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Zhonglai Wang1 School of Mechatronics Engineering, University of Electronic Science and A Simulation Method to Estimate Technology of China, Chengdu, Sichuan 611731, China; The State Key Laboratory of Advanced Design Two Types of Time-Varying and Manufacturing for Vehicle Body, Failure Rate of Dynamic Changsha 410082, China e-mail: [email protected] Systems Xiaoqiang Zhang School of Mechatronics Engineering, The failure rate of dynamic systems with random parameters is time-varying even for lin- University of Electronic Science and ear systems excited by a stationary random input. In this paper, we propose a simulation- Technology of China, based method to estimate two types (type I and type II) of time-varying failure rate of Chengdu, Sichuan 611731, China dynamic systems. The input stochastic processes are discretized in time and the trajecto- e-mail: [email protected] ries of the output stochastic process are calculated. The time of interest is partitioned into a series of time intervals and the saddlepoint approximation (SPA) is employed to Hong-Zhong Huang estimate the of failure in each interval. Type I follows the commonly used def- School of Mechatronics Engineering, inition of failure rate. It is estimated at discrete time intervals using SPA and the correla- University of Electronic Science and tion information from a properly selected time-dependent copula function. Type II is a Technology of China, proposed new concept of time-varying failure rate. It provides a way to predict the failure Chengdu, Sichuan 611731, China rate considering a virtual “good-as-old” repair action of repairable dynamic systems. e-mail: [email protected] The effectiveness of the proposed method is illustrated with a vehicle vibration example. [DOI: 10.1115/1.4034300] Zissimos P. Mourelatos Mechanical Engineering Department, Keywords: time-varying failure rate, dynamic systems, simulation-based method, Oakland University, repaired samples, saddlepoint approximation, time-dependent copula function Rochester, MI 48309 e-mail: [email protected]

1 Introduction further studied for time-dependent reliability analysis based on total probability theorem [8]. Also, a time-dependent reliability The failure rate quantifies the probability of failure over a small analysis is performed in Ref. [9] using a nested based pre- time interval dt after time t under the condition that the system did diction model where time-independent reliability methods such as not fail before time t. In , it is common to the first order reliability method (FORM) [10] and the saddlepoint derive the failure rate from the cumulative distribution function approximation method (SPA) [11] can be used. The out-crossing (CDF) [1–4]. Sufficient samples are needed to estimate the rate method has been widely used for time-dependent reliability CDF accurately with statistical methods, especially for a small problems and many advanced methods have been proposed probability of failure. For example, about 234,000 data samples [12–15] under the assumption of independent out-crossings fol- are needed to estimate a 0.00171 probability of failure with 10% lowing the Poisson distribution, which may result in a poor accu- error under 95% confidence. Estimation of the CDF using physics racy. A parallel system reliability formulation, the so-called PHI2 of failure (POF) has attracted more attention for structural systems method, using the out-crossing rate is proposed in Ref. [16]. A by considering the time-dependent properties. The time horizon is Monte Carlo based set theory method, similar with the PHI2 partitioned into many time intervals for discrete CDF estimation method, is reported in Ref. [17]. Studies reveal that the PHI2 and then a continuous CDF is fitted. Accordingly, discrete and based method shows relatively poor accuracy when dealing with continuous time-varying failure rates could be derived from the nonmonotonic problems [6,18]. In order to avoid the calculation discrete and continuous CDFs, respectively. of out-crossing rate, a time-dependent reliability analysis method To estimate a time-varying failure rate, time-dependent reliabil- based on stochastic process discretization is presented in ity analysis must be conducted over the planning horizon. Many Ref. [19]. methods have been proposed for time-dependent reliability analy- Most of time-dependent reliability analysis methods provide sis of structural systems. Because of the independent non-negative only the reliability over a specific time interval. Design under increments of the Gamma process, a combination of Gamma pro- uncertainty considering maintenance and lifecycle cost requires cess and of live load models is used in Ref. [5]. A distri- however, the reliability at all times within the planning horizon. bution of extreme values approach is employed in Ref. [6]to Singh et al. [20] proposes such a method by estimating the failure transform the time-dependent reliability problem to a time- rate of random dynamic systems using importance . independent reliability problem. An equivalent time-invariant Their method is suitable for low dimensional limit state functions. composite limit state is defined and used in Ref. [7] for time- Wang et al. [21] also presents a method to estimate the failure rate dependent reliability analysis. The composite limit state method is at all times within the planning horizon using an improved subset simulation with splitting by partitioning the original high dimen- 1Corresponding author. sional random process into a series of correlated, short duration, Contributed by the Design Automation Committee of ASME for publication in low dimensional random processes. However in estimating the the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 22, 2016; final manuscript received July 5, 2016; published online September 14, 2016. Assoc. time-varying failure rate, the high computational effort is still a Editor: Xiaoping Du. challenge.

