Use of generalized refined composite multiscale fractional dispersion entropy to diagnose the faults of rolling bearing

Abstract

A typical symptom of vibration signals collected from rolling bearings with local faults is the existence of periodic transients, which makes the intrinsic structure of vibration signals become more and more regular. Generally, the vibration always contains multiple intrinsic oscillatory modes on different scales, which generally is caused by the interaction and coupling of machine components. Therefore, it is necessary to detect the behavior change of complexity of vibration signals in the view of multiple scales for fault information representation. The complexity and nonlinear failure symptom of rolling bearing can be evaluated by the recently proposed nonlinear dynamic tools, such as multiscale entropy (MSE) and its variants. Recently, the improved MSE method, multiscale dispersion entropy (MDE) and its improvement refined composite MDE (RCMDE) are developed to measure the complexity of time domain data. However, the intrinsic shortages of coarse graining approach that used in MDE and RCMDE have limited their application to fault feature representation. In this paper, an improved RCMDE approach named generalized refined composite multiscale fluctuation-based fractional dispersion entropy (GRCMFDE) is proposed to enhance MDE and RCMDE in complexity measurement of time series. GRCMFDE was compared with MPE, MDE, RCMDE by analyzing synthetic simulation signals to verify its advantages. After that, an intelligent fault diagnosis method was proposed by combining GRCMFDE with supervised multi-clustering feature selection and gray wolf optimized SVM for fault classification of rolling bearing. Lastly, the proposed fault diagnostic method was applied to two experimental data set analysis by comparing with multiscale permutation entropy, MDE- and RCMDE-based fault diagnostic methods and the comparison results indicate that the proposed method can effectively diagnose the fault locations and severities of rolling bearing and get a higher fault identifying rate than the comparative methods.

Appendix A: Reviews of DE and MDE

For a given univariate time series $X = \left\{ {x_{1} ,x_{2} , \ldots ,x_{N} } \right\}$ with a length of N, the computation steps of DE are given as follows [12, 13].

• (1) The original time series $X$ is mapped to $Y = \left\{ {y_{1} ,y_{2} , \ldots ,y_{N} } \right\}$ by using normal cumulative distribution function (NCDF) shown as

  $$ y_{j} = \frac{1}{{\sigma \sqrt {2\pi } }}\int\limits_{ - \infty }^{{x_{j} }} {e^{{\frac{{ - \left( {t - \mu } \right)^{2} }}{{2\sigma^{2} }}}} } {\text{d}}t. $$

  (4)

  It is obvious that $y_{i} \in (0,1)$. $\sigma$ is standard deviation and $\mu$ is mean of $X$.

• (2) By using the linear transform shown in Eq. (2), all elements of $Y$($y_{i}$, $j = 1,2, \ldots ,N$) are mapped to c classes (which is an integer), i.e.,

  $$ z_{j}^{c} = R\left( {c \cdot y_{j} + 0.5} \right) $$

  (5)

  where R represents the rounding operation and $z_{j}^{c}$ represents the jth member of the classified time series. Although the step (2) is linear, the whole mapping way is nonlinear for the use of NCDF in step (1).

• (3) For the given time delay d and embedding dimension m, ${\text{z}}_{i}^{m,c}$ can be obtained according to $z_{i}^{m,c} = \left\{ {z_{i}^{c} ,z_{i + d}^{c} , \ldots ,z_{i + (m - 1)d}^{c} } \right\}$, $i = 1,2, \ldots ,N - (m - 1)d$. Each time series ${\text{z}}_{i}^{m,c}$ can be mapped to a dispersion pattern $\pi_{{v_{0} v_{1} \ldots v_{m - 1} }}$, where $z_{i}^{c} = v_{0}$, $z_{i + d}^{c} = v_{1}$,…, $z_{i + (m - 1)d}^{c} = v_{m - 1}$. The signal has m members, and they all belong to the integer interval from 1 to c. The number of possible dispersion patterns is equal to $c^{m}$.

• (4) The relative frequency for each dispersion patterns $\pi_{{v_{0} v_{1} \ldots v_{m - 1} }}$ can be estimated by

  $$ p\left( {\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right) = \frac{{{\text{Number}}\left\{ {i\left| {i \le N - (m - 1)d,\;z_{i}^{m,c} \;{\text{has}}\;{\text{type}}\;\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right.} \right\}}}{N - (m - 1)d} $$

  (6)

  where $p\left( {\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right)$ stands for the number of dispersion patterns of $\pi_{{v_{0} v_{1} \ldots v_{m - 1} }}$ assigned to ${\text{z}}_{i}^{m,c}$ divided by the total number of embedded signals for embedding dimension m.

• (5) The DE of $X$ is computed by

  $$ {\text{DisEn}}(X,m,c,d) = - \sum\limits_{\pi = 1}^{{c^{m} }} {p\left( {\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right)\ln \left( {p\left( {\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right)} \right)} $$

  (7)

It can be found from the algorithm of DE that when all probability of distribution patterns $p\left( {\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right)$ are equal, DE gets the largest entropy value $\ln (c^{m} )$ and a typical example is Gaussian white noise. In contrast, when the probability of distribution pattern $p\left( {\pi_{{v_{0} v_{1} \ldots v_{m - 1} }} } \right)$ is unitary, i.e., only one value is not equal to zero, DE get the smallest value, which indicates that the time series is a completely predictable data and a typical example is the periodic signal with low frequency.

Based on DE, the computation steps of MDE are given as follows.

• (1) For a given univariate data $W = \left\{ {w_{1} ,w_{2} , \ldots ,w_{L} } \right\}$ with a length of L, firstly, it is divided into $\tau$ non-overlapping segments with length $\left\lfloor {{L \mathord{\left/ {\vphantom {L \tau }} \right. \kern-0pt} \tau }} \right\rfloor$. Then, the average of each segment is computed to derive the coarse-grained time series. This process is named coarse graining and is given as follows:

  $$ y_{j}^{(\tau )} = \frac{1}{\tau }\sum\limits_{a = (j - 1)\tau + 1}^{j\tau } {w_{a} } ,\quad 1 \le j \le \left\lfloor {{L \mathord{\left/ {\vphantom {L \tau }} \right. \kern-0pt} \tau }} \right\rfloor $$

  (8)

  $\tau$ is called scale factor. $y_{{}}^{(1)}$ is the original data $W$ and when $\tau > 1$, $W$ is divided into $\tau$ coarse-grained time series $y^{(\tau )}$ with a length of $\left\lfloor {{N \mathord{\left/ {\vphantom {N \tau }} \right. \kern-0pt} \tau }} \right\rfloor$ (which stands for the largest integral smaller than $\{ x(i),i = 1,2, \cdots ,N\}$).

• (2) The DE of $y^{(\tau )}$ is estimated by

  $$ {\text{MDE}}(W,\tau ,m,c,d) = {\text{DE}}\left( {y^{(\tau )} ,m,c,d} \right) $$

  (9)

The DEs over different time scales are depicted as a function of scale factor, and this process is called MDE analysis.