A intelligent rolling bearing fault diagnosis method is proposed on Empirical Mode Decomposition (EMD) – Teager Energy Operator (TEO) and Mahalanobis distance. EMD can adaptively decompose vibration signal into a series of Intrinsic Mode Functions (IMFs), that is, zero mean mono-component AM-FM signal. TEO can estimate the total mechanical energy required to generate signals, so it has good time resolution and self-adaptive ability to the transient of the signal, which shows the advantage to detect the signal impact characteristics. With regards to the impulse feature of the bearing fault vibration signals, TEO can be used to detect cyclical impulse characteristic caused by bearing failure, gain the instantaneous amplitude spectrum of each IMF component, then identify the characteristic frequency of the interesting and single IMF component in bearing faults by means of Teager energy spectrum. The amplitude of the Teager energy spectrum in inner race fault frequency, outer fault frequency and the ratio of the energy of the resonance frequency to the total energy were extracted as the feature vectors, which were used as training samples and test samples separately for fault diagnosis. Then the Mahalanobis distances between the real measure and different type overalls of fault sample are calculated to classify the real condition of rolling bearing. Experimental results was concluded that this method can accurately identify and diagnose different fault types of rolling bearing

Chen  Lu

Hongmei Liu

Jiameng Hu

Jing Wang

English

Vibroengineering PROCEDIA

  ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533 59 
Rolling Bearing Fault Diagnosis Based on EMD-TEO and 
Mahalanobis Distance 
Lu Chen, Hu Jiameng, Liu Hongmei and Wang Jing 
ROLLING BEARING FAULT DIAGNOSIS BASED ON EMD-TEO AND MAHALANOBIS DISTANCE.  
LU CHEN, HU JIAMENG, LIU HONGMEI AND WANG JING 
Lu Chen1, Hu Jiameng2, Liu Hongmei3 and Wang Jing4 
1, 2 School of Reliability and Systems Engineering, Beihang University, Beijing, 100191, China 
1, 2, 3, 4 Science & Technology on Reliability & Environmental Engineering Laboratory, Beijing 
100191, China 
 
E-mail: hujiamengbrave@163.com 
Abstract. A intelligent rolling bearing fault diagnosis method is proposed on Empirical Mode Decomposition 
(EMD) – Teager Energy Operator (TEO) and Mahalanobis distance. EMD can adaptively decompose vibration 
signal into a series of Intrinsic Mode Functions (IMFs), that is, zero mean mono-component AM-FM signal. 
TEO can estimate the total mechanical energy required to generate signals, so it has good time resolution and 
self-adaptive ability to the transient of the signal, which shows the advantage to detect the signal impact 
characteristics. With regards to the impulse feature of the bearing fault vibration signals, TEO can be used to 
detect cyclical impulse characteristic caused by bearing failure, gain the instantaneous amplitude spectrum of 
each IMF component, then identify the characteristic frequency of the interesting and single IMF component in 
bearing faults by means of Teager energy spectrum. The amplitude of the Teager energy spectrum in inner race 
fault frequency, outer fault frequency and the ratio of the energy of the resonance frequency to the total energy 
were extracted as the feature vectors, which were used as training samples and test samples separately for fault 
diagnosis. Then the Mahalanobis distances between the real measure and different type overalls of fault sample 
are calculated to classify the real condition of rolling bearing. Experimental results was concluded that this 
method can accurately identify and diagnose different fault types of rolling bearing. 
1. Introduction 
Bearings are an important element in a variety of industrial applications such as machine tool and 
gearbox. An unexpected failure of the bearing may cause significant economic losses. For that reason, 
faults detection of bearing has been the subject of intensive research [1]. 
Signals generated by the rolling bearing failure often contain a large number of non-stationary 
components. The global frequency domain information can be obtained by fourier transform for 
signals containing non-stationary composition, and the fault information is difficult to be extracted 
effectively from spectrum overwhelmed by noise. Local information of non-stationary signals can be 
extracted by wavelet transform. But the information of high frequency is lost which is caused by lack 
of the decomposition high-frequency part, so that the fault characteristics of high frequency is difficult 
to be extracted. Wavelet packet transform makes up for the lack of wavelet transform for high-
frequency decomposition. It can have a whole multi-level decomposition in the full-band signal, 
coupled with good time-frequency characteristics. However, in terms of feature extraction of low 
signal to noise ratio of the vibration signal, it has no breakthrough.  
In this paper, we present a intelligent diagnosis method based on Empirical Mode Decomposition 
(EMD) Teager Energy Operator (TEO) and Mahalanobis distance. EMD is a fundamentally new 
approach to the decomposition of nonlinear and nonstationary signal presented originally by Huang 
[2]. EMD can decompose adaptively signal into a finite and usually small number of Intrinsic Mode 
Functions (IMFs) which are usually monocomponent Amplitude Modulation-Frequency Modulation 
(AM-FM) signals. Signal components with amplitude modulation characteristics exist in high 
frequencies, therefore, IMFs including high frequency are selected for further analysis. In this paper, 
the appropriate and interested IMF1 is selected to gain feature vectors for the input of TEO because 
                                                     
