Life Cycle Vibration Analysis Based on EMD of Rolling Element Bearing under ALT by Constant

Stress*

Dong Xu, Yongcheng Xu, Xun Chen & Wei Zha College of Mechatronical Engineering and Automation

National University of Defense Technology Hunan Changsha 410073, China

Kylin@ustc.edu

Xinglin Li Hangzhou Bearing Test & Research Center (HBRC)

Zhejiang Hangzhou 310022, China line 4: e-mail address if desired

Abstract—Bearing is one of critical rotation components in mechanical systems. Many grievous accidents of mechanical systems are chiefly caused by bearing failure. So it’s significant meaningful for practical application to research failure feature detection and residual life prediction of key rotation components such as bearings. This paper has designed experimental scheme of Accelerated Life Test of rolling element bearings and gained all vibration signals of bearings through their total life, from normal to fault. Then we extracted failure feature from those signals and analyzed its evolving rule in the bearing life cycle based on Empirical Mode Decomposition(EMD). Finally, we have analyzed bearing failure feature in each stage through its life cycle by comparing Frequency Spectrums with Hilbert Marginal Spectrums.

Keywords- EMD; fault diagnosis; IMF; Hilbert transform; fault prediction

I. INTRODUCTION Bearings are key rotation components in mechanism system.

Rolling element bearings are usually adopted when there is no space or cost to use hydraulic System. Because of the feature of bearings structure, rolling element bearing is one of easier fault parts in mechanism system, the total system performance is affected by the condition of rolling element bearings. In the past, the common method to deal with fault is to change a new one. Along with the increasing of system complexity more and more difficulties are appearing when bearings are assembled and disassembled. The use life and function precision of other parts in relation to bearings have been affected by assembly and un-assembly of them. For the increasing of use life and the developing of reliability and availability, bearings are the bottleneck of improving the use life of mechanism system. Moreover, other faults in mechanism system usually are felt through move parts, so we need to distinguish faults which occur in bearings or not. Hence the research of feature extraction and prediction of remains life of typical key move parts such as bearings have a critical practical value for safely working of the complex mechanism system.

At present, there are many methods for fault prediction. Such as the trend of history data fitting curve[1], time series model of AR & ARMA[2,3], ferrography analysis[4], grey

model based on grey system theory[5], learning and training of neural network[6], wavelet decomposition[7,8], fuzzy model based on expert system of fuzzy knowledge base[9], Markov model[10], machine learning of support vector machine[11], Kalman filter[12], pattern recognition in artificial intelligence[13] and so on. In addition, there are also some methods by information entropy and combined methods for prediction. But no matter any methods, statistics or signal analysis, the key is to detect Anomaly Characteristics. Only when anomaly feature is distinguished from normal, prediction can be underway.

Nowadays there many fault diagnosis methods for rolling element bearings, the core of most of above methods is means for fault diagnosis such as time series model, ferrography analysis, wavelet decomposition and so on. Because of nonlinear and nonstationary when bearings run, the performance of many fault diagnosis methods is not all roses.

In this paper, we designed experiment of Constant Stress Accelerated Life Testing for rolling element bearings and obtained vibration signals in each phases of rolling element bearings life cycle. Then we decomposed the vibration signals by the method of Empirical Mode Decomposition(EMD)[14] which was proposed by Norden E. Huang in 1998. For the adaptability of EMD, the result of decomposition is dependent on signal itself. It shows that EMD be suitable for the analysis of nonlinear and nonstationary signals. Intrinsic Mode Functions(IMFs) which are decomposed by EMD basically satisfy the demand of completeness and orthogonality.

II. EMD Vibration signals of bearings are typically nonlinear and

nonstationary. They do not satisfy the qualification of directly using Hilbert Transformation and Fourier Transformation to deal with them. EMD is a signal processing method proposed by Norden E. Huang for decomposing signals and signals decomposed are called Intrinsic Mode Functions(IMFs) who are linear and stationary and satisfy basal demands of signal processing methods. IMFs need to satisfy two conditions[15]:

(1). Number of extremum points and zero-crossing points should be equal or no more than one point;

Supported by National Natural Science Fund(50705096) and Ministries Level Pre-research Fund(06KG0187)

978-1-4244-4905-7/09/$25.00©2009 IEEE 1177

(2). Average value of envelopes of local maximum and minimum should be zero at any time.

