Bearing performance degradation condition recognition based on a combination of improved pattern spectrum entropy and fuzzy C-means

Abstract

In allusion to performance degradation condition recognition issue for rolling bearing, a method based on improved pattern spectrum entropy (abbreviated as IPSE) and fuzzy C-means algorithm (abbreviated as FCM) is proposed in this paper. Basic pattern spectrum analysis is improved by introducing morphological corrosion operator and IPSE is proposed as the degradation feature parameter in describing bearing performance degradation degree. Simulation analysis shows that IPSE value will increase monotonously along with the deepening of the degradation degree. IPSE and degradation degree has a stable relevance. On this basis, in consideration of the fuzzy character of performance degradation condition boundary, FCM is introduced in degradation condition recognition so that the degradation condition could be recognized effectively in line with maximum subordination degree principle. Rolling bearing fatigue life enhancement testing was carried out in Hangzhou Bearing Test & Research Center, the whole life data was gathered and applied using the proposed technique. The classification coefficient reaches 0.9849 and average fuzzy entropy gets 0.0239 for training set clustering, meanwhile, the whole recognition ratio reaches 90% for testing set. The analysis shows that the technique has a good clustering effect and an acceptable recognition result.

Keywords

Degradation feature extraction mathematics morphology fuzzy c-means degradation condition recognition rolling bearing

1 Introduction

Rolling bearing is a key mechanical part of ‘rotor-bearing’ system in rotating machinery which usually fails during operation [1 –3]. These failures will result in high industrial cost or even disaster. Therefore, considerable attention has been paid to the condition monitoring and fault prediction for bearings [4, 5]. Vibration based methods usually have extensively prospect among the techniques [6 –8].

At present, breakdown maintenance and scheduled maintenance are still the main mode for equipment maintenance [9], insufficient or excess maintenance are inevitable and the demand of modern equipment maintenance cannot be fulfilled on account of neglecting actual state changing and health condition. Condition based maintenance (CBM as abbreviation) is an initiative equipment maintenance technology [10, 11] which takes equipment real-time condition as the foundation, the running status can be recognized combining the structuring and dynamics characters, and predicting the evolutionary trend of equipment failure [12]. Three key techniques support CBM including degradation feature extraction, performance degradation condition recognition and remaining service life prediction [13].

Scientific degradation feature is able to represent performance degradation degree accurately and stably [14]. Some studies have proposed different methods using various signal processing means including time domain analysis, frequency domain analysis and time-frequency domain analysis. Time domain degradation feature extraction takes time domain statistical magnitude as the feature parameter, such as root-mean-square (RMS as abbreviation), peak value, kurtosis and so on [15]. Frequency method takes frequency domain statistical magnitude or characteristic frequency energy value as the feature parameter. Time-frequency method is usually employed combining with theory of spectrum or entropy such as sample entropy [16], Feature-Weighted Entropy [17], LCD relative spectral entropy [18], local characteristic-scale decomposition-approximate entropy [19] and improved multiscale fuzzy entropy [20]. However, time domain and frequency domain degradation feature only reflect feature from a single angle and hardly to analyze non-stationary and aperiodic signals. The time-frequency domain degradation feature is heavily influenced by the performance of time-frequency analysis method. With the increasing complexity for mechanical equipment in operational states such as rolling bearing, its internal feature and complexity cannot be rigorously analyzed through the degradation feature extraction methods mentioned above, in addition to this, quantitative analysis of the mentioned methods also has some deficiency [21].

In machinery equipment’s whole lifetime period, its performance condition will go through a span from normal to invalidation, and its vibrating signal usually possesses non-stationary and non-linear characteristics. The proportion of random component will alter along with the deepening of the degradation degree accordingly [22]. The evolution process may be an extraordinary effective breakthrough point for extracting performance degradation feature. Mathematical morphological particle analysis is apt to reflect shape changing information on different scales by which the signal’s essence can be dissected at different levels easily [23, 24]. Moreover, information entropy is able to characterize the uniformity for probability distribution in quantity from the angle of essence. It is applicable that combining the above-mentioned theories.

Mathematical morphological particle analysis has been widely used in different image analyzing fields currently [25, 26]. There are also some studies focused on machinery equipment’s fault feature extraction in recent years [27, 28], beyond that, there were rare studies in degradation feature extraction method based on mathematical morphological particle analysis. Mathematical morphological particle analysis is able to structure pattern spectrum on different scales and the pattern spectrum distribution may be an effective way to reflect the signal essence. Therefore, mathematical morphological particle analysis is employed as the basic signal processing theory.

