A novel rolling bearing fault diagnosis method based on improved local characteristic-scale decomposition (LCD), Teager energy operator (TEO) and softmax classifier is proposed in this paper. First, vibration signals are decomposed into several intrinsic scale components (ISCs) by using improved LCD; second, TEO and fast Fourier transform (FFT) are respectively used to extract instantaneous amplitude (IA) and frequency spectra of ISC1s, and then FFT is again employed to obtain spectra of IA; third, energy ratio of the resonant frequency band against the total, frequency entropy (FE) in the spectra of ISC1s and several amplitude ratios in the frequency spectra of demodulated ISC1s are extracted as fault feature vectors, and principal components analysis (PCA) is applied for dimensionality reduction; finally, these feature vectors are taken as inputs to train and test softmax classifier. As a new non-stationary signal analysis tool, LCD can decompose adaptively a signal into series of ISCs in different scales and give good results in situations where other methods failed. However, there are two main issues in this method, end effect and mode mixing, possibly leading to unexpected results. In this paper, a slope-based method and noise assisted analysis are applied to restrain the problems respectively. Experimental results show the proposed method performs effectively for bearing fault diagnosis

Chen Lu

Hongmei Liu

Lianfeng Li

Wanlin Zhao

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 229 
Rolling bearing fault diagnosis using improved 
LCD-TEO and softmax classifier 
Hongmei Liu1, Lianfeng Li2, Wanlin Zhao3, Chen Lu4 
School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China 
Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China 
2Corresponding author 
E-mail: 1liuhongmei@buaa.edu.cn, 2lilianfeng@buaa.edu.cn, 3rsezwl@buaa.edu.cn, 4luchen@buaa.edu.cn 
(Accepted 28 August 2015) 
Abstract. A novel rolling bearing fault diagnosis method based on improved local 
characteristic-scale decomposition (LCD), Teager energy operator (TEO) and softmax classifier 
is proposed in this paper. First, vibration signals are decomposed into several intrinsic scale 
components (ISCs) by using improved LCD; second, TEO and fast Fourier transform (FFT) are 
respectively used to extract instantaneous amplitude (IA) and frequency spectra of ISC1s, and then 
FFT is again employed to obtain spectra of IA; third, energy ratio of the resonant frequency band 
against the total, frequency entropy (FE) in the spectra of ISC1s and several amplitude ratios in the 
frequency spectra of demodulated ISC1s are extracted as fault feature vectors, and principal 
components analysis (PCA) is applied for dimensionality reduction; finally, these feature vectors 
are taken as inputs to train and test softmax classifier. As a new non-stationary signal analysis 
tool, LCD can decompose adaptively a signal into series of ISCs in different scales and give good 
results in situations where other methods failed. However, there are two main issues in this 
method, end effect and mode mixing, possibly leading to unexpected results. In this paper, a 
slope-based method and noise assisted analysis are applied to restrain the problems respectively. 
Experimental results show the proposed method performs effectively for bearing fault diagnosis. 
Keywords: improved LCD, TEO, softmax classifier, bearing, fault diagnosis. 
1. Introduction 
Rolling bearings take an important role in modern manufacturing industry, which makes its 
health monitoring and fault diagnosis great significant. When a rolling bearing operates with  
faults, its dynamic behavior is complex and non-linear, and the corresponding vibration signal 
presents non-linear and non-stationary characteristics, hence the key of bearing fault diagnosis is 
how to extract fault information from the non-stationary signals. 
So far, researchers have proposed various feature extraction methods. Most of traditional 
