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The University ofNew South Wales PHM Montreal
Machine Diagnostics using Advanced
Signal Processing
Em. Prof. R.B.Randall
School of Mechanical and Manufacturing Engineering
The University of New South Wales,
Sydney 2052, Australia
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Presentation Layout
Background to separation of measured
response signalsmachine diagnostics andoperational modal analysis
Introduction to the cepstrum
First separationdiscrete frequency from
stationary random and cyclostationary randomcomponents, including use of cepstrum, and
application to bearing and gear diagnostics
Second separationforcing functions from
transfer functions, including use of cepstrum
Conclusion
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Separation of measured
response signals
Two important situations in which one
only has access to response signals are:
1. machine condition monitoring (MCM), where
a change in condition could be indicated by a
change in either the forcing function or
structural properties
2. Operational modal analysis (OMA), where oneseeks to extract structural dynamic
properties in the presence of forcing function
effects. Also useful in machine diagnostics.
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INTRODUCTION TO THE CEPSTRUM
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CEPSTRUM TERMINOLOGY
SPECtrum CEPStrum
FREQUency QUEFRency
HARmonic RAHmonic
MAGnitude GAMnitude
PHASe SAPHeFILter LIFter
Low pass filter Short pass lifter
Frequency analysis Quefrency alanysis
Ref: Bogert , Healy and Tukey (yes th e one o f FFT fame, bu t two y ears
earlier)The Quefrency Alanys is of Time Series For Echoes;
Cepstrum , Pseudo-autcovar iance, Cross -cepstrum and Saphe
Cracking. Proc. Symp. On Time Series Analysis , Wiley, 1963.
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Complex Cepstrum
)(log)( 1 fXC )(exp)()()( fjfAtxfX where
BUT phase must be a continuous function
of frequency, ie unwrapped
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Echoes overlaporiginal signal
Echoes give delta
functions in cepstrum
Echoes give added periodic
function in log amplitude
and phase spectra
Delta functions
removed
Overlapping echoes
removed
Smoothed log amplitudeand phase
ECHO REMOVAL USING THE CEPSTRUM
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APPLICATION OF CEPSTRUM
TO MACHINE DIAGNOSTICS
A. Detection of periodic structure in spectrum
Harmonics (Faults in gears, bearings, blading)
Sidebands (Faults in gears, bearings, blading)
Echoes, reflections
B. Separation of Source and Transmission Path Effects
(SIMO)
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Th U i it f
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LATER DEVELOPMENT
Original
condition
After 4 years, 2nd
harmonic of
gearmesh has
increased and 2nd
ghost componentreduced
(indicating wear),
but no further
sideband growth
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Use of cepstrum to detect missing blades in a
steam turbine (French Electrical Authority EDF)
Missing blade causes misdirected steam jet to impinge on local stator
area once per rev; picked up by casing mounted accelerometer.
Increased shaft speed harmonics in mid frequency range.
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Separation of Source and Transmission
Path Effects (SIMO only)
h(t)
H(f)
x(t)
X(f)
y(t)
Y(f)
Input System Output
HXY
fHfXfY
fHfXfYthtxty
logloglog
)()()(
)()()()(*)()(
222
}{log}(log}{log 111 HXY Thus, source and transmission path effects are additive in
cepstrum. Moreover, they are often separated
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PHM M l
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INSENSITIVITY OF CEPSTRUM TO TRANSFER PATH
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PHM M t l
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yNew South Wales PHM Montreal
BACKGROUND TO CM
Most condition monitoring involves separation of
signals from different sources A typical case is separation of gear signals from
bearing signals in a gearbox
Gear signals are deterministic (when tooth contact
maintained) Bearing signals are stochastic because of random
slip
This permits their separation, even when the gear
signals are much stronger
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PHM M t lM th d f th S ti
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yNew South Wales PHM Montreal
Linear prediction - gives simultaneous prewhitening. Some
choice of what is removed by order of filter. Self adaptive noise cancellation (SANC) - copes with some
speed variation. Removes all deterministic components.
