MEAN residual lifetime (MRL) may be predicted according. Degradation models for reliability estimation and mean residual lifetime

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1 Degradation models for reliability estimation and mean residual lifetime Christophe Letot 1, Pierre Dehombreux 1 1 Service de Génie Mécanique & Pôle Risques, Faculté Polytechnique de Mons 53 Rue du Joncquois, 7 Mons - Belgique christophe.letot@umons.ac.be, pierre.dehombreux@umons.ac.be, Abstract This paper discusses the interest of some degradation models and measurements in order to improve the reliability assessment of an equipment. Classical methods used to estimate the reliability are based on a set of failure times on which a reliability model is fitted. Usual models are exponential, Weibull and lognormal laws. If an equipment is subjected to a degradation process on which a threshold may be fixed, it is useful to track this degradation to retrieve the health state of the system at some time points. From the set of degradation points, a degradation model may be fitted. This degradation model is then used to predict the failure time. The interest of this approach is that no failure is needed to retrieve the reliability information. Moreover it allows the estimation of the mean residual lifetime of the equipment as the degradation curve may be extrapolated to its critical threshold. Keywords reliability, mean residual lifetime, degradation, physical model I. INTRODUCTION AND REVIEWING MEAN residual lifetime MRL) may be predicted according to three approaches figure 1). Fig. 1. The three approaches to estimates MRL and their links The first method considers Physics-of-Failures PoF) models for which the degradation law is supposed to be known. It consists of studying the effects of dispersion on some parameters of the law. The second method condition monitoring) supposes that a degradation measurement may be tracked with time this could be a crack size, a loss of efficiency in a gas turbine, a vibration measurement,...) ; based on the degradation data, this approach fits a degradation law. The last approach is the statistical reliability study based on the collection of failure times. Improving the reliability estimation with degradation information may be achieved by linking the reliability approach with either the monitoring approach if the degradation may be tracked) or either with the Physics-of- Failures approach. The first link is achieved by using a proportionnal hazard model developped by Cox [2]. The Cox model is mainly used in the fields of medecine or maintenance to study the influence of covariate data on different subjects. The expression of this model is composed of a base risk function which is corrected by a term that includes the covariates. The main difficulty for the Cox model is the estimation of the parameters when the covariates are time dependant. The second method is achieved by using First Hitting Threshold Time FHTT) models [3]. The goal of this method is to determine the time of reaching a critical threshold for stochastic degradation curves. FHTT models may be also used with monitored data as illustrated in figure 1. In this paper, we will focus our work on the FHTT approach applied to PoF models. FHTT models have been already used in various applications. Whitmore [4] used such models in a survival analysis study. Van Noortwijk [5] used a gamma FHTT process to represent a loss of resistance for a dike. Lawless [6] also used a gamma FHTT process to model the evolution of cracks. Lee and Whitmore [3] summarized the recent advancements for FHTT models. Maintenance decision policies based on the FHTT theory are presented in [12] and [13]. A. Reliability notions II. BACKGROUND INFORMATION Reliability R) is the ability of a system or component to perform its required functions under stated conditions for a specified period of time. Reliability is measured by a probability value that decreases as time goes on. If T is a stochastic failure time of an equipment, the mathematical