methods such as time domain analysis, frequency analysis, and energy analysis are merely 
applicable to a stationary signal. Thus, time-frequency analysis has been increasingly applied in 
the analysis of non-stationary signals. The commonly used time-frequency method includes 
wavelet analysis [1], empirical mode decomposition (EMD) [2, 3], local mean decomposition 
(LMD) [4], etc. Nevertheless, wavelet transform is a non-adaptive data analysis method in nature. 
EMD can decompose a complex signal into some intrinsic mode functions (IMFs) and a residual, 
but it has some drawbacks, such as envelope overshoot or shortage, end effect, mode mixing. 
Though there are performance enhancements than EMD, LMD has problems of mode mixing, 
distorted components and being time-consuming. A novel time-frequency analysis method named 
LCD, was recently presented by Junsheng Cheng [5]. LCD is an adaptive non-stationary signal 
decomposition tool and suitable for processing vibration signals of bearings.  
LCD has advantages over EMD and LMD in processing nonlinear and non-stationary signals 
because it can reduce invalid components and mode mixing; however, end effect and mode mixing 
also appears in LCD. Exploring appropriate methods to resolve end effect becomes a significant 
pursuit to LCD researchers. If the signal is predictable in physics, the best method may be a 
model-based method, e.g. AR (autoregressive) or ANN (artificial neural networks) extension. 
However, these methods are not computationally acceptable. A simpler method proposed by 
ROLLING BEARING FAULT DIAGNOSIS USING IMPROVED LCD-TEO AND SOFTMAX CLASSIFIER.  
HONGMEI LIU, LIANFENG LI, WANLIN ZHAO, CHEN LU 
230 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
Rilling [6] is to ‘mirrorize’ the extrema closest to the edge. Dätig proposed a method based on the 
slopes of maxima and minima near both ends of the data series [7]. Mode mixing is another 
drawback of LCD, which is defined as a single ISC either consisting of signals of widely disparate 
scales [8]. Usually, mode mixing is a consequence of the intermittency or noises [9]. To solve the 
mode mixing of EMD, a subjective intermittence test algorithms was proposed by Huang [10]. A 
masking signal method was presented by Deering [11]. However, it is difficult to choose the 
masking signal. A noise-assisted signal analysis approach called ensemble EMD (EEMD) was 
proposed by Huang [8]. Admittedly, EEMD is a great breakthrough for the good performance. 
Generally, fault information is modulated at the high frequency band. TEO [12] can obtain IA 
of signals. Because of its simple calculation, avoidance of negative frequency, and satisfactory 
demodulation effectiveness, TEO was used for computing the IA. 
The rest of the paper is organized as follows. Section 2 presents the proposed method in detail; 
in Section 3, this method is applied to the experimental data; and the conclusions are given in 
Section 4. 
2. Methodology 
2.1. An improved LCD method 
In this paper, slope-based method (SBM) is used to restrain end effect while noise-assisted 
analysis approach is applied to weaken mode mixing in the process of LCD. 
For the signal ݔ(ݐ) depicted in Fig. 1(a), let (ݑ(݅), ܷ(݅)) and (ݒ(݅), ܸ(݅)) be the coordinates 
of the ݅th maximum Max(݅) and the ݅th minimum Min(݅). Fig. 1(b) considers the case when a 
maximum is the first extremum in the front of a signal. Then slope ݏଵ and slope ݏଶ are defined as: 
ݏଵ =
ܷ(2) − ܸ(1)
ݑ(2) − ݒ(1) ,   ݏଶ =
ܸ(1) − ܷ(1)
ݒ(1) − ݑ(1). (1)
Then, the time gaps between the first two successive maxima and minima are determined as 
Δݐ୫ୟ୶(1) = ݑ(2) − ݑ(1) and Δݐ୫୧୬(1) = ݒ(2) − ݒ(1). The new boundary extrema Min(0) and 
Max(0) are newly defined and shifted according to the corresponding time gaps Δݐ୫୧୬(1) and 
Δݐ୫ୟ୶(1), and gradients ݏଵ and ݏଶ. The coordinates of new extrema are located at: 
ݒ(0) = ݒ(1) − Δݐ୫୧୬(1), ܸ(0) = ܷ(1) − ݏଵ൫ݑ(1) − ݒ(0)൯, 
ݑ(0) = ݑ(1) − Δݐ୫ୟ୶(1), ܷ(0) = ܸ(0) − ݏଶ(ݒ(0) − ݑ(0)). (2)
 
