ( xx, ( x Pr X ( x Pr X ( xpr X ( x Pr X ( x (Advance online publicaion: 8 May 06

3 Engineering Leers, 4:, EL_4 08 x( Pr X ( x (3 Given predicion reliabiliy level p (0<p< performance degradaion X( =x based on curren ime, we can obain he coninuous operaion ime L= - of he operaed unil as: xl Z p (4 L where Z -p can be obained hrough checing Normal disribuion probabiliy able. Secondly, we focus on he residual life predicion mehod under he degradaion model M. From he Equaion (, given μ σ, we now ha he PDF of X( follows as ( x f( x exp (5 Considering ha he drif parameer μ is rom variable, by using he oal law of probabiliy, he PDF of Y( can be reconsruced in model M as g( x f( x d ( x exp ( ( Then, we now ha he PDF of Y( follows as Y ( ~ N, (6 (7 Y s N s s s (8 ( ~ (, ( ( where Y ( s Y ( Ys (. Similarly, suppose ha a produc has operaed unil ime wihou failure, Y( is he corresponding degradaion quaniy a ime, he condiional reliabiliy can be formulaed as RY ( x Pr TY ( x Pr Y ( x x( (9 ( ( Given predicion reliabiliy level p (0<p< performance degradaion Y( =x based on curren ime, we can obain he coninuous operaion ime L = - of he operaed unil as: xl Z p (0 L L where Z -p can be obained hrough checing Normal disribuion probabiliy able. Finally, we focus on he residual life predicion mehod under he degradaion model M 3. If he drif parameer μ he diffusion coefficien σ are rom variables, by using he similarly mehod, he PDF of Z( can be reconsruced in model M 3 as hx ( ( 0 ( x ( exp dd ( ( x ( ( ] ( ( ( Noe ha Z wih degrees of freedom β. Tha is o say ( has a T disribuion Z(~ T ( ( ( where T β is he T disribuion funcion wih degrees of freedom β. Similarly, supposing ha a produc has operaed unil ime wihou failure, Z( is he corresponding degradaion quaniy a ime, he condiional reliabiliy can be formulaed as R Z( x Pr T Z( x Pr Z( x ( ( x T (3 ( ( Given predicion reliabiliy level p (0<p< performance degradaion Z( =x based on curren ime, we can obain he coninuous operaion ime L = - of he operaed unil as: L( x L L M p (4 where M -p can be obained hrough checing T disribuion probabiliy able. The residual life L can be obained by resolving he above equaion. IV. PARAMETERS ESTIMATION VIA BAYESIAN MCMC METHOD Bayesian inference is an efficien approach o evaluae he unnown parameers of a given model. When i is difficul o obain he analyical poserior disribuion, MCMC mehod can be used. I can generae samples from he poserior disribuion, hese samples can also be used o esimae he desired feaures of he poserior disribuion. Suppose he degradaion pah of produc is governed by M 3. To achieve parameers esimaion, we assume ha n unis are esed, X i ( ij denoes he cumulaive degradaion values of produc i a ime ij, for i =,,, n; j = 0,,,, m. Le Z i( ij Z i( ij Z i( i( j, ij ij, i( j i0 0, Zi( i0 0 From he Equaion (, he join densiy can be obained as m m ( f( Zi ( Z ( m i i A Zi i (5 ( A ( where (Advance online publicaion: 8 May 06

4 Engineering Leers, 4:, EL_4 08 Z ( Z (, Z (,, Z (, i i i i i i im ip ip pq i ( i, i,, im, [ Apq] ip iq p q Due o he independence assumpion of he degradaion measuremens of differen produc, he log-lielihood funcion can be expressed as n m m l(,,, Z log ( log ( log log A i m log ( Zi i A ( Zi i (6 From Equaion (6, we now ha he model no only has four parameers, bu is also very complicaed from a compuaional viewpoin. For his reason, he MCMC wih he Gibbs sampling echniques is employed in his sudy o esimae model parameers. Le ( j j,z denoe he full condiional poserior disribuion of j, where j (,, j, j,, n Z is he observed daa. The algorihm of parameers esimaion via he Gibbs sampling can be summarized as follow: (0 (0 (0 (0 Sep : Iniialize (,,, ; Sep : Se ; ( Sep 3: Generae from condiional disribuion (, 3,,,Z; ( Sep 4: Generae from condiional disribuion (, 3,,,Z; ( Sep 5: Generae from condiional disribuion j j j j j ( (,,,,Z ; (,,,,,,Z; Sep 