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ACKNOWLEDGEMENT
I would like to convey my most sincere thanks to my project supervisors, Prof Xu
Jianxin and A/Prof Sanjib Kumar Panda for their encouragement and advices. Their
profound knowledge and experiences in the field of machine learning techniques and
machine system are my source of inspiration. I also appreciate the opportunity to work in
the field of Empirical Mode Decomposition, which is the most versatile and powerful
signal processing tool I have read.
I would like to thank Mr Woo Ying Chee and Mr Mukaya Chandra from the
Electrical Machine and Drives Laboratory, who helped in setting up the Machine Fault
Simulator, DAQ measurement systems and preparing workstations for MATLAB
simulations used in the project, and their help in booking meeting room for project
briefing.
Lastly, I would like to thank Dr N. Rehman and Dr D.P. Mandic for making the
Noise-Assisted Multi-variate Empirical Mode Decomposition MATLAB code publicly
available at http://www.commsp.ee.ic.ac.uk/~mandic/research/emd.htm, and Dr G.
Rilling and Dr P. Flandrin for making the mono-variate Empirical Mode Decomposition
MATLAB code publicly available at http://perso.ens-lyon.fr/patrick.flandrin/emd.html.
i
TABLE OF CONTENTS
CHAPTER 1: INTRODUCTION ................................................................................................. 1
1.1
Objectives ............................................................................................................. 2
1.2
Thesis Organization.............................................................................................. 3
1.3
Fault Mode Statistics Survey................................................................................ 6
1.4
Literature Survey .................................................................................................. 8
1.4.1
Vibration Signature Analysis ........................................................................ 8
1.4.2
Motor Current Signature Analysis ................................................................ 9
CHAPTER 2: MECHANICS OF MACHINE FAULT MODES ...................................................... 14
2.1
Eccentricity......................................................................................................... 15
2.1.1
Static Eccentricity ....................................................................................... 15
2.1.2
Dynamic Eccentricity.................................................................................. 16
2.1.3
Mixed Eccentricity Motor Current Signature ............................................. 17
2.2
Unbalanced Rotor Fault ..................................................................................... 18
2.2.1
2.3
Unbalanced Rotor Fault Motor Current Signature...................................... 20
Bearing Faults .................................................................................................... 20
2.3.1
Bearing Faults Vibration Signatures ........................................................... 22
2.3.2
Bearing Faults Motor Current Signatures ................................................... 25
2.4
Bearing General Roughness ............................................................................... 26
2.5
Broken Rotor Bar Motor Current Signature ....................................................... 27
2.6
Shorted Stator Winding Fault Motor Current Signature .................................... 28
2.7
Healthy Machine Signature ................................................................................ 29
2.8
Dynamic Estimation of the Machine Slip .......................................................... 30
CHAPTER 3: MOTOR CURRENT SIGNATURE AND VIBRATION SIGNATURE ANALYSIS ......... 32
3.1
Motor Current signal Analysis or Vibration Analysis? ...................................... 32
3.2
Challenges of Motor Current Signature Analysis .............................................. 33
3.3
Discussions: Proposed Ensemble Spectrum Approach ...................................... 35
CHAPTER 4: ARTIFICIAL INTELLIGENCE TECHNIQUES FOR MACHINE FAULT DIAGNOSIS .... 37
4.1
k-Nearest Neighbour (k-NN) ............................................................................. 37
4.1.1
4.2
k-NN Algorithm .......................................................................................... 38
Self-Organizing Map (SOM) ............................................................................. 38
ii
4.2.1
Structure and Operation of SOM ................................................................ 39
4.2.2
SOM Algorithm .......................................................................................... 41
4.3
Support Vector Machine (SVM) ........................................................................ 42
4.3.1
Multi-Class SVM (M-SVM) ....................................................................... 44
4.3.2
M-SVM Algorithm ..................................................................................... 45
4.4
Empirical Mode Decomposition (EMD) ............................................................ 46
4.4.1
EMD Algorithm .......................................................................................... 47
4.4.2
Mode Mis-alignment ................................................................................... 48
4.4.3
Multi-variante EMD.................................................................................... 57
4.4.4
Mode Mixing .............................................................................................. 57
4.4.5
Noise-assisted Multi-variate EMD (N-A M-EMD) .................................... 57
CHAPTER 5: A STUDY ON AUTOMATIC DIAGNOSIS OF BEARING AND UNBALANCED ROTOR
FAULTS .............................................................................................................................. 59
5.1
Fault Diagnosis using Time-Domain Vibration Signatures ............................... 59
5.1.1
Similarity measures by Cross-Correlation Operator ................................... 60
5.2
Machine Fault Simulator .................................................................................... 66
5.3
Machine Signatures Collection .......................................................................... 69
5.4
Experimental Results.......................................................................................... 70
5.5
Discussions: Difficulty in choosing a suitable value for k ................................. 71
5.6
Discussions: Larger Training Samples ............................................................... 72
5.7
Visualization of Classification Results by k-NN ............................................... 72
5.8
Fault Diagnosis using Frequency-Domain Vibration Signatures ....................... 75
5.8.1
Discrete Wiener Filter ................................................................................. 76
5.8.2
Frequency Analysis of Vibration Signatures .............................................. 78
5.8.2.1 Frequency content of Healthy Machine .................................................... 78
5.8.2.2 Frequency content of Unbalanced Rotor fault .......................................... 78
5.8.2.3 Frequency content of Bearing fault ........................................................... 78
5.8.2.4 Discussion on vibration frequency analysis .............................................. 78
5.8.3
Feature Extraction of Frequency domain information ................................ 83
5.8.4
Cluster Analysis of Vibration Feature Vectors ........................................... 84
5.8.5
Further Feature Extraction .......................................................................... 85
5.8.6
Multi-class SVM (M-SVM) for Classifying Machine Fault Data .............. 86
iii
5.9
Discussions: Frequency-domain Analysis of Vibration Signatures ................... 88
CHAPTER 6: A STUDY ON MOTOR CURRENT SIGNATURE USING EMPIRICAL MODE
DECOMPOSITION ................................................................................................................ 89
6.1
Fourier Transform .............................................................................................. 89
6.2
Wavelet Transform ............................................................................................. 91
6.3
Hilbert-Huang Transform................................................................................... 92
6.3.1
Hilbert Spectrum ......................................................................................... 93
6.3.2
Marginal Hilbert Spectrum ......................................................................... 94
6.4
Discussion: EMD as a suitable Analysis Tool ................................................... 94
6.5
N-A M-EMD Experiment Results...................................................................... 95
6.5.1
Discussions: IMF Derived by EMD ........................................................... 97
6.5.2
Discussions: Filter-bank Property of EMD Algorithm ............................... 99
6.5.3
Discussions: Significance of IMF1, IMF2, IMF3, IMF4......................... 101
6.5.4
Discussions: Significance of IMF10, IMF11 ........................................... 103
6.5.5
Discussions: Significance of IMF5, IMF6, IMF7, IMF8, IMF9 ............. 104
6.6
Visualization of the Comparison results by SOM ............................................ 107
6.7
Discussions: Discovery of Unique Features by SOM ..................................... 107
CHAPTER 7: CONCLUSION ................................................................................................ 111
REFERENCES .................................................................................................................... 113
APPENDIX A: INTRINSIC MODE FUNCTIONS DERIVED BY N-A MEMD ALGORITHM FOR
MACHINE SPEED AT 20HZ ................................................................................................. 124
APPENDIX B: HILBERT SPECTRUM AND MARGINAL HILBERT SPECTRUM OF MACHINE
SIGNATURE (AT MACHINE SPEED OF 20HZ) INTRINSIC MODE FUNCTION 5 TO 9 .............. 131
APPENDIX C: PSEUDO CODE FOR 2-CLASS SVM LEARNING ............................................. 135
APPENDIX D: PSEUDO CODE FOR MULTI-CLASS SVM LEARNING .................................... 140
APPENDIX E: PSEUDO CODE FOR SOM LEARNING............................................................ 147
APPENDIX F: PSEUDO CODE FOR K-NN LEARNING............................................................ 160
iv
SUMMARY
Induction machine are used widely in industrial process e.g., steel mills, chemical
plants etc. it is therefore vital to condition monitor the health of the machine to prevent
unexpected and untimely failure. Their untimely downtime have significant economic
and social impact, such as, disruption to production process, spoilage to work-inprogress, costly plant process re-start etc. It is therefore useful to investigate automatic
machine fault detection and diagnosis techniques. This creates the motivation for this
study. Why condition monitoring? Incipient machine faults can be detected by continuous
monitoring [1]. As such, condition-based maintenance has become a new maintenance
methodology that has rapidly been adopted by the industry as the standard operating
procedure. In the past, it is essentially a routine periodic machine shutdown for servicing
and inspection. This method has proved to be inefficient. In condition-based
maintenance, the machine is carefully and continuously condition monitored for
symptoms of failure. Based on such continuous tracking of the machine health-states,
imminent failures is detected and planned shutdown made, only when necessary. This
way, machine downtime and maintenance costs are reduced, and asset security and
reliability increased, both achieving efficiency and profitability for the organization. With
this in view, this project investigates the various machine condition monitoring
techniques, with the objective to implement effective automatic fault detection and
diagnosis methods, to reveal developing incipient faults, so that timely intervention is
made to prevent sudden catastrophic failures.
v
LIST OF TABLES
Table 1.1: Summary of percentage of each of the failure mode. ........................................ 6
Table 1.2: Fault statistics on 8 surveyed articles. (*MC denotes Most Common fault) ..... 7
Table 1.3: Percentage of each of the failure mode derived from Table 1.2. ....................... 7
Table 5.1: Similarity Measure for v1, v2 and v3 ................................................................. 66
Table 5.2: Training and test sets for k-NN classification. ................................................ 69
Table 5.3: Fault classification confusion matrix. .............................................................. 70
Table 5.4: Classification result summary.......................................................................... 70
Table 5.5: Tabulated results of error rate (%) with various k-neighbor values. ............... 71
Table 5.6: Distribution of spectrum of machine vibration signatures. ............................. 83
Table 5.7: Fault classification confusion matrix of vibration signature. .......................... 87
Table 5.8: M-SVM classification of vibration signatures result summary. ...................... 88
Table 6.1: Similarity measures of same-indexed pair of machine current IMFs at 20Hz. 98
Table 6.2: Similarity measures of same-indexed pair of machine current IMFs at 30Hz. 98
Table 6.3: Similarity measures of same-indexed pair of machine current IMFs at 40Hz. 98
Table 6.4: Frequency band for HTY30 (IMF 5-9) machine current signature. .............. 100
vi
LIST OF FIGURES
Figure 1.1: Approaches of this project to investigate machine fault diagnosis. ............... 13
Figure 2.1: Perturbation force (Fc) created by unbalanced mass (m) rotating at Ω. ........ 19
Figure 2.2: Unbalanced rotor fault signatures at various machine speeds. ....................... 19
Figure 2.3: Bearing assembly. .......................................................................................... 21
Figure 2.4: Defective rolling elements (adopted from [93]). ............................................ 21
Figure 2.5: Raceway faults (adopted from [93]). .............................................................. 22
Figure 2.6: Rolling element pitch, diameter and contact angle of a bearing. ................... 23
Figure 2.7: Rolling element fault signatures at various machine speeds. ......................... 24
Figure 2.8: Inner raceway fault signatures at various machine speeds. ............................ 24
Figure 2.9: Outer raceway fault signatures at various machine speeds. ........................... 25
Figure 2.10: Healthy machine signatures at various machine speeds. .............................. 29
Figure 4.1: Connections between the input and output neurons of SOM. ........................ 39
Figure 4.2: Linear decaying learning rate versus learning steps of a SOM. ..................... 40
Figure 4.3: Support vectors, decision boundary and margin of 2-class SVM. ................. 43
Figure 4.4: Multi-class SVM using one-versus-all learning strategy. .............................. 45
Figure 4.5: Mono-variate EMD of BRG signature. .......................................................... 49
Figure 4.6: Mono-variate EMD of UBR signature. .......................................................... 50
Figure 4.7: Mono-variate EMD of HTY signature. .......................................................... 51
Figure 4.8: Extremum of two signals x1(t) and x2(t). ........................................................ 52
Figure 4.9: Mean of 3D ―tube‖ of complex signal signals zb(t)........................................ 53
Figure 4.10: Evolution of bi-variate complex signal for (|HTY40(t)|,|BRG40(t),t). ......... 53
Figure 4.11: Mean of complex signal signals zt(t). ........................................................... 54
Figure 4.12: Evolution of tri-variate complex signal for
(|HTY40(t)|,|BRG40(t)|,|BRB40(t)|).................................................................................. 55
Figure 4.13: Mode mixing. ............................................................................................... 56
Figure 4.14: N-A M-EMD on signal with Guassian white noise added. .......................... 58
Figure 5.1: Template matching using cross-correlation of machine signatures. .............. 60
Figure 5.2: Deterministic signals v1(t), v2(t) and v3(t). ...................................................... 61
Figure 5.3: Normalized cross-correlation sum coefficients of a pair signature. ............... 64
Figure 5.4: Machinery Fault Simulator (MFS) by SpectraQuest®, Inc............................. 66
Figure 5.5: Schematic of machine signature acquisition using DAQ by Dewetron®. ...... 67
Figure 5.6: Bearing fault simulation using MFS. ............................................................. 68
Figure 5.7: Unbalanced rotor fault simulation using MFS. .............................................. 68
Figure 5.8: Error rate (%) versus k-neighbor values. ........................................................ 71
Figure 5.9: Unbalanced rotor fault misclassified as healthy machine. ............................. 73
Figure 5.10: Healthy machine signatures correctly classified. ......................................... 73
Figure 5.11: Unbalanced rotor signature correctly classified. .......................................... 74
vii
Figure 5.12: Unbalanced rotor signature correctly classified. .......................................... 75
Figure 5.13: Wiener Filter................................................................................................. 77
Figure 5.14: Filtered machine signatures at fr=15Hz and 31Hz. ...................................... 77
Figure 5.15: Frequency content of HTY signatures at fr=15Hz........................................ 79
Figure 5.16: Frequency content of HTY signatures at fr=31Hz........................................ 79
Figure 5.17: Frequency content of UBR signatures at fr=16Hz........................................ 80
Figure 5.18: Frequency content of UBR signatures at fr=32Hz........................................ 80
Figure 5.19: Frequency content of BRG signatures at fr=15Hz........................................ 81
Figure 5.20: Frequency content of BRG signatures at fr=32Hz........................................ 82
Figure 5.21: 11-dimensional feature vector at fr=15Hz and 31Hz.................................... 84
Figure 5.22: Semantic map of vibration signatures from two SOM different simulations.
........................................................................................................................................... 85
Figure 5.23: 2-dimensional feature vector. ....................................................................... 86
Figure 5.24: M-SVM classification (Gaussian kernel hsvm=8.0, slack factor C=0.1) of
vibration signature. ........................................................................................................... 87
Figure 6.1: Fourier Series (a finite sum of a 10Hz square wave with n=3 and n=10). ..... 90
Figure 6.2: Different wavelet basis functions. .................................................................. 92
Figure 6.3: A 7-channel Motor Current Signature decomposition by N-A MEMD. ........ 96
Figure 6.4: EMD as filter-banks for HTY30 (IMF 5 – 9) machine current signature. ..... 99
Figure 6.5: IMF1, IMF2, IMF3, IMF4 of the machine signatures at 20Hz. ................... 101
Figure 6.6: IMF1, IMF2, IMF3, IMF4 of the machine signatures at 30Hz. ................... 102
Figure 6.7: IMF1, IMF2, IMF3, IMF4 of the machine signatures at 40Hz. ................... 102
Figure 6.8: IMF10 and IMF11 (residue) of the machine signatures at 20Hz. ................ 103
Figure 6.9: IMF10 and IMF11 (residue) of the machine signatures at 30Hz. ................ 103
Figure 6.10: IMF10 and IMF11 (residue) of the machine signatures at 40Hz. .............. 103
Figure 6.11: IMF5-9 of the HTY machine signatures at 20Hz. ...................................... 104
Figure 6.12: IMF5-9 of the BRG machine signatures at 20Hz. ...................................... 105
Figure 6.13: IMF5-9 of the BRB machine signatures at 20Hz. ...................................... 105
Figure 6.14: IMF5-9 of the UBR machine signatures at 20Hz. ...................................... 106
Figure 6.15: IMF5-9 of the SWF machine signatures at 20Hz. ...................................... 106
Figure 6.16: Feature map using fea_IMF vector at fs=20Hz. ......................................... 108
Figure 6.17: Feature map using fea_IMF vector at fs=30Hz. ......................................... 109
Figure 6.18: Feature map using fea_IMF vector at fs=40Hz. ......................................... 110
viii
LIST OF SYMBOLS &ACRONYMS
ai: non-zero Lagrange multipliers
BMU: Best Matching Unit of SOM
bo: SVM bias
BRG: Bearing fault
BRB: Broken rotor bar
C: slack factor of SVM
Db: is the diameter of the rolling element
Dc: is the rolling element pitch
di , dj: data label for OVA learning strategy for M-SVM
"EMD": Empirical Mode Decomposition
fr: rotor frequency
FR: rotor mechanical frequency
fs: fundamental supply frequency
g0: radial air-gap length in the case of a uniform air-gap
hsvm: Guassian kernel width
h: inverter harmonic order
HTY: healthy machine
IMF: Intrinsic Mode Function
isD: instantaneous values of direct-axis component of monitored stator current
isQ: instantaneous values of quadrature-axis component of the monitored stator current
k-NN: k-Nearest Neighbor
ix
Lm: three-phase magnetizing inductance
Lr: three-phase self-inductance of the rotor winding
Ls’: stator transient inductances
m: rotor unbalanced mass
MFS: machine fault simulator
MMF: magneto motive force
MCSA: Motor Current Signature Analysis
M-SVM: multi-class Support Vector Machine
N: is the number of rotor bars
NB: is the number of rolling elements
nd: eccentricity order number, (static eccentricity=0, dynamic eccentricity=1)
N-A MEMD: noise-assisted MEMD
OVA: one-versus-all learning strategy for M-SVM
P: number of pole-pairs
P0: average air-gap permeance
r: distance between the centre of rotation and the centre of gravity of the rotor
Rr: resistance of rotor phase winding
s: machine slip
SOM: Self-Organizing Map
SWF: Shorted stator winding fault
SVM: support vector machine
Tr: open-circuit rotor time constant given by Lr/Rr
UBR: Unbalanced rotor fault
x
w_som(i,j): SOM output neuron‘s weight vector
w_svmi: SVM weight term
wk[i]: Wiener filter weights at instant ith
xi: support vectors for a SVM
zb(t): complex signal in space (|x1(t)|, |x2(t)|,t)
zt(t): complex signal in space (|x1(t)|, |x2(t)|, |x3(t)|)
β: rolling element contact angle
γ: ―nudge-to-zero‖ constant for Wiener filter
η0: initial learning rate of SOM
θ: angular position of r
θr: angular position of the rotor with respect to the stator reference
μ: constant to adjust the rate of convergence of the weights for Wiener filter
ν: order of the stator time harmonics present in the power supply
ρ: degree of eccentricity
Φ: particular angular position along the stator inner surface
Φn: phase delay
ψrd: instantaneous values of direct-axis component of the rotor flux linkage
ψrq: instantaneous values of quadrature-axis component of the rotor flux linkage
Ω: rotor shaft rotational speed
ω1: angular stator frequency
ωsl: angular slip frequency
ωv: frequency of the kth vibration due to bearing defect
xi
LIST OF RELEVANT PUBLICATIONS
1. W.-Y. Chen, J.-X. Xu, S.K. Panda, ―A Study on Automatic Machine Condition
Monitoring and Fault Diagnosis for Bearing and Unbalanced Rotor Faults‖, IEEE
International Symposium on Industrial Electronics (ISIE‘2011), Poland, Gdansk, 2830 Jun 2011, Accepted for publication
2. W.-Y. Chen, J.-X. Xu, S.K. Panda, ―Application of Artificial Intelligence Techniques
to the Study of Machine Signatures‖, IEEE International Symposium on Industrial
Electronics (ISIE‘2012), China, Hangzhou, 28-31 May 2012, Manuscript submitted
for publication
xii
CHAPTER 1: INTRODUCTION
The field of machine condition monitoring and fault diagnosis is vast. A literature
survey; which is presented subsequently, has shown wide ranging diagnostic techniques.