Journal of Mechanical Design Copyright VC 2016 by ASME DECEMBER 2016, Vol. 138 / 121404-1

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Copula functions have been widely used for modeling the cor- provides a new way to predict the failure rate by essentially build- relation of random variables and in economic field ing a relationship between the system failure rate and the capacity [22,23]. Because of their great capacity to represent correlation, for maintenance of repairable dynamic systems. Copulas have been recently introduced in the structural safety The contributions of this article are: (1) a new time-varying field. Noh et al. [24] uses Gaussian Copula to describe the correla- failure rate (type I) estimation method is proposed for dynamic tion of input random variables for reliability-based design optimi- systems; (2) a time-dependent copula function approach is pro- zation and further identifies the joint CDFs from provided data posed to establish the failure correlation as time progresses with- using a copula function [25]. In order to express the multidimen- out additional computational effort; and (3) a new concept of sional correlation, the vine-copula has been also introduced in time-varying failure rate (type II) is presented for repaired sam- structural reliability analysis [26]. Tang et al. [27] studies the ples and its expression is derived. effect of different copula models on the structural system reliabil- The reminder of the paper is organized as follows. Section 2 ity results. reviews the SPA method and the method to construct a time- The above studies have played an important role in fostering dependent copula function. We use both methods in the proposed applications of Copula functions for time-independent problems time-varying failure rate approach. Section 3 provides details on in the structural safety field. For dynamic systems, the correlation the proposed approach and Sec. 4 uses a vehicle vibration exam- of performance between adjacent time intervals can be established ple to illustrate its effectiveness. Finally, Sec. 5 summarizes and accounting therefore, for time. In order to address the time- concludes. dependent probability of failure or failure rate, of dynamic systems, a simulation-based method is proposed in this article to 2 Saddlepoint Approximation and Copula Function estimate two types of time-varying failure rate. A simulation approach is used because the failed samples still 2.1 Saddlepoint Approximation. The SPA is an attractive remain in the operating queue, resulting in correlated failures in method to perform structural reliability analysis because of its time. The input stochastic processes are discretized using a small high computational accuracy in approximating the tails of a distri- time step into a set of high-dimensional random variables and the bution and its comparable efficiency with FORM [11,28]. If Y is a trajectories of the output stochastic process are calculated accu- with probability density function (PDF) fY ðyÞ, the rately with simulation. The planning horizon is then partitioned generating function (MGF) of Y is given by ð into a series of time intervals and the saddlepoint approximations 1 nY ny (SPA) method is used to calculate the probability of failure in MðnÞ¼E½e ¼ e fY ðyÞdy (1) each time interval. With the same trajectories used in SPA, a 1 time-dependent copula is built to provide the correlation between the maximum response in each time interval and the maximum where E½ denotes expectation. The cumulative generating func- responses up to that time interval. A discrete time-varying tion (CGF) KðnÞ is failure rate in each time interval is calculated using an estimated probability of failure from the SPA and the correlation informa- KðnÞ¼log ½MðnÞ (2) tion from the estimated time-dependent copula. A continuous mean time-varying failure rate is also derived from the fitted CDF where log ½ is the natural logarithm. The inverse Fourier trans- from the discrete cumulative of failure based on a form is used to obtain fY ðyÞ from KðnÞ as validated . This time-varying failure rate, ð i1 called Type I failure rate, is calculated using the classical 1 fY ðÞy ¼ exp½KðÞn ny dn (3) definition. 2pi i1 During simulation, we assume that after failure, performance is restored to the state just before failure occurred (good-as-old Analytical forms of the CGF are only available for certain dis- repair assumption). In order to compare the failure evolution of tributions [29]. If there is no analytical form, an approximation the repaired and nonrepaired samples, we propose the concept of can be obtained using samples. It has been shown that an accurate time-varying failure rate for repaired samples, called type II fail- approximation of the CGF can be obtained using the following ure rate, derive its expression and compare it with type I. Type II four cumulants [30]

8 > s1 > k1 ¼ > N > s > 2 > Nss2 s1 <> k2 ¼ NsðÞNs 1 3 2 (4) > 2s 3Nss1s2 þ N s3 > k ¼ 1 s > 3 N ðÞN 1 ðÞN 2 > s s s > 4 2 2 2 > 6s1 þ 12Nss1s2 3NsðÞNs 1 s2 4NsðÞNs þ 1 s1s3 þ Ns ðÞNs þ 1 s4 : k4 ¼ NsðÞNs 1 ðÞNs 2 ðÞNs 3

P Ns j r j where sr ¼ j¼1ðy Þ ðr ¼ 1; 2; 3; 4Þ and y ðj ¼ 1; …; NsÞ is the If the derivative of KðnÞ with respect to n is set equal to y as output of the performance function. NsðNs 4Þ is the sample size. Using the above four cumulants, the CGF is approximated as K0ðnÞ¼y (6)

1 2 1 3 1 4 the solution to Eq. (6) is the saddlepoint n , which is also the KðÞn ¼ k1n þ k2n þ k3n þ k4n (5) sp 2 3! 4! extreme point of KðnÞny in Eq. (3).

