New Methods for Modeling Reliability Using Degradation Data

Transcription

1 Papr Nw Mthods for Modling Rliability Using Dgradation Data José G. Ramírz, W.L. Gor & Associats, Inc., Elkton, MD Gordon Johnston, SAS Institut, Inc., Cary, NC ABSTRACT Enginrs and rsarchrs us lif tsts in a wid varity of industrial applications to dtrmin how wll critical componnts and matrials will prform undr diffrnt oprating conditions. By stimating th failur tim distribution, dcisions can b mad that influnc th dsign, improvmnt, and storag of products so thy mt customr rquirmnts. Manufacturr warranty dcisions may also b influncd by knowldg of product rliability prformanc. Traditional lif tsts ar oftn not th most fficint way to obtain rliability information bcaus fw, if any, actual failurs ar obsrvd. On th othr hand, it is oftn possibl to obtain psudofailur data using dgradation masurmnts. Consquntly, thr is growing intrst in studying th dgradation of product prformanc or matrial proprtis ovr tim. In this papr w dscrib two mthods for th analysis of such data. Th first uss th SAS/QC RELIABILITY procdur with psudo-failur tims to prdict a givn prcntil of th failur tim distribution, and th scond uss th SAS/STAT NLMIXED procdur to fit concav dgradation modls, and som SAS macros to stimat th failur distribution. Ths mthods ar illustratd with a dgradation study on th strngth of an lctronic matrial, which was subjctd to four diffrnt tmpraturs and masurd ovr tim. W compar th two approachs, and show th advantags of using concav dgradation modls in this stting. 1. INTRODUCTION show how dgradation data can b analyzd using traditional rliability mthods, whil in Sction 4 w show how to fit dgradation modls using non-linar mixd modls, w thn us ths modls to obtain an stimat of th failur tim distribution. In sction 5 w compar th two approachs. 2. EXPERIMENTAL DESIGN For th acclratd dgradation study (ADT) four diffrnt tmpraturs wr slctd. Two tmpraturs that ar common storag conditions: rfrigration (9 C) and room tmpratur (22 C), and two tmpraturs (40 C & 50 C) chosn to acclrat th dgradation procss. Sampling in tim was lss frqunt at low tmpraturs, mor frqunt at high tmpraturs. Th study was dsignd to last a yar but nough information to answr th abov qustions was obtaind bfor that. Thr lots of matrial wr usd in th study, and six units pr lot wr randomly slctd for a givn tmpratur and tim intrval. All th sampls wr sald in mtalizd Mylar bags to avoid issus with humidity. This xprimnt was dsignd to track svral ky rsponss of th dilctric as a function of tim and tmpratur, but in this papr will concntrat on on of th strngth proprtis of this product. An intrnal tst was dvlopd to masur th strngth of th matrial. An nginring tam was put togthr to dtrmin th shlf lif of a dilctric usd in lctronic componnts. Th study and undrstanding th shlf lif of this product, as wll as how it dgrads with tim and tmpratur, will allow us to dfin intrnal practics and mak customr rcommndations basd on rliability data. It was th goal of th tam to answr th following qustions: 1. What is th propr tmpratur for storing th matrial?, and 2. How long will th matrial last at a givn tmpratur? An xprimnt was dsignd to answr ths qustions using dgradation data, and is dscribd in Sction 2. In Sction 3 w Strngth Tst: To masur th strngth th matrial is hld across a jig using doubl-stick tap. A ball prob is lowrd until contact is mad with th sampl. Onc in contact, th prob is acclratd through th sampl at 10 mm/sc. Th forc rquird by th prob to ruptur th matrial is thn rcordd. Th proprtis of th matrial chang quit rapidly with tmpratur. To adjust for matrial variability th initial strngth of ach sampl was masurd, and usd as a baslin. Th rspons is th amount of strngth loss, in prcnt, from th initial condition. 1

