Integrated Equipment Health Prognosis Considering Crack Initiation Time Uncertainty and Random Shock

Prognostic models are typically categorized into three groups: data-driven methods, physics-based methods and integrated methods.

Data-driven models are built purely on data such as sensor-collected condition monitoring data. They typically require large amount of data to achieve reasonable accuracy. In Ref. [3], a proportional-hazards model with time dependent stochastic covariates was used as lifetime model to predict the failure rate and to optimize the unit replacement policy. The set of failure time and covariates was assumed to be a joint nonhomogeneous Markov process. The maximum likelihood was applied to estimate the model parameters including the coefficients of covariates and the transition probabilities of the Markov process. In Ref. [4], the degradation model was selected to be exponential. Two error terms were considered: one was treated as a multiplicative random variable, and the other was treated as a multiplicative Brownian motion process. A Bayesian procedure was applied to update the parameters in the exponential model. As a research mainstream of data-driven models, various machine learning techniques were investigated in the literature. The authors of Ref. [5] developed neural networks to predict bearing failure time, which aimed to train a relationship between the bearing service time and the corresponding vibration spectrum. The authors of Ref. [6] developed a neural network to predict RUL using both failure and suspension condition monitoring histories. An extended recurrent neural network was proposed to predict the health condition of gears in Ref. [7]. The incorporation of Elman context layer in the proposed networks enhanced its ability to model nonlinear time series. Data-driven models are straightforward to establish given sufficient and well-distributed data. Therefore, data of good quality availability is a prerequisite for data-driven models, which is rare for costly industrial equipment.

In contrast, physics-based models resort to physical laws governing the defect growth, where the values or distributions of the model parameters in the physical models, e.g., material dependent parameters, are kept constant and will not be updated based on condition monitoring information. A commonly used law to describe crack propagation is Paris’ law, which was originated in Ref. [16]. Many publications are devoted to devising numerical algorithms to calculate the quantity (e.g., stress intensity factor) needed in Paris’ law. Readers can refer to Refs. [8‐10] for these approaches in which finite element modeling was discussed for stress analysis near crack tip. In the physics-based prognostic models, component failure time is defined as when the defect size exceeds a critical value. Authors of Ref. [11] investigated several factors that influence the crack growth trajectory in the gear tooth, including backup ratio, initial crack location, fillet geometry, rim/web compliance, gear size and pressure angle. The research work in Ref. [12] took account of the variation of moving load on gear tooth into crack growth prediction by breaking the tooth engagement into multiple steps. Authors in Ref. [13] applied Paris’ law to predict fatigue crack growth with utilization of transmission error to estimate the current crack size, which improved predictive accuracy. The service life of gears was divided into crack initiation and crack propagation periods in Ref. [14]. The strain-life method was used to determine the time required for crack initiation, while Paris’ law was used to obtain the time required for the crack to grow from initial crack size to the critical value. Kacprzynksi et al. [15] developed a prognostic tool which predicts gear failure probability by fusing physics-of-failure models and diagnostics information. The results showed variance reduction in failure probability when diagnostics information was present. Physical models can achieve high predictive accuracy if appropriately built. However, it demands comprehensive physics theory and intensive computation to build physical models of high-fidelity. Furthermore, physical models are unavailable for complex systems which limit its usages in real-world applications. In addition, physical models are deterministically used in the above mentioned literature, which means they are unable to address the uncertainty in failure.