Performance degradation condition recognition is able to recognize equipment’s running condition by way of analyzing its degradation feature and it is a kind of fuzzy pattern recognition issue in essence [29–30]. Fuzzy C-means algorithm (FCM as abbreviation) is one of the typical fuzzy clustering method which is able to solve the classification issue for objects with blurred boundaries, achieving an effective result on condition of less prior knowledge [31 –33].

This paper concentrated on performance degradation condition recognition for rolling bearing. The study begins with mathematical morphological particle analysis, a degradation feature extraction method is proposed based on improved pattern spectrum entropy (IPSE as abbreviation), and simulated signal is adopted to verify its validity and stability. On this basis, performance degradation condition recognition based on IPSE - FCM is proposed, accelerated bearing life tests over the whole lifetime were implemented and the whole lifetime data is applied in evaluating the IPSE - FCM technique.

2 Degradation feature extraction based on IPSE

2.1 Theory of mathematical morphological particle analysis

Mathematical morphological particle analysis (abbreviated as MMP) is an effective process widely used in image analyzing field, image particle and shape character can be analyzed with MMP by using different structuring element that has diverse size and shape.

MMP is defined as a collection of image transformation operation. The transformation expression is shown in (1) which should satisfy three prerequisites [34]. $Ψ = {ψ_{λ}}_{λ \geq 0}$ (1)

Among the four kinds of basic morphological operators including opening operator (∘), closing operator (•), corrosion operator (Θ) and dilation operator (⊕), the morphological opening operator (∘) is able to meet the three prerequisites in MMP and morphological opening operator is brought into MMP analysis.

Pattern spectrum (abbreviated as PS) is a sort of curve describing morphological particle distribution based on MMP presented above. Shape alteration information on diverse structuring element scales will be presented using PS [35]. The calculation of PS can be described as follows:

Supposing f (n) is the time domain function, g (m) is a convex structuring element. The PS of f (n) can be calculated in (2). $PS (f, λ, g) = {\begin{matrix} - \frac{dA (f \circ λ g)}{d λ} & λ \geq 0 \\ - \frac{dA (f • (- λ) g)}{d λ} & λ < 0 \end{matrix}$ (2)

Where λ denotes the analytical scale, A (f) denotes the limited area for f (n) in definition domain. The PS exists in both positive-negative regions. PS represents structuring information when defined with opening operator in positive definition region and it means the background information when defined with closing operator in negative definition region. Because the two parts provide equivalent information, the research studies on PS are mainly performed in the positive region.

For 1-D discrete signal such as rolling bearing vibrating signal, the value of λ is consecutive integer and calculation of PS in (3) is apt to be simplified as follows:

$\begin{matrix} {PS}_{+} (λ, g) & = & A [f \circ λ g - f \circ (λ + 1) g] \\ 0 \leq λ \leq λ_{max} \\ {PS}_{-} (λ, g) & = & A [f • (- λ) g - f • (- λ + 1) g] \\ λ_{min} \leq λ \leq 0 \end{matrix}$ (3)

Considering the non-expandable character for opening operator, PS for 1-D discrete signal is a set of non-negative real number sequences.

2.2 Improvement on PS

Considering the feature of different morphological operators, morphological corrosion operator is introduced into mathematical morphological particle analysis and a new spectrum named improved pattern spectrum (abbreviated as IPS) is proposed.

Supposing f (n) is the time domain function, g (m) is a convex structuring element. The improved pattern spectrum of f (n) can be calculated in (4). $IPS (f, λ, g) = {\begin{matrix} - \frac{dA (f Θ λ g)}{d λ} λ \geq 0 \\ - \frac{dA (f \oplus (- λ) g)}{d λ} λ < 0 \end{matrix}$ (4)

Where λ denotes the analytical scale, A (f) denotes the limited area for f (n) in definition domain.

As described above, the IPS also exists in both positive-negative regions. IPS represents structuring information when defined with corrosion operator in positive definition region and it means the background information when defined with dilation operator in negative definition region. Because the two parts provide equivalent information, the research studies on IPS are mainly performed in the positive region.

