Irregular Time‐Varying Stress Degradation Path Modeling: a Case Study on Lithium‐ion Cell Degradation

The aim of this paper is to investigate the issue of degradation modeling and reliability assessment for products under irregular time‐varying stresses. Conventional degradation models have been extensively used in the relevant literature to characterize degradation processes under deterministic stresses. However, the time‐varying stress, which may affect degradation processes, widely exists in field conditions. This paper extends the general degradation‐path model by considering the effects of time‐varying stresses. The new degradation‐path model captures influences of varying stresses on performance characteristics. A nonlinear least square method is used to estimate the unknown parameters of the proposed model. A bootstrap algorithm is adopted for computing the confidence intervals of the mean time to failure and percentiles of the failure‐time distribution. Finally, a case study of lithium‐ion cells is presented to validate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Real‐time Reliability Evaluation with a General Wiener Process‐based Degradation Model

The aim of this paper is to investigate the issue of real‐time reliability evaluation based on a general Wiener process‐based degradation model. With its mathematical tractability, the Wiener process with a linear drift has been widely used in the literature, to characterize the dynamics of the degradation process or its transformation. However, the nonlinear degradation process, which can't be properly linearized, exists in practice. The dynamics of such a degradation process can't be accurately captured by linear models. Here, a general Wiener process‐based degradation model is proposed, which covers a variety of Wiener process‐based models as its limiting cases. A two‐stage method is presented to estimate the unknown parameters. Two real‐time reliability evaluation procedures are presented for different conditions: one is the analytical evaluation procedure, and the other is the simulated evaluation procedure. It is shown that when new degradation information is available, the evaluation results can be adaptively updated. Moreover, to check out the proposed degradation model, a graphical method is provided. Finally, the validity of the proposed evaluation method is illustrated by a numerical example and two real‐world examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

Using degradation data to assess reliability: a case study on train wheel degradation

Degradation experiments are usually used to assess the lifetime distribution of highly reliable products, which are not likely to fail under the traditional life tests or accelerated life tests. In such cases, if there exist product characteristics whose degradation over time can be related to reliability, then collecting ‘degradation data’ can provide information about product reliability. In general, the degradation data are modeled by a nonlinear regression model with random coefficients. If we can obtain the estimates of parameters under the model, then the failure‐time distribution can be estimated. In order to estimate those parameters, three basic methods are available, namely, the analytical, numerical and the approximate. They are chosen according to the complexity of the degradation path model used in the analysis. In this paper, the numerical and the approximate methods are compared in a simulation study, assuming a simple linear degradation path model. A comparison with traditional failure‐time analysis is also performed. The mean‐squared error of the estimated 100pth percentile of the lifetime distribution is evaluated for each one of the approaches. The approaches are applied to a real degradation data set. Copyright © 2008 John Wiley & Sons, Ltd.     

A Novel Multi‐hidden Semi‐Markov Model for Degradation State Identification and Remaining Useful Life Estimation

With the increase of product reliability, collecting time‐to‐failure data is becoming difficult, and degradation‐based method has gained popularity. In this paper, a novel multi‐hidden semi‐Markov model is proposed to identify degradation and estimate remaining useful life of a system. Multiple fused features are used to describe the degradation process so as to improve the effectiveness and accuracy. The similarities of the features are depicted by a new variable combined with forward and backward variables to reduce computational effort. The degradation state is identified using modified Viterbi algorithm, in which linear function is adopted to describe the contribution of each feature to the state recognition. Subsequently, the remaining useful life can be forecasted by backward recursive equations. A case study is presented, and the results demonstrate the validity and effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Reliability modeling of multi‐state degraded repairable systems and its applications to automotive systems