2 Figur 1 shows th prcnt rduction in strngth as a function of tim for th four diffrnt tmpraturs. Th graph shows th typical concav dgradation bhavior at high tmpraturs; i.., 40ºC and 50ºC. STR_pct A SAS macro was writtn to intrpolat th psudo-failur tim at which a givn sampl was lss than, or gratr than, a critical dgradation lvl. This macro taks a SAS data st with th dgradation information and crats anothr SAS data st with psudo-failur tims and cnsoring information. Th macro uss linar intrpolation btwn th two tims at which th crossing may hav occurrd Figur 2 shows th rsulting psudo-failur tims for a cutoff point of 35% rduction in initial strngth. Not that at high tmpraturs, 40ºC and 50ºC, th psudo-failur tims ar lss than 30 tim units, whil at 9ºC all th psudo-failur tims ar cnsord; i.., non of th sampls had rsponss lss than th cutoff point Tim TEMP Figur 1. Prcnt rduction in strngth as a function of tim for four tmpraturs. Sinc this is a dstructiv tst a sampl was dfind as a collction of siblings from th sam locations within a lot. This within sampl variability is considrd to b ngligibl. 3. FAILURE-TIME ANALYSIS OF ADT DATA Traditional failur-tim analysis of acclratd dgradation data can b prformd using psudo-failur tims. Rmmbr that in ADT studis w monitor th dgradation or prformanc of a matrial or product ovr tim. In othr words, w do not rcord actual failur tims but th valu of a rspons ovr tim. By slcting a critical dgradation point for th rspons of intrst psudo-failur tims can b computd by obsrving whn a givn sampl rachs this critical point. It is important to not that ths psudo-failur tims ar dpndnt on th chosn cutoff point; i.., changing th cutoff point changs th psudo-failur tim for a givn sampl. Th dcision on what th critical dgradation point is should b basd on nginring considrations. On also nds to considr th dirction of th dgradation procss ovr tim; i.., dos th rspons incras or dcras with tim? Fig psudoft Tmpratur ºC Figur 2. Psudo-Failur tims for a 35% rduction in initial strngth. Onc th psudo-failur tims ar calculatd PROC RELIABILITY can b usd to stimat th failur tim distribution F(t). Th first stp in th analysis is to invstigat which distribution fits th data bst. Figur 3 shows Wibull probability plots for ach tmpratur. Not that for ach tmpratur th psudo-failur tims sm to follow a straight lin, indicating that th Wibull distribution dos a good job in fitting th data for all tmpraturs. Basd on this w can now stimat th lif of th dilctric for a givn prcntil. Th analysis using psudo-failur tims has basically two stps: 1. For ach sampl on calculats th psudo-failur tim at which th sampl crosss th givn cutoff point. 2. Ths psudo-failur tims ar thn usd to stimat th failur distribution F(t). 2

