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					  WAVELET TRANSFORMS AND ITS APPLICATION TO FAULT
                   DETECTION

                          ASHISH K DARPE
           DEPARTMENT OF MECHANICAL ENGINEERING, IIT DELHI



1. INTRODUCTION
Machine vibration analysis is one of the important tools for faults identification. There
are two types of conventional analysis methods: time domain and the frequency domain.
The frequency domain analysis is more attractive one because it can give more detailed
information about the signal and its component frequencies whereas; the time domain
analysis can give overall qualitative information. Generally, the machine vibration signal
is composed of three parts: stationary vibration, random vibration, and noise.
Traditionally, Fourier transform (FT) was used to perform such analysis. It is well
known, however, that the Fourier analysis has some inherent limitations in the analysis of
the non-linear phenomena and some short duration transient signals. Over the past 10
years, the wavelet theory has become one of the emerging and fast-evolving
mathematical and signal processing tools for its many distinct merits. Different from the
fast-Fourier transform (FFT), the wavelet transform can be used for multi –scale analysis
of the signal through dilation and translation, so it can extract the time-frequency features
of the signals effectively.
Wavelets are mathematical functions that cut up data into different frequency
components, and then study each component with a resolution matched to its scale. They
have advantages over traditional Fourier methods in analyzing physical situations where
the signal contains discontinuities and sharp spikes. Wavelet functions are distinguished
from other transformations in that they not only dissect signals into their component
frequencies, they also vary the scale at which the component frequencies are analyzed.
Therefore wavelets, as component pieces used to analyze a signal, are limited in space. In
other words, they have definite stopping points along the axis of a graph--they do not
repeat to infinity like a sine or cosine wave does. The ability to vary the scale of the
function as it addresses different frequencies also makes wavelets better suited to signals
with spikes or discontinuities than traditional transformations such as the FT.


                                             1
2. APPLICATION OF WAVELET TRANSFORMATION.


        Wavelet transformation has been applied in wide variety of engineering analysis.
It includes machinery fault diagnosis, in particular, time–frequency analysis of signals,
fault feature extraction, singularity detection for signals, de-noising and extraction of the
weak signals, compression of vibration signals and system identification. By combining
the time domain and classical Fourier analysis, the wavelet transform provides
simultaneously spectral representation and temporal order of the signal decomposition
components, which is also applied to perform the analysis of NDE ultrasonic signal
received during the inspection of reinforced composite materials. Wavelet transform can
be used to process acoustic emission signals to find faults such as micro-failure modes
and their interaction in composites.
        For rotating machines fault detection, wavelet transform has been used for
isolation and identification of electrical faults in induction machine, bearing and gear
related faults. Recently wavelet transform has also been applied to detect the stable and
propagating cracks.


3. WAVELET THEORY
In this section a brief theory of wavelet transformation has been explained. To develop a
better insight a comparison between wavelet transform and fourier transform has been
made.

3.1 Fourier Transforms

The Fourier transforms utility lies in its ability to analyze a signal in the time domain for
its frequency content. The transform works by first translating a function in the time
domain into a function in the frequency domain. The signal can then be analyzed for its
frequency content because the Fourier coefficients of the transformed function represent
the contribution of each sine and cosine function at each frequency. An inverse Fourier
transform does just what you'd expect; transform data from the frequency domain into the
time domain.




                                             2
    F () = F {f (t)} = -   e
                                     -it
                                            f (t) dt                            (3.1)


3.2 Discrete Fourier Transforms

The discrete Fourier transform (DFT) estimates the Fourier transform of a function from
a finite number of its sampled points. The sampled points are supposed to be typical of
what the signal looks like at all other times.

The DFT has symmetry properties almost exactly the same as the continuous Fourier
transform. In addition, the formula for the inverse discrete Fourier transform is easily
calculated using the one for the discrete Fourier transform because the two formulas are
almost identical.

3.3 Windowed Fourier Transforms

If f(t) is a non periodic signal, the summation of the periodic functions, sine and cosine,
does not accurately represent the signal. You could artificially extend the signal to make
it periodic but it would require additional continuity at the endpoints. The windowed
Fourier transform (WFT) is one solution to the problem of better representing the non
periodic signal. The WFT can be used to give information about signals simultaneously
in the time domain and in the frequency domain.