In this paper, we presented a continuous‐time Markov process‐based model for evaluating time‐dependent reliability indices of multi‐state degraded systems, particularly for some automotive subsystems and components subject to minimal repairs and negative repair effects. The minimal repair policy, which restores the system back to an “as bad as old” functioning state just before failure, is widely used for automotive systems repair because of its low cost of maintenance. The current study distinguishes with others that the negative repair effects, such as unpredictable human error during repair work and negative effects caused by propagated failures, are considered in the model. The negative repair effects may transfer the system to a degraded operational state that is worse than before due to an imperfect repair. Additionally, a special condition that a system under repair may be directly transferred to a complete failure state is also considered. Using the continuous‐time Markov process approach, we obtained the general solutions to the time‐dependent probabilities of each system state. Moreover, we also provided the expressions for several reliability measures include availability, unavailability, reliability, mean life time, and mean time to first failure. An illustrative numerical example of reliability assessment of an electric car battery system is provided. Finally, we use the proposed multi‐state system model to model a vehicle sub‐frame fatigue degradation process. The proposed model can be applied for many practical systems, especially for the systems that are designed with finite service life.     

Real‐time Reliability Evaluation for an Individual Product Based on Change‐point Gamma and Wiener Process

Reliability evaluation based on degradation data has received significant attentions in recent years. However, existing works often assume that the degradation evolution over time is governed by a single stochastic process, which may not be realistic if change points exist. Here, for cases of degradation with change points, this paper attempts to capture the degradation process with a multi‐phase degradation model and find the method to evaluate the real‐time reliability of the product being monitored. Once new degradation information becomes available, the evaluation results are adaptively updated through the Bayesian method. In particular, for a two‐stage degradation process of liquid coupling devices (LCDs), a model named as change‐point gamma and Wiener process is developed, after which issues of real‐time reliability evaluation and parameters’ estimation are addressed in detail. Finally, the proposed method is illustrated by a case study of LCDs, and the corresponding results indicate that trustful evaluation results depend on the fitting accuracy in cases of multi‐phase degradation process. Copyright © 2013 John Wiley & Sons, Ltd.     

Degradation‐shock‐based Reliability Models for Fault‐tolerant Systems

Reliability modeling of fault‐tolerant systems subject to shocks and natural degradation is important yet difficult for engineers, because the two external stressors are often positively correlated. Motivated by the fact that most radiation‐induced failures are contributed from these two external stressors, a degradation‐shock‐based approach is proposed to model the failure process. The proposed model accommodates two kinds of failure modes: hard failure caused by shocks and soft failure caused by degradation. We consider a generalized m–δ shock model for systems with fault‐tolerant design: failure occurs if the time lag between m sequential shocks is less than δ hours or degradation crosses a critical threshold. An example concerning memory chips used in space is presented to demonstrate the applicability of the proposed model. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of Methods to Estimate the Time‐to‐failure Distribution in Degradation Tests

Degradation tests are alternative approaches to lifetime tests and accelerated lifetime tests in reliability studies. Based on a degradation process of a product quality characteristic over time, degradation tests provide enough information to estimate the time‐to‐failure distribution. Some estimation methods, such as analytical, the numerical or the approximated, can be used to obtain the time‐to‐failure distribution. They are chosen according to the complexity of the degradation model used in the data analysis. An example of the application and analysis of degradation tests is presented in this paper to characterize the durability of a product and compare the various estimation methods of the time‐to‐failure distribution. The example refers to a degradation process related to an automobile's tyre, and was carried out to estimate its average distance covered and some percentiles of interest. Copyright © 2004 John Wiley & Sons, Ltd.     

Analyzing experiments with degradation data for improving reliability and for achieving robust reliability

Statistically designed experiments provide a proactive means for improving reliability; moreover, they can be used to design products that are robust to noise factors which are hard or impossible to control. Traditionally, failure‐time data have been collected; for high‐reliability products, it is unlikely that failures will occur in a reasonable testing period, so the experiment will be uninformative. An alternative, however, is to collect degradation data. Take, for example, fluorescent lamps whose light intensity decreases over time. Observation of light‐intensity degradation paths, given that they are smooth, provides information about the reliability of the lamp, and does not require the lamps to fail. This paper considers experiments with such data for ‘reliability improvement’, as well as for ‘robust reliability achievement’ using Taguchi's robust design paradigm. A two‐stage maximum‐likelihood analysis based on a nonlinear random‐effects model is proposed and illustrated with data from two experiments. One experiment considers the reliability improvement of fluorescent lamps. The other experiment focuses on robust reliability improvement of light‐emitting diodes. Copyright © 2001 John Wiley & Sons, Ltd.     