3 P r c n t In vrsion 8 of SAS, PROC RELIABILITY is usd to prdict th lif of th dilctric at a givn tmpratur for a givn prcntil, assuming a Wibull distribution and an Arrhnius rlation btwn tim and tmpratur. For this application w wantd to know th prdictd tim at which 1% and 95% of th population would fail, whn th matrial was stord at 9ºC. A 95% confidnc intrval for th prdictd tims was rquird to assss our prdictions Wibull Plot For psudoft tmp Figur 3. Wibull probability plots for psudo-failurs. Th rsults indicat that for 9ºC storag tmpratur, 1% of th dilctric matrial will surviv for 313 tim units, with a 95% confidnc intrval [ 262; 373 ]. Th prdictd failur tim for 95% of th population is 742 tim units, with a 95% confidnc intrval [ 664; 830 ]. Th output also shows th paramtr stimats for th Arrhnius-Wibull modl. Modl Information Input Data St Analysis Variabl Rlation Cnsor Variabl Distribution WORK.SUGI26_FT psudoft Arrhnius (Activation Enrgy) cnsor Wibull Algorithm convrgd. Summary of Fit Obsrvations Usd 71 Uncnsord Valus 53 Right Cnsord Valus 18 Missing Obsrvations 2 Maximum Logliklihood Wibull Paramtr Estimats Asymptotic Normal Standard 95% Confidnc Limits Paramtr Estimat Error Lowr Uppr Intrcpt tmp EV Scal Wibull Shap Obsrvation Statistics psudoft cnsor tmp PCNTL STDERR LOWER UPPER % % 3

4 A vry usful tool for assssing th fit of th modl, and also to prdict failur tims, is a rlation plot (s Nlson (1990), Chaptr 3). This plot, Figur 4, shows again th good fit obtaind by using th Wibull distribution. In th right-hand-sid plot th vrtical lin shows wr th 1% and 95% lins intrsct th 9ºC tmpratur. By projcting onto th vrtical axis on can rad-off th 313 and 742 failur tim prdictions for ths two prcntils. put "Dsign Tmpratur for &pct% Lif at &D_Tim Days= " D_TEMP "C"; run ; giving a prdictd storag tmpratur of 10ºC. Dsign Tmpratur for 1% Lif at 280 Days= 10 C NOTE: DATA statmnt usd: ral tim 5 sconds cpu tim 0 sconds ANALYSIS OF ADT DATA USING CONCAVE DEGRADATION MODELS Dgradation data can provid mor information than th traditional cnsord-failur data. Figur 1 shows that th prcnt rduction in strngth dcrass with tim until it rachs an asymptot or limit dgradation, for 40ºC and 50ºC. This typ of dgradation ovr tim can b approximatly dscribd by a concav modl of th typ: Prcnt tmp β 1 β 2 * AF( Tmp) * tim Dgradation( tim) = * 1 (1) Prcntils 1 95 tmp Figur 4. Rlation plot for dilctric strngth data. Th paramtr stimats from th Arrhnius-Wibull modl can b usd to stimat th tmpratur ndd so that a givn prcntil of th population will surviv a givn numbr of tim units. For xampl, lt us suppos that w want to prdict what th rquird tmpratur would b so that 1% of th population of our dilctric matrial will surviv at last 280 tim units. Again, w ar assuming that a failur is dfind as th point in tim at which a sampl was blow th cutoff point of 35% rduction in initial strngth. Ths typ of calculations ar usful whn, for xampl, on nds to figur out appropriat storag condition for a givn product. Th rquird tmpratur, so that a givn prcntil of th population will surviv a givn numbr of tim units, can b asily computd using th valus of intrcpt ( ), tmpratur (0.8537), and Wibull shap (6.5848) obtaind from th PROC RELIABILITY fit. /**Tmpratur for Arrhnius-Wibull Fit for a givn prcntil**/ %lt b0 = ; %lt b1 = ; %lt wshap = ; %lt D_tim = 280 ; %lt pct = 1 ; data _null_ ; D_TEMP=round(11605*&b1/(log(&D_Tim/(-log(1- &pct/100))**(1/&wshap)) - &b0) , 1.0) ; β 2 β 1 is th rat raction at Hr dscribs th asymptot; a baslin tmpratur (9ºC in our cas); and AF(Tmp) is th Arrhnius acclration factor for a givn tmpratur, Tmp, with rspct to th baslin tmpratur (9ºC in our cas), and activation nrgy β 3 : β Tmp AF( Tmp) = Traditionally PROC NLIN can b usd to fit th non-linar concav dgradation modl of quation (1). Howvr, in this typ of studis th sampls ar not idntical; i.., thr is sampl-tosampl variability that nds to b takn into account to gt a bttr stimat of th standard dviations of paramtr stimats. Ths non-linar dgradation modls ar vry similar in natur to non-linar growth curvs in which diffrnt subjcts ar obsrvd ovr tim, and th subjct-to-subjct variation nds to b takn into account. In vrsion 8 of SAS w hav availabl a nw procdur, PROC NLMIXED, to fit non-linar mixd modls in which fixd and random ffct can ntr non-linarly into a givn modl. For th prcnt rduction in strngth, th paramtrs of Equation (1) wr (2) 4