Various machine operation quantities may be used for monitoring the health of a motor,
e.g., partial discharge, thermo-graphic monitoring of hot-spots, chemical content; such as,
oil degradation detection, wear debris detection, machine axial leakage flux, acoustic,
torque, machine power efficiency, machine vibration signal, and motor current signature
[2, 3]. Among these, the technique by analyzing machine stator current is known as
Motor Current Signature Analysis (MCSA) is the state-of-the-art technique [102]. It is a
popular research area where many algorithms have been proposed, but a single effective
method that is able to detect and diagnosis multiple classes of machine fault still elude
researchers.
The current harmonics that is present in the motor current is mainly created by the
machine asymmetries and vibrations due to machine faults. Hence, this project focuses on
two fault detection techniques, namely, vibration signature and MCSA. There are a
number of issues to address in the formulation of a reliable fault detection and diagnosis
scheme [4]:
definition of a single diagnostic procedure for any type of faults
insensitive to and independent of operating conditions
reliable fault detection for position, speed and torque controlled drives
reliable fault detection for drives in time-varying conditions
quantify a stated fault threshold independent of operating conditions
1
1.1
Objectives
With the above issues in mind, this project aims to accomplish two main
objectives, namely,
Objective 1:
To investigate and formulate an automatic machine condition monitoring
scheme to detect and diagnose the most common machine fault modes,
namely, bearing and unbalanced rotor fault, that is insensitive to machine
operating speed
Objective 2:
To investigate and study the use of MCSA to cover a wider range of
machine fault modes; apart from bearing and unbalanced rotor faults, to
include broken rotor bars and shorted winding faults as well, where
vibration analysis is difficult to diagnose, and to discover unique nonlinear
and non-stationary features for automatic fault classifications
In these studies, computational intelligence are applied. Of particular interests, are
the Self-Organizing Map (SOM), multi-class Support Vector Machine (M-SVM), kNearest Neighbor (k-NN) case-based learning and the Empirical Mode Decomposition
(EMD).
On the first objective, this project has formulated and implemented a simple and
effective data-based scheme, using time-domain vibration data, for the continuous
automatic condition monitoring and diagnosis of bearing and unbalanced rotor faults is
proposed. The key idea is to use a novel normalized cross-correlation sum operator as
2
similarity measure, and in combination with the use k-NN algorithm for the effective
automatic classification of machine faults. This technique is both noise-tolerant and shiftinvariant., It also has a low error rate and insensitive to machine operating speed, as
shown subsequently in this thesis. Further, the diagnosis of these two mechanical faults
using vibration frequency-domain information is also shown, where SOM is used to
discover cluster information on the extracted features in an unsupervised fashion, and an
M-SVM is next used to derive the clusters globally optimal separating hyperplanes for
the automatic classification of the fault modes.
On the second objective, this project use EMD technique to study the motor
current signatures harmonic contents of a healthy machine (HTY), a machine with
bearing fault (BRG), unbalanced rotor fault (UBR), broken rotor bar fault (BRB) and
shorted stator winding fault (SWF). In this project, new unique non-linear and nonstationary features are discovered for these fault modes at machine operating speed of
20Hz and 30Hz. However, it is also observed in this project that uniqueness of these
features is not obvious at higher speed of 40Hz. With the newly discovered unique
features at 20Hz and 30Hz, future works on automatic fault classifications by a single
effective fault detection and diagnosis scheme based on EMD technique is achievable.
1.2
Thesis Organization
This thesis consists of seven chapters.
3
Chapter 1: Introduction on the issues of formulating a reliable machine fault diagnostic
scheme, and the rationale for condition monitoring using MCSA and vibration
analysis, and sets the stage for stating the objectives of this research. Fault
statistics and literature survey are also carried out to compile the fault
statistics and identifies the most common failure modes. This allows research
effort to be directed at the most common failure modes. Fault diagnostic
technique literature survey is next conducted, to understand how various novel
diagnostic techniques are formulated and the difficulties encountered. This
identifies niche research area where this project adds values.
Chapter 2: Mechanics of machine fault elucidates the origin of different type of
machine faults, presents the various fault vibration signatures and the
expected motor current fault spectrum for MCSA.
Chapter 3: Motor Current Signature and Vibration Signature Analysis explain the
difficulties, challenges and issue of vibration analysis and MCSA techniques
and a new approach is proposed.
Chapter 4: Application of Artificial Intelligence (AI) techniques for fault diagnosis
presents the various AI techniques used in this project.
Chapter 5: A study on Automatic Diagnosis of Bearing and Unbalanced Rotor faults
presents the results of the data-based machine fault detection and diagnosis
4
scheme using time-domain vibration data. It explains how cross-correlation
sum operation in time-domain data series is a suitable similarity measure for
the vibration signatures for the purpose of automatic pattern classification
using k-NN classifier, and presents the fault classification error rate and
confusion matrix. It also presents feature extraction using vibration frequencydomain information, fault-class clusters study and discovery using
unsupervised learning by SOM, the clusters globally optimal separating
hyperplane derived from a M-SVM using one-versus-all learning strategy, and
the M-SVM classification error rate and confusion matrix.
Chapter 6: A study on
Motor Current Signature
using Empirical
Mode
Decomposition explains the disadvantages of the traditional analysis tool for
MCSA using Fourier-based and Wavelet transform, the rationale for using
EMD techniques as an effective tool for the analysis of machine current
signatures, with a view to discover new information that Fourier and Wavelet
transform may not be able to reveal.
Chapter 7: Conclusion
5
In the next, a fault mode statistics survey is carried out with a view to identify the
most commonly occurring machine fault modes. The survey article in 1985 [5] reveals
that bearing fault is the most common machine fault mode. The followings further survey
and present the situation in the 1990s.
1.3
Fault Mode Statistics Survey
In [1], a detail survey on fault statistics was done in 2008. In this comprehensive
survey, several sources; including the private communication between the author and an
original equipment manufacturer, referenced about 80 journal papers published in IEEE
and IEE on the subject over the past 26 years since 2008, were used. The table below
summaries the survey result.
Table 1.1: Summary of percentage of each of the failure mode.
Failure Modes
Bearing
Stator Related
Rotor Related
Others
%
52.5%
22.0%
13.0%
12.5%
A majority of the failure mode is due to bearing (52.5%). If bearing fault is
combined with stator related faults, this together accounted for more than 87.5% of the
total faults. Further fault information from referenced articles [6-13] is conducted. The
following table summarizes the findings.
6
Table 1.2: Fault statistics on 8 surveyed articles. (*MC denotes Most Common fault)
Referenced articles
Bearing
Stator Related
Rotor Related
Others
[6]
45%
-
[7]
50%
40%
10%
-
[8]
10%
-
[9]
45%
-
[10]
52%
25%
6%
17%
[11]
41%
37%
10%
12%
[12]
50%
40%
10%
-
[13]
40%
-
By taking the average across the rows of Table 1.2, the following table is derived.
Table 1.3: Percentage of each of the failure mode derived from Table 1.2.
Failure Modes
Bearing
Stator Related
Rotor Related
Others
%
46.1%
35.5%
9.2%
14.5%
Table 1.3 presents a similar failure mode statistics as in Table 1.1. Bearing fault
accounted for more than half the total faults, and the second most common fault is the
stator related faults. These two faults together accounts for more than at least 65% of the
total faults. This finding is consistent with that in Table 1.1. As such, from Table 1.3,
bearing faults accounts for about half of the fault modes, hence it is worthwhile to focus
research effort on bearing faults. If the fault coverage is extended to include rotor and
stator related faults e.g., unbalanced rotor, broken rotor bars and shorted stator windings,
about 85% of all fault modes is covered. With this information, objective 2, presented
above, is set.
7
1.4
Literature Survey
This section presents the literature survey to shed lights on the various techniques
used and progress made. The survey focuses on machine vibration signature and MCSA,
as these techniques are able to detect bearing, unbalanced rotor, broken rotor bars and
shorted stator windings fault modes [1].
1.4.1 Vibration Signature Analysis
Vibration signature analysis is the most commonly monitored operation parameter
for detection and diagnosis of mechanical fault modes e.g., bearing defects and
eccentricities [14]. Using Wavelet techniques to preprocess the vibration signal is
popular. The articles [15-17] presented such a study where the wavelet coefficients were
the feature vectors. It is interesting to note that, in Elsevier collection of articles, the use
of Morlet wavelet basis function is common, whereas in IEEE collection of articles,
Daubechies wavelet is popular. Higher order spectral analysis using Bispectral transform,
is used for noise suppression, detection of non-Gaussian data, and to detect nonlinearity
of the fault information in [18]. Envelope analysis is used in [19, 20] for feature
extraction. Article [21] showed that combining features extracted from Mel-frequency
Cepstral Coefficients (MFCC) and Kurtosis, are effective for diagnosing bearing faults.
Independent component analysis can be used to extract features from vibration signatures
for bearing fault diagnosis [22]. After feature extraction, AI techniques e.g., neural
network, SVM, SOM, are next used to predict fault modes. Excellent examples of works
done are [23-26].
8
However, the most interesting technique is the use of EMD for the analysis of
machine vibration signatures, where the basis function is derived based on empirical data
in terms of Intrinsic Mode Functions (IMFs). Articles [27-31] presented such a study.
The IMFs thus derived are the feature vectors for fault diagnosis.
1.4.2 Motor Current Signature Analysis
A survey on MCSA technique reveals that the approaches are numerous and
wide-ranging. The articles maybe broadly categorized into: reviews, model construction
for fault modes, feature extraction techniques, and the use of computational intelligence
for machine fault diagnosis.
Over the years, a series of reviews have been made by [1, 4, 7, 32-37]. They offer
a good overview on how progress has been made. A notable change is the progress from
the use of traditional Fourier transform e.g., Fast Fourier Transform (FFT), to analyze
motor current signatures, and the increasingly popular use of Wavelet transform e.g.,
Discrete Wavelet Packet transform, to identify fault spectrum and extract unique features
for fault diagnosis. FFT is the traditional tool for MCSA where by locating individual
fault spectrum, the machine fault is diagnosed. This approach is successful for broken
rotor bars and eccentricities faults [11, 38, 39].
In [40, 41], a good comparative study of the various techniques for MCSA for
broken rotor bar and air-gap eccentricities is presented. In [42], a good review is given on
the various diagnosis methods for stator voltage asymmetry and rotor broken bars.
However, these techniques are mainly Fourier based. It is worth to note that, as the fault
spectrum is functions of machine slip, it is particularly difficult to locate the fault
9
spectrum in low slip situation. Further, since motor current signature is non-linear and
non-stationary in nature [43], as such, Wavelet multi-resolution decomposition approach
is popular. In [44-52], Wavelet transform is used to decompose the motor current into
various approximate and detail levels wavelet coefficients, and features are extracted
from these coefficients for fault diagnosis. Diagnoses of bearing, broken rotor bar,
eccentricities faults were reported. However, careful selection of a wavelet basis function
is not trivial [40, 49], as wavelet decomposition is a convolution computation of machine
signature with wavelet basis function and hence a different choice of basis function
produces different results.
Beside wavelet technique, other high resolution frequency-domain techniques e.g.,
Eigen-analysis Multiple Signal Classification (MUSIC) spectrum Estimator, Welch, Burg
[53] are used. In [54], these techniques are applied for the detection of rotor cage faults.
In [8], a detail study using different auto-regressive parametric methods e.g., YuleWalker, and the possibility of using a lower sampling rate were explored. The article [55],
showed that a sliding window ROOT-MUSIC algorithm for bearing fault diagnosis is
possible, and in [56] a novel combination of maximum covariance method for frequency
tracking and Zoom-FFT technique, to selectively increase the frequency resolution of the
frequency range of interest for fault diagnosis were demonstrated. Other methods
incorporating temporal information of the motor current using higher order statistic, such
as, spectral kurtosis is used in [57].
With the popular use of inverter speed controller for machine, the effect of PWM
inverter harmonics on MCSA was investigated in [58, 59]. In [60], inverter input and
output current were studied with a view to detect the twice fundamental frequency
10
harmonics for diagnosis of rotor faults. It is shown that detection of these harmonics is
possible using inverter input current near zero frequency.
To extend the type of fault coverage, stator winding faults are investigated as well.
In [61], a novel diagnostic indicator for stator winding fault, that does not involve ground
fault, is formulated using positive and negative sequence line-voltage and line-current
information. The key idea is that various indicators were determined at various machine
speeds, and a kind of lookup table was created for diagnosis at different machine
operating speed. However, the use of line-voltage, made the method invasive, where
potential transformer (PT) is required. The article [62] presented a method using
Extended Park‘s Vector Approach (EPVA), where instead of observing the ovality of the
signature in the D-Q plane, the frequency-domain information revealed the presence of
fault for stator winding. This approach may be used for bearing fault as well [10].
Survey also revealed other innovative approaches. Instead of using steady-state
information, transient start-stop information may be used for diagnosis as well, as shown
in [63, 64]. In [6], monitoring instantaneous power factor and motor efficiency [65] are
possible approaches too. An interesting approach is presented in [66], where a noise
cancellation technique was used for diagnosing general roughness fault. The scheme
assumed that all frequencies that are not related to the bearing faults, e.g., supply
frequency, supply unbalance harmonics, the eccentricity harmonics, the slot harmonics,
saturation harmonics and interferences from environmental sources are regarded as noise
and are estimated by a Wiener Filter. All these noise components are then cancelled out
by their estimate in a real-time manner. The remaining components are hence related to
11
bearing fault, and the RMS value of this noise-cancelled signal is next calculated online
as fault index, with impending fault as an increase in fault index.
Model-based approach aims to construct a mathematical model of the machine
and thereby using the model to analysis and predict fault mode [67-74]. Finite element
analysis is popular for simulating and studying of fault mode; especially for broken rotor
bar. Winding-Function model is specially formulated for modeling air-gap eccentricity as
shown in [75, 76].
Data-based approach collects real machine fault data rather than using
sophisticated mathematical model to calculate them, and uses these data as examples for
fault diagnosis. These examples are collected from fault simulator. With the fault data
available, AI techniques e.g., SOM, neural network, fuzzy logic, M-SVM etc., are used to
automatically classify the faults [77-81]. Other modeling approaches are possible, such as,
[12] use Autoregressive (AR) Spectrum Estimation; a form of parametric spectrum
estimation technique to model a healthy motor signature, and deviation from this baseline
indicates a bearing general roughness fault. However, this method requires the use of
notch-filter to remove the dominant fundamental frequency and a series of filter banks to
remove the harmonics of other possible faults e.g., unbalance voltage source, cyclical
load torque, eccentricities, broken rotor bars, rotor slotting effects etc. This adds to the
complexity of this method. Recently, the use of Independent Component Analysis (ICA)
has achieved remarkable results, where the diagnostic procedure is independent of
machine operation speed for the diagnosis of bearing and broken rotor bars [82-84].
12
EMD is applied for the diagnosis of shorted stator winding fault and broken rotor
bars in [85, 86]. However, in each of the study, only one fault mode was covered. This
runs the risk of mis-diagnosing a fault, as another fault signatures not covered in the
study, may produce similar features. This project aims to widen the scope of motor
current signature study to cover more fault modes. Figure 1.1 illustrates the approaches of
this project to investigate the automatic fault diagnosis of AC synchronous machine. In
the next, the mechanics of machine fault mode is presented.
AC synchronous machine fault diagnosis
Vibration signatures
Motor current signatures
Unbalanced Rotor Bar fault
Broken Rotor Bar fault
Bearing fault
Unbalanced Rotor Bar fault
Bearing fault
Shorted Winding fault
Time-domain analysis
Normalized Cross-correlation
Time-domain analysis
Empirical Mode Decomposition
Wiener filter
Frequency-domain analysis
Fast Fourier Transform
Frequency-domain analysis
Hilbert Hwang Transform
Figure 1.1: Approaches of this project to investigate machine fault diagnosis.
