Rolling bearing degradation condition clustering using multidimensional degradation feature and Gath

Abstract

In allusion to the issue of rolling bearing degradation feature extraction and degradation condition clustering, a logistic chaotic map is introduced to analyze the advantages of C₀ complexity and a technique based on a multidimensional degradation feature and Gath–Geva fuzzy clustering algorithmic is proposed. The multidimensional degradation feature includes C₀ complexity, root mean square, and curved time parameter which is more in line with the performance degradation process. Gath–Geva fuzzy clustering is introduced to divide different conditions during the degradation process. A rolling bearing lifetime vibration signal from intelligent maintenance system bearing test center was introduced for instance analysis. The results show that C₀ complexity is able to describe the degradation process and has advantages in sensitivity and calculation speed. The introduced degradation indicator curved time parameter can reflect the agglomeration character of the degradation condition at time dimension, which is more in line with the performance degradation pattern of mechanical equipment. The Gath–Geva fuzzy clustering algorithmic is able to cluster degradation condition of mechanical equipment such as bearings accurately.

Keywords

Gath–Geva fuzzy clustering rolling bearing degradation character feature extraction

Highlights

Performance degradation feature extraction based on C₀ is proposed.

A Curved time parameter is proposed for describing the performance degradation process.

Sequence dispersion is proposed to comprehensively evaluate the clustering effect.

The Gath–Geva method is introduced in performance degradation condition clustering.

The effectiveness and superiority is proved by analysis on whole life data from intelligent maintenance system (IMS).

1. Introduction

Port crane is important material handling equipment in ports whose power mainly comes from its hoisting mechanism which is used for containers moving. Rolling bearing is the key rotating supporting component in hoisting mechanism. In case of sudden failure, economic losses even casualties will be caused. It is meaningful for reducing sudden failure probability and improving reliability by analyzing rolling bearing monitoring signals and assessing performance degradation condition accurately. Port cranes usually work in heavy-duty, high-speed environment, and the signals usually show nonlinear, nonstationary, and nonperiodic character. It is a hot topic that how to get an effective degradation condition assessment technique.

Degradation condition assessment mainly comprises two key procedures, including degradation feature extraction and assessment model constructing (Han et al., 2019). The purpose of degradation feature extraction is calculating health indicators reflecting performance degradation pattern. The merits of the health indicators will determine the accuracy of degradation condition assessment. At present, some typical degradation features are studied based on time domain analysis, frequency domain analysis, and time–frequency domain analysis methods; some health indicators are proposed as degradation features such as root mean square (RMS), variance, average frequency, and so on (Araneo et al., 2017; Xiao et al., 2015). Considering that most engineering vibration signals are nonlinear and unstable, a complexity analysis method based on information entropy and fractal theory are usually introduced to provide an effective way of degradation feature extraction. Some commonly used methods include fuzzy entropy (Wang et al., 2018a, 2018b), sample entropy (Yang et al., 2018), approximate entropy (Saif et al., 2018), and fractal dimension (Wang et al., 2015). Along the train of thought, the C₀ complexity analysis method which is proposed based on the Fourier transform provides a new idea for degradation feature analysis. The C₀ complexity method has achieved good effects on electroencephalogram and speech signals; the advantage of this method lies in convenient calculation and low signal requirement (Cai et al., 2018; Ye et al., 2018a, 2018b). Ye et al. (2018a, 2018b) proved that spectrum entropy and C₀ complexity curves are able to describe dynamic character of continuous chaotic systems accurately and effectively. Xu and Zhang (2015) proposed an endpoint detection algorithm based on the combination of improved C₀ complexity and Mel-scale frequency cepstral coefficients similarity. At present, there are few studies on degradation analysis of mechanical equipment using the C₀ algorithm which contains potential value in nonlinear signal analysis.

