Optimal design of an accelerated degradation experiment with reciprocal Weibull degradation rate

 Cuthbert Cross
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1 Unversty of South Florda Scholar Coons Graduate Theses and Dssertatons Graduate School 4 Optal desgn of an accelerated degradaton experent wth recprocal Weull degradaton rate Indra Polavarapu Unversty of South Florda Follow ths and addtonal works at: Part of the Aercan Studes Coons Scholar Coons Ctaton Polavarapu Indra "Optal desgn of an accelerated degradaton experent wth recprocal Weull degradaton rate" (4). Graduate Theses and Dssertatons. Ths Thess s rought to you for free and open access y the Graduate School at Scholar Coons. It has een accepted for ncluson n Graduate Theses and Dssertatons y an authorzed adnstrator of Scholar Coons. For ore nforaton please contact
2 Optal Desgn of An Accelerated Degradaton Experent wth Recprocal Weull Degradaton Rate y Indra Polavarapu A thess sutted n partal fulfllent of the requreents for the degree of Master of Scence n Industral Engneerng Departent of Industral and Manageent Systes Engneerng College of Engneerng Unversty of South Florda Major Professor: Okogaa Geoffrey Ph.D. Rao A.N.V Ph.D. Qang Huang Ph.D. Date of Approval: Septeer 4 Keywords: hghly relale products lfe testng degradaton testng recprocal weull dstruton degradaton falure Copyrght 4 Indra Polavarapu
3 ACKNOWLEDGMENT I would lke to express y grattude to all those who gave e the posslty to coplete ths thess. I a deeply ndeted to y advsor Dr. O. Geoffrey Okogaa whose help stulatng suggestons and encourageent helped e n all the te of research and splfed y task n copleton of ths thess. Hs understandng encouragng and personal gudance have provded a good ass for ths work. Specal thanks to Dr.A.N.V.Rao for hs wde knowledge and expertse and hs portant support as a cottee eer throughout ths work. I warly thank y other cottee eer Dr.Qang Huang for hs valuale advce and frendly support. Durng ths work I have collaorated wth any frends and colleagues for who I have great regard and I wsh to extend y warest thanks to all those who have helped e wth y work n the Departent of Industral & Manageent Systes Engneerng. I would lke to thank all the staff at The Insttute on Black Lfe for ther help support and frendshp. Grateful thanks to y faly for eng there fro the very start standng the whole way through stors and sunshne. Wthout ther encourageent and understandng t would have een possle for e to fnsh ths work.
4 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ABSTRACT v v CHAPTER INTRODUCTION. Background. ADTs Versus Other Testng Methods.3 Relalty Analyss 3.4 Applcatons 4.4. Lght Ettng Dodes 5.4. Logc Integrated Crcuts Power Crcuts 5.5 Motvaton for Ths Research 6.6 Organzaton of Ths Thess 7 CHAPTER LITERATURE REVIEW 8. Introducton 8. Degradaton Models 9.. Lnear Degradaton 9.. Convex Degradaton 9..3 Concave Degradaton.3 General Degradaton Path Model.4 Degradaton and Falure Types 3.4. Soft Falures: Specfed Degradaton Level 3.4. Hard Falures: Jont Dstruton of Degradaton and Falure Level 3.5 Constant Stress Degradaton Models 4.5. The Arrhenus Rate Degradaton Model 4.5. Inverse Power Relatonshp Eyrng Relatonshp 5.6 Acceleraton Model 5.6. Elevated Teperature Acceleraton 6.6. Nonlnear Degradaton Path Reactonrate Acceleraton 6
5 .6.3 Lnear Degradaton Path Reactonrate Acceleraton Degradaton wth Parallel Reactons 8.7 Estaton of Accelerated Degradaton Model Paraeters 8 CHAPTER 3 PROBLEM STATEMENT The Optzaton Prole 3 CHAPTER 4 METHODOLOGY 5 4. Degradaton Model wth Rando Coeffcent 5 4. Assuptons The MeanTeToFalure The Coputaton of [ φ ] 4.5 The Coputaton of [ φ ] φ 7 E 3 φ ( n) 4.6 The Cost Functon TC { f l } = The Optzaton Model Algorth 35 CHAPTER 5 EXAMPLE Sulaton Experent Optal Test Plan Based on the LED Data Optal Paraeters Based on the ADT Experent Senstvty Analyss Test Plans under a Varety of C Test Plans for Dfferent Values of and a Varety of Conatons of { S } = Senstvty Analyss of Msspecfyng u andσ 53 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH Future Research Drectons 57 REFERENCES 59
6 LIST OF TABLES Tale 5. The Standardzed Saple Degradaton Paths at Stress Level S 4 Tale 5. The Standardzed Saple Degradaton Paths at Stress Level S 4 Tale 5.3 The Standardzed Saple Degradaton Paths at Stress Level S 3 4 Tale 5.4 Optal Degradaton Test Plans under Varous Cost Condtons (C s C p C C d C ) 48 Tale 5.5 The Optal Test Plans for Soe Values of C 5 Tale 5.6 The Optal Test Plans for Varous Values of and Conatons of Stress Levels 5 3 { n } * * * Tale 5.7 The Optal Soluton ( f l ) for the Case that { ε } 3 = are = Changed over the Ranges ±.5% 53 3 { n } * for the Case that { ε } 3 = are = Changed over the Ranges ± 5% 54 * * Tale 5.8 The Optal Soluton ( f l ) 3 { n } * * * Tale 5.9 The Optal Soluton ( f l ) for the Case that { ε } 3 = are = Changed over the Ranges ± % 54
7 LIST OF FIGURES Fgure. Possle Shapes of Degradaton Curves 8 Fgure. Propagaton Delay Growth Curve wth a Plateau (Maxu Degradaton) 9 Fgure.3 Equal LogSpacng Plan for Measureent Fgure 5. The Standardzed Saple Degradaton Paths under (a) S () S and (c) S 3 43 Fgure 5. The Plots of ω j (t) versus t.5 under (a) S () S and (c) S 3 44 Fgure 5.3 The Weull Proalty Plot for { } 3 5 ˆ β 46 Fgure 5.4 The Noral Proalty Plots for Resduals under (a) S () S (c) S 3 47 j j= v
8 OPTIMAL DESIGN OF AN ACCELERATED DEGRADATION EXPERIMENT WITH RECIPROCAL WEIBULL DEGRADATION RATE INDIRA POLAVARAPU ABSTRACT To eet ncreasng copetton get products to arket n the shortest possle te and satsfy heghtened custoer expectatons products ust e ade ore roust and fewer falures ust e oserved n a short developent perod. In ths crcustance assessng product relalty ased on degradaton data at hgh stress levels ecoes necessary. Ths assessent s accoplshed through accelerated degradaton tests (ADT). These tests nvolve over stress testng n whch nstead of lfe product perforance s easured as t degrades over te. Due to the role these tests play n deternng proper relalty estates for the product t s necessary to scentfcally desgn these test plans so as to save te and expense and provde ore accurate estates of relalty for a gven nuer of test unts and test te. In ADTs several decson varales such as nspecton frequency the saple sze and the ternaton te at each stress level are portant. In ths research an optal plan s developed for the desgn of accelerated degradaton test wth a recprocal Weull degradaton data usng the ean te to falure (MTTF) as the nzng crtera. A non lnear nteger prograng prole s developed under the constrant that the total experental cost does not exceed a pre v
9 deterned udget. The optal conaton of saple sze nspecton frequency and the ternaton te at each stress level s found. A case exaple ased on Lght Ettng Dode (LED) exaple s used to llustrate the proposed ethod. Senstvty analyses on the cost paraeters and the paraeters of the underlyng proalty dstruton are perfored to assess the roustness of the proposed ethod. v
10 CHAPTER INTRODUCTION. Background Today s anufacturers are facng new pressures to develop hghly sophstcated products to atch rapd advances n technology ntense gloal copetton and ncreasng custoer expectatons. As a result anufacturers ust produce coponents n record te whle provng productvty relalty and overall qualty of the coponent. It s a sgnfcant challenge to desgn develop test and anufacture hghly relale products wthn short turn around tes and rean wthn the strngent condtons posed y oth nternal and external crcustances. Estatng the tetofalure dstruton or longter perforance of coponents of hgh relalty products s partcularly dffcult. Most odern products are desgned to operate wthout falure for several years. Thus few of such unts wll fal or degrade to a sgnfcant aount n a test of any practcal length ased on noral use condtons. For exaple durng the desgn and constructon of a councaton satellte there ay e only 6 onths avalale to test the coponents whch are eant to e n servce for 5 to years. The coponents used n suarne cales are often requred to operate for 5 years under the sea. Very few test unts are avalale that wll actually reflect the lfe profles of these coponents. For these reasons Accelerated tests (ATs) are used wdely
11 n anufacturng ndustres partcularly to otan tely nforaton on the relalty of products.. ADTs Versus Other Testng Methods Tradtonally relalty assessent of new products has een ased on accelerated lfe tests (ALTs) that record falure and censorng tes of products sujected to elevated stress. However ths approach ay offer lttle help for hghly relale products whch are not lkely to fal durng an experent of reasonale length. An alternatve approach s to assess the relalty fro the changes n perforance (degradaton) oserved durng the experent f there exsts a qualty characterstc of the product whose degradaton over te can e related to relalty. Accelerated degradaton tests copared to other tests have the advantage of analyzng perforance efore the ateral or the coponent fals. Degradaton tests deterne how uch lfe there s left n a ateral or n coponents and such knowledge enales lfe extenson. Extrapolatng perforance degradaton to estate when t reaches falure level enales analyss of degradaton data. However such analyss s correct only f a good odel for extrapolaton of perforance degradaton and a sutale perforance falure have een estalshed. Soe of the general assuptons of accelerated degradaton odels are Degradaton s not reversle. A odel apples to a sngle degradaton process echans or falure ode. If there are sultaneous degradaton processes and falure odes each requres ts own odel. Degradaton of specen perforance efore the test starts s neglgle. Perforance s easured wth neglgle rando error.
12 The falure process at hgh stress levels are the sae as at the desgn or use stress levels. Accelerated degradaton tests are usually very expensve and thus t s essental to plan the carefully. Good test plans yeld etter results for a gven cost and te paraeters on the other hand poor test plans waste te and resources and ay not even yeld the desred nforaton. When conductng an ADT the followng ssues are usually of nterest. How long should the test e run? or How any unts should e tested at each stress level? Thus to address the ssues a scentfc plan s needed to ake the ost effcent use of test resources and ultately to otan an accurate estate of the lfe profle of an entty under the noral use condtons..3 Relalty Analyss Most thngs have a lfe span defned n soe for or another. These lfe tes when easured present us wth data sets that are used for scentfc or other purposes. It s natural to study the lfe te dstruton of an entty through a set of easured data. An area of research whch s stll vary uch actve s the theory of relalty. A generc defnton of relalty s: Relalty s the proalty that a product or a syste wll perfor ts ntended functon wthout falure for a specfed perod of te under specfc operatng condtons. To express ths relatonshp atheatcally we defne the contnuous rando varale to e the te to falure of the coponent or syste. Thus relalty at te t can e expressed as: ( ) = Pr{ T t} ; R t T Where : R () t R( ) = L R() t = t 3
13 In relalty analyss the ajor ssue s the proalty dstruton of the lfe tes of the entty under study. For ths purpose the standard ethod s to take a set of oserved lfe tes T T... T n or censored soetes where we assue that the oserved lfe te s a functon of an unknown paraeter θ whch can e expressed as T ~ F (.; θ). Where F (.; θ) s the proalty densty functon of an unknown paraeter θ. Fro the lkelhood functon constructed fro ths saple we can ake an nference wth respect to the unknown paraeter θ. When the for of F (.; θ) s known and the coplete dstruton of F s deterned y a fnte densonal paraeter θ then we have a classcal paraetrc odel; f F s copletely unknown except for soe qualtatve descrptons such as contnuty or soothness then we have a nonparaetrc odel; fnally f F s unknown ut the paraeter θ has soe structure to explore then we have a separaetrc odel. When the paraeter θ exhts soe structure we wll naturally eed our nference prole nto tradtonal and tetested odels for statstcal analyss. These nclude technques such as experental desgn regresson logstc regresson accelerated lfe testng etc. These ethods ncorporate varous stuatons that ay e encountered n practce. There s no need however to restrct the nference to the classcal frequentsts paraetrc setup. We can f the stuaton requres use the Bayesan ethod or even the eprcal Bayes technques..4 Applcatons Applcatons of accelerated degradaton tests nclude lght ettng dodes logc ntegrated crcuts power supples etc. 4
14 .4. Lght Ettng Dodes (LED) Lght ettng dodes are wdely used n any felds rangng fro consuer electroncs to optcal fer transsson systes. The LED has any features such as less power consupton sall volue good vsual effects and long lfe. Nowadays they are used as electronc oards on hghways and streets and as soke censors on celngs etc. Because of ther hgh relalty t s dffcult to otan the product lfe te nforaton under noral stress levels n a relatvely short te. Thus ADTs are used to otan the relalty nforaton of LED products [6 7]..4. Logc Integrated Crcuts Soe logc ntegrated crcuts are used as coponents of suarne cales. The portant paraeter n deternng the relalty nforaton of logc ntegrated crcuts s propagaton delay [8]. The logc ntegrated crcuts ght not functon f the propagaton delay of a logc gate ncreases (degrades). For exaple a logc crcut whch s desgned to have a axu propagaton delay of nsec fro nput to the output requres that the coned propagaton delay of the ndvdual logc gates n the crtcal path does not exceed nsec. These logc ntegrated crcuts are requred to operate for at least 5 years under the sea wthout falure. So accelerated degradaton tests are eployed to predct the lfe of logc ntegrated crcut and to study the assocated propagaton delay..4.3 Power Crcuts For power supples falures are coon due to low DC output. Power supply unts show downward drft n ther DC output. Accelerated degradaton tests are used to easure the DC output and to ontor the devce for relalty nforaton. 5
15 Nelson [3 pp 5548] lsts other applcatons of accelerated degradaton tests whch nclude: etals se conductors and cro electroncs delectrcs and nsulatons food and drugs and plastcs and polyers..5 Motvaton for Ths Research An Accelerated Degradaton test s a echans desgned to shorten the lfe of products y sujectng the test unts to hgher levels of stresses that are ore severe than the noral use stress levels. The nforaton fro hgh stress levels s extrapolated through a reasonale statstcal ethod to otan estates of lfe or longter perforance at the noral use stress level. Tradtonal approaches are ased on lfe tests that record only tetofalure. Such analyses have een extensvely studed and developed over the past few decades and any artcles have een pulshed n ths area. Due to the fact that tradtonal lfe testng of hghly relale products does not gve good relalty estates relalty assessent usng degradaton data has ecoe ncreasngly portant. In the lterature ost degradaton odels assues that the degradaton paths or transfored degradaton paths are lnear and are developed for only the noral use stress level [38]. Most of the lterature focuses on estatng the paraeters n the lnear degradaton odel and the lfe dstruton. Research aout accelerated test plannng s also reported. But when carryng out the accelerated degradaton tests several decson varales such as nspecton frequency the saple sze and the ternaton te at each stress level are portant. The prary ojectve for ths research s to deterne the optal paraeters of an ADT wth respect to products whose degradaton rates follow the recprocal weull 6
16 dstruton. Ths s accoplshed y takng the MLE s of the varance of the confdence nterval of the MTTF under the constrant that the total experental cost does not exceed a predeterned udget. A non lnear prograng prole s developed to deterne the optal value of the decson varales such as saple sze nspecton frequency and the ternaton te at each stress level..6 Organzaton of the Thess In Chapter a revew of the lterature s dscussed. Chapter 3 dscusses the prole stateent assuptons ade and notatons used. An optzaton prole s proposed. In Chapter 4 an optal plan for solvng the optzaton prole s presented. To llustrate the optal plan an exaple of LED degradaton s presented n chapter 5.Fnally the conclusons of the study and suggestons for further research are presented n chapter 6. 7
17 CHAPTER LITERATURE REVIEW. Introducton Durng the 99 s Nelson[3] (chapter ) provded a farly thorough survey on ADT whch ncluded areas of applcatons statstcal odels descres asc deas on ADT odels and usng a specfc exaple shows how to analyze a type of degradaton data. Carey and Koeng[4] (99) have descred a dataanalyss strategy and a odelfttng ethod to extract relalty nforaton fro oservatons on the degradaton of ntegrated logc devces that are coponents n a new generaton of suarne cales. Most falures can e traced to an underlyng degradaton process. Meeker and Escoar (998) gave soe exaples of three general shapes for degradaton curves n artrary unts of degradaton and te: lnear convex and concave whch are shown n fg... The dashed horzontal lne represents the degradaton level at whch falure would occur. Fgure. Possle Shapes of Degradaton Curves 8
18 . Degradaton Models.. Lnear Degradaton Meeker and Haada [7] uses lnear degradaton n soe sple wear processes lke autoole tre wear. Let D (t) represents the aount of autoole tre tread at te t and dd ( t) wear rate dt = C then D (t) = D () + C t. The paraeters D () and C could e taken as constant for ndvdual unts ut rando fro unttount... Convex Degradaton The convex degradaton approach s used n odels for whch the degradaton rate ncreases wth the level of degradaton such as n odelng the growth of fatgue cracks. Let a (t) denote the sze of a crack at te t. The Pars odel[8] s gven as da( t) dt = [ ( )] (.) C k a Where a = sze of the crack C and = ateral propertes and k(a) = stressntensty factor 9
19 s often used to descre the growth of fatgue cracks. Lu and Meeker [8] use ths odel n whch k ( a) = a for descrng the growth rate of fatgue cracks wthn a certan sze range. Then ( () ) = a a t (.) /( { [ a()] C( ) t} ) where a () =.9 nches s the ntal crack length at t =. Dvdng oth sdes of Eq. (.) y a () yelds /( ) a (t)/a () = / { [ a ()] C( ) t} (. )..3 Concave Degradaton Meeker and LuValle [9] descre odels for growth of falurecausng conductng flaents of chlornecopper copounds n prntedcrcut oards. They consder degradaton fro a frstorder checal reacton. These flaents cause falure when they reach fro one copperplated throughhole to another. Let A (t) e the aount of chlorne avalale for reacton at te t and A (t) e the aount of falure causng chlornecopper copound at te t. Under approprate condtons copper cones wth chlorne A to produce the chlornecopper copound A wth a constant rate k. The equatons for the rate for ths process are da dt = ka and da dt = ka Let c and A () e the ntal aounts. Assung A () = we get
20 A (t) = A ( )[exp(kt)]. (.3) Ths functon s llustrated y the concave curve n Fg... The asyptote at A ( ) = A () reflects the aount of chlorne avalale for the reacton producng copounds. LuValle and Meeker [9] also suggest other ore elaorate odels for ths falure process. Carey and Koeng [4] use slar odels to descre degradaton of electronc coponents..3 General Degradaton Path Model Lu & Meeker (993) use the followng odel [Eq.3] for the analyss of degradaton data at a fxed level of stress (.e. no acceleraton) and to estate a tetofalure dstruton. They denote the true degradaton path of a partcular unt(a functon of te) y D(t) t >. In applcatons values of D(t) are sapled at dscrete ponts n tet t.the oserved saple degradaton path for unt at te t j s a unt s actual degradaton path plus error and s gven y y = + =.. n j = (.3) j D j j Where D j = D (t j β ) s the actual path of the th unt at te t j (the tes need not e the sae for all unts) ( σ ) j ~ N s the devaton fro the assued odel for unt at te t j β = ( β. β k ) s a vector of k unknown paraeters for unt. The devatons are used to descre the easureent error. The total nuer of nspectons on unt s denoted y. Te t could e realte operatng te or soe surrogate lke les for autoole tres or loadng cycles n fatgue tests. Typcally sall paths are descred wth a odel that has up to four ponts (.e. k= 3 4). Soe of the