Recently, integrated prognostics methods [17‐26] were developed to achieve real-time RUL prediction during the system operations by combining both condition monitoring (CM) data and physics of failure. The integrated methods usually have an updating process by assimilating observations, during which the uncertainty is expected to shrink so that the confidence increases in the predicted results. The integrated prognostics methods are more advantageous than physics-based methods in that the model parameters are able to be adjusted for a specific component in a specific working condition, and more advantageous than data driven methods in that massive data trending is not necessary. Bayesian framework allows for uncertainty quantification; hence, it is widely used in integrated prognostics methods. A problem of crack propagation in a fuselage panel of aircraft was considered in Ref. [17]. Bayesian inference was used to characterize parameters in Paris’ law and error term. The work in Ref. [18] extended the methods in Ref. [17] to consider the correlation between model parameters. In authors’ prior work [19], integrated prognostics methods were proposed for gear health prediction and uncertainty quantification. Physical models include Paris’ law, fracture mechanics model and one-stage gearbox dynamics model. Through Bayesian updating, the uncertainty in RUL prediction was reduced as crack measurements became available. To increase the efficiency of uncertainty quantification, the authors proposed to use polynomial chaos expansion to accelerate the Bayesian process in prognostics [20]. Furthermore, an approach was devised to deal with time-varying operating conditions to make them applicable in various loading environment [21]. Bayesian framework was also used for bearing health prognostics. The bearing spall propagation was investigated in Ref. [22] where Bayesian inference was applied to reduce prediction uncertainty. In recent years an increasing volume of literature was published that treated prognostic model as a dynamic system mainly because of its natural interface with real-time data. In Ref. [23], a health indicator extracted from vibration signature was fed into Paris’ law to update the RUL of bearings. The update was implemented using a Kalman filter. Prognostic models in Refs. [24‐26] are established in a particle filtering framework, in which the problem of non-linear state transition and non-Gaussian noise can be tackled.

A large grain of inherent uncertainty imposes major challenges in prognostic methods development. There are various sources for uncertainties, such as micro-structure of material, operating conditions, working environment, measurement as well as human factors. Many research efforts go to addressing how to identify, capture and manage these multiple uncertainties to make the RUL prediction more accurate, precise and reliable. The existing prognostic approaches usually start the prediction at an assumed time instant when a fault at certain severity is detected. This is based on an assumption that the starting point of prognosis is accurate. However, due to the limitations of the fault detection and diagnostic technologies, there is a large variation in the accuracy of fault detection. This variation affects the prediction accuracy accordingly: an early starting point of prognostics will lead to underestimated RUL, and late starting point will lead to overestimated RUL. In authors’ prior work [19‐21], it was proposed to update the uncertain model parameters in Paris’ law to make RUL prediction more accurate by feeding CM data to a Bayesian framework. These model parameters actually determine the “slope” of degradation path in a scale of damage size versus time. What the Bayesian updating process does is adjusting the “slope” of the degradation path to maximize the likelihood of crack observations when an uninformative prior is given. Apart from the “slope”, another factor that controls the degradation path is the “intercept” with time axis. Hence, to better characterize the degradation path, it is needed to explicitly consider the uncertainty in CIT, which determines its “intercept” with time axis.

Among existing approaches to real-time prognostics, most of them investigated components/systems undergoing a gradual degradation with time. Such gradual degradation is governed by a dynamic model, either data driven or physics-of-failure based, which usually leads to a continuous degradation. Very few of them have considered the degradation path containing shock, which appears as a sudden jump in the degradation. The shock will cause sudden damage increase and then accelerate the degradation rate. In reliability engineering, researchers have investigated the ways of modeling shock process in system reliability analysis [27‐29]. The purpose of these studies was to investigate reliability properties and/or maintenance strategies considering the shock effects. While in the present study, we look into the RUL prediction in a different perspective to consider the shock in the degradation for a specific unit under monitoring, so that the prognostic solution can be given in real time. This purpose is also the main focus of prognostic algorithm development in CBM/PHM.

In the crack propagation problem, model parameters in the Paris’ law include material dependent coefficients, which should not be affected by external forcing factors, i.e., overload causing shock. Hence, additional uncertainty source other than model parameters is needed to account for the effect of shock on the degradation prediction. Because sudden damage increment results in a discontinuity in the degradation path, the lifetime is shortened accordingly. Note that, if the slope of degradation path is given as fixed, the degradation path with a shortened lifetime can be considered as equivalent to a gradual degradation path with an earlier CIT. Therefore, the variation in CIT provides a degree of freedom in translational adjustment for the degradation path. With both slope and translational adjustments, the proposed method is expected to reduce the uncertainty in RUL prediction and to capture the effect of shock on the degradation as well. In different applications, the shock occurrence time might be known or unknown. If known, the corrective action can be taken right after the shock occurs; otherwise additional work is needed. In this study, both the two situations will be considered.