In the health condition assessment procedure, degradation condition division is a difficult step; the condition numbers and boundaries are difficult to determine, and it is apparent that subjective division methods are not scientific. In the physical degradation process, signals within the same condition are usually continuous at time dimension, so it is pivotal to divide degradation conditions as its physical feature; in the meantime, it will lay an effective foundation for the on-site online health condition assessment.

An unsupervised clustering method is an effective technique in degradation condition division and some typical clustering methods include K-means clustering (Li et al., 2019), fuzzy C-means clustering (FCM) (Wang et al., 2017), Gath–Geva (GG) fuzzy clustering (Wang et al., 2018a, 2018b), and Gustafson–Kessel (GK) clustering (Chen et al., 2018) are usually applied. The GG clustering algorithm has some improvement comparing with FCM and GK clustering methods (Abonyi et al., 2005; Yu et al., 2017). The hot spots of the GG clustering algorithm mainly focus on fault diagnosis, and the clustering number and time continuity constraints in degradation process are less considered.

In summary, in allusion to degradation feature extraction and degradation condition division, a technique based on the multidimensional degradation feature and GG fuzzy clustering algorithm is proposed. The multidimensional degradation feature includes C₀ complexity, RMS, and curved time (CT) parameter which is more in line with the performance degradation process. The GG algorithm is introduced to cluster different conditions in the degradation process. The bearing lifetime vibration signal from the IMS bearing test center is used to verify the method.

The article is organized as follows: Section 2 introduces the basic theory in this article. Section 3 expounds C₀ complexity and its analysis on the logistic chaotic map. The procedure of degradation condition clustering technique is expounded in Section 4. In Section 5, the technique is verified and the results are discussed. Finally, the conclusion is given in Section 6.

2. Basic theory

2.1. C₀ complexity

The C₀ complexity algorithm is a nonlinear analysis method which has the advantage of fast calculation speed (Guo et al., 2017). The main idea lies in describing signal complexity with the proportion of nonregular components quantitatively in the sequence.

Assuming one-dimensional time series is x(t), t = 0, 1, 2, …, M−1, the basic procedures are as follows:

Making the discrete Fourier transform on x(t)

F_{M} (k) = \sum_{t = 0}^{M - 1} x (t) e^{- 2 π i (t k / M)}, k = 0,1,2, \dots, M - 1

(1)

Defining $W_{M} = e^{- 2 π i (t k / M)}$ and then simplifying $F_{M} (k)$ as follows

F_{M} (k) = \sum_{t = 0}^{M - 1} x (t) W_{M}^{t k}

(2)

Calculating mean square value of $F_{M} (k)$

G_{M} = \frac{1}{M} \sum_{t = 0}^{M - 1} {| F_{M} (k) |}^{2}

(3)

Retaining the spectrum, which is larger than the mean square value $G_{M}$ , and setting the rest components as zero

{\bar{F}}_{M} (k) = {\begin{matrix} F_{M} (k) & {| F_{M} (k) |}^{2} > G_{M} \\ 0 & {| F_{M} (k) |}^{2} \leq G_{M} \end{matrix}

(4)

Processing the Fourier inverse transform on ${\bar{F}}_{M} (k)$

\bar{x} (K) = \frac{1}{M} \sum_{t = 0}^{M - 1} {\bar{F}}_{M} (k) W_{M}^{- t k}

(5)

Calculating C₀ complexity as

C_{0} = \frac{\sum_{t = 0}^{M - 1} {| x (t) - \bar{x} (t) |}^{2}}{\sum_{t = 0}^{M - 1} {| x (t) |}^{2}}

(6)

Generally speaking, C₀ complexity is introduced for describing complexity of time sequence in quantity. The value of C₀ is proportional to the degree of fluctuation pattern. In this article, C₀ complexity is introduced to describe the performance degradation feature of rolling bearing.