Discrete/random separation (DRS)more efficient than
SANC, but may require order tracking. Removes alldeterministic components.
Time Synchronous Averaging (TSA)minimum disruption of
residual signalrequires separate angular sampling for
each harmonic familyDoes not remove modulation
sidebands.
New cepstral methodremoves selected uniformly spaced
frequency components, including sidebandsCan leave
some if required.
Methods for the Separation
of Deterministic and Random Signals
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Autoregressive (AR) model used for Linear
Prediction
Raw
Signal
Predicted
SignalResidual
Signal
1. In an Autoregressive model (AR), we try to capture the
information about the deterministic part using linear
prediction.
2. The value Y for sample number n is expressed as a linear
combination of previous p elements, i.e
Yn=a(2)Yn-1+a(3)Yn-2+a(4)Yn-3+...........+a(p+1)Yn-p
Residual signal is whitened (noise and impulses)
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New South Wales PHM MontrealResidual Analysis of local
gear faults by Linear Prediction
AR MethodConventional Method
(removal of toothmesh harmonics)
W. Wang and A. K. Wong (2002) Autoregressive Model-Based Gear Fault Diagnosis,
Trans. ASME, Jou rnal of Vibrat ion and Acous t ics, 124, pp. 172- 179.
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New South Wales PHM MontrealSANC
(Self Adaptive Noise Cancellation)
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SEPARATION USING SANC
(b)
(a)
(c)
Signal from rig(normal gear
signal with
bearing fault)
Gear signal(discrete
frequency)
Bearing outer
race fault signal(stochastic)
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NEW METHOD OF DISCRETE - RANDOM
SEPARATION
input data estimation of a H1 type
filter
filtering
estimated periodic part
+-
estimated random part
Uses only
FFTs
Uses only
FFTs
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0.06 0.08 0.1 0.12 0.14-10
0
10
20
30
40
PSD
Inputsignal(dB)
Time delay = 50; Filter length = 4096; Overlap = 50%
0.06 0.08 0.1 0.12 0.14
0.2
0.4
0.6
0.8
1
SeparationFilter
Window = Parzen; Resolution = single
0.06 0.08 0.1 0.12 0.14-10
0
10
20
30
40
PSD
Perio
dicpart(dB)
Normalised frequency
0.06 0.08 0.1 0.12 0.14
0
10
20
30
40
PSD
Randompart(dB)
Normalised frequency
DRS applied to a helicopter gearbox signal
Input
spectrum
Discrete
frequencypart
Generated
filter
discrete = 1
noise = 0
Noise part
bysubtraction
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Before TSA, signal must be order tracked to
give integer number of samples per revolution
and defined start point:
One sample spacing corresponds to 360 of
phase at sampling frequency and 140 of phaseat highest valid frequency
Just 0.1% speed fluctuation gives extra sample
in typical 1024-point time record
Sampling frequency may have to be changed foreach gear in the signal
Only removes harmonicsnot sidebands
TIME SYNCHRONOUS AVERAGING (TSA)
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PHM Montreal
COMPARISON OF TSA WITH DRS(N. Sawalhi & R.B. RandallCM-MFPT Edinburgh 2008)
Originalspectrum
TSA
Method 1
TSA
Method 2
DRS
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PHM Montreal
Editing Cepstrum
Previously thought it was necessary to use ComplexCepstrum to edit time signal, eg echo removal
Not possible to unwrap phase of excitation or
response signals, therefore complex cepstrum
excluded
Real cepstrum used to edit spectrum, eg remove
particular harmonic/sideband families, or reveal
system resonances
New proposed method uses the real cepstrum to edit
the amplitude of force or response signals andcombines with original phase to generate edited time
signals
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Editing cepstrum to remove
specrum components
Original baseband
spectrum
All harmonics (+ sidebands)
of 50 Hz shaft removed by
editing the 20 ms rahmonics
from the cepstrum andforward transforming to the
log spectrum
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NEW CEPSTRAL METHOD
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Application to UNSWGearbox Rig