For 1-D discrete signal such as rolling bearing vibrating signal, the value of λ is consecutive integer. Calculation of IPS in (5) is apt to be simplified as follows:

$\begin{matrix} {IPS}_{+} (λ, g) & = & A [f Θ λ g - f Θ (λ + 1) g] \\ 0 \leq λ \leq λ_{max} \\ {IPS}_{-} (λ, g) & = & A [f \oplus (- λ) g - f \oplus (- λ + 1) g] \\ λ_{min} \leq λ \leq 0 \end{matrix}$ (5)

Considering that morphological corrosion operator has non-expandable character like morphological opening operator, so the IPS for 1-D discrete signal is a set of non-negative real number sequences.

2.3 Degradation feature extraction method of IPSE

In order to describe performance degradation degree, we propose a degradation feature extraction method named IPSE which is based on IPS curve and information entropy theory [36]. The improved pattern spectrum entropy (abbreviated as IPSE) is defined in (6). ${\begin{array}{l} I P S E = - (\sum_{i = 1}^{λ_{\max} - 1} p_{i} \cdot \ln (p_{i})) / \ln (λ_{\max} - 1) \\ p_{i} = I P S_{+} (i) / \sum_{i = 1}^{λ_{\max} - 1} I P S_{+} (j) \\ \sum_{i = 1}^{λ_{\max} - 1} p_{i} = 1 \end{array}$ (6)

Where IPS₊ (i) denotes improved mathematical morphological spectrum, λ_max refers to the largest analytical scale, IPSE is just the normalized improved mathematical morphological spectrum entropy by which the sparse degree for IPS can be reflected.

In conclusion, a performance degradation feature extraction method based on IPSE is proposed in this section, the detailed operation course is shown in Fig. 1.

Fig. 1

Flow chart of IPSE method.

As described in Fig. 1, condition monitoring signal acquisition is the basic step, and we monitor and collect signals in different degradation condition. In order to extract the performance degradation character contained in the signal, we analyze the signal and calculate IPSE for each monitoring signal, which is taken as the performance degradation parameter. After processing the whole lifetime monitoring signal respectively, we can obtain performance degradation evolution tendency described by IPSE, which is an important ocular evidence for performance degradation condition recognition.

2.4 Simulation analysis

A simulation signal is designed for imitating the performance degradation of a bearing. By using degradation feature extraction method based on IPSE, we draw a conclusion that is conducive to our view. The simulation signal is shown in (7) [37]. $\begin{matrix} x (t) & = & cos (w_{1} t) + 0.2 t^{2} cos (w_{2} t + 2) + n (t) \\ w_{1} & = & 2 π * 50 \end{matrix}$ (7) $w_{2} = 2 π * 10$

Where 0.2t² cos(w₂t + 2) denotes the fault component, cos(w₁t) denotes the inherent component, the coefficient 0.2t² is used to represent the degradation course for fault condition with time, n (t) refers to the added gaussian noise. The noise sequence is generated by function WGN and noise power is set as 0, 1, 3 and 6 dbW for simulating diverse noise intensity.

Time domain wave with 1dbW noise intensity is shown in Fig. 2. The sampling point is N = 10240 and sampling frequency is f = 1024 Hz. In order to describe ceaseless deepen performance degradation process, x (t) is divided into ten groups and each group contains 1024 sampling points respectively.

Fig. 2

Time-domain waveform of simulation signal.

Degradation feature extraction for each group simulated signal is processed. The largest analytical scale is set as λ_max = 40 and the unit structuring element is set as g = [0 0 0] to reduce calculation error for the signal. Taking noise-free (noise intensity is set as 0dbW) signal as an example, we obtained ten groups of IPS curves in Fig. 3. It is clear that with the increasing of analytical scale, the curves present monotonous descend tendency. IPS curve in different group is apt to be distinguished distinctly.

Calculating IPSE for different group simulation signal in four noise background, the result is shown in Fig. 4 which presents the relevance between IPSE and degradation degree.

Fig. 3

IPS Curve of ten groups simulation signals.

Fig. 4

IPSE of different group simulation signal in four noise background.

It is obvious that the four IPSE curves reflect an incremental tendency and the relevance between IPSE and degradation degree looks accordant and preferable. Besides, noise intensity affects the initial value of IPSE. The manifestation is apt to be explained that in initial stage of performance degradation, the amplitude is low and noise may change shape composition proportion deeply, and then the value of IPSE will change distinctly.