2.2. GG fuzzy clustering

Fuzzy maximum likelihood distance is introduced in GG fuzzy clustering, and the basic procedures are as follows:

Supposing that the clustering sample set is X = {X₁, X₂, …, Xn} and the number of feature vector is n. First, a clustering number is initialized as c (c ≥ 2), and the membership matrix is set as U = [u_ip]_c×n, u_ip ∈ [0, 1], i = 1, 2, …, c, p = 1, 2, …, n, where u_ip represents the degree of the pth sample belonging to the ith clustering.

Setting terminal parameter ε(ε > 0) and initializing membership matrix U.

Calculating clustering centers $v_{i}^{l}$

v_{i}^{l} = \frac{\sum_{p = 1}^{n} {(u_{i k}^{l - 1})}^{m} x_{p}}{\sum_{p = 1}^{n} {(u_{i p}^{l - 1})}^{m}}, 1 \leq i \leq c, l = 1,2, \dots, n

(7)

Calculating fuzzy maximum likelihood distance as the following formula

D_{i p A_{i}}^{2} (x_{p}, v_{i}) = \frac{{(\det (A_{i}))}^{0.5}}{α_{i}} \exp (0.5 {(x_{k} - v_{i}^{l})}^{T} A_{i}^{- 1} (x_{k} - v_{i}^{l}))

(8)

α_{i} = \frac{1}{n} \sum_{k = 1}^{n} u_{i k}

(9)

A_{i}^{(l)} = \frac{\sum_{k = 1}^{n} {(u_{i k}^{l - 1})}^{m} {(x_{k} - v_{i}^{(l)})}^{T}}{\sum_{k = 1}^{n} {(u_{i k}^{l - 1})}^{m}}

(10)

where

α_{i}

means the prior probability and A_i is the covariance matrix of the ith clustering.

Setting the objective function J and achieving the optimal classification by minimizing the target

J = \sum_{i = 1}^{c} \sum_{p = 1}^{n} {(u_{i p})}^{2} D_{i p A}^{2}

(11)

Updating the membership matrix

u_{i j}^{(l)} = \frac{1}{{(\sum_{j = 1}^{c} (D_{i p A_{i}} (x_{p}, v_{i})) / D_{i p A} (x_{p}, v_{i}))}^{2}}

(12)

where i = 1, 2, …, c and k = 1, 2, …, n; the updating will stop when ||U^(l) − U^(l−1)|| < ε.

2.3. Clustering evaluation indexes

It is a key issue that how to evaluate the degradation condition clustering result. Considering maximum membership degree principle is usually adopted in this type of clustering methods; therefore, three evaluation indexes are introduced toward membership degree matrix U are proposed in this section.

Among the indexes, the classification coefficient α and average fuzzy entropy β are introduced, and the definitions are as follows

α = \frac{1}{n} \sum_{i = 1}^{c} \sum_{j = 1}^{n} μ_{i j}^{2}

(13)

β = \frac{1}{n} \sum_{i = 1}^{c} \sum_{j = 1}^{n} μ_{i j} \ln μ_{i j}

(14)

The better the clustering effect, the closer the classification coefficient to 1 and the closer the average fuzzy entropy to 0.

Furthermore, sequence dispersion index (SD) is proposed in this section to measure the continuity at time dimension during the same degradation condition. The definition is as follows.

Assuming I is the label sequence in clustering result, and n is the number of samples of this cluster. m − 1 is defined as the D-value between the maximum and minimum values of I. The sequence dispersion of this cluster is defined as the following formula

b = \frac{(m - n)}{m}

(15)

Obviously, the value of b will be zero when I is a continuous sequence. The more discontinuous the sequence I, the greater the value of b.

Assume that the entire sequence is divided into number of c classes. The SD index of the clustering result is calculated as follows

γ = \sum_{i = 1}^{c} b_{i}

(16)

From the definition above, the clustering method will get a good effect if SD is closer to zeros. The larger the index, the lower the time aggregation and the worse the clustering effect.

