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 EngineeringSeoul National University, Korea

*[email protected]

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 teeth2rev/s

20 teeth3rev/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 teeth2rev/s

20 teeth3rev/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, fin 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 FaultyDistributed 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 LocalFaulty


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 filter5) Spectral Kurtosis (SK)6) Cyclo‐stationary analysis (Frequency‐frequency analysis)

Wavelet Transform

Time

Freq

uency

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 104

-1

-0.5

0

0.5

1

shift

ed

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-1

-0.5

0

0.5

1 SALAAM with switching the 1st 5000 samples with the tail segment

Orig

inal

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

0 1000 2000 3000 4000 50000

1000

2000

3000

4000abs(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

Orig

inal

…

• 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 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.

…


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 LocalFaulty


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 experienceMode : a particular form or variety of somethingDecomposition (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 signal2) 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‐drivenmethod.

• 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

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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 tt

F̂ w F w H w f̂ t f t h t

Slide Courtesy of Jongmoon Ha

Relationship with the Fourier transform (FT)

Fourier Transform of h(t)

2

2

, 00, 0

, 0

i

i

i e for wH w for w

i e for w

1H w

, 02

0, 0

, 02

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 tt


Fourier Transform of

2

2

, 0ˆ 0, 0

, 0

i

i

F w i F w e for wF w for w

F w i F w e for w

F̂ w F w H w f̂ t f t h t

2

2

, 00, 0

, 0

i

i

i e for wH 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

, 0ˆ 0, 0

, 0

F w for wiF w for w

F w for w

ˆ

2 00 0

Z w F w iF w

F w for wfor w

Recall:

2

2

, 0ˆ 0, 0

, 0

i

i

F w i F w e for wF 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

ˆtan Im ln

f tt z t

f t

Instantaneous phase/frequency

ˆ i tz t f t if t A t e

Analytic Signal

Amplitude Phase

Im ln Im ln

Im ln

i tz t A t e

A t j t t


Instantaneous phase

( )d t

w tdt

Instantaneous frequency

( ) Rei w t dt

f t A t e

1( ) Re j

n i w t dtj

jf t A t e

1

2

⋮

1

( ) Re jn

iw tj

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 frequencyThe 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 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.

• …


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 impulseTransmissionpath 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

Transmissionpath 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 2.4105

-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 2.4105

-2

0

2

정상

반절삭

대각절삭

표면손상

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." 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.

…


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 peakednessWestfall, 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."Mechanical Systems and Signal Processing 20.2 (2006): 308-331.

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 kurtogramFast kurtogram paving of the (frequency/frequency resolution) plane.


Procedures of Fault Diagnosis Using SK

2017/2/25 ‐ 49 ‐

Find the frequency that hasmaximum 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." 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.

• …


6) Cyclo‐stationary : In search of hidden periodicities

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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*

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• 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)

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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 ·∈


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∑ ·∈


Cyclic power

·


Interpretation of the cyclic modulation spectrum



, ; Δ ; ; Δ ·∈


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∑ ·∈


Cyclic power

·


Physical interpretation of the spectral frequency and the cyclic frequency



Spectral Correlation Density & Spectral Coherence

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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

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• 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)

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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



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• 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



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• 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

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• 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

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• 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

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BACK‐UP

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Procedures for EMD (1)

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Results of EMD

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6) Cyclo‐stationary

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• 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

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cos 2 2 · cos 2

12 cos 2 2 ·

12 cos 2 2 ·

14

14

14

14

SIGNAL PROCESSING TECHNIQUES USED FOR GEAR FAULT DIAGNOSIS

Engineering

Transcript of SIGNAL PROCESSING TECHNIQUES USED FOR GEAR FAULT DIAGNOSIS