Tutorial Diagnostics Randall.pdf

8/13/2019 Tutorial Diagnostics Randall.pdf

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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

The University of


12/84


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.

The University of


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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

The University of

PHM M l


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INSENSITIVITY OF CEPSTRUM TO TRANSFER PATH


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The University of

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

The University of

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

The University of

PHM Montreal


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New South Wales PHM Montreal

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)

The University ofN S th W l PHM Montreal


19/84

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.

The University ofN S th W l PHM Montreal


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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


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


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


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


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


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)


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)


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


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


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


AMPLITUDE MODULATION


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AMPLITUDE MODULATION


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


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)


Case History Helicopter Gearbox Rig


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Case HistoryHelicopter Gearbox Rig

100.00 HZ

5.73 HZ

Planetary

Bearing

Blind analysis


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


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


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


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


Findings Agree With Analysis Results


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Findings Agree With Analysis Results

Planetary Bearing Inner Race

Rollers


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


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


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


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


Envelope Analysis after MED and SK


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Envelope Analysis after MED and SK

Harmonics at BPFI, sidebands at shaft speed


Trending Fault Development


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g p


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


SPECTRUM COMPARISON


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Dominated by gears, but differences

at high and low frequencies


Removal of gear signals by DRS


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Totalsignal

Deterministic

part (gears)(note scale)

Random

part

(bearings)


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


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.

The University of


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

New South Wales PHM MontrealAnalytical Expression for the Cepstrum

(Oppenheim & Schafer)


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(Oppenheim & Schafer)

The University of

New South Wales PHM MontrealCurve-fitting poles and zeros of transfer

function in the response cepstrum


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function in the response cepstrum

The University of


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

The University of


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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The University of


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

The University of


Transperth B Series Railcar


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Excited by burst random input from shakerSupported on elastomeric mounts

The University of


TYPICAL CYCLIC SPECTRA


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The University of


Transperth B Series Railcar

OMA R lt


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OMA Results

12Hz 16Hz

21Hz 26Hz

OM EM

The University of

New South Wales PHM MontrealPotential application to machine structural

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

The University of


CONCLUSION


84/84

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

Tutorial Diagnostics Randall.pdf

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Transcript of Tutorial Diagnostics Randall.pdf