121404-2 / Vol. 138, DECEMBER 2016 Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Daniels [31] uses a power series expansion to evaluate the inte- gral in Eq. (3) resulting in the following expression of the PDF of Y

"#1 1 2 ½KðÞnsp nspy fY ðÞy ¼ } e (7) 2pK nsp

The saddlepoint approximation approach can also be used to Fig. 1 Schematic of the time-varying failure rate estimation process approximate the CDF FY ðyÞ. A reasonable and widely used for- mula is provided by Lugannini and Rice [32] 1 1 motion are discretized in time and expressed in a state-space FY ðÞy ¼ PrfgY y ¼ UðÞw þ /ðÞw (8) w v form. The discretized equations are time integrated using for example, a Runge–Kutta method or Newmark-beta method [41] where UðÞ and /ðÞ are the CDF and PDF of a standard normal and are expressed as distribution, and w and v can be expressed by Xðt þ DtÞ¼hðXðtÞ; Z; MðtÞ; tÞ (13) 1=2 w ¼ sgnðnspÞf2½nspy KðnspÞg (9) c where Xðt þ DtÞ2< is the vector of uncertain states xsðt þ DtÞ, s ¼ 1; 2; …; c at time t þ Dt, Dt is the integration time step and and hðÞ indicates a function of the arguments. Z 2

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Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use no int int Ecn ¼ max gðXðtÞ; tÞSt j En1 ðn ¼ 1; …; N; T0 ¼ 0Þ r C Pr En ; PrfgEn1 t2½Tn1;Tn k ðÞTn1; Tn ¼ (22) PrfgE ðÞTn Tn1 (15) n1

where which we call type II failure rate. It provides a new way to predict the failure rate by essentially building a relationship between the no system failure rate and the capacity for maintenance of repairable En1 ¼ max gðXðtÞ; tÞ < St ðn ¼ 1; …; N; T0 ¼ 0Þ (16) t2½0;Tn1 dynamic systems. Also, the calculation of type II failure rate does not require additional computational effort. is the safe event over the time interval ½0; Tn1. The subscript “c” Section 3.2 provides step-by-step details of the proposed int in Ecn (Eq. (15)) stands for “conditional,” while the subscript “n” approach. is the time interval counter. The mean time-varying failure rate over the time interval 3.2 Procedure. The proposed method consists of two stages. ½Tn1; Tn is provided by In the first stage, we calculate the probabilities of failure over a series of time intervals. The second stage builds the time- int Pr Ecn dependent copula function and estimates the type I and type II kðÞTn1; Tn ¼ (17) Tn Tn1 failure rates. Figure 2 summarizes all steps of the proposed method. Details are provided below. int If Tn Tn1 is sufficiently small, kðTn1; TnÞ is the instantane- Stage 1: Estimation of probabilities of failure PrfEn g over the ous failure rate at time instant Tn1. N time intervals ½Tn1; Tnðn ¼ 1; …; N; T0 ¼ 0Þ. int int Using the definition, PrfEcn g is equal to In this stage, we calculate the probability PrfEn g over the time int int PrfEn En1g=PrfEn1g and Eq. (17) can be rewritten as interval ½Tn1; Tnðn ¼ 1; …; N; T0 ¼ 0Þ using the SPA. PrfEn g is used in Eqs. (20) and (21). The process consists of the following int Pr En En1 six steps. kðÞTn1; Tn ¼ (18) Step 1: Construct the time-dependent performance function PrfgEn 1 ðÞTn Tn 1 LðtÞ¼gðXðtÞ; tÞ. Step 2: Characterize each input stochastic process. where PrfEn1g is calculated recursively using the copula func- tion as We use time series model to characterize a stochastic process. Many time series models are available including autoregressive PrfE g¼1 PrfE gðn ¼ 2; …; N; PrfE g¼0Þ (19) (AR), , autoregressive moving average, and autor- n1 n1 0 egressive integrated moving average models [42–44]. In this where work, we use AR models. An input stochastic process XðtÞ is represented as a collection of random variables XðtiÞ at the discrete times int int PrfEn1g¼½PrfEn2gþPrfEn1gCðPrfEn1g; PrfEn2gÞ ti ¼ iDt; i ¼ 0; 1; 2; …; N1; t0 ¼ 0; tN1 ¼ T. The time step Dt is (20) much smaller than DT ¼ Tn Tn1 resulting in N1 N. For an AR(p) model of order p, the discretized sample function (trajec- tory) is represented as int int Note that PrfEn En1g is expressed in terms of PrðEn Þ and PrfEn1g as xðtiÞl ¼ u1ðxðti1ÞlÞþu2ðxðti2ÞlÞþ int int þ u ðxðt ÞlÞþe (23) PrfEn En1g¼CðPrðEn Þ; PrðEn1ÞÞ (21) p ip i