For contrast, common pattern spectrum and pattern spectrum entropy is put into signal analysis. First, each set of simulation signal is processed and ten groups of PS curves in different noise background can be obtained. In the processing course, The largest analytical scale is also set as λ_max = 40 and the unit structuring element is set as g = [0 0 0]. Ten groups of PS curves with noise-free environment are shown in Fig. 5. Compared with Fig. 3, it is obvious that the PS curves present a certain undulated tendency along with the increasing of analytical scale. In addition, PS amplitude contrast among curves is not consistent, for example, when analytical scale is less than 10, the higher the group, the smaller the PS amplitude. Relatively, when analytical scale is greater than 10, the higher the group, the greater the PS amplitude.

Fig. 5

Pattern Spectrum Curve of the tenth simulation data.

Calculating entropy based on pattern spectrum and pattern spectrum entropy (abbreviated as PSE) can be obtained. The relevance between PSE and degradation degree is shown in Fig. 6. It is apparent that PSE curves showed some fluctuations though the main trend is ascending. Therefore, the PSE method is weaker in stability and monotonicity compared with IPSE in Fig. 4.

Fig. 6

PSE of different group simulation signal in four noise background.

In conclusion, it is evident that IPSE has a favorable relevance with performance degradation degree, the calculation course is steady and anti-noise, performance feature extraction based on IPSE is valid.

3 Degradation condition recognition based on IPSE - FCM

In the performance degradation process for machinery equipment such as rolling bearing, it will go through a series of performance degradation conditions from normal to failure [38], and these conditions usually have fuzzy character, showing in uncertainty for number and boundary of degradation conditions. The fuzzy recognition can be settled effectively by cluster analysis method based on fuzzy mathematics theory [39]. In many clustering algorithms, Fuzzy C-means algorithm [40] has a thorough theory basis and perfect good clustering effect, non-identifying data can be trained and recognized without prior knowledge, this method has been used in many application areas successfully [41 –43], so we take FCM as the basic recognition model in our paper.

In order to enhance the completeness of the degradation feature vectors, on the basis of IPSE feature presented above, we chose another two verified degradation feature parameters including Root Mean Square (RMS as abbreviation) [44, 45] and mathematical morphological fractal dimension (MMFD as abbreviation) [46] constituting three dimension degradation feature vectors V = [IPSE, MMFD, RMS].

At present, there is no definite way for dividing the performance degradation condition, and we adopt the widely cited means in [47] that dividing the performance degradation condition into four operating states including normal, slight degradation, severe degradation and failure.

Classification coefficient and average fuzzy entropy are employed in clustering effect evaluating and the definitions are as follows [48]:

Classification coefficient (abbreviated as CC) is defined as the quadratic mean of membership matrix. The calculating formula is as follows in which μ_ik represents the membership value. The closer CC to 1, the better the clustering effect. $α = \frac{1}{n} \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{ik}^{2}$ (8)

Average fuzzy entropy (abbreviated as AFE) is defined as the information entropy of the membership distribution. The calculating formula is as follows in which μ_ik represents the membership value. The closer AFE to 0, the better the clustering effect. $β = - \frac{1}{n} \sum_{i = 1}^{c} \sum_{k = 1}^{n} u_{ik} ln u_{ik}$ (9)

Moreover, recognition rate is used in evaluating the recognition effect for unknown samples.

In conclusion, we propose a performance degradation condition recognition method based on IPSE - FCM, this method is able to extract the feature and solve the performance degradation issue for a bearing with the feature of blurred boundaries. The detailed operation course is shown in Fig. 7.

Fig. 7

Flow chart of IPSE - FCM method.

As can be seen in Fig. 7, this method consists of three steps as follows:

Condition monitoring signal acquisition

Monitoring and collecting whole lifetime vibration signal, dividing the signal into training set and testing set as follows: supposing sampling interval of original signal is I, sampling length is L, sampling time is T, so the collected whole lifetime vibration signal is a T group of disperse sequence and interval of group is I, group length is L. Carrying out re-sampling with the method of taking one separating two, we can obtain two group data with length of M, M = floor (T/2). One of the two groups is selected as training set and conducting RMS analysis on another group, choosing its subset with length of N as the testing set.

Performance degradation feature extraction

Calculating degradation feature vector V_i = [IPSE, MMFD, RMS] for each group sampling data, obtaining training set degradation feature vector V_T and testing set degradation feature vector V_C, and proceeding normalized preprocessing.