With the WFT, the input signal f(t) is chopped up into sections, and each section is
analyzed for its frequency content separately. If the signal has sharp transitions, one can
window the input data so that the sections converge to zero at the endpoints. This
windowing is accomplished via a weight function that places less emphasis near the
interval's endpoints than in the middle. The effect of the window is to localize the signal
in time.

3.4 Fast Fourier Transform

To approximate a function by samples, and to approximate the Fourier integral by the
discrete Fourier transform, requires applying a matrix whose order is the number sample
points n. Since multiplying an n X n matrix by a vector costs on the order of 2n arithmetic


                                                 3
operations, the problem gets quickly worse as the number of sample points increases.
However, if the samples are uniformly spaced, then the Fourier matrix can be factored
into a product of just a few sparse matrices, and the resulting factors can be applied to a
vector in a total of order nlogn arithmetic operations. This is the so-called fast Fourier
transform or FFT.

Example: Figure 3.2 shows the FFT of the rub signal shown in Fig 3.1.




 Figure 3.1. Simulation signals of the rub-impact fault (ordinates are in number of
      rotations)(a) signal of the slight rub fault;(b)signal of the severe rub fault.
                                     (Peng et al., 2003).




 Figure 3.2. FFT spectrum of the rub-impact signal: (a) spectrum of the slight rub-
     impact fault (b) spectrum of the severe rub-impact fault. (Peng et al., 2003).


3.5 Similarities between Fourier and Wavelet Transforms
The fast Fourier transform (FFT) and the discrete wavelet transform (DWT) are both
linear operations that generate a data structure that contains log2 n segments of various
lengths, usually filling and transforming it into a different data vector of length 2n.

The mathematical properties of the matrices involved in the transforms are similar as
well. The inverse transform matrix for both the FFT and the DWT is the transpose of the
original. As a result, both transforms can be viewed as a rotation in function space to a


                                               4
different domain. For the FFT, this new domain contains basis functions that are sines
and cosines. For the wavelet transform, this new domain contains more complicated basis
functions called wavelets, mother wavelets, or analyzing wavelets.

Both transforms have another similarity. The basis functions are localized in frequency,
making mathematical tools such as power spectra (how much power is contained in a
frequency interval) and scalograms useful at picking out frequencies and calculating
power distributions.

3.6 Dissimilarities between Fourier and Wavelet Transforms

The most interesting dissimilarity between these two kinds of transforms is that
individual wavelet functions are localized in space. Fourier sine and cosine functions are
not. This localization feature, along with wavelets' localization of frequency, makes many
functions and operators using wavelets "sparse" when transformed into the wavelet
domain. One way to see the time-frequency resolution differences between the Fourier
transform and the wavelet transform is to look at the basis function coverage of the time-
frequency plane. Figure 3.3 shows a windowed Fourier transform, where the window is
simply a square wave. The square wave window truncates the sine or cosine function to
fit a window of a particular width. Because a single window is used for all frequencies in
the WFT, the resolution of the analysis is the same at all locations in the time-frequency
plane.




 Figure. 3.3. Fourier basis functions, time-frequency tiles, and coverage of the time-
                                   frequency plane
An advantage of wavelet transforms is that the windows vary. In order to isolate signal
discontinuities, one would like to have some very short basis functions. At the same time,



                                            5
in order to obtain detailed frequency analysis, one would like to have some very long
basis functions. A way to achieve this is to have short high-frequency basis functions and
long low-frequency ones. This happy medium is exactly what one gets with wavelet
transforms. Figure 3.4 shows the coverage in the time-frequency plane with one wavelet
function, the Daubechies wavelet.




Figure 3.4. Daubechies wavelet basis functions, time-frequency tiles, and coverage of
                            the time-frequency plane.
       One thing to remember is that wavelet transforms do not have a single set of basis
functions like the Fourier transform, which utilizes just the sine and cosine functions.
Instead, wavelet transforms have an infinite set of possible basis functions. Thus wavelet
analysis provides immediate access to information that can be obscured by other time-
frequency methods such as Fourier analysis.

For signal shown in Figure 3.1, we have the following scalograms.




 Figure 3.5. Scalogram of the rub-impact signals (a) scalogram of the slight rub (b)
             scalogram of the severe rub-impact fault (Peng et al., 2003).