 
a) 
 
b) 
Fig. 1. a) The time series ݔ(ݐ); b) the illustration of the slope-based method 
Here, we used LCD to analyze a simulated signal: 
ݔ(ݐ) = ݔଵ(ݐ) + ݔଶ(ݐ) = cos(2ߨ(30 + 5ݐ)ݐ) + cos ൬4ߨݐ −
200ߨ
180 ൰. (3)
The result of LCD to ݔ(ݐ) using SBM and mirror method (MM) is shown in Fig. 2, from which 
we can see that LCD with SBM can decompose this signal more efficiently and accurately into a 
ROLLING BEARING FAULT DIAGNOSIS USING IMPROVED LCD-TEO AND SOFTMAX CLASSIFIER.  
HONGMEI LIU, LIANFENG LI, WANLIN ZHAO, CHEN LU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 231 
set of ISCs than with MM. 
To restrain mode mixing, white noises are added to homogenize the scale in time-frequency 
space. To improve the decomposition performance of LCD, the ideas of EEMD are learnt and 
referred to in this study.  
To verify the validity of this method, we analyzed a simulated signal  
ݕ(ݐ) = sin(2ߨ80ݐ) + ݕଶ(ݐ)  shown in Fig. 3(a). Here, ݕଶ(ݐ)  is a high-frequency intermittent 
signal with amplitude 0.1 and frequency 1600 Hz riding on the third peak of the sinusoidal signal. 
According to Fig. 3(b), the first ܫܵܥ1(ݐ) is the mixture of both low-frequency sinusoidal and 
high-frequency intermittent signal, which means mode mixing occurs. Fig. 3(c) shows the noises 
assisted LCD can decompose the original signal more effectively (ܰ݁ = 100). 
 
a) 
 
b) 
 
c) 
Fig. 2. a) The signal ݔ(ݐ); b) The result of LCD using MM; c) The result of LCD using SBM 
 
a) 
 
b) 
 
c) 
Fig. 3. a) The signal ݕ(ݐ); b) The result by original LCD; c) The result with added noises 
2.2. Fault diagnosis scheme based on improved LCD, TEO and softmax classifier 
In this part, a new fault diagnosis approach based on improved LCD, TEO demodulation and 
softmax classifier is proposed. The scheme of fault identification is depicted in Fig. 4. The main 
steps are described briefly below: 
(1) Decompose preprocessed signals into several ISCs using improved LCD method. 
2(
)
x
t
1(
)
x
t
()xt
( )
ISC1
ISC2
ISC3
( )x t
( )
( )x t
ISC1
ISC2
( )
ROLLING BEARING FAULT DIAGNOSIS USING IMPROVED LCD-TEO AND SOFTMAX CLASSIFIER.  
HONGMEI LIU, LIANFENG LI, WANLIN ZHAO, CHEN LU 
232 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
(2) Obtain IA of ISC1s by calculating TEO, then extract frequency spectra of IA of ISC1s 
by FFT. 
(3) Compute amplitude ratios at the inner race, outer race and ball fault frequency and their 
second harmonic frequency in the frequency spectrum of demodulated ISC1s, and apply FFT to 
obtain frequency spectra of ISC1s, calculate its frequency entropy (FE) and energy ratios of 
resonant frequency band (RFB) against the total. Then, an 8-Dimension feature vector is obtained. 
(4) Use PCA for dimensionality reduction. 
(5) Train softmax classifier using train data and identify fault modes of test data. 
 