Degradation condition recognition

According to FCM theory, taking training set degradation feature vectors as data set X, the performance degradation condition is divided into four operating states including: normal, slight degradation, severe degradation and failure, the number of category is set as C = 4. Carrying through fuzzy analysis using FCM algorithm, getting clustering center for four states: Z = [Z₁, Z₂, Z₃, Z₄]. According to clustering center Z and testing set degradation feature vectors V_C, calculating subordination degree between each testing sample and clustering centers, and then setting up subordination matrix U. the degradation for testing sample is able to be recognized based on maximum subordinate degree principle.

Evaluation

Evaluating the clustering effect for training set with classification coefficient and average fuzzy entropy, moreover, evaluating degradation condition recognition effect for testing set with recognition rate.

4 Application and discussion on rolling bearing

4.1 Rolling bearing fatigue life enhancement experiment

Considering the difficulty for entire life-time vibrating signal gathering, accelerated bearing life tests over the whole lifetime were operated at the Hangzhou Bearing Test & Research Centre to prove the technique proposed above.

The experimental rig is shown in Fig. 8 which is mainly composed of rig head, drive system, loading system, electric control system. Four testing bearings were concurrently set up in bearing pedestal inside the rig head and the load is alterable provided by loading system. Four vibrating sensors and thermocouples are set up for monitoring vibration and temperature information of the four bearings respectively and one vibrating sensor is installed for monitoring the whole vibrating signal of the rig head, Sensors layout is shown in Fig. 9. The vibrating and temperature signal are gathered through DH-5920 dynamic signal acquisition instrument.

Fig. 8

ABLT-1A bearing experiment platform.

Fig. 9

Layout of sensors and thermocouples.

The load is set as overload through the loading system to shorten test time and accelerate the test process. In order to achieve a smooth loading process, at the beginning, only 5 kg weight is loaded and 5 kg is added every 15 minutes. The maximum load is set as 50 kg with which each bearing is loaded about 6.3 kN radial load (reaching 95% of the rated load).

The sampling frequency is set as 25.6 kHz and sampling time is one second with the interval time of ten minutes. The type of the tested bearings is 6204. During the fatigue life enhancement experiment, the threshold for the time-to-failure is set on the monitoring program firstly. According to experience and experiment condition, we set two criteria to stop the test automatically, the first one is 2.5 times of the RMS value of normal condition and the second criterion is the temperature of 70 degrees. We will select the faulty bearing and replace it with a new bearing after the test stop automatically.

We collected five groups of complete life-time dataset over the whole accelerated bearing life test. The final fault types include inner race fault, outer race fault and rolling element fault. One of the typical dataset is adopted in this paper in which the final failure time is at 9700th minute and fault type is inner race fault shown in Fig. 10.

Fig. 10

Live-action of inner race defect.

4.2 Degradation feature extraction based on IPSE

Carrying out re-sampling with the method of taking one separating two and getting two groups data named A and B with length of 485 respectively. Calculating degradation feature vector V_i = [IPSE, MMFD, RMS] for the data group in A and B. The training set degradation feature vector V_T and testing set degradation feature vector V_C can be obtained. In the calculating process of IPSE, the largest analytical scale is set as λ_max = 5 and the unit structuring element is chosen as g = [0 0 0]. In the calculating process of MMFD, we select unit structuring element g = [0 0 0] and the analytical scale is set as [2,4,8,16,32 , 2,4,8,16,32].

The IPS curves and IPSE are shown in Figs. 11 and 12 respectively, for contrast, PS curves and PSE are shown in Figs. 13 and 14. It is clear that the IPS curves have the same evolutionary trend with the degradation process and IPSE curve presents a monotonous ascending principle. Comparatively, the PS curves’ evolutionary trend is not consistent in different analytical scale, and the consistency and volatility for PSE curve in Fig. 14 is worse than IPSE.

Fig. 11

IPS curves in four anlytical scale.

Fig. 12

The IPSE tendency for training set.

Fig. 13

PS curves in four anlytical scale.

Fig. 14

The PSE tendency for training set.