                                              6
3.7 The Wavelet Transform
The driving force behind wavelet transforms (WTs) is to overcome the disadvantages
embedded in short time Fourier transform (STFT), which provides constant resolution for
all frequencies since it uses the same window for the analysis of the inspected signal x (t).
On the contrary, WTs use multi-resolution, that is, they use different window functions to
analyse different frequency bands of the signal x (t). Different window functions
ψ(s,b,t);which are also called son wavelets, can be generated by dilation or compression
of a mother wavelet ψ(t) , at different time frame. A scale is the inverse of its
corresponding frequency. WTs can be categorised as discrete WTs or continuous WTs.
For vibration-based fault diagnosis, usually continuous WTs are employed. A continuous
type of wavelet transform (CWT) that is applied to the signal x(t) can be defined as,
                
            1            t b
w(a, b) 
            a
                 f (t ) 
                        
                              dt
                            a 
                                                                          (3.2)

Where

•    a is the dilation factor,

•    b is the translation factor and

•    ψ(t) is the mother wavelet.

•     1/a is an energy normalization term that makes wavelets of different scale has the
     same amount of energy.




                                             7
3.8 Wavelet Definition
A wavelet is a waveform of effectively limited duration that has an average value of zero.




Figure 3.7 (a) Morlet wavelet of arbitrary width and amplitude, with time along the
x-axis. (b) Construction of the Morlet wavelet (blue dashed) as a Sine curve (green)
               modulated by a Gaussian (red). (www.paos.colorad.edu)


3.9 Wavelet Properties
1.The definition domain is compact support ,which ensures that the function is fast
decaying, and so time localization can be obtained.
2.The admissibility condition
C =  (( () 2)/ ) d <                                   (3.3)
Where  () =   (t) e-it dt .

This condition means that the waveform of the mother wavelet function must be
oscillating i.e. the average value of the wavelet in the time domain must be zero.

3.10 Scaling of Wavelet

•    Scaling a wavelet simply means stretching (or compressing) it.

•    Low frequencies (high scales) correspond to a global information of a signal (that
     usually spans the entire signal), whereas high frequencies (low scales) correspond
     To detailed information of hidden pattern in the signal (that usually last for
     relatively short time.)




                                             8
In terms of mathematical functions, if f(t) is a given function f(st) corresponds to a
contracted (compressed) version of f(t) if s > 1 and to an expanded (dilated) version of
f(t) if s < 1 Figure 3.8 shows scaling of wavelet at four different scales.




                 Figure 3.8 Scaling of Wavelet (www.users.rowan.edu)


3.11 COMPUTATION OF THE CWT

Interpretation of the equation (3.2) will be explained in this section. Let x(t) is the signal
to be analyzed. The mother wavelet is chosen to serve as a prototype for all windows in
the process. All the windows that are used are the dilated (or compressed) and shifted
versions of the mother wavelet. There are a number of functions that are used for this
purpose. The Morlet wavelet is mostly used for vibration based fault diagnosis, and it is
used for the wavelet analysis of the vibration signals due to rubbing which are presented
in the later chapters. Once the mother wavelet is chosen the computation starts with
s=1(scale) and the continuous wavelet transform is computed for all values of s, smaller
and larger than ``1''.The wavelet is placed at the beginning of the signal at the point


                                              9
which corresponds to time=0. The wavelet function at scale ``1'' is multiplied by the
signal and then integrated over all times. The result of the integration is then multiplied
by the constant number 1/sqrt{s}. This multiplication is for energy normalization
purposes so that the transformed signal will have the same energy at every scale. The
final result is the value of the transformation, i.e., the value of the continuous wavelet
transform at time zero and scale s=1. In other words, it is the value that corresponds to
the point t =0, s=1 in the time-scale plane. The wavelet at scale s=1 is then shifted
towards the right by “b” amount to the location t=b , and the above equation is computed
to get the transform value at t=b , s=1 in the time-frequency plane. This procedure is
repeated until the wavelet reaches the end of the signal. One row of points on the time-
scale plane for the scale s=1 is now completed. Then, “s” is increased by a small value.
Note that, this is a continuous transform, and therefore, both time (b) and “s” must be
incremented continuously. However, if this transform needs to be computed by a
computer, then both parameters are increased by a sufficiently small step size. This
corresponds to sampling the time-scale plane. The above procedure is repeated for every
value of “s”. Every computation for a given value of “s” fills the corresponding single
row of the time-scale plane. When the process is completed for all desired values of s, the
CWT of the signal has been calculated. The Figures 3.9 below illustrate the entire process
step by step.