13
CHAPTER 2: MECHANICS OF MACHINE FAULT
MODES
Machine is a moving magnetic apparatus. Its various parts are subject to kinetic
energy, magnetic energy, operational thermal stress and harsh environmental conditions
e.g., humidity and chemical corrosion, causing wear and tear, and ultimately numerous
types of fault may develop. Faults are categorized as electrical or mechanical faults.
Electrical failure modes are mostly insulation failure of core, stator winding, rotor
winding and rotor bar breakage. Mechanical failure modes are mostly bearing failure,
rotor eccentricities e.g., unbalanced rotor; caused for example by wear and tear,
accumulation of deposits and temperature changes etc., and creating unequal distribution
of rotor mass.
An ideal healthy machine has physical constructional symmetry, such as, an
equally spaced and constant air-gap length, equal rotor resistances in the rotor and stator
windings and a balanced rotor. However, there are inherent construction asymmetries and
imperfections in an actual healthy machine, for example, the air-gap length is not
perfectly spaced and as the rotor rotates the air-gap length varies, and the rotor and stator
winding resistances for each phase are not the same. These minor physical asymmetries
generate unequal magnetic flux and as a result magnetic force induced vibrations are
caused. Hence, a healthy machine is expected to generate some low magnitude
vibrations.
A faulty machine has much more severe physical and electrical asymmetries,
generating larger unequal magnetic flux, and the resultant magnetic force creates larger
vibrations. The resulting vibrations, which are a series of movement of the machine parts
14
cause variations in magnetic permanence of the air-gap. The stator windings, acting like a
transducer, pick up these stray magnetic fluxes and induce current harmonics into the
stator current. In the next presents the detail mechanics of some major fault modes,
namely, 1) eccentricity, 2) unbalanced rotor, 3) bearing general roughness, 6) bearings
faults, 5) broken rotor bar, and 6) shorted stator winding fault.
2.1
Eccentricity
Induction machine may fail due to air-gap eccentricity. Air-gap eccentricity
occurs due to shaft deflection, inaccurate positioning of the rotor with respect to stator,
bearing wear, stator core movement etc. Air-gap eccentricity creates unbalanced radial
forces and hence unbalanced magnetic pull that may cause rotor-to-stator rub, and
ultimately results in damage of the stator core and stator windings.
2.1.1 Static Eccentricity
There are two types of air-gap eccentricity: static and dynamic eccentricity. In
static air- gap eccentricity, the position of the minimal radial air-gap length is fixed in
space. For example, static air-gap eccentricity may be caused by the ovality shaped of the
core, or by incorrect positioning of the stator or rotor during commissioning stage. The
air-gap of static eccentricity is independent of θr, the angular position of the rotor with
respect to the stator reference, and is given by [87]
g ( ) g 0 g 0 cos
(0,1)
(2.1)
where ρ is the degree of eccentricity and g0 is the radial air-gap length in the case of a
uniform air-gap, and Φ is the particular angular position along the stator inner surface.
15
2.1.1.1
Static Eccentricity Motor Current Signature
With the motion of the air-gap given in (2.1), it can be shown that the harmonic
frequency components in the stator currents of an induction machine with static air-gap
eccentricity are given by [87],
1 s
Fstatic _ ECC kN
f s
P
k 1,2,3...
(2.2)
where fs is the fundamental supply frequency, k an integer, N number of rotor slots, nd
eccentricity order number, (static eccentricity is nd =0, dynamic eccentricity is nd =1), P is
the number of pole-pairs, ν is the order of the stator time harmonics that are present in the
power supply driving the motor, taking the values ±1, ±3, ±5, …etc, and s the slip.
2.1.2 Dynamic Eccentricity
In the case of dynamic eccentricity, the centre of the rotor is not at the centre of
rotation and the position of minimum air-gap rotates and varies with the rotor. This
misalignment maybe caused for example by, a bent rotor shaft, bearing wear or
misalignment, mechanical resonance at critical speed, etc. The air-gap of dynamic
eccentricity is given by [59]
g ( , r ) g 0 g 0 cos( r )
(0,1)
(2.3)
where ρ is the degree of eccentricity, θr is the angular position of the rotor with respect to
the stator reference, g0 is the radial air-gap length in the case of a uniform air-gap, and Φ
16
is the particular angular position along the stator inner surface. The corresponding
permeance variation due to dynamic eccentricity is [59],
Pg ( , r ) P0 Pn cos(n( r ) n )
(2.4)
n
where P0 is the average air-gap permeance and Φn is the phase delay.
2.1.2.1
Dynamic Eccentricity Motor Current Signature
Due to permeance variation as a result of eccentricity, side-band components
appear around the slot harmonics in the stator line current frequency spectrum. The
frequency components in the stator currents of an induction machine with to dynamic airgap eccentricity are given by [3, 88],
1 s
Fdyn _ ECC (kN n d )
f s
P
k 1,2,3...
(2.5)
where fs is the fundamental supply frequency, k an integer, N number of rotor slots, nd
eccentricity order number, (static eccentricity is nd =0, dynamic eccentricity is nd =1), P is
the number of pole-pairs, ν is the harmonic of the stator magneto motive force (MMF)
time harmonics, given by ±1, ±3, ±5, …etc., and s the slip.
2.1.3 Mixed Eccentricity Motor Current Signature
However, in a practical machine, both static and dynamic eccentricities are
present. This mixed eccentricity creates the following harmonics in the machine current
[4],
Fmix _ ECC 1 k
1 s
fs
P
k 1,2,3...
(2.6)
17
With the use of VSI, additional harmonics are introduced [59, 89],
Fvsi _ ECC h f s k f r
h, k 1,2,3...
(2.7)
where h is the inverter harmonic order, fs is the fundamental supply frequency and fr is
the rotor frequency. It is hence possible to detect air-gap eccentricity by monitoring
motor current.
2.2
Unbalanced Rotor Fault
Unbalance rotor is a type of eccentricity fault where the off-center rotation of the
rotor is caused by unbalanced mass rather than bent rotor shaft. It is the most common
source of excessive vibration [90, 91]. Possible causes are, asymmetrical mass
distribution of the rotating element as a result of wear, erosion, material buildup, thermal
expansion or contraction, causing shaft bending or misalignment. As a result, the centre
of gravity of the rotating element does not coincide with the centre of rotation, and at the
point of unbalanced mass creates a synchronous radial perturbation force (Fc), causing a
forced vibration. This phenomenon is described by the following expression, assuming a
rigid isotropic rotor system [14],
Fc m r 2 e j ( t )
(2.8)
where m is the unbalanced mass, r is the distance between the centre of rotation and the
centre of gravity of the rotor, Ω is the shaft rotational speed, θ is the angular position of r
and j is the complex operator. Figure 2.1 illustrates this.
18
Fc
Machine centerline and
center of rotation
m
r
θ
Rotor
Ω
Machine center of gravity
Figure 2.1: Perturbation force (Fc) created by unbalanced mass (m) rotating at Ω.
Figure 2.1 shows some sample vibration signatures of unbalanced rotor fault at
various machine speeds. A characteristic oscillatory sine wave is observed.
Figure 2.2: Unbalanced rotor fault signatures at various machine speeds.
19
2.2.1 Unbalanced Rotor Fault Motor Current Signature
With the motion of the rotor described by Eq. (3.8), the expected current
harmonics for a machine with unbalanced rotor is given by [92],
FUBR 1 k
1 s
fs
P
k 1,2,3...
(2.9)
where fs is the supply frequency and s is the machine slip.
2.3
Bearing Faults
Machines with rolling element bearings have moving bearings e.g., spherical
balls, tapered rollers or cylindrical rollers, to support the rotating shaft. The machine
bearing assembly consists of four basic components, namely, the outer race, inner race,
rolling elements which are the ball bearings inside the cage, and the retainer or the cage.
The balls are bounded by the cage, which ensures a uniform distance between the balls
and prevents ball-to-ball contacts. These rolling elements are always in metal-to-metal
contact with the inner and outer raceway, and as a result are subject to constant wear and
tear. Figure 2.3 is an illustration of the construction of the bearing assembly and the
various type of bearing faults.
20
a
d
b
e
c
f
g
h
a. Outer raceway fault
b. Rolling element fault
c. Inner raceway fault
d. Retainer
e. Inner raceway
f. Shaft
g. Rolling element
h. Outer raceway
Figure 2.3: Bearing assembly.
Bearing and raceway wear and tear present initially as general roughness and
progresses to metal fatigue, and ultimately spall and chip on the surface of the rolling
elements [93]. Figure 2.4 and 2.5 show severely chipped rolling elements and spalled
raceway faults. These defective surfaces on these components are a source of machine
vibration. A chipped rolling element spins as it revolves around the raceway. When it is
in contact with the defective surface of the raceway, an impact pulse is produced, creating
a free vibration. In the absence of significant damping medium in the bearing assembly,
the impact pulses decay exponentially.
Figure 2.4: Defective rolling elements (adopted from [93]).
21
Spalled
Spalled
Figure 2.5: Raceway faults (adopted from [93]).
2.3.1 Bearing Faults Vibration Signatures
There are four basic motions that describe the dynamics of faulty bearing
movement, namely, cage, outer race, inner race or rolling elements. Each fault generates
a unique natural frequency. The following equations show the natural frequencies
associated with each of the bearing single-point defect, where the cage fault, outer race
fault, inner race fault and rolling element fault frequencies are FC, FO, FI and FB
respectively [59],
1 Db cos
FR 1
2
Dc
(2.10)
FO
N B Db cos
FR 1
2
D
c
(2.11)
FI
N B Db cos
FR 1
2
Dc
FC
(2.12)
22
2
Dc Db cos
FB
FR 1
Db Dc
(2.14)
where FR is the rotor mechanical frequency, NB is the number of rolling elements, Dc is
the rolling element pitch, Db is the diameter of the rolling element, and β is the rolling
element contact angle. Figure 2.6 illustrates this. Therefore, these fault frequencies are
functions of the bearing geometry, the number of rolling elements, and the bearing
rotational speed.
Contact Angle (β)
Rolling Element
diameter (Db)
Pitch (Dc)
Figure 2.6: Rolling element pitch, diameter and contact angle of a bearing.
Figure 2.7, 2.8 and 2.9 show the fault vibration signatures, measured in
acceleration (m/s2), of a faulty rolling element, inner raceway and outer raceway at
different machine operating speeds.
23
Figure 2.7: Rolling element fault signatures at various machine speeds.
Figure 2.8: Inner raceway fault signatures at various machine speeds.
24
Figure 2.9: Outer raceway fault signatures at various machine speeds.
2.3.2 Bearing Faults Motor Current Signatures
Bearing defect causes minute radial movement of the rotor, and hence is a kind of
dynamic eccentricity. However, the difference between dynamic eccentricity and bearing
fault is the characteristic of the mechanical oscillations. For the former, an eccentric rotor
causes a non-uniform sinusoidal air-gap, but for the latter, bearing defect causes an
instantaneous mechanical impulse displacement in the air-gap, giving rise to vibrations
that cause air-gap permeance variation that is a complex sum of an infinite number of
rotating eccentricities [94]. With instant eccentricities generated by the bearing fault, the
air-gap is given by [59],
g ( , r ) g 0 g 0 cos( v t )
k
(2.15)
25
where ωv is the frequency of the kth vibration due to bearing defect, and the permeance
variation are of the machine is [59],
Pg ( , r ) P0 Pn, k cos(n(n v t n, k )
k
(2.16)
n
Therefore, the periodical changes in the machine permeance, in turn, creates
harmonics in the stator current shown below [59],
FBRG f s k f v
k 1,2,3...
(2.17)
where fs is the supply frequency and fv is FC, FO, FI or FB. With the use of inverter, there
is interaction between the inverter harmonics and bearing fault induced harmonics and
the expected bearing fault spectrum is [59],
FBRG h f s k f v
k 1,2,3...
(2.18)
where h is the inverter harmonic order.
2.4
Bearing General Roughness
It was reported in the literature that, there should be a distinction between bearing
fault and general roughness of the bearing [12]. General roughness of bearing is an early
sign of impending bearing fault. However, it does not show a distinctive failure mode like
bearing fault whereby clear and visible point-faults are developed e.g., cracks, pitting and
other localized damages. Normally, general roughness is simulated by de-greasing the
bearing, thus causing great friction of bearing movement and hence general roughness.
Whereas for bearing faults, it is simulated by creating holes e.g., drilling, on the raceway
assembly. Different level of fault severity can be created by creating holes of various
diameters. General roughness is reported to cause a general increase in the noise level in
the motor signature, and does not exhibit any particular frequency spikes. Generalized
26
roughness fault is subtle and does not have clear distinguishable defects. Therefore, it
does not have a unique fault frequency, but rather, it manifests as a general and
unpredictable increase in magnitude and broadband changes in vibration and stator
current frequencies. However, general roughness is considered as a kind of eccentricity
fault [95]. Therefore, instead of measuring the vibration frequencies, the machine
condition can be monitored by stator current harmonics to detect harmonics created by
the minute variations in the machine permeance.
2.5
Broken Rotor Bar Motor Current Signature
Rotating machine subjects its rotor to prolong kinetic, electrical and thermal
stresses, and breakage results as material fatigue develops over time. Breakage increases
the rotor resistance of the broken rotor bars and causes electrical asymmetry and
distortion of the rotor bar currents, and hence distorts the three-phase magnetic field. The
rotor MMF is distorted as well. This distorted rotor MMF consists of a forward and
backward rotating wave with respect to the rotor fixed reference frame. The former is the
main magnetic field and the latter is due to the rotor electrical asymmetry caused by the
breakage. The backward travelling wave induces a stator voltage harmonic component at
the frequency (1−2s)fs where fs is the stator supply frequency and s is the slip. This stator
voltage harmonics in turns creates a stator current harmonic of the same frequency [72],
FBRB 1 2 s f s
k 1,2,3...
(2.19)
The interaction of these side band currents with flux and the speed ripple creates
additional harmonics at frequency [72],
FBRB _ lu 1 2 k s f s
k 1,2,3...
(2.20)
27
where fs is the supply frequency, s is the machine slip, the difference gives the lower
sideband and the sum gives the upper sideband. These frequencies are a function of the
machine slip (s). Therefore, these frequencies are dynamic in nature and vary as the
operational condition of the motor varies. At higher slip, and spectra are further away
from fs, and at lower slip the spectrum lines are close to fs and are difficult to detect.
2.6
Shorted Stator Winding Fault Motor Current Signature
The most common kind of fault related to stator winding of induction motors are:
phase-to-ground, phase-to-phase and short-circuit of coils of the same or different phase
(―turn-to-turn‖ fault). These insulation faults maybe caused by hot spots in the stator
winding, oil contamination, moisture, dirt, electrical discharges, slack core lamination,
cooling system failure [2].
In this fault mode, the winding turn-to-turn fault is the subject of interest for
condition monitoring, as such short-circuit fault involving different phase is difficult
detected by the usual machine protection relays. If it persists undetected, causes heating
and ultimately progress rapidly to phase-ground or phase-phase faults with little
warnings, damaging the machine core permanently. Short-circuited turns on the stator of
the induction motor causes asymmetry of the three-phase stator winding, and the effect is
the presence of three-phase negative sequence currents. The diagnosis of shorted turns
using MCSA is to detect the frequency components in Eq. (2.21) since the rotating flux
induces corresponding harmonics in the stator winding [96],
1 s
FSWF kN
1 f s
P
k 1,2,3...
(2.21)
28
where N is the number of rotor bars, P is the number of pole-pairs, s is the induction
machine slip, fs is the supply frequency.
2.7
Healthy Machine Signature
Serving as a reference, healthy machine signatures are recorded. A healthy
machine has a low level of vibration, as observed in Figure 2.10, and may contain
harmonics due to inherent physical constructional asymmetries and imperfections.
Figure 2.10: Healthy machine signatures at various machine speeds.
29
2.8
Dynamic Estimation of the Machine Slip
From the preceding sections, it is clear that some of the fault spectrums are
functions of machine slip (s). Hence, to locate the fault spectrum, machine slip has to be
determined. This section presents a method for determining machine slip. It is possible to
construct a slip-sensing device by using the instantaneous values of the stator voltages
and currents of the induction motor. This method is referenced from Peter Vas,
“Parameter Estimation, Condition monitoring, and diagnosis of Electrical Machines”.
Using the dynamic model of an induction machine, the angular slip frequency ωsl
is obtained from the stator voltages and currents of the machine. The instantaneous
machine slip, s, can thus be obtained as,
s
sl
,
1
(2.22)
where ωsl is the angular slip frequency and ω1 is the angular stator frequency. Using the
following expression, the angular slip frequency can be obtained,
sl
Lm
Tr
rd i sQ rq i sD
,
2
2
rd rq
(2.23)
where isD is the instantaneous values of direct-axis component of monitored stator
current, isQ is the instantaneous values of quadrature-axis component of the monitored
stator current, ψrd is the instantaneous values of direct-axis component of the rotor flux
linkage, ψrq is the instantaneous values of quadrature-axis component of the rotor flux
linkage, Tr is the open-circuit rotor time constant given by Lr/Rr, where Lr is three-phase
30
self-inductance of the rotor winding, Rr is the resistance of rotor phase winding and Lm is
the three-phase magnetizing inductance.
The instantaneous values of direct- and quadrature-axis components of the rotor
flux linkage ψrq is calculated as,
L
d '
'
r r u s ( R s L s p )i s .
dt
Lm
(2.24)
where Ls’ is the stator transient inductances. With an integrator 1/s, ψrq is calculated.
Therefore, with the instantaneous values of the terminal quantities, usD, usQ, isD
and isQ measured, the machine parameters Lr, Rr, Lm and Ls’ known, the rotor flux linkage
ψrq is calculated, and in turn the instantaneous machine slip (s) is determined using Eq.
(2.23). Similar schemes based on slip determination have indeed been implemented in
[97, 98].
31
CHAPTER 3: MOTOR CURRENT SIGNATURE
AND VIBRATION SIGNATURE ANALYSIS
MCSA technique is a popular method of monitoring the condition of a motor
where the stator current is used for fault diagnosis. It has been used in many industrial
cases since 1980s with good results [11, 38]. Its history dates back to early 1970s when
the US Nuclear Regulatory Commission needs to check the conditions of motors located
inside the nuclear reactors using non-intrusive techniques. Oak Ridge National Labs
initiated research for this technology. It was found that motor current was always
modulated by any faults conditions inside the motor. This is because a faulty machine
creates asymmetries. These physical asymmetries generate vibrations. The resulting
vibrations, which are a series of minute movement of the machine parts cause variations
in magnetic permeance of the air-gap. The stator windings, acting like a transducer, pick
up these stray fluxes and hence induce current harmonics into the stator current.
Therefore by doing spectrum analysis of the motor current signals, faults can be detected
online without disturbing its operation e.g., shutdown.
3.1
Motor Current signal Analysis or Vibration Analysis?
MCSA is the state-of-the-art techniques [102]. MCSA technique has many
advantages. It is non-invasive, where stator current is measured simply by using current
transformers (CT) and no other special equipment is needed. By simply processing the
motor current signals, fault diagnostic information is extracted. Numerous faults can be
diagnosed using MCSA: damaged rotor bar, such as, broken rotor bars, static or dynamic
32
eccentricities, for example, due to unbalanced rotor, bearing defects, stator winding
shorted [1, 3].
As such, condition of a machine can be monitored and diagnosed at a remote
location, which may be located in a hazardous environment or inaccessible location, and
the machine does not need to be physically disassembled for diagnosis where individual
parts are inspected for signs of faults. Motor current is highly sensitive to changes in the
magnetic flux of the machine, any minute changes, due for example to machine faults,
induces harmonics in the motor current. Therefore, prognosis of incipient fault is also
made possible by MCSA. This way fault is pre-empted and spare parts ordered well in
advance before the actual faults develop, and allows for speedy repairs and shortened
shutdown time. These advantages have greatly motivated researches in MCSA. However,
there are a few issues with MCSA as presented below.
3.2
Challenges of Motor Current Signature Analysis
Fault signals are nonlinear and non-stationary. From Eq. (2.2), (2.5), (2.6), (2.9),
(2.10-2.12), (2.14), (2.19-2.21), fault frequencies are functions of the machine slip and
require the calculation of instantaneous machine slip. Instantaneous machine slip can be
calculated using Eq. (2.23) and Eq. (2.24), as shown above. However, the method is
invasive as it requires the installation of Potential Transformer (PT); a direct connection
to live electric bus-bar is needed for voltage sensing, and requires accurate machine
parameters. It is also particularly difficult to locate the fault spectrum in low slips
situations, where the fault spectrum is very close to the fundamental supply frequency.
33
For bearing faults, as shown in Eq. (2.10-2.14), accurate dimensional
measurements of the machine parts e.g., contact angle, rolling element diameter etc., is
crucial for successful location of the fault spectrum. Therefore, a method that is machine
parameter-free is desirable.
Fault signatures have extremely low signal-to-noise ratio. Typically the fault
signal magnitude is in the order of 10-4 volts and the is very close to the noise floor, and a
suitably high sampling rate is needed to ensure sufficient harmonic information is
captured, and yet at the same time a sufficiently high frequency resolution is required.
When high frequency resolution is required, this posed a constraint that limits low
sampling rate to be used, in order to have better frequency resolution when using FFT
analysis.
Fault spectrum contaminated by noises is a challenge. Such as by machine
inherent harmonics that arise from constructional asymmetries, the use of variable speed
drive introduces additional harmonics into the stator current, contaminated by harmonics
from supply unbalance e.g., unequal loading of single-phase loads on a three-phase
supply source, and by harmonics due to nonlinear load e.g., ―dirty‖ sources from
nonlinear load such as IT digital devices.
An inspection of the expressions given for the fault spectrum of eccentricities
fault and shorted stator windings, show that these expressions produce similar harmonics.
These expressions are reproduced below. The harmonics of is machine eccentricities is
given by,
1 s
FECC (kN nd )
f s
P
k 1,2,3...
(3.1)
34
and for unbalanced rotor fault,
FUBR 1 k
1 s
fs
P
k 1,2,3...
(3.2)
and shorted stator windings fault,
1 s
FSWF kN
1 f s
P
k 1,2,3...
(3.3)
Hence, it is difficult to differentiate these set of harmonics caused by three
different fault modes, where for example, shorted stator winding fault may be confused
with those that may arise due to inherent eccentricities [96, 99, 100], especially under
extreme low signal-to-noise ratio situation and presence of varying load torque effect
[101]. There exist risks of mis-diagnosing shorted stator windings with eccentricities e.g.,
unbalanced rotor.
3.3
Discussions: Proposed Ensemble Spectrum Approach
[102] presents a good comparison on the fault diagnostic technique based on
vibration and current signature analysis, while vibration signature is a good indicator for
diagnosing machine faults, current signature offers numerous advantages and with
suitable analytic tools both are effective tools for machine diagnosis.
As the fault mode survey in the preceding section has shown, bearing fault and
unbalanced rotor form the majority of the machine fault modes, and it is difficult to detect
and differentiate unbalanced rotor and bearing faults using motor current signatures
spectrum, thus vibration signatures are used instead for diagnosis of these two fault
modes.
35
Since machine instantaneous slip calculation for the purpose of determining
individual fault spectrum is difficult and invasive, it is proposed that a range of spectrum
is considered instead. This way, ensembles of frequency lines are together considered to
detect the fault. By identifying the unique fault spectrum range, machine faults is thus
detected and diagnosed. To do this, the traditional Fourier and Wavelet based frequency
domain analysis tools is not used, since motor current signature is nonlinear and nonstationary. EMD is used instead where no a prior assumption is made about the machine
signatures, but by using empirical data only.
36
CHAPTER 4: ARTIFICIAL INTELLIGENCE
TECHNIQUES FOR MACHINE FAULT DIAGNOSIS
An approach to machine fault diagnosis is the data-based method, whereby
machine fault diagnosis is to associate an unseen machine diagnostic parameters e.g.,
vibration signals, stator motor current spectrum, torque variations etc., with the various
known fault parameters, stored a priori in a database. This fault data base is next used as
training samples for machine learning, after which the fault knowledge is stored in the
neuronal weights for subsequent automatic fault detection and diagnosis. As such,
machine fault diagnosis is solved as a pattern recognition problem. In this chapter,
various AI techniques used in the project are presented.
4.1
k-Nearest Neighbour (k-NN)
k-NN algorithm is an empirical classification method, where it uses actual data of
the subject matter, in this case machine signatures, to serve as model to solve the class
prediction problem. Given labeled training data set D={(y1,ω1) (y2,ω2) (y3,ω3)… (yn,ωn)},
where yn are the templates and ωn its class, 1-NN algorithm finds a template ŷ among D
set, that is closest; based on some similarity measure S(y,ŷ), to the unseen test vector y,
and assign y to the class ω which ŷ belongs. Hence, by using a suitable similarity measure,
the predicted class ω, is given by,
yˆ arg max [S1 ( y, yˆ1 ) S2 ( y, yˆ 2 ) Sn ( y, yˆ n )], yˆ
(4.1)
n
37
k-NN classifier has been used widely in many applications, such as, in handwritten
character recognition [103]. It has also been used successfully for sub-space learning and
dimensionality reduction [104]. It is proposed to use k-NN classifier for machine fault
diagnosis. It is a type of classifier based on non-parametric statistical pattern recognition,
where the probability density function assumes arbitrary distributions.
4.1.1 k-NN Algorithm
Given data set D={(y1,ω1) (y2,ω2) (y3,ω3)… (yn,ωn)}, where yn are the templates
and ωn its class, the k-NN algorithm is,
1: initialize programme parameters and set a value for k, the number of nearest
neighbors to feature vector
2: select a feature vector from the test set
3: calculate similarity metric for all the prototypes with the feature vector
4: choose the k prototypes that is most similar to the feature vector
5: within the chosen k prototypes count the votes
6: decide the ―winner‖
7: update confusion matrix and repeat 2 for the next feature vector
4.2
Self-Organizing Map (SOM)
SOM is a type of neural network that is trained using unsupervised learning to
produce a low-dimensional; normally two-dimensional, discretized representation of the
input space of the training samples, called a map. This makes SOM useful for visualizing
38
low-dimensional views of high-dimensional data. SOM is different from other artificial
neural networks, in the sense that they use a neighborhood function to preserve the
topographical properties of the input space [105, 106]. It was invented by Professor
Teuvo Kohonen, Helsinki University of Technology, Finland, during the 1960s, and it is
sometimes called a Kohonen map.
4.2.1 Structure and Operation of SOM
SOM consists of two layers; an input layer and an output layer in a planar 2D grid
of NxM arrays. All input nodes are connected to all the output nodes. It is a feed-forward
network. The Figure 5.1 shows the basic SOM architecture.
Figure 4.1: Connections between the input and output neurons of SOM.
At the start of the SOM algorithm, each element of the weight vector is initialized
randomly to a real number between 0 and 1. Each data set is arranged column-wise in a
matrix. Other SOM parameters are also specified, namely, the SOM size (NxM), the
39
initial learning rate, the initial effective width of the SOM, and number of learning
iterations. This project uses a special linear learning function rather than an exponential
learning function to reduce the learning time [107],
(n) 0 * 1
n
iteration
(4.2)
where η(n) is the learning rate as a function of learning step n, iteration the number of
defined learning steps, and η0 the initial learning rate. Figure 4.2 shows this linearly
decaying learning rate.
Figure 4.2: Linear decaying learning rate versus learning steps of a SOM.
40
4.2.2 SOM Algorithm
The SOM algorithm is,
1: initialize weight vectors of each output neuron randomly
2: normalization of the input vectors
x'
x mean( x)
std ( x)
(4.3)
3: select an input vector, x(:,n), randomly
4: WHILE learning iteration is less than specified, DO for each output neuron‘s weight
vector, w_som(i,j)
calculate the Euclidean distance between the input and weight vector
locate the output neuron that has the smallest distance, the Best Matching Unit
(BMU),
eud arg min zi w _ somij (n) ,
n 1,2,3.....
(4.4)
i
5: update BMU and the neurons in the neighborhood of BMU, using the adaptive
updating rule,
w _ somij i j (n) ( z (n) w _ somij (n)) (n), n 1,2,3.....
w _ somij (n 1)
n 1,2,3.....
w _ somi j ( z (n) w _ somij (n)) (n),
for BMU ' s neighbours
for BMU
(4.5)
where Hij(n) is the neighborhood function and η(n) is the decaying learning rate
6: increment learning iteration and repeat 3
41
4.3
Support Vector Machine (SVM)
SVM algorithm seeks to maximize the geometric margins of the separating hyper-
planes using support vectors, which in turns results in a globally optimal hyper-plane
where the probability of misclassification error is lowered. As the separating hyper-plane,
is derived in a structured way by constrained quadratic programming method, therefore a
global minimum is guaranteed. SVM is able to work on very small sample size without
having to deal with the problem of having to determine the statistical properties of the
training sample and over-fitting problem. Based on Cover‘s Theorem, the probability that
different classes are linearly separable increases, when data points in the input space are
nonlinearly mapped to a higher dimensional feature space. Therefore, to lower error rate,
a suitable transformation of the input data set into higher dimensional space is needed.
This is achieved by using a kernel for transformation. Kernel trick is used to calculate
such a kernel transformation rather than the actual full expansion of the transformation, in
order to avoid the curse of dimensionality. Such use of kernel has made SVM a very
versatile algorithm, whereby the user chooses a suitable kernel to suit the application.
Common examples of kernels are Gaussian and Polynomial functions.
To further enhance classification accuracy, slack variables are introduced, and the
constant C is associated with these slack variables. After learning off-line, the SVM
performs classification task in a computationally efficient way using support vectors,
which are a sub-set of the training samples. Figure 4.3 illustrates the separating hyperplane i.e., the decision boundary and support vectors for two-class problem that is
linearly non-separable. Figure 4.3 bottom-panel is a zoom-in view of the top-panel.
42
Class 2
Class 1
Class 1
region
Zoom-in view
Class 2 training
Class 1 training
Test Sample
Decision Boundary, g(x)=0
Margin
Support Vectors
Figure 4.3: Support vectors, decision boundary and margin of 2-class SVM.
Class 1 data set is ―x‖ and class 2 data set is ―+‖. The support vectors are those data
points with ―○‖. Notice the support vectors are at both sides of the decision boundary,
and these support vectors define the decision boundary. The following presents the
formulation of a 2-class SVM.
43
4.3.1 Multi-Class SVM (M-SVM)
SVM is originally designed as a binary classifier. However, it can be easily
extended to a multi-class setting. There are numerous methods for solving a multi-class
problem, with one-versus-all (OVA) learning strategy being the easiest to implement and
produces reasonable results [108]. There are other methods, such as, one-versus-one,
hierarchical tree-based methods etc., as shown in [109]. In OVA approach, K-number of
SVMs is required to classify K-classes of data. Each SVM is trained using one class (Si)
against the rest of the classes (Sj), i.e., the class label (di and dj) for training is defined as,
Si {( xi , di )}ii 1n ,
di {11 ,12 ,... 1n }
S j {( x j , d j )} jj 1m ,
i j
d j {11 ,12 ,... 1m }
j i
(4.12)
After training, each SVM classifies the input feature vector by returning a
discriminant value. There are thus K discriminant values, one for each class, and are
calculated using the following,
gk (X )
a d K(X , x )
i
i
i
L {ai , d i , xi }
(4.13)
iL
where L is a sub-set of the training samples, consists of xi support vectors, ai are the nonzero Lagrange multipliers, and the associated di labels, and a suitable kernel K(X,xi). The
absolute value of the discriminant value g(X) is used. The input feature vector belongs to
the class where the SVM that gives the largest discriminant value, namely,
k arg max {abs( g k ( X ))}
k
k
(4.14)
44
The following figure illustrates this approach,
SVM-1
abs( g1 ( X ))
abs( g 2 ( X ))
SVM-2
X
max()
i
abs( g k ( X ))
SVM-k
Figure 4.4: Multi-class SVM using one-versus-all learning strategy.
4.3.2 M-SVM Algorithm
Given the data sets as in Eq. (4.12), the M-SVM algorithm does,
1: load and define the training set and test set
2: FOR Kth SVM for a K-class problem, training one SVM versus the rest, DO
3: form the Gram matrix (K) and the Hessian matrix (H) for optimization
4: define the quadprog(·) parameters and calculate the Lagrange Multipliers
5: find support vector to calculate the weight term w_svmi and bias bo
6: save the support vectors, labels, Lagrange multipliers αi, and the bias bo
7: use the support vectors, labels, Lagrange multipliers αi and bias bo, to calculate the
discriminant values for classifications (Eq. (4.13)),
N
g k ( X ) i d i K ( x, xi )
i 1
8: Repeat 2 for other SVM to obtain another gk(X)
9: Predict class by using Eq. (4.14),
k arg max {abs( g k ( X ))}
k
k
45
4.4
Empirical Mode Decomposition (EMD)
Empirical Mode Decomposition (EMD) is an adaptive signal analysis method
invented by Dr Norden Huang [110]. In EMD, the decomposition does not use a priori
basis function, but in an adaptive fashion, based on actual empirical data [110]. In this
sense, it allows the data to speak for themselves, rather than using a pre-defined basis
function, and the resulting transformation a mere convolution computation. EMD uses a
iterative sifting process to reduce a time-domain signal to its lowest frequency, call the
‗Intrinsic Mode Function‘ (IMF). The sifting process derived a series of IMFs, and stops
according to a certain stoppage criteria, and when the resulting IMF is monotonic and
there is only one extremum in the data span. By superimposing these IMFs thus derived,
the original signal is re-constituted. Upon deriving the IMFs further analysis maybe
carried out using Hilbert-Huang transform, where the amplitude / instantaneous
frequency information can be graphed on a Frequency versus Time plot call the Hilbert
Spectrum and an Amplitude versus Frequency plot call the Marginal Hilbert Spectrum.
Each IMF satisfies the following two conditions [110],
C1) in the given time series, the number of extrema and the number of zero crossings
must either equal or differ at most by one
C2) at any data point, the mean value of the envelope defined using the local maxima and
the envelope defined using the local minima is zero
46
4.4.1 EMD Algorithm
Given an arbitrary time series data x(t), the algorithm of EMD does,
1:
start the shifting process by identify all local extrema
connect them by a cubic spline to form the upper and lower envelope
calculate the mean m1 of the two envelopes
2: subtract m1 from x(t), and the proto-mode (proto-IMF) h1 is thus obtained,
h1 x(t ) m1
(4.15)
3: check if h1 meet condition C1 and C2. If not, h1 is used as the data for the next
iteration,
h11 h1 m1
(4.16)
Thus at k-iteration, the IMF is given by,
h1k h1( k 1) m1k
(4.17)
4: check if stoppage criteria is met. One possible stoppage criteria is [110],
2
T
SDk
h
t 0
k 1
(t ) hk (t )
T
h
2
k 1
(t )
(4.18)
t 0
5: after the stoppage criteria is met, store h1k in c1, and the first IMF is derived as c1,
given by,
c1 h1k ,
r1 x(t ) c1
(4.19)
where r1 is the residue. This residue may contains long-period variations i.e.,
possible sinusoids in the signal, and use as data for the next iteration, as follows,
rn rn 1 c n
(4.20)
6: Go back to 1 if rn has more than one extremum
47
Eq. (4.18) is the normalized square difference between two successive sifting
operations. The obtained SD value has to be smaller than a predetermined value. This is
usually done by assigning a small value in the order of 10-3 to SD at the start of the EMD
programme. It is observed that from the above algorithm, the number of extrema in the
time series decrease after each extraction of an IMF, and thus guarantee that the time
series is completely decomposed in a finite number of steps. The decomposition process
stops when the residue rn becomes monotonic or when it has only one extremum where
no further IMF can be extracted. Thus the original data x(t) is decomposed into n-number
of IMFs, whose summation re-constitutes the original data x(t), that is [110],
n
x(t ) c j rn
(4.21)
j 1
4.4.2 Mode Misalignment
When there are more than one time series e.g., x1(t), x2(t), x3(t) ..., a mono-variate
EMD analyses on these time series one by one may produce for each time series IMFs
that does not have the same scale even though they are same indexed. This is expected,
since EMD decomposition is an automatic adaptive and time variant filtering [111]. This
problem is known as the mode misalignment [112]. In the study of machine fault
signatures harmonic, a number of signatures from different fault modes are decomposed
by EMD, and their IMFs thus derived are compared, to identify uniqueness for each fault
class. In order to examine and compare the spectrum content of each signature, it is
desirable to compare the same indexed IMF across different signatures. When using this
approach to real-world machine fault signature, the mode misalignment problem is
encountered. This complicates the study of the machine signature harmonic. The
48
followings illustrate this when some machine signatures; recorded at a sampling rate of
20kHz, are decomposed by standard EMD one by one.
