Time frequency method for bearing fault diagnosis

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(1)Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey. School of Engineering and Sciences. Time-Frequency Analysis Method for Rolling Bearings Fault Diagnosis A thesis presented by. Israel Benjamin Ruiz Quinde Submitted to the School of Engineering and Sciences in partial fulfillment of the requirements for the degree of Master of Science In Manufacturing Systems. Monterrey, Nuevo León, May 20 th 2019.

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(6) @2019 by Israel Benjamin Ruiz Quinde All rights reserved.

(7) Dedication. A mis padres por su amor y apoyo constante en todo momento, y por ser parte fundamental de mi desarrollo académico y profesional. A mis hermanos que siempre estuvieron dispuestos a ayudarme en las situaciones más difı́ciles y por alentarme a conseguir un grado académico más. A mis amigos por sus consejos, apoyo y por permitirme compartir con ellos gratas experiencias durante mi estadı́a. A Dennis por su amor y paciencia a pesar de la distancia. Finalmente, a Dios por permitirme vivir esta importante experiencia académica y ser mi soporte en cada momento.. 4.

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(9) Acknowledgements I would like to express my deepest gratitude to Tecnológico de Monterrey for allowing me to be part of this postgraduate program, and for giving me a scholarship during my studies. I thank to the Automotive Consortium research group for the new knowledge acquired during the project Monitoring and Diagnosis of High Speed Machining Spindles. I want to express my most sincere thanks to my thesis advisor, Dr. Rubén Morales Menéndez for his suggestions, remarkable support and patience. I thank to my co-advisor Dr. Antonio Jr. Vallejo Guevara for his comments and suggestions during weekly work meetings. Finally, thanks to CONACyT for the maintenance support during my studies.. 5.

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(11) Time-Frequency Analysis Method for Rolling Bearings Fault Diagnosis By Israel Benjamin Ruiz Quinde Abstract. Spindle bearings are some of the most critical and vulnerable components in rotating machines. Friction, load forces and vibrations actuating over bearings can produce wear, fatigue and impending cracks on these which may end in a full damage of the spindle over time. The Condition-Based Maintenance (CBM) have arose as a strategy to address this problem, in which, analysis of vibration signals can be performed in real time to anticipate the damage of the machine. A wide range of strategies based on digital processing techniques have been developed for vibration analysis. Wigner-Ville Distribution (WVD) is probably the most used non-linear timefrequency distribution for signal processing in fault diagnosis, however, the presence of cross terms can lead to misleading interpretations of their Time-Frequency Representations (TFR). Signal decomposition methods such as Variational Mode Decomposition (VMD) and Local Mean Decomposition (LMD) have been developed to reduce the complexity of vibration signals allowing to reconstruct them only with their main components. Moreover, this can reduce the cross terms in WVD. However, after the signal decomposition procedure, the identification of the relevant components, which contain the fault information, is commonly based in visual inspection and identification of the bearing housing resonance band. A methodology which combines the great characteristics of the VMD and the WVD is proposed to get more reliable and illustrative results of bearing fault diagnosis from TFR of the vibration signals. Kullback-Leibler Divergence (KLD) was included in the analysis to guide the selection of the effective components with the most relevant information about the fault in an automatic way. After applying the proposed method, in some cases, the amplitude of the fault frequencies in the spectrum were increased around 53% for Outer Race (OR) signals, 45% for Inner Race (IR) signals and 73% for Rolling Element (RE) signals, regarding the amplitude of the found peaks by using the traditional envelope-FFT method. An automatic fault diagnosis method based on an Artificial Neural Network (ANN) and WVD was also presented to avoid the visual inspection. The LMD was used as the signal decomposition method. The TFR, obtained by computing the WVD over the effective Product Functions (PF), were used to build the feature vectors. A classification accuracy in average = 98.2% was obtained by testing the proposed methodology with experimental data.. 6.

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(13) List of Figures 2.1 2.2. Rolling bearings components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 VMD algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34. 3.1 3.2 3.3 3.4 3.5 3.6 3.7. Test Rig used by CWRU, [Smith and Randall, 2015] . . Schematic sketch of IMS test rig, [Qiu et al., 2006]. . . . Searching frequency algorithm. . . . . . . . . . . . . . . Analyzed signals from CWRU, category: N1-DE. . . . . Run to fault experiment from group of signals in Tests 1. Run to fault experiment from group of signals in Tests 2. Analyzed signals from IMS. . . . . . . . . . . . . . . .. . . . . . . .. 36 37 39 41 42 42 43. 4.1 4.2 4.3 4.4. Flowchart of proposed methodology. . . . . . . . . . . . . . . . . . . . . . . . . . Trend in data. Left plot: signal in time and right plot : spectrum of signal. . . . . . OR fault signal from CWRU bearing database. . . . . . . . . . . . . . . . . . . . . Sifting process in LMD. Envelope or AM signals (left) and purely FM signals (right) are illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PF (left plot) and their respectively spectra (right plot) obtained by LMD algorithm. Spectrum of x(t) (left), and of each BLIMF obatined by VMD method (right). . . . Envelope signal of: PF (left), BLIMF (right) signals . . . . . . . . . . . . . . . . . 2-D and 3-D TFR of SPWVD applied to the envelope of effective BLIMF. . . . . . Searching frequency algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46 47 48. 4.5 4.6 4.7 4.8 4.9 5.1 5.2 5.3 5.4 5.5 5.6. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. VMD performance in the time domain. . . . . . . . . . . . . . . . . . . . . . . . Spectrum segmentation performed by textitVMD. . . . . . . . . . . . . . . . . . Spectrum of each BLIMF for an RE fault signal (X048 DE time). . . . . . . . . . Spectrum of each BLIMF for an OR fault signal, (X130 DE time). . . . . . . . . Spectrum of each BLIMF for an IR fault signal, (X171 DE time). . . . . . . . . . KLD values of each BLIMF for the three fault conditions, RE (top-left), OR (topright), IR (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. . . . . .. 48 48 49 51 52 53 56 57 58 59 60. . 60.

(14) 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20. Spectrum of the sum of the envelopes of effective BLIMF. Normal condition (left), RE signal (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrum of the sum of the envelopes of effective BLIMF. OR signal (left), IR signal (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal decomposition by LMD of X171 signal. . . . . . . . . . . . . . . . . . . . Spectrum of the sum of the envelopes of effective PF, IR signal. . . . . . . . . . Spectrum of each BLIMF, X056 signal. . . . . . . . . . . . . . . . . . . . . . . Fault diagnosis for signal X056, using 5 revolutions of shaft speed. . . . . . . . . SPWVD of the effective BLIMF. . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between normalized values of KR and KLD to select the effective components in Tests: 1454 (left plot), 1760 (middle plot), 454 (right plot). . . . . OR signal IMS database (early stage) rev = 5. . . . . . . . . . . . . . . . . . . . OR signal IMS database (early stage), rev = 10. . . . . . . . . . . . . . . . . . . IR signal IMS database (critical stage), r = 10. . . . . . . . . . . . . . . . . . . RE signal IMS database (early stage), r = 10.. . . . . . . . . . . . . . . . . . . . Spectrum of each BLIMF signal IMS database (RE early stage), r = 10. . . . . . Methodology using LMD for faulty signals in IMS database . . . . . . . . . . . .. . 61 . . . . . .. 61 63 63 65 65 66. . . . . . . .. 66 67 68 68 69 69 70. B.1 Zero-padding the FFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 C.1 Obtained PF using [Liu et al., 2017b] objective function. . . . . . . . . . . . . . C.2 Obtained PF using modified objective function. . . . . . . . . . . . . . . . . . . C.3 XCR between original signal and the reconstructed for different values of α and K. IMS signal at left plot, and CWRU signal at right plot. . . . . . . . . . . . . . . C.4 OI between original signal and the reconstructed for different values of α and K. IMS signal at left plot, and CWRU signal at right plot. . . . . . . . . . . . . . . . C.5 OI between original signal and the reconstructed for different K with an α = 500. C.6 BLIMF obtained by VMD with alpha = 2800 and K = 2. . . . . . . . . . . . . . C.7 Spectrum of original signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.8 Decomposed signals by into 2 BLIMF. . . . . . . . . . . . . . . . . . . . . . . . C.9 Window functions in frequency with different sizes (Lh). . . . . . . . . . . . . . C.10 Smoothed WVD applied to RE signal from IMS. . . . . . . . . . . . . . . . . . . C.11 Marginal in frequency of WVD using Lh = 69 (left), Lh = 7 . . . . . . . . . . . C.12 Density histograms for vibration signal from CWRU database, and for each PF. . C.13 Comparison between real data distribution and a normal standard distribution. . . C.14 Kernel-based estimation of pdf for vibration signal from CWRU database, and for each PF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. . 90 . 90 . 91 . . . . . . . . . .. 91 92 92 92 93 94 95 96 97 98. . 98.