5 stimatd using PROC NLMIXED. Th PROC NLMIXED output givs paramtr stimats for th paramtrs β 1, β 2, and β 3 ; stimat for th ovrall varianc. Ths ar summarizd in th tabl blow. th variancs and covarianc for β1 and β 2 ; as wll as an Paramtr Estimat Standard Error DF t Valu Pr > t Alpha Lowr Uppr Gradint mb < sb mb < sb c E < b < s < β 2, From ths valus w gt an stimat of th asymptot, qual to -xp(-965) = 90% rduction in strngth, and an activation nrgy, β 3, of Th stimatd valus along with Equation (1) can b usd to obtain prdictd curvs for %rduction in strngth, for ach of th four tmpraturs. Figur 5 shows th fittd curvs using th prdictd valus from th concav dgradation mixd modl fittd with PROC NLMIXED. Not how th 50ºC curv has an asymptotic bhavior of about 90% rduction in strngth. STR_pct Prdictd of Mkr and Escobar (1998). Bootstrap confidnc limits for F(t) using th stimatd paramtrs from modl (1) can also b computd by following th procdur dscribd in Algorithm 13.3, pag 332, of Mkr and Escobar (1998), as outlind in th following stps. 1. Simulat a larg numbr of bootstrap sampls of dgradation data using th sam xprimntal dsign as in th original xprimnt. 2. Fit th modl (1) to ach sampl using PROC NLMIXED and comput F(t) for ach sampl using th procdur dscribd prviously. 3. For ach t, sort th stimatd valus of F(t) in incrasing ordr. 4. Th uppr and lowr bootstrap confidnc limits ar th uth and lth ordrd valus of F(t), whr th indics u and l ar computd from formulas givn by Mkr and Escobar (1998) A SAS macro was writtn to comput F(t) and bootstrap confidnc limits, using DATA stp programs, IML, and PROC NLMIXED Tim Figur 5. Concav dgradation curvs from PROC NLMIXED Computing th Failur Tim Distribution F(t). On can comput th failur tim distribution F(t) of th psudoliftims by invrting Equation (1) and using algorithm 13.1, pag Figur 6 displays th rsulting stimatd failur tim distribution F(t) along with 95% bootstrap confidnc bands for a cutoff point of 35% strngth rduction and a tmpratur of 9ºC. Th 1% prcntil of th stimatd failur distribution is 386 tim units, with a 95% confidnc intrval [ 199, 479 ], and th 95% prcntil of th stimatd failur distribution is 714 tim units, with a 95% confidnc intrval [ 608, 796 ]. 5

6 Proba b ility Dgradation Modl Estimat of Psudo-liftim CDF Bootstrap 95% Confidnc Limits Mthod Prcntil Lowr 95% CI Estimat Uppr 95% CI Psudo-Failurs 1% Dgradation Modl 1% Psudo-Failurs 95% Dgradation Modl 95% Not that th intrvals for th dgradation analysis ar widr than thos obtaind using th psudo failur tims bcaus thy contain th sampl-to-sampl variation. This is spcially apparnt for th 1 prcntil whr a lot of cnsoring is prsnt. Samp l siz: 1373 Tmpratur: 09 dgrs Figur 6. Dgradation modl stimat of psudo-liftim distribution with two-sidd 95% Bootstrap confidnc bands. 