6: Generae from condiional disribuion Sep 7: Se, repea Seps 3-6,,,, N ; Sep 8: Esimae he desired feaures based on simulaion ( ( ( N samples of,,,. Using he Bayesian sofware pacage WinBUGS (see in Ref. [5] carrying ou he Gibbs sampling, he esimaor of he model parameers can be obained. V. NUMERICAL EXAMPLE In his secion, a numerical example abou faigue cracs daa is given o demonsrae he validiy of he proposed model mehod. The faigue crac daase is presened in Ref. [], degradaion samples are colleced. The observed measuremen variable is he crac lengh over ime, all samples are measured every 0.0million cycles. The produc is defined o be failed if he lengh of crac crosses.6 inch, he esing sopped a 0.0 million cycles. In he original degradaion daa, he degradaion pah of each sample over ime is nonlinear funcion. Here, a proper ransformaion as y=(x-0.9/x is adoped o mae i approximaely linear. The ransformed crac lengh daa are lised in Table I par of samples are depiced in Fig.. Based on his ransformaion mehod, he failure hreshold value becomes inch. Crac lengh(inches Uni Uni3 Uni9 Uni5 Uni TABLE I THE CRACK LENGTH DATA (INCHES Uni Crac lengh Uni3 Uni5 Uni Uni Millions of cycles Fig.. The developmen of crac sizes over ime. A. Populaion reliabiliy assessmen Based on he above daa, we can esimae he reliabiliy wih Equaions (4, (7 (0 of differen degradaion models M, M M 3, respecively. In order o judge which model is more flexible, now we compare he resuls obained wih he above hree models. By using MCMC mehod, he esimaion of he unnown parameers in hose models can be obained as follows: ˆ 3.377, ˆ ˆ 3.377, 0.649, 0.06 ˆ 0.593, ˆ 43., ˆ., ˆ (Advance online publicaion: 8 May 06

5 Engineering Leers, 4:, EL_4 08 Correspondingly, we esablish he reliabiliy curves wih hree models respecively, as shown in Fig.. From Fig., i can be found ha reliabiliy of he faigue crac under model M is no falling before he 0.0 million cycles. In fac, when he running ime arrived a 0.0 million cycles, some unis have failed, he oher unis are also gradually close o fail, hus we can conclude ha degradaion assessmen wih M is no so consisen wih acual degradaion daa. On he conrary, he reliabiliy curve under he model M model M 3 can well reflec he acual degradaion siuaion of producs. Reliabiliy curve of M TABLE Ⅱ RESULT COMPARE OF UNIT TV PV RE ME 0.36 PV RE ME Reliabiliy curve of M Reliabiliy curve of M 3 PV M RE Reliabiliy ME TV= True value of RL, ME= mean error under M, RE=relaive error under M, PV= predic value under M, ME= mean error under M, RE=relaive error under M, PV3= predic value under M 3 ME3= mean error under M 3, RE3=relaive error under M 3, PV3= predic value under M 3,RE= PV-TV /TV, ME is he arihmeic mean of RE. TV TABLE RESULT COMPARE OF UNIT PV M 3 RE M ME Millions of cycles Fig. The reliabiliy curve of hree degradaion models B. Individual residual life predicion Se he hreshold ξ=0.4375, from Table, we can find ha here are wo unis (i.e. uni uni failed, he corresponding failure imes are 0.0, 0.0, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09 million cycles respecively. Given predicion reliabiliy level p=0.95, we can obain he residual life of produc. In order o verify he superioriy of he proposed model, we conduc some comparaive sudies by comparing he resuls obained wih model M, M, M 3 he rue residual life, he corresponding resul are shown as in Table Table. From he Table 3 Table 4, we can find ha he mean error of uni is 0.09, in he differen degradaion models M, M M 3. The mean error of uni is 0.36, in he differen degradaion models M, M M 3. Obviously, rom effec model M 3 has smalles predicion error in he above hree degradaion models. VI. CONCLUSION In his paper, hree differen Wiener process models are proposed o characerize he degradaion pah; he corresponding reliabiliy assessmen model residual life predicion mehod are esablished. A case sudy of