(15) C.15 Histogram of each BLIMF (left). Kernel-based estimation of pdf for each BLIMF (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 D.1 D.2 D.3 D.4 D.5 D.6. Methodology for automatic diagnosis. . . . . . . . . . . . . . . . . . . . . . . . . 102 Representation of WVD segmentation. . . . . . . . . . . . . . . . . . . . . . . . . 102 Marginal distribution in frequency of SPWVD for a RE signal from CWRU database.104 Classification based on ANN in high frequencies (without demodulation). . . . . . 104 Classification based on ANN in low frequencies (demodulation). . . . . . . . . . . 106 SPWVD of the 4 bearing conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 107. E.1 WVD results for each bearing condition in High Frequency (HF). . . . . . . . . . E.2 Comparison between normalized values of KR and KLD to select the effective components, signals: X058 (left plot), X120 (middle plot) and X301 (right plot). . E.3 Analyzed signals from CWRU, category: N1-DE. Group a. . . . . . . . . . . . . E.4 Analyzed signals from CWRU, category: N1-DE. Group b. . . . . . . . . . . . . E.5 Analyzed signals from CWRU, category: N1-DE. Group c. . . . . . . . . . . . . E.6 Analyzed signals from CWRU, category: N1-DE. Group d. . . . . . . . . . . . . IR signal IMS database (early stage), r = 5. . . . . . . . . . . . . . . . . . . . . RE signal IMS database (early stage), r = 5. . . . . . . . . . . . . . . . . . . . . OR signal IMS database (medium stage). . . . . . . . . . . . . . . . . . . . . . . OR signal IMS database (critical stage). . . . . . . . . . . . . . . . . . . . . . . IR signal IMS database (medium stage). . . . . . . . . . . . . . . . . . . . . . . IR signal IMS database (critical stage). . . . . . . . . . . . . . . . . . . . . . . . RE signal IMS database (medium stage). . . . . . . . . . . . . . . . . . . . . . . RE signal IMS database (critical stage). . . . . . . . . . . . . . . . . . . . . . . Comparison between normalized values of KR and KLD to select the effective components in OR signals. Tests: 653 (left plot) and 710 (right plot). . . . . . . . F.9 Kurtosis measurements for two signals with: several peaks (left), two highest peaks (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.10 Comparison between normalized values of KR and KLD to select the effective components, RE signals. Tests: 1667 (left plot) and 1753 (right plot). . . . . . . . F.11 Comparison between normalized values of KR and KLD to select the effective components in IR signals. Tests: 1820 (left plot) and 1990 (right plot). . . . . . .. F.1 F.2 F.3 F.4 F.5 F.6 F.7 F.8 F.12. 9. . 109 . . . . .. 110 112 113 114 115. . . . . . . . .. 117 118 118 119 119 120 120 121. . 121 . 123 . 123 . 123.

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(17) List of Tables 2.1. Comparison between research studies on WVD, VMD and LMD. . . . . . . . . . . 26. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9. Characteristics of used bearings. CWRU database. . . . . . . . . Frequency factors (multiple of shaft speed in Hz). . . . . . . . . Characteristics of used bearings. IMS database. . . . . . . . . . Frequency factors (multiple of shaft speed in Hz) . . . . . . . . Experimental signals from different bearing conditions, CWRU. Characteristic fault frequencies for CWRU . . . . . . . . . . . . Tested signals, hardly and non-diagnosable CWRU. . . . . . . . Proposed categories for IMS database. . . . . . . . . . . . . . . Expected fault frequencies for IMS . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 36 36 37 37 39 39 40 41 42. 5.1 5.2 5.3 5.4. Selected BLIMF based on KLD . . . . . . Detected peaks in each BLIMF . . . . . . KLD values of each BLIMF, X056 signal. Selected BLIMF by KLD criteria. . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 57 62 64 67. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. A.1 Acronyms Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.2 Variables Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 B.1 B.2 B.3 B.4 B.5 B.6 B.7. Vector of frequencies of the FFT without zero padding . . . . . . . . . . . . . Vector of frequencies of the FFT with zero padding . . . . . . . . . . . . . . . Frequency resolution for CWRU database, fs = 12, 000, fr = 29.95Hz . . . . Frequency resolution for CWRU database, fs = 12, 000Hz, fr = 28.8Hz . . . Frequency resolution for IMS database, fs = 20, 000Hz, fr = 33.3Hz . . . . . Mean computational time of VMD-FFT for 5 and 10 revolutions of IMS signals Computational cost of WVD and VMD working together (5 revolutions). . . . .. . . . . . . .. . . . . . . .. 84 84 85 86 86 87 88. C.1 Impulse duration of fault components. . . . . . . . . . . . . . . . . . . . . . . . . 93 C.2 Sizes of window functions in frequency, CWRU. . . . . . . . . . . . . . . . . . . . 95 C.3 Sizes of window functions in frequency, IMS. . . . . . . . . . . . . . . . . . . . . 95 11.

(18) C.4 XMR for each window type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 D.1 D.2 D.3 D.4 D.5 D.6. SE feature vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Categories for fault diagnosis. . . . . . . . . . . . . . . . . . . . . . . True positive rates per each bearing condition. High frequency analysis. Results of ANN-based classifier, CWRU database . . . . . . . . . . . . True positive rates per each bearing condition. Low frequency analysis. Results of ANN-based classifier, CWRU database . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 103 103 105 105 106 106. E.1 Performance of KLD-based method for select effective signal components . . . . . 111 E.2 Detected frequency components . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 F.1 F.2 F.3 F.4. Frequencies of detected peaks in IMS signals (5 revolutions). . Amplitude in detected peaks IMS signals (5 revolutions). . . . Frequencies of detected peaks IMS signals (10 revolutions). . . Amplitudes of detected peaks in IMS signals (10 revolutions).. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 121 122 122 122. G.1 Performance of statistical parameters. (CWRU database) . . . . . . . . . . . . . . 127. 12.

(19) Contents 1. 2. 3. 4. Introduction 1.1 Motivation . . . . . . 1.2 Problem Description 1.3 Research Question . 1.4 Solution Overview . 1.5 Main Contribution . 1.6 Organization . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. State of the Art 2.1 Literature Review . . . . . . . . . . . . . . 2.2 Theoretical Background . . . . . . . . . . . 2.2.1 Condition Based Monitoring (CBM) 2.2.2 Bearing Faults . . . . . . . . . . . 2.2.3 Time-Frequency Analysis . . . . . 2.2.4 Kullback-Leibler Divergence (KLD). . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. Experimental System 3.1 Experimental Databases . . . . . . . . . . . . . . . 3.1.1 Case Western Reserve University (CWRU) 3.1.2 IMS Database . . . . . . . . . . . . . . . . 3.2 Design of Experiments . . . . . . . . . . . . . . . 3.2.1 Tested signals for CWRU database . . . . . 3.2.2 Tested signals for IMS database . . . . . . Proposal 4.1 Proposed Methodology for Visual Diagnosis . . . . 4.1.1 Trend Removal 0 . . . . . . . . . . . . . 4.1.2 Signal Decomposition 1 . . . . . . . . . 4.1.3 Selection of the effective signal component. 13. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . . . . . . 2 . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . .. . . . . . .. 17 18 18 19 20 20 20. . . . . . .. 23 23 28 28 28 30 33. . . . . . .. 35 35 35 36 38 38 39. . . . .. 45 45 46 47 49.

(20) 4.1.4 4.1.5 4.1.6 5. 6. Extract envelope 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Time-frequency analysis 4 . . . . . . . . . . . . . . . . . . . . . . . . . 51 Fault diagnosis 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. Results 5.1 CWRU Results . . . . . . . . . . . . . . . . . 5.1.1 Methodology with VMD . . . . . . . . 5.1.2 Methodology with LMD . . . . . . . . 5.1.3 Hardly and non-diagnosable N1 signals 5.2 IMS results . . . . . . . . . . . . . . . . . . . 5.3 Discussions . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 55 55 55 60 64 65 68. Conclusions 71 6.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. Bibliography. 75. A Acronyms and variables definitions. 81. B Time-frequency resolution B.1 Time resolution . . . . . . . B.2 FFT Resolution . . . . . . . B.3 Selection of the signal length B.4 Estimate of fault frequencies B.5 Computational cost . . . . .. . . . . .. 83 83 83 85 87 87. . . . .. 89 89 90 93 96. C Parameters of used algorithms C.1 LMD Parameters . . . . . C.2 VMD parameters . . . . . C.3 Smoothing the WVD . . . C.4 Data analysis with KLD . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. D Automatic diagnosis. 101. E CWRU results. 109. F IMS results 117 F.0.1 Kurtosis as indicator of fault . . . . . . . . . . . . . . . . . . . . . . . . . 118 14.

(21) G Statistical parameters 125 G.1 Other statistical parameters for fault diagnosis . . . . . . . . . . . . . . . . . . . . 125 G.1.1 Statistical parameters for selecting effective components . . . . . . . . . . 127 H Code 129 H.1 Visual Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 H.2 Automatic Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Curriculim Vitae. 147. 15.

