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Seoul National University2/25/2017 1
SIGNAL PROCESSING 
TECHNIQUES
USED FOR GEAR FAULT 
DIAGNOSIS
Jungho Park, Ph. D. cand...
Seoul National University
Significance
2/25/2017 2
• One of the most widely used mechanical elements, gear
• One of the ke...
Seoul National University
Fault Detection of a Gear
2/25/2017 3
• Fault detection of a gear is usually performed by vibrat...
Seoul National University
Fault Detection of a Gear
2/25/2017 4
• Fault detection of a gear is usually performed by vibrat...
Seoul National University
Fourier Analysis
2/25/2017 5
“An arbitrary function, 
continuous or with 
discontinuities, defin...
Seoul National University
Frequency Analysis
2/25/2017 6
Z Hz
Y Hz
X Hz
freq.
Amp.
X Y Z
Fourier Transform : 
Inverse Four...
Seoul National University
Gear Fault Diagnosis Using Frequency Analysis
2/25/2017 7
• Normal Gear signals
– Consist of 3 h...
Seoul National University
Non‐stationary Gear Signals
2/25/2017 8
• Normal gear signals
– No harmonics with 10% fluctuatin...
Seoul National University
Signal Processing for Advanced Fault Diagnosis
2/25/2017 9
1) Wavelet transform (Time‐frequency ...
Seoul National University
A Drawback of Fourier Analysis
2017/2/25 ‐ 10 ‐
0 0.5 1 1.5 2 2.5 3 3.5
x 10
4
-1
-0.5
0
0.5
1
s...
Seoul National University
Short Time Fourier transform (STFT)
2017/2/25 ‐ 11 ‐
0 0.5 1 1.5 2 2.5 3 3.5
-1
-0.5
0
0.5
1
SAL...
Seoul National University
Short Time Fourier transform (STFT)
2017/2/25 ‐ 12 ‐
• Multiple FT over smaller windows translat...
Seoul National University
1) Wavelet Transform
2017/2/25 ‐ 13 ‐
• Wavelet, a small wavelike signal, is used as 
a basis fu...
Seoul National University
Papers on Wavelet for Fault Diagnosis
2017/2/25 ‐ 14 ‐
• Wang, W. J., and P. D. McFadden. "Appli...
Seoul National University
Application of Wavelet (1) : Planetary Gear
2017/2/25 ‐ 15 ‐
• Wavelet transform is applied to t...
Seoul National University
Results (Methodology from Reference*)
2017/2/25 ‐ 16 ‐
WT
Coeff.
FT fp
*Wang, Changting, Robert ...
Seoul National University
Application of Wavelet (2) : Spur Gear
(Simulated signals)
2017‐02‐25 17
• Normal gear signals
–...
Seoul National University
Results
2017‐02‐25 18
STFT
WT
• Hard to differentiate between 
normal and fault using STFT.
• Go...
Seoul National University
Advantages
2017/2/25 ‐ 19 ‐
• Effective in extracting transient features.
• Adaptive in resoluti...
Seoul National University
Research Direction (1)
2017/2/25 ‐ 20 ‐
• Wavelet + Machine learning algorithm
– Abbasion, Saeed...
Seoul National University
Research Direction (2)
2017/2/25 ‐ 21 ‐
Chen, Jinglong, et al. "Wavelet transform based on inner...
Seoul National University
2) EMD (Empirical mode decomposition)
2017/2/25 ‐ 22 ‐
Empirical : based on testing or experienc...
Seoul National University
Principles of EMD
2017/2/25 ‐ 23 ‐
Signals
Low frequency High frequency
Seoul National University
Procedures
2017/2/25 ‐ 24 ‐
1. Identify local maxima and minima in the signal
2. Deduce an upper...
Seoul National University
Advantages/Disadvantages of EMD
2017/2/25 ‐ 25 ‐
• EMD is a model‐free, and fully 
data‐driven m...
Seoul National University
Mode‐mixing Problems in EMD
2017/2/25 ‐ 26 ‐
• Mode‐mixing : a single IMF with oscillations of d...
Seoul National University
EEMD (Ensemble Empirical Mode Decomposition)
2017/2/25 ‐ 27 ‐
• Ensemble average :  Mean of a qu...
Seoul National University
Procedures of EEMD
2017/2/25 ‐ 28 ‐
1. Initialize the number of ensemble M, and m = 1.
2. Perfor...
Seoul National University
Comparison btw. EMD and EEMD
2017/2/25 ‐ 29 ‐
Lei, Yaguo, et al. "A review on empirical mode dec...
Seoul National University
Papers on EMD for Fault Diagnosis
2017/2/25 ‐ 30 ‐
• Loutridis, S. J. "Damage detection in gear ...
Seoul National University
3) Hilbert Spectrum 
2017/2/25 ‐ 31 ‐
Hilbert Transform
         
1ˆ , whereHT f t f t...
Seoul National University
Hilbert Transform
2017/2/25 ‐ 32 ‐
Definition
         
1ˆ , whereHT f t f t f t h t h...
Seoul National University
Analytic Signal
2017/2/25 ‐ 33 ‐
Definition
Relationship with the Fourier transform (FT)
   ...
Seoul National University
Properties
Properties of Analytic Signal & Relation with EMD
2017/2/25 ‐ 34 ‐
       2 2...
Seoul National University
Comparison with Fourier and Wavelet
2017/2/25 ‐ 35 ‐
Fourier Wavelet Hilbert
Basis a priori a pr...
Seoul National University
Examples
2017/2/25 ‐ 36 ‐
Peng, Z. K., W. Tse Peter, and F. L. Chu. "A comparison study of impro...
Seoul National University
Papers on HHT for Fault Diagnosis
2017/2/25 ‐ 37 ‐
• Huang, Norden E., et al. "The empirical mod...
Seoul National University
4) AR‐MED filter
2017/2/25 ‐ 38 ‐
• Combination of AR filter and MED filter
• AR filter : Autore...
Seoul National University
AR filter
2017/2/25 ‐ 39 ‐
• AR filter : Autoregressive model‐based filtering technique
• AR mod...
Seoul National University
MED filter
• MED : Minimum Entropy Deconvolution
• The filter searches for an optimum set of fil...
Seoul National University
Application of AR‐MED filter
2017/2/25 ‐ 41 ‐
-0.1
0
0.1
-1
0
1
-0.2
0
0.2
2 2.05 2.1 2.15 2.2 2...
Seoul National University
Papers on AR or MED filter for Fault Diagnosis
2017/2/25 ‐ 42 ‐
• Sawalhi, N., R. B. Randall, an...
Seoul National University
5) Spectral kurtosis
2017/2/25 ‐ 43 ‐
• Kurtosis* : 
• Spectral kurtosis (SK) extends the concep...
Seoul National University
Definition of SK (1)
2017/2/25 ‐ 44 ‐
Antoni, Jérôme. "The spectral kurtosis: a useful tool for ...
Seoul National University
Definition of SK (2)
2017/2/25 ‐ 45 ‐
Antoni, Jérôme. "The spectral kurtosis: a useful tool for ...
Seoul National University
Estimation of SK : (1) STFT 
2017/2/25 ‐ 46 ‐
• STFT (short‐time Fourier transform) of the proce...
Seoul National University
Estimation of SK : (2) Kurtogram
2017/2/25 ‐ 47 ‐
• In STFT, non‐stationarity of the signals sho...
Seoul National University
Fast kurtogram
2017/2/25 ‐ 48 ‐
• Calculation of the whole plane ( , ∆ ) is a formidable task in...
Seoul National University
Procedures of Fault Diagnosis Using SK
2017/2/25 ‐ 49 ‐
Find the frequency that has
maximum kurt...
Seoul National University
Papers on SK for Fault Diagnosis
2017/2/25 ‐ 50 ‐
• Antoni, Jérôme. "The spectral kurtosis: a us...
Seoul National University
6) Cyclo‐stationary : In search of hidden periodicities
2017/2/25 ‐ 51 ‐
0
Stationary signals 
…...
Seoul National University
Cyclo‐stationary*
2017/2/25 ‐ 52 ‐
• Cyclo‐stationary at the 1st order (periodic waveforms with ...
Seoul National University
, ; Δ 	 ; ; Δ ·
∈
Cyclic Decomposition of Energy Flow
: Extraction of Cyclic Trends (1)
2017/2/2...
Seoul National University
, ; Δ 	 ; ; Δ ·
∈
Cyclic Decomposition of Energy Flow
: Extraction of Cyclic Trends (2)
2017/2/2...
Seoul National University
, ; Δ 	 ; ; Δ ·
∈
Cyclic Decomposition of Energy Flow
: Extraction of Cyclic Trends (2)
2017/2/2...
Seoul National University
Spectral Correlation Density & Spectral Coherence
2017/2/25 ‐ 56 ‐
Spectral Correlation
, lim
→
...
Seoul National University
Physical Meaning of SCD and SC
2017/2/25 ‐ 57 ‐
• Spectral Correlation Density
– Non‐zero value ...
Seoul National University
Examples : Planetary Gear (1, simulated signals)
2017/2/25 ‐ 58 ‐
ACC.
• Inherent modulated acce...
Seoul National University
Examples : Planetary Gear (2, simulated signals)
2017/2/25 ‐ 59 ‐
• Inherent modulated accelerat...
Seoul National University
Examples : Planetary Gear (3, simulated signals)
2017/2/25 ‐ 60 ‐
• For a faulty case, more ener...
Seoul National University
Papers on Cyclostationary for Fault Diagnosis
2017/2/25 ‐ 61 ‐
• Capdessus, C., M. Sidahmed, and...
Seoul National University
Other Techniques
2017/2/25 ‐ 62 ‐
• Time‐frequency analysis
– Wigner–Ville Distribution (WVD)
– ...
Seoul National University
THANK YOU
2/25/2017 63
Seoul National University
BACK‐UP
2/25/2017 64
Seoul National University
Procedures for EMD (1)
2/25/2017 65
Seoul National University
Procedures for EMD (2)
2/25/2017 66
Seoul National University
Procedures for EMD (3)
2/25/2017 67
Seoul National University
Procedures for EMD (4)
2/25/2017 68
Seoul National University
Procedures for EMD (5)
2/25/2017 69
Seoul National University
Procedures for EMD (6)
2/25/2017 70
Seoul National University
Results of EMD
2/25/2017 71
Seoul National University
6) Cyclo‐stationary
2017/2/25 ‐ 72 ‐
• Ensemble average :  Mean of a quantity that is a function...
Seoul National University
6) Cyclo‐stationary
2017/2/25 ‐ 73 ‐
cos	 2
2
· cos	 2
1
2
cos 2
2
·
1
2
cos 2
2
·
1
4
1
4
1
4
1...
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SIGNAL PROCESSING TECHNIQUES USED FOR GEAR FAULT DIAGNOSIS

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The slides are about signal processing techniques widely used for gear fault diagnosis (also the techniques could be used for other various rotating machine diagnosis such as bearing, rotor, motor, etc.). The techniques include wavelet transform, EMD (empirical mode decomposition), HHT (Hilbert-Huang transform), AR-MED filter, Spectral kurtosis, and cyclo-stationary analysis.

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SIGNAL PROCESSING TECHNIQUES USED FOR GEAR FAULT DIAGNOSIS

  1. 1. Seoul National University2/25/2017 1 SIGNAL PROCESSING  TECHNIQUES USED FOR GEAR FAULT  DIAGNOSIS Jungho Park, Ph. D. candidate* Lab for System Health Risk Management Department of Mechanical Engineering and Aerospace Engineering Seoul National University, Korea *hihijung@snu.ac.kr
  2. 2. Seoul National University Significance 2/25/2017 2 • One of the most widely used mechanical elements, gear • One of the key research issues in the fault diagnostics.  – Nonlinear : 6, Rotating machine/bearing/gears : 13, Structures/Energy  Harvesting : 4, Uncertainty/Bayesian methods : 3, Acoustics/waves : 3,  Control/image processing : 3, Machine Tools : 1. (MSSP, Dec. 2016)  • Can be applied to other rotating machine diagnostics (rotor, bearing, motor, …)
  3. 3. Seoul National University Fault Detection of a Gear 2/25/2017 3 • Fault detection of a gear is usually performed by vibration signals. – Frequency of vibration signals are determined by speed and tooth number • In an ideal case, fault detection could be done by calculating P2P (peak‐ to‐peak), RMS, or kurtosis of the measured vibration signals. • In a practical case, however, it is not possible due to noises from other  elements or environments.  FREQUENCY ANALYSIS 30 teeth 2rev/s 20 teeth 3rev/s 30 teeth (=0.5s) 60hz 30 teeth (=0.5s)
  4. 4. Seoul National University Fault Detection of a Gear 2/25/2017 4 • Fault detection of a gear is usually performed by vibration signals. – Frequency of vibration signals are determined by speed and tooth number • In an ideal case, fault detection could be done by calculating P2P (peak‐ to‐peak), RMS, or kurtosis of the measured vibration signals. • In a practical case, however, it is not possible due to noises from other  elements or environments.  FREQUENCY ANALYSIS 30 teeth 2rev/s 20 teeth 3rev/s 30 teeth (=0.5s) 60hz 30 teeth (=0.5s)
  5. 5. Seoul National University Fourier Analysis 2/25/2017 5 “An arbitrary function,  continuous or with  discontinuities, defined in a finite  interval by an arbitrarily  capricious graph can always be  expressed as a sum of sinusoids” J.B.J. Fourier 0 cos 2 sin 2
  6. 6. Seoul National University Frequency Analysis 2/25/2017 6 Z Hz Y Hz X Hz freq. Amp. X Y Z Fourier Transform :  Inverse Fourier Transform :  to extract coeff.  related with  frequency, f in the x(t)
  7. 7. Seoul National University Gear Fault Diagnosis Using Frequency Analysis 2/25/2017 7 • Normal Gear signals – Consist of 3 harmonics (GMF = 500Hz) • Faulty Gear signals – 1) Distributed and 2) local fault* (Fault frequency = 50Hz) – Induce side‐bands near the GMF  Good indicators for gear faults sin 2 . sin 2 . sin 2 *Randall, R. B. "A new method of modeling gear faults." Journal of mechanical design 104.2 (1982): 259-267. Normal Faulty Distributed Local Time Freq.
  8. 8. Seoul National University Non‐stationary Gear Signals 2/25/2017 8 • Normal gear signals – No harmonics with 10% fluctuating speeds with 75Hz • Faulty gear signals – Distributed and local fault (Fixed Fault frequency = 50Hz) – Difficult to differentiate using side‐bands behaviors sin 2 sin 2 : Frequency Modulated Normal Distributed Local Faulty
  9. 9. Seoul National University Signal Processing for Advanced Fault Diagnosis 2/25/2017 9 1) Wavelet transform (Time‐frequency analysis) 2) EMD (Empirical mode decomposition) 3) Hilbert Spectrum  4) AR‐MED filter 5) Spectral Kurtosis (SK) 6) Cyclo‐stationary analysis (Frequency‐frequency analysis) Wavelet Transform Time Frequency Cyclo‐stationary analysisEMD Hilbert‐Huang Transform(HHT)
  10. 10. Seoul National University A Drawback of Fourier Analysis 2017/2/25 ‐ 10 ‐ 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 -1 -0.5 0 0.5 1 shifted 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 -1 -0.5 0 0.5 1 SALAAM with switching the 1st 5000 samples with the tail segment Original 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 abs(fft) of SALAAM with shifting the 1st 5000 samples to the tail sine functions• In Fourier analysis, sin/cos functions are used for  basis function.  • Fourier analysis could not represent time‐domain  information. (Only frequency information) Time-domain Frequency-domain Time Freq.
  11. 11. Seoul National University Short Time Fourier transform (STFT) 2017/2/25 ‐ 11 ‐ 0 0.5 1 1.5 2 2.5 3 3.5 -1 -0.5 0 0.5 1 SALAAM with switching the 1st 5000 samples with the tail segment Original … • Multiple FT over smaller windows translated in time  Could represent time-domain information • However, as window size is predetermined, resolution is limited  (poor time or frequency localization)  Time Time Freq. ,
  12. 12. Seoul National University Short Time Fourier transform (STFT) 2017/2/25 ‐ 12 ‐ • Multiple FT over smaller windows translated in time  Could represent time-domain information • However, as window size is predetermined, resolution is limited (poor time or frequency localization)  25ms 125ms 375ms 1000ms
  13. 13. Seoul National University 1) Wavelet Transform 2017/2/25 ‐ 13 ‐ • Wavelet, a small wavelike signal, is used as  a basis function, instead.  • Changing the variables (a and b), WT could  represent time‐frequency information  without much loss of resolution. Ψ Time achanges b changes Scale , : Time
  14. 14. Seoul National University Papers on Wavelet for Fault Diagnosis 2017/2/25 ‐ 14 ‐ • Wang, W. J., and P. D. McFadden. "Application of wavelets to gearbox vibration signals for fault detection." Journal of sound and vibration 192.5 (1996): 927-939. • Lin, Jing, and Liangsheng Qu. "Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis." Journal of sound and vibration234.1 (2000): 135-148. • Lin, Jing, and M. J. Zuo. "Gearbox fault diagnosis using adaptive wavelet filter." Mechanical systems and signal processing 17.6 (2003): 1259-1269. • Peng, Z. K., and F. L. Chu. "Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography. "Mechanical systems and signal processing 18.2 (2004): 199-221. • Peng, Z. K., W. Tse Peter, and F. L. Chu. "A comparison study of improved Hilbert–Huang transform and wavelet transform: application to fault diagnosis for rolling bearing." Mechanical systems and signal processing 19.5 (2005): 974-988. • Rafiee, J., et al. "A novel technique for selecting mother wavelet function using an intelligent fault diagnosis system." Expert Systems with Applications 36.3 (2009): 4862-4875. • Yan, Ruqiang, Robert X. Gao, and Xuefeng Chen. "Wavelets for fault diagnosis of rotary machines: a review with applications." Signal Processing 96 (2014): 1-15. • Sun, Hailiang, et al. "Multiwavelet transform and its applications in mechanical fault diagnosis–A review." Mechanical Systems and Signal Processing 43.1 (2014): 1-24. • Chen, Jinglong, et al. "Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review." Mechanical Systems and Signal Processing 70 (2016): 1-35. …
  15. 15. Seoul National University Application of Wavelet (1) : Planetary Gear 2017/2/25 ‐ 15 ‐ • Wavelet transform is applied to the  planetary gear in wind turbine simulator. • The acceleration signals are acquired  from both normal and fault gears in a  constant speed. (fault case : a crack in  the planet gear of the planetary gear)
  16. 16. Seoul National University Results (Methodology from Reference*) 2017/2/25 ‐ 16 ‐ WT Coeff. FT fp *Wang, Changting, Robert X. Gao, and Ruqiang Yan. "Unified time–scale–frequency analysis for machine defect signature extraction: theoretical framework." Mechanical Systems and Signal Processing 23.1 (2009): 226-235.
  17. 17. Seoul National University Application of Wavelet (2) : Spur Gear (Simulated signals) 2017‐02‐25 17 • Normal gear signals – No harmonics with 10% fluctuating speeds with 75Hz • Faulty gear signals – Distributed and local fault (Fixed Fault frequency = 50Hz) – Difficult to differentiate using side‐bands behaviors sin 2 sin 2 : Frequency Modulated Normal Distributed Local Faulty
  18. 18. Seoul National University Results 2017‐02‐25 18 STFT WT • Hard to differentiate between  normal and fault using STFT. • Good localization of impact signals  using WT.           
  19. 19. Seoul National University Advantages 2017/2/25 ‐ 19 ‐ • Effective in extracting transient features. • Adaptive in resolution (both in frequency and time) • Adaptive in wavelet functions     
  20. 20. Seoul National University Research Direction (1) 2017/2/25 ‐ 20 ‐ • Wavelet + Machine learning algorithm – Abbasion, Saeed, et al. "Rolling element bearings multi-fault classification based on the wavelet denoising and support vector machine." Mechanical Systems and Signal Processing 21.7 (2007): 2933-2945. – Hu, Qiao, et al. "Fault diagnosis of rotating machinery based on improved wavelet package transform and SVMs ensemble." Mechanical Systems and Signal Processing 21.2 (2007): 688-705. – Wu, Jian-Da, and Chiu-Hong Liu. "An expert system for fault diagnosis in internal combustion engines using wavelet packet transform and neural network." Expert systems with applications 36.3 (2009): 4278-4286. – Saravanan, N., and K. I. Ramachandran. "Incipient gear box fault diagnosis using discrete wavelet transform (DWT) for feature extraction and classification using artificial neural network (ANN)." Expert Systems with Applications 37.6 (2010): 4168-4181. – Konar, P., and P. Chattopadhyay. "Bearing fault detection of induction motor using wavelet and Support Vector Machines (SVMs)." Applied Soft Computing11.6 (2011): 4203-4211. – Li, Ning, et al. "Mechanical fault diagnosis based on redundant second generation wavelet packet transform, neighborhood rough set and support vector machine." Mechanical systems and signal processing 28 (2012): 608-621. – Shen, Changqing, et al. "Fault diagnosis of rotating machinery based on the statistical parameters of wavelet packet paving and a generic support vector regressive classifier." Measurement 46.4 (2013): 1551-1564.
  21. 21. Seoul National University Research Direction (2) 2017/2/25 ‐ 21 ‐ Chen, Jinglong, et al. "Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review." Mechanical Systems and Signal Processing 70 (2016): 1-35. SGWT : Ψ , MWT : Ψ=(Ψ1, … , ΨT)T WT : Ψ ,
  22. 22. Seoul National University 2) EMD (Empirical mode decomposition) 2017/2/25 ‐ 22 ‐ Empirical : based on testing or experience Mode : a particular form or variety of something Decomposition (decompose) : to separate into constituent parts or  elements or into simpler compounds Empirical Mode Decomposition, Patrick Flandrin, CNRS & École Normale Supérieure de Lyon, France Definition by *
  23. 23. Seoul National University Principles of EMD 2017/2/25 ‐ 23 ‐ Signals Low frequency High frequency
  24. 24. Seoul National University Procedures 2017/2/25 ‐ 24 ‐ 1. Identify local maxima and minima in the signal 2. Deduce an upper and a lower envelope by interpolation (cubic splines) 1) subtract the mean envelope from the signal 2) iterate until #{extrema} = #{zeroes} ±1 3. subtract the so‐obtained Intrinsic Mode Function (IMF) from the signal 4. Iterate on the residual Click to see the figures of details for EMD
  25. 25. Seoul National University Advantages/Disadvantages of EMD 2017/2/25 ‐ 25 ‐ • EMD is a model‐free, and fully  data‐driven method. • EMD can deal with non‐ stationarities and nonlinearities. • Differently from wavelet, EMD is  a self‐adaptive signal processing  method, which is based on the  local characteristics of time‐ domain signals. (Wavelet uses pre‐defined basis  functions.) • Lack of theoretical backgrounds • End‐effects : When the end  points are not extrema, the spline  could swing wildly.   Solutions : Mirror images, adding  characteristics waves, … • Mode‐mixing : a single IMF with  oscillations of disparate scales, or  a component of a similar scale  residing in different IMFs  EEMD (ensemble empirical mode  decomposition)
  26. 26. Seoul National University Mode‐mixing Problems in EMD 2017/2/25 ‐ 26 ‐ • Mode‐mixing : a single IMF with oscillations of disparate scales, or a  component of a similar scale residing in different IMFs Lei, Yaguo, et al. "A review on empirical mode decomposition in fault diagnosis of rotating machinery." Mechanical Systems and Signal Processing 35.1 (2013): 108-126.
  27. 27. Seoul National University EEMD (Ensemble Empirical Mode Decomposition) 2017/2/25 ‐ 27 ‐ • Ensemble average :  Mean of a quantity that is a function of the  microstate of a system (from                 )  ≜ lim → ∑ Concept of  Ensemble : Ensemble of  white noises : Ensemble average
  28. 28. Seoul National University Procedures of EEMD 2017/2/25 ‐ 28 ‐ 1. Initialize the number of ensemble M, and m = 1. 2. Perform the mth trial on the signal added white noise. 1) Add a white noise to the investigated signal (where nm(t) indicates the mth added white noise series, and xm(t) represents the noise‐added signal of the mth trial.) 2) Decompose the noise‐added signal xm(t) into P IMFs ci,m(I = 1,2,…, P) using the EMD method (where ci,m is the ith IMF of the mth trial, and P is the number of IMFs.) 3) If m<M then go to step 1) with m = m+1. Repeat steps 1) and 2)  again and again, but with different white noise series each time. 3. Calculate the ensemble mean  of the M trials for each IMF 4. Report the mean  (I = 1,2,…,P) of each of the P IMFs as the final IMFs. xm(t) = x(t) + nm(t) ∑ , , 1,2, … , , 1,2, … ,
  29. 29. Seoul National University Comparison btw. EMD and EEMD 2017/2/25 ‐ 29 ‐ Lei, Yaguo, et al. "A review on empirical mode decomposition in fault diagnosis of rotating machinery." Mechanical Systems and Signal Processing 35.1 (2013): 108-126. EMD EEMD
  30. 30. Seoul National University Papers on EMD for Fault Diagnosis 2017/2/25 ‐ 30 ‐ • Loutridis, S. J. "Damage detection in gear systems using empirical mode decomposition." Engineering Structures 26.12 (2004): 1833-1841. • Yu, Dejie, Junsheng Cheng, and Yu Yang. "Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings." Mechanical systems and signal processing 19.2 (2005): 259-270. • Yu, Yang, and Cheng Junsheng. "A roller bearing fault diagnosis method based on EMD energy entropy and ANN." Journal of sound and vibration 294.1 (2006): 269-277. • Liu, Bao, S. Riemenschneider, and Y. Xu. "Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum." Mechanical Systems and Signal Processing 20.3 (2006): 718-734. • Lei, Yaguo, Zhengjia He, and Yanyang Zi. "Application of the EEMD method to rotor fault diagnosis of rotating machinery." Mechanical Systems and Signal Processing 23.4 (2009): 1327- 1338. • Shen, Zhongjie, et al. "A novel intelligent gear fault diagnosis model based on EMD and multi- class TSVM." Measurement 45.1 (2012): 30-40. • Lei, Yaguo, et al. "A review on empirical mode decomposition in fault diagnosis of rotating machinery." Mechanical Systems and Signal Processing 35.1 (2013): 108-126. • Jiang, Hongkai, Chengliang Li, and Huaxing Li. "An improved EEMD with multiwavelet packet for rotating machinery multi-fault diagnosis." Mechanical Systems and Signal Processing 36.2 (2013): 225-239. …
  31. 31. Seoul National University 3) Hilbert Spectrum  2017/2/25 ‐ 31 ‐ Hilbert Transform           1ˆ , whereHT f t f t f t h t h t t            ˆF w F w H w     ˆf t f t h t  Slide Courtesy of Jongmoon Ha Relationship with the Fourier transform (FT) Fourier Transform of h(t)   2 2 , 0 0, 0 , 0 i i i e for w H w for w i e for w                1H w    , 0 2 0, 0 , 0 2 for w H w for w for w              w H(w) i -i w |H(w)| 1 w ∠H(w) /2 ‐ /2
  32. 32. Seoul National University Hilbert Transform 2017/2/25 ‐ 32 ‐ Definition           1ˆ , whereHT f t f t f t h t h t t       Relationship with the Fourier transform (FT) Fourier Transform of            2 2 , 0 ˆ 0, 0 , 0 i i F w i F w e for w F w for w F w i F w e for w                   ˆF w F w H w     ˆf t f t h t    2 2 , 0 0, 0 , 0 i i i e for w H w for w i e for w              Amplitudes are left unchanged Phases are shifted by  π/2 Recall: Slide Courtesy of Jongmoon Ha
  33. 33. Seoul National University Analytic Signal 2017/2/25 ‐ 33 ‐ Definition Relationship with the Fourier transform (FT)      ˆz t f t if t       ˆZ w F w iF w        , 0 ˆ 0, 0 , 0 F w for w iF w for w F w for w               ˆ 2 0 0 0 Z w F w iF w F w for w for w        Recall:           2 2 , 0 ˆ 0, 0 , 0 i i F w i F w e for w F w for w F w i F w e for w              w F(w) or i (w) w |Z(w)| Slide Courtesy of Jongmoon Ha
  34. 34. Seoul National University Properties Properties of Analytic Signal & Relation with EMD 2017/2/25 ‐ 34 ‐        2 2ˆA t z t f t f t   Instantaneous amplitude        1 ˆ tan Im ln f t t z t f t               Instantaneous phase/frequency          ˆ i t z t f t if t A t e     Analytic Signal Amplitude Phase             Im ln Im ln Im ln i t z t A t e A t j t t               Slide Courtesy of Jongmoon Ha Instantaneous  phase   ( ) d t w t dt   Instantaneous  frequency     ( ) Re i w t dt f t A t e         1 ( ) Re j n i w t dt j j f t A t e         1 2 ⋮   1 ( ) Re j n iw t j j f t A e   
  35. 35. Seoul National University Comparison with Fourier and Wavelet 2017/2/25 ‐ 35 ‐ Fourier Wavelet Hilbert Basis a priori a priori adaptive Frequency convolution over global  domain, uncertainty convolution over global  domain, uncertainty differentiation over  local domain, certainty Presentation energy in frequency  space energy in time‐ frequency space energy in time‐ frequency space Nonlinearity no no yes Nonstationarity no yes yes Feature extraction no discrete, no;  continuous, yes yes Theoretical base complete  mathematical theory complete  mathematical theory empirical Huang, Norden E., and Zhaohua Wu. "A review on Hilbert‐Huang transform: Method and its applications to geophysical studies." Reviews of Geophysics 46.2 (2008).
  36. 36. Seoul National University Examples 2017/2/25 ‐ 36 ‐ Peng, Z. K., W. Tse Peter, and F. L. Chu. "A comparison study of improved Hilbert–Huang transform and wavelet transform: application to fault diagnosis for rolling bearing." Mechanical systems and signal processing 19.5 (2005): 974-988. Different frequency resolution at each frequency The estimated frequency can reflect the real  frequency pattern of the analysed signal, but  only in a mean sense.
  37. 37. Seoul National University Papers on HHT for Fault Diagnosis 2017/2/25 ‐ 37 ‐ • Huang, Norden E., et al. "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 454. No. 1971. The Royal Society, 1998. (google citation : 13579) • Peng, Z. K., W. Tse Peter, and F. L. Chu. "A comparison study of improved Hilbert–Huang transform and wavelet transform: application to fault diagnosis for rolling bearing." Mechanical systems and signal processing 19.5 (2005): 974-988. • Peng, Z. K., W. Tse Peter, and F. L. Chu. "An improved Hilbert–Huang transform and its application in vibration signal analysis." Journal of sound and vibration 286.1 (2005): 187-205. • Yan, Ruqiang, and Robert X. Gao. "Hilbert–Huang transform-based vibration signal analysis for machine health monitoring." IEEE Transactions on instrumentation and measurement 55.6 (2006) • Rai, V. K., and A. R. Mohanty. "Bearing fault diagnosis using FFT of intrinsic mode functions in Hilbert–Huang transform." Mechanical Systems and Signal Processing 21.6 (2007): 2607-2615. • Cheng, Junsheng, Dejie Yu, and Yu Yang. "Application of support vector regression machines to the processing of end effects of Hilbert–Huang transform." Mechanical Systems and Signal Processing 21.3 (2007): 1197-1211. • Huang, Norden E., and Zhaohua Wu. "A review on Hilbert‐Huang transform: Method and its applications to geophysical studies." Reviews of Geophysics 46.2 (2008). • Li, Hui, Yuping Zhang, and Haiqi Zheng. "Hilbert-Huang transform and marginal spectrum for detection and diagnosis of localized defects in roller bearings." Journal of Mechanical Science and Technology 23.2 (2009): 291-301. • …
  38. 38. Seoul National University 4) AR‐MED filter 2017/2/25 ‐ 38 ‐ • Combination of AR filter and MED filter • AR filter : Autoregressive filter • MED filter : Minimum Entropy Deconvolution filter • Widely used for fault diagnosis of rolling element bearings  ② Periodic part ③ Fault impulse Transmission path effect AR  filter MED filter ① Noise ∗ Removes  periodic parts ∗ Enhance  impulsiveness
  39. 39. Seoul National University AR filter 2017/2/25 ‐ 39 ‐ • AR filter : Autoregressive model‐based filtering technique • AR model of order  :  • The output variable ( ) depends linearly on its own previous values  ( ) and on a stochastic term ( ). ( is a constant.)  AR filter could well predict deterministic patterns of signals. Inverse AR model of  undamaged gears : Input signals with the effect of  gear fault : AR prediction of undamaged gear  signal : Prediction error (AR residual) Endo, H., R. B. Randall, and C. Gosselin. "Differential diagnosis of spall vs. cracks in the gear tooth fillet region: Experimental validation." Mechanical Systems and Signal Processing 23.3 (2009): 636-651.
  40. 40. Seoul National University MED filter • MED : Minimum Entropy Deconvolution • The filter searches for an optimum set of filter coefficients that recover the  output signal (of an inverse filter) with the maximum value of kurtosis (using iterative optimization process) Barszcz, Tomasz, and Nader Sawalhi. "Fault detection enhancement in rolling element bearings using the minimum entropy deconvolution." Archives of acoustics 37.2 (2012): 131-141. ∑ ∑Objective function :  kurtosis where( ) ② Periodic part ③ Fault impulse Transmission path effect AR  filter MED filter ① Noise ∗ Removes  periodic parts ∗ Enhance  impulsiveness
  41. 41. Seoul National University Application of AR‐MED filter 2017/2/25 ‐ 41 ‐ -0.1 0 0.1 -1 0 1 -0.2 0 0.2 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 10 5 -5 0 5 -0.2 0 0.2 -1 0 1 -1 0 1 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 105 -2 0 2 정상반절삭대각절삭표면손상 반절삭 대각절삭 표면손상
  42. 42. Seoul National University Papers on AR or MED filter for Fault Diagnosis 2017/2/25 ‐ 42 ‐ • Sawalhi, N., R. B. Randall, and H. Endo. "The enhancement of fault detection and diagnosis in rolling element bearings using minimum entropy deconvolution combined with spectral kurtosis." Mechanical Systems and Signal Processing 21.6 (2007): 2616-2633. • Endo, H., and R. B. Randall. "Enhancement of autoregressive model based gear tooth fault detection technique by the use of minimum entropy deconvolution filter." Mechanical Systems and Signal Processing 21.2 (2007): 906-919. • Endo, H., R. B. Randall, and C. Gosselin. "Differential diagnosis of spall vs. cracks in the gear tooth fillet region: Experimental validation." Mechanical Systems and Signal Processing 23.3 (2009): 636-651. • Randall, Robert B., and Jerome Antoni. "Rolling element bearing diagnostics—a tutorial." Mechanical Systems and Signal Processing 25.2 (2011): 485-520. • Jiang, Ruilong, et al. "The weak fault diagnosis and condition monitoring of rolling element bearing using minimum entropy deconvolution and envelope spectrum." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science (2012): 0954406212457892. • Barszcz, Tomasz, and Nader Sawalhi. "Fault detection enhancement in rolling element bearings using the minimum entropy deconvolution." Archives of acoustics 37.2 (2012): 131-141. …