21 paraeters n β could e rando fro unttount and soe of the could e odeled as constant across all unts. The scales of y and t can e chosen to splfy the for of D(t β ). The choce of a degradaton odel requres not only the specfcaton of the for of D(t β ) functon ut also the specfcaton of whch of the paraeters n β are rando and whch are fxed as well as the jont dstruton of the rando coponents n β. Lu & Meeker (993) descre the use of a general faly of transforatons to a ultvarate noral dstruton wth ean vector µ β and covarance atrx β. It s generally reasonale to assue that the rando coponents of the vector β are ndependent of the j. We also assue that j are ndependent and dentcally dstruted. Because the y j are taken serally on a unt however there s potental for autocorrelaton aong the j. Especally f there are any closely spaced readngs. In any practcal applcatons nvolvng nference on the degradaton of unts fro a populaton or process however f the odel ft s adequate and f the testng and easureent processes are n control then the autocorrelaton s typcally weak and oreover t s donated y the unttount varalty n the β values and thus can e gnored. Also t s well known that pont estates of regresson odels are not serously affected y autocorrelaton ut gnorng autocorrelaton can result n standard errors that are serously ased. Ths however s not a prole when confdence ntervals are constructed y usng an approprate sulatonased ootstrap ethod. In ore coplcated stuatons t ay also happen that σ wll depend on the level of the acceleraton varale. Often however approprate odelng (for varance stratfcaton
22 e.g. transforaton of the degradaton response) wll allow the use of a spler odel ased on constantσ..4 Degradaton and Falure Types.4. Soft Falures: Specfed Degradaton Level For soe products there s a gradual loss of perforance (e.g. decreasng lght output fro a fluorescent lght ul). Then falure would e defned n an artrary anner at a specfed level of degradaton such as 6 % of ntal output. Tseng Haada and Chao (995) explan ths wth an exaple and defned ths as soft falure. A fxed value of D f s used to denote the crtcal level for the degradaton path aove (or elow) whch falure s assued to have occurred. The falure te T s defned as the te when the actual path D(t) crosses the crtcal degradaton level c and t c s used to denote the planned stoppng te n the experent. Inferences are ade on the falurete dstruton of a partcular product or ateral. For soft falures t ay e possle to contnue oservaton eyond D f..4. Hard Falures: Jont Dstruton of Degradaton and Falure Level For soe products a falure event s defned as when the product stops workng (e.g. when the resstance of a resstor devates too uch fro ts nonal value causng the oscllator n an electronc crcut to stop oscllatng or when an ncandescent lght ul urns out). These are called hard falures. In general wth hard falures falure tes correspond to a partcular level of degradaton. But the level of degradaton at whch falure occurs s rando fro unt to unt or fro te to te. Ths could e odeled y usng a dstruton to descre unttount varalty n D f or ore generally the jont dstruton of β and the stochastc ehavor of D f. 3
23 .5 Constant Stress Degradaton Models Nelson (99) refly descres soe asc degradaton odels for constant stress. The followng are the ost wdely used constant stress degradaton odels..5. The Arrhenus Rate Model The Arrhenus rate relatonshp s wdely used for teperatureaccelerated degradaton. Ths odel s ostly used n pharaceutcals nsulatons delectrcs plastcs polyers Adhesves attery cells and ncandescent lap flaents. Arrhenus law: Accordng to the Arrhenus rate law the rate of a sple (frstorder) checal reacton depends on teperature as follows rate = A exp[ E /( kt)] (.5) where: E s the actvaton energy of the reacton usually n electronvolts. k s Boltzann s constant electronvolts per C. T s the asolute Kelvn teperature; t equals the teperature n Centgrade plus 73.6 degrees; the asolute Rankne teperature equals the Fahrenhet teperature plus Fahrenhet degrees. A s a constant that s characterstc of the product falure echans and the test condtons. The product s assued to fal when soe crtcal aount of the checal has reacted (or dffused); Crtcal aount = (rate) (te to falure) or Te to falure = (crtcal aount) / (rate) 4
24 Therefore the nonal te τ to falure ( lfe ) s nversely proportonal to the rate. Ths yelds the Arrhenus lfe relatonshp τ = Aexp[ E /( kt)].5. Inverse Power Relatonshp The nverse power relatonshp s wdely used to odel product lfe as a functon of an acceleratng stress. Ths s ostly used n electrcal nsulatons delectrcs n voltageendurance tests all and roller earngs ncandescent laps and flash laps etc. The relatonshp s soetes called the nverse power law or sply the power law. Suppose that the acceleratng stress varale V s postve. The nverse power relatonshp etween nonal lfe τ of a product and V s τ γ ( V ) A/ V = ; (.5) Here A and γ are paraeters characterstc of the product specen geoetry and farcaton the test ethod etc. The paraeter γ s called the power or exponent..5.3 Eyrng Relatonshp An alteratve to the Arrhenus relatonshp for teperature acceleraton s the Eyrng relatonshp. The Eyrng relatonshp for nonal lfe τ as a functon of asolute teperature T s τ = ( A/ T )exp[ B /( kt)] ; (.53) here A and B are constants that are characterstc of the product ad test ethod and k s the Boltzann s constant..6 Acceleraton Model To otan tely nforaton fro laoratory tests soetes t s requred to use soe for of acceleraton. In soe falure echanss such as the weakenng of an adhesve echancal ond or the growth of a conductng flaent through an nsulator 5
25 the checal or other degradaton process can e accelerated y ncreasng the level of acceleraton varales lke teperature hudty voltage or pressure. If an adequate physcallyased statstcal odel s avalale to relate falure te to levels of acceleratng varales the odel can e used to estate lfete or degradaton rates at product use condtons. Lu Meeker & Escoar [5] entoned the followng acceleraton odels..6. Elevated Teperature Acceleraton The Arrhenus odel whch descres the effect of teperature on the rate of a sple frstorder checal reacton s Ea Ea 65 R( tep) = γ exp = γ exp (.6) k B ( tep ) tep Where tep s teperature n C and k B = /65 s the Boltzann s constant n unts of electron volts per C. The preexponental factor γ and the reacton actvaton energy E a n unts of electron volts are characterstcs of the partcular checal reacton. Takng the rato of the reacton rates at teperatures tep and Acceleraton Factor tep U cancels γ gvng an AF(tep tep U E a ) = R( tep) R( tep U ).6. Nonlnear Degradaton Path and Reactonrate Acceleraton The sple checal degradaton path odel wth a teperature acceleraton factor affectng the rate of reacton s gven y D( t; tep) = D { exp[ RU AF( tep) t]} (.6) 6