Figure 4.5: Mono-variate EMD of BRG signature.
Figure 4.4 shows the decomposition of BRG signature with five IMFs. Subplot 1 shows
the original signature, labeled as IMF0. IMF1 and IMF2 are the high-frequency
components and IMF3 is the fundamental supply frequency. IMF4 and IMF5 are the lowfrequency ―long-term trend‖.
49
Figure 4.6: Mono-variate EMD of UBR signature.
Figure 4.5 shows the decomposition of UBR signature with six IMFs. Subplot 1 shows
the original signature. IMF1 and IMF2 are the high-frequency components and IMF3 is
the fundamental supply frequency. IMF4 to IMF6 are the low-frequency ―long-term
trend‖. Even though the same numbers of IMFs are derived from two different EMD
sifting processes, it is difficult to conclude that IMF1 and IMF2 from two different
signatures are of the same scale. Figure 4.6 shows the EMD of a HTY signature with six
IMFs. This time, IMF4 is the low-frequency component, and in this case, mode
misalignment is clear with compare to the previous two cases.
50
Figure 4.7: Mono-variate EMD of HTY signature.
4.4.3 Multi-variate EMD (M-EMD)
A possible solution is the use of multi-variate EMD (M-EMD). This project uses
the M-EMD algorithm by Rehman et al. (2010) [112, 113] to study the harmonics of
machine signature. In the standard ―mono-variate‖ EMD setting, the local mean is
calculated by taking the interpolation between the local maxima and minima, i.e., the
decomposition extract oscillatory components in the time series. Figure 4.7 illustrates the
cubic spline interpolation of the extremum of an arbitrary signal x1(t). The mean of x1(t)
is the average of the extremum envelope.
51
|x1(t)|
Maxima envelope
Mean of x1(t)
x1(t)
Minima envelope
time
Figure 4.8: Extremum of two signals x1(t) and x2(t).
However, in a multi-variate setting, a different approach is required. Given N set
of signals, a possible approach to construct the multi-variate EMD problem is to express
the given set of signals in multi-dimensional space as complex signals, and seek to
extract rotating components from the complex signals as intrinsic modes, and thus
separating the slowly rotating components from rapidly rotating ones. That is, in a bivariate setting (i.e., N=2), given two time series x1(t) and x2(t), the bi-variate complex
signal zb(t) is [114],
z b (t ) x1 (t ) j x2 (t )
(4.22)
Figure 4.8 illustrates the complex signal zb(t) is the complex space (|x1(t)|, |x2(t)|,t). The
mean, shown in green line, is the average value in the 3D ―tube‖ enclosed by the
extremum shown in blue line.
52
|x1(t)|
3D extremum “tube”
time
Mean of zb(t)
zb(t)
|x2(t)|
Figure 4.9: Mean of 3D ―tube‖ of complex signal signals zb(t).
For this project, the complex signal is shown in Figure 4.9, in the complex space for
machine signatures HTY40 and BRG40 for duration of 0.06 second.
Figure 4.10: Evolution of bi-variate complex signal for (|HTY40(t)|,|BRG40(t),t).
53
For a tri-variate setting (i.e., N=3), given three time series x1(t), x2(t) and x3(t), the trivariate complex signal zt(t), also known as quaternion signal, is [115],
z t (t ) ix1 (t ) j x2 (t ) k x3 (t )
(4.23)
Figure 4.10 illustrates the complex signal zt(t) is the complex space (|x1(t)|, |x2(t)|, |x3(t)|).
The mean, shown in green line, is the average value in the 3D space enclosed by the
extremum of zt(t).
|x1(t)|
|x3(t)|
Mean of zt(t)
zt(t)
|x2(t)|
Figure 4.11: Mean of complex signal signals zt(t).
For this project, the complex signal is shown below in the complex space for machine
signatures HTY40, BRG40 and BRB40 for duration of 0.06 second.
54
Figure 4.12: Evolution of tri-variate complex signal for (|HTY40(t)|,|BRG40(t)|,|BRB40(t)|).
For a 5-class study of the machine signature harmonics, i.e., N>3, a multi-variate
setting is needed. A possible approach is to generate N-dimensional multiple envelops of
the signal by projections along different directions in N-dimensional spaces, and these
projections are next averaged to obtain the local mean. Given a set of N component
multi-variate signal [116],
x(t )Tt1 x1 (t ), x 2 (t ), x N (t ),
(4.24)
Generate a set of direction vectors xθk for projections based on quasi-Monte Carlo-based
low-discrepancy sequences [116],
xk x k 1 , x k 2 , x k N ,
(4.25)
where the angles of projection is given by angles,
k k 1 , k 2 , k N 1 ,
(4.26)
55
Next the algorithm calculate a projection pθk(t) for the input signal x(t) along direction
vector xθk for all k i.e., the whole set of direction vectors (from k=1 to K). From the each
of the k-set of projections, the envelope eθk(t) (from k=1 to K) of each projections is
determined, and the mean is defined as [116],
1 K
m(t ) e k (t ).
K k 1
(4.27)
From m(t), the usual sifting process proceeds to determine the IMFs.
4.4.4 Mode Mixing
There is usually noise n(t) being coupled into the measurement of data i.e., x(t) +
n(t). This causes mode mixing. Mode mixing is when there exists a signal of similar scale
in different IMF. This is due to the presence of transient signals, or signal ―intermittency‖.
Figure 4.12 illustrates this by using a synthetic time series s(t); denoted by IMF0, which
contains three tones, one high frequency (IMF1) and two low frequency (IMF2 and
IMF3) tones.
Figure 4.13: Mode mixing.
56
In the absence of white Gaussian noise, the three IMFs are derived from s(t) using
standard EMD, which correspond to each of the component of s(t), as shown in Figure
4.12 right-panel. However, when white Gaussian noise is added, the three component
tones cannot be recovered. Fig 4.12 left-panel shows the noise-perturbed IMFs, where it
is observed that multi-tone is present in the IMFs. This phenomenon is mode mixing.
4.4.5 Noise-assisted Multi-variate EMD (N-A M-EMD)
To deal with this problem, a noise-assisted EMD approach is adopted. For a
mono-variate EMD, a different white Gaussian noise is added to s(t) for each complete
iteration of the sifting process i.e., from Eq. (4.15) to Eq. (4.20), thus yielding a series of
N set IMFs corresponding to each sifting process. The ensemble of these N set of IMFs is
next averaged, i.e., the ensemble averaging, to obtain the final IMFs that is free from
mode mixing. That is [110],
1 N
IMFk , j
N N
k 1
IMF j lim
(4.28)
However, a practical value for N is a few hundreds [110],
1 N
IMF j IMFk , j
N k 1
(4.29)
Similarly, M-EMD is also plagued by noise. Therefore, a noise-assisted M-EMD
(N-A M-EMD) algorithm by Rehman et al. (2010) [112, 113], is used in this study. In NA M-EMD algorithm, all the signatures are submitted for decomposition in a single
application of the algorithm. To resolve the mode mixing problem due to the presence of
noise, two additional ‗channels‘ of white Gaussian noise are added. This two noise
57
channels are discarded after the EMD sifting processes. Figure 4.13 demonstrates the
result of such decomposition by N-A M-EMD where mode mixing is resolved.
Figure 4.14: N-A M-EMD on signal with Guassian white noise added.
58
CHAPTER 5: A STUDY ON AUTOMATIC
DIAGNOSIS OF BEARING AND UNBALANCED
ROTOR FAULTS
This chapter presents the diagnostic scheme proposed using time-domain and
frequency-domain vibration analysis, to diagnose two of the most common mechanical
faults, namely, bearing and unbalanced rotor faults.
5.1
Fault Diagnosis using Time-Domain Vibration Signatures
Fault diagnosis may be solved as a pattern recognition problem. There are mainly
four approaches to pattern recognition, namely, template matching, statistical
classifications, syntactic or structural matching, and neural networks. In the template
matching approach, a database of templates or prototypes is stored. A classifier next
assigns an unseen input feature vector, by comparing the feature vector with the set of
prototypes, and assigns the feature vector to a particular class based on certain similarity
measure. K-NN algorithm is used to accomplish such machine learning, and in
combination with the use of a normalized cross-correlation sum operator as similarity
measure, for the effective diagnosis of bearing and unbalanced rotor faults, using timedomain vibration signatures only with low error rate.
59
Figure 5.1 illustrates the proposed scheme.
Machine Fault
Detection and Diagnosis
k-NN algorithm
Signature
Template
Database
Bearing
fault
signatures
Unbalanced
Rotor fault
signatures
Healthy
Machine
signatures
Time-domain
vibration
signature
Figure 5.1: Template matching using cross-correlation of machine signatures.
5.1.2 Similarity measures by Cross-Correlation Operator
To automatically classify the fault signatures between the three classes, namely,
the unbalanced rotor fault, bearing fault and healthy machine, a suitable similarity
measure needs to be defined. There are many ways to define the similarity between a
template (q) and a feature vector (p) of dimension N. Such as, by Euclidean distance
[116]
S e ( p, q ) ( p q ) T ( p q )
(5.1)
where the objective is to find the minimum Euclidean distance among the templates, and
to associate the feature vector to that template with minimum distance.
60
There is also the simple cross-correlation between q and p [117],
Ssc ( p, q) pT q
(5.2)
A value of 1 indicates perfect matching, and a value close to 0 indicates
otherwise. The geometrical interpretation is that, the angle between two perfectly
matching vector pair is 0o, and its cosine is 1; and an orthogonal vector pair at 90o and its
cosine is 0. The objective is thus to find the maximum correlation value among the
templates for a feature vector, and classify the feature vector to that template. However,
these similarity measures are suitable for synchronized static pattern, and are not suitable
for machine fault signatures in this study that have time lags among the samples, as these
are recorded at different time snap-shots. By using the aforementioned similarity
measures on these signatures, large errors are made, leading to inaccurate classification.
The following experiment which uses deterministic signals illustrates this.
Figure 5.2: Deterministic signals v1(t), v2(t) and v3(t).
61
Given three deterministic signals, as shown in Figure 5.2, namely, a half squarewave v1(t), a half sine-wave v2(t), and a half sine-wave v3(t), that is time shifted by 30ms,
and v1(t) and v2(t) are templates for matching to feature vectors, v3(t). As observed from
Figure 5.2, even though v2(t) is identical to v3(t), their Euclidean distance at 0.3568, is
further than the distance between v1(t) and v2(t) at 0.1109. This is due to the time shift. In
this case, if Euclidean distance is a measure of similarity, then a half square-wave is
associated with a half sine-wave. The situation is not improved by using ―simple‖ crosscorrelation. v2(t) is identical to v3(t) in shape, but their ―simple‖ cross-correlation value is
0. This reflects the fact that both signals are uncorrelated, but actually they are time
shifted. If ―simple‖ cross-correlation is used as similarity measure, then a half squarewave is again associated with a half sine-wave.
Thus, the use of cross-correlation sum operator is proposed. Cross-correlation
sum operation between two signals is not just simply template matching, but also taking
the temporal structure of the two signals into consideration, hence shift invariance. It is
also noise tolerance, as will be shown. Feature extraction is not required, and original
vibration data are used without synchronization. This allows the original data to speak for
itself. The following elaborates the advantages of this similarity measure, where it is both
shift-invariant and noise-tolerant.
For two bounded signals p(t) and q(t), the cross-correlation is defined as,
1
R pq ( ) lim
T 2T
T
p(t ) q(t ) dt
(5.3)
T
where τ is the time lag of the two signals. To implement the cross-correlation similarity
measure for discrete signals, the cross-correlation sum operator is used [117],
62
R pq ( )
N 1
p (t ) q (t ).
(5.4)
t 0
The Normalized Cross-Correlation sum operator is given by normalizing Eq. (5.4) with
the l2-norm of the respective template and feature vector,
N 1
p (t ) q (t )
R pq ( )
t0
p (t ) q (t )
.
(5.5)
The cross-correlation sum Eq. (5.5), is computed at every time interval τ, and produces a
vector of cross-correlation sum coefficients rpq(τ) of 2N-1 dimension with value between
[-1 1] as in Eq. (5.6) [117],
rpq ( ) [r1
r2 r2 N 1 ], rpq 1x 2 N 1 .
(5.6)
This way, the time lag between p(t) and q(t) is considered, hence shift-invariant.
A value of 1 indicates perfect match, a value of 0 indicates no relationship, and a value of
–1 indicates direct opposite. Figure 5.3 shows the superimposed plot of the normalized
cross-correlation sum coefficients between a pair of unbalanced rotor signatures (denoted
by UBR/UBR), and between an unbalanced rotor and a healthy signature (denoted by
UBR/HTY).
63
Figure 5.3: Normalized cross-correlation sum coefficients of a pair signature.
The cross-correlation sum between two similar signatures (UBR/UBR); the blue
plot produces higher cross-correlation sum coefficients than between two dissimilar
signatures (UBR/HTY), the magenta plot. Therefore, the similarity measure between any
two signatures is the maximum of vector rpq(τ), from Eq. (5.6), that is,
S xc ( p, q) arg max[r1 r2 rj ] .
(5.7)
j
Cross-correlation sum operator is able to work on noisy signals. The following
illustrates this.
64
Let us add zero-mean additive noise n(t), to deterministic signals p(t) and q(t), and
normalized the expression Rpq(τ) by 1/N , where N is the length of the signal,
1
N
R pq ( )
N 1
[ p (t ) n (t )][q (t ) n (t )]
t 1
1 N 1
[ p (t ) q (t ) n (t ) q (t )
N t 1
p (t ) n (t ) n (t ) n (t )]
E [ p (t ) q (t )] E [n (t ) q (t )]
E [ p (t ) n (t )] E [n (t ) n (t )]
(5.8)
where E[·] is the expectation operator. Since the noise is zero-mean, p(t) and q(t)
deterministic, and with large N, the terms involving the noise in Eq. (5.8) are destroyed.
That is,
E [n (t )] E [q (t )] 0
E [ p (t )] E [n (t )] 0
E [n (t )] E [n (t )] 0 .
(5.9)
Ultimately, Eq. (5.8) is reduced to Eq. (5.10), which is the same expression as in Eq.
(5.4), except with an additional normalizing constant term 1/N. Therefore Rpq(τ), is
recovered without the noise term n(t), hence noise tolerance.
R pq ( ) E [ p (t ) q (t )]
1 N 1
p (t ) q (t )
N t 1
(5.10)
By using Eq. (5.5), the time shift of the two signals is accounted for. The
similarity measure indicates that v2(t) and v3(t) have a perfect match with a maximum
cross-correlation sum value of 1, hence v3(t) is correctly classified as v2(t). As such, to
65
take into consideration the temporal characteristic of these machine signatures, Eq. (5.7)
is used as a similarity measure in this study. Table 5.1 shows the resulting similarity
measures Se(p,q), Ssc(p,q) and Sxc(p,q).
Table 5.1: Similarity Measure for v1, v2 and v3
Se(p,q)
5.2
Ssc(p,q)
Sxc(p,q)
v1(t) / v2(t )
0.1109
0.9192
v1(t) / v3(t )
0.3115
0
0.9192
v2(t) / v3(t))
0.3568
0
1.0000
Machine Fault Simulator
To simulate the two fault modes in this study, a machine fault simulator (MFS), as
shown in Figure 5.4, by SpectraQuest®, is used. It is a 1/2 HP 239V/50Hz/3-phase AC
variable speed synchronous machine.
Tachometer Vibration sensor
Machine driver
VSI Speed
Controller
Figure 5.4: Machinery Fault Simulator (MFS) by SpectraQuest®, Inc.
Figure 5.5 shows the schematic of the experimental setup.
66
VSI
230Vac
CPU
Dewetron DAQ
CT
Figure 5.5: Schematic of machine signature acquisition using DAQ by Dewetron®.
The template database contains 270 signatures of bearing fault (denoted by BRG),
unbalanced rotor fault (denoted by UBR), and healthy machine (denoted by HTY) with
90 samples for each class. 10 samples are recorded at each machine operating speed of
15Hz, 17Hz, 19Hz, 21Hz, 23Hz, 25Hz, 27Hz, 29Hz and 31Hz. BRG class consists of a
mixture of rolling element fault, inner raceway fault and outer raceway fault. Similarly,
the test set contains 270 unseen signatures with 90 samples for each class. 10 samples are
recorded at each machine operating speeds of 16Hz, 18Hz, 20Hz, 22Hz, 24Hz, 26Hz,
28Hz, 30Hz and 32Hz.
67
Vibration sensor
Faulty
bearing
Loader
Figure 5.6: Bearing fault simulation using MFS.
Figure 5.6 shows the experimental setup to simulate the various bearing fault. The
acceleration of the machine vibration is measured by a stud-mounted piezoelectric
accelerometer of 10kHz bandwidth above the bearing journal. Vibration signatures are
recorded by DEWETRON® digital data acquisition system with a sampling frequency of
5kHz using Hanning windowing. Each data block of the vibration signature is 4000 in
length, representing a snap-shot time window of 800ms.