Estimation of time‐to‐failure distribution derived from a degradation model using fuzzy clustering

Some life tests are terminated with few or no failures. In such cases, a recent approach is to obtain degradation measurements of product performance that may contain some useful information about product reliability. Generally degradation paths of products are modeled by a nonlinear regression model with random coefficients. If we can obtain the estimates of parameters under the model, then the time‐to‐failure distribution can be estimated. In some cases, the patterns of a few degradation paths are different from those of most degradation paths in a test. Therefore, this study develops a weighted method based on fuzzy clustering procedure to robust estimation of the underlying parameters and time‐to‐failure distribution. The method will be studied on a real data set. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

This study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.     

An analytical method for reliability analysis of hardware‐software co‐design system

Hardware‐software co‐design systems abound in diverse modern application areas such as automobile control, telecommunications, big data processing, and cloud computing. Existing works on reliability modeling of the co‐design systems have mostly assumed that hardware and software subsystems behave independently of each other. However, these two subsystems may have significant interactions in practice. In this paper, an analytical approach based on paths and integrals is proposed to analyze reliability of nonrepairable hardware‐software co‐design systems considering interactions between hardware and software during the system performance degradation and failure process. The proposed approach is verified using the Markov‐based method. As demonstrated by case studies on systems without and with warm standby sparing, the proposed approach is applicable to arbitrary types of time‐to‐failure or degradation distributions. Effects of different transition and fault detection/recovery parameters on system performance are also investigated through examples.     

A New Model for Repairable Systems with Nonmonotone Intensity Function

In this study, a new model based on the mixture of bounded Burr XII failure intensity and bounded intensity process is proposed to describe the failure intensity of minimally repaired systems with approximate bathtub behavior. The estimates of the model parameters are easily obtained using the maximum likelihood estimation method. The confidence intervals for the model parameters are also provided. Other existing models, such as superposed power law process, log‐linear process–power law process, and bounded bathtub intensity process are used to compare with our proposed model. Through numerical examples, the results show that our proposed model performs well regarding the Akaike information criterion value and the mean of square errors. Copyright © 2014 John Wiley & Sons, Ltd.     

The transformed gamma process for degradation phenomena in presence of unexplained forms of unit‐to‐unit variability

The transformed gamma process is a suitable model for degradation phenomena where damages accumulate gradually over time in a sequence of tiny increments. Attractiveness of the transformed gamma process mainly lies in the fact that it provides a relatively easy way for dealing with phenomena in which the degradation increments over disjoint time intervals are not independent. The transformed gamma process is also a very flexible model. In fact, it is indexed by 2 functions, the “age function” and the “state function,” whose mathematical form can be chosen ad hoc for modeling the dependence of the future degradation increment of a unit on its current age and state, respectively. In this paper, the transformed gamma process is adopted to describe the degradation paths of degrading units in the presence of an unexplained form of unit‐to‐unit variability. The degradation path of each unit is described via a transformed gamma process. Heterogeneity among paths of different units is accounted for by assuming that the scale parameters of the age and state functions vary randomly from unit to unit. Under these assumptions, a quite mathematically tractable model is obtained. The main properties of the proposed model are discussed, and inferential procedures based on the maximum likelihood criterion are implemented. A simple test is presented to check the goodness of fit of the proposed model. Three applicative examples, based on real degradation data, are developed.     

Residual Useful Life Estimation by a Data‐Driven Similarity‐Based Approach

This paper is to provide a general data‐driven, similarity‐based approach for residual useful life estimation for industrial components or systems. Degradation signal data from failure dynamic systems are used to create a database of reference trajectory patterns. Variable weights are assigned to associated points before pointwise similarity computation. Distance score and weight are computed by calculating the fuzzy similarity between test trajectory pattern and reference trajectory patterns. Remaining useful life then could be predicted through aggregating those failure references' residual lifetimes in a weighted sum and updated as time goes by. The potentiality of this approach is illustrated on a problem of contact resistance degradation data. The results indicate that there is no significant difference between Gaussian function and General bell shaped function. Copyright © 2016 John Wiley & Sons, Ltd.     