26 Here R U s the rate reacton at use teperature tep U R U AF(tep) s the rate reacton at teperature tep and D s the asyptote. When degradaton s easured on a scale decreasng fro zero D < then the falure occurs at the sallest t such that D(t) D f. Equatng D(T ; tep) to D f and solvng for T gves the falure te at teperature tep as T ( tep) = D log R U D AF( tep) f = T ( tep U ) AF( tep) Where T ( tep ) =  (/R U ) log (D f /D ) s falure te at use condtons. U Here the lfe/teperature odel nduced y the sple degradaton process and the Arrhenusacceleraton odel results n a Scale Accelerated Falure Te (SAFT) odel. Under the SAFT odel the degradaton path of a unt at any teperature can e used to deterne the degradaton path that the sae unt would have had at any other specfed teperature y scalng the te axs y the acceleraton factor AF (tep).6.3 Lnear Degradaton Path Reactonrate Acceleraton Consder the odel wth nonlnear degradaton path and reacton rate acceleraton along wth the crtcal level D f. When D(t) s sall relatve to D D( t; tep) D { exp[ RU AF ( tep) t]} = (.63) D + RU AF( tep) t = RU AF( tep t ) If falure occurs when D(T) D f then D(T;tep)= D f and the falure te s gven as D f T ( tepu ) + T ( tep) = = where T(tep + U ) = D f / RU s falure te at use R AF( tep) AF( tep) U condtons. Ths s also an SAFT odel. 7
27 .6.4 Degradaton wth Parallel Reactons A ore coplcated degradaton path odel wth two parallel onestep falure causng checal reactons s gven y : D( t; tep) = D + D { exp[ R { exp[ R U U AF ( tep) t]} AF ( tep) t]} (.64) Where R U and R U are the usecondton rates of the two parallel reactons contrutng to falure. Ths degradaton odel does not lead to an SAFT odel ecause the teperature affects the degradaton processes dfferently nducng nonlnearty nto the acceleraton functon relatng tes at two dfferent teperatures..7 Estaton of Accelerated Degradaton Model Paraeters Lu and Meeker (993) use a twostage ethod to estate the paraeters of the xedeffects accelerated degradaton odel. Stage  For each unt ft the degradaton odel to the saple path and otan the estate of the odel paraeters of each unt. Stage  Cone the estate of the odel paraeters of each unt n the frst stage to produce estates of the populaton paraeters. In another research Lu Meeker & Escoar [5] suggest that n soe cases an approxate axu lkelhood (ML) s faster than n nonlnear least squares estatons requred for the twostage ethod. ML estaton also has the advantages of desrale largesple propertes and easy to use saple paths for whch all of the paraeters cannot e estated. The twostage estaton s useful for otanng startng values for the ML approach for odelng especally when another dstruton other than a jont noral dstruton for the rando effects s eng consdered. 8
28 Meeker and Escoar (993) have gven an updated lterature survey on varous approaches used to assess relalty nforaton fro degradaton data. Boulanger and Escoar[7] address the prole of desgnng a class of ADTs. They assue that each unt s sujected to an elevated constant stress level over the duraton of the experent. The perforance degradaton of each test unt at a stress level could e descred y a growth curve whch levels off to a plateau (axu degradaton) after a perod of te. Fgure. shows the degradaton aount over te. The odel s gve y: y(t) = α[exp((βt) γ )] + є(t) Where y(t) = oserved change of propagaton delay up to te t α β = plateau where the degradaton wll level off = rando coeffcent γ = a constant whch s equal to.5 and є (t) = easureent error. Fgure. Propagaton Delay Growth Curve wth a Plateau (Maxu Degradaton) 9
29 The authors consder α as lognorally dstruted and stress dependent. The desgn prole they consder s to nze the varance of the estate of the ean of the logarth of the plateau ln(α) at use condton. Ther ojectve here s to provde soe gudelnes n desgnng a useful plan for a specal class of accelerated degradaton tests. They frst deterne the optal stress levels and the proporton of unts allocated to each stress level and then deterne optal tes to easure the perforance degradaton of unts at each stress level. The test stress levels are chosen to e 448 K and 373 K whch are the hghest teperatures the plastc package can wthstand and the nu teperature the easureent equpent can detect any degradaton at the end of the experent respectvely. Equal logspacng plan shown n Fg..3. s used for easureent ecause the process shows a great deal of degradaton at the early stage and then stalzes. Fgure.3 Equal LogSpacng Plan for Measureent Although the result s nterestng t s not practcal snce an approprate ternaton te for an experent s usually not known n advance. In the lterature ost of the degradaton odels are lnear or can e transfored to lnear odels. Also ost of the lterature focuses on estatng the paraeters n the degradaton odel and the lfe dstruton. Yu and Tseng[8] proposed an ntutvely onlne and realte rule to deterne an approprate ternaton te for an ADT. Park and Yu[9] develop optal accelerated degradaton test plans under the assuptons of
30 destructve testng and the sple constant rate relatonshp etween stress and product perforance. The authors deterne stress levels the proporton of test unts allocated to each stress level and easureent tes such that the asyptotc varance of the axu lkelhood estator of the MTTF at the use condton s nzed. Yu and Tseng [ 4] descre a ethod for conductng a degradaton experent effcently consderng several factors such as the nspecton frequency ternaton te and the saple sze. They consder a typcal degradaton path of an LED whch s ln(ln(y(t))) = ln(α) + β ln(t) + є(t) Where y (t) = standardzed lght ntensty of an LED devce at te t α β = paraeters of the degradaton path and є (t) = easureent error of the devce at te t. Based on data (t k y j (t k )) s the for stress level k = where k s the ndex that represents the th easureent for unt j the degradaton paraeters α β for unt j can e otaned. The falure te of unt j can e found usng ln( D) τ = α β assung D s the crtcal value for the standardzed lght ntensty when an LED fals. In these papers they deal wth the optal desgn for a degradaton experent under the constrant that the total experental cost does not exceed a predeterned udget. The optal decson varales are otaned y nzng the varance of the estated pth percentle of the lfete dstruton. But these three decson varales have a
31 great deal of nfluence upon the experental cost and the precson of selectng the ost relale product usng degradaton data. A nonlnear nteger prograng prole s developed to deterne the optal conaton of saple sze nspecton frequency and ternaton te. As s evdent fro the aove revew an extensve lterature on the desgn of degradaton tests and accelerated degradaton tests exsts. But when desgnng a degradaton test the dstruton of the degradaton rate of the product/coponent at whch t degrades s very portant. The Weull and lognoral dstrutons are two ost popular lfete odels n relalty analyss that have een used for ths purpose. An ncorrect choce of the dstruton ay lead to serous as.