2 To whom any correspondence should be addressed. 
ROLLING BEARING FAULT DIAGNOSIS BASED ON EMD-TEO AND MAHALANOBIS DISTANCE.  
LU CHEN, HU JIAMENG, LIU HONGMEI AND WANG JING 
60 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533  
IMF1 obtain more information of signal characteristics. 
TEO is a nonlinear operator, which can track the energy of the interesting Intrinsic Mode Functions 
(IMFs), gain the instantaneous amplitude spectrum of each IMF component and identify the 
characteristic frequency of bearing faults by means of Teager energy spectrum [3]. 
Moreover, the feature frequencies of the inner race and outer race faults, the ratio of the resonance 
frequency band energy to the total energy in the Teager energy spectrum were extracted as feature 
vectors, which were used as training samples and test samples separately for fault diagnosis. TEO also 
enable the extraction of bearing fault feature frequencies for fault detection. In end, the extracted 
features are classified using the Mahalanobis distances for fault diagnosis of rolling bearing. The result 
shows that the proposed method is effective. 
2. Methodology 
2.1. EMD Algorithm 
Norden E. Huang and his colleagues developed a method of empirical mode decomposition (EMD) in 
1998 [4]. This method decomposes signals into IMFs. The IMF must satisfy two requirements as 
below: In a series of data, the number of extreme value points and the zero-crossing point must be 
equal or differ by one point at most. In any point of time axis, the average value of the upper and 
lower envelope is zero. 
The basic algorithm of EMD is as follows, therein ( )x t  is the original signal sequence  
(1) Initialize ( )h x t← . 
(2) Search all the maximum and minimum values of h. 
(3) The upper and lower envelope is fitted by cubic-spline, and the mean value m of the upper and 
lower envelope is calculated. 
(4) The mean value of envelope is subtracted from the sequence h, h h m← − . 
(5) Repeat step 2) ~ step 4) ,until h is an IMF. 
(6) I I h← − , repeat the steps from (1) to (5) until the last sequence m cannot be decomposed 
anymore. 
The result of EMD can be written as below: 
  