he faigue crac daa is given o validae he effeciveness of he PV RE ME PV RE ME TV= True value of RL, ME= mean error under M, RE=relaive error under M, PV= predic value under M, ME= mean error under M, RE=relaive error under M, PV3= predic value under M 3 ME3= mean error under M 3, RE3=relaive error under M 3, PV3= predic value under M 3,RE= PV-TV /TV, ME is he arihmeic mean of RE. proposed model mehod. Main conclusions of his sudy are summarized below: ( Rom effec Wiener process is well fied o describe degradaion. ( Since he lielihood funcion is so complicaed, insead of direcly maximizing he lielihood funcion, MCMC mehod can be used o esimae he unnown parameers. (3 The residual life predicion mehod based on curren performance degradaion is obained wih he proposed model. Compared wih fixed effec model M, he rom effec model M 3 has smalles predicion error. In his paper, we consider only he case when he performance degradaion is governed by rom effec Wiener process. In pracice, i is also possible ha he degradaion pahs of he produc follow gamma process, Marov process, or oher sochasic process. (Advance online publicaion: 8 May 06

6 Engineering Leers, 4:, EL_4 08 REFERENCES [] W.Q. Meeer, L.A. Escobar, Saisical mehod for reliabiliy daa, New Yor: John Wiley & Sons, 998. [] W. Elson, Acceleraed Tesing: Saisical Models, Tes Plans, Daa Analysis, New Yor: John Wiley & Sons, 990. [3] M.J. Zuo, R.Y. Jiang, R.C.M. Yam, Approaches for reliabiliy modeling of coninuous sae devices, IEEE Transacions on Reliabiliy, vol.48, pp.9-8, 999. [4] L. Tan, J. Yang, Z. Cheng, B. Guo, Opimal replacemen policy for cold sby sysem, Chinese Journal of Mechanical Engineering, vol.4, pp.36-3, 0. [5] M.Y. Lee, J. Tang, A modified EM-algorihm for esimaing he parameers of inverse Gaussian disribuion based on ime-censored Wiener degradaion daa, Saisica Sinica, vol.7, pp , 007. [6] C. Su, X.Q. Zhou, Condiion based mainenance opimizaion for wind urbines based on semi-marov decision process, Journal of Mechanical Engineering, vol.48, pp.44-49, 0. [7] C. Su, Y. Zhang, Sysem reliabiliy assessmen based on Wiener process compeing failure analysis, Journal of Souheas Universiy, vol.6, pp.405-4, 00. [8] J. Tang, T. Su, Esimaing failure ime disribuion parameers based on inermediae daa from a Wiener degradaion mode, Naval Research Logisics, vol.55, pp.65-76, 008. [9] S.H. Ren, H.F. Zuo, F. Bai, Real ime performance reliabiliy predicion for civil aviaion engines based on Brownian moion wih drif, Journal of Aerospace Power, vol.4, pp , 009. [0] X.S. Si, W.B. Wang, C.H. Hu, D.H. Zhou, M.G. Pech, Remaining useful life esimaion based on a nonlinear diffusion degradaion process, IEEE Transacion on Reliabiliy, vol.6, pp.50-67, 0. [] C.Y. Peng, S. T. Tseng, Mis-specificaion analysis of linear degradaion models, IEEE Transacions on Reliabiliy, vol.58, pp , 009. [] X. Wang, Wiener processes wih rom effecs for degradaion daa, Mulivariae Analysis. vol.0, pp , 00. [3] C. Pae, W.J. Padge, Sochasic degradaion models wih several acceleraing variables, IEEE Transacions on Reliabiliy, vol.55, pp , 006. [4] Z. Wang, J. Yang, G. Wang, G. Zhang, Applicaion of hree-parameer Weibull mixure model for reliabiliy assessmen of NC machine ools: a case sudy, Proceedings of he Insiuion of Mechanical Engineers, Par C: Journal of Mechanical Engineering Science, vol.5, pp.78-76, 0. [5] I. Nzoufras, Bayesian modeling using WinBUGS, New Yor: John Wiley & Sons, 009. [6] H. Assareh K. Mengersen, Bayesian esimaion of he ime of a decrease in ris-adjused survival ime conrol chars, IAENG Inernaional Journal of Applied Mahemaics, vol.4, no.4, pp , 0. [7] J. Lai, L. Zhang, C.F. Duffield, L. Aye, Engineering reliabiliy analysis in ris managemen framewor: developmen applicaion in infrasrucure projec, IAENG Inernaional Journal of Applied Mahemaics, vol.43, no.4, pp. 4-49, 03. (Advance online publicaion: 8 May 06