Fig. 4. Flowchart of fault diagnosis 
3. Case study 
3.1. Data preparation 
All vibration data are from Bearing Data Center of Case Western Reserve University. The 
drive end accelerometer data under the normal, inner race fault, outer race fault located at 6 o’clock 
and ball fault with fault diameters of 7 mils are selected and each datum is divided into 20 
segments, and each segment has 6000 points. 
3.2. Features extraction based on improved LCD-TEO and FE 
The original signals are decomposed into ISCs firstly. The ISC1 is extracted for the first few 
ISCs always contain the majority of fault information. Fig. 5 depicts the spectra of demodulated 
ISC1s. The three fault feature frequencies of inner race, outer race and ball fault are worked out. 
Next, several amplitude ratios mentioned above are computed. 
However, the identification performance only using feature variables comprised of amplitude 
ratios at several feature frequencies is just moderate, especially for the ball fault. The spectra of 
ISC1s are shown in Fig. 6, and two new feature variables, energy ratio of resonant frequency band 
against the total and frequency entropy are added into the feature vector. At last, a complete 8-D 
feature vectors, are extracted as intrinsic information of bearing fault. Here, the basic principle of 
entropy is briefly introduced. Entropy is defined to measure complexity and self-similarity of the 
series, and can reflect complexity changes of intrinsic oscillation. 
3.3. Feature reduction using PCA 
For a diagnosis model, the huge amount of features will probably decrease the robustness, and 
even cause a decline in the diagnostic accuracy. Therefore, high dimensional features are projected 
into a three-dimensional space by PCA.  
ROLLING BEARING FAULT DIAGNOSIS USING IMPROVED LCD-TEO AND SOFTMAX CLASSIFIER.  
HONGMEI LIU, LIANFENG LI, WANLIN ZHAO, CHEN LU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 233 
With feature visualization, we analyzed these obtained features. As shown in Fig. 7, the 
extracted feature vectors by the improved LCD method has good clustering results, therefore this 
method has high robustness and diagnosis accuracy. 
 
a) The normal 
 
b) Inner race fault 
 
c) Outer race fault 
 
d) Ball fault 
Fig. 5. Spectra of IA of ISC1s 
 
a) The normal 
 
b) Inner race fault 
 
c) Outer race fault 
 
d) Ball fault 
Fig. 6. Frequency spectra of ISC1s 
 
Fig. 7. Features clustering chart with improved LCD 
3.4. Pattern recognition using softmax classifier 
3-Dimension fault feature vectors of bearings are obtained by feature extraction and dimension 
reduction method. Softmax classifier is employed to recognize automatically faults. Identification 
results are listed partly in Table 1. 
ROLLING BEARING FAULT DIAGNOSIS USING IMPROVED LCD-TEO AND SOFTMAX CLASSIFIER.  
HONGMEI LIU, LIANFENG LI, WANLIN ZHAO, CHEN LU 
234 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
The proposed approach can identify effectively fault patterns of bearing and has the high accuracy. 
Table 1. Results of fault identification 
Class Number ଵܻ ଶܻ ଷܻ ܲ ܴ Class Number ଵܻ ଶܻ ଷܻ ܲ ܴ 
1 
11 4.783 0.430 0.121 1 
1 3 
51 0.797 0.448 0.162 3 
3 12 5.110 0.462 0.130 1 52 0.797 0.449 0.162 3 … … 1 … … 3 
20 4.862 0.437 0.123 1 60 0.793 0.449 0.161 3 
2 
31 0.066 0.303 1.200 2 
2 4 
71 0.611 4.533 0.196 4 
4 32 0.066 0.303 1.198 2 72 0.604 4.505 0.190 4 … … 2 ... … 4 
40 0.065 0.303 1.202 2 80 0.608 4.521 0.188 4 
ଵܻ- ଷܻ are feature variables after PCA; ܲ represents the predicted label; ܴ stands for the diagnosis decision 
4. Conclusions 
In this study, the slope-based method is applied to restrain end effect and the noise-assisted 
analysis is utilized to weaken mode mixing in the decomposition process of LCD. Therefore, a 
novel bearing fault diagnosis method based on improved LCD, TEO and softmax classifier is 
proposed. Experimental results show that the proposed method produces the satisfactory 
diagnostic performance. Further works can be focused on studying its adaptability to different 
working conditions and different fault severities. 
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. Z132013B002). 
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Rolling bearing fault diagnosis using improved LCD-TEO and softmax classifier

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Rolling bearing fault diagnosis using improved LCD-TEO and softmax classifier

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