5. PSEUDO-FAILURES OR DEGRADATION MODELS? Figur 5 shows that many of th sampls had not rachd th 35% cutoff rduction in initial strngth at th nd of th xprimnt, spcially at th lowr tmpraturs. Thrfor ths units wr tratd as cnsord data in th traditional failur-tim analysis of psudo-liftims dscribd in Sction 3. In othr words, th distribution of th dgradation paths with dgradation lss than th 35% cutoff, was ignord in th analysis, thus information was lost. Th dgradation analysis dscribd in Sction 4 uss all of th information in th dgradation data, sinc it dirctly modls th dgradation procss. Th dgradation analysis approach should provid mor accurat rsults, spcially whn xtrapolation byond th cutoff point is rquird. On th othr hand, th dgradation analysis approach is mor computationally intnsiv. Fitting th modl of Equation (1) by maximum liklihood rquirs th solution of a difficult optimization problm. PROC NLMIXED automats th stimation procdur to a grat xtnt, but thr still can b problms in finding good initial paramtr stimats so th fitting algorithm will convrg. Bootstrap confidnc limits rquir fitting th modl to a larg numbr of simulatd data sts, which can b tim-consuming. Th following tabl shows th 1% and 95% stimats, along with 95% confidnc intrvals for th two approachs at a storing tmpratur of 9ºC. Tim As a gnral rcommndation w suggst using: 1) Th psudo failur tim analysis whn a quick and approximat analysis is dsird, and thr is littl cnsoring involvd, and 2) Th dgradation analysis approach whn a mor dtaild analysis of th dgradation data is rquird, at th xpns of a mor computationally burdnsom approach. S sction 13.8 of Mkr and Escobar (1998) for a usful discussion on th limitations of th psudo-failur tim analysis as compard to th dgradation analysis. REFERENCES Mkr, W.Q., and Escobar, L.A. (1998), Statistical Mthods for Rliability Data, Nw York: John Wily & Sons. Nlson, W. (1990) Acclratd Tsting: Statistical Modls, Tst Plans, and Data Analysis, Nw York: John Wily & Sons. SAS Institut Inc. (2000), SAS/QC Usr's Guid, Vrsion 8, Cary, NC: SAS Institut Inc ACKNOWLEDGEMENTS W would lik to thank th following W.L. Gor and Associats, Inc., associats: Michal Coll and Lroy Chadwick for running th ADT study and collcting th data; and Josph Korlski for supporting this work. CONTACT INFORMATION W wlcom commnts and qustions. José G. Ramírz Jramirz@wlgor.com (410) Gordon Johnston Gordon.johnston@sas.com (919)

7 SAS and all othr SAS Institut Inc. product or srvic nams ar rgistrd tradmarks of SAS Institut Inc. in th USA and othr countris. indicats USA rgistration. Othr brand and product nams ar rgistrd tradmarks or tradmarks of thir rspctiv companis. APPENDIX /** Figur 3 **/ /** Wibull Probability Plots **/ symbol1 c=rd v=plus; symbol2 c=blu v=x; symbol3 c=grn v=squar; symbol4 c=brown v=circl; proc rliability data=sugi26_ft ; distribution wibull; pplot psudoft=tmp / ovrlay noconf cfram = ligr; run; symbol ; titl ; /** Figur 4 **/ /** Arrhnius-Wibull Fit **/ proc rliability data=sugi26_ft ; distribution wibull ; modl psudoft*cnsor(1) = tmp / rlation = arrhnius2 obstats( q= control=cntrl) ; rplot psudoft*cnsor(1) = tmp / noconf pplot fit = modl lrclpr rlation = arrhnius2 plotfit luppr = 1000 slowr = 0 srf = 9 plotdata cfram = ligr ; run ; titl ; /** PROC NLMIXED Cod **/ proc nlmixd data=sugi26_str tch=quanew updat=bfgs inhss bopt; parms / data=brt_init ; tmp0 = 09; AF = xp( b3 * ( 11605/(tmp ) /(tmp ) )); Dt = -xp( b2 ) * ( 1 - xp( - xp( b1 ) * AF * days ) ); prdict Dt out = Brittl_Prd; modl brit_pct ~ normal( Dt, s2 ); random b1 b2 ~ normal( [ mb1, mb2 ], [ sb1, c21, sb2 ] ) subjct = sampl out=brt_ran; id af ; run; 7