On the practical application, the first condition is satisfied more easily than the second one. Therefore, we should define stopping criteria for the second condition.

The algorithm steps as following:

(1). Find the extremum points of the original signal ( )x N and then connect them by spline curves to obtain envelope curves. After this we can figure out average curves of the envelopes maximum and minimum;

1m

(2). Take away which are values of corresponding to

1 ( )m N

1m ( )x N from ( )x N , the remnant is , that is

1 ( )h N

1 1( ) ( ) ( )x N m N h N− = ;

(3). when satisfy the two conditions of IMFs, if not, repeat step 1 and step 2 until satisfy the two conditions of IMFs and mark it as ;

1 1( ) ( )imf N h N= 1 ( )h N

1 ( )kh N

1 1( ) ( )kimf N h N=

(4). Separate from 1 ( )imf N ( )x N and mark it as ; 1 1( ) ( ) ( )r N x N imf N= −

(5). If the spline curve of is a monotone function or it satisfy the stopping criteria, terminate decompostion process. Otherwise, repeat steps above and obtain :

1 ( )r N

2 ( ) ( )nimf N imf N

2 1 2

1

( ) ( ) ( )

( ) ( ) ( )n n n

r N r N imf N

r N r N imf N−

= −

= −

( )nr N is either a monotone function or satisfying the stopping criteria, terminate decompostion process and mark

. Then we can represent ( ) ( )nres N r N= ( )x N by

1

( ) ( ) ( )n

ii

x N imf N res=

= +∑ N .

In this paper, the method of EMD is discretized and it will be suit data process well in practical using. We use it to decompose vibration signals which were obtained in our experiment.

III. EXPERIMENTAL SCHEME Experimental object: 6307 deep groove ball bearings;

Experimental stress: radial load =11 ; rF .13kN

Experimental method: assemble four 6307 deep groove ball bearings on the common axis of rotation and then install it into ABLT-1A which is a automatic control machine of fatigue life and reliability enhancement experiment of rolling element bearings. Test bearings are assembled on the position of 14, 15 and 16 in Fig1. Then make sure the radial load of each bearing correctly by hydraulic system. Install accelerometers on appropriate positions and then start it and do our experiment following the flow chart in Fig2.

Figure 1. Structure Principle of ABLT-1A automatic control machine of

fatigue life and reliability enhancement experiment of rolling element bearings

Figure 2. Experiment and Data Acquisition Flow

Statistical data from large-sized bearing factory showed that failure process of rolling element bearings could be divided into four stages in vibration signal field. There are superfrequency stage, natural frequency stage, fault frequency stage and wide band random vibration stage. The flow of our experiment is organically integrated with failure process of bearings and we measure vibration signals through the total life cycle of bearings. Following this we can obtain the feature of vibration signals of bearings in each stage. Based on the constant stress , the machine stopped because the kurtosis was beyond the value set before on 90h. After disassembled the fault type and position we detected was fatigue failure on the inner raceway in Fig3.

rF

Figure 3. Fatigue Failure on the Inner Raceway

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n

iDoD

ioD

oiD

d

mD

Figure 4. 6307 deep groove ball bearings

IV. CHARACTERISTIC FREQUENCY CALCULATION For more effectively analyzing the physical meaning of

feature frequency in each stage, we are necessary for figuring out several more important characteristic frequency of 6307 deep groove ball bearings. In Fig4, the ball number is 8. The diameter of inner ring is 35 and the diameter of outer ring is . The width of bearing

ziD mm

oD 80mm B is . The diameter of raceway bottom of inner ring D is and the diameter of raceway bottom of outer ring is . The diameter of ball d is . The contact angle of 6307 deep groove ball bearings

21mm

io 45.006mm

oiD 72.007mm13.494mm

α is . So the average diameter . Spindle speed n is

.