UNSW Spur Test Rig Inner race fault
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Ti D i Si l
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Time Domain Signals
0 1 2 3 4-40
-20
0
20
0 1 2 3 4-20
0
20
0 1 2 3 4-20
0
20
Shaft rotation
Acceleratio
n(m/s2)
(a)
(b)
(c)
Raw signal
Residual signal
(af ter removingsynchronous
average)
Residual signal
after edit ingthe Cepstrum
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Po er Spectra
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Power Spectra
0 1000 2000 3000 4000 5000 6000-20
0
20
40
60Harmonic Spacing at : 320.016 Hz (Gear mesh frequnecy)
0 1000 2000 3000 4000 5000 6000-20
0
20
40
60
PowerSpectrumM
agnitude(dB)
0 1000 2000 3000 4000 5000 6000-20
0
20
40
60
Frequency (Hz)
(a)
(b)
(c)
Raw signal
Residual signal
(af ter remo ving
synchronous
average)
Residual signalafter edit ing
the Cepstrum
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Envelope Spectra
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Envelope Spectra
0 50 100 150 200 250 300 350 400 450 5000
1
2
3
4
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
1.5
2
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
1.5
2
Frequency (Hz)
Harmonic Spacing at : 71.0525 Hz (BPFI)
(b)
(a)
(c)
BPFI
BPFI
(320 Hz)Gear mesh frequnecy
Squared envelope spectrum (1-20 kHz)
10 Hz (Shaft speed)
Raw signal
Residual signal
(af ter removing
synchronous
average)
Residual signalafter edit ing
the Cepstrum
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Application to UNSW Fan Test rig
Outer Race Fault
Accelerometer Position: Accelerometer
attached using magnetic base
Defective bearing
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P S (F ll R )
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Power Spectra (Full Range)
0 0.5 1 1.5 2 2.5 3
x 104
-20
0
20
0 0.5 1 1.5 2 2.5 3
x 104
-20
0
20
PowerSpectru
mM
agnitude(dB)
0 0.5 1 1.5 2 2.5 3
x 104
-20
0
20
Frequency (Hz)
Original
TSA
method
Cepstrum
method
Remaining periodic components at low frequency are from bearing fault
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Power Spectra (0-5 kHz)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-20
-10
0
1020
30
Shaft Harmonics at: 39.8 Hz
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-20
-10
0
10
20
30
PowerSpectrumM
agnitude(dB)
BPFO Harmonics at 231.6 Hz
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-20
-10
010
20
30
Frequency (Hz)
Blade pass frequency (19X)
Original
TSA
Cepstrummethod
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Power Spectra (5-10 kHz)
5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000-20
-10
0
10
20
30Carrier Frequency at : 7586.41 Hz, Sideband Spacing at : 757.369 Hz (Blade Pass Frequency)
5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000-20
-10
0
10
20
30
PowerSpectrumMa
gnitude(dB)
5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000-20
-10
010
20
30
Frequency (Hz)
Original
TSA
Cepstrum
method
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C i th l t
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Comparing the envelope spectrum
using three methods
0 100 200 300 400 500 600 700 800 900 10000
0.5
1
1.5
2
2.5
Frequency (Hz)
Harmonic Spacing at : 39.9919 Hz (Shaft speed)
0 100 200 300 400 500 600 700 800 900 10000
0.5
1
1.5
2
2.5
Frequency (Hz)
Squared envelope spectrum ( Bandpass 1000 Hz - 45000 Hz)
0 100 200 300 400 500 600 700 800 900 10000
0.5
1
1.5
2
2.5
(a)
BPFO
2 X BPFO
BPFO
(b)
(c)
TSA
DRS
Cepstrum
method
The University ofNew South Wales PHM MontrealSEMI-AUTOMATED METHOD
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for Bearing Diagnostics
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Semi Automated
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Semi-Automated
Bearing Analysis Procedure
1
2
3
4
5
Order trackingRemove speed
fluctuation
DRS, SANC or Linear Prediction -
Remove discrete frequenciesMEDRemove smearing effect of
signal transfer path
SKDetermine optimum band for
filtering and demodulationEnvelope analysisDetermine fault
characteristic frequencies
