Faults Diagnosis of Rolling Bearing Based on Empirical Mode Decomposition: Optimized Threshold De-noising Method

Faults Diagnosis of Rolling Bearing Based on Empirical Mode Decomposition:

Optimized Threshold De-noising Method

Abstract— The faults of rolling bearings frequently occur in rotary machinery, therefore the rolling bearings fault diagnosis is a very important research project. The vibration signal is usually noisy and the information about the fault in the early stage of its development can be lost. A threshold de-noising method based on Empirical Mode Decomposition (EMD) is presented in this paper. Firstly, the signal is decomposed into a number of IMFs using the EMD decomposition .Secondly the algorithm based on the energy to determine the trip point is designed for IMF selection, then, by comparing the energy of the selected IMFs with excluded IMFs, singular selected IMFs are dealt with soft threshold function, and finally the de-noised signal is obtained by summing up the selected IMFs, it is proved that the best IMFs can be summed up and properly de-noised by the proposed method .The results show the effectiveness of the proposed technique in revealing the bearing fault impulses and its periodicity for both simulated and real rolling bearing vibration signals.

Keywords— bearing fault detection; EMD; threshold De- noising; IMF selection; Singular IMF.

I. INTRODUCTION

Mechanical system having rotating components such as bearings and/or gears provides a good example of condition monitoring. Specifically, bearing systems experience overload, misalignment, fatigue, looseness, and contamination, which can become major causes of cracks or spalls on the surface of the inner or outer-race. Typically, fault-induced signals from rotating machinery involve

periodical impulses that are masked by environmental noises, along with the high frequency dynamics of structural components of rotating components [1]. The spectral signatures of good and defective bearings have been ascertained, and widely explored in a variety of literatures.

Empirical mode decomposition (EMD) is a recently proposed method to analyze non-linear and non-stationary time series by decomposing them into intrinsic mode functions (IMFs)[8]. One of the most popular applications of such a method is noise elimination. EMD based de-noising methods require a robust threshold to determine which IMFs are noise related components [6].

In this study, we propose optimized threshold de-noising method based on EMD [9]. Firstly, the algorithm based on the energy to determine the trip point is designed for IMF selection, then, by comparing the energy of the selected IMFs with excluded IMFs, singular selected IMFs are dealt with soft threshold function, and finally the de-noised signal is obtained by summing up the selected IMFs, it is proved that the best IMFs can be summed up and properly de-noised by the proposed method.

II. THE EMD ALGORITHME

The EMD method is a sifting process based on the local oscillation of the signal. By this method, the signal will be decomposed into a finite sum of IMFs and a residue, where the IMF should satisfy [5]:

1) In the whole data set, the number of extrema and the number of zero-crossings must either equal or differ at most by one.

Rabah ABDELKADER^1,2,Ziane DEROUICHE^{1 ,}ZERGOUG Mourad²

1Laboratoire Signaux, Systèmes et Données.Département électronique. Faculté de Génie Electrique. USTO. Alegria

2Research Center in Industrial Technologies CRTI P.O.Box 64, Cheraga 16014, Algiers, Algeria Email : r.abdelkader@crti.dz

2) At any point, the mean value of the envelope defined by local maxima and the envelope defined by the local minima is zero.

For a given signal

x

, EMD algorithm can be described as follows:

1) Identify all the local extrema, and then connect all the local maxima (resp. minima) by a cubic spline line as the upper envelope.

2) Compute the envelope mean

^{m t}  

3) Calculate difference between the signal

^{x t}  

^and

m

1 ,is the first component,

h

₁^:

 

1 1

x t  m  h

(1) If

h

₁ is an IMF, then

h

₁ is the first IMF component of

^{x t}  

^.

4) If

h

₁ is not an IMF,

h

₁ is treated as the original signal.

5) Repeat the steps (1)–(3) ; then treat

h

₁as the

x

, get

h

1,1^.If

h

_1,1is an IMF, stop; Otherwise, continue:

1 1,1 1,1

1, 1 1, 1,

...

k k k

h m h

h

_

m h

 

 

(2) The stopping condition is: (SD) is calculated from the two consecutive sifting results, namely

h

i_1

  n

and

h n

i

 

as,

   

 

2 1

2

0 1

N i i

n i

h n h n

SD h n



 

  

(3) When the value of SD resides within a predefined range, the sifting process is terminated, and

h n

i

 

is termed as

c n

1

 

.