Besides, the tendencies of RMS and MMFD for training set are shown in Figs. 15 and 16 respectively. As we can see, the RMS curve shows an ascending tendency along with the degradation process because it reflects the energy of bearing vibration. However, MMFD puts up a decreasing tendency. The reason lies that the fractal complexity gradually becomes small and the value of MMFD will decline along with the degradation process. The result is accordant with other studies and the two vectors are able to reflect the whole tendency of the bearing degradation.

Fig. 15

Feature indicator variation tendency for RMS.

Fig. 16

Feature indicator variation tendency for MMFD.

Above all, we extract three degradation features including IPSE, RMS and MMFD. Thus MMFD has an opposite tendency contrast with the others. IPSE is able to present the rule from the angle of complexity for pattern spectrum distribution. RMS is the effective indicator from the angle of energy accumulation in time domain and MMFD has the ability to reflect the fractal evolution rule from the angle of nonlinearity.

According to machinery industrial criterion, we choose RMS of vibrating signal as degradation condition separation reference [49], calculating RMS of data set B and analyzing its variation tendency, and then dividing the sequence into four stages including normal, slight degradation, severe degradation and failure state shown in Fig. 15. The first stage is normal status preserved approximately before the 360th sampling group in which the curve remains relatively stable and rolling bearing is in normal condition. The second stage is slight degradation status preserved from the 360th to the 420th sampling group in which a slight increase appears in the curve and rolling bearing is in slight degradation condition. The third stage is severe degradation status preserved from the 420th to the 460th sampling group in which a substantial change arises in the curve and rolling bearing is in severe degradation condition. The fourth stage is failure status from the 460th to the 485th sampling group when the curve keep stabilized at the extreme value, a relatively large fluctuation appears later, this may be the quick smooth effect after sudden appearance of the local abrading, and we consider the bearing is invalid in this stage.

At the same time, we select five groups of data in each degradation state from testing set and composing the testing degradation feature vectors.

In order to reduce some influence caused by feature parameters’ units and variation tendency, we operated preprocessing course including normalized and tendency conversion. The rule of normalized is shown in (10) where x represents the old value and X means the normalized value which is limited to section [0 1]. $X = \frac{x - min (x)}{max (x) - min (x)}$ (10)

Tendency conversion is operated after normalization for MMFD vectors. The conversion rule is presented in (11) . ${MMFD}_{new} = 1 - MMFD$ (11)

4.3 Degradation condition recognition based on IPSE-FCM

Programming clustering for training set feature vector V_T using FCM method. With respect to the parameters, vector V_T is set as data set X, the number of category is set as C = 4 and weighting index is set as m = 2. Conducting fuzzy analysis after parameter setting, clustering centers of four statuses and subordinate matrix U are calculated. Clustering center Z = [Z₁, Z₂, Z₃, Z₄] is shown in Table 1.

Table 1
The clustering centers of the four statuses

Clustering Center IPSE MMFD RMS

Normal Condition Z₁ 0.1333 0.0786 0.0382

Slight Degradation Condition Z₂ 0.2078 0.1483 0.0811

Severe Degradation Condition Z₃ 0.3166 0.2076 0.1393

Failure Condition Z₄ 0.7019 0.6116 0.4693

Clustering Center	IPSE	MMFD	RMS
Normal Condition Z₁	0.1333	0.0786	0.0382
Slight Degradation Condition Z₂	0.2078	0.1483	0.0811
Severe Degradation Condition Z₃	0.3166	0.2076	0.1393
Failure Condition Z₄	0.7019	0.6116	0.4693

The clustering result is able to be deduced according to subordinate matrix. Three-dimensional clustering effect is shown in Figs. 17 and 18 describes clustering effect from the perspective of IPSE. For contrast, k-means clustering method is employed with the same basic parameters and the effect is shown in Figs. 19 and 20. It is evident that the training set is clustered into four categories in the absence of prior knowledge and the four clustering conditions are mainly distributed as the time order for FCM method, thus reflecting the whole degradation process from normal to failure for rolling bearings approximately. However, there is too much overlap between normal and slight degradation state in Fig. 6 for k-means method, presenting a poor effect for degradation states clustering.

Fig. 17

Three-dimensional classification effect of FCM.

Fig. 18

Clustering effect from the perspective of IPSE.

Fig. 19

Three-dimensional classification effect of K-means.

Fig. 20

Clustering effect from the perspective of IPSE.