                                            10
(a)




(b)


11
                                            (c)

        Figure 3.9 Computation of the CWT(a) scale s=1(b) scale s=5 (c) scale
                           s=20(www.users.rowan.edu)


3.12 Different Shapes of Wavelet

Wavelet transforms comprise an infinite set. The different wavelet families make
different trade-offs between how compactly the basis functions are localized in space and
how smooth they are. In Figure 3.10, several different wavelet families are illustrated.




                  Figure. 3.10. Several different families of wavelets.




                                             12
3.13 Scalogram
For all (t), x (t)  L2 (R), the inverse wavelet transform of x (t) is defined as


x(t) = (1/ C)   a-2 wx (a,b;) ab (t) da db             (3.4)


Since the wavelet transform does not lose any information, and the energy is preservative
for the transform. So the following equation is tenable:

< x(t) , x(t) >= x(t)2 dt = (1/ C)  a-2 da  wx (a,b;)2 db         (3.5)


wx (a,b;)2 a b/ C a2 represents the total energy of a domain centered at(a ,b )with
scale interval a and time interval b. and wx (a,b;)2 is defined as the wavelet
scalogram. It shows how the energy of the signal varies with time and frequency. The
wavelet scalogram has been widely used for the analysis of non-stationary signal.
In order to illustrate the characteristics of the scalogram , a signal X (t) is considered .
Figure3.11 shows the signal X (t)in the time domain and it can be expressed by


      X= 6 sin 2(10t)+ 6 sin 2(20t)         0  t < 0.5                          (3.8)
      X= 6 sin 2(20t)+ 6 sin 2(30t)         0.5  t < 1




                                Figure 3.11 Test signal X(t)



                                                  13
                     Figure 3.12 FFT of numerical test signal X(t)




                         Figure 3.13 Scalogram for signal X (t)


Although we know that the signal X (t) contains three components whose frequencies are
10, 20 and 30 Hz, respectively, from the spectrum (Figure 3.12) we cannot determine
whether the component exists during all the signal lifespan or only part of the lifespan.



                                             14
However, through the signal scalogram, using Morlet Wavelet, we are not only able to
recognize, the three components but also able to know the time at which they exist. It is
to be noted that for Scalogram scale a is inversely proportional to the frequency, this
means that low scale value shows high frequency and vice-versa. In figure (4) vertical-
axis is represented by scale factor “a” which is inversely proportional to frequency.
Horizontal-axis as time. Bright region in scalogram shows the presence of that particular
frequency at the particular time.
   a=8 represents frequency =10 Hz which is present in the time span 0 < t< 0.5.
   a= 4 represents frequency =20 Hz which is present in the time span 0 < t< 1
   a= 3 represents frequency =30 Hz which is present in the time span 0.5 < t < 1


The above analysis of simulated test signals X(t) with the help of FFT and CWT
(Continuous Wavelet Transform) using Morlet Wavelet helps to understand the
applicability of Wavelet transformation for rubbing. Such representation is very helpful
to reveal the time intervals as well as the frequency range of the occurrence of some
transient disturbances due to typical faults like rub in rotors and structures.


3.14 Frequency –Scale relation for Scalogram
As explained above scale is inversely related the frequency. Though there is no exact
mathematical relation for this, with approximation it can de stated as
                       Fa=  x Fc/ s         (3.9) ( Matlab 6.5 Help)
Fa = pseudo frequency ( for the scale value s )

 = sampling frequency

s   = Scale

Fc = central frequency of mother wavelet in Hz.

Centralfrequency of the Morlet wavelet used in this project is .0.8125 Hz ( Matlab 6.5 )




                                              15
4. SOME CASE STUDIES


To elucidate the effectiveness of wavelet Scalogram, a few case studies are now
discussed.
       a) Rotor-Stator Rub Detection (Experimental Study)
       b) Crack detection in a rotating shaft (Analytical Study)
4.1 Rub Detection in a rotor bearing system.
In the first case experimental results related to rotor-stator rub are discussed in which
wavelet transform tool available in Matlab software is used and results presented. