SIGNAL PROCESSING TECHNIQUES USED FOR GEAR FAULT DIAGNOSIS

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

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, …)

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)

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)

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

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)

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.

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

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)

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.

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.
,

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

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

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
• 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.
…

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)

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.

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

Results
2017‐02‐25 18
STFT
WT
• Hard to differentiate between
normal and fault using STFT.
• Good localization of impact signals
using WT.
         

Advantages
2017/2/25 ‐ 19 ‐
• Effective in extracting transient features.
• Adaptive in resolution
(both in frequency and time)
• Adaptive in wavelet functions
    

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.

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 ,

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
*

Principles of EMD
2017/2/25 ‐ 23 ‐
Signals
Low frequency High frequency

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

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)

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.

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

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, … ,

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

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.
…

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

Hilbert Transform
2017/2/25 ‐ 32 ‐
Definition
         
1ˆ , whereHT f t f t f t h t h t
t
     
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:

Analytic Signal
2017/2/25 ‐ 33 ‐
Definition
     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)|

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

 
     
    
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

 

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).

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.

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
• 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.
• …

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

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.

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

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

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."
• 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.
…

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

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
/
/

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

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."
SK of measurements on a gearbox
submitted to an accelerated fatigue test

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

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.

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.

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."
• 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.
• …

6) Cyclo‐stationary : In search of hidden periodicities
2017/2/25 ‐ 51 ‐
0
Stationary signals
…
• 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

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

, ; 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.

, ; Δ ; ; Δ ·
∈
2017/2/25 ‐ 54 ‐
∑ ·∈
Cyclic power
·
Interpretation of the cyclic modulation spectrum

, ; Δ ; ; Δ ·
∈
2017/2/25 ‐ 55 ‐
∑ ·∈
Cyclic power
·
Physical interpretation of the spectral frequency and the cyclic frequency

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

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

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

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

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

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).
• …

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
…

THANK YOU
2/25/2017 63

BACK‐UP
2/25/2017 64

Procedures for EMD (1)
2/25/2017 65

2/25/2017 66

2/25/2017 67

2/25/2017 68

2/25/2017 69

2/25/2017 70

Results of EMD
2/25/2017 71

6) Cyclo‐stationary
2017/2/25 ‐ 72 ‐
• 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
→
∑

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

SIGNAL PROCESSING TECHNIQUES USED FOR GEAR FAULT DIAGNOSIS

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