Fuzzy analysis of bearing accelerated degradation testing with uncertainty

2.1. Degradation index

Motivated by [4], the combination of PCA and GMM is used for obtaining the CVs from bearing features in abovementioned domains. The main equations are:

where, $\mathbf{X}$ is the $p$-by-$n$ raw matrix for all extracted features, $\mathbf{W}$ is the $n$-by-$n_l$ eigenvector whose eigenvalues are in descending order with the cumulated percentage over 85 %, $\mathbf{Y}$ is the $p$-by-$n_l$ lower dimensional feature set, $M$ is the number of Gaussian component which is set to 3 according to experience, ${\omega }_i$ is the weight of component $i$, and $g\left(y|{\mu }_i,{\Sigma }_i\right)$ is its Gaussian density function. Details about the estimation algorithms are referred to the references.

Once the GMM models for both baseline (normal) state and degraded state are obtained. The CV at state $k$ can be calculated by:

2.2. Degradation modeling and failure time

Fuzzy regression method is a powerful tool to model the degradation path containing the epistemic uncertainty by means of fuzzy theory. The model for the case study is selected as [10]:

where, $a$ and $b$ are constant, ${\epsilon }_1$ is the error term which follows normal distribution. Then, the (1-$\alpha \text{'}$) 100 % confidence interval for parameters are regarded as the fuzzy estimators $\stackrel{-}{\theta }$ for parameters $\widehat{\theta }$, which is:

and symmetric triangular membership function is used for fuzzy calculation for all $\alpha$-cuts, $\alpha \in \left[\text{0, 1}\right]$. Then, fuzzy parameters are given as $\stackrel{-}{\theta }\left[\alpha \right]$. When the failure threshold $C$ is available, the corresponding fuzzy failure time $T\left[\alpha \right]$ is computed by the determined regression model.

2.3. Acceleration factors

Linear mixture model, i.e. acceleration model, is applicable to find the relationship between environmental variables, e.g. load, speed, and bearing fuzzy failure time [11]. The model is:

where, ${\lambda }_i$ is the coefficient, $i=$ 1, 2,…, 5, $S$ and $L$ donate speed and load for bearing operating conditions, $T\left[\alpha \right]$ is the fuzzy failure time under alpha-cuts.

When the optimal parameters for Eq. (5) are estimated, failure time at both normal and the ith stress level can be given as $\mathrm{}{\widehat{T}}_0\left[\alpha \right]$ and $\mathrm{}{\widehat{T}}_i\left[\alpha \right]$. Then, the Acceleration Factor (AF) is:

which can convert fuzzy failure times under accelerated levels into normal ones, i.e. $T_0\left[\alpha \right]$.

2.4. Fuzzy reliability analysis for bearings

Studies have suggested that the failure time of bearing follows Weibull distribution [12]:

where, $\eta$ is scale parameter, and $\beta$ is shape parameter. In general, reliability analysis, $\beta >$ 1 is used for describing the wearout phase in the Bathtub-Shaped curve which should be satisfied since the bearings experienced a harsh operating environment. For fuzzy failure times, the fuzzy parameters, i.e. ${\eta }_{\alpha }$, ${\beta }_{\alpha }$, are obtained by life distribution fitting for $T_0\left[\alpha \right]$.

Thus, the fuzzy reliability for time interval [$t_1$, $t_2$] is:

If other life distribution is satisfied, e.g. Log-normal distribution, Eq. (8) can be rewritten to:

where, $\mathrm{\Phi }\left(\bullet \right)$ is the standard normal cumulative distribution function, while $\mu$ and $\sigma$ are the mean and standard deviation for standard normal distribution.

3.1. Experiment platform

Details about the experiment platform from IEEE PHM 2012 prognostic challenge are given in [8]. The aim of this competition is to estimate the remaining useful life time for 11 bearings by the learning set of 6 bearings under three operating conditions. However, this paper concentrates on the reliability analysis for bearings under accelerated operating levels with uncertainty from the perspective of population but not individual. Therefore, we ignore the bearings whose lifetimes are significant lower than others, i.e. 1_2, 1_4, 2_1, 2_7, for the following analysis. Meanwhile, only vertical vibration signals is used since it can better reflect the health state [13].

3.2. Fuzzy analysis results

Following the chart flow in Fig. 1, we extracted all the features in four domains and then reduced them by PCA with the cumulated percentage over 85 %. The CV results for the bearings are shown in Fig. 2.

Fig. 2The degradation processes for bearings under three accelerated levels with their fitted paths

From Fig. 2, it is reasonable to set 0.2 as the failure threshold from the learning set (solid line), which then is used to compute the fuzzy failure times by Eqs. (3), (4) with $C=$0.2. The results are given in Tables 1, 2. The statistical analysis for the error term$\mathrm{}{\epsilon }_1$ to check the assumption of normal distribution and the fuzzy estimation for model parameters are also presented.

Note that: The degradation paths for 1_3, 2_3, 2_5 and 3_2 are not compromised with the normal assumption for error item of Eq. (3), although the $R^2$ and $R_{adj}^2$ suggest good fitting. Hence, we eliminate bearing 1_3 and recalculate bearing 2_3, 2_5, 3_2 by cutting the tail of degradation path, which are given in brackets ([]).

Eqs. (5), (6) are used to calculate the fuzzy acceleration factors (Fig. 3(a)) and convert the failure times into normal stress levels (Table 2). Anderson-Darling test is then used to select the applicable life distribution for failure times, and results suggest that both Weibull and Log-normal are suitable. The fuzzy parameters for the two distributions are shown in Fig. 3(b).

Fig. 3a) Fuzzy acceleration factors under three different operating conditions, b) fuzzy parameters for two life distributions: Log-normal (.a, .b) and Weibull (.c, .d)

a)

b)

Fig. 4Fuzzy reliability under two life distributions at time interval [7, 8] (h)

As described in the challenge details, most of the bearings exprience 1 (h) to 7 (h). Thus, we analyze the reliability under normal condition in interval [7, 8] (h) through Eqs. (8), (9) and the results are given in Fig. 4, which are [0.9177, 1] and [0.9405, 0.9989] for Log-normal and Weibull distributions when $\alpha =\text{0.5}$. For traditional analysis, the reliability is 0.9894 and 0.9792.