Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 1 (June 2014), 97-102

Intelligent Fault Classification of Rolling Bearing at Variable Speed Based on
Reconstructed Phase SpaceWeigang Wen

School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University
Beijing 100044, China
Weidong Cheng

School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University
Beijing 100044, China
E-mail: wgwen@bjtu.edu.cn

Abstract
Rolling bearing is a common mechanical part which is subject to be damaged. It is important tomonitor the condition of bearing. An effective mean is to extract faulty features of bearing fromthe vibration signal. In this paper, a method is introduced to realize intelligent classification ofbearing state. The vibration signal is reconstructed into phase space by estimating the time delay
and embedded dimension of time series. After reconstruction, fault classification isaccomplished through normalized principal component analysis. It is testified that this method
is effective for classifying fault of bearing by experiment and data analysis.
Keywords: Fault classification; Time delay; embedded dimension; phase space reconstruction;
principal components analysis.

1. Introduction
The rolling bearings are support elements inmany kinds of rotating machinery. Manymachines are working at variable rotating speed.
The bearing faults always cause the structural
damage of machine. The measure to identify therolling bearing fault and judge its fault type
intelligently is crucial in state monitoring ofmachine. The common way to diagnose therolling bearing faults is the method of vibration
signal analysis. Many techniques have been
proposed to obtain features of vibration signals
for fault diagnosis. A lot of researches focus on
extracting the fault information from thevibration signal by time and frequency domain
signal processing techniques. These methods
include FFT, Envelope analysis, EMD, wavelet
and wavelet packet method, 1-4 etc. Especially,

the order tracking technique was proposed tosolving the rolling bearing fault diagnosis atvariable shaft rotational speed.5, 6
But because of instantaneous variations inrolling ball movement, friction, damping andmachine load, the rolling bearing is often
characterized by non-linear behaviors. Thenon-linear analysis can provide a goodalternative method to draw fault features fromthe vibration signals. A lot of non-linear
methods, such as correlation dimension,
Lyapunov exponent and approximate entropy,
have been investigated.7,8 The results haveshown that non-linear fault diagnosis is aneffective method. But in real application, it isquite a difficult problem on how to decide thethreshold of the non-linear features. Moreoverthe non-linear features are not only sensitive tofault, especially when the state of the bearing is

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W. Wen and W. Cheng

non-stationary, such as the variable rotating
speed, the changing load, and the noise in thesampled signal, etc.
This paper mainly discusses the intelligent fault
diagnosis of rolling bearing at variable speed.
The paper is organized as follows: Section 2addresses the main principles of methods
including method of reconstructed phase space(RPS) and the normalized principal component
analysis (PCA). The section involves the wholescheme of fault diagnosis. In Section 3, thevibration signal of different bearing status atvarious speeds is collected and analyzed using
these methods. Moreover, selection of timedelay, embedding dimension are discussed inthis section. In Section4, the conclusions arepresented finally.
2. Main Principles and Methods

dimension m and time delay τ can be obtained
simultaneously, there is no need to set thethreshold value manually, and small sample datais needed. The principle of C-C method is
described as follows.9
The N sample time series x is divided into t timeseries which are disjoint:{x(1), x(t+1), x(2t+1), ……}{x(2), x(t+2), x(2t+2), ……}…………
{x(t), x(2t), x(3t), ……}S(m, N, r, t) of each time series is defined as:, , ,, , ,1, , ,(2)Where C(m, N, r, t) is the correlation integral oftime series:, , ,(3)2.1. Reconstructing phase space
In order to extract the non-linear feature ofvibration signal, the dynamic system embedded
in time series need to be reconstructed. Thedynamics can be built on the time series byreconstructing phase space which is same asoriginal system according to Takens’s theorem.
RPS method is very useful and has been widely
applied to a variety of nonlinear signals
processing applications. The basic idea of theRPS is that a scalar time series x(t) may be used
to construct a time sequence of vectors that isequivalent to the original dynamics from a
topological point of view. If the proper
embedded dimension m and time delay τ can bechosen, based on N sample time series x={x(1),x(2), …, x(N-1), x(N)}, the phase space can beobtained:,,…,1(1)Where0,1,0,00,1 ,

‖ ‖

is

1, ,(4)and→ ∞, there is:infinite norm. When, ,, ,The maximum difference of S is defined as:∆,, ,, ,(5)The local maximum time corresponds to the zeroof, ,and the minimum of ∆, .
The time delay τ is the first local maximum time.
The statistics is defined as:∆(6)andare the averages
Where ∆of ∆and. The time window
corresponds to the minimum ofpractical application,

Where the vector ofis a point in the RPS,
t=1, 2, …, N-(m-1)τ. The sufficient condition for
topological equivalence of RPS with originalsystem is that m is greater than twice thedimension of the original dynamic system. Whenthe embedded dimension is not known, as is thecase for most real systems, there are manymethods to calculate m and τ. C-C method isadopted in this paper. The advantages of thismethod are that its robustness to noise forvibration signal, the value of embedded,. In

2 , σ is thestandard deviation of time series,

500 andN=3000 is applied for vibration signal timeseries.
2.2. Normalized principal component
analysis
The fault of rolling bearing causes the change ofdynamics of mechanical system. Phase trajectoryof dynamical systems could be obtained by