Equations (18)–(21) are used sequentially for all time intervals. where xðtiÞ is the value of the sample function at time ti, l is the 2 For each cycle (new time interval), the SPA estimates the CDF of temporal mean of the stochastic process XðtÞ, ei Nð0; reÞ is int ; ; ; maximum response to calculate PrfEn g, and a copula function Gaussian white noise, and u1 u2 … up are feedback parameters int 2 provides the dependence between En and En1. We can then cal- to be estimated. For the AR(2) model, the re can be culate a discrete mean time-varying failure rate for each time determined by interval. 2 Based on the statistics of maximum response, we use a vali- re ¼ cð0Þð1 u1q1 u2q2 upqpÞ (24) dated Weibull distribution to fit the discrete cumulative probabil- ities of failure with a continuous CDF and thus obtain a where cð0Þ is the variance of the stochastic process and qi ði ¼ continuous time-varying failure rate. This failure rate is calculated 1; 2; …; pÞ is the value of the function at time lag using the common definition as the conditional probability of fail- s ¼ iDt. More details are provided in Ref. [43]. ure over a small time interval dt after time t under the condition Step 3: Partition time of interest [0, T]. that the system did not fail before time t. We call this definition The time of interest ½0; T is partitioned into a series of N time type I failure rate. intervals ½Tn1; Tn; n ¼ 1; 2; …; N using a time step DT ¼ int The copula CðPrfEn1g; PrfEn2gÞ in Eq. (20) represents the Tn Tn1 which is a multiple of Dt. probability that samples failed in both time intervals ½0; Tn2 and Step 4: Calculate K trajectories of the limit state function over ½Tn2; Tn1. It appears because the failed samples still remain in each time interval ½Tn1; Tn. the operating queue during the simulation. The copula can be We generate K sample trajectories of the limit state function viewed as an index describing the failure correlation as time pro- LðtÞ¼gðXðtÞ; tÞ by solving the discretized dynamic equations of gresses. This is similar to the failed items still remaining in the motion Xðt þ DtÞ¼hðXðtÞ; Z; MðtÞ; tÞ, Tn1 t Tn, K times operating queue after engineering maintenance. During simula- using the Dt time step. Each trajectory ljðtÞ; j ¼ 1; …; K is repre- j tion, we assume that after failure, performance is restored to the sented by NP ¼ðTn Tn1=DtÞþ1 values l ðtiÞ at time instants state just before failure occurred (good-as-old repair assumption). ti ¼ Tn1 þði 1ÞDt; i ¼ 1; …; NP. In order to compare the failure evolution of the repaired and non- Step 5: Calculate extreme values over each time interval repaired samples, we propose the concept of time-varying failure ½Tn1; Tn. j rate for repaired samples. A discrete mean time-varying failure For the nth time interval, the extreme value rn of the jth trajec- rate over the time interval ½Tn1; Tn is defined as tory from step 4, is

121404-4 / Vol. 138, DECEMBER 2016 Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 2 Flowchart of the proposed method