0o

( ) /iD 2 58.507m io oD D m= + = m3000 / minr

Fault characteristic frequencies of inner, outer, ball and cage contact with inner and outer are respectively[16]:

8*3000 13.494(1 cos ) (1 ) 246.132*60 120 58.507i

m

zn dF HzD

α= + = + =

8*3000 13.494(1 cos ) (1 ) 153.872*60 120 58.507o

m

zn dF HzD

α= − = − =

2 2 2* 3000*58.507 13.494(1 ( ) cos ) (1 ( ) )*60 60*13.494 58.507

205.26

mb

m

n D dFd D

Hz

α= − = −

=3000 13.494(1 cos ) (1 ) 30.77

2*60 120 58.507pim

n dF HzD

α= + = + =

3000 13.494(1 cos ) (1 ) 19.232*60 120 58.507po

m

n dF HzD

α= − = − =

We don’t consider the frequency of cage contact with inner or outer because the probability that it happens is low and it belong to unexpected event.

V. EXPERIMENTAL RESULTS COMPARING

A. Experimental Results By the analysis and calculation above, we gained fault

characteristic frequencies of 6307 deep groove ball bearings. The sample frequency is 2560Hz and it twice more than the largest characteristic frequency. We used EMD to deal with vibration signals we gained before and each IMF we obtained after EMD processing satisfy the demand of orthogonality, completeness and stationarity. Therefore we can use many

analysis methods satisfied by those conditions to deal with IMFs. The physical meaning of characteristic frequencies in FFT spectrum is the value they contribute to vibration in whole time and they cannot reflect the real-time change of characteristic frequencies of signals. But the spectrum of FFT can be regarded as a reference. On the other side each IMF dealt with EMD satisfies the demand of Hilbert Transform and can be translated into the Hilbert Marginal Spectrum(HMS). There are not obvious characteristic frequencies under the condition of non-failure in HMS even power frequency or doubling. They are obvious failure feature frequencies only when there have faults and characteristic frequencies such as power frequency and doubling will be also obvious because of faults. Figures can show us the visualized trend of change. But they cannot be compared quantificationally because natural frequencies and other non-failure feature frequencies change as well as failure feature frequencies. Therefore normalization processing should be implemented for characteristic frequencies and failure feature frequencies.

B. Result Comparison Fig5 are IMFs which are the results of bearing vibration

signals measured in the beginning dealt with EMD. From the figure we can observe that the original signal was decomposed by EMD into IMFs from high frequency to low. The last one is a residual signal after EMD processing. We didn’t do it again because there is less meaning to deal with IMFs in time domain.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-505

imf1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-202

imf2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-101

imf3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-101

imf4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.5

00.5

imf5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.5

00.5

imf6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.2

00.2

imf7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.05

00.05

imf8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.05

00.05

imf9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.05

00.05

imf10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.02

00.02

imf11

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-505

signal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.05

00.05

res

Figure 5. EMD processing of vibration signals

Fig6-Fig9 are frequency domain results of IMFs dealt with FFT, and IMFs in each figure are the results of bearing vibration signals in each stage dealt with EMD in total life cycle. Fig10-Fig13 show that the results that FFT deal with IMFs which are enveloped by Hilbert Transform. Two types of data processing all can give voice to the feature of frequency in each IMF. But envelopment analysis depresses the influence of amplitude modulation for frequency modulation and the result of it will be more obvious and correct.

0 200 400 600 800 1000 1200 14000

5001000

imf1

0 200 400 600 800 1000 1200 14000

5001000

imf2

0 200 400 600 800 1000 1200 14000

200400

imf3

0 200 400 600 800 1000 1200 14000

100200

imf4

0 200 400 600 800 1000 1200 14000

200400

imf5

0 200 400 600 800 1000 1200 14000

10002000

sign

al

0 200 400 600 800 1000 1200 14000

50

res

Figure 6. IMFs in the beginning dealt with FFT

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0 200 400 600 800 1000 1200 14000