2 expression of its reliability is : Rt) = P T > t) 1) The complementary function of the reliability is the failure function F ). The failure function represents the repartition of failure times for a set of identical equipements that have been tested. The failure function is thus used to fit a reliability model on failure data. F t) = 1 Rt) = 1 P T > t) = P T < t) 2) The repartition of failure times may be represented by the density function f). The density function is the derivative of the failure function. df t) ft) = = drt) 3) dt dt From the above equations we define the failure rate h) as to be the ratio of the density function by the reliability function. It indicates the risk of failure at time t according that the equipment has already survived until t. B. Reliability models ht) = ft) Rt) From a set of failure times, we may fit a reliability model. Usual models are : Exponential 4) Rt) = exp λt) 5) λ is the scale parameter. Exponential models have a constant failure rate and are memoryless. They are well suited when the failures occur randomly. Lognormal Rt) = Φ nor lnt) µ ) 6) σ µ is the mean parameter and σ is the standard deviation. Lognormal models allows to get only positive failure times by the logarithmic transformation of time. Weibull law Rt) = exp t γ η )β ) 7) η is the scale parameter, β is the shape parameter and γ is the location parameter usually γ = ). Weibull models are well suited for most cases as they are quite flexible with both parameters η and β. They are mainly used for components that undergo some degradation. C. Model fitting From a collection of N failure data, a non parametric failure function ˆF ) may be calculated. There are several methods to get this function. The basic idea is to sort the failure times by ascending order and to count the number i) of failures occured after each time inspection. Here are some examples : Kaplan-Meier ˆF t i ) = i 8) N This basic formulation presents the disadvantage of having a probability equal to 1 for the last observed failure time. Product limit or medium rank) ˆF t i ) = i 9) N + 1 The denominator is increased by one to solve the problem of having a probability = 1 for the last observed failure time. Rank adjust or median rank) ˆF t i ) = 1 + N i + 1 ) 1 F.5,k,l 1) i where F.5,k,l is the Fisher-Snedecor distribution with k = 2N i + 1) and l = 2i degrés de liberté. An approached formulation may be used : ˆF t i ) = i.3 N ) From this non parametric failure function ˆF t i ) we can adjust a reliability model. The methods to fit these models are : Regression methods It consists of linearising the failure function to obtain the form Y = AX + B. Example for the weibull law : )) 1 ln ln 1 ˆF = βlnt) βlnη) 12) t)

3 Identifying the parameters, we obtain { β = A η = exp B β ) 13) Maximum likelihood methods This method consists of finding the parameters of the law that maximize the likelihood to find back the failure times. The likelihood function has the form [7] : Degradation value Degradation following a Wiener process Lp) = n ft i p) 14) i=1 Using the logarithmic transformation, the parameters that maximize the likelihood are found by taking the partial derivative for each parameter. δ ln Lp 1,..., p k ) δp i = ; i = 1,..., k 15) Example for the exponential law : Lλ, t i ) = δ lnλ, t i ) δ λ = n λ exp λt i ) 16) i=1 n i=1 1 λ = 1 n ) 1 λ t i = 17) n t i 18) i=1 III. FHTT APPROACH AND MEAN REMAINING USEFUL LIFE First Hitting Threshold Time models are composed by : A stochastic degradation function Zt) that represents the evolution of a degradation process physical wear, chemical corrosion, a loss of quality or performance) A critical threshold z c that corresponds to the failure of the equipment or the process critical stress, size of cracks, critical limit of performance,...) Units of time Fig. 2. Example of Gaussian stochastic degradation curves for 2 items From the above statements, the first hitting time T c ) to reach the critical threshold is : T c = inf{t Zt) z c } 19) This formulation involves that we observed a truncated stochastic process as we do not take care of the evolution of the process when it goes beyond the threshold. For each item, the observed field is thus a random interval [, T ci ] as at time T ci the failure occurs for item i and ZT ci ) = z ci ). Choosing a stochastic degradation model is function of the nature of the observed process and available data. Indeed for a process that mainly undergoes shocks without any loss of degradation between two consecutive shocks, a gamma model of the degradation is well suited. The equation 19 may be linked to the reliability : Rt) = P T c > t) 2) ZT c ) = z c T c = Z 1 z c ) 21) Rt) = P Z 1 z c ) > t) 22) For example, if we consider the following degradation process : with : Zt) = Z) + mt + σw t) 23) m the mean trend and σ the drift, Wt) is a Wiener process caracterised by : W) = ; Wt) is a continuous function respect to time ;