j j are calculated as in Ref. [37]. The Clayton copula, one of the rn ¼ max ðl ðtiÞÞ; j ¼ 1; …; K; n ¼ 1; …; N (25) i¼1;…;NP bivariate Archimedean copulas, is selected for the vehicle vibra- tion example (Fig. 5 in Sec. 4). The set of the K extreme values is represented as In order to distinguish time intervals ½0; Tn1 and ½Tn1; Tn,we use subscripts Rðn 1Þ and n to represent the time intervals 1 K j rn ¼frn rn g (26) ½0; Tn1 and ½Tn1; Tn. Therefore, rRðn1Þ and rRðn1Þ represent the jth extreme value and the set of K extreme values over the int j Step 6: Calculate PrfEn g over each time interval ½Tn1; Tn with time interval ½0; Tn1. Similarly for the time interval ½Tn1; Tn, rn SPA. is the jth extreme value and rn is the set of K extreme values. The failure threshold St replaces y in Eq. (6) resulting in A semiparametric estimation method is used to determine the copula-based model parameters based on rRðn1Þ and rn in two 0 K ðnÞ¼St (27) steps. The first step estimates the marginal distribution functions using a nonparametric estimation method. The marginal distribu- int Equations (8)–(10) are then used to calculate PrfEn g. tion function of extreme values URðn1Þ over the time interval Stage 2: Estimation of type I and type II failure rates over the ½0; Tn1 is empirically estimated as [45] planning horizon ½0; T. XK In this stage, we build time-dependent copula functions using _ 1 U rj ¼ Iri rj (28) the same K trajectories from step 4 of stage 1, and use them RðÞn1 RðÞn1 K þ 1 RðÞn1 RðÞn1 sequentially to estimate type I and type II failure rates throughout i the planning horizon using Eqs. (18)–(22). The process consists of the following two steps. where Step 1: Construct time-dependent copula functions. 8 In this step, we characterize the dependence of extreme values < i j i j IðrRðn1Þ rRðn1ÞÞ¼1ifrRðn1Þ rRðn1Þ between time intervals ½Tn1; Tn and ½0; Tn1. Note that in stage (29) 1, we do not account for the dependence of extreme values of the : i j i j IðrRðn1Þ rRðn1ÞÞ¼0ifrRðn1Þ > rRðn1Þ performance function LðtÞ among different time intervals. We only consider the correlation of the extreme values between time intervals ½0; Tn1 and ½Tn1; Tn. This captures the correlation Similarly, the marginal distribution function of extreme values between the two time intervals by simply changing the number n. Un over the time interval ½Tn1; Tn is estimated as A suitable copula is selected among commonly used bivariate copulas such as Clayton, Gumbel, Frank, Gaussian, Ali-Mikhail- XK _ j 1 i j Haq (AMH), Farlie–Gumbel–Morgenstern (FGM), Arch 12, or Un r ¼ Ir r (30) n K þ 1 n n Arch 14 and a Bayesian approach where the normalized weights i

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Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use where ( Iðri rj Þ¼1ifri rj n n n n (31) i j i j Iðrn rnÞ¼0ifrn > rn ^ The second step estimates the copula parameter hn1 using the maximum likelihood method. Given the nonparametric estimators _ _ ^ URðn1Þ and Un, the copula parameter hn1 is estimated as [46] XK ^ ðjÞ ðjÞ hn1 ¼ arg max log cðU ; Un ; hn1Þ (32) h Rðn1Þ n1 j¼1

For the vehicle vibration example of Sec. 4, the Clayton copula is employed to describe the correlation as [25] 1=hn1 P P hn1 hn1 CU ðn1Þ;Unjhn1 ¼ U ðn1Þ þðUnÞ 1 (33)