6) Once

c n

1

 

is obtained, it is then subtracted from the original data to get a residue

r n

1

 

,

     

1 1

r n  x n  c n

(4) The residue

r n

1

 

is treated as a new signal, and sifting process as described above is carried out on

r n

1

 

to obtain the next residue signal

r n

2

 

. Therefore, the residue signal thus obtained can be expressed in general as,

 

1

   

j j j

r n  r

_

n  c n

(5)

If

r n

j

 

becomes a constant or monotonic function, the process of decomposing the signal into IMFs is terminated.

The whole process is ended with

r

_Nhas at most one extremum, thus, we have:

1 N

j N

j

x c r



  

(6) The signal

x

is decomposed into a finite sum of IMFs and a residue.

A. IMF Selection

The energy density of IMF and its corresponding averaged period are defined as follows [9]:

   

²

1

1

^N

n n

i

E c i

N



 

(7)

max n

T N

 N

(8)

E T

n n

 const

(9) Where

c

_n ^{is the}

n

-th IMF,

E

_nis the energy density, and

N

is the length of the data;

T

_nis the average period,

N

_max ^is

the maximum numbers of

c

_n^.

The product of the energy density and the corresponding average period is a constant.

The

E

_n,

T

_n^and

E T

_{n n}of each IMF are calculated in accordance with equations (7)-(9). Because the

E T

_{n n}of white noise is a constant and the high-frequency IMF is usually the noise. So there will be a trip point in the curve of

E T

_{n n}.Excluding all IMFs before the trip point, the summation of left IMFs are the de-noised signal [6].

B. De-noising Method

The signal is over decomposes because the noise introduce some false local extrema and the sampling frequency is 48000Hz, we have taken a very large sampling frequency to get many IMFs which represent only the noise. For noise reduction, the EMD can be combined with a filtering method such as Savitzky-Golay smoothing [6] or nonlinear transformation such as the soft-thresholding [3].

C. EMD-Soft thresholding

Smooth version of the input data can be obtained by thresholding the IMFs before signal reconstruction. The



_jis

the threshold value and this can be determined in many ways [7]. Donoho and Johnstone [8] proposed a universal threshold for removing added Gaussian noise



_jgiven by:

 

ˆ 2 log

j j

L

  

(10)

ˆ

_j

MAD

_j

0.6745

 

(11)

Where

 ˆ

_j is the noise level of the

j

-th IMF.

MAD

jrepresents the absolute median deviation of the

j

-th IMF and is defined by:

     

 

j j j

MAD median IM t median IM t

(12) Instead of using a global thresholding, level-dependent thresholding uses a set of thresholds, one for each IMF (scale level). The soft-thresholding method shrinks the IMF samples by



_j towards zero as follows [3] .

 

   

 

   

ˆ 0

j j j j

j j j

j j j j

IM t if IM t

c t if IM t

IM t if IM t

 



 

  

 

   



(13)

D. The Problem

As can be seen from [9], the averages

E T

_{n n}of IMFs under different noise intensity are difference. View from the portrait, the average

E T

_{n n}of IMFs increases totally with the increased noise intensity, and there are individual circumstances, the average

E T

_{n n}values don‘t obey the law of gradual increase with the decomposition numbers, there are singular IMFs. The judgment standard of trip point is not given. So there are two problems need to be solved in practical applications:

1) The quantitative indicators of the trip point under different noise intensity.

2) The processing method of the singular IMFs.

III. OPTIMIZD THRESHOLD DENOISING METHOD Aiming to the two problems mentioned above, an optimized threshold de-noising method based on EMD is put forward. The specific steps of the method are as follows [9]:

1) Obtain the IMFs by EMD.

2) Calculate the energy density, the average period and the product for each IMF by using the ―(7)‖-―(9)‖.

3) Determine the trip point. Definite

 

1 1 1, 2,3...., 1

n n n n n

Q E T_{ } E T n N (14) Where

N

is the total number of IMFs. After large

simulation experiments, the first IMF satisfying the condition

Q

_n

 2

is considered as the trip point.

4) Calculate the average value of all IMFs before the trip point:

1 n

ave n n

i

ET mean E T



 

  

  

(15) 5) Definite the IMF component meeting the condition

E T

m m

 ^2* ET

ave

 m   n ^1, n  ^2,..., N  ¹ 

as the singular IMF. If there is a singular IMF existing, do the soft threshold function [10]. The threshold is estimated by the following formula:

 

 

 

 

0.6745 * 2 ln ln 1

j j

median abs IMF L

  j



(16)

Where

L

the length of the signal is,

j

means the

j

-th IMF.

6) Summing up all IMFs after the

n

-th IMF and the

―trend‖ component.

IV. EXPERIMENTAL VALIDATION

The practical experimental data comes from Case Western Reserve University bearing Data center [14]. Bearing test rig device is shown in Fig 1. The vibration signals measured from three bearings under three different conditions:

Fig 2-6 illustrates the vibrations signals and its IMFs of the bearing with outer race defect and inner race defect. Fig.

7-8 shows the filtered signals obtained by the proposed method and the envelope spectra for a healthy bearing, bearing with outer and inner race defects, are shown in Fig 9- 11 respectively. Envelope spectra (Figs. 10 and 11) show outer and inner race defect frequencies at 107and 162 Hz respectively.

In order to evaluate the efficacy of EMD based on optimized threshold de-noising method along with kurtosis and crest factor values, kurtosis value has been widely used for the detection of faults in rotating machines as well as cutting tool wear monitoring in the machining process.

Kurtosis increases as spikiness of vibration increases.

However, as the damage increases the vibration signal becomes random and the values of kurtosis reduce to more normal bearing-like levels. Thus, the kurtosis measurement reduces ability to detect the bearing defect at later stages of damage development [3].

Table 1-2 shows a comparison of the change in scalar indicator (kurtosis, crest factor, RMS, maximum value) before and after application of proposed method. For the bearing with an outer race defect, the kurtosis of the original signal is 7.40;

the kurtosis of the de-noising signal is 40.07.Similar observations can be made from the results corresponding to the bearings with an inner race defect, the kurtosis value is 6.80 for noisy signal and 41.10for filtered signal. The value of crest factor is close to 13.42 for inner race fault and 11.72for outer race fault.

In summary, the results demonstrate that the proposed method can recover faulty bearing signals from large noise and increase the kurtosis of the analyzed signal to a remarkable degree. At the same time, the resultant signals preserve the important features of faulty bearings so that the impacts caused by the faulty elements of the bearings can be easily determined.

TABLE I. Signal with Inner Race Defect Indicator

scalar

Original signal

De-noised signal

kurtosis 6.8086 41.1085 crest factor 4.9678 13.4287

TABLE II. Signal With Outer Race Defect Indicator

scalar

Original signal

De-noised signal

kurtosis 7.4001 40.0746

crest factor 5.6022 11.7299

Fig. 1. The practical experimental data test system diagram.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

original signal no defect

time (s)

magnitude

Fig. 2. Signal vibration with no-defect

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-6 -4 -2 0 2 4 6 8

original signal with outer race defect

time (s)

magnitude

Fig.3. The original signal with outer race defect

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

original signal with inner race defect

time (s)

magnitude

Fig .4. The original signal with inner race defect

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-5 0 5

IMF 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-2 0 2

IMF 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.5 0 0.5

IMF 3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.2 0 0.2

IMF 4

time (s)

Fig .5. The first four IMFs of signal with outer race defect

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-2 0 2

IMF 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-1 0 1

IMF 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-1 0 1

IMF 3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.5 0 0.5

IMF 4

time (s)

Fig .6.The first four IMFs of signal with inner race defect

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

denoising signal with outer race defect

time(s)

magnitude

Fig .7. The de-noised signal with outer race defect

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

denoising signal with inner race defect

time(s)

magnitude

Fig .8. The de-noised signal with inner race defect

0 100 200 300 400 500 600 700 800 900 1000

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

frequence(Hz)

Magnitude

signal no defect

Fig .9. The envelope of signal no defect

200 400 600 800 1000 1200 1400 1600 1800

0 5 10 15 20 25

frequence(Hz)

Magnitude

signal with outer race defect

BPFO=107Hz

Fig .10. The envelope of signal with outer race defect

200 400 600 800 1000 1200 1400 1600 1800 2000 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

frequence(Hz)

Magnitude

signal with inner race defect

BPFI=162 Hz

Fig 11. The envelope of signal with inner race defect I. CONCLUSION

In this paper, the empirical mode decomposition based on the optimized threshold de-noising method is proposed for bearing vibration signal. The results presented in this study demonstrate that the EMD de-noising based on optimized threshold can be used to identify early damage in bearing element. We note that the proper choice of the thersholding parameters in the post processing is important .This method is very simple, and it is capable of reducing noise and preserving signal information.

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