3. Degradation feature analysis based on C₀ complexity

A logistic chaotic map sequence is used as the simulation signal for analysis of C₀ complexity. The simulation signal expression is as follows (Jabbari and Mohasefi, 2019)

x (t + 1) = λ x (t) (1 - x (t))

(17)

where x(t) is the condition variable with section (0,1), t is the iteration number, and λ represents a nonlinear parameter. The equation will exhibit different dynamic limit behaviors as λ is taken as a different value. An initial value is set as x = 0.4 in this article, and the bifurcation diagram is shown in Figure 1. It is clear that the system is in periodic condition and the number of cycles is also constantly changing when λ varies within section (3, 3.57). When λ ∈ (3.57, 4.0), the system enters a chaotic state and some narrow intervals are mixed with periodic cycles which reduces the complexity of the sequence.

Figure 1.

Logistic chaotic map bifurcation diagram.

C₀ complexity under different λ is calculated and compared with commonly used entropy analysis methods. The simulation computer processor is AMD A8-7600 Radeon R7 and memory is 8.00 GB; the parameter setting and calculating time are shown in Table 1. The complexity trend of different complexity methods is shown in Figure 2.

Table 1.

Algorithm parameters and calculating results.

Analysis method	Parameter setting	Cycle number of λ	Calculating time (s)
C₀ complexity	r = 5	2000	2.8362
Fuzzy entropy	Dim = 2, r = 0.5, n = 2	2000	199.5029
Approximate entropy	Dim = 4, r = 0.5	2000	158.9880
Sample entropy	Dim = 4, r = 0.5	2000	148.3839

Figure 2.

Trends for different complexity methods: (a) C₀ complexity, (b) fuzzy entropy, (c) approximate entropy, and (d) sample entropy.

From the figure above, it is evident that C₀ complexity and fuzzy entropy are able to reflect the increscent complexity during the interval (3.45, 3.57) as well as the downtrend of the complexity caused by the mixed periodic cycle; however, the approximate entropy and sample entropy algorithms cannot reflect this phenomenon effectively. Considering the calculation speed, C₀ complexity is much faster than the other three methods because it involves only the Fourier transform and inverse transform operations. Therefore, the C₀ complexity parameter has a good correlation with the sequence complexity. In addition, this method has few parameters and fast calculation speed.

4. Flowchart of degradation condition clustering

A degradation condition clustering technique based on multidimensional degradation features and GG fuzzy clustering is proposed. The flowchart is shown in Figure 3.

Figure 3.

Flowchart for degradation condition clustering.

From the flowchart above, a lifetime vibration signal of mechanical equipment is obtained firstly, Multidimensional degradation features are extracted, respectively, and an RMS indicator is calculated to characterize signal energy accumulation. C₀ complexity is introduced to reflect complexity changing information. To consider the agglomeration degree of the same degradation condition on time dimension, the CT parameter is introduced to reflect time character during the performance degradation process. The value of the CT indicator is calculated by normalizing lifetime time T and mapping it as the following formula

CT = e^{T} - 1

(18)

where CT means “curved time” which has a gentle early stage and an intense late stage, thereby reflecting the time distribution character of mechanical equipment degradation more accurately. The comparison of the lifetime parameter T with CT is shown in Figure 4.

Figure 4.

Comparison of T and CT. Note: CT: curved time.

After feature extraction, GG fuzzy clustering is performed on the degradation division so that different degradation condition is able to be clustered with the fuzzy matrix calculated by the GG fuzzy clustering method. Finally, the clustering effect is evaluated by the three indicators expounded above.

5. Instance analysis

5.1. Bearing lifetime dataset

The whole lifetime dataset used in this section is from the IMS Center at the University of Cincinnati (Antoni and Borghesani, 2019). The test rig is shown in Figure 5. The testing bearing is Rexnord ZA-2115 double-row roller bearing, and the number of rollers is 16. The roller group pitch diameter is 75.501 mm, the roller diameter is 8.4074 mm, and the contact angle is 15.17°.