4 Wt) has independant steps according to Wt)-Ws) = N,t-s). Zt) is thus a Gaussian degradation process with a linear trend. An example of degradation curves following this process is showed in figure 2 with the following parameters m =.5, σ =.5, z c = 1 and a step increment of.1 units of time. One can proove that the failure times are distributed according to an inverse Gaussian law with density function [14] : ft, z c, σ, m) = z [ c t 3/2 exp z ] c mt) 2πσ 2σ 2 t 24) If the degradation relationship Zt) is continuous respect to time, we can define a damage index DI at time t i by : DIt i ) = Zt i) Zt ) 25) ZT c ) Zt ) The mean residual lifetime at time t MRLt)) may be obtained from the reliability information by [8] : t Rt)dt MRLt) = 26) Rt) At t=, we obtain the mean time to failure MTTF) : MRL) = Rt)dt Rt)dt = = MTTF 27) R) 1 The figure 3 presents the evolution of the mean residual life obtained from the failure rate function respect to time for a modified Weibull extension distribution with a reliability function [9] : [ ) )] t β Rt) = exp η 1 exp 28) α The figure 4 shows a comparison of the mean residual life function obtained by eq. 26. MRLt) beta =.45 beta =.5 beta =.6 Mean residual life with time t Fig. 4. MRLt) for a Weibull extension distribution α = 1, η = 2) by eq. 26 IV. THEORITICAL ILLUSTRATIONS This approach is applied to two examples. The first one considers the evolution of crack growth according to Paris law. The second one considers the degradation of electrical resistances according to an Arrhenius type formulation and its consequences on the output voltage threshold. A. Paris degradation law The first application concerns the crack growth for mechanical equipments. The degradation model is the simplified Paris relationship [1] which links the evolution of a crack size a with the number of cycles N by : da dn = C Km 29) with K = K max K min. K is the stress intensity factor MPa m). This factor may have various forms depending of the shape function F a) : K = σ πa F a) 3) For this example we consider F a) = 1. K = σ max σ min ) π a 31) The sollicitation is supposed to be composed of tensile strength and compressive strength with σ mean =. σ min corresponds to compressive strength and is fixed to as a compressive strength tends to close the crack. with σ = σ max and K = σ πa 32) Fig. 3. MRLt) and FRt) for a Weibull extension distribution α = 1, η = 2) [9] da dn = C σm π a) m/2 33) 1 dn = C σm π a) m/2 da 34)

5 ac N c = C σ m π a) m/2 ) 1 da 35) a N c = 1 C σ m π m/2 ac a a m/2 da 36) N c = a m/2)+1 c a m/2)+1 m/2) + 1) C σ m π m/2 ) 37) Thus the number of cycles to critical failure N c ) may be calculated by the above equation if we know the parameters. For example : a =.1 m a c =.8 m σ = 1 MPa m = 3 C = m cycle MPa m) m We obtain N c = cycles Then we introduce some dispersion on the parameter C according to a Weibull law distribution. The scale parameter is equal to the nominal value of C η C = m cycle MPa m) m while the shape parameter is arbitrary fixed to β C = 4 to represent the dispersion. Crack length a Paris law da/dn..e+ 5.e+5 1.e+6 1.5e+6 2.e+6 2.5e+6 Number of cycles N Fig. 5. Paris degradation curves da/dn The figure 5 represents the evolution of the crack size a with the number of cycles N for 2 items with the above parameters a, a c, σ, m, C Weibullβ C, η C )). The reliability function may be expressed by : RN) = P N c > N) 38) ) a m/2)+1 c a m/2)+1 RN) = P m/2) + 1) C σ m π m/2 ) > N 39) ) RN) = P C < a m/2)+1 c a m/2)+1 m/2) + 1) σ m π m/2 4) ) N and the failure function by : F N) = 1 RN) 41) C is a stochastic value following a Weibull distribution W β c, η c ) : P C < u) = 1 exp u η c ) βc ) 42) So the reliability function RN) is : 1 exp a m/2)+1 c a m/2)+1 m/2) + 1) σ m π m/2 ) N η C ) βc 43) Simulations have been achieved on 5 items. Results are presented on the figure 6. The red stair curve is the non parametric failure function obtained by the product limit PL) approach on simulated failure times eq. 9). The long dash dot cyan curve is a Weibull model adjusted from the non parametric failure function by the regression method eq. 12). The dash blue curve is a lognormal model also fitted on the non parametric failure function by regression. The dot brown curve is a Weibull model fitted by the maximum likelihood estimation MLE) method. Finally the dash dot green curve is the predicted failure function from eq. 41 and eq. 43. FN) Failure functions. 1.6e+6 1.7e+6 1.8e+6 1.9e+6 2.e+6 2.1e+6 2.2e+6 2.3e+6 Number of cycles non parametric failure function weibull PL model lognormal PL model Weibull MLE model predicted degradation model Fig. 6. Failure function for different reliability models The predicted degradation model best fits the non parametric failure function. The weibull PL model fits the data with a correlation factor of 8% while the lognormal model fits the data with a correlation factor of 94% table I). The reliability obtained by the predicted degradation model allows us to predict the mean residual lifetime for a given damage index. For a DI =.5 we can find the mean number of cycles N i corresponding to this DI by eq. 25. DIN i ) = an i) a) an c ) a) 44)