Vibration sensor
Rotor with screws
attached
Figure 5.7: Unbalanced rotor fault simulation using MFS.
68
Figure 5.7 shows the experiment setup to simulate unbalanced rotor fault. Two
rotors with screws attached are mounted on the shaft. The weight of these screws
generates unequal centrifugal forces when the rotor spins, thus creating vibrations due to
rotor unbalance.
5.3
Machine Signatures Collection
The example database contains 270 signatures of bearing fault (BRG), unbalanced
rotor fault (UBR), and healthy machine (HTY) with 90 samples for each class, where 10
samples are recorded at each machine operating speed of 15Hz, 17Hz, 19Hz, 21Hz,
23Hz, 25Hz, 27Hz, 29Hz and 31Hz. BRG class consists of a mixture of rolling element
fault, inner raceway fault and outer raceway fault. The test set contains 270 unseen
signatures with 90 samples for each class. To demonstrate the ability of cross-correlation
sum operator to work on unseen data, 10 samples are recorded with a different machine
operating speeds for the training set at 16Hz, 18Hz, 20Hz, 22Hz, 24Hz, 26Hz, 28Hz,
30Hz and 32Hz.
Table 5.2: Training and test sets for k-NN classification.
Training Set
fs (Machine Operating Frequency, Hz)
Sample size
Healthy (HTY)
15/17/19/21/23/25/27/29/31
90
Bearing fault (BRG)
15/17/19/21/23/25/27/29/31
90
Unbalanced rotor (UBR)
15/17/19/21/23/25/27/29/31
90
fs (Machine Operating Frequency, Hz)
Sample size
Healthy (HTY)
16/18/20/22/24/26/28/30/32
90
Bearing fault (BRG)
16/18/20/22/24/26/28/30/32
90
Unbalanced rotor (UBR)
16/18/20/22/24/26/28/30/32
90
Test Set
69
5.4
Experimental Results
The classification result is presented in Table 5.3 and 5.4. An overall error rate of
0.74% is achieved. For UBR class the error rate is 2.22%, and for both BRG and HTY
classes the error rate is 0%. The average computation time per test sample is 0.2s with a
feature vector dimension of 4000.
Table 5.3: Fault classification confusion matrix.
Predicted Class
UBR
BRG
HTY
Error Rate (%)
Ground
UBR
88
0
2
2.22 %
Truth
BRG
0
90
0
0%
HTY
0
0
90
0%
Table 5.4: Classification result summary.
k (nearest neighbor):
1
Similarity measure:
Eq. (6.7)
Overall error rate (%):
0.74 %
Feature vector dimension:
4000
Template sample size:
270
Test sample size:
270
Computation time per test sample:
0.2 s
The value of k is varied to locate an optimal value for k. Odd values of k are used
deliberately so as to avoid a tie in the voting.
70
Table 5.5 tabulates the classification error rate at different k values.
Table 5.5: Tabulated results of error rate (%) with various k-neighbor values.
k
Error Rate (%)
1
0.74 %
3
1.85 %
5
1.48 %
7
0.74 %
9
1.11 %
11
0.74 %
Error Rate (%) Vs k-neighbor
2.00%
1.80%
1.85%
1.60%
Error Rate (%)
1.40%
1.48%
1.20%
1.00%
1.11%
0.80%
0.60%
0.74%
0.74%
0.74%
0.40%
0.20%
0.00%
1
3
5
7
9
11
k-neighbor
Figure 5.8: Error rate (%) versus k-neighbor values.
5.5
Discussions: Difficulty in choosing a suitable value for k
The best error rate is achieved at k equals 1, 7 and 9 where the error rate is 0.74%.
As more nearest neighbors are being enclosed by the Voronoi cell, the error rate
increases. A higher value of k provides a smoother decision boundary and provides
71
probabilistic information, thus tends to give better generalization for classification of
unseen input feature vectors. However, with a higher k, the Voronoi cell is increased;
encompassing other categories in the process, and this destroys the locality of the
estimation. As a result the error rate increases. Evidently it also increases the
computational burden. In this project, at k equals 7 and 11, the error rate reduces as more
prototype of the same class is enclosed. However, from Figure 5.8, a good value of k is
equal to 1 which produces the lowest error rate with least computational burden.
5.6
Discussions: Larger Training Samples
k-NN works well for multi-modal distribution, provided that the training sample
is sufficiently large, so that the probability distribution function is as continuous as
possible. With larger training samples, computational burden and memory requirement
are increased as well. This is especially so when the dimension of the feature vector is
high. For this project, even though the dimension of the feature vector is high at 4000, the
average execution of 0.2 second for one feature vector.
5.7
Visualization of Classification Results by k-NN
To provide visual inspection of the classification, plots of the classification results
are made, as shown in Figure 5.9, 5.10, 5.11 and 5.12. Subplot 1 shows the test vector to
be classified. The symbol ―=>‖ denotes ―classified as‖. Subplot 2 shows the template that
the test vector is associated with. Subplot 3 is a graph of maximum cross-correlation sum
values (denoted by max xc) versus the location of the prototype in the database. It shows
the ―winner‘s‖ max xc value, and its location among the 270 templates in the database.
72
Figure 5.9: Unbalanced rotor fault misclassified as healthy machine.
In Figure 5.9, a UBR signature is misclassified as HTY signature, as the
―winner‘s‖ max xc value is 0.13547 and belongs to HTY class.
Figure 5.10: Healthy machine signatures correctly classified.
73
Fig. 5.10 shows a correctly classified HTY signature. The ―winner‘s‖ value is
0.11365. This value is the highest among the templates, hence the ―winner‖ belongs to
HTY signature and class prediction is HTY. Visual inspections of subplot 1 and 2
verified this.
Figure 5.11: Unbalanced rotor signature correctly classified.
Figure 5.11 shows a correctly classified UBR signature. The ―winner‘s‖ value is
0.17628. This value is the highest among the templates and belongs to UBR class.
Therefore, the test vector is predicted to be UBR class.
74
Figure 5.12: Unbalanced rotor signature correctly classified.
Fig. 5.12 shows a correctly classified BRG signature. In this case, k is 11 and
subplot 4 shows the voting scheme. With 11 against 0 for HTY class, the predicted class
is therefore HTY class.
5.8
Fault Diagnosis using Frequency-Domain Vibration Signatures
This section presents vibration analysis in frequency-domain. To study vibration
spectrum in frequency-domain, noise in time-domain signature is removed so as to reveal
the spectrum that are accountable for the vibration. There are various ways to de-noise
signal e.g., ensemble synchronized averaging, band specific filtering etc. In this project
an adaptive filter is used. An adaptive filter is preferred instead of using a low-pass or
75
high-pass filter, because these band specific filters simply remove all the stop-band
frequencies, and this may also remove frequencies that may contain fault information.
5.8.1 Discrete Wiener Filter
The vibration signatures are filtered by a Discrete Wiener filter. Discrete Wiener
filter is an adaptive filter that adjusts its filter coefficients, or filter weights wk[i] i.e., the
weights at instant ith, to produce an estimated output ŷ[k], that is free from uncorrelated
components i.e., the correlated component estimate, and this estimated output is an
optimal estimate of the original signal, in the least-mean-square (LMS) error sense. This
LMS error ek, the uncorrelated components, is computed iteratively using successive
samples of the input signal y[k], by Widrow-Hoff LMS algorithm, show below [117],
N 1
yˆ [k ] wk [i ]x[k i ],
(5.11)
i 0
ek y [k ] yˆ [k ],
wk 1 [i ] wk [i ] 2 ek x[k i ].
(5.12)
(5.13)
where γ is the ―nudge-to-zero‖ constant to prevent drift of the signal from optimal or to
grow in magnitude during adaptation, especially for long signals, and μ is a constant to
adjust the rate of convergence of the weights to optimal values.
76
Figure 5.13 shows the scheme [117],
y [k ] yˆ [k ] ek [k ]
Delay
N 1
yˆ [k ] wk [i]x[k i]
ek [k ] y [k ] yˆ [k ]
i 0
Wiener
Filter
Figure 5.13: Wiener Filter.
The feedback error signal ek, is used to adjust the Wiener filter‘s coefficients and
generates an estimated signal ŷ[k], that is free from uncorrelated components and noise.
Therefore, by using an adaptive filter, the temporally correlated deterministic portion of
the signal is preserved, and is separated from the uncorrelated stochastic portion of the
signal and noise. In the first 100ms, the filtered signal is flat and appears compressed.
This is due to the adaptation algorithm finding convergence. After about 100ms, the
algorithm converges. Figure 5.14 shows this effect.
Figure 5.14: Filtered machine signatures at fr=15Hz and 31Hz.
77
5.8.2 Frequency Analysis of Vibration Signatures
From the mechanics of the vibration signals, an unbalanced rotor fault generates a
low frequency of one-times (i.e., commonly refers as 1X) rotor shaft rotational speed due
to the forced-vibration created by the unbalanced mass [14], and bearing faults generate a
high frequency ―ring‖ due to the free-vibration created by the impact of single-defect
surface [90, 93]. To examine the frequency content of the vibration signatures, the timedomain signals are transformed into the frequency-domain representation by discretetime Fourier transform is given by [117],
X ( )
x[n] e j n .
(5.14)
n
Figure 5.15 to 5.20 shows the ―before‖ and ―after‖ filtering of the machine
vibration signatures. Subplot 1 is the original spectrum and subplot 2 is the filtered
spectrum. The interesting frequency range is highlighted in red.
5.8.2.1
Frequency Content of Healthy Machine
After filtering healthy machine vibration signature, the interesting frequency is in
the range of 100-900Hz. The vibration is mainly due to metal-to-metal sliding contacts of
the bearing assembly and other inherent constructional asymmetries. The dominant
frequencies are clearly visible in the range 200-400Hz.
78
Figure 5.15: Frequency content of HTY signatures at fr=15Hz.
However, as the speed increases, additional spectrum appears in the range of 600900Hz and become dominant, as higher rotational speed causes more free-vibrations due
to metal-to-metal contacts. However, the magnitude is low in the order of 10-3.
Figure 5.16: Frequency content of HTY signatures at fr=31Hz.
79
5.8.2.2
Frequency Content of Unbalanced Rotor Fault
The 1x rotor rotational speed is clearly visible in the 0-60Hz range. This dominant
frequency is responsible for the time-domain signal characteristic sinusoidal oscillation.
Figure 5.17: Frequency content of UBR signatures at fr=16Hz.
Figure 5.18: Frequency content of UBR signatures at fr=32Hz.
80
5.8.2.3
Frequency Content of Bearing Faults
The high frequencies ―rings‖ created by the free-vibration from single-defect
impacts on metallic surface are clearly visible in the range of 800-1100Hz and 19002100Hz. These are unique to bearing faults. Figure 5.19, 5.20 show this.
Figure 5.19: Frequency content of BRG signatures at fr=15Hz.
As the rotational speed of the machine increases to 32Hz, the spectrum in the
range 800-1100Hz and 1900-2100Hz becomes dominant, as the ―rings‖ impact on the
metallic surfaces increases in force and magnitude.
81
Figure 5.20: Frequency content of BRG signatures at fr=32Hz.
5.8.2.4
Discussions on Vibration Frequency Analysis
From the above figures, it is observed for healthy machine, that the regions of
interest are 0-60Hz and above 600Hz, in that there is very low magnitude or no spectra in
these regions. For unbalanced rotor fault, the regions of interest are the low frequency 060Hz, mainly due to the forced-vibrations as a result of the unbalanced mass at 1X
synchronous machine speed. For bearing fault, the regions of interest are in the high
frequencies of 800-1100Hz and 1900-2100Hz, mainly due to the significant freevibrations as a result of the impact of the single-point defect. Table 5.6 summaries this
observation.
82
Table 5.6: Distribution of spectrum of machine vibration signatures.
Fault Class
Frequency Range (Hz)
Healthy machine
Bearing fault
Unbalanced rotor
0 – 60
800 – 1100
1900 – 2100
LOW
HIGH
LOW
0 – 60
800 – 1100
1900 – 2100
LOW
HIGH
HIGH
0 – 60
800 – 1100
1900 – 2100
HIGH
LOW
LOW
In the next, feature extraction is carried out to condense the 2500-dimensional
frequency vector into a compact 11-dimensional feature vector.
5.8.3 Feature Extraction of Frequency domain information
Feature extraction aims to reduce the dimensionality of the input vector, such that
the reduced dimension vector contains all the necessary data for classification, and is a
compact and faithful representation of the original input vector. Using the region of
interest, the spectrum of each signature is divided into 11 equal segments. The rootmean-square (RMS) value of each segment is computed. The resulting 11 values thus
obtained, are arranged in an 11-dimensional vector as in Eq. (5.15). This is the feature
vector for the vibration signatures.
SOM _ feature_ vector fea1
fea2
fea11 .
(5.15)
Figure 5.21 shows some of the feature vectors.
83
Figure 5.21: 11-dimensional feature vector at fr=15Hz and 31Hz.
As observed, unbalanced rotor fault produces a feature vector that has a high
value in the first dimension (fea1) and very low value for the remaining 10 dimensions
i.e., fea2, fea3…..fea11, whereas the healthy machine has a very low value for fea1 and
fea9, and bearing fault has a very low value for fea1 and high value for fea9. With feature
extracted, SOM is next used to visualize this 11-dimensional data and to discover
clustering for classification in an unsupervised way.
5.8.4 Cluster Analysis of Vibration Feature Vectors
SOM is used to study the cluster of vibration feature vectors in an unsupervised
way. The input to the SOM consists of 540 11-dimensional feature vectors from
unbalanced rotor signatures, healthy signatures and bearing faulty signatures. These
vectors are arranged columnwise to form an 11x540 input matrix, are normalized to value
of [0 1] and submitted to the SOM for learning. After the learning process, a map is a
84
created that groups the input feature vectors according to their topological characteristics,
and hence clustering of vectors from the same class is formed.
Figure 5.22: Semantic map of vibration signatures from two SOM different simulations.
Figure 5.22 shows two ―fault‖ maps obtained by two separate simulation runs,
each with the neuronal weights initiated differently and randomly. Each map shows three
distinct clusters, each belonging to a particular machine condition status class, namely
unbalanced rotor, healthy machine and bearing fault. All the 540 feature vectors have
been packed into a 10x10 grid, in three distinct clusters. Even though the clusters in both
figures are oriented differently, its non-overlapping distinctness in clusters is preserved.
Therefore SOM has discovered three distinct clusters.
5.8.5 Further Feature Extraction
Instead of using an 11-dimensional feature vector, further feature extraction is
carried out to derive a set of unique 2-dimensional feature vectors (fea_2D_SVM), using
only the first (fea1) and ninth feature (fea9) only.
85
fea _ 2 D _ SVM fea1
fea9 .
(5.16)
Figure 5.23 shows the plot of fea1 versus fea9. Three distinct clusters are observed. The
left panel shows the zoom-in view.
Figure 5.23: 2-dimensional feature vector.
5.8.6 Multi-class SVM (M-SVM) for Classifying Machine Fault Data
M-SVM is next used to classify the 2-dimensional machine fault feature vector
(fea_2D_SVM). Even though SOM is able to perform clustering operation as
demonstrated in the preceding section, its decision boundary obtained by neighborhood
function learning is not optimal. M-SVM is a classifier whose separating hyper-plane is
derived based on structural risk minimization and the optimization of the support vectors
from the different classes, and thus optimal. Figure 5.24 shows the hyper-plane using a
Guassian kernel width (h) of 8.0 and a slack factor (C) of 0.1. An error rate of 1.48% is
achieved. Table 5.7 and 5.8 show the fault classification result and confusion matrix. The
training set is shown as circle ―○‖ and the test set as ―+‖. The training samples with black
circle are the support vectors.
86
Figure 5.24: M-SVM classification (Gaussian kernel hsvm=8.0, slack factor C=0.1) of vibration
signature.
Table 5.7: Fault classification confusion matrix of vibration signature.
Predicted Class
UBR
BRG
HTY
Error Rate (%)
Ground
UBR
84
4
0
4.44 %
Truth
BRG
0
90
0
0%
HTY
0
0
90
0%
87
Table 5.8: M-SVM classification of vibration signatures result summary.
hsvm (kernel width):
8.0
C (slack factor):
0.1
b1 from class 1 versus 2 and 3:
1.019
b2 from class 2 versus 1 and 3:
1.134
b3 from class 3 versus 1 and 2:
-0.920
Number of Support Vectors:
Error rate (overall):
Feature vector dimension:
1.48%
2
Template sample size:
270
Test sample size:
270
Computation time per test sample:
5.9
342
40.38 s
Discussions: Frequency-domain Analysis of Vibration Signatures
By selecting interesting features in frequency-domain, the 2400-dimensional
feature vector is reduced to 2-dimensional feature vector. Such dimensionality reduction
has not degraded the diagnostic information. This is shown by M-SVM automatically
classifying the 2-dimensional feature vectors with an error rate of 1.48%. Bearing fault
generates high frequency vibrations while unbalanced rotor creates dominant 1x machine
rotation speed frequency. Therefore, by examining vibration signals in these frequency
ranges, bearing and unbalanced rotor faults can be effectively detected and diagnosed.
88
CHAPTER 6: A STUDY ON MOTOR CURRENT
SIGNATURE USING EMPIRICAL MODE
DECOMPOSITION
This chapter presents the study and experiments carried out using EMD
technique, to discover new harmonic information about faulty machine signatures. This
approach is unlike all previous methods which use Fourier transform harmonic analysis
and wavelet decomposition as shown in the literature survey, and also in [86, 118], where
EMD is used to study one fault mode only. The objective is to identify harmonics that are
unique to each fault signature so as to validate that different machine fault indeed
generates a unique harmonic in the motor current. In the next, rationale for the use of this
technique is explained.
6.1
Fourier Transform
Fourier transform is the most well-known and most commonly used tool for
signal analysis. The main assumption is that any arbitrary square-integrable signal, x(t)
consists of an infinite sum of simple sinusoidal basis functions, as follows [117],
xˆ (t )
c e
n
n
i
2nt
T
,
(6.1)
Figure 7.1 illustrates Eq. (6.1). It shows a 10Hz square wave (black plot) being
approximated by the sum of 3 terms sinusoids (blue plot) and 10 terms sinusoids (red
plot). The ‗wiggles‘ represents information distortions.
89
Figure 6.1: Fourier Series (a finite sum of a 10Hz square wave with n=3 and n=10).
Each of these basis functions is a complex exponential, each of a frequency that is
an integer multiple of 1/T i.e., a family of harmonics. This assumes periodicity of the
signal, and in the case of non-periodicity signal, the integral is extended from positive
infinity to negative infinity. cn is a series of weighting coefficients; the Fourier
coefficients, where the optimal value for cn, is the correlation between x(t) and e-i2πnt/T
over the interval [0 T]. Thus [117],
F x(t ) cn
T
i
1
x
(
t
)
e
T 0
2nt
T
dt.
(6.2)
which gives the Fourier Transform.
90
6.2
Wavelet Transform
Similarly, wavelet transform decomposes a square-integrable signal x(t) into a
family of wavelet bases ψs,τ(t) and weighting coefficients W(s,τ).
xˆ (t )
1
C
1 t
s s
W (s, )
0
ds d .
(6.3)
where Cψ is a constant depending on the base function. Thus, the wavelet transform is,
1
W ( s, )
s
x(t )
*
t
d ,
s
x(t ) L2 ( R)
(6.4)
where the wavelet basis function ψa,b(t) is,
s , (t )
1 t
s s
.
s 0,
(6.5)
In the above expression, s is the scale parameter inversely related to frequency, τ
is the translation in time, and ψ(t) is called the prototype (―mother‖) wavelet and ψs,τ(t)
the daughter wavelets. A unique property of a wavelet is that is has to be zero-mean on
the real line that is localized in time and frequency. In order to fulfill the zero-mean
condition, a wavelet thus needs to be oscillatory. Also, the wavelet transform has better
localization at low frequency than higher frequency components. Therefore, wavelet
transform is functions of time and frequency, describing the information on x(t) at various
time window and frequency bands, allowing information of x(t) on frequency and
amplitude variations at different time to be captured, that is, it reveals non-stationary
information of x(t). Figure 6.2 illustrates some of the different types of wavelet basis
functions.
91
Figure 6.2: Different wavelet basis functions.
6.3
Hilbert-Huang Transform
In order to study the non-linearity and non-stationary of x(t), the time-frequency-
energy model of x(t) is required. Fourier transform is unable to reveal such information.
A possible way to describe non-stationary of x(t) is to find instantaneous frequency and
instantaneous amplitude. Hilbert transform is able to yield this information. Hilbert
transform is another integral transform.
Instead of using a sinusoidal basis function for convolution with x(t), a kernel
1/π(t-η) and Cauchy Principal integral is used,
x(t )
1
P x(t )
1
d ,
t
x(t ) L2 ( R)
(6.6)
Cauchy Principle Integral is an integration technique (by Cauchy) that solves the
singularity problem at point t=η.
From the Hilbert transform Eq. (6.6), the analytic function z(t) is obtained,
z(t ) x(t ) i x(t ),
(6.7)
92
With Euler‘s identity, the analytic function z(t) is expressed alternatively as,
(6.8)
z (t ) a(t ) e i (t ) .
where a(t) is the instantaneous amplitude, θ(t) is the instantaneous phase function and ω
is the instantaneous frequency,
a(t ) x(t ) 2 2 x(t ),
x(t )
,
x(t )
(t ) tan 1
d
. (6.9)
dt
In Hilbert-Huang transform, IMFs is the x(t). Using instantaneous amplitude and
instantaneous frequency information, the x(t), can be expressed as a function of
instantaneous frequencies ωj(t) and amplitudes aj(t) of the IMFs thus derived using the
aforementioned EMD algorithm, as
n
i ( t ) dt
x(t ) Re a j (t )e j
j 1
(6.10)
where Re[·] is the real part of the term aj(t)e iωj(t)dt.
6.3.1 Hilbert Spectrum
With Hilbert-Huang transform and the instantaneous quantities aj(t) and ωj(t), a
two-dimensional graph of Frequency versus Time is plotted with the amplitude aj(t)
shown as colour bar. This graph is the Hilbert spectrum, and shows how the frequency of
the IMF varies with time.
93
6.3.2 Marginal Hilbert Spectrum
By summing the various amplitude aj(t) of the IMF of the same frequency across
different time span, as follows,
T
h( ) H ( , t )dt.
(6.11)
0
and a graph of Amplitude versus Frequency can be plotted. This graph is the Marginal
Hilbert spectrum. Marginal Hilbert spectrum shows the distribution of various
frequencies and the along the x-axis and amplitude of each frequency on the y-axis.
Nonlinear and non-stationary information about each IMF is observed by analyzing the
Hilbert spectrum and Marginal Hilbert spectrum.
6.4
Discussion: EMD as a suitable Analysis Tool
From Eq. (6.2), it is clear Fourier decomposition depends heavily on the pre-
defined sinusoidal basis function and has a strong a priori assumptions about x(t) of being
linear, periodic and stationary. From the nature of the inner product of two functions,
Fourier transform is the project of x(t) onto the sinusoidal basis function, and hence there
is no guarantee that the particular choice of sinusoidal basis function produces good
transformation. Motor current signal is non-stationary, i.e., fault frequencies are a
function machine slip, speed and loading conditions, thus changes as the loading
conditions vary. To analyze the motor current spectrum using FFT is very difficult to
locate the fault frequencies precisely and accurately, unless the machine operating
conditions are held constant in steady state conditions. Also, during FFT transformation,
the temporal relationship between the frequency and time is lost.
94
Similarly, from Eq. (6.4), wavelet transform at scale s and time τ is a projection of
x(t) onto the wavelet with scale s and time shift τ, and hence shows how similar x(t) is to
that particular wavelet. Therefore, the decomposition relies on the good choice of the a
priori assumed wavelet basis function. Good decomposition results is obtained if the
wavelet function with similar features as x(t) is used, hence skillful and careful selection
of the wavelet basis function is not trivial [52].
Many real-world
data
series
is
non-linear
and
non-stationary
[119].
Transformation of x(t) based on these simplistic assumptions clearly lead to information
loss. EMD method, in contrast, is not constrained by this, as it is free of predefined basis
function, but derived them directly from actual empirical data i.e., x(t), in the form of
iterating sifting processes (Eq. (4.15-4.20)), to derive the IMFs in an adaptive fashion. As
such, the IMFs thus derived are the basis functions, and allow for the direct reconstitution
of x(t) by Eq. (6.10).
6.5
N-A M-EMD Experiment Results
In this experiment, motor current signatures of healthy machine (denoted by
HTY), Unbalanced rotor fault (UBR), Bearing fault (BRG), Broken rotor bar (BRB) and
Shorted stator winding fault (SWF) at machine operating speed of 20Hz, 30Hz and 40Hz;
almost spanning the entire normal machine operation speed, are studied. To avoid loss of
harmonic information in the time-domain signature, a higher sampling rate of 100kHz is
used. Three sets of signatures at machine operating speed of 20Hz, 30Hz and 40Hz, and
two sets of white Guassian noise, are submitted to N-A MEMD algorithm, to extract
95
harmonic information for a 5-class signature simultaneously. Figure 6.3 illustrates the
design of the multi-variate EMD problem.
HTY
channel
1
BRG
channel
2
n1
channel
6
BRB
channel
3
n2
channel
7
UBR
channel
4
IMF 1
IMF 1
IMF 1
IMF 2
SWF
channel
5
IMF 1
IMF 2
IMF 1
IMF 1
IMF 1
IMF 2
IMF 2
IMF 2
IMF 3
IMF 2
IMF 3
.
.
.
.
.
IMF 2
IMF 3
.
.
.
.
.
IMF n
IMF 3
.
.
.
.
.
IMF n
IMF 3
.
.
.
.
.
IMF n
.
.
.
.
.
IMF 3
IMF 3
.
.
.
.
.
.
.
.
.
.
IMF n
IMF n
IMF n
IMF n
Figure 6.3: A 7-channel Motor Current Signature decomposition by N-A MEMD.
Two channels of white Gaussian noise, channel 6 and 7, denoted by n1 and n2,
are added to this 7-channel presentation of multi-variate EMD. Appendix A shows the
EMD of HTY20, BRG20, BRB20, UBR20, SWF20 and added noise n1, n2
96
6.5.1 Discussions: IMF Derived by EMD
It is observed that the decomposition has derived eleven IMFs. With mode mixing
and mode misalignment problems resolved, same-indexed IMFs across different motor
current signatures are compared. To study the intrinsic modes derived by EMD
algorithm, a pair of IMF from different machine signatures is compared, such as, between
HTY20 IMF1 and BRG20 IMF1, HTY20 IMF1 and UBR20 IMF1, HTY20 IMF1 and
SWF IMF1 etc. To have an objective comparison between the same-indexed IMFs across
different signatures, the normalized cross-correlation sum between these same-indexed
pairs of IMFs are computed to determine their similarity to each other, as shown in Eq.
(6.11),
N 1
R pq ( )
p (t ) q (t )
t 0
(6.11)
p (t ) q (t )
Next a vector rpq, with each element of the vector a normalized cross-correlation
value, evalauated at every time interval τ, is form as shown in Eq (6.12),
rpq ( ) [r1 r2 r2 N 1 ], rpq 1 x 2 N 1
(6.12)
The similarity measure between any two pair of same-indexed IMFs motor
current signatures is the maximum of the rpq vector, that is,
S ( p, q) arg max[r1 r2 rk ]
k
(6.13)
Using Eq. (6.11), (6.12) and (6.13), the following cross-correlation results is tabulated.
97
Table 6.1: Similarity measures of same-indexed pair of machine current IMFs at 20Hz.
HTY/BRG:
HTY/BRB:
HTY/UBR:
HTY/SWF:
BRG/BRB:
BRG/UBR:
BRG/SWF:
BRB/UBR:
BRB/SWF:
UBR/SWF:
IMF2
0.7260
0.7177
0.7234
0.7280
0.8934
0.9211
0.9206
0.9970
0.9964
0.9985
IMF3
0.9640
0.9575
0.9621
0.9632
0.9751
0.9851
0.9862
0.9983
0.9977
0.9997
IMF4
0.8661
0.8671
0.8682
0.8664
0.8874
0.9043
0.8874
0.9949
0.9979
0.9987
IMF5
0.0854
0.1194
0.1206
0.1174
0.8518
0.8936
0.8784
0.9863
0.9899
0.9885
IMF6
0.0655
0.0981
0.0954
0.0747
0.6599
0.6451
0.6904
0.7383
0.8143
0.9263
IMF7
0.1535
0.2442
0.1743
0.1806
0.3290
0.3417
0.3800
0.5646
0.6784
0.8792
IMF8
0.5187
0.5746
0.5874
0.5969
0.6546
0.6713
0.6435
0.7322
0.7861
0.9052
IMF9
0.7185
0.7741
0.7771
0.7871
0.9081
0.9289
0.9166
0.9727
0.9747
0.9893
IMF10
0.9873
0.9866
0.9875
0.9871
0.9958
0.9970
0.9966
0.9992
0.9995
0.9998
Table 6.2: Similarity measures of same-indexed pair of machine current IMFs at 30Hz.
HTY/BRG:
HTY/BRB:
HTY/UBR:
HTY/SWF:
BRG/BRB:
BRG/UBR:
BRG/SWF:
BRB/UBR:
BRB/SWF:
UBR/SWF:
IMF2
0.7125
0.7131
0.7825
0.7039
0.9775
0.7111
0.9352
0.7052
0.9653
0.7035
IMF3
0.9495
0.9501
0.9613
0.9438
0.9983
0.9451
0.9840
0.9449
0.9845
0.9500
IMF4
0.9522
0.9530
0.9444
0.9497
0.9706
0.9474
0.9571
0.9514
0.9939
0.9481
IMF5
0.1039
0.1535
0.2107
0.1315
0.9080
0.1600
0.8794
0.1790
0.9591
0.1823
IMF6
0.0707
0.1752
0.2359
0.1476
0.7278
0.1101
0.7214
0.1312
0.8257
0.1431
IMF7
0.3328
0.4391
0.5588
0.4465
0.3886
0.3662
0.3593
0.4968
0.4209
0.4607
IMF8
0.6430
0.7347
0.9350
0.7307
0.8438
0.6470
0.8451
0.7407
0.9313
0.7426
IMF9
0.6111
0.7472
0.9644
0.7679
0.5799
0.5685
0.6141
0.7341
0.9612
0.7549
IMF10
0.9906
0.9924
0.9999
0.9925
0.9975
0.9909
0.9978
0.9923
0.9999
0.9924
Table 6.3: Similarity measures of same-indexed pair of machine current IMFs at 40Hz.
HTY/BRG:
HTY/BRB:
HTY/UBR:
HTY/SWF:
BRG/BRB:
BRG/UBR:
BRG/SWF:
BRB/UBR:
BRB/SWF:
UBR/SWF:
IMF2
0.8143
0.9674
0.9538
0.9794
0.8147
0.9013
0.8516
0.8909
0.9141
0.9919
IMF3
0.9593
0.9897
0.9847
0.9931
0.9622
0.9583
0.9607
0.9624
0.9704
0.9973
IMF4
0.9638
0.9904
0.9837
0.9960
0.9647
0.9669
0.9677
0.9868
0.9895
0.9949
IMF5
0.7301
0.9259
0.8533
0.9392
0.8087
0.8053
0.7688
0.9315
0.9443
0.9462
IMF6
0.6426
0.8156
0.7687
0.8588
0.6701
0.6108
0.6220
0.8317
0.8212
0.8906
IMF7
0.3727
0.7053
0.5129
0.6225
0.3321
0.3323
0.3320
0.6995
0.6210
0.7575
IMF8
0.7563
0.9610
0.9527
0.9627
0.7693
0.7890
0.7703
0.9599
0.9600
0.9740
IMF9
0.6736
0.9668
0.9795
0.9855
0.6522
0.6453
0.6691
0.9770
0.9704
0.9885
IMF10
0.9956
0.9996
0.9995
0.9998
0.9962
0.9966
0.9962
0.9999
0.9998
0.9997
98
6.5.2 Discussions: Filter-bank Property of EMD Algorithm
From Table 6.1, 6.2 and 6.3, it is observed that IMF5 to IMF9 are unique having
low maximum cross-correlation sums, whereas IMF2-4 and IMF10 closely resemble with
each other as their maximum cross-correlation sums having high score, and hence not
unique. Since IMF 5 to 9 are unique, analysis is focus on these intrinsic modes. Figure
6.4 shows the Hilbert spectrum and Marginal Hilbert Spectrum of machine signatures at
30Hz machine speed in subplot 1 and subplot 2 to 6 respectively, with amplitude in logscale.
Figure 6.4: EMD as filter-banks for HTY30 (IMF 5 – 9) machine current signature.
99
It is also observed that the sifting process the EMD algorithm has in fact separated
the signatures into a few frequency bands, acting essentially like a ―wavelet-like‖ filter
[120], with the highest frequency band associated with lower indexed IMF. Table 6.4
summaries the frequency bands associated with each IMF.
Table 6.4: Frequency band for HTY30 (IMF 5-9) machine current signature.
IMF
Frequency range (Hz)
5
1000 - 9000
6
900 - 4000
7
300 - 2000
8
200 - 1000
9
150 - 600
The Hilbert spectrum shows the spectrum activities as function of time, where
each color dot in the plot represents a frequency at a particular point in time. It shows that
the machine signature is a complicated signal producing a wide range of frequencies.
However, the main spectra activity is around 100Hz to 800Hz from IMF8 and IMF9.
With the effective separation of the harmonic into separate frequency bands, it allows the
identification of interesting unique spectrum for the study of signature, and to discard the
common intrinsic modes that show the same features across different signatures. The
following section elaborates further on this.
100
6.5.3 Discussions: Significance of IMF1, IMF2, IMF3, IMF4
IMF1 is the added white Guassian noise and is discarded. IMF2, IMF3 and IMF4
are the common intrinsic modes to all the signatures, since their maximum crosscorrelation sums of these pairs of IMFs have almost the same values with high scores.
They are visually indistinguishable and closely resemble the same-indexed IMFs of other
signatures. They are the common intrinsic modes for the signatures, due primarily to the
harmonics from supply voltage inverter high frequency switchings, high-frequency noise
and the interactions between harmonics.
Figure 6.5: IMF1, IMF2, IMF3, IMF4 of the machine signatures at 20Hz.
101
Figure 6.6: IMF1, IMF2, IMF3, IMF4 of the machine signatures at 30Hz.
Figure 6.7: IMF1, IMF2, IMF3, IMF4 of the machine signatures at 40Hz.
102
6.5.4 Discussions: Significance of IMF10, IMF11
IMF10 is the fundamental supply frequency by the Voltage Supply Inverter (VSI).
EMD has neatly separated the fundamental supply frequency; which is the dominant
frequency of the machine signature analysis. This separation is achieved by the adaptive
sifting process and no notch filter is required. This is a marked difference from all
surveyed literatures where notch filtering is used to remove this dominant frequency.
Figure 6.8: IMF10 and IMF11 (residue) of the machine signatures at 20Hz.
Figure 6.9: IMF10 and IMF11 (residue) of the machine signatures at 30Hz.
Figure 6.10: IMF10 and IMF11 (residue) of the machine signatures at 40Hz.
103
The residue correctly shows a ―long term trend‖ of zero as the preceeding IMF10
is a sinosoid of zero-mean. With the observations discussed thus far, IMF1, IMF2, IMF3,
IMF4, IMF10 and IMF11,which is the residue, are discarded.
6.5.5 Discussions: Significance of IMF5, IMF6, IMF7, IMF8, IMF9
IMF5, IMF6, IMF7, IMF8, IMF9 are of interest, as they show significant
differences in their maximum cross-correlation sum values. Therefore, they are of interest
to the study of identifying uniqueness of different machine fault signatures. The figures
below show some of these IMFs, especially IMF5 to 9 at 20Hz machine speed. Their
corresponding Hilbert spectrum and Marginal Hilbert spectrum are shown in Appendix
B.
Figure 6.11: IMF5-9 of the HTY machine signatures at 20Hz.
104
Figure 6.12: IMF5-9 of the BRG machine signatures at 20Hz.
Figure 6.13: IMF5-9 of the BRB machine signatures at 20Hz.
105
Figure 6.14: IMF5-9 of the UBR machine signatures at 20Hz.
Figure 6.15: IMF5-9 of the SWF machine signatures at 20Hz.
106
6.6
Visualization of the Comparison results by SOM
In this section, the difference between the pair of machine signature is visualized
using SOM. It is well-known that SOM preserves the topological order of multidimensional feature vectors, where feature vectors of the same type clusters together, and
presents such visualization of the clusters in a 2D-grid. Using this unique capability of
SOM, the Euclidean distances among the different pair of same-indexed IMF comparison
results are visualized. The SOM input consists of a 5x10 matrix, with each column is a
feature vector, fea_IMF, created by using the maximum cross-correlation sum of sameindex IMF5 to IMF9 of different signatures, as follows,
fea _ IMF IMF 5
IMF 6
IMF 7
IMF 8
IMF 9
T
(6.14)
In the next, a 20x20 SOM leans the topological order of the feature vectors, and
display their Euclidean distance relationships on a 2D grid. Each dot on the grid
represents a SOM neuron. Figure 6.16, 6.17 and 6.18 show this.
6.7
Discussions: Discovery of Unique Features by SOM
It is observed from Figure 6.16 that HTY/UBR is close to HTY/SWF. However,
all other vectors are further apart and equally spaced. HTY/BRG and HTY/BRB are far
apart. Therefore, at low machine speed of 20Hz, motor current is unable to generates
unique sigantures that allows the diagnosis of four separate machine fault modes under
this study, namely, bearing, unbalanced rotor, broken rotor bars and shorted stator
windings.
107
Figure 6.16: Feature map using fea_IMF vector at fs=20Hz.
It is observed from Figure 6.17, at higher machine speed of 30Hz, HTY/UBR and
HTY/SWF features are farther now than at 20Hz. This indicates that UBR and SWF
signatures are more distinct to HTY signature than at 20Hz. This allows for the diagnosis
of unbalanced rotor fault and shorted stator windings. HTY/BRB and HTY/BRG are
clearly apart and hence allow for the diagnosis for bearing and broken rotor bar faults. All
other feature vectors are far apart and almost equally spaced. This indicates they are
unique.
108
Figure 6.17: Feature map using fea_IMF vector at fs=30Hz.
At even higher speed of 40Hz, HTY/SWF and HTY/UBR vectors are far apart as
at 30Hz, as shown in Figure 6.18. HTY/BRB and HTY/BRG are still clearly apart and
hence allow for diagnosis for bearing and broken rotor bar faults.
109
Figure 6.18: Feature map using fea_IMF vector at fs=40Hz.
EMD has discovered unique non-linear and non-stationary features at machine
operating speed of 30Hz and 40Hz. Unbalanced rotor and shorted stator windings faults
produce similar harmonics at lower speed machine of 20Hz, blurring the uniqueness of
the signatures of unbalanced rotor fault and shorted stator windings, to allow for fault
diagnosis purposes. Therefore, there is a potential risk that an unbalanced rotor fault
maybe wrongly diagnosed as shorted stator windings, and vice versa.
110
CHAPTER 7: CONCLUSION
This project has demonstrated a simple and effective data-based scheme, using
time-domain vibration data, for the continuous automatic condition monitoring and
diagnosis of the two most common machine fault modes, namely, bearing and
unbalanced rotor faults. The key idea is to use a novel normalized cross-correlation sum
operator as similarity measure and the use of k-NN algorithm, for the automatic
classification of machine faults. This technique is both noise-tolerant and shift-invariant.
Experiments conducted showed that a low error rate of 0.74% is achieved and insensitive
to a wide range of machine operating speed from 15Hz to 32Hz. As such, objective 1
mentioned in Chapter 1 has been achieved.
Further, this project also showed the successful diagnosis of these two mechanical
faults using vibration frequency-domain information, where SOM is used to discover
cluster information on the extracted features in an unsupervised way, and an M-SVM is
next used to derive the clusters globally optimal separating hyper-plane for the automatic
classification of the fault modes. A low error rate of 1.48% is achieved and insensitive to
a wide range of machine operating speed from 15Hz to 32Hz. This has achieved
objective 1 mentioned in Chapter 1.
This project also study of motor current signature harmonic content using EMD
technique. A wide range of fault modes are studied, namely, bearing fault, unbalanced
rotor fault, broken rotor bar fault and shorted stator winding fault, which together
accounts for more than 85% of all machine fault mode. This project has demonstrated the
111
use of the unique filter bank property of EMD algorithm, to effectively separate the
various harmonics presence in the machine signatures, namely, the high-frequency
switching harmonics by VSI and high-frequency noise, and the low-frequency
fundamental supply frequency. By such separation of the common modes, unique nonlinear and non-stationary features are discovered at machine operating speed of 30Hz and
40Hz. This has achieved objective 2 mentioned in Chapter 1.
However, it is also observed in this project that the uniqueness of the signatures is
not clearly present in lower machine operating speed of 20Hz. With these unique features
at machine speed of more than 30Hz extracted, automatic fault classifications by a single
effective fault detection and diagnosis scheme based on EMD technique can be achieved.
This will lay the foundation for future works for using EMD technique for machine fault
detection and diagnosis.
112
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APPENDIX A: INTRINSIC MODE FUNCTIONS DERIVED BY N-A MEMD
ALGORITHM FOR MACHINE SIGNATURES AT MACHINE SPEED OF 20HZ
Figure A 1: EMD of HTY20 signature.
124
Figure A 2: EMD of BRG20 signature.
125
Figure A 3: EMD of BRB20 signature.
126
Figure A 4: EMD of UBR20 signature.
127
Figure A 5: EMD of SWF20 signature.
128
Figure A 6: EMD of added noise n1.
129
Figure A 7: EMD of added noise n2.
130
APPENDIX B: HILBERT SPECTRUM AND
MARGINAL HILBERT SPECTRUM OF MACHINE
SIGNATURE (AT MACHINE SPEED OF 20HZ)
INTRINSIC MODE FUNCTION 5 TO 9
Figure B 1: Hilbert Spectrum and Marginal Hilbert Spectrum for (IMF 5-9) of the BRG
machine signatures at 20Hz.
131
Figure B 2: Hilbert Spectrum and Marginal Hilbert Spectrum for (IMF 5-9) of the BRB
machine signatures at 20Hz.
132
Figure B 3: Hilbert Spectrum and Marginal Hilbert Spectrum for (IMF 5-9) of the UBR
machine signatures at 20Hz.
133
Figure B 4: Hilbert Spectrum and Marginal Hilbert Spectrum for (IMF 5-9) of the SWF
machine signatures at 20Hz.
134
APPENDIX C: PSEUDO CODE FOR 2-CLASS SVM
LEARNING
% S U P P O R T V E C T O R M A C H I N E (SVM)
% - 2 class problem
% - Training SVM
% Written By : Chen Wee Yuan (HT080482M)
% Date : 1 Dec 2009
clear all; clc; close all;
%new test data
gp1=[1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 ...
4 4 4 4 4 5 5 5 5 5 12 12 12 12 12 ...
13 13 13 13 13 14 14 14 14 14 15 15 15 15 15 ...
16 16 16 16 16;
11 12 13 14 15 ...
11 12 13 14 15 ...
11 12 13 14 15 ...
11 12 13 14 15 ...
11 12 13 14 15 ...
6 7 8 9 10 ...
6 7 8 9 10 ...
6 7 8 9 10 ...
6 7 8 9 10 ...
6 7 8 9 10];
[r cgp1]=size(gp1);
figure;
for i=1:cgp1
plot(gp1(1,i), gp1(2,i),''); grid on; hold on;
text(gp1(1,i), gp1(2,i),'1','Color',[0 1 0]); grid on; hold on;
end
gp2=[1 1 1 1 2 2 2 2 3 3 3 3 ...
4 4 4 4 5 5 5 5 6 6 6 6 ...
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 ...
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 ...
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ...
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 ...
11 11 11 11 12 12 12 12 13 13 13 13 14 14 14 14 15 15 15 15 ...
16 16 16 16;
17 18 19 20 17 18 19 20 17 18 19 20 17 18 19 20 17 18 19 20 ...
20 17 18 19 ...
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ...
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ...
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ...
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ...
4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1];
[r cgp2]=size(gp2);
for i=1:cgp2
plot(gp2(1,i), gp2(2,i),''); grid on; hold on;
text(gp2(1,i), gp2(2,i),'2','Color',[1 0 1]); grid on; hold on;
end
axis( [0 18 0 22]);
135
title('2-class SVM training samples');
%define gp1 and gp2 labels
d1=ones(cgp1,1);
d2=-1*ones(cgp2,1);
di=[d1;d2];
trainx=[gp1 gp2];
ntrainx=[cgp1 cgp2];
[trainx,ps]=mapminmax(trainx);
save trainx;
save ntrainx;
%set SVM parameter - soft-margin p, C
%beta1=1;beta0=1;
%gaussian
rl=3.1; C=1;
%polynomials
%p=2.5; C=1; rl=p;
kernelpara=[rl C]; save kernelpara;
%map and scale train data to [-1,1]
xi=trainx;
%xi_test=mapminmax('apply',testx,ps); %use the same to scale test data
%calculate the Gram Matrix%
tic;
[r1 c1]=size(xi);
for i=1:c1
for j=1:c1
%K(i,j)=(xi(:,i)'*xi(:,j)+1)^p;
%polynomial kernel
%K(i,j)=xi(:,i)'*xi(:,j);
%linear kernel hard-margin
%K(i,j)=tanh(beta0*xi(:,j)'*xi(:,i)+beta1);
K(i,j)=exp(-1*(xi(:,j)-xi(:,i))'*(xi(:,j)-xi(:,i))/2*rl^2); %exp kernel
end
end
%check Gram Matrix for Mercer‘s condition where negative eigenvalues exist
[rx cx]=size(K);
ev=eig(K);
for i=1:rx
if ev(i)< 0
fprintf('one of the eigenvalues is negative, ...\n');
fprintf('this kernel candidate is not admissable\n');
flag=1;
break
end
end
%if flag==1;break;end
% calculate the Hessian Matrix for maximization problem %
for i=1:c1
for j=1:c1
%H(i,j)=di(i)*di(j)*xi(:,i)'*xi(:,j);
%linear kernel hard margin
%H(i,j)=di(i)*di(j)*(xi(:,i)'*xi(:,j)+1)^p; %polynomial kernel
%H(i,j)=di(i)*di(j)*tanh(beta0*xi(:,j)'*xi(:,i)+beta1);
H(i,j)=di(i)*di(j)*exp(-1*(xi(:,j)-xi(:,i))'*(xi(:,j)-xi(:,i))/2*rl^2); %exp kernel
end
end
136
% solving the dual problem for Langrange multipliers (alpha) %
f=-1*ones(c1,1);
%maximization problem
A=zeros(1,c1);
b=zeros(1,1);
Aeq=di';
beq=zeros(1,1);
lb=zeros(c1,1);
%ub=ones(c1,1)*1e6; %for hard-margin
ub=ones(c1,1)*C;
%for soft-margin
a0=0.1*randn(c1,1); %randomly initialise initial alphas
options=optimset('LargeScale','off','MaxIter',3000);
[a,fval,exitflag,output,alpha]=quadprog(H,f,A,b,Aeq,beq,lb,ub,a0,options);
% finds the support vectors xsv %
for ( all the given datapoints )
{ find the non-zero Langrange Multipliers (alpha) }
end
% support vector xsv for weights and bias calculations: find the weight term-(wok) using support vector
wok=0;
for i=1:c1
%wok=wok+a(i)*di(i)*(xsv{1}'*xi(:,i)+1)^p; %polynomial kernel
%wok=wok+a(i)*di(i)*(xsv{1}'*xi(:,i));
%linear kernel hard-margin
%wok=wok+a(i)*di(i)*tanh(beta0*xsv{1}'*xi(:,i)+beta1);
wok=wok+a(i)*di(i)*exp(-1*(xsv{1}-xi(:,i))'*(xsv{1}-xi(:,i))/2*rl^2); %exp kernel
end
bo=1/dsv{1}-wok;
%find the bias bo using ds
sim_time=toc; sim_time;
save xsv; save dsv; save asv; save bo;
% S U P P O R T V E C T O R M A C H I N E (SVM)
% - 2 class problem
% - Testing SVM constructed
% Written By : Chen Wee Yuan (HT080482M)
% Date : 1 Dec 2009
clear all; clc; close all;
load xsv; load dsv; load asv; load bo;
load ntrainx; load trainx; load kernelpara;
gridsize_x=1.5; gridsize_y=1.5;
[rt ct]=size(xi);
%set SVM parameter - soft-margin p, C
%beta1=1;beta0=1; %p=6;C=1;
rl=kernelpara(1,1); C=kernelpara(1,2);
%test data
gpt1=[3.3 14.5; 13.5 8.9];
gpt2=[3.2 8.3 8.6 9.3 13.3;19.5 19.3 10.4 2.3 2.3];
testx=[gpt1 gpt2];
di_test=[1;1;-1;-1;-1;-1;-1];
xi_test=mapminmax('apply',testx,ps);
137
tic;
% test SVM using test_set
[r2 cw]=size(xi_test);
ker1=0;
[rsv csv]=size(xsv);
number_sv=csv;
for j=1:cw
for i=1:csv
sv=xsv{i};
%ker1(i,j)=xi_test(:,j)'*xi(:,i);
%linear kernel hard-margin
%ker1(i,j)=(xi_test(:,j)'*xi(:,i)+1)^p; %polynomial kernel
%ker1(i,j)=tanh(beta0*xi_test(:,j)'*xi(:,i)+beta1);
ker1(i,j)=exp(-1*(xi_test(:,j)-sv)'*(xi_test(:,j)-sv)/2*rl^2); %exp kernel
end
end
dv=cell2mat(dsv)';
av=cell2mat(asv)';
gx=(av.*dv)'*ker1+bo*ones(1,cw);
for i=1:cw
if gx(i)>0;
gx(i)=1;
elseif gx(i)[...]... investigate and formulate an automatic machine condition monitoring scheme to detect and diagnose the most common machine fault modes, namely, bearing and unbalanced rotor fault, that is insensitive to machine operating speed Objective 2: To investigate and study the use of MCSA to cover a wider range of machine fault modes; apart from bearing and unbalanced rotor faults, to include broken rotor bars and shorted... related to 11 bearing fault, and the RMS value of this noise-cancelled signal is next calculated online as fault index, with impending fault as an increase in fault index Model-based approach aims to construct a mathematical model of the machine and thereby using the model to analysis and predict fault mode [67-74] Finite element analysis is popular for simulating and studying of fault mode; especially... illustrates the approaches of this project to investigate the automatic fault diagnosis of AC synchronous machine In the next, the mechanics of machine fault mode is presented AC synchronous machine fault diagnosis Vibration signatures Motor current signatures Unbalanced Rotor Bar fault Broken Rotor Bar fault Bearing fault Unbalanced Rotor Bar fault Bearing fault Shorted Winding fault Time-domain analysis... classification of the fault modes On the second objective, this project use EMD technique to study the motor current signatures harmonic contents of a healthy machine (HTY), a machine with bearing fault (BRG), unbalanced rotor fault (UBR), broken rotor bar fault (BRB) and shorted stator winding fault (SWF) In this project, new unique non-linear and nonstationary features are discovered for these fault modes at machine. .. Chapter 1: Introduction on the issues of formulating a reliable machine fault diagnostic scheme, and the rationale for condition monitoring using MCSA and vibration analysis, and sets the stage for stating the objectives of this research Fault statistics and literature survey are also carried out to compile the fault statistics and identifies the most common failure modes This allows research effort to... multiple classes of machine fault still elude researchers The current harmonics that is present in the motor current is mainly created by the machine asymmetries and vibrations due to machine faults Hence, this project focuses on two fault detection techniques, namely, vibration signature and MCSA There are a number of issues to address in the formulation of a reliable fault detection and diagnosis scheme... detection and diagnosis scheme [4]: definition of a single diagnostic procedure for any type of faults insensitive to and independent of operating conditions reliable fault detection for position, speed and torque controlled drives reliable fault detection for drives in time-varying conditions quantify a stated fault threshold independent of operating conditions 1 1.1 Objectives With the above... failure modes Fault diagnostic technique literature survey is next conducted, to understand how various novel diagnostic techniques are formulated and the difficulties encountered This identifies niche research area where this project adds values Chapter 2: Mechanics of machine fault elucidates the origin of different type of machine faults, presents the various fault vibration signatures and the expected... current fault spectrum for MCSA Chapter 3: Motor Current Signature and Vibration Signature Analysis explain the difficulties, challenges and issue of vibration analysis and MCSA techniques and a new approach is proposed Chapter 4: Application of Artificial Intelligence (AI) techniques for fault diagnosis presents the various AI techniques used in this project Chapter 5: A study on Automatic Diagnosis of. .. harmonics for diagnosis of rotor faults It is shown that detection of these harmonics is possible using inverter input current near zero frequency To extend the type of fault coverage, stator winding faults are investigated as well In [61], a novel diagnostic indicator for stator winding fault, that does not involve ground fault, is formulated using positive and negative sequence line-voltage and line-current ... mathematical model of the machine and thereby using the model to analysis and predict fault mode [67-74] Finite element analysis is popular for simulating and studying of fault mode; especially... approaches of this project to investigate the automatic fault diagnosis of AC synchronous machine In the next, the mechanics of machine fault mode is presented AC synchronous machine fault diagnosis. .. voltages and currents of the induction motor This method is referenced from Peter Vas, “Parameter Estimation, Condition monitoring, and diagnosis of Electrical Machines” Using the dynamic model of