where hn1 > 0. Its corresponding density is provided by P cU ðn1Þ; Un; hn1 P ðhn1þ1Þ ¼ðhn1 þ 1Þ U ðn1ÞUn 1=hn12 P hn1 hn1 U ðn1Þ þðUnÞ 1 (34) Fig. 3 Vehicle vibration model ^ Each copula parameter hn1ðn ¼ 2; …; NÞ is estimated by repeating this procedure for time intervals ½0; Tn1 and ½Tn1; Tnðn ¼ 2; …; NÞ. Failure occurs if the magnitude of the vertical acceleration Step 2: Estimate type I and type II time-varying failure rates exceeds a threshold, jSðd; X; tÞj St where St ¼ 3:5. over the planning horizon ½0; T. A third-order autoregressive time-series model AR(3) repre- The discretized time-varying failure rate over the planning hori- sents the stochastic road elevation process MðtÞ as zon ½0; T is determined from type I failure rates kðTn1; TnÞ; n ¼ 1; 2; …; N. The failure rate kðTn1; TnÞ over mðtiÞ¼1:2456mðti1Þ0:2976mðti2Þ0:1954mðti3Þ each time interval ½Tn1; Tn is calculated using Eqs. (18)–(21). 2 int int þ eið0; 0:5132 Þ PrfEcn g is calculated from the joint probability PrfEn En1g using the established copula function from step 1 of stage 2, the 2 int where eið0; 0:5132 Þ is a Gaussian white noise with a standard PrfEn g from stage 1, and the safe event probability PrfEn1g calculated using Eq. (19). Finally, Eq. (17) provides the failure deviation of 0.5132. The coefficients 1:2456, 0:2976, and 0:1954 are the three estimated feedback parameters. rate kðTn1; TnÞ. For dynamic systems, the extreme values of performance are The equations of motion usually Weibull distributed. The parameters of Weibull distribu- m x€ b b x_ b x_ k k x k x k m b m_ tion are estimated by fitting the discrete cumulative probabilities of u u þð t þ sÞ u s s þð t þ sÞ u s s ¼ t þ t failure using regression. A continuous time-varying failure rate is then derived from the estimated Weibull distribution. and Finally, the discretized time-varying failure rate for repaired m x€ þ b ðx_ x_ Þþk ðx x Þ¼0 samples over the planning horizon ½0; T, is determined from type s s s s u s s u r II failure rates k ðTn1; TnÞ; n ¼ 1; 2; …; N. The latter are calcu- lated in Eq. (22), using the established copula function from step are transformed to a state-space form and the Runge–Kutta method is used to integrate them in time. The time step is Dt ¼ 1 of stage 2, PrfEintg from stage 1, and the failure event probabil- n 0:01 s. The same time step is used for the discretization of the sto- ity PrfE g from Eq. (20). n1 chastic road elevation. The planning horizon is T ¼ 300s resulting in T=Dt ¼ 4 A Vehicle Vibration Example 300=0:01 ¼ 30; 000 random variables. The time interval is DT ¼ The vehicle vibration example of Fig. 3 is used to illustrate the Tn Tn1 ¼ 2s resulting in ðDT=DtÞ¼ð2=0:01Þ¼200 random proposed method. The vehicle travels over a stochastic terrain variables per time interval. We use 1000 trajectories to estimate with a speed of 20 mile per hour (mph). There are two random the probabilities of failure for each time interval ½Tn1; Tnðn ¼ variables representing the random parameters of the system and a 1; …; 150; T0 ¼ 0Þ using SPA. random process MðtÞ representing the road excitation. The two Figure 4 shows the cumulative probability of failure by inte- random variables are the damping coefficient bs and the stiffness grating the probabilities of failure over each time interval without/ 2 ks. Both are normally distributed as bs Nð7000; 1400 ÞN=m=s with considering the correlation among them based on Monte 2 and ks Nð40; 000; 4000 Þ N=m, respectively. The sprung mass Carlo simulation (MCS) with 1,000,000 samples. We observe that ms, unsprung mass mu, tire stiffness kt, and tire damping bt are the cumulative probability of failure from MCS without consider- deterministic parameters with ms ¼ 100 kg, mu ¼ 100 kg, ing correlation is greater than that with correlation. The reason is 4 3 kt ¼ 40 10 N=m, and bt ¼ 4 10 N=m=s. that an item could fail several times during the planning horizon We use the state-space approach to determine the vertical accel- but the population of items is fixed if the correlation is not consid- eration response Sðd; X; tÞ¼x€sðtÞ of the sprung mass in g’s. ered, leading to an increase of probability of failure. The

121404-6 / Vol. 138, DECEMBER 2016 Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 4 Estimated cumulative probability of failure from MCS with/without considering correlation

Fig. 6 Copula weights results for first (r1 and r2)(a) and last (rR149 and rR150)(b) data pairs

rRðn1Þ and rn. Figure 5 shows the scatter diagram for data pairs rR1 and r2 (first pair), and rR149 and r150 (last pair), respectively. To select the proper copula among Clayton, Gumbel, Frank, Gaussian, Ali-Mikhail-Haq (AMH), Farlie–Gumbel– Morgenstern (FGM), Arch 12, and Arch 14, a Bayesian approach is used based on 1000 simulated trajectories. We find that the Clayton copula is suitable for this example. Figure 6 shows the estimated relative weights of the Bayesian approach for data pairs rR1 and r2, and rR149 and r150 verifying the selection of Clayton copula. After selecting the Clayton copula, we use the same 1000 tra- jectories to build the time-dependent copula functions among dif- ferent data pairs and to subsequently estimate the time-varying failure rate. Figure 7 shows the Clayton copula parameter for all data pairs showing an increasing trend as time progresses which indicates that the extreme values over the time interval ½Tn1; Tn are not only correlated with those over the previous time interval ½Tn2; Tn1 but also with those over the time intervals Fig. 5 Scatter diagram for first (rR1 and r2)(a) and last (rR149 ½0; T1; ½T1; T2; …; ½Tn3; Tn2. This is in contrast to an intuitive and r150)(b) data pairs expectation that the extreme values over the time interval ½Tn1; Tn are correlated only with those over the previous time interval ½Tn2; Tn1. difference with and without correlation varies depending on the Figure 8 shows the cumulative probability of failure from int correlation. Therefore, the correlation can be used as an index to Eq. (20) using the probabilities of failure PrðEn Þ and the time- illustrate failure dependence for dynamic systems as time dependent Clayton copula function. Four different runs are per- progresses. formed to illustrate its variability. The figure also shows the aver- Because there are 150 time intervals, ðT=DTÞ1 ¼ age cumulative probability of failure of the four runs. The ð300=2Þ1 ¼ 149 copula functions must be estimated to repre- estimated cumulative probability of failure from MCS with sent the time-dependent correlation. A data pair is defined by 1,000,000 trajectories is used as a reference for accuracy

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Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 7 Parameters of time-dependent Clayton copula functions Fig. 10 RE of cumulative probability of failure. RE rela- tive error.