1
( )=
n
i n
i
x t IMF r
=
+∑  (1) 
2.2. Teager Energy Operator (TEO) 
The TEO, (.)ψ  is defined for continuous-time signal ( )x t  as[5]: 
  [ ] [ ]
2
( ) ( ) ( ) ( ),x t x t x t x tψ = −ɺ ɺɺ  (2) 
where ( )x tɺ  and ( )x tɺɺ  represent the first and second time derivatives of ( )x t  respectively. In fact, the 
output of the Teager Energy Operator tracks the total energy required to produce a signal.  
  ( ) ( )cosx t A tω ψ= +  (3) 
where A represents the amplitude of the  vibration, ψ  represents the initial phase. 
At any time the mechanical energy of the vibration system is the sum of the potential energy of the 
spring and the kinetic energy of the mass: 
  ( ) ( )
2 21 1
2 2
E k x t m x t   = +   ɺ  (4) 
The total energy can be written as 
ROLLING BEARING FAULT DIAGNOSIS BASED ON EMD-TEO AND MAHALANOBIS DISTANCE.  
LU CHEN, HU JIAMENG, LIU HONGMEI AND WANG JING 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533 61 
  2 2
1
2
E mA ω=  (5) 
Equation (5) shows the instantaneous total energy of the harmonic vibration and the square of the 
amplitude is inversely proportional to the square of the frequency. From the above equations, we can 
get:  
  ( ) ( ) 2 2cosx t A t Aψ ψ ω ψ ω   = + =       (6) 
Contrasting to (5) and (6), there is only a difference of a constant m/2 between the output of the 
TEO and the instantaneous total energy of simple harmonic motion. 
In the discrete case, the time derivatives may be approximated by time differences. 
The discrete-time counterpart of TEO becomes [6]: 
  [ ] [ ]
2
( ) ( ) ( 1) ( 1)x n x n x n x nψ = − − +  (7) 
For the discrete-time signal, TEO only need 3 samples to calculate the energy of the signal at any 
time. Therefore, TEO has a good time resolution for the instantaneous change of the signal, and can 
detect the transient components of the signal. 
2.3. Diagnosis approach for rolling bearing 
 
 
Figure 1. Flow chart of the proposed method. 
 
The fault diagnosis method for the rolling bearing is as shown in Figure1: 
(1) Each signal is decomposed by EMD, and a finite number of IMF components can be obtained. 
Select the appropriate and interesting Intrinsic Mode Functions (IMFs) for the input of TEO. 
(2) TEO gain the instantaneous amplitude spectrum of each IMF component and identify the 
characteristic frequency of bearing faults by means of Teager energy spectrum for fault detection. 
Moreover, the feature frequencies of the inner race and outer race faults, the ratio of the resonance 
frequency band energy to the total energy in the Teager energy spectrum were extracted as feature 
vectors, which were used as training samples and test samples separately. 
(3) The Mahalanobis distances between the test sample and different type overalls of fault samples 
are calculated to classify the real condition of a rolling bearing. 
(4) Identify the condition of the rolling bearing. 
3. Case study 
3.1. Experiment setup 
This paper uses the data from bearing data center of Case Western Reserve University for testing and 
verification. As shown in Figure 2, test rig consists of a 2hp (horsepower) motor (left), a torque 
converter / encoder (in), a dynamometer (right) and the control circuit (not shown). A deep groove ball 
bearing of 6205-2RS JEM SKF is mounted in the motor. An acceleration sensor installed on the 
magnetic base shell is used to collect vibration datas. Vibration signals collected by sensors include 
normal signal, inner race fault signal, outer race fault signal, and the sampling frequency is 12khz. The 
experimental datas were collected from 30 groups of samples. It was composed of three types of 
conditions (normal, inner race fault, outer race fault), each of which includes 10 groups of samples. 
Here, the first six groups were selected as training samples, and the remaining four groups as testing 
ROLLING BEARING FAULT DIAGNOSIS BASED ON EMD-TEO AND MAHALANOBIS DISTANCE.  
LU CHEN, HU JIAMENG, LIU HONGMEI AND WANG JING 
62 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533  
samples. 
 
 
Figure 3. Experimental setup. 
3.2. Feature extraction based on EMD-TEO 
Figure 3 and Figure 4 show the IMFs obtained by EMD. The original signal can be decomposed into 4 
IMFs and one residue. The amplitudes of IMF3 and IMF4 are more little comparing with others. So 
IMF3 and IMF4 can be abandoned. To select the interested IMF1 component for the input of TEO 
according to the objective of fault diagnosis because IMF1 contains the highest signal frequencies. 
The transient caused by the bearing faults is clearly captured. From Figures 3 and 4, it can be easily 
proven that the EMD decomposes vibration signal very effectively on an adaptive method. 
The amplitude of the Teager energy spectrum in inner race fault frequency, outer fault frequency 
and the ratio of the energy of the resonance frequency to the total energy are extracted as the feature 
vectors. For avoiding the error of the frequency leak, the weighted of Teager spectrum in fault 
frequency and f∆ = 2Hz around were extracted as the feature vectors.  
 