The remainder of this paper is organized as follows. Section 2 introduces the integrated prognostics framework that gives a global view of the structure of the proposed method. In Section 3, the Bayesian updating procedure is presented to deal with gradual degradation considering uncertainty in both CIT and material parameters. Section 4 investigates integrated prognostics for degradation with shock. Two cases are considered: a). shock degradation with known shock occurrence time; b) shock degradation with unknown shock occurrence time. The formula to compute RUL is also given in this section. In Section 4, examples are given to show the effectiveness and efficacy of the methods. Section 5 concludes the work.

The update process is executed every time when observation on crack size is available. The posterior distribution in the last iteration will be used as the prior distribution in the next iteration. By assimilating the observations, the variances of uncertainty parameters are expected to be reduced and the mean to approach the actual value. The shrinkage of uncertainty leads to a narrower distribution, which appears to have a peak covering a very small range of parameter values. When we implement the Bayesian inference Eq. (4), importance sampling technique is utilized, where the density is represented by samples from the associated distribution. This discretization of the continuous function excludes the samples with low density, especially when sample size is limited. When the shock occurs, there is a sudden increase in crack size, which results in a noticeable difference between the predicted crack size using degradation model and the observation. In order to attribute this difference to an earlier CIT, it needs a fair extent of adjustment of CIT in its marginal posterior distribution. Hence, the coverage of prior distribution must be large enough to give moderate density for the virtual CIT \(t_{0\_virtual}\), otherwise the samples could not carry useful information. Therefore, the posterior distribution before shock occurs is adjusted in a way that its variance of CIT marginal is artificially increased to make sure the vicinity of the targeted virtual CIT \(t_{0\_virtual}\) has moderate chance to be sampled.

Let the inspection times be a set \(\{ t_{u} : u = 1, 2, \cdots , U\}\). At these inspection times, the observed crack sizes forma set \(\{ a_{u}^{obs} : u = 1, 2, \cdots , U\}\) and the uncertain parameters are denoted by \(\left\{ {\left( {t_{0}^{(u)} ,\xi^{(u)} } \right): u = 1, 2, \cdots , U} \right\}\). Assume that the measurement errors are independent among different inspection times; hence, the likelihood function to observe a crack history up to an inspection time \(t_{s}\) iswhere \(l \left( {\left. {a_{u}^{obs} } \right|t_{0}^{(s - 1)} ,\xi^{(s - 1)} } \right)\) was defined in Eq. (3). \(a_{1:s}^{obs}\) refers to the set of observation history \(\{ a_{u}^{obs} : u = 1, 2, \cdots , s\}\) up to the inspection time \(t_{s}\). The update process at \(t_{s}\) can be obtained accordingly by

The inspection times divide the whole lifetime into several disjoint intervals. If the degradation is gradually happening without shock, the update of uncertain parameters is to directly assign the posterior distribution of uncertain parameters at current inspection time to the next inspection time. However, if the shock occurs at a known time instant during interval [\(t_{v - 1} , t_{v}\)), the posterior distribution will be adjusted before being assigned to the prior distribution for next update. As discussed earlier, the adjustment is to increase the variance of marginal distribution for CIT to cover \(t_{0\_virtual}\). Denote the adjusted distribution as \(\tilde{f}_{post} \left( {t_{0}^{(v - 1)} ,\xi^{(v - 1)} } \right)\). In addition, the likelihood also needs modification. Because of the virtual degradation path, the actual observations on crack sizes before the shock occurs will have adverse effects on the likelihood to observe the crack sizes after the shock. Hence, only the observations after shock will be used to define the likelihood function in Bayesian formula for the updates executed after shock occurrence time. The update process considering the shock occurring at a known time instant thus is modified to be

If the shock occurs at an unknown time, we could do the proposed adjustment above at every inspection time to ensure that the targeted virtual CIT will be covered within the distribution samples. However, this approach will result in a large amount of uncertainty in RUL prediction, which provides little useful information in maintenance decision making. Hence, an additional step of shock detection is proposed to add into the update process to deal with the case when shock occurrence time is unknown.

Shock occurrence will cause sudden increase of the crack size. The amount of increase is assumed to be far out of the range of measurement error. A large adjustment of CIT from the last inspection time is expected. The average of predicted crack size using the distribution of CIT at the last inspection time ought to deviate a lot from the observed crack size at the inspection time right after shock occurs. Therefore, a shock is said to be detected if the amount of such deviation exceeds a predefined threshold \(\delta\). The criterion is that, shock occurs during \(\left[ {t_{v - 1} , t_{v} } \right)\) if

The symbol of \({\mathbb{E}}\) denotes the operator for expectation. After identifying the shock occurrence time, update process Eqs. (7–8) will apply.

The ultimate goal of updating uncertain parameters is to predict the RUL of the cracked gear more accurately. The RUL is defined as the difference between final failure time and current inspection time. The failure time is the instant when the observed crack size reaches or exceeds a predefined critical value \(a_{C}\) for the first time. With the updated parameter distribution obtained through Bayesian inference, the RUL prediction is given accordingly at the inspection time. Paris’ law can be written in its reciprocal form as in Eq. (10),

Let the current inspection cycle be \(N_{u} (t_{0} )\) and the crack increment be \(\Delta a\). The RUL is calculated by discretizing Eq.(10) in the following way,where \(a_{i/2} = a_{0} + \Delta a(N_{i} + N_{i + 1} )/2\). The RUL is the summation \(\mathop \sum \limits_{i} \Delta N_{i} (\xi )\) from the current inspection cycle to the cycle when failure occurs. Therefore, the total failure time is expressed as \({\text{F}}\left( {t_{0} ,\xi } \right) = N_{u} (t_{0} ) + \mathop \sum \limits_{i} \Delta N_{i} (\xi |t_{0} )\). This expression shows that the uncertainty in total failure time prediction is determined by the uncertainty in both the CIT and the physical model parameters, \(\left( {t_{0} ,\xi } \right)\). Thus more accurate values of \(\left( {t_{0} ,\xi } \right)\) will produce more accurate \({\text{F}}\left( {t_{0} ,\xi } \right)\). An update of \({\text{F}}\left( {t_{0} ,\xi } \right)\) will be triggered after an update of \(\left( {t_{0} ,\xi } \right)\) to adjust the lifetime prediction.

In this case, the prior information for uncertain parameters is assumed as: \(m\sim N\left( {1.4, 0.2} \right)\), \(t_{0} \sim N ( 2\times 1 0^{6} , 2 \times 1 0^{5} )\). The measurement error is \(e\sim N(0, 0.15)\). The values of the constants are set to be: \(C{ = 9} . 1 2\times 1 0^{ - 11}\), \(a_{0} = 0.1\) mm, \(a_{C} = 5.2\) mm. The history of SIF is adopted from Ref. [19] in which the input torque is 320 N⋅m and the effect of dynamic load is considered. The characteristics of two actual degradation paths are shown in Table 2.

We have two test degradation paths, which share the real value of \(m = 1.6\), but have different CITs. The characteristics of these two paths are given in Table 2. The two degradation paths are shown in Figures 5 and 6. The crack observations and updated results for the two paths are listed in Tables 3 and 4, respectively. The updated PDFs for distributions of \(t_{0}\) and \(m\) are displayed in Figures 5 and 6 for path #1, and in Figures 7 and 8 for path #2.

The update results show that, as more crack observations are fed into Bayesian inference, the means of \(m\) and \(t_{0}\) progressively approach their actual values. Meanwhile, the standard deviations are shrinking, which makes the distribution shape become narrower. This is beneficial for improving the performance of prognostic algorithm since narrower distribution indicates reduced uncertainty, and it is useful in making more accurate and cost-effective maintenance decisions. Based on the updated uncertain parameters, the mean of the failure time approaches its actual value accordingly. The actual values are denoted using star marks in the Figures 7, 8, 9 and 10.