On the basis of clustering center Z obtained above, calculating subordinate matrix for testing set degradation feature vector V_C, and twenty groups of testing samples are able to be recognized based on maximum subordinate degree principle and the calculated subordinate matrix and recognition results marked in bold letter are shown in Table 2 which are judged by maximum subordination. It is clear that two testing samples are mistakenly recognized and the whole recognition ratio reaches 90%.

Table 2

Recognition results for testing set based on IPSE-FCM

Number	Subordinate matrix				Recognition result	Actual condition
	Z ₁	Z ₂	Z ₃	Z ₄
1	0.97	0.03	0.00	0.00	Normal	Normal
2	0.92	0.07	0.01	0.00	Normal	Normal
3	0.91	0. 09	0.00	0.00	Normal	Normal
4	0.85	0.15	0.01	0.00	Normal	Normal
5	0.92	0.06	0.01	0.01	Normal	Normal
6	0.19	0.80	0.01	0.00	Slight	Slight
7	0.10	0.88	0.01	0.01	Slight	Slight
8	0.05	0.94	0.01	0.00	Slight	Slight
9	0.11	0.87	0.02	0.00	Slight	Slight
10	0.10	0.88	0.02	0.00	Slight	Slight
11	0.03	0.05	0.68	0.24	Severe	Severe
12	0.03	0.05	0.49	0.43	Severe	Severe
13	0.04	0.05	0.44	0.47	Severe	Severe
14	0.04	0.03	0.72	0.21	Severe	Severe
15	0.02	0.03	0.17	0.78	Failure	Severe
16	0.00	0.00	0.01	0.99	Failure	Failure
17	0.03	0.04	0.11	0.82	Failure	Failure
18	0.03	0.04	0.22	0.71	Failure	Failure
19	0.02	0.04	0.88	0.06	Severe	Failure
20	0.01	0.02	0.08	0.89	Failure	Failure

Moreover, we calculate the subordinate matrix based on the k-means clustering center got above. The whole recognition ratios reaches 75% and four groups belong to normal are recognized as slight degradation state and one group in severe state is mistaken as failure condition.

The quantitative results are shown in Table 3. It is evident that the proposed method has a good effect in degradation states clustering for training set and the recognition result is acceptable.

Table 3

The quantitative results for FCM method and k-means

Technique	Clustering		Recognition
	Classification coefficient	Average fuzzy entropy	Recognition rate
FCM	0.9849	0.0239	90%
K-means	0.7357	0.4592	75%

In conclusion, the proposed IPSE feature has a stable and monotonous character in depicting the degradation process for rolling bearing and it is effective for degradation feature extraction. The employed FCM method is able to realize degradation states clustering and the recognition effect is acceptable.

5 Conclusion

A performance degradation condition recognition method based on IPSE - FCM is proposed and approved with rolling bearing whole lifetime data, it is clear that the method based on IPSE has a better effect than traditional method. The simulation and application show that IPSE has a favorable relevance with performance degradation degree. In conclusion, IPSE is brought into degradation feature extraction field and an application-oriented method for performance degradation condition recognition is proposed, the IPSE - FCM method is employed on the basis of degradation condition division that describes a standard for different condition. The analysis shows that the method has a good clustering effect and an acceptable recognition result.

However, it cannot be denied that some deficiencies exist in this technique. For example, the degradation condition division method has some subjectivity and the fuzzy character of different condition boundary is not thoroughly considered. That will be the emphasis in the following studies.

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Bearing performance degradation condition recognition based on a combination of improved pattern spectrum entropy and fuzzy C-means

Abstract

Keywords

1 Introduction

2 Degradation feature extraction based on IPSE

2.1 Theory of mathematical morphological particle analysis

4.1 Rolling bearing fatigue life enhancement experiment

Table 1 The clustering centers of the four statuses Clustering Center IPSE MMFD RMS Normal Condition Z1 0.1333 0.0786 0.0382 Slight Degradation Condition Z2 0.2078 0.1483 0.0811 Severe Degradation Condition Z3 0.3166 0.2076 0.1393 Failure Condition Z4 0.7019 0.6116 0.4693

References

Table 1
The clustering centers of the four statuses

Clustering Center IPSE MMFD RMS

Normal Condition Z₁ 0.1333 0.0786 0.0382

Slight Degradation Condition Z₂ 0.2078 0.1483 0.0811

Severe Degradation Condition Z₃ 0.3166 0.2076 0.1393

Failure Condition Z₄ 0.7019 0.6116 0.4693