Table 1 RE of cumulative probability of failure

RE of cumulative probability of failure (%)

Time (s) Run 1 Run 2 Run 3 Run 4 Average Weibull

2 100.1 49.76 22.97 32.23 10.26 41.61 4 39.85 59.38 0.93 23.36 10.95 25.02 6 18.84 32.55 7.95 26.57 8.08 18.42 8 15.45 33.60 2.43 35.62 11.05 14.56 10 1.52 17.51 2.79 30.67 10.36 10.51 12 5.70 15.28 0.43 29.73 9.93 8.92 14 5.19 4.35 0.72 34.19 8.16 7.08 16 1.45 0.11 0.47 29.21 7.76 6.25 … … ……… … … 288 5.14 3.75 0.01 4.18 1.42 1.46 290 4.95 3.62 0.13 4.48 1.49 1.37 Fig. 8 Estimated cumulative probability of failure 292 4.83 4.13 0.44 4.32 1.15 1.31 294 4.72 4.12 0.30 3.96 1.06 1.22 296 4.45 3.98 0.46 3.54 0.89 1.15 298 4.03 4.05 0.33 3.30 0.74 1.08 300 3.91 4.11 0.31 3.20 0.67 1.03

function. We see that the Weibull distribution is a good fit. The goodness of fit indices sum of squares due to error, R-square, adjusted R-square, and root mean squared error are equal to 0.01009, 0.9999, 0.9999, and 0.008229, respectively, indicating that the Weibull distribution is suitable for the lifetime of the dynamic system. The two Weibull parameters are a ¼ 0:8828 and b ¼ 775:9. Figure 10 and Table 1 show the relative error (RE) of the cumu- lative probability of failure from SPA/Copula and Weibull distri- bution methods with respect to MCS. The relative error decreases, reaching an asymptotic value of approximately 1%. Figure 11 shows the corresponding mean failure rate for each time interval ½Tn1; Tn from four individual runs using Eqs. (17)–(21). It also shows the average curve of the four individual Fig. 9 Kolmogorov–Smirnov goodness of fit runs. The solid curve shows the smooth time-varying failure rate from the estimated CDF of Weibull distribution. The estimated comparison. To obtain a smooth estimation of the failure rate, we mean time-varying failure rate from MCS with 1,000,000 trajecto- use a Weibull distribution to fit the cumulative probabilities of ries is used as a reference for accuracy comparison. We observe failure calculated from Eq. (20). We observe that the cumulative that the estimated time-varying failure rate oscillates around the probability of failure from Weibull distribution and the average reference provided by MCS with 1,000,000 trajectories. cumulative probability of failure from SPA are very close to the Figure 12 and Table 2 show the relative error (RE) of type I MCS estimate. failure rate using the SPA/Copula and Weibull distribution meth- To test the suitability of the Weibull distribution, Fig. 9 shows ods with respect to MCS. The relative error from Weibull distribu- the Kolmogorov–Smirnov goodness of fit. The Weibull CDF is tion is less than approximately 10%, which is deemed acceptable. transformed to a and the ln½1=ð1 FðtÞÞ is plotted Figure 13 shows the estimated type II failure rate over each against lnt where FðtÞ is the cumulative probability of failure time interval ½Tn1; Tn from four individual runs using Eq. (22).It using the conditional probabilities and the time-dependent copula also provides the average of the four individual runs. Type II

121404-8 / Vol. 138, DECEMBER 2016 Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 13 Estimated type II failure rate kr(t)

Fig. 11 Estimated type I failure rate k(t)

Fig. 14 RE of type II failure rate kr(t)