  43. 43. Seoul National University 5) Spectral kurtosis 2017/2/25 ‐ 43 ‐ • Kurtosis* :  • Spectral kurtosis (SK) extends the concept of kurtosis to that of a function  of frequency that indicates how the impulsiveness of a signal. 3 0.01 0.01 0.01 10*Note that kurtosis is not related to peakedness Westfall, Peter H. "Kurtosis as peakedness, 1905–2014. RIP." The American Statistician 68.3 (2014): 191-195. To make kurtosis of  normal distribution 0
  44. 44. Seoul National University Definition of SK (1) 2017/2/25 ‐ 44 ‐ Antoni, Jérôme. "The spectral kurtosis: a useful tool for characterising non-stationary signals." Mechanical Systems and Signal Processing 20.2 (2006): 282-307. • 2n‐order instantaneous moment  , , ≜ , d , · ≜ , , d , · • Spectral moments (by ensemble averaging) • 2n‐order time‐averaged moment (for practical cases where experiments are limited) , ≜ lim → 1 , d / /
  45. 45. Seoul National University Definition of SK (2) 2017/2/25 ‐ 45 ‐ Antoni, Jérôme. "The spectral kurtosis: a useful tool for characterising non-stationary signals." Mechanical Systems and Signal Processing 20.2 (2006): 282-307. • Spectral cumulant (combinations of several moments of different orders) 2 , 0. ≜ 2, 0. • Spectral kurtosis • Spectral kurtosis could be estimated in some different approaches – STFT (short‐time Fourier transform) based SK – Kurtogram (The map formed by the STFT‐based SK as a function of  and  ) – Adaptive SK – Protrugram
  46. 46. Seoul National University Estimation of SK : (1) STFT  2017/2/25 ‐ 46 ‐ • STFT (short‐time Fourier transform) of the process  , ≜ • 2n‐order empirical spectral  moment of  , • STFT‐based estimator of the SK 2 ≜ , ≜ 2 Antoni, Jérôme, and R. B. Randall. "The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines." Mechanical Systems and Signal Processing 20.2 (2006): 308-331. SK of measurements on a gearbox  submitted to an accelerated fatigue test
  47. 47. Seoul National University Estimation of SK : (2) Kurtogram 2017/2/25 ‐ 47 ‐ • In STFT, non‐stationarity of the signals should have slow temporal  evolution, as compared to the window length. • Kurtogram : map formed by the STFT‐based SK as a function of  and  – A band‐pass filter has better chance to select only one impulsive source (the  strongest one) in the case where several such sources are present in the signal. Kurtogram of a rolling element bearing  signal with an outer race fault
  48. 48. Seoul National University Fast kurtogram 2017/2/25 ‐ 48 ‐ • Calculation of the whole plane ( , ∆ ) is a formidable task in kurtogram. • Fast kurtogram – Based on the multirate filter‐bank structure (MFB) and quasi‐analytic filters. – The complexity of calculation is reduced to  log . (same as FFT) Result of SK using kurtogram Fast kurtogram paving of the  (frequency/frequency resolution) plane.
  49. 49. Seoul National University Procedures of Fault Diagnosis Using SK 2017/2/25 ‐ 49 ‐ Find the frequency that has maximum kurtosis using SK.* Band‐pass filter the signals with the  frequency. Envelope analysis freq. Amp. X 2X 3X Detection of fault frequency *AR‐MED filter could be used before SK.
  50. 50. Seoul National University Papers on SK for Fault Diagnosis 2017/2/25 ‐ 50 ‐ • Antoni, Jérôme. "The spectral kurtosis: a useful tool for characterising non-stationary signals." Mechanical Systems and Signal Processing 20.2 (2006): 282-307. • Antoni, Jérôme, and R. B. Randall. "The spectral kurtosis: application to the vibratory surveillance and diagnostics of rotating machines." Mechanical Systems and Signal Processing 20.2 (2006): 308-331. • Wang, Yanxue, et al. "Spectral kurtosis for fault detection, diagnosis and prognostics of rotating machines: A review with applications." Mechanical Systems and Signal Processing 66 (2016): 679-698. • Antoni, Jerome. "Fast computation of the kurtogram for the detection of transient faults." Mechanical Systems and Signal Processing 21.1 (2007): 108-124. • Barszcz, Tomasz, and Robert B. Randall. "Application of spectral kurtosis for detection of a tooth crack in the planetary gear of a wind turbine." Mechanical Systems and Signal Processing 23.4 (2009): 1352-1365. • Eftekharnejad, Babak, et al. "The application of spectral kurtosis on acoustic emission and vibrations from a defective bearing." Mechanical Systems and Signal Processing 25.1 (2011): 266-284. • Wang, Dong, W. Tse Peter, and Kwok Leung Tsui. "An enhanced Kurtogram method for fault diagnosis of rolling element bearings." Mechanical Systems and Signal Processing 35.1 (2013): 176-199. • …
  51. 51. Seoul National University 6) Cyclo‐stationary : In search of hidden periodicities 2017/2/25 ‐ 51 ‐ 0 Stationary signals  … • Ensemble average :  Mean of a quantity that is a function of the  microstate of a system (from                 )  • Stationary signals are random signals of zero cycle with 0 ensemble avg. • Periodic signals are deterministic signals (don’t need an ensemble)  ≜ lim → ∑ + Cyclo‐stationary stationary periodic
  52. 52. Seoul National University Cyclo‐stationary* 2017/2/25 ‐ 52 ‐ • Cyclo‐stationary at the 1st order (periodic waveforms with stationary  random noise) • Cyclo‐stationary at the 2nd order (stochastic processes with periodic  amplitude or/and frequency modulation) *J. Antoni, F. Bonnardot, A. Raad, and M. El Badaoui, "Cyclostationary modelling of rotating machine vibration signals," Mechanical Systems and Signal Processing, vol. 18, pp. 1285-1314, 11// 2004. ≜ , ≜ ∗ , Example of CS2Example of CS1
  53. 53. Seoul National University , ; Δ ; ; Δ · ∈ Cyclic Decomposition of Energy Flow : Extraction of Cyclic Trends (1) 2017/2/25 ‐ 53 ‐ The mean instantaneous power ∑ ·∈ The instantaneous power spectrum Cyclic power · Cyclic modulation spectrum Interpretation of the instantaneous power spectrum Antoni, Jérôme. "Cyclostationarity by examples." Mechanical Systems and Signal Processing 23.4 (2009): 987-1036.
