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) iL 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 )Tt1  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],   xk  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 ( )  t0 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 2nt 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 2nt 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. 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Mandic, ―Filter Bank Property of Multivariate Empirical Mode Decomposition‖, IEEE Transactions on Signal Processing, vol. –, no. –, Month 2011 123 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