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(23) Chapter 1 Introduction Rotating machines always generate mechanical vibrations even in good conditions, and periodic events in operation of machine such as: rotating shafts, meshing gear-teeth and so on, are often linked to such vibrations. Therefore, the frequency with such events repeat could give an important information of the source which generates them, [Randall, 2011]. Several digital processing techniques based on vibration analysis have been developed, especially for the fault detection in the spindle and its rolling bearings. Within this group, the timefrequency signal representations have allowed the analysis of the complex vibration signals, which present non-linear and non-stationary characteristics while the failure occurs in the rotating machine. Among those techniques, Short Time Fourier Transform (STFT), Wavelet Transform (WT) and Hilbert-Huang Transform (HHT) are the most commonly used time-frequency analysis tools in the fault diagnosis systems. The Wigner-Ville Distribution (WVD), is another time-frequency representation that has a some good properties for analyzing non-stationary, even better than the aforementioned. The most relevant characteristics of this distribution in contrast to other Time-Frequency Representations (TFR), besides offering a good localization in time-frequency domain are: energy conservation, which means there is no leakage of information in the analyzed signal, and the real-valued property which avoids to work with the imaginary part of the spectrum. However, the WVD is a non-linear transformed and as a result of that, interference terms (known as cross-terms), are introduced in the TFR of the vibration signal generated by WVD, making it difficult to interpret, [Cohen, 1989]. To address this issue, there are some techniques with which WVD could be combined, to increase its effectiveness. In literature review, different techniques have been proved to reduce the effect of cross-terms over WVD, such as those based on window functions to filter the TFR. However, windowing or techniques such as has resulted in unwanted feature in reducing the time-frequency concentration of the vibration signal energy. Signal decomposition methods such as Empirical Mode Decomposition (EMD), Local Mean 17.

(24) 18. CHAPTER 1. INTRODUCTION. Decomposition (LMD) and Variational Mode Decomposition (VMD), have also arisen to perform a more selective analysis in multicomponents signals, by estimating the contained components in signal including noise, [Wang and Markert, 2016]. These methods have also been used to reduces cross terms in WVD, [Li et al., 2006]-[Xie et al., 2012]. With a high resolution TFR that represents the vibration signals, methods for detecting abrupt changes on them: Kolmogorov Distance (KD), Rènyi entropies [Sucic et al., 2014], or a method which represents the entire TFR of a specific signal in only one measure such as Shannon Entropy [Xie et al., 2012], can be included to provide a useful criterion for analyzing and comparing different probability distributions, and also to get a measure which quantifies the energy variation in the TFR.. 1.1. Motivation. Machining processes which include operations such as turning, milling, drilling and grinding are widely used in manufacturing systems, and due to the industrial requirements of improving production rates and especially maintaining high availability in their operations, machinery health monitoring becomes a priority. The working conditions under which rotating machines, used in machining operations, play an important role in machinery diagnosis, so there is an increased requirement for effective and efficient techniques that monitor machines in real time and detect the presence of defects in its early stage or its progression over the time. This allows the operator to schedule maintenance activities, before the fault causes a completely damage of the components in the machine, and therefore an unexpected machine downtime; which, leads to high maintenance costs and production delays.. 1.2. Problem Description. The rotating part of a machine is generally supported by bearings. Spindle bearings are the most critical and also the most vulnerable components in rotating machines. Friction, load forces and vibrations actuating over bearings can produce wear, fatigue and impending cracks on these. A survey for induction machines done in [R. Hyers and Syrett, 2006] reports that about 40% failures are related to bearings, 38% to the stator and 10% to the rotor. Therefore, the capability of detecting possible faults in rolling bearings at very early stages and in the most accurate way, represents a fundamental task. A number of time-frequency domain methods have been developed for fault diagnosis in rotating machines. One of these techniques is the WVD, which can achieve an excellent representation of dynamical signal. However, the primary disadvantage of the WVD is that cross-terms which.

(25) 1.3. RESEARCH QUESTION. 19. do not permit a straightforward interpretation of the energy distribution in the TFR of the analyzed signals. Some signal decomposition methods such as EMD and LMD have been proposed in the literature to reduce cross terms and perform a more discriminatory analysis of the different components in the signal. However, the performance of these methods can be affected by some limitations such as: mode mixing problem, which can appear in the decomposition procedure when the signal components cannot be separated to a single one, end defects and the sensitivity to noise and sampling frequency, [Dragomiretskiy and Zosso, 2014]. VMD is another signal decomposition method, but in contrast with the aforementioned, this has an analytic definition and operates in a non-recursively way to separate the signal components. However, these methods, generally choose the optimal segment of frequency to perform the bearing fault diagnosis based on the a priori knowledge of the resonance band of bearing housing or by a simple visual inspection, choose the segment of frequencies with the highest concentration of energy in the TFR of the analyzed signal, [Du et al., 2016]. There are some parameters in WVD and in its smoothed versions such as: resolution, window functions in time and frequency and also regarding signal decomposition methods such as: orthogonality, sifting criteria, the number of desired components, etc that can be studied to have better results in the fault diagnosis.. 1.3. Research Question. There are many problems when dealing with vibration analysis for spindle faults detection. Traditional signal processing can not deal with complex characteristics of signals, in particular when they are influence by high background noise and signal transmission paths. This makes the diagnosis difficult to extract bearing weak fault features and decreases its level of accuracy in the detection process. Based on a brief State of the Art review, the following hypothesis arise: i WVD combined with a signal decomposition method performs an efficient transient detection because its characteristics in the time-frequency analysis. ii The fault diagnosis based on a time-frequency method is possible even when signals are immersed in high levels of noise. iii The fault diagnosis based on a time-frequency method can be used to work independently of the knowledge of the analyzed signal. iv The performance of the bearing fault diagnosis can be improved by choosing the proper frequency bands in which fault components are located, using a frequency searching method and.

(26) 20. CHAPTER 1. INTRODUCTION statistical parameters that faithfully measures the changes in energy signal such as kurtosis, root, entropy, etc.. v Fault diagnosis can be automatized by using feature extraction methods in time-frequency based on the aforementioned tools and a ANN as classifier.. 1.4. Solution Overview. In this work a comparison between LMD and VMD methods is performed to define the the best method for early bearing fault detection. By using an automatic selection of the proper signal component based on a statistics parameter such as entropy and a peak searching algorithm, the time consumption in the analysis is reduced. Two techniques are proposed for selecting the effective component, the first one is the KullbackLeibler Divergence (KLD) or relative entropy, unlike kurtosis, this can measure the correlation between the distributions of original signal and each of its components, which can be obtained by VMD or LMD, identifying the components which are non-relevant to discard them, [Si et al., 2017]. The second one is a simple peak searching algorithm. This ensures that the analyzed signal components have peaks in the vicinity of the characteristic fault frequencies. Therefore, the signal is reconstructed only with the components that are enough to represent the main characteristics of the whole signal and specially keep the transitory events (transients). An envelope-WVD analysis is included to analyze the fault components in the spectrum.. 1.5. Main Contribution. This research project has the following contributions: i A methodology based on WVD and signal decomposition methods to detect faults in bearings by the analysis of vibration signals; especially, in those which are influence by high background noise. ii An algorithm for selecting the optimal frequency band in which fault components are present for the signal decomposition methods such as LMD and VMD.. 1.6. Organization. This research work is organized as follows:.

(27) 1.6. ORGANIZATION. 21. Chapter 2 presents the state of the art of different methods for signal analysis in bearing fault detection. This chapter includes theoretical background of the main bearing faults and timefrequency analysis methods such as WVD, LMD and VMD. Chapter 3 describes the experimental dataset used for testing the proposed method. This chapter describes the design of experiments for each database. The used signals are presented. Chapter 4 presents the proposed methodology for bearing fault diagnosis. A detailed description of each step is showed. The main tools for signal processing are presented. The used statistical parameters for selecting the effective components after signal decomposition are defined. Chapter 5 shows the results obtained by the proposed methodology. Signals from each bearing database are processed to test the effectiveness of the method. A comparison is performed between the proposed method using WVD and FFT with the traditional envelope-FFT method. Chapter 6 shows the contributions, conclusions, and future work. Appendix A describes all the acronyms and the description of the used variables in this work. Appendix B presents an analysis about the time-frequency resolution of the analyzed signals, and how it affects to the computational cost of the used algorithms. Appendix C shows the used criteria for setting the main parameters in the VMD, LMD and the characteristics of the window functions for smoothing the WVD. Appendix D presents a complementary study to the proposed methodology for performing an automatic bearing fault diagnosis. The results are presented for the analyzed signals in their modulated and demodulated representations. Appendix E shows more results using signals from Case Western Reserve University (CWRU) bearing database. Appendix F presents more results using signals from the Intelligent Maintenance System (IMS) bearing database. Appendix G describes a the performance of some of the statistical parameters commonly used in bearing fault diagnosis to identified the relevant components in a signal decomposition method..