32 CHAPTER 3 PROBLEM STATEMENT Wth regard to hghly relale products t s portant to consder the ssues of how to plan tests that provde the ost effcent use of resources especally as t relates to conductng an accelerated degradaton test. The prole s to nze the expected value of the MeanTeToFalure of a product suject to the constrant that the total cost of the test does not exceed a predeterned test udget. 3. The Optzaton Prole The followng decson varales are portant n conductng an ADT effcently (Yu & Chao [ ]). These varales not only affect the experental cost ut also affect the precson of estatng the MTTF ( φ ) whch can e defned as the expected or the ean value of the falure te. The pertnent questons regardng the test plan are How s an approprate nspecton frequency (f ) deterned? How any tes (l ) should the product s perforance e nspected at each stress level? How any devces (saple sze n) should e taken for testng at each stress level? In order to easure the precson of estatng the MTTF the expected wdth of the confdence nterval values of the MeanTeToFalure( φ ) s coputed. The expected value of a realvalued rando varale gves the ean or central tendency of the 3
33 dstruton of the varale. The unased axu lkelhood estator of the MTTF can e acheved y estatng the asyptotc varance. The asyptotc varance can e otaned y nzng ether the ean square error or the expected value of the range of MTTF (MTTF ax MTTF n ). Snce we do not have a pror estate of the ean square error we wll estate the asyptotc varance y nzng the MTTF. Thus the optal decson prole ased on the expected range of the MTTF s forulated as follows: Where as TC ({ f l} n) ({ f l } n) = Mnze [ φ ] E (3.) φ Suject to TC { f l } ( n) C (3.) = f l n є N = { 3 } (3.3) = 3. = denote the total cost of conductng an ADT. ˆ φ denote an estator of [ φ φ ] denote a (p) % CI of [ φ ] φ ( n) = ( f l n) φ ased on a test plan { f l } φ fro the test plan { } = E denotes the expected wdth of the ( p)% CI of φ C s the total cost of the udget. p s the percentle of the lfe te dstruton of the product at noral use condton. 4
34 CHAPTER 4 METHODOLOGY 4. Degradaton Model wth Rando Coeffcent Let η() t denote the qualty characterstc (degradaton path) of the product at te t. Assue that there exsts a sutale functon ω (.) such that α ω ( η( t)) = βt t (4.) Where α > s a fxed and known constant; β > s a rando coeffcent that vares fro unt to unt. 4. Assuptons The ADT uses stress levels S S S S satsfyng (S ) S S S where S s the use condton. Due to the easureent errors the actual degradaton path cannot e oserved drectly. Let y j (t k ) denote the saple degradaton path of the j th devce at te t k under the stress level S.The path can e expressed as follows: α ω y ( t )) = β t + ( t ) (4.) ( j k j k j k The unts put nto test are randoly selected fro the saples and are randoly assgned to test stress levels. At each stress level n devces are randoly selected for testng. 5
35 Suppose that under stress level S the nspectons are ade l tes for every f unts of te (e.g. f hours or f days) untl te t l = f l t u where l s a postve nteger and t u s a unt of te. Assue that β j follows a recprocal weull dstruton then β  follows a weull dstruton wth scale paraeter θ and shape paraeter δ (whch s denoted y β  ~ Weull (θ δ ) The shape paraeter δ does not depend on the stress level and the scale paraeter or the characterstc lfe θ s a functon of transfored levels of stress: ln ( θ ) γ + γ X = (4.) Where γ and γ are unknown paraeters to e estated fro the data X =X (S ) and X (.) s a sutale transforaton. Two falar exaples for X (.) are as follows: X (S ) = / S f an Arrhenus odel s assued = ln S f an nversepower odel s assued Soe other relatonshps whch are coonly used are entoned n the lterature revew n chapter. In order to solve the optzaton prole the MTTF has to e coputed frst. 4.3 The MeanTeToFalure The product lfe te (τ ) s sutaly defned as the te when η crosses the crtcal level D. Fro Eq (3.) τ can e expressed as ω( D) τ = β / α Takng natural logarth on oth sdes ln( τ ) = [ ln( ω( D )) ln β ] (4.3) α 6
36 Snce β follows recprocal weull dstruton ln β follows the extree value dstruton wth scale paraeter u and locaton paraeter (whch s denoted y ln β ~ Extree (u ) and n equaton (4.) wth α fxed t can e shown that τ follows the Weull dstruton wth scale paraeter ( θ *( D / α ω( ))) and shape paraeterαδ. Let τ denote the product s lfete under S. Thus we have τ / ( θ * ( ω( ))) αδ ) α = Weull D The MTTF φ of the product s lfete dstruton under S s Where u = γ + γ X ) ( S / ( θ ( ω( ))) Γ + α φ = D αδ / α u φ ( ( )) Γ = ω D exp + (4.3) α α Here the prole s to desgn an effcent ADT such that φ can e estated precsely. The optzaton prole can e solved y usng the followng steps. 4.4 The Coputaton of [ φ φ ] l For j n and ased on the oservatons {( t y ( t )} k j k ) k = the leastsquares estator (LSE) βˆ of β condtonal on j j β j can e coputed y nzng l = LS( β j ) k = { ( ( )) } α ω y t + β t j k j k Thus we otan l ω( yj ( t k )) t l α t k α k = k ˆ β = (4.4) j k = 7