Engineering‐driven performance degradation analysis of hydraulic piston pump based on the inverse Gaussian process

As a key aircraft component, hydraulic piston pumps must be developed with high reliability. However, collecting failure time data of such pumps for reliability analysis is a big challenge. To save testing time, performance degradation data obtained from degradation tests can be used for quick reliability estimation of hydraulic piston pumps. This paper proposes an engineering‐driven performance degradation analysis method considering the nature of mechanical wear of hydraulic piston pumps. First, the failure mechanism of a type of hydraulic piston pump is investigated. By taking into account the close relationship between the degradation rate and the failure mechanism, an inverse Gaussian (IG) process model with a variable rate is developed to describe the degradation behavior of the pump. Under this model, a Bayesian statistical method is developed for degradation data analysis. The corresponding procedure for model parameter estimation and reliability evaluation is also presented. The proposed degradation analysis method is illustrated using a real experimental data. The results show that the engineering‐driven approach is quite effective in evaluating the lifetime of the hydraulic piston pump and will improve the overall reliability of aircraft operation in the field.     

Computational techniques for optimization of problems involving non‐linear transient simulations

Many optimization problems in engineering require coupling a mathematical programming process to a numerical simulation. When the latter is non‐linear, the resulting computer time may become unaffordably large because three sequential procedures are nested: the outer loop is associated to the optimization process, the middle one corresponds to the time marching scheme and the innermost loop is required for solving iteratively the non‐linear system of equations at each time step. We propose four techniques for reducing CPU time. First, derive the initial values of state variables at each time (innermost loop) from those computed at the previous optimization iteration (outermost loop). Second, select time increment on the basis of those used for the previous optimization iteration. Third, define convergence criteria for the simulation problem on the basis of the optimization process, so that they are only as stringent as really needed. Finally, computations associated to the optimization are shown to be greatly reduced by adopting Newton–Raphson, or a variant, for solving the simulation problem. The effectiveness of these techniques is illustrated through application to three examples involving automatic calibration of non‐linear groundwater flow problems. The total number of iterations is reduced by a factor ranging between 1·7 and 4·6. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimal Bayesian maintenance policy for a gearbox subject to two dependent failure modes

Most maintenance optimization models of gear systems have considered single failure mode. There have been very few papers dealing with multiple failure modes, considering mostly independent failure modes. In this paper, we present an optimal Bayesian control scheme for early fault detection of the gear system with dependent competing risks. The system failures include degradation failure and catastrophic failure. A three‐state continuous‐time–homogeneous hidden Markov model (HMM), namely the model with unobservable healthy and unhealthy states, and an observable failure state, describes the deterioration process of the gear system. The condition monitoring information as well as the age of the system are considered in the proposed optimal Bayesian maintenance policy. The objective is to maximize the long‐run expected average system availability per unit time. The maintenance optimization model is formulated and solved in a semi‐Markov decision process (SMDP) framework. The posterior probability that the system is in the warning state is used for the residual life estimation and Bayesian control chart development. The prediction results show that the mean residual lives obtained in this paper are much closer to the actual values than previously published results. A comparison with the Bayesian control chart based on the previously published HMM and the age‐based replacement policy is given to illustrate the superiority of the proposed approach. The results demonstrate that the Bayesian control scheme with two dependent failure modes can detect the gear fault earlier and improve the availability of the system.     

Remaining useful life prediction for hard failures using joint model with extended hazard

In this paper, we investigate a joint modeling method for hard failures where both degradation signals and time‐to‐event data are available. The mixed‐effects model is used to model degradation signals, and extended hazard model is used for the time‐to‐event data. The extended hazard is a general model which includes two well‐known hazard rate models, the Cox proportional hazards model and accelerated failure time model, as special cases. A two‐stage estimation approach is used to obtain model parameters, based on which remaining useful life for the in‐service unit can be predicted. The performance of the method is demonstrated through both simulation studies and a real case study.