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MINIMUM ENTROPY DECONVOLUTION (MED)
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Wiggins Minimum Entropy Deconvolution:
Basic idea is to maximize K by varying g(l):
*g ~wy
liylgiwL
l
1
2
1 1
24 /
N
i
N
i
iwiwlgK
solve for minimum value of)(lg
K
R A Wiggins
MINIMUM ENTROPY DECONVOLUTION (MED)
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SPECTRA KURTOSIS
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SPECTRAL KURTOSISGives kurtosis (impulsiveness) for each frequency line
in a time-frequency diagram
),( ftH
Short Time FourierTransformation
STFT
t
f
SK
f
2))((
)()( 2
2
ymean
ymeanykurtosis
y = autospectrum value
(ie amplitude squared)
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Fast Kurtogram
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g
Filter combinations for
1/3 binary tree
Normal
kurtogram
Fast
kurtogram
J. Antoni (2006) Fast computation of the kurtogram for the detection of transient faults,
Mechanical Systems and Signal Process ing, 21(1), pp. 108124
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HILBERT TRANSFORM
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HILBERT TRANSFORM
Relationship between the real and imaginary parts
of the Fourier transform of a one-sided function
)()( fXtx )()()( txtxtx oe
)(txe
)(txo
)(Re)( fXtxe
)(Im)( fXtxo
)sgn()()( ttxtx eo )sgn()()( tfXfX RI
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ANALYTIC SIGNAL
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ANALYTIC SIGNAL
Complex time signal with one-sided spectrum
Real and imaginary parts related by a Hilbert transform
Real
Imag
Real
Imag
Vector sum at
Time zeroProjection on real
axis at time zero
Projection on imag.axis at time zero:
gives Hilbert
transform of real part
-f
f
-f CkCk/2 Ck/2
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AMPLITUDE MODULATION
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AMPLITUDE MODULATION
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HILBERT TECHNIQUE FOR ENVELOPE ANALYSIS
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HILBERT TECHNIQUE FOR ENVELOPE ANALYSIS
Note that the ideal
bandpass filter
removes adjacent
discrete peaks
Note that 1- sided
spectrum values
must be complex
It is normally better
to analyze thesquared envelope
rather than the
envelope
The University ofNew South Wales PHM MontrealAdvantage of using
one sided spectrum
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one-sided spectrum
Spectrum Spectrum Convolution EnvelopeSpectrum
Analytic signal
difference
frequencies only
Real signal
also sum
frequencies
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Advantages of Squared
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Advantages of Squared
rather than Rectified Envelope
Squared signal contains only DC component plus (double) frequencyRectified signal has sharp cusps requiring harmonics to infinity which
alias into measurement range (ie avoid taking square root)
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Case History Helicopter Gearbox Rig
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Case HistoryHelicopter Gearbox Rig
100.00 HZ
5.73 HZ
Planetary
Bearing
Blind analysis
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Time domain after filtration
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Order tracked signal
Residual signalafter DRS and linear
prediction
Filtered signal usingSK
Time (s)
Accel
eration
Kurtosis =(-0.61)
Kurtosis =(2.2)
Kurtosis =(14.1)
Time domain after filtration
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SK analysis showing the
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y g
maximum excited bands
10
0
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New cepstral pre-whitening technique
Based on the new method of editing a time signal
by editing the spectrum amplitude in the real
cepstrum, then combining with the original phase
to return to the time domain
Extreme case is where real cepstrum is set tozero (spectrum amplitude set to one, ie whitened).