(28) 22. CHAPTER 1. INTRODUCTION. Appendix H presents the developed Matlab code for the proposed methodology..

(29) Chapter 2 State of the Art 2.1. Literature Review. Condition-Based Maintenance (CBM) of rotating machines can be achieved by the effective analysis of parameters such as vibration, wear debris in oil, acoustic emission, etc, acquired from the machine. Fortunately, the changes in these parameters are associated to the development of faults, allowing so the diagnosis can be achieved to anticipate the damage of machine and thus affect its availability. Besides the instrumentation required for carrying out the machine monitoring, methods of analysis based on signal processing techniques must be applied to gain more benefits of vibration or audio measurements, which are normally buried in noise. Between non-linear distributions, WVD is a technique which found successful applications in the research area of fault diagnosis, [Zhou et al., 2011], [D’Elia et al., 2012]. Most of these studies address and propose methodologies for reducing the inference terms in WVD to highlight the presence of impacts related to incipient faults. One of the first WVD approach for fault detection in rotating machine was assessed by [Shin and Jeon, 1993], in which one of its variants, the Pseudo Wigner-Ville Distribution (PWVD), was used for the analysis of practical examples of an actual transient signal, illustrating its dynamic features jointly in time-frequency plane. Several methods have been proposed for reducing cross terms, among them, are those who are based on signal decomposition. In [Li et al., 2006] a study was developed for the diagnosis of rolling bearings using EMD as the method to reduce cross terms in WVD. The original signal was decomposed in a set of Intrinsic Mode Functions (IMF) and theWVD of the first IMF was use to identify the fault frequencies. In signal decomposition procedure, some drawbacks may occur, regarding distortions which can be introduced in the obtained components. In some cases the EMD can induce end defects in their resulting components, thus affecting the selection process of the suitable IMF for fault diagnosis. To attend this drawback, [Xie et al., 2012] proposed a methodology based on another 23.

(30) 24. CHAPTER 2.. signal decomposition termed LMD, to obtain a more faithful separation between components in the original signal, at the same time it helps to reduce the cross terms in WVD. In this case, first Product Function (PF) obtained in the decomposition, was processed by WVD. Then, feature vectors were built for each machinery condition and were used as the input to a Support Vector Machine (SVM) classifier to automatize the diagnosis. A classification accuracy in average = 95.83% was reported. In the studies carried out in [Liu et al., 2017a] and [Liu et al., 2017b], important contributions were exposed, regarding the setting parameters that must be considered in LMD to obtain reliable results in the decomposition process. Issues, such as end defects, mode mixing problems and sifting stopping criteria were assessed. Furthermore, to reduce mode mixing problems caused by the uniform time-frequency distribution of a signal, [Zhang et al., 2017] proposed the Optimize Ensemble Local Mean Decomposition (OELMD). The effect of three critical parameters: noise, bandwidth amplitude, and ensemble number were optimized. Due to LMD and EMD do not have a bandwidth criteria for signal decomposition, this can lead to a wrong separation of resultant components when signals are too close in frequency. The Variational Mode Decomposition (VMD) was introduced by [Dragomiretskiy and Zosso, 2014] to attend this and other drawbacks such as: mode mixing and end defects in recursively decomposition methods. Regarding fault diagnosis, [Tang et al., 2016] presented a method using VMD and a feature extraction method based on Permutation Entropy (PE) for representing each Band-Limited IMF (BLIMF) obtained by VMD as single values to train a SVM classifier. Using an experimental bearing database, classification accuracy of 100 % was reported. [Ma et al., 2018b] introduced a method based on VMD and the Synchro-Extracting Transform (SET) for improving time-frequency analysis and estimating accurately the instantaneous frequency of signals from rolling bearing under fluctuated shaft speeds. kurtosis and Mutual Information (MI) were included in the method to select the optimal frequency band for the fault diagnosis. Experimental results showed an error of 0.98% between estimate and true components. [Ma et al., 2018a] presented a method based on VMD and Teager Energy Operator (TEO) as demodulation technique. The effective BLIMF was selected based on the correlation coefficient criterion. Penalty term aplha and the number of BLIMF is adaptively determined. Experimental results show a successful diagnosis in rolling bearings with an incipient fault. [Chen et al., 2019] proposed a method using energy entropy of four modal components in VMD to build features vectors in a SVM-based classifier. The recognition rate of different faults in bearings reached a 98.75 %. In [Li et al., 2017c], the possible information loss problem or over decomposition in VMD was overcame. The appropriate number of BLIMF was determined by the criterion of approximate complete reconstruction. Regarding other feature extraction methods used in joint with WVD, [Li et al., 2013] proposed a fault detection and classification system for rolling bearings, using the Relative Crossing Information (RCI) to extract feature spectrum from the time-frequency distribution obtained by PWVD, and.

(31) 2.1. LITERATURE REVIEW. 25. Synthesizing Symptom Parameters (SSP) to express the characteristic of feature spectra. A Fuzzy Logic System (FLS) based on sequential inference and possibility theory was used as a classifier to identify the conditions of the machine in automatic way. Selecting the optimal frequency band, in which is found the most relevant information about the fault characteristic frequency, is a critical part of the diagnosis. From the statement of when rolling bearing experiments a failure, the high frequencies in the resonance band of its structure are excited, the analysis can be addressed by a priori knowledge of that frequency band. [Du et al., 2016] proposed a method based on the SPWVD to obtain the time-frequency spectrum of each IMF and by visual inspection, define the resonance band as the zone with the greatest luminosity in the energy distribution. On the other hand [Xiang and Yan, 2016] chose the PF components which present the highest correlation coefficient between them and the original signal. [Chen et al., 2017] proposed a method for fault diagnosis using texture features extracted from TFR of signals, by using Adaptive Optimal Kernel (AOK) function and uniform Local Binary Patterns (uLBP). The algorithm showed a better resolution of fault components in different noisy conditions and an average accuracy in fault classification of 90.19% for signals with medium noise level (Signal to Noise Ratio, SNR = 15 dB). A summary of WVD, VMD and LMD methods approach research applied to rotating machine diagnostic is shown in Table 2.1, where the techniques as well as the studied defects are presented considering a chronological order to compare new trends. According to the above mentioned, a methodology is developed to analyze faults in bearings, from their early stages and which can deal with signals buried in high level of noise, by using experimental data. Moreover, a complementary study for each used technique is performed to reduce the mainly drawbacks found in the state of the art..

(32) 26. Table 2.1: Comparison between research studies on WVD, VMD and LMD. Sensor. Defects. Case Study. Technique. Additional Analysis. Classifier and efficiency. [Shin and Jeon, 1993]. Vibration. Actual Fan, Pump start-up. Test Rig, 500 RPM. PWVD. Simulation. Does not apply. [Baydar and Ball, 2001]. Vibration and AE. Pitting Gear. CWRU 1459/395 RPM. SPWVD. Does not apply. Does not apply. [Li et al., 2006]. Vibration. IR. Simulation/ Test Rig 1500 RPM. EMD, WVD. Does not apply. Does not apply. [Xie et al., 2012]. Vibration. IR, OR, RE. CWRU web data 1720-1797 RPM. LMD-WVD. SE. SVM 98.34%. [Li et al., 2013]. Vibration. IR, OR, RE. Test Rig 1800 RPM. PWVD. RCI-SSP. Fuzzy Logic Not specified. [Du et al., 2016]. Vibration. IR, RE. Test Rig 1600 RPM. SPWVD. EMD. Does not apply. [Xiang and Yan, 2016]. Vibration. Unbalance, Oil Whirl, Pitting Gear. Simulation/ Test Rig 834/1500 RPM. WVD-LMD. Correlation Coefficient. Does not apply. [Tang et al., 2016]. Vibration. IR, OR, RE. CWRU web data 1797 RPM. VMD. PE. SVM 100 % (DE signals). [Li et al., 2017c]. Vibration. IR, OR, RE. Test Rig 360 RPM. VMD. Correlation coefficient. Does not apply. [Chen et al., 2017]. Vibration. IR, RE, OR. CWRU web data 1720-1797 RPM. AOK-TFR. uLB, WVD, S transform distribution. SVM 90.19% with noise SNR=15dB% 98.39% without added noise. [Liu et al., 2017a]. Vibration. OR. Simulation/ Test Rig 1500 RPM. LMD. Fast Kurtogram, Square Envelope. Does not apply. [Zhang et al., 2017]. Vibration. OR, IR. Simulation/ Test Rig 1750 RPM. OELMD. Relative RMSE. Does not apply. [Liu et al., 2017b]. Vibration. Pitting Gear. Test Rig 3000 RPM. LMD. Kutosis, RMSE. Does not apply. [Ma et al., 2018b]. Vibration. OR. Test Rig 360-1200 RPM. VMD-SET. Kurtosis, MI. Does not apply. CHAPTER 2.. References.

(33) References. Sensor. Defects. Case Study. Technique. Additional Analysis. Classifier and efficiency. [Ma et al., 2018a]. Vibration. OR. Test Rig 1: 2000 RPM Test Rig 2: 620 RPM CWRU: 1720-1797 RPM. VMD. Correlation coefficient. Does not apply. [Chen et al., 2019]. Vibration. IR, RE, OR. CWRU web data 1730 RPM. VMD. Energy entropy. SVM 98.75 %. 2.1. LITERATURE REVIEW. Table 2.1: Comparison between research studies on WVD, VMD and LMD (Continued).. 27.

(34) 28. 2.2. CHAPTER 2.. Theoretical Background. Most of the time, vibration signals from rolling bearings have non-stationary behavior and when the analysis is performed only in frequency domain, it is not possible to appreciate changes in frequency with time. Time-frequency analysis methods emerged as a powerful tool to address these problems. The basics of used techniques for signal processing and the bearing fault diagnosis are presented in this section.. 2.2.1 Condition Based Monitoring (CBM) CBM is founded on having the ability to monitor the existing condition of devices while functioning. This means that details must be obtained externally regarding internal effects as the rotating machines are operating. One of the techniques for obtaining information about internal conditions is the vibration analysis. In regular conditions a machine has a specific vibration signature. Fault progressing adjusts that signature in a manner that can be linked to the fault [Randall, 2011]. The philosophy of CBM consists of scheduling maintenance activities only once a failure is detected. The International Organization for Standardization (ISO) has published some guides for rotating machinery. Among them are: ISO 172431-1, Evaluation of machine tool spindle vibrations by measurements on spindle housing, which recommends the use of vibration velocity RMS for monitoring long term spindle condition, while for short term spindle condition, the vibration acceleration RMS, [ISO, 2014]. ISO 10816-1, Evaluation of machine vibration by measurements on non-rotating parts, which gives the following criteria when acquiring vibration signals to obtain suitable records. ISO 2954, Requirements for instruments for measuring vibration severity, recommends the use of vibration acceleration is recommended for high-speed machines and for rolling element bearing, [ISO, 2012].. 2.2.2. Bearing Faults. In rotating machines the rolling bearings are one of elements most widely used and their failure probably the most frequent known reasons for machine breakdown. Therefore, vibration signals generated by faults in them have been widely studied. The rolling bearing are composed by: Inner Race (IR), Outer Race (OR) Rolling Elements (RE) and cage. The IR is straight installed on the shaft, and the OR rests on the bearing casing or machine pedestal. A schematic diagram of a rolling bearing is shown in Fig. 2.1..

(35) 2.2. THEORETICAL BACKGROUND. 29. Figure 2.1: Rolling bearings components. When the RE strike a local fault on the OR or on the IR, a shock caused by impact is introduced exciting high-frequency resonances of the whole structure between the output of the transducer and the analyzed bearing, [Randall, 2011]. The series of broadband burst, excited by the shocks, are modulated in amplitude, wherein envelope signals contain information about fault frequencies: d 1 − cosφ , D nfr d BP F I = 1 + cosφ , 2 D " 2 # Dfr d BSF = 1− cosφ 2d D fr d FTF = 1 − cosφ 2 D nfr BP F O = 2. (2.1a) (2.1b) (2.1c) (2.1d). where fr is the shaft speed of the rotating machine, n and d are the number of the RE (balls) and their diameters respectively, D is the pitch circle diameter and φ is the contact angle measures from the center line of the RE and the bearing axis. Depending on where the fault is located, the following terms can be defined: frequency from the OR as (BPFO, Ball Pass Frequency Outer), from the IR as (BPFI, Ball Pass Frequency Inner), of the RE as (BSF, Ball Spin Frequency) and from the Cage as Fundamental Train Frequency (FTF). A fault in RE generate two impacts every shaft rotation, this is due to the fact a defect strikes both the IR and the OR. Sometimes energy of the fault components is dispersed in their sidebands which appear as a result of modulation effects that occurs when fault frequencies are modulated by bearing housing’s natural frequency, [Sinha, 2015] . In case of IR signals, sidebands are spaced at fr regarding IR component and its harmonics. In RE signals, sidebands are spaced at Cagef , eqn (2.1d), regarding RE component and its harmonics. OR does not show sidebands, [Randall, 2011]..

(36) 30. CHAPTER 2.. Sometimes energy of the fault components is dispersed in their sidebands which appear as a result of modulation effects that occurs when fault frequencies are modulated by bearing housing’s natural frequency, [Sinha, 2015] . In case of IR signals, sidebands are spaced at fr regarding IR component and its harmonics. In RE signals, sidebands are spaced at Cagef , eqn (2.1d), regarding RE component and its harmonics. OR does not show sidebands, [Randall, 2011].. 2.2.3. Time-Frequency Analysis. One of the most commonly practices used in signal processing field is to transform the signals and represent them in other domains, with the aim to have a better understanding of the information contained in them. Time-frequency analysis methods emerged as a powerful tool to describe signal behavior over time and frequency domains simultaneously. There is a fundamental trade off among linear TFR such as: STFT and WT, this is, a good time resolution requires a short window weighting function and a good frequency resolution requires a long window weight function. In this case, it is impossible to simultaneously achieve both good time and frequency resolution. Then, to improve the time-frequency resolution, the quadratic TFR, WVD, can be considered. This nonlinear representation belongs to Cohen’s class, which is the class of time-frequency energy distributions covariant by translations in time and in frequency [Cohen, 1989]. Wigner-Ville Distribution (WVD) For a signal x(t) the WVD is defined as: Z W V Dx (t, f ) =. ∞. τ ∗ τ −j2πf τ x t+ e dτ x t− 2 2 −∞. (2.2). where, x∗ (t) is the complex conjugate of x(t). As it can be seen in eqn (2.2), WVD definition describes a bilinear energy distribution, because it involves a product of a signal with itself. Moreover, these results in a non-linear combination of the components contained in x(t). In contrast with linear TFR, time windows are not used. WVD can also be seen as the Fourier Transform of the local auto-covariance or autocorrelation of x(t). In eqn (2.2), the product in the integral at any time τ is the product of the signal τ /2 in the past with its own future of the same τ /2 duration. Essentially, the left part of the signal (at any time τ ) is folded to the right part of the signal to find the overlap between the past and future values of the signal, and it is performed over its entire time duration. Considering a signal with two frequency components x(t) = x1 (t) + x2 (t), then WVD is: W V Dx (t, f ) = W V Dx1 x1 (t, f ) + W V Dx2 x2 (t, f ) + W V Dx1 x2 (t, f ) + W V Dx2 x1 (t, f ) (2.3).

(37) 2.2. THEORETICAL BACKGROUND. 31. where W V Dx1 x1 , W V Dx2 x2 are auto-terms and W V Dx1 x2 , W V Dx2 x1 are cross-term which satisfy: W V Dx1 x2 (t, f ) = W V Dx∗2 x1 (t, f ).. (2.4). Although the signal term is well localized in both domains, numerous other terms are present at position where the energy distribution should be null. This could lead to misleading interpretation of T F R. Cross terms appear halfway between each pair of auto terms, [Boashash, 2015]. WVD possess great properties for signal processing. Among them, the two most relevant are: the distribution keeps an energy conservation, and it is a real-valued function of time and frequency. WVD preserves time and frequency shifts and satisfy the marginal properties, that is, the energy spectral density and the instantaneous power can be obtained as the marginal distributions of W V Dx in time or frequency. Smoothed Pseudo WVD The two main disadvantages of the WVD are: the possibility of having negative energy values and the interference terms in the TFR. Thus, to facilitate its interpretation, a method which has little effect on true signal components, but it highly attenuates the interference terms, must be included. One way to successfully achieve this is to filter the definition in eqn (2.2) with lowpass filters, [Ioana and Quinquis, 2005]. The filter can be unidimensional or bidimensional. In practice, the use of a separable time-frequency window is preferred. The SPWVD uses independent windows to smooth in time and frequency. In the continuous time case it is defined as, eqn (2.5) [Boashash, 2015]: Z SP V W Dx (t, f ) =. ∞. τ ∗ τ −j2πf τ g(t)H(f )x t + x t− e dτ 2 2 −∞. (2.5). where g(t) and H(f ) are the smoothing windows applied in time and frequency domains respectively. Signal Decomposition Methods Signal decomposition is the task of separating a set of source signals from a set of mixed signals. Some methods have been developed for performing this task. Among them, LMD and VMD methods have received wide attention in signal processing field..

(38) 32. CHAPTER 2.. Local Method Decomposition (LMD) LMD is a self-adaptive method which has been wide used in electroencephalogram signal analysis, and first proposed by Smith [Smith, 2006]. This method involves progressively separating a FM signal from an AM or envelop signal. In the decomposition procedure a number of PF can be formed, by multiplying the purely FM signals with their respectively envelope signals. The following steps summarize LMD algorithm, [Smith, 2006], [Liu et al., 2017b] : 1. Find local extremes in original discrete signal x(n) and compute the raw version of local mean, m0 (n), and local magnitude, a0 (n), components. For two successive extrema ni and ni+1 in x(n): m0 (n) = [x(ni ) + x(ni+1 )]/2. (2.6). a0 (n) = [x(ni ) − x(ni+1 )]/2. (2.7). Both signals are smoothed by using a moving averaging method and m(n) and local magnitude a(n) are formed. 2. Get the estimated FM signal s11 (t) via: s11 = (x(n) − m11 (n))/a11. (2.8). If the next computation of the envelope a12 (t) = 1, s11 is a purely FM signal and the first product function P F1 (n) is obtained. However, if a12 (t) 6= 1, the previous steps are repeated r times until the local envelope a1r (n) of s1r = 1. lim a1r (n) = 1. n→∞. (2.9). 3. The next PF are formed using a residual signal u(n), the whole process is repeated q times until uq (n) is a constant or contains no more oscillations. A more detailed description of the algorithm is presented in [Liu et al., 2017b]. P Fq (n) and their envelopes, aq (n), can be used for performing the fault diagnosis either in high or low frequencies respectively. Variational Mode Decomposition (VMD) VMD is a non-recursive decomposition signal method developed by [Dragomiretskiy and Zosso, 2014], in which the modes are concurrently extracted. This method introduces a penalty term,.

(39) 2.2. THEORETICAL BACKGROUND. 33. alpha, for estimating the bandwidth of each mode (called BLIMF). The VMD decomposition is a constrained optimization problem that minimizes the sum of the estimated mode bandwidths: min. nP. {uk }{wk }. k. ∂t. . δ(t) +. s.t.. P. k. j πt. . −jw t 2 o ∗ uk (t) e k 2. (2.10). uk = f. where uk is the k th mode component, wk is the center frequency of the k th mode component, f is the original signal, δ is the Delta Dirac distribution, and ∗ is the convolution operator. Introducing a quadratic penalty term alpha and Lagrange multipliers lambda, the constrained variational problem is transformed into an unconstrained problem. A further description of the mathematical foundations of this algorithm can be found in [Dragomiretskiy and Zosso, 2014]. The solution of the original minimization problem of eqn (2.10) is found introducing the alternate direction method, using Parseval/Plancherel Fourier isometry under the L2 norm. In spectral domain ûn+1 can be defined as: k. ûn+1 = arg min k uˆk ,uk X.  . α kjw (1 + sgn(w + wk )) ûk (w + wk )k22 + fˆ(w) −. . X i. ûi (w) +. λ(w) 2. 2.  . 2. . (2.11) Let be τ the time-step of the dual ascent, a predefined convergence tolerance level and function G, eqn (2.12): G=. X. uˆk n+1 − uˆk n. 2 2. / kûnk k22. (2.12). k. the VMD algorithm can be summarized as shown in Fig. 2.2.. 2.2.4 Kullback-Leibler Divergence (KLD) Relative entropy or KLD measures the distinguishability of two probability distributions. Considering two random variables x and y, and let p and q be the probability distributions associated to these variables respectively, the KLD between p and q can be defined as, [Roldán, 2014]: Z D [p(x) k q(y)] =. p(x)ln. p(x) dx q(y). (2.13). KLD is always a positive value and vanishes if and only if p(x) = q(x). Moreover, it is not symmetric, i.e.: D [p(x) k q(y)] 6= D [q(x) k p(y)]. (2.14).

(40) 34. CHAPTER 2.. Figure 2.2: VMD algorithm. In a signal decomposition KLD can be used as an indicator of the correlation between each of the signal components with the original, and to distinguish the true components from the false..

(41) Chapter 3 Experimental System 3.1. Experimental Databases. To test the validity of the methodology, two different open-access experimental databases were used. The experiments were accomplished with the intention of studying the behavior of vibration signals when rotating machines present a failure related to rolling bearings, by inducing localized defects in them. They were performed under different speed and load conditions, and the data were recorded using different sampling frequencies, and changing the position of the sensors in the test rig. Accelerometers were used for vibration acquisition. First data set is provided by the Case Western Reserve University (CWRU) and has become a standard reference in the bearing fault diagnosis. A helpful benchmark study was achieved in [Smith and Randall, 2015] in which every data record was categorized according to the experimental conditions and based on the success in the diagnosis achieved by three established algorithms such as: Envelope Analysis, Cepstrum Prewhitenning and Fast Kurtogram. The second dataset is provided by the Intelligent Maintenance System (IMS), University of Cincinnati, [Lee et al., 2009]. Unlike aforementioned, this is about a run-to-failure experiment involving rolling bearings on a constant loaded shaft.. 3.1.1. Case Western Reserve University (CWRU). The experimental setup in CWRU consists of a 2 HP electric motor, a torque transducer and an encoder. The electric motor was coupled to a dynamometer to apply torsional loads of 0, 1, 2, and 3 HP, Fig. 3.1. Defects of 0.007, 0.014, 0.021, and 0.028 inches were seeded in RE, IR, and OR by ElectroDischarge Machining (EDM) in Drive End (DE) and Fan End (FE) bearings. Dimensions of used bearings are shown in Table 3.1.. 35.

(42) 36. CHAPTER 3.. Figure 3.1: Test Rig used by CWRU, [Smith and Randall, 2015] Table 3.1: Characteristics of used bearings. CWRU database. Inside Diameter. Outside Diameter. Thickness. Ball Diameter. Pitch Diameter. (inches). (inches). (inches). d (inches). D (inches). Drive end. 0.9843. 2.0472. 0.5906. 0.3126. 1.537. Fan end. 0.6693. 1.5748. 0.4724. 0.2656. 1.122. Location. Using Table 3.1 and eqns (2.1a)-(2.1d), frequency factors can be defined for each condition as multiples of shaft speed to compute fault frequencies in a direct way, Table 3.2. Every test consisted on a faulty bearing (DE or FE), shaft speed varies between 1,720 - 1,797 RPM with a load applied of 0-3 HP. Signals are recorded using accelerometers placed in vertical position on DE and FE, Fig. 3.1, with sampling frequencies of 12,000 samples/second and at 48,000 samples/second in other cases. Table 3.2: Frequency factors (multiple of shaft speed in Hz). Location. Model. BPFI. BPFO. BSF. FTF. Drive End. SKF6205-2RSJEM. 5.4152. 3.5848. 4.7135. 0.3983. Fan End. SKF6203-2RSJEM. 4.9469. 3.0530. 3.9874. 0.3817. 3.1.2 IMS Database This system includes 4 test Rexnord ZA-2115 double row bearings on one shaft driven by an AC motor, Fig (3.2). It has three different datasets: in set 1, 8 high precision accelerometers were.

(43) 3.1. EXPERIMENTAL DATABASES. 37. installed on the bearing housings at both vertical and horizontal directions (2 for each bearing), while in the two remainders only one accelerometer was used for each bearing. Experiments were performed with a shaft speed = 2,000 RPM and constant radial load = 6,000 lb. Data were recorded with a sampling frequency = 20 KHz. Each file name indicates the date when the data was recorded and has a length of 2,048 sample points.. Figure 3.2: Schematic sketch of IMS test rig, [Qiu et al., 2006]. Dimensions of used bearings and frequency factors computed using eqns (2.1a)-(2.1d), for each condition are shown in Table 3.3 and 3.4 respectively. Table 3.3: Characteristics of used bearings. IMS database. Ball diameter. Pitch diameter. d(mm). D(mm). 8.4074. 71.5001. Number of rollers. Contact angle φ[deg]. 16. 15.17. Table 3.4: Frequency factors (multiple of shaft speed in Hz) BPFI (Hz). BPFO (Hz). BSF (Hz). FTF (Hz). 8.91. 7.09. 4.19. 0.44.