Figure 5.

Map of accelerating test rig.

One lifetime dataset is introduced for instance analysis which contains 984 groups of samples, and the final fault occurs at the outer ring. A time domain waveform is shown in Figure 6 in which the sampling interval has been filtered. It seems that the amplitude begins to increase at about the 700th group sample, and a quantitative feature needs to be extracted to describe the degradation process.

Figure 6.

Time domain wave of the lifetime dataset.

5.2. Multidimensional degradation feature

Degradation feature analysis is performed on each group to calculate a multidimensional feature vector [RMS, C₀, CT]. The parameter r is set as 10 for the C₀ complexity method. The normalized feature vector is shown in Figure 7. It is clear that C₀ complexity gradually decreases as the performance degradation deepens, whereas the trend of the RMS curve is opposite. The reason lies that the RMS indicator represents energy accumulation, whereas C₀ complexity describes the complexity degree. It shows that complexity degree of the signal will gradually decrease as the performance degradation degree increases, and signal energy will increase with the degradation degree. The two curves show a phased character which reflects different conditions of bearing performance degradation. In addition, the “CT” parameter curve marked in green is more effectively related to the overall trend of performance degradation through the mapping operation.

Figure 7.

Multidimensional degradation features.

5.3. Degradation condition clustering

GG fuzzy clustering after the normalization process on a feature vector [C₀, RMS, CT] is performed. The degradation condition is divided into four categories in this article, and the parameters are set as c = 4, m = 2, and ε = 0.00001. The contour diagram is shown in Figure 8. It is clear that the contour line is arbitrary which means the clustering technique has a low requirement for data source distribution and has a good adaptability.

Figure 8.

Contour map for Gath–Geva clustering.

The clustering result is shown in Figure 9. The entire degradation process is divided into four conditions marked as normal, slight, severe, and failure; each condition has preferable continuity on the time scale. It is apparent in Figure 9(a) that the bearing is kept in normal condition for a long time before the 520th group and C₀ complexity is maintained at around 0.7. When degradation occurs slightly at the 520th group, C₀ is very sensitive and drops rapidly. The bearing performance was seriously degraded and the value of C₀ was maintained at around 0.4 with little value rebounded in the third condition basically. In the failure condition, C₀ is generally low and some discrete points of numerical anomalies appear.

Figure 9.

Clustering effect for Gath–Geva clustering: (a) two-dimensional clustering effect and (b) three-dimensional clustering effect.

5.4. Influence analysis of CT feature

To analyze the influence of the CT feature clearly, The GG fuzzy clustering model and parameter are kept unchanged; two different degradation feature vector schemes are introduced for comparative analysis including two-dimensional features [C₀, RMS] and three-dimensional features [C₀, RMS, T], where T is the time parameter without mapping. The clustering effect diagram is shown in Figure 10, and the quantitative evaluation is shown in Table 2. Compared with the clustering effect in Figure 9, the continuity of the same condition is intermittent which is not in line with the real physical process so that the same condition will continue a consequent stage whenever the indicators fluctuate at a certain interval has preferable continuity on the time scale.

Figure 10.

Gath–Geva clustering effect for different features: (a) two-dimensional feature [C₀, RMS] and (b) three-dimensional feature [C₀, RMS, T]. Note: RMS: root mean square.

Table 2.

Quantitative evaluation for different features.

Degradation feature	α	β	γ
[C₀, RMS]	0.9887	0.0075	1.3014
[C₀, RMS, T]	0.9915	0.0078	1.6765
[C₀, RMS, CT]	0.9895	0.0018	0.4621

Note: RMS: root mean square; CT: curved time.

It is clear that three schemes have similar values on the classification coefficient (α); the proposed method have the smallest value in fuzzy entropy parameter (β) and sequence dispersion (γ), indicating that the clustering time aggregation and clustering effect are good because of the CT parameter. The reason lies that the CT parameter is able to enhance the condition relevance in time dimension and imply a gentle tendency in early stage and a dramatic trend in the later stage.