6 .5 = an i) ) We find that the mean crack length an i ) for DI =.5 is.45 m. The mean number of cycles N i may be obtained by eq. 37 N i = an i) m/2)+1 a) m/2)+1 m/2) + 1) C σ m π m/2 ) 46) We find N i = Mcycles. Finally eq. 26 gives us the mean residual lifetime. MRL1.813 Mcycles) = Mcycles Rt)dt R1.813 Mcycles) 47) The MRL for an observed DI =.5 before the crack reaches the threshold worth cycles. TABLE I RESULTS OBTAINED FOR THE 4 RELIABILITY MODELS Model scale shape correlation MRL1.813 Mcycles) Weibull PL Mcycles %.143 Mcycles Lognormal PL cycles.37 9%.138 Mcycles Weibull MLE Mcycles Mcycles Degradation %.126 Mcycles We can see that the degradation reliability model is the pessimistic model for the MRL calculation. B. Electrical resistance degradation This second example concerns the increase of electrical resistance values over time [11]. For an electrical system, this increase of resistance values will have an effect of the output voltage. It is well known that the inital value of resistances may differs by some percentages of the nominal value. The goal is to determine the impact of this initial dispersion on the evolution of the output voltage. Lets consider the following basic electric diagram figure 7). Fig. 7. Electrical diagram Fig. 8. Failure : V out > 5.5 V The link between the input voltage and the output voltage is V out t) = R 1 t) R 1 t) + R 2 t)) V in 48) We consider the input voltage V in = 1 V fixed in time. Nominal initial values for R 1 and R 2 are both fixed to 5 Ω and supposed to be affected by the same dispersion. The expected initial nominal value for V out is thus equal to 5 V. The maximum output voltage allowed is fixed to 5.5 V. A weibull distribution is chosen to consider some dispersions on the initial values of the resistances. P R 1, R 2 < X) = 1 exp ) ) X β η 49) η = 5 Ω corresponds to the expected initial value and β is fixed to 4 to have a dispersion on R 1,2 that ranges from [4, 5.5[ with a high probability. Resistances values will degrade over time due to temperature effect. This is known as the resistance drift R R and can be modelized by an Arrhenius type formulation. Rt) R = Rt) R R = α t exp E ) a k T 5) where : E a is the activation energy =.9 ev. α is the pre-exponential constant. α R1 = Ω/time unit and α R2 = Ω/time unit. k is the Boltzmann s constant = ) ev/k. T the temperature = 293 K. Equation 5 may be transformed in Rt) = R 1 + α t exp )) Ea k T Subsituting the parameters by their values we have : { R 1 t) = R t) R 2 t) = R t) 51) 52) First of all we will established the mathematical relationship that gives the failure time = time from which V out > 5.5V ). From equation 48, introducing the ratio R1t)/R2t) = At) : V out t) = V out t) = R1t)/R2t) R1t)/R2t)) + 1) V in 53) At) At) + 1) V in 54) At T c The threshold = 5.5V is reached

7 V out T c ) = Isolating T c, we obtain : T c = R T c) R T c) R T c) R T c) + 1) V in = 55) R 1 R R 1 R 2 V in.5 R 1 V in R 2 R 1 R 2.1) 56) R 1 /R 2 = A being the initial ratio of the resistances, T c = A 1 V in ) A V in 1).1) 57) Equation 57 allows to predict the failure times by the knowledge of the initial values of R 1,2 and the ratio V in /. Table II shows a comparison between simulated times and predicted times with equation 57 for 1 items. TABLE II SIMULATED AND PREDICTED TIMES COMPARISON Rt) = P T c > t) 58) ) A 1 V in V P max ) A V in 1).1) > t 59) Isolating the factor A in the left hand term : ) 1 +.5t P A < V 6) in 1)1 +.5t) Remembering that the initial values of resistances R 1 and R 2 are distributed according to a Weibull law with shape parameter β = 4 and scale parameter η = 5, the ratio of the initial resistances A is distributed according to a Weibull law with shape parameter β A = 4 and a scale parameter η A = R 1 /R 2. A is a stochastic value following a Weibull distribution W β A, η A ) ) ) u βa P A < u) = 1 exp 61) Thus equation 6 becomes : η A n o Tsimul Tpred Degradation curves Rt) = 1 exp t V in 1)1 +.5 t) η A ) βa 62) Equation 62 is different from usual Weibull expressions as the time factor also appears at the denominator. Simulations have been achieved on 5 items figure 1). The red stair curve is the non parametric failure function obtained by the product limit approach on simulated failure times eq. 9). The dash blue curve is a Weibull model adjusted from the non parametric failure function by the regression method eq. 12). The dot yellow curve is a lognormal model also fitted on the non parametric failure function by regression. The dash dot green curve is the predicted failure function from eq. 62. Voutt) Ft) Failure functions t Fig. 9. Degradation curves from random initial values of resistances The reliability function may be also predicted as well Units of time non parametric failure function Weibull PL model lognormal PL model Weibull MLE model predicted model Fig. 1. Failures functions for different models