Both discrete frequencies and resonances
removed. Uniform spectrum weighting means
that impulsive frequency bands dominate time
signals
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Example of application to the
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Example of application to the
helicopter gearbox signal
0 5 10 15 20 25
0
20
40
60
0 5 10 15 20 25-40
-20
0
20
40
P
owerSpectrumM
agnitude(dB)
0 5 10 15 20 25-40
-20
0
20
40
Frequency (kHz)
(a)
(b)
(c)
Original spectrum
Whitening using low
order AR model
Cepstral whitening
Spectra
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ENVELOPE SPECTRA
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ENVELOPE SPECTRA
0 20 40 60 80 100
0.005
0.01
0.015
0.02
0.025FTF: Harmonic Spacing at : 9.81 Hz
Frequency (Hz)100 200 300 400 500
2
4
6
8
10
12
14x 10
-3BPFI: Harmonic Spacing at 117.57 Hz
Frequency (Hz)
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
Frequency (Hz)100 200 300 400 5000
0.05
0.1
0.15
Frequency (Hz)
Low frequency (FTF) High frequency (BPFI)
DRS - SK
Cepstrum
whitened
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Findings Agree With Analysis Results
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Findings Agree With Analysis Results
Planetary Bearing Inner Race
Rollers
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T di b d SK Oil W D b i
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Trending based on SK vs. Oil Wear Debris
Accumulated oil wear debris Kurtosis of filtered signal
Measurement (hours)
37 .160 37 ....160
Mass(mg)
kurto
sis
0
500
0
16
1
2
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Second Case History
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High Speed Bearing Test Rig
FAG Test Rig L17 .. High Speed ( 12,000rpm)
Spall in the inner race
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The Effect of using The MED Technique
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The Effect of using The MED Technique
The SK before using the MED The SK after using the MED
8
6
4
2
1
0
2
1.5
1
0.50
0.5
2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18
Frequency [kHz]
1
N
umberofFilters/Octave 1
2
3
4
6
12
24
2
4
3
6
12
24
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The Effect of using the MED Technique
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The Effect of using the MED Technique
(a) Raw signal
(b) Residual of
linear prediction
filtering
(c) Signal b filtered
using MED
(d) Signal c filteredusing optimal SK
filter
Time (s)
Acceleration(m/s2)
Kurtosis =-0.38
Kurtosis =1.05
Kurtosis =11.44
Kurtosis =11.58
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Envelope Analysis after MED and SK
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Envelope Analysis after MED and SK
Harmonics at BPFI, sidebands at shaft speed
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Trending Fault Development
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g p
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Third Case HistoryRadar Tower Bearing
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Very slow speed(12 sec period)
118 square rollers
in alternate
directions so each
race strikes everysecond roller
Bearing and ring
gear changed,
pinion unchanged
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SPECTRUM COMPARISON
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Dominated by gears, but differences
at high and low frequencies
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Removal of gear signals by DRS
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Totalsignal
Deterministic
part (gears)(note scale)
Random
part
(bearings)
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Increased kurtosis from SK filtration
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c eased u tos s o S t at o
(gearmesh signals removed)
Note that extremelyhigh kurtosis
indicates that it is not
an absolute measure
of severity.
Fault could have been
detected at a very
early stage
Envelope spectrum
showing harmonics
of (half) ballpassfrequency modulated
at rotation speed
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GEAR DIAGNOSTICS - METHODS
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UNIFORM ERRORS
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Tooth deflection under load
For constant load is same for each tooth pair.
Therefore toothmesh frequency and harmonics are
affected. This is load sensitive, so spectrum
comparisons must be for same load.
Mean geometric profile errors
From initial manufacture and wear. By definition thisis same for each tooth pair. Therefore toothmesh
frequency and harmonics are affected. This is only
weakly load sensitive.