(44) 38. CHAPTER 3.. 3.2. Design of Experiments. A group of the signals from CWRU and IMS are of special interest for this research work, because they have experimental signal that represent the condition of early failure in bearings. The description of the signals used and some established considerations for these databases are described.. 3.2.1. Tested signals for CWRU database. The group of signals from categories Y1, Y2, P1, P2 and N1, according to the study performed in [Smith and Randall, 2015], are selected for testing the robustness of the used algorithms. They were classified based on the results obtained by other techniques such as Envelope FFT, Cepstrum Pre-whitening and Benchmark Method and are listed below: • Y1: Data clearly diagnosable and showing classic characteristics for the given bearing fault in both time and frequency domain • Y2: Data clearly diagnosable but showing non-clasic characteristics in either or both time and frequency domain • P1: Data probably diagnosable, e.g., envelope spectrum shows discrete components at expected fault frequencies but they are not dominant • P2: Data potentially diagnosable, e.g., envelope spectrum shows smeared components that appear to coincide with the expected fault frequencies • N1: Data not diagnosable for the specified bearing fault but with other identifiacable problems (e.g. Looseness) A first group is formed by signals which represent the 3 bearing fault conditions IR, OR and RE and 1 representing the normal state . This signals were measured from the position DE, Table 3.5. The notation of these files DESF SS FS represents: the sampling frequency (SF), The shaft speed (SS), and the failure size seeded in the bearing (FS). The signals in time domain can be in Fig. 3.3. The sampling frequencies for all the tested signals is 12KHz. The characteristic fault frequencies for rolling used in these experiments were computed for each shaft speed and are shown in Table 3.6 using eqns (2.1a)-(2.1d). These values are used to decided the bearing condition. A group of signals labeled as hardly and not diagnosable (P2, N1) by other techniques such as Envelope FFT, Cepstrum Pre-whitening and Benchmark Method, [Smith and Randall, 2015], were also tested, Table 3.7. These are presented in Fig. 3.4 in the time domain..

(45) 3.2. DESIGN OF EXPERIMENTS. 39 X130 Amplitude (g). Amplitude (g). X048 5 0 -5. 0. 0.05. 0.1. 5. 0. -5. 0.15. 0. 0.05. Amplitude (g). Amplitude (g). 5. 0. -5 0.05. 0.15. X097. X171. 0. 0.1. time (s). time (s). 0.1. 0.15. 5. 0. -5 0. 0.05. time (s). 0.1. 0.15. time (s). Figure 3.3: Searching frequency algorithm. Table 3.5: Experimental signals from different bearing conditions, CWRU. File Name. Signal. Bearing condition. Category. DE12k 0 028. X048 DE time. RE. Y2. DE12k 2 014. X171 DE time. IR. P1. DE12k 0 007. X130 DE time. OR. Y1. NBD 0 1797. X097 DE time. Normal. -. Table 3.6: Characteristic fault frequencies for CWRU. 3.2.2. Shaft speed (RPM). Shaft speed (Hz). BPFO. BSF. BPFI. FTF. 1797. 29.95. 107.36. 141.17. 162.19. 11.93. 1772. 29.53. 105.87. 139.21. 159.93. 11.76. 1754. 29.23. 104.80. 137.79. 158.30. 11.64. 1730. 28.83. 103.36. 135.91. 156.14. 11.48. Tested signals for IMS database. Due to IMS database involves a run-to-fault experiment, to know the different conditions that showed the used bearings through the tests, the data records need to be individually analyzed..

(46) 40. CHAPTER 3. Table 3.7: Tested signals, hardly and non-diagnosable CWRU. File Name. Type of fault. # of Test. Evelope. Cepstrum PW. Benchmark. (suggested) DE12k 0 028 IR. IR. X056. N1. N1. N1. DE12k 1 028 IR. IR. X057. N1. N1. N1. DE12k 2 028 IR. IR. X058. N1. N1. N1. DE12k 3 028 IR. IR. X059. N1. N1. N1. DE12k 0 007 B. RE. X118. N1. N1. N1. DE12k 2 007 B. RE. X120. N1. N2. N1. DE12k 2 014 B. RE. X187. N1. P2. N1. DE12k 3 014 OR C. OR. X200. N1. N1. N1. FE12k 0 007 B. RE. X284. N1. Y2. P2. FE12k 0 021 B. RE. X290. N1. N1. N1. FE12k 2 021 B. RE. X292. N1. P2. N1. FE12k 3 007 OR Or. OR. X301. N1. P2. N1. FE12k 3 007 OR Op. OR. X307. N1. N2. N1. According to [Li et al., 2017b] to categorize IMS database, the kurtosis index can be computed for all the data records (tests) to know the degradation process that suffers the used bearing over its whole life. Figures 3.5 and 3.6 show the result of the experiment labeled as Test1 and Test 2 respectively. In Test 1, the notations bix and biy represent the 2 different places (x, y) where sensors were positioned in the ith bearing. One channel represents one sensor. In test 2 only one sensor per bearing was considered for recording. The index of kurtosis resulted different between channels, and it increases over number of tests (except for CH3 in Test 2). The fault was more representative in some rolling bearings than others. Based on Figs. 3.5, 3.6 and [Li et al., 2017b] research work, the indexes of tests that are proposed for test the methodology are shown in Table 3.8. At the end of the test, there was a clear evidence of IR, RE and OR defects in bearings 3 and 4 of test 1 and bearing 1 of test 2, [Ju et al., 2019]. The signals in time domain are presented in Fig 3.7. Signals from channels 5, 7 and 8 of Test 1 and from channel 1 of Test 2 were analyzed. Five revolutions of the signal length was considered for the analysis. The sampling frequencies for all the tested signals is 20KHz. The theoretical fault.

(47) time (s) X120-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X200-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X284-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. -5 0. 0.05. 0.1. 0.15. time (s) X187-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X059-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X292-DE-time 5 0 -5 0. 0.1. 0.15. X057-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X058-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X290-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s) X307-DE-time 5 0 -5 0. 0.05. time (s) Amplitude (g). time (s). 0.05. Amplitude (g). 0.15. 0. Amplitude (g). 0.1. X056-DE-time 5. Amplitude (g). 0.05. 41. Amplitude (g). 0. Amplitude (g). -5. Amplitude (g). 0. Amplitude (g). X118-DE-time 5. Amplitude (g). Amplitude (g). Amplitude (g). Amplitude (g). Amplitude (g). 3.2. DESIGN OF EXPERIMENTS. 0.1. 0.15. time (s). X301-DE-time 5 0 -5 0. 0.05. 0.1. 0.15. time (s). Figure 3.4: Analyzed signals from CWRU, category: N1-DE. Table 3.8: Proposed categories for IMS database. Type of fault. # Test / Channel. # Test - early. # Test - medium. # Test -critical. stage of fault. stage of fault. stage of fault. IR. Test1 / CH5-CH6. 1,760. 1,820. 1,990. OR. Test2 / CH1. 454. 653. 710. RE. Test1 / CH7-CH8. 1,454. 1,667. 1,754.

(48) 42. CHAPTER 3.. Figure 3.5: Run to fault experiment from group of signals in Tests 1.. IMS Test 2, Run to failure 15. Kurtosis. 10. CH1 b1 CH2 b2 CH3 b3 CH4 b4. CH3 b3. CH1 b1. 5. CH4 b4. CH2 b2. 0 100. 200. 300. 400. 500. 600. 700. 800. 900. Index of Test Figure 3.6: Run to fault experiment from group of signals in Tests 2. frequencies computed for this group of signals are shown in Table 3.9 Table 3.9: Expected fault frequencies for IMS BPFO (Hz). BPFI (Hz). BSF (Hz). 230.66. 296.93. 131. 1000.

(49) 3.2. DESIGN OF EXPERIMENTS. 43. IR signals, CH5, b3y Critical stage - Test: 1990. Medium stage - Test: 1820 2. 2. 1.5. 1.5. 1.5. 1 0.5 0 -0.5 -1. Magnitude. 2. Magnitude. Magnitude. Early stage - Test: 1760. 1 0.5 0 -0.5 -1. 0. 0.05. 0.1. 0. 0.15. -1. -2. -2. -2. 0 -0.5. -1.5. -1.5. -1.5. 1 0.5. 0.05. 0.1. 0. 0.15. 0.05. Time. 0.1. 0.15. Time. Time OR signals, CH1-b1. Early stage - Test: 454. 2. Medium stage - Test: 653. 1. 1. 0. 0. -1. -1 -2. 2. Magnitude. Magnitude. Magnitude. 2. 0.05. 0.1. Time. 0.15. 1 0. -1. -2 0. Critical stage - Test: 710. 0. 0.05. 0.1. Time. -2. 0.15. 0. 0.05. 0.1. Time. 0.15. RE signals CH7/CH8-b4x/b4y. Magnitude. Magnitude. 1. 2. 0. -1. -2. Medium stage - Test: 1667. 1. 0. -1. -2 0. 0.05. 0.1. Time. 0.15. Critical stage -Test: 1753. 2. Magnitude. Early stage - Test: 1454. 2. 1. 0. -1. -2 0. 0.05. 0.1. 0.15. 0. Time. Figure 3.7: Analyzed signals from IMS.. 0.05. Time. 0.1. 0.15.

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(51) Chapter 4 Proposal A methodology based on signal decomposition method and WVD is developed to detect faults in rolling bearings such as; IR, OR and RE. A scheme of the proposed methodology is shown in Fig 4.1. This was adapted to be applied with VMD and LMD methods to verify and compare their performances, in separate components of experimental signals, and specially to help identify small transients which may be an indicative of an early bearing fault. Choosing many components may not have much impact on the processing of the original signal, because the original signal remains without any improvement. Select the improper components can hide relevant peaks in the frequencies of interest and lead to a wrong diagnosis. A criteria based on relative entropy is used to identify the effective signal components related to the fault. A threshold for KLD is established to separate non-relevant components and attenuate them. MATLAB software was used with the Signal Processing Toolbox 2018b. VMD and LMD MATLAB functions were obtained from [Dragomiretskiy and Zosso, 2014] and [Liu et al., 2017a] respectively.. 4.1. Proposed Methodology for Visual Diagnosis. The methodology uses the envelope of resultant effective component (sum of the envelopes of the effective components) to perform a visual diagnosis based on the peaks in its spectrum that match with the fault characteristic frequencies previously computed, eqn (2.1a)-(2.1c). A spectrum searching algorithm is also presented to give information of the relevant peaks found in the vicinity of the theoretical fault frequencies. To explain the methodology performance, signals from CWRU bearing database were used.. 45.

(52) 46. CHAPTER 4. PROPOSAL. Figure 4.1: Flowchart of proposed methodology.. 4.1.1. Trend Removal. 0. Before signal decomposition, the acquired vibration signal is preprocessed by a trend removal. Measured vibration signals can show overall patterns that are not intrinsic to the real data. These.

(53) 4.1. PROPOSED METHODOLOGY FOR VISUAL DIAGNOSIS. 47. trends can sometimes hinder the data analysis and must be removed. A trend component is reflected in the frequency domain as a peak in 0 Hz having amplitude equal to the constant value in the signal (DC component). A trend typically indicates a systematic increase or decrease in the data and can appear as a result of sensor drift [George et al., 2017]. A better illustration about trends in data is shown in Fig. 4.2. Raw Signal. 0.4. Original signal. 0.3. 0.03. 0.2. Original signal. Magnitude. 0.025. 0.1. X(t). Spectrum of x(t). 0.035. 0. -0.1. 0.02. 0.015. Detrended signal. 0.01 -0.2. Signal without trend. -0.3. 0.005. 0. -0.4 0. 0.005. 0.01. 0.015. 0.02. 0.025. 0.03. 0.035. 0.04. 0.045. 0.05. 0. 100. 200. 300. 400. 500. 600. Frequency(Hz). Time(s). Figure 4.2: Trend in data. Left plot: signal in time and right plot : spectrum of signal. Signals can present linear or non-linear trends. For a better analysis, non-linear behavior of trend is assumed. To eliminate the non-linear trend, it is modeling by a n-order polynomial, eqn (4.1), fitted to the original signal and then it is subtracted from original data set. In this document, a 6th order polynomial is considered enough. p(x) = p1 xn + p2 xn−1 + · · · + pn x + pn+1. 4.1.2. Signal Decomposition. (4.1). 1. Data X234 DE from CWRU bearing database were used to illustrate the proposed methodology, Fig. 4.3. This experimental signal corresponds to an OR fault signal with a sampling frequency of 12 KHz and a shaft speed of 1,797 RPM. • LMD. When vibration signal is decomposed by LMD, a series of PF are obtained. Each PF is composed by an envelope signal and a purely frequency modulated signal, eqn (??), as shown in Fig 4.4. Moving average ma is used to smooth the local and amplitude mean in each iteration, eqns (2.6)-(2.7). Then, PF are built by multiplying each envelope signal by its respectively purely FM component. Resulting PF and the effect of signal decomposition method are illustrated in Fig. 4.5, where each spectrum belongs to each generated PF..

(54) 48. CHAPTER 4. PROPOSAL Raw Signal. 6. 4. X(t). 2. 0. -2. -4. -6 0. 0.05. 0.1. 0.15. 0.2. 0.25. 0.3. 0.35. Time(s). Figure 4.3: OR fault signal from CWRU bearing database. Amplitude Signal1 (ma). FM Signal1 (ma). 3 2 1. 1 0 -1 -2 0.05. 0.1. 0.15. 0.2. 0.25. 0.3. 0. Amplitude Signal2 (ma) 0. 0.05. 0.1. 0.15. 0.2. 0.25. 0.3. Amplitude Signal3 (ma) 0.04 0.02 0. 0.05. 0.1. 0.15. 0.2. 0.25. 0.3. Amplitude Signal4 (ma) 0.04 0.02 0. 0.05. 0.1. 0.15. 0.2. 0.25. 0.1. 0.15. 0.2. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 2 0 -2 0. 0.05. 0.1. 0.15. 0.2. FM Signal3 (ma). 2 0 -2 0. 0.05. 0.1. 0.3. 0.15. 0.2. FM Signal4 (ma). 2 0 -2 0. 0.05. 0.1. Amplitude Signal5 (ma). 10 -3. 14 12 10 8 6 4. 0.05. FM Signal2 (ma). Amplitude of FM signals. Amplitude of AM signals. 0 0.25 0.2 0.15 0.1 0.05. 0.15. 0.2. FM Signal5 (ma) 1 0 -1. 0. 0.05. 0.1. 0.15. 0.2. 0.25. 0.3. 0. 0.05. 0.1. Time(s). 0.15. 0.2. Time(s). Figure 4.4: Sifting process in LMD. Envelope or AM signals (left) and purely FM signals (right) are illustrated. PF1(ma) KLD=0.0075828 2 0 -2. Spectrum of PF1 0. 0.05. 0.1. 0.15. 0.2. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 0.25. 0.3. 0.1 0.08 0.06 0.04 0.02. PF2(ma) KLD=0.20568. 0. Acceleration(g). 0. 0.05. 0.1. 0.15. 0.2. PF3(ma) KLD=0.28986 0.05 0 -0.05 0. 0.05. 0.1. 0.15. 0.2. PF4(ma) KLD=0.36611 0.02 0 -0.02 -0.04 0. 0.05. 0.1. 0.15. 0.2. PF5(ma) KLD=0.49581. Magnitude of PFs(f). 0.2 0 -0.2. 1000. 0. 1000. 0. 1000 10. -3. 10. -3. 0 0.05. 0.1. 0.15. 0.2. Residual (ma). 10 -3 8 4 0 -4. 2.5 2 1.5 1 0.5 0. 0. 0.05. 0.1. 0.15. 0.2. 4000. 5000. 6000. 2000. 3000. 4000. 5000. 6000. 2000. 3000. 4000. 5000. 6000. 5000. 6000. 5000. 6000. Spectrum of PF4. 2.5 2 1.5 1 0.5. 0.01 0 -0.01 0. 3000. Spectrum of PF3. 10 -3. 4 2. 2000. Spectrum of PF2. 10 -3. 12 10 8 6 4 2. 1000. 2000. 3000. 4000. Spectrum of PF5 1000. 2000. 3000. 4000. Frequency (Hz). Time(s). Figure 4.5: PF (left plot) and their respectively spectra (right plot) obtained by LMD algorithm..

(55) 4.1. PROPOSED METHODOLOGY FOR VISUAL DIAGNOSIS. 49. • VMD. When signal is decomposed by VMD, eqn (2.11), a series of BLIMF are obtained. By using LMD or VMD, original signal suffers a segmentation of its spectrum, but in different ways, Fig. 4.6, Fig. 4.5 (right plot). Spectrum of x(t). Spectrum of BLIMF (VMD). 1 0.9. 0.8. 0.8. 0.7. 0.7. Magnitude. Magnitude. 1 0.9. 0.6 0.5 0.4. BLIMF 5. BLIMF 1 BLIMF 4. BLIMF 6. 0.6 0.5. BLIMF 3. 0.4. 0.3. 0.3. 0.2. 0.2. BLIMF 7 BLIMF 2. 0.1. BLIMF 8. 0.1. 0. 0 0. 1000. 2000. 3000. 4000. 5000. 6000. 0. 1000. 2000. Frequency(Hz). 3000. 4000. 5000. 6000. Frequency(Hz). Figure 4.6: Spectrum of x(t) (left), and of each BLIMF obatined by VMD method (right).. In this signal, the levels of decomposition for VMD and LMD were set to 5 and 8 respectively. The penalty term α VMD was set to 500. The criteria to select these parameters are commented in Appendix C.. 4.1.3. Selection of the effective signal component. 2. Two main arguments can be considered in the vibration signals analysis for bearing diagnosis, first, the resonance frequency band of a bearing need to be analyzed when a fault occurs, which can be found above 500 Hz according to [Sinha, 2015]. Second, only the components with the most relevant information about the fault needs to be analyzed or highlighted over others. The identification of the effective signal component and thus the optimal frequency band is performed using KLD to identify the components which are related to the fault. • As KLD is based on the entropy concept in information theory, this parameter allows to verify which distribution, between all of the PF, perseveres the most information from original signal. The smaller KLD value, a high correlation between a the signal component and original signal. Since KLD is not symmetric, eqn (2.14), it can be defined between the probability distributions of the original signal x(n) and the ith signal component S in two senses as: δi (p(x), q(Si )) =. X. p(x)ln. p(x) q(Si ). (4.2a). δi (q(Si ), p(x)) =. X. p(x)ln. q(Si ) p(x). (4.2b).