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Intelligent fault classification

phase spacer construction of vibration signal
time series. The different features of signals
generated by topologically different systems
could be drawn out through RPS. In comparison
with the eigen-value, such as correlation
dimension, Kolmogorov entropy and Lyapunov
exponents, the phase trajectory could describe
completely the reconstructed dynamical system,
especially for the pseudo-random time series ofrolling bearing vibration signal. In this paper, theprincipal components of RPS are used todescribe the characteristics of multi dimension
phase trajectory in phase space.10 The advantageof this method is that it is applicable to timeseries contaminated by noise which really exist
in mechanical signal.
From the N sample time series and the RPS X(t),the phase trajectory of the dynamics system canbe drawn by the matrix Y:12(7)/⋮Wherematrix A is:

1 . And the con-variance(8)
1,2, ⋯and theThen the eigen-values
eigen-vectors
1,2, ⋯of matrix
can be calculated.
andare theprincipal components. The eigen-values arelisted in descending order:⋯The principal component spectrum can be drawn
from . The principal component spectrum ofnoise is horizontal. The characteristics of thedynamics system can be extracted from the
maximal principal components excluding thehorizontal minimal components. The scale ofvibration signal is different at various rotating
speed. In order to classify the fault intelligently,
the principal components must be normalized.
The sum ofis:∑(9)classification of fault in rolling bearing withvarious speeds, loads and noise.
3. Experiments and Discussions
In order to testify the validity of method
proposed above, experimental data werecollected from rolling bearing of anexperimental mechanical system. The SKF
rolling bearings was applied to support therotating shaft. An accelerometer was mounted on
the housing near the rolling bearing to acquire
the vibration signals of the bearing. Single point
faults were introduced to the test bearings using
electro-discharge machining. The vibration
signal was sampled from bearing at variousspeeds. The class of data corresponds to thebearing conditions respectively: normal bearing,and outer race fault, etc.
The vibration signal time series of bearing withdifferent faults and various rotating speeds areshown in Fig. 1. The sample rate is 24k Hz. (a)is the normal bearing at 1000rpm, (b) is thenormal bearing at 2500rpm, (c) is the normal
bearing at 4500rpm, (d) is the outer race faulty
bearing at 2000rpm, (e) is the outer race faulty
bearing at 3000rpm, (f) is the outer race faulty
bearing at 4500rpm.
1050-50

2000

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(a)

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15
10
50-5-100(b)30
20
100-10The normalized principal components are:-20(10)-300,, ⋯ can be used as classification
components. So the normalized principal
components analysis can be utilized for the
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99(c)W. Wen and W. Cheng40

200-20-40
0

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(d)

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3002001000-100-2000(e)outer race faulty bearing at 2000rpm, (e) is thePCS of outer race faulty bearing at 3000rpm, (f)is the PCS of outer race faulty bearing at4500rpm. The similarity of same bearing at
various speeds is shown in the PCS, shown in(a)-(c) normal bearing and (d)-(f) faulty bearing.
And the components following the second
component increase along with increasing speed.
The difference between the normal bearing andfaulty bearing is clear. The first component is farlarger than the other components in the PCS ofnormal bearing. And the second component isalmost equal to the first component in the PCSof outer race faulty bearing. The features of thedynamics system can be drawn from themaximal principal components.

300

1200
100

0.8

0

0.6

-100

0.4

-200

0.2
-300
0

2000

4000(f)

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00Fig.1. The vibration signal time series of bearing withdifferent faults and various rotating speeds

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At first, the phase space is reconstructed. Theembedded dimension m and time delay τ areobtained simultaneously through C-C method.
The result of C-C method is shown in Fig. 2.
Here the structure of RPS is that embedded
dimension m is set to 26 and time delay τ is set
to 6.
S(t)平均值ΔS(t)平均值
Scorinf

0.12

0.6
0.4
0.2
00

5

1
0.8

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-0.02

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180Fig.2. The result of C-C method(b)

(c)

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Then the principal components can becalculated. The principal component spectrum ofvibration signal is shown in Fig. 3. (a) is the PCSof normal bearing at 1000rpm, (b) is the PCS ofnormal bearing at 2500rpm, (c) is the PCS ofnormal bearing at 4500rpm, (d) is the PCS of0.2

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100005(d)Intelligent fault classification

extract the features from the phase trajectory offaulty bearing. It is proved that this method iseffective for classifying fault of bearing byexperiment and data analysis. But in theexperiment, the classification of the faulty
bearing through principal components is difficultat the speed lower than about 1000rpm. Andvibration signals adopted here are only single
point fault. So the work in the next step could befocused on classification for the more variablerotating speed and the multiple points bearing
fault.

1
0.8
0.6
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005(e)

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Acknowledgements

0.200510This project is supported by the Fundamental
Research Funds for the Central Universities(2011JBM093) and National Natural Science
Foundation of China (51275030).

15(f)Fig.3. The principal component spectrum

12 samples of normal and fault bearing atdifferent speed are tested respectively. Afternormalization, principal component analysis
method is implemented in 3 dimension space.
The plot of the first three principal components(PCs) of the clustering results is shown in Fig. 4.
Almost all samples are correctly classified to thecorresponding clusters by the method.

References
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The third PC

1

4.
0.5

5.

01
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The second PC

0 0

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1.5

6.

The first PCFig. 4. Scatter plot of principal components forclustering result
7.

4. Conclusions
A method that integrates RPS and NPCA ispresented in this paper to implement detection
andclassification
ofbearingfaults.
Reconstruction of vibration signal in phase space
is made through C-C method. Then normalized
principal components analysis is applied to8.

9.

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