37 and σ can e estated y σ = j= ( ˆ β ) n = LS j (4.4) n( l ) By consderng the frstorder Taylor seres expanson aout of followng approxate forula for ln βˆ j : βˆ j ln β j we can otan the ˆ β ˆ j ln β ln + j β j (4.43) β j where condtonal on ˆ β j β j E = β j Var ˆ β β l α / = t k k and [ j j ] σ =.Hence t s β j seen that ˆ β j β j as l k = t α k (4.44) Fro equaton (4.44) t s seen that the asyptotc dstruton of uncondtonal follows an extree value dstruton wth ( ˆ ˆ ). ln βˆj u Thus ( uˆ ˆ ) axu lkelhood estators (MLEs) of ( u ) can e otaned drectly (Lawless98) y: the conventonal e uˆ n x j = exp n j ˆ = ˆ and 8
38 9 ˆ exp exp = = = = = = n j j n j j n j j j x n x x x where. ˆ ln n j x j j = β Here û and ˆ can e solved y usng soe nuercal ethods( e.g. Newton s ethod) wth an teratve procedure. Based on the asyptotcally effcent property of axu lkelhood estate (Lawless 98) the jont densty of û and ˆ follows an asyptotcally varate noral dstruton as follows. ~ ˆ ˆ u N u (4.45) where [ ] [ ] [ ] [ ] = Var Cov u Cov u u Var ˆ ˆ ˆ ˆ ˆ ˆ = I  (u ) denotes the covarance atrx of û and ˆ. The fsher nforaton atrx I(u ) can e expressed as follows: ( ) ( ) ( ) ( ) ( ) ln ln ln ln = u L E u u L E u u L E u u L E u I where ( ) = = j j n j u x u x u L exp exp
39 By usng the technque of ntegraton y parts Var [ û ] Cov [ û ˆ ] and Var [ ˆ ] can e otaned as follows. Var 6 π [ uˆ ] = + ( γ ) (4.46) n π 6 [ ˆ 6 ] = ( γ ) ˆ Cov u (4.47) n π Var [ ˆ 6 ] = (4.48) n π Where Γ (x) s the gaa functon and γ =.577 s the Euler s constant. In a real stuaton the experent s only conducted up to te t l.thus the paraeters u ) can e slghtly calrated y the condtonal expectaton technque. ( Assung ( u l l ) denotes the paraeters after refned calraton the approxate relatons etween ( ) l l u and u ) can e expressed (Hong Fwu Yu []) as follows: ( u l u + γ ( l ) where: l + 6σ θ Γ + δ α α l α tu f k = t k To assure that k = l t α k s suffcently large t s reasonale to set 6σ θ Γ + δ α α l t f u = t k α k / s < s < (4.49) 3
40 Ths equaton ndcates that the slower the qualty characterstc of a product degrades the longer the degradaton test should last. Thus ˆ can e further estated as follows: ˆ = (4.4) ˆ ˆ = Based on these estators { } can e otaned as follows (Lawless [6]): u = the LSEs ˆ γ ˆ ) of γ ) n Equaton (4.) ˆ γ = ˆ γ ( γ T T ( X X ) X Y ( γ where X T = X ( S )... X ( S )... X ( S ) and Y T ( u uˆ... uˆ ) ˆ =.Thus u can e estated y = ˆ γ + ˆ γ X ( ) (4.4) uˆ S The approxate dstruton of û and ˆ s as follows 6 Π + H u ˆ ~ N u * γ ( γ ) (4.4) n Π 6 G = S = where H = X ( S ) + X ( S ) X ( S ) X ( ) ( = S = and G = X S ) ( X ( )) ˆ 6 ~ N (4.43) n Π The approxate ( p) % and ( p ) % confdence nterval (CI) of u and can e otaned as follows: 3
41 [ u u ] = Ζ exp uˆ uˆ ) Ζ exp p / ˆ / var( ˆ u p u uˆ var( uˆ ) and [ ] = Ζ exp ˆ p / var ˆ () ˆ Ζ ˆexp p / var ˆ () ˆ Where: Ζ s the( p ) th percentle of standard noral dstruton and Ζ / s p / p the( p ) th percentle of standard noral dstruton. p and p are the percentle values for scale(u ) and shape() paraeters respectvely. Now susttutng ( u ) and ( u ) ( p)( p )% CI for φ as follows: [ φ φ ] = ω( ) nto Equaton (4.3) we otan an approxate u u ( ) Γ + α ( ( )) Γ ω α D exp D exp + (4.44) α α α α 4.5 The Coputaton of E [ φ ] φ By takng the natural logarth of oth sdes of Eq (4.3) we have ln ˆ φ ( ) = f l n [ ln( ω( D) ) + u ] + ln Γ + α α (4.5) 3
42 33 The asyptotc dstruton of ( ) n l f ˆ ln φ follows a noral dstruton ( ) ν N where ( ) ( ) ( ) [ ] ( ) ( ) 6 ln ln ln ˆ var ln ˆ ln Γ + + Γ + Γ + + = Γ = γ π α α α α α ν α ω α n E E u and E u D Hence the asyptotc ean of { } ( ) n l f ˆ = φ can e expressed as follows: { } ( ) [ ] + = exp ˆ ν φ n l f E (4.5) Therefore [ ] ( ) + exp ν ν φ φ E where ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) Γ + Γ + + Γ + + Γ + Γ + Γ + + = Γ + Γ = n n E E E E u u and E E u u D ln ln * 6 ln ln ln ln ˆ var ˆ var ln ln ˆ ˆ ln α α γ π α α α α α α ν ν α α ω α
43 The Cost Functon { } ( ) n l f TC = The total cost { } ( ) n l f TC = of conductng an ADT s dvded nto three parts (Yu and Tseng [ ]). The cost of conductng an ADT s { } = + p s l f C l f C ax where C s denotes the operator s salary per unt of te and C p denotes the unt cost of power of the testng equpent. The easureent cost s = l n C where C denotes the unt cost of easureent. The cost to test the devces s n C d where C d denotes the unt cost per devce. Therefore the total cost of the experent s { } ( ) { } n C l n C l f l f n l f TC d + + = = = = p s + C ax C (4.6) 4.7 The Optzaton Model Fro the foregong results the optzaton prole can e expressed as follows ( ) ( ) { } ( )( ) n N n l f p p p where p p p s k f t C n C l n C l f C l f C s t Mn l k u d p s 3... : (4.73) * 6 (4.7) ax.. (4.7) exp = < < = Γ = = = δ θ σ ν ν ε α α α
44 4.8 Algorth Due to the coplexty of the ojectve functon t s dffcult to fnd an effcent ethod to solve the optzaton odel n equaton (4.7). The ojectve functon can e ( n) expressed as a functon of p and { f l } =. Hence wth splcty structure of the constrant and the nteger restrcton on the decson varales an approxate soluton can e otaned y the followng steps. ( ( n ) = [ φ φ ] Let p { f l } = E. Partton the nterval ( ) Set p equally nto l (say l =) suntervals. p p ( k) = k k =.( l ). For each ( ) l p k the correspondng ( ) optal conaton { f k) l ( k) } n( k) Gven { f }. = ( Deterne the correspondng { } Deterne the correspondng n. ( ) [ ] = can e otaned as follows. l = y Equaton (4.49). Copute V { f} = φ fro the test plan { f l} E φ = ({ } n ( k) ) The optal soluton f ( k) l ( k) { } f = Where = satsfy V { f } = n n () ( ) ( n) = can e deterned f ( ) V { f} =... ( ) (4.8) = f f f f f 6σ θ Γ + δ = α tu π s G 35