5.5. Influence analysis of clustering methods

Keeping three-dimensional feature vectors [C₀, RMS, CT] unchanged, GK clustering and FCM clustering are used for comparative analysis. The clustering effect of two clustering methods is shown in Figure 11. The quantitative comparison is shown in Table 3.

Figure 11.

Clustering effect for different clustering methods: (a) GK clustering effect and (b) FCM clustering effect. Note: GK: Gustafson–Kessel; FCM: fuzzy C-means.

Table 3.

Quantitative evaluation results for different methods.

Clustering method	α	β	Γ
GK clustering	0.9041	0.2038	0.1110
FCM clustering	0.8309	0.3277	0.0662
GG clustering	0.9895	0.0018	0.4621

Note: GK: Gustafson–Kessel; FCM: fuzzy C-means; GG: Gath–Geva.

In the table above, the proposed GG clustering technique has the largest classification coefficient (α) and the lowest average fuzzy entropy (β), showing a good clustering effect. In addition, it is clear that sequence dispersion of the three methods is relatively low because of the “CT” parameter, indicating a good time aggregation effect. However, as to the index of average fuzzy entropy (β), the chosen two contrastive methods are higher than GG clustering which means that the membership matrix is easy to cause condition misjudgment. For example, in Figure 11(a), some sampling groups of a slight degradation condition from 500th to 600th are misjudged as a normal condition by the GK clustering algorithm. Above all, the introduced GG clustering algorithm has a good clustering effect because of fuzzy maximum likelihood distance introduced in this clustering technique.

In summary, the C₀ complexity parameter has a good performance degradation indication capability. Combining with the proposed multidimensional degradation feature, the GG clustering algorithm has a good clustering effect for degradation condition division.

6. Conclusions

GG fuzzy clustering for rolling bearing degradation condition division using multidimensional degradation features is used in this article. By means of instance data analysis, the following conclusions are obtained:

C₀ complexity is able to reflect the irregular components proportion and describe the regularity in the process of performance degradation effectively; in addition, it is sensitive to complexity changing and the calculation speed is fast. The analysis on the logistic chaotic map sequence and the instance signal show a good effect.

The proposed “CT” parameter is able to reflect the temporal character of the dataset, and it is more in line with the performance degradation regular of the mechanical device by the mapping operation of the exponential function.

The GG clustering method is able to cluster data samples with arbitrary shapes; by using the multidimensional features, the clustering accuracy will be improved as well as the time aggregation degree within the same condition. The proposed SD parameter is able to reflect time aggregation effect of clustering.

The research results show that the degradation process of rolling bearing performance shows obvious stage, and different degradation condition shows certain agglomeration inside. More scientific degradation feature extraction method is apt to be studied based on the multidimensional features. At the same time, an effective real-time degradation condition assessment technique is also necessary in the following studies.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National High Technology Research Development Plan (2013AA041106), National Natural Science Foundation (31300783), and China Postdoctoral Science Foundation (2014M561458).

ORCID iD

Bing Wang

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Rolling bearing degradation condition clustering using multidimensional degradation feature and Gath–Geva fuzzy clustering algorithm

Abstract

Keywords

Highlights

1. Introduction

2. Basic theory

2.1. C0 complexity

2.2. GG fuzzy clustering

2.3. Clustering evaluation indexes

3. Degradation feature analysis based on C0 complexity

4. Flowchart of degradation condition clustering

5. Instance analysis

5.1. Bearing lifetime dataset

5.2. Multidimensional degradation feature

5.3. Degradation condition clustering

5.4. Influence analysis of CT feature

5.5. Influence analysis of clustering methods

6. Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

2.1. C₀ complexity

3. Degradation feature analysis based on C₀ complexity