# wavelets

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```					 WAVELET TRANSFORMS
AND ITS APPLICATION TO
FAULT DETECTION

Ashish K Darpe
Department of Mechanical
Engineering, IIT Delhi
"Diagnostic Maintenance and Machine Condition   1
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Introduction
► Signal Analysis of Vibration Data – KEY for
Fault Detection & Monitoring
► Time Domain & Fourier Analysis has some
► Wavelet Transforms scores over traditional
techniques for transient signals

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HUMAN BODY

Thermometer
Glucometer
B P Apparatus

Angiogram, Echocardiogram

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MACHINE

Thermocouple
Wear / Oil Analysis
Vibration Meter

New Signal Analysis Techniques
Wavelets

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Different Techniques
to analyse raw vibration data

Waterfall, Trend Plot, Acoustic Emission,
Wavelet Transform

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Topics of Discussion

► Wavelet Transforms – Why & When?
► Basic Theory
► Simple Examples
► Case Studies-
FFT not able to detect
CWT proved very effective

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Fourier Analysis
8

T
6

4
Amplitude

2

0

-2

-4
0   0.05    0.1       0.15      0.2      0.25    0.3   0.35
Time(sec)

 To find different frequency components
 Amplitudes of different components
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Fourier Analysis

Breaking down a periodic signal into its constituent sinusoids of
different frequencies

N 1                           2nk
1                                     j
F (k )   f (n)e                                   N
N n 0

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Decomposition of time domain signal in
frequency domain
1.2

2
1.0
+
1               =                                                                          0.8
Amplitude

Amplitude
0.6
0

0.4

-1
0                                                               40                    0.2

0.05                                                   30
0.0
Frequency

0.1                         20                                                0   10   20         30         40   50   60
Time     0.15             10                                                                   Frequency (Hz)
0
0.2

Drawback: Time information is lost
Problem not serious for stationary signals
Important for signals having non-stationary characteristics
Ex. Drift, trends, abrupt changes, beginnings & end of events

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Time domain & its
1

0.8
Frequency Domain
0.6

0.4
Representation
0.2
Amplitude

0

-0.2
0.7
-0.4

-0.6
0.6
-0.8

-1                                                                                            0.5
0    0.1   0.2   0.3   0.4      0.5     0.6   0.7   0.8   0.9   1
Time (Sec)
0.4

Amplitude
A 20Hz sinusoidal signal                                                            0.3

0.2

0.1

0
0   50   100   150   200   250 300     350   400   450   500
Frequency (Hz)

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1

0.8

0.6

0.4

0.2
Amplitude

0

-0.2                                                                                          Short Duration Transient Signal
-0.4

-0.6                                                                                  0.05

-0.8
0.04
-1
0.03
0   0.1   0.2   0.3   0.4      0.5     0.6   0.7   0.8   0.9   1
Time (Sec)
0.02

Pure Sine Wave
+               Amplitude
0.01

0

-0.01

-0.02

-0.03

-0.04

-0.05
0   0.2   0.4   0.6      0.8   1   1.2        1.4
Time (sec)

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1
Fourier Transform fails to
0.5
detect clearly, event of
0
disturbance is lost
0.7
-0.5

0.6
-1

0.5
-1.5

0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9    1                 0.4

Amplitude
0.3
Resultant Sine wave +
Transient disturbance                                                                0.2

0.1

0
0   50   100   150   200 250 300       350   400   450   500
Frequency (Hz)

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Original signal

Signal with disturbance

Wavelet Transform Locates the
disturbance in Time-Frequency
Representation

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Short Time Fourier Transform

Analyzing a small section of the signal at a time with Fourier Transform
Same Basis Functions (sinusoids) are used
Window size is fixed (uniform) for all frequencies
so all spectral estimates have same (constant) bandwidth

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Short Time Fourier Transform
►     Maps a signal into a two-
dimensional function of time
and frequency

►     Technique is called windowing
the signal

►     A compromise between the
time- and frequency- based
views of the signal

►     Provides some info @ both
when & at what frequencies a
signal event occurs

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Can we have something better?
►       NEED?
    Varying window size
►   To determine more accurately either time or frequency

Wavelet Analysis – A windowing technique with
variable sized regions
Allows use of long time intervals where we need
more precise low-frequency information
& use of shorter regions where we want high-
frequency information

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Wavelet Transform
Fourier Transform –
signal broken into sinusoids
that are global functions

Wavelet Transform –
signal broken into a series of
local basis functions
called wavelets, which are
scaled and shifted versions of
the original (or Mother) wavelet

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Comparison of Transforms
Event (time)
Frequency
information lost
information not
available

Simultaneous
High resolution
in both Time &
Freq. domains
NOT possible
Short data window of time T – B/W of each spectral coeff is 1/T - wide
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Wavelet
►   Sine waves – basis functions for Fourier Analysis extends from
+ to -
►   Wavelets have limited duration that has an average value of
zero
►   Sinusoids are smooth & predictable, Wavelets tend to be
irregular & asymmetric

Morlet wavelet (blue dashed) as a Sine curve (green)
modulated by a Gaussian (red)

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Wavelet
        2 
 (t )  e t cos         t
2

Morlet Wavelet                    
      ln 2 


►   Wavelet means a small wave

►   The function that defines a            

wavelet integrates to zero
►   It is local in the sense that it       
 (t )dt  0


decays to zero when sufficiently
far from its center

►   It is square integrable, i.e., it           
Mother Wavelet
has finite energy


|  (t ) | dt  
2

Scaling &
shifting

Son/daughter wavelets
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Wavelets
Signals with sharp sudden changes could be better
analyzed with an irregular wavelet than with a
smooth sinusoid

In other words, local features can be better
captured with wavelets which have local extent

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Scaling

Scaling a wavelet means stretching (or compressing) it

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Scaling

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Shifting

Shifting a wavelet means delaying or hastening its onset

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Continuous Wavelet Transform

Ensures energy stays
same for all s&b

Sum over all time of the signal multiplied by scaled and shifted versions of the wavelet
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Continuous Wavelet Transform
290Hz

120Hz

50Hz

20Hz

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Process of CWT
Sweep over the
entire span of the
signal

Dilate the mother wavelet
Redo the above sweeping

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Relation between scale & frequency
Fc
Fa 
s
Fa = pseudo frequency ( for the scale value s )
 = sampling time
s = Scale
Fc = central frequency of mother wavelet in Hz.

Central frequency of the Morlet wavelet is 0.8125Hz
It is the freq. that maximizes the FFT of the wavelet or is the
leading dominant frequency of the wavelet
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Case Studies
a) Rotor Stator Rub
►

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Rotor-Stator Rub Test Setup
Rotor-stator
arrangement

Rotor Disc         Casing (Stator)

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Experimental Results

NO RUB

RUB

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CWT of the Signals
NO RUB

RUB

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PARTIAL/INTERMITTENT RUB

NO RUB

Partial
RUB

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CWT of Partial Rub

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ROTOR RUB DETECTION
►  Localized (in time) rubbing is detected
using wavelet transform
► Intermittent rub is better detected
► High frequency components are also
localized in a cycle of rotation

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Case Studies - b) Rotor Crack

Breathing behaviour of crack
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Finite Element Model

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Cross coupled Stiffness Variation

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Response of Cracked Rotor w/o Torsional
Excitation

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Response of Cracked Rotor with Transient
Torsional Excitation at =00 during 5th cycle

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CWT of the Torsional Vibration

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CWT of Lateral Response of Cracked Rotor
with Transient Torsional Excitation

at =00
during
5th cycle

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Response of Cracked Rotor with Transient
Torsional Excitation at =1800 during 5th cycle

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CWT of Lateral Vibration Response

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CWT of Lateral Vibration
Response

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Sensitivity of CWT coefficients to
crack depth

5% crack depth
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Novel way to detect crack
► Short duration transient excitation can be
applied so that the rotor is not stressed
► Good use of the advantages of Wavelet
Transform for bringing out transient
response features of crack
► Good use of nonlinear nature of crack
breathing making the detection foolproof
► Highly sensitive to depth of crack

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WAVELET TRANSFORM
► Wavelet  Transform is an excellent tool for
detection of non-stationary vibration signals
► Features that are obscured during Fourier
Transformation are revealed with better
clarity
► Time information is preserved
► Standard functions available in Matlab

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Thank You !!

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 views: 7 posted: 12/10/2011 language: pages: 49