0 200 400 600 800 1000 1200
-1
0
1
o
ri
g
in
a
l
0 200 400 600 800 1000 1200
-1
0
1
IM
F
1
0 200 400 600 800 1000 1200
-0.5
0
0.5
IM
F
2
0 200 400 600 800 1000 1200
-0.5
0
0.5
IM
F
3
0 200 400 600 800 1000 1200
-0.1
0
0.1
IM
F
4
0 200 400 600 800 1000 1200
-0.1
0
0.1
re
s
id
u
a
l
 
 
0 200 400 600 800 1000 1200
-1
0
1
o
ri
g
in
a
l
0 200 400 600 800 1000 1200
-1
0
1
IM
F
1
0 200 400 600 800 1000 1200
-0.5
0
0.5
IM
F
2
0 200 400 600 800 1000 1200
-0.2
0
0.2
IM
F
3
0 200 400 600 800 1000 1200
-0.2
0
0.2
IM
F
4
0 200 400 600 800 1000 1200
-0.1
0
0.1
re
s
id
u
a
l
 
Figure 3. IMFs of the inner race vibration 
signal. 
 Figure 4. IMFs of the outer race vibration 
signal. 
3.3. Pattern Recognition 
3.3.1. Fault detection based on the TEO. Cyclical shock will generate when the fault bearing works. 
The frequency of this cyclical shock reflects the reason of the bearing fault. According to the 
parameters of the rolling bearings, fault characteristic frequencies of each element were calculated 
respectively. The results are as follows: 
inner
f =157.94Hz, 
outer
f =104.56Hz. 
The TEO can be applied to each IMF component, resulting in Instantaneous amplitude signal. 
Figure 5 and Figure 6 respectively are instantaneous amplitude of IMF1 component for the inner race 
and outer race fault bearing. After the EMD, in addition to the effects of low frequency noise, when 
bearing fault impact other parts, the peak of high-frequency vibration sequence is obvious. The above 
results show that the instantaneous amplitude spectrum of IMF1 is to the best effect. 
ROLLING BEARING FAULT DIAGNOSIS BASED ON EMD-TEO AND MAHALANOBIS DISTANCE.  
LU CHEN, HU JIAMENG, LIU HONGMEI AND WANG JING 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533 63 
Therefore, the IMF1 is selected to identify the characteristic frequency of bearing faults through the 
Teager energy spectrum. In Figure 7, TEO highlights the cyclical impact characteristics, the fault 
characteristic frequency(157Hz, close to 
inner
f ) and the frequency doubling of the rolling bearing with 
inner race fault can be clearly identified. Therefore, the IMF1 is related to the inner race defect of the 
bearing and can be selected as a suitable diagnostic feature in the analysis of the inner race defect. In 
Figure 8, the fault characteristic frequency (107Hz, close to 
outer
f ) and the frequency doubling of the 
ball bearing with outer race fault can be clearly identified. Therefore, the IMF1 is related to the inner 
race defect of the bearing and can be selected as a suitable diagnostic feature in the analysis of the 
outer race defect. 
 
0 200 400 600 800 1000 1200
-0.5
0
0.5
1
1.5
2
2.5
 
 
0 200 400 600 800 1000 1200
-2
0
2
4
6
8
10
12
 
Figure 5. Instantaneous amplitude of IMF1 in the 
inner race fault condition. 
 Figure 6. Instantaneous amplitude of IMF1 for 
the outer race fault condition. 
 
0 500 1000 1500 2000 2500 3000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
X: 157
Y: 0.1796
 
 
0 500 1000 1500 2000 2500 3000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
X: 107
Y: 1.246
 
Figure 7. Teager energy spectrum of the inner 
race fault condition. 
 Figure 8. Teager energy spectrum of the outer 
race fault condition. 
3.3.2. Intelligent fault diagnosis based on Mahalanobis distance. Calculate and compare the 
mahalanobis distance ( ) ( )2 1 2 3normal; : inner race fault; : outer rac, 1,2,3, e fa: ultiX i G G GGd =  from 
the unknown test sample X to the three overall 
i
G , then identify the test sample X belong to the 
i
G  of 
the smallest distance. The criterion is that if ( ) ( )22 , min ,i jX G d Xd G= , the sample is determined 
from the overall 
i
G  [7]. Finally, it is distinguished its category of the fault according to the 
"proximity" principle. The results are shown in Table 1. 
 