8 We can see that the predicted model best fits the data. For the adjusted models, the lognormal model has a better correlation factor 98%) than the weibull law 87%) table III). Residual lifetime estimation from a damage index is obtained as follow. For a DI =.5, we can find the mean time t i corresponding to this DI by eq. 25. DIt i ) = V outt i ) V out t ) V out t c ) V out t ) 63) DIt i ) = V outt i ) 5 64) We obtain V out t i ) = 5.25 V. By eq. 57 we obtain the time t i corresponding to the DI =.5 A 1 V in V ) + 1 outti ) t i = V.5 A in V 1).1) 65) outti ) We find t i = units of time UT) and finally eq. 26 gives us the expected mean remaining useful life at time t i 23.67UT MRL23.67UT ) = Rt)dt 66) R23.67UT ) The MRL at time t i = for the device to reach the voltage threshold of 5.5 V worth UT. TABLE III RESULTS OBTAINED FOR THE 4 RELIABILITY MODELS Model scale shape correlation MRL23.67) Weibull PL 6.5 UT % 33.1 UT Lognormal PL 4 UT.16 UT 98% UT Weibull MLE 59.3 UT UT Degradation % UT Once again, we see that the degradation reliability model is the pessimistic model for the MRL calculation. REFERENCES [1] Byington, Kalgren, Prognostic Techniques and Critical Data Elements for Avionics, Prognostics and Health Management for Electronics Workshop University of Maryland, 24. [2] Cox, Regression Models and Life Tables with Discussion), Journal of the Royal Statistical Society, Series B 34 :187-22, 2. [3] Mei-Ling Ting Lee, G.A. Whitmore, Threshold regression for survival analysis : Modeling event times by a stochastic process reaching a boundary, Statistical Science,4 :51-513,26. [4] G.A. Whitmore, Assessing lung cancer risk in railroad workers using a first hitting time regression model, Environmetrics 155) : , 24. [5] J.M. Van Noortwijk, Gamma processes for time-dependant reliability of structures, 2. [6] Jerry Lawless, Covariates and random effects in a gamma process model with application to degradation and failure, Lifetime data Analysis,1 : ,24. [7] O. Basile, P. Dehombreux, F. Riane, Evaluation of the uncertainty affecting reliability models Journal of Quality and Maintenance Engineering, Vol. 13, No 2, 27. [8] Peter J. Smith, Analyse of failure and survival data, Texts in Statistical Science, Chapman & Hall/CRC 22 [9] M. Xie, T.N. Goh, Y. Tang, On changing points of mean residual life and failure rate function for some generalized Weibull distributions, Reliability Engineering and System Safety, 84, , Elsevier, 24. [1] R. Jones, L. Molent, S. Pitt, Similitude and the Paris crack growth law, International Journal of Fatigue 3, , 28 [11] Johannes Adrianus Van Den Bogaard Product lifecycle optimization using dynamic degradation models, Thesis, Technische Universiteit Eindhoven 25 [12] A. Grall, M. Fouladirad, Maintenance decision rule with embedded online Bayesian change detection for gradually non-stationary deteriorating systems, Proc. IMechE Vol. 222, 28. [13] Estelle Deloux, Politiques de maintenance conditionnelle pour un systeme à dégradation continue soumis à un environnement stressant, Thesis, École Nationale Supérieure des Techniques Industrielles et des Mines de Nantes, 28. [14] Mikhail Nikulin & al. Statistique des essais accélérés, Lavoisier, Paris, 27. V. CONCLUSION This paper discussed the interest of having the knowledge of the degradation process for a device to retrieve the reliability. We focused on theoretical examples that showed that the predicted model of the reliability best fits the simulated failure times. From the reliability model based on the degradation process, the estimation of the mean remaining useful life may then be achieved. The calculation of the MRL for the different reliability models showed that the degradation model is the pessimistic one for both applications. Future works for this research consists of applying the FHTT theory to experimental degradation data and to establish a maintenance decision rule based on degradation reliability models.