5001000

imf1

0 200 400 600 800 1000 1200 14000

5001000

imf2

0 200 400 600 800 1000 1200 14000

500

imf3

0 200 400 600 800 1000 1200 14000

200400

imf4

0 200 400 600 800 1000 1200 14000

100200

imf5

0 200 400 600 800 1000 1200 14000

10002000

sign

al

0 200 400 600 800 1000 1200 14000

50100

res

Figure 7. IMFs on19min past 41h dealt with FFT

0 200 400 600 800 1000 1200 14000

200400

imf1

0 200 400 600 800 1000 1200 14000

200400

imf2

0 200 400 600 800 1000 1200 14000

500

imf3

0 200 400 600 800 1000 1200 14000

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imf4

0 200 400 600 800 1000 1200 14000

500

imf5

0 200 400 600 800 1000 1200 14000

5001000

sign

al

0 200 400 600 800 1000 1200 14000

50100

res

Figure 8. IMFs on39min past 65h dealt with FFT

0 200 400 600 800 1000 1200 14000

500010000

imf1

0 200 400 600 800 1000 1200 14000

500010000

imf2

0 200 400 600 800 1000 1200 14000

20004000

imf3

0 200 400 600 800 1000 1200 14000

10002000

imf4

0 200 400 600 800 1000 1200 14000

500

imf5

0 200 400 600 800 1000 1200 14000

500010000

sign

al

0 200 400 600 800 1000 1200 14000

5001000

res

Figure 9. IMFs on27min past 89h dealt with FFT

The Envelopment Spectrums(ES) represent more than failure feature frequencies because of characterization of IMFs and Hilbert Transform.

0 200 400 600 800 1000 1200 14000

1000

2000

imf1

0 200 400 600 800 1000 1200 14000

1000

2000

imf2

0 200 400 600 800 1000 1200 14000

500

1000

imf3

0 200 400 600 800 1000 1200 14000

200

400

imf4

0 200 400 600 800 1000 1200 14000

500

1000

imf5

Figure 10. ES of IMFs in the beginning

0 200 400 600 800 1000 1200 14000

1000

2000

imf1

0 200 400 600 800 1000 1200 14000

1000

2000

imf2

0 200 400 600 800 1000 1200 14000

500

1000

imf3

0 200 400 600 800 1000 1200 14000

500

1000

imf4

0 200 400 600 800 1000 1200 14000

200

400

imf5

Figure 11. ES of IMFs on 19min past 41h

0 200 400 600 800 1000 1200 14000

500

1000

imf1

0 200 400 600 800 1000 1200 14000

500

imf2

0 200 400 600 800 1000 1200 14000

500

1000

imf3

0 200 400 600 800 1000 1200 14000

500

imf4

0 200 400 600 800 1000 1200 14000

500

1000

imf5

Figure 12. ES of IMFs on 39min past 65h

0 200 400 600 800 1000 1200 1400012

x 104

imf1

0 200 400 600 800 1000 1200 1400012

x 104

imf2

0 200 400 600 800 1000 1200 14000

500010000

imf3

0 200 400 600 800 1000 1200 14000

20004000

imf4

0 200 400 600 800 1000 1200 14000

5001000

imf5

Figure 13. ES of IMFs on 27min past 89h

Characteristic frequencies are everydayness and the bearing is health in Fig6 and in Fig10. A lot of frequencies and harmonics appear around 1000Hz in IMF1 of Fig7 and Fig11 and the value of them cannot be recognized. The bearing is under the condition of initial damage. But these cannot show that the failure feature frequency of initial damage is about 1000Hz, it may be more high frequency than 1000Hz and 1000Hz may be only one of its harmonics. In Fig8 and Fig12, the value of amplitude in IMF1 is contracted and many harmonics appear around 246Hz which is the failure feature frequency of inner raceway we have calculated before. In Fig9 and Fig13, characteristic frequencies suddenly are very obvious in each IMF of Fig9 and Fig13. There are obvious failure feature frequencies in IMF1, IMF2 and IMF3. The machine stopped because the value of kurtosis was beyond stopping criteria.

0 200 400 600 800 1000 1200 14000

50

100

150

200

250

300

350Hilbert边际谱采样频率Fs=2560Hz

频率(Hz)

幅值

Failure Frequency Point

Figure 14. FigHMS in the beginning

0 200 400 600 800 1000 1200 14000

100

200

300

400

500

600

700

800Hilbert边际谱采样频率Fs=2560Hz

频率(Hz)

幅值

Failure Frequency Point

Figure 15. HMS on 19min past 41h

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0 200 400 600 800 1000 1200 14000

100

200

300

400

500

600

700Hilbert边际谱采样频率Fs=2560Hz

频率(Hz)

Figure 16. HMS on 39min past 65h

0 200 400 600 800 1000 1200 14000

2000

4000

6000

8000

10000

12000

14000Hilbert边际谱采样频率Fs=2560Hz

频率(Hz)

Figure 17. HMS on 27min past 89h

Fig14-Fig17 are Hilbert Marginal Spectrums of bearing vibration signals in each stage dealt with EMD in total life cycle. This method highlights characteristic frequencies when some faults arise. In Fig14, the failure feature frequency is submerged by other frequencies and there are not obvious feature frequency components. In Fig15 and Fig16, the failure

feature frequency of bearing is relatively obvious and it is in a stabilizing phase. Fig17 represents the HMS of bearing fault in the later period. Feature frequencies in HMS are very obvious. Along with the fault being aggravated, feature frequency is realization of periodicity and the amplitude of it is larger and larger. Therefore it shoots way out of strong noise frequencies and the period looks like piecewise linear and stationary.

In the later period of bearing fatigue failure, the failure feature frequency is more and more obvious. The viscosity of lubricating oil in degrading and the amplitude value of fault is growing and the temperature around the fault bearing is rising speedly. The contact area of failure surface is spreading quickly. All these will lead the bearing to accelerating damage. For analyzing the vibration signal of bearing quantificationally, the vibration signal in the beginning and on 27min past 89h are normalized, then deal with them and the result is showed in Table1.

By observing the data in the table, we can find that values around 246Hz are very different in the beginning and on 27min past 89h. It shows that the bearing failure feature frequency is about 246Hz. By comparing with failure feature frequencies we have calculated before, we know the fault type is inner raceway fatigue failure and the result we observe after disassembling has proved it. Furthermore, the amplitude value of each later characteristic frequency is larger than the former, this shows that amplitude values of natural frequencies and their relative frequencies are increasing with fault growing. There are many harmonic frequencies appear in the stage of fault growing.

Failure Frequency Point

Failure Frequency Point

TABLE I. NORMALIZAITON OF VIBRATION SIGNALS IN THE BEGINNING AND ON 27MIN PAST 89H

The Beginning 27min past 89h Frequency Amplitude Amplitude

/Maximum Amplitude

/Mean Amplitude Amplitude /Maximum

Amplitude /Mean

49.75 96.5 99.5 146.25 149.25 152.25 196 198.75

158.3 41.6 57.5 52.8 106.9 736.6 140.1 167.0

0.215 0.058 0.080 0.070 0.145 1.000 0.190 0.227

14.77 3.89 5.37 4.92 9.97 68.71 13.07 15.60

746.0 252.4 503.5 965.0 412.5 970.5 1725.2 468.6

0.270 0.091 0.182 0.349 0.149 0.352 0.625 0.170

24.03 8.13 16.22 31.09 13.29 31.28 55.60 15.10

242.75 245.75 248.5

28.0 89.8 29.4

0.038 0.122 0.040

2.61 8.38 2.75

383.5 581.0 700.0

0.139 0.211 0.253

12.35 18.72 22.60

295.5 298.25 304.25 305 342.25 345.25 392 395 441.75

114.0 64.9 56.7 653.8 27.5 53.1 42.3 38.3 27.2

0.155 0.088 0.077 0.888 0.037 0.072 0.057 0.052 0.037

10.63 6.06 5.29 60.99 2.54 4.95 3.94 3.57 2.54

2761.8 350.2 468.0 436.0 1078.5 1921.5 1440.0 280.0 203.0

1.000 0.125 0.170 0.158 0.391 0.696 0.521 0.102 0.073

89.01 11.28 15.08 14.05 34.75 61.92 46.40 9.11 6.53

Attention: frequencies we compare are less than 450Hz.

VI. CONCLUSIONS This paper has designed experimental scheme of total life

cycle of bearings under constant stress and dealt with the vibration signal we gained from experiment. Then we analyzed IMFs which were the results of vibration signals dealt with EMD.

By dealt with vibration signals by HHT and analyzing the results, we can summarize several conclusions as following:

(1). By analyzing EMD and discretization, we know the second condition of IMF is very harsh. So we must choose proper stopping criteria in practical using. Otherwise the loop will be iterated many times or all along and we cannot get correct IMFs or the result of decomposition is a constant

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amplitude harmonic frequency, and it loses the physical meaning of the original signal.

(2). There are usually several neighboring IMFs that contain feature frequencies because of nonlinear and nonstationary, but they must not appear in nonadjacent IMFs at the same time.

(3). the vibration signal in the beginning and on 27min past 89h are normalized and we observed the failure feature frequency and knew that the fault of the bearing is inner raceway fatigue failure by comparing the feature frequency with those we have calculated before.

REFERENCES

[1] Yun Jiang, Xue Yang & Qiming Ruan(2001). Review of prediction methods of fault rules and trend of machine, Chinese Journal of Mechatronics, No.3, pp.14-17.

[2] Zhongsheng Chen, Yongmin Yang & Zheng Hu(2005). Early Fault Forecasting based on Almost Periodic Time-Varying Autoregressive Model, Chinese Journal of Mechanical Engineering, No.1, pp.184-188.

[3] Jie Han & Ruilin Zhang(1997). Methods of Failure Mechanism and Diagnosis in rotation machine. Beijing: Publishing Company of Mechanical Industry.

[4] Dexiu Tang(2007). Application of Ferrography in the Research of Vehicle Failure Prediction, Chinese Journal of Southwest University(Natural Science), No.1, pp.111-115.

[5] Juhua Chen & Yizhi Guo(2002). Application of GM Fuzzy Optimal Method in Fault Forecasting for a Few Sample Mechanical Systems, Chinese Journal of China Mechanical Engineering, No.19, pp.1658-1660.

[6] Zhiying Lu, Zhichao Zhao & Wei Hao(2004). Multi-model Ensemble Forecast Method Based on ANN, Chinese Journal of Computer Applications, No.4, pp.50-51.

[7] Junsheng Cheng, Dejie Yu, Yu Yang & Qianwang Deng(2004). Application of Scale-Wavelet Power Spectrum to Fault Diagnosis of Rolling Bearings, Chinese Journal of Vibration Engineering, No.1, pp.82-85.

[8] Zheng Li, Zhengjia He, Yangyang Zi &Yangxue Wang(2008). Customized Wavelet Denoising Using Intra- and Inter-scale Dependency for Bearing Fault Detection, Journal of Sound and Vibration. No.2008, pp.342-359.

[9] Qiang Wang, Jian Zheng, Junqi Qin & Changzhi Jia(2002). The Modeling Method Study of Fault Fuzzy Forecast System, Chinese Journal of Computer Measurement & Control, No.1, pp. 23-25.

[10] Yuanchao Li, Xiaodong Wu, Zhengquan Li, Shuangquan Liu & Yuping Wang(2006).Application of Gray Markov Prediction Model in Prediction of Progressive-cavity Pumps Lifetime, Chinese Journal of Natural Gas Industry, No.10, pp.92-94.

[11] Dejie Yu, Miaofeng Chen, Junsheng Cheng & Yu Yang(2006). Fault Diagnosis Approach for Rotor Systems Based on Support Vector Machine Predictive Model, Chinese Journal of China Mechanical Engineering, No.7, pp.696-699.

[12] Minze Chen & Donghua Zhou(2003). Fault Prediction Techniques for Dynamic Systems, Chinese Journal of Control Theory & Applications, No.6, pp.819-823.

[13] Fengyu Wang, Dongxiang Jiang, Ji Han & Haijun Sun(2000). Application of Pattern Recognition on Faults Prediction, Chinese Journal of Vibration Engineering, No.5, pp.669-672.

[14] Norden E. Huang, Zheng Shen & Steven R. Long(1998). The Empirical Decomposition and the Hilbert Spectrum for Nonlinear and Non-stationary Time Series Analysis, Proceedings of the Royal Society of London, A.pp.903-995.

[15] Gabriel Rilling, Patrick Flandrin & Paulo Goncalvès(2003). On Empirical Mode Decomposition and Its Algorithms, Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image, Grado. Italy.

[16] Zejiu Liu, Shiquan He & Xinglin Li(2006). Application Handbook of Rolling Element Bearing. Beijing: Publishing Company of Mechanical Industry.

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