Fig. 12 RE of type I failure rate k(t). RE means relative error. Weibull distribution is also used to fit the SPA/Copula average failure rate results. Figure 14 shows the relative error of type II failure rate using SPA/Copula with MCS. The relative error of average and Weibull Table 2 RE of type I failure rate k(t) distribution is around 20%, which is greater than that of type I failure rate. The reason is that the denominator PrfEn 1g in RE of the type I failure rate k(t) (%) Eq. (21) is much smaller than PrfEn1g and more samples are Time (s) Run 1 Run 2 Run 3 Run 4 Average Weibull needed to improve the computational accuracy. Comparison of type I and type II failure rates in Figs. 11 and 13 2 100.1 49.76 22.97 32.23 10.26 25.16 show that type II failure rate is larger. This means that the repaired 4 15.45 68.26 22.89 15.23 30.46 10.52 samples have a greater risk of failure after a good-as-old mainte- 6 22.37 19.95 25.40 32.94 2.49 5.82 nance is performed. 8 5.35 36.89 33.75 62.97 32.06 3.21 10 51.74 43.73 4.19 11.95 6.04 4.75 12 27.63 3.72 11.92 25.02 2.70 0.80 5 Summary and Conclusions 14 2.10 61.78 7.74 61.43 2.55 3.95 16 51.33 33.59 9.46 7.84 4.84 2.28 We propose a simulation-based method to estimate type I and … …………… …type II failure rates of dynamic systems in this paper. The trajecto- 288 43.15 35.92 15.89 139.1 40.56 13.89 ries of the output random process are simulated at discrete times. 290 31.44 21.06 7.40 67.61 16.16 2.79 The time of interest is partitioned into a series of time intervals in 292 20.41 118.0 126.0 29.87 43.57 16.20 order to study the time-dependent failure correlation. The SPA is 294 17.18 3.420 30.33 72.82 15.77 11.93 296 50.68 26.22 34.44 82.17 35.27 14.18 used to estimate accurately the small probabilities of failure over 298 82.44 19.31 26.94 45.39 30.05 11.32 each time interval using a small number of simulated trajectories 300 20.77 18.89 3.64 16.52 13.13 9.85 (1000 trajectories for probability of failure 0.0017). A time- dependent copula function to describe the failure correlation, is built with the same trajectories. Using a time-dependent copula function, we propose methods for two types of time-varying fail- failure rate from MCS with 1,000,000 trajectories is used as a ref- ure rate (type I and type II). Type I failure rate is based on the erence for accuracy comparison. The estimated time-varying fail- commonly used definition of failure rate, while type II is a new ure rate oscillates around the MCS reference with 1,000,000 concept to describe the failure evolution of repaired samples con- trajectories. Type II failure rate is similar with that of type I. The sidering a good-as-old virtual maintenance. A vehicle vibration

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Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 10/05/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use example demonstrates that the proposed approach is computation- [20] Singh, A., Mourelatos, Z. P., and Nikolaidis, E., 2011, “Time-Dependent Reli- ally efficient and accurate for both type I and type II failure rate ability of Random Dynamic Systems Using Time-Series Modeling and Impor- tance Sampling,” SAE Paper No. 2011-01-0728. estimation of dynamic systems. [21] Wang, Z., Mourelatos, Z. P., Li, J., Singh, A., and Baseski, I., 2014, “Time- Future research will concentrate on improving the accuracy of Dependent Reliability of Dynamic Systems Using Subset Simulation With type II failure rate and on using the concept of type II failure rate Splitting Over a Series of Correlated Time Intervals,” ASME J. Mech. Des., in maintenance scheduling. 136(6), p. 061008. [22] Patton, A. J., 2012, “A Review of Copula Models for Economic Time Series,” J. Multivar. Anal., 110, pp. 4–18. [23] Abegaz, F., and Naik-Nimbalkar, U. V., 2008, “Dynamic Copula-Based Mar- Acknowledgment kov Time Series,” Commun. Stat.: Theory Methods, 37(15), pp. 2447–2460. This research was partially supported by the National Natural [24] Noh, Y., Choi, K. K., and Du, Li., 2009, “Reliability-Based Design Optimiza- tion of Problems With Correlated Input Variables Using a Gaussian Copula,” Science Foundation of China under the Contract No. 11472075, Struct. Multidiscip. Optim., 38(1), pp. 1–16. the Automotive Research Center (ARC) in accordance with Coop- [25] Noh, Y., Choi, K. K., and Lee, I., 2010, “Identification of Marginal and Joint erative Agreement No. W56HZV-04-2-0001 U.S. Army Tank CDFs Using Bayesian Method for RBDO,” Struct. Multidiscip. Optim., 40, Automotive Research, Development and Engineering Center pp. 35–51. [26] Jiang, C., Zhang, W., Han, X., Ni, B. Y., and Song, L. J., 2015, “A Vine- (TARDEC), the China Postdoctoral Science Foundation under the Copula-Based Reliability Analysis Method for Structures With Multidimen- Contract No. 2013T60838, the Science Fund of State Key Labora- sional Correlation,” ASME J. Mech. Des., 137(6), p. 061405. tory of Advanced Design and Manufacturing for Vehicle Body [27] Tang, X. S., Li, D. Q., Zhou, C. B., Phoon, K. K., and Zhang, L. M., 2013, No. 31315007, and the Fundamental Research Funds for Central “Impact of Copulas for Modeling Bivariate Distributions on System Reliability,” Struct. Saf., 44, pp. 80–90. Universities under Contract No. ZYGX2014J071. [28] Yuen, K. V., Wang, J., and Au, S. K., 2007, “Application of Saddlepoint Approximation in Reliability Analysis of Dynamic Systems,” Earthquake Eng. Eng. Vib., 6(4), pp. 391–400. 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