Uniform wear
Gives change in harmonics of toothmesh frequency
under constant load conditions. First indication at
second harmonic of gearmesh frequency
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Variations Between Teeth
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Variations Between Teeth
At rotational harmonics other than toothmesh.Harmonic spacing indicates which gear has causedchange. Can be further subdivided:
Slow variations, e.g. runout, distortion. Low
harmonics and sidebands around toothmesh areaffected.
Local faults, e.g. cracks, spalls. Wide distribution ofharmonics results.
Random errors, e.g. Random tooth spacing error.
Wide distribution of harmonics results. Systematic errors, e.g. Ghost components, from
gear cutting machine.
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Operational modal analysis
i th t
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using the cepstrum
Forcing and transfer function effects additive in cepstrumfor a single input
They are also separated for a smooth flat input spectrum
(impulsive or random)
Pole/zero parameters can be extracted from responseautospectra, and used to update and scale FRFs
For multiple inputs, New blind source separation
techniques give the possibility of extracting the responses
to a particular input
Cepstral techniques then give the scaled FRFs for the
resulting SIMO system
The University of
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(Oppenheim & Schafer)
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(Oppenheim & Schafer)
The University of
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function in the response cepstrum
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function in the response cepstrum
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TRUNCATION OF OUT-OF-BAND MODES
FRFs regenerated from in-band poles and zeros only
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Point 1 (driving point).Poles and zeros balanced
Point 5, typical point.No. of zeros approx.
half no. of poles
Point 8 (end-to-end).
No zeros
FRFs regenerated from in-band poles and zeros only
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FRF RECONSTRUCTION
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When generating FRFs from in-band poles andzeros only there are two missing factors
One is an equalisation curve depending on the ratio
of poles to zeros
The other is an overall scaling factor, as this is
contained in the zero quefrency component
Neither changes greatly with small changes in pole
and zero positions, and so can be determined from
an earlier measurement, a similar measurement or a
finite element model
The University of
New South Wales PHM MontrealUSE OF CYCLOSTATIONARITY TO
OBTAIN SIMO FROM MIMO (David Hanson)
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eg Burst random signal
- zero mean (1st order)- periodic autocovariance (2nd order)
Spectral correlation is 2D FT of 2D autocovariance
It is also the correlation of the spectrum with itself
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OBTAIN CEPSTRUM FROM CYCLIC SPECTRUM
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Starting with the system equation:
( ) ( ) ( )Y f H f X f and defining the cyclic spectral density of the response as:
*( ) lim ( ) ( )Wy WW
S f Y f Y f
E
we get*( ) ( ) ( ) ( )Y xS f H f H f S f
Taking the log and inverse Fourier transform to obtain the cepstrum2
( ) ( ) ( ) ( )j
y h h xC C C e C
Impulsive force has flat spectrum and short cepstrum, so:2
0( ) ( ) ( ) ,j
y h hC C C e
and if the system is minimum phase
( ) 0hC so
0( ) ( ),
y hC C
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Transperth B Series Railcar
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Excited by burst random input from shakerSupported on elastomeric mounts
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TYPICAL CYCLIC SPECTRA
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Transperth B Series Railcar
OMA R lt
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OMA Results
12Hz 16Hz
21Hz 26Hz
OM EM
The University of
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dynamicsgas turbine engine
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y g g
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 104
-60
-40
-20
0
20
PowerSpec
trumM
agnitude(dB)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 104
-50
-40
-30
-20
-10
0
Frequency (Hz)
Total (Raw)
signal
Residual
sign al after
edi t ing the
Cepst rum
Removal of discrete frequenciesuseful for OMA
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CONCLUSION
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Diagnostics involves separating the different signal
components, eg discrete frequency from random Several viable methods available with different pros
and cons
Many other techniques available for enhancing
various features of faults, for example in bearings and
gears
Another useful separation is of forcing function from
transfer function for each source and path
Blind determination of transfer functions (system
identification) useful to detect faults due to structuralchange rather than forcing function
Cepstrum useful for many of these functions