  54. 54. Seoul National University , ; Δ ; ; Δ · ∈ Cyclic Decomposition of Energy Flow : Extraction of Cyclic Trends (2) 2017/2/25 ‐ 54 ‐ The mean instantaneous power ∑ ·∈ The instantaneous power spectrum Cyclic power · Cyclic modulation spectrum Interpretation of the cyclic modulation spectrum Antoni, Jérôme. "Cyclostationarity by examples." Mechanical Systems and Signal Processing 23.4 (2009): 987-1036.
  55. 55. Seoul National University , ; Δ ; ; Δ · ∈ Cyclic Decomposition of Energy Flow : Extraction of Cyclic Trends (2) 2017/2/25 ‐ 55 ‐ The mean instantaneous power ∑ ·∈ The instantaneous power spectrum Cyclic power · Cyclic modulation spectrum Physical interpretation of the spectral frequency   and the cyclic frequency  Antoni, Jérôme. "Cyclostationarity by examples." Mechanical Systems and Signal Processing 23.4 (2009): 987-1036.
  56. 56. Seoul National University Spectral Correlation Density & Spectral Coherence 2017/2/25 ‐ 56 ‐ Spectral Correlation , lim → ∆ ; ∆ ; , lim → 1 ∆ ; ∆ ; Spectral Correlation Density lim → lim → 1 ∆ ; /2 ∆ ; /2 d lim → 1 ∆ ; /2 ∆ ; /2 Spectral Coherence /2, /2 2 2 2 2
  57. 57. Seoul National University Physical Meaning of SCD and SC 2017/2/25 ‐ 57 ‐ • Spectral Correlation Density – Non‐zero value of  is relation with carrier frequency  and periodic  modulation at frequency  in signal  of a sinusoidal component  • Spectral Coherence – Normalization of the correlation coefficients by energy Spectral Correlation Density and Spectral Coherence
  58. 58. Seoul National University Examples : Planetary Gear (1, simulated signals) 2017/2/25 ‐ 58 ‐ ACC. • Inherent modulated acceleration signals in a planetary gear • Hard to differentiate faulty gears due to side‐bands near the main  frequencies even in normal conditions 
  59. 59. Seoul National University Examples : Planetary Gear (2, simulated signals) 2017/2/25 ‐ 59 ‐ • Inherent modulated acceleration signals in a planetary gear • Hard to differentiate faulty gears due to side‐bands near the main  frequencies even in normal conditions  Normal Fault
  60. 60. Seoul National University Examples : Planetary Gear (3, simulated signals) 2017/2/25 ‐ 60 ‐ • For a faulty case, more energies are extracted. (which is expected, as  fault signals are added in the normal signals.) • Need to discover more features. Normal Fault
  61. 61. Seoul National University Papers on Cyclostationary for Fault Diagnosis 2017/2/25 ‐ 61 ‐ • Capdessus, C., M. Sidahmed, and J. L. Lacoume. "Cyclostationary processes: application in gear faults early diagnosis." Mechanical systems and signal processing 14.3 (2000): 371-385. • Antoniadis, I., and G. Glossiotis. "Cyclostationary analysis of rolling-element bearing vibration signals." Journal of sound and vibration 248.5 (2001): 829-845. • Antoni, Jérôme, et al. "Cyclostationary modelling of rotating machine vibration signals." Mechanical systems and signal processing 18.6 (2004): 1285-1314. • Bonnardot, Frédéric, R. B. Randall, and François Guillet. "Extraction of second-order cyclostationary sources—application to vibration analysis." Mechanical Systems and Signal Processing 19.6 (2005): 1230-1244. • Antoni, J. "Cyclic spectral analysis of rolling-element bearing signals: facts and fictions." Journal of Sound and vibration 304.3 (2007): 497-529. • Antoni, Jérôme. "Cyclic spectral analysis in practice." Mechanical Systems and Signal Processing 21.2 (2007): 597-630. • Raad, Amani, Jerome Antoni, and Ménad Sidahmed. "Indicators of cyclostationarity: Theory and application to gear fault monitoring." Mechanical Systems and Signal Processing 22.3 (2008): 574-587. • Antoni, Jérôme. "Cyclostationarity by examples." Mechanical Systems and Signal Processing 23.4 (2009): 987-1036. • Feng, Zhipeng, and Fulei Chu. "Cyclostationary Analysis for Gearbox and Bearing Fault Diagnosis." Shock and Vibration 2015 (2015). • …
  62. 62. Seoul National University Other Techniques 2017/2/25 ‐ 62 ‐ • Time‐frequency analysis – Wigner–Ville Distribution (WVD) – Adaptive Optimal Kernel – Cohen class distributions – Affine class distributions • Auto‐regressive moving average (ARMA) • Local mean decomposition (LMD) • Stochastic Resonance • Principal Component Analysis …
  63. 63. Seoul National University THANK YOU 2/25/2017 63
  64. 64. Seoul National University BACK‐UP 2/25/2017 64
  65. 65. Seoul National University Procedures for EMD (1) 2/25/2017 65
  66. 66. Seoul National University Procedures for EMD (2) 2/25/2017 66
  67. 67. Seoul National University Procedures for EMD (3) 2/25/2017 67
  68. 68. Seoul National University Procedures for EMD (4) 2/25/2017 68
  69. 69. Seoul National University Procedures for EMD (5) 2/25/2017 69
  70. 70. Seoul National University Procedures for EMD (6) 2/25/2017 70
  71. 71. Seoul National University Results of EMD 2/25/2017 71
  72. 72. Seoul National University 6) Cyclo‐stationary 2017/2/25 ‐ 72 ‐ • Ensemble average :  Mean of a quantity that is a function of the  microstate of a system (from                 )  • Stationary signals are random signals of zero cycle with 0 ensemble avg. • Periodic signals are deterministic signals (don’t need an ensemble)  input System output lim → ∑
  73. 73. Seoul National University 6) Cyclo‐stationary 2017/2/25 ‐ 73 ‐ cos 2 2 · cos 2 1 2 cos 2 2 · 1 2 cos 2 2 · 1 4 1 4 1 4 1 4

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