45 and x G denotes the largest nteger not greater than x. In the nzaton ( n) process of (4.6) any { f l } would not e taken nto consderaton. * * * Fnally an approxate optal soluton p { f l } = that does not satsfy the cost constrant * ( ( n ) = can e deterned f * * * * * * * ( p ({ f l } n ) = n ( p( k) ){ f ( k) l ( k) } n ( k ( )) = = k l 36
46 CHAPTER 5 EXAMPLE The relalty of electronc devces s of a crtcal concern especally for ltary aerospace and councaton applcatons. LEDs (lght ettng dodes) are consdered a good lght source for optcal lnks wth good teperature dependence sall power consupton and hgh relalty. Snce LEDs are desgned to e n servce for several years wthout falure t s hard to oserve falures under noral operatng condtons n a short te. The relalty perforance of LEDs (Lght Ettng Dodes) has nearly always een superor to that of ncandescent neon and other type laps. In addton today s LED s have uch hgher relaltes than early LED devces. Iproved assely growth ethods and structures along wth new aterals have allowed for the developent and ass producton of extreely relale hgh rghtness LEDs n all colors ncludng whte. The expected useale lfete of an LED s usually estated y the extrapolaton of easured data or y estatng the value fro accelerated testng. Accelerated testng nvolves sujectng the LED to ore extree condtons (.e.: hgher teperature and/or hgher currents) than would e expected under noral operatng condtons. Ths s necessary snce t s often dffcult and practcal to actually test an LED for hours or over years. The an concern wth accelerated testng of LEDs s understandng how to accurately translate these results to noral operatng condtons. 37
47 The lfete of an LED s defned as the te t takes for the lght output to reach 5% of ts ntal value. The average lfete specfed y LED anufacturers s hours. Ths does not ean that the LED wll cease to operate after K hours; n fact ost LEDs wll functon for thousands of hours eyond the specfed lfete value. It eans that after hours the LED wll e half as rght as ts ntal lunosty level. In ths chapter the applcalty of the proposed odel s deonstrated y a nuercal exaple. 5. Sulaton Experent The purpose of the sulaton experent s to generate the data that would e used to estate the relalty of LEDs (type GaAlAs) at noral operatng condton wth teperature S = 78 K (5 C) y usng the degradaton data otaned at the three accelerated stress levels S = 98 K (5 C) S = 338 K (65 C) S 3 = 378 K (5 C). The data for twenty fve LEDs were sulated at each of these three teperatures. The duraton of each cycle (Sulaton run) s 336 hours and the total nuer of cycles s 9. Each cycle represents an nspecton nterval. Let ω (t) denote the oserved standardzed lght ntensty of the jth LED at te t j under S. The data s sulated n Matla y assung the rando varale β follows a recprocal Weull dstruton. By usng the Arrhenus relatonshp etween teperature and te the degradaton data was generated at the three stress levels S S and S 3. The data represents the standardzed lght ntensty of each coponent at a partcular te.65 t.the resultng data s gven n tales Fgure 5. shows the sulated saple j 38
48 degradaton paths of 5 LEDs. Fgure 5. s the plots of ω (t) versus j.65 t under S S and S 3. It s seen fro the fgure that there exsts a lnear relatonshp etween ω (t) andt. 65 s gven y: j.65 ω ( t) = β t + ( t) (5.) j j j Where (t) s the error ter. j 39
49 4 Tale 5.: The Sulated Standardzed Saple Degradaton Paths at Stress Level S Te (hr) w (t) w (t) w 3(t) w 4(t) w 5(t) w 6(t) w 7(t) w 8(t) w 9(t) w (t)
50 4 Tale 5.: The Sulated Standardzed Saple Degradaton Paths at Stress Level S Te (hr) w (t) w (t) w 3(t) w 4(t) w 5(t) w 6(t) w 7(t) w 8(t) w 9(t) w (t)
51 4 Tale 5.3: The Sulated Standardzed Saple Degradaton Paths at Stress Level S 3 Te (hr) w 3(t) w 3(t) w 33(t) w 34(t) w 35(t) w 36(t) w 37(t) w 38(t) w 39(t) w 3(t)
52 wj(t) E44E46E48E4 E3 .E3 .4E te(hour) E 4E 6E 8E E .E .4E (a) te(hour) () 5E E 8 w3j(t) .5E 8 E E 83E 8 t e (h o u r) (c) Fgure 5.. The Standardzed Saple Degradaton Paths under (a) S () S and (c) S 3 43
53 E E wj(t) 6E48E4 E3 .E3 .4E3 t^.65 (a) wj(t) E 4E E 8E E .E .4E t^ () 5E9 E w3j(t) .5E8 E8 .5E83E8 t^.65 (c) Fgure 5.. The Plots of ω j (t) versus t.65 under (a) S () S and (c) S 3 44
54 Based on the oservatons {( ( ))} 3 k j t k t ω the LSEs βˆ j and σˆ k = j can e coputed. To ake sure of the approprateness of the Weulldstruton Weull proalty plots were constructed for each hgher stress level (Fgure 5.3). All of the trends appear lnear aout the reference lne. Fgure 5.4 shows the noral proalty plots for the resduals under S S and S 3. The plots ndcate that the dstruton assuptons for β and (t) are reasonale. Fro Equatons and 4. we have ˆ σ = ˆ =.68 and the Arrhenus relatonshp s gven y: ˆ γ = ˆ γ + (5.) S uˆ Where ˆ γ =.9977 and ˆ γ = To otan the optal test plan for the ADT of LED we need the actual values ofσ and ( γ γ ). For convenence these estates are treated as the true values to evaluate the optal test plan of LED data. 45
55 (a) () (c) Fgure 5.3 The Weull Proalty Plot for { β } 3 ˆ j 5 j= 46
56 (a) () (c) Fgure 5.4 The Noral Proalty Plots for Resduals under (a) S () S (c) S 3 47