ROLLING BEARING FAULT DIAGNOSIS BASED ON EMD-TEO AND MAHALANOBIS DISTANCE.  
LU CHEN, HU JIAMENG, LIU HONGMEI AND WANG JING 
64 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533  
Table 1. The results of fault diagnosis 
 
1
N  
22
N  
3
N  
4
N  
1
I   
2
I   
3
I   
4
I   
1
O   
2
O   
3
O   
4
O  
1
G   0.0132 0.0254 0.0175 0.0569 1.2433 1.2719 1.1806 1.2031 1.3593 1.4025 1.3061 1.3446 
2
G   1.3475 1.3988 1.3120 1.3624 0.0543 0.0481 0.0395 0.0456 0.0897 0.0847 0.0796 0.0872 
3
G   1.4302 1.5895 1.5037 1.5496 0.1654 0.1876 0.1004 0.1263 0.0174 0.0152 0.0160 0.0229 
Min 0.0132 0.0254 0.0175 0.0569 0.0543 0.0481 0.0395 0.0456 0.0174 0.0152 0.0160 0.0229 
Mod 
N
F   
N
F  
N
F  
N
F  
I
F   
I
F  
I
F  
I
F  
O
F   
O
F  
O
F  
O
F  
( )1, 2, 3, 4iN i = : Mahalanobis distance of the four testing samples under normal condition 
( )1, 2, 3, 4iI i = : Mahalanobis distance of the four testing samples under inner race fault 
( )1, 2, 3, 4iO i = : Mahalanobis distance of the four testing samples under outer race fault 
N
F : normal 
I
F : inner race fault 
O
F : outer race fault 
4. Conclusions 
In this paper, a method for fault diagnosis of rolling bearing is presented based on a newly developed 
signal processing technique named as EMD-TEO and Mahalanobis distance. The EMD-TEO can 
extract the transient impact feature of the faulty bearing and fault detection, and the Mahalanobis 
distance can classify the bearing failure mode. EMD acts as a multi-band filter and can adaptively 
decompose complex signal into a series of IMFs, that is, zero mean mono-component AM-FM signal. 
TEO tracks the modulation energy of the interesting Intrinsic Mode Functions (IMFs) for feature 
vectors. And TEO can be used to detect period impulse characteristic of bearing localized damage, and 
then identify the characteristic frequency of bearing faults by means of Teager energy spectrum. The 
experimental results show the robustness and the effectiveness of this proposed method. 
Acknowledgements 
This research was supported by the National Natural Science Foundation of China (Grant Nos. 
61074083, 50705005 and 51105019), as well as the Technology Foundation Program of National 
Defense (Grant No. Z132010B004).  
References 
[1] Li H, Fu Lihui and Zhang Y P 2009 International Conference on Measuring Technology and 
Mechatronics Automation 
[2] Huang N E, Shen Z and Long S R 1999 Annual Review of Fluid Mechanics 31 417-457 
[3] Wang T J, Feng Z P, Hao R J and Chu F L 2012 Journal of vibration and shock 31  
[4] Huang N E, Shen Z and Long S R et al 1998 Proc. R. Soc. Lond. A 454 903–95 
[5] Potamianos A and Maragos P. 1994 Signal Processing 37 95–120 
[6] Maragos P, Kaiser J F and Qu T F 1993 IEEE Transactions on Signal Processing 41 3024–51 
[7] Zhang D X and Wu X P 2009 Anhui University Journal:Natural Science  


Rolling Bearing Fault Diagnosis Based on EMD-TEO and Mahalanobis Distance

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Rolling Bearing Fault Diagnosis Based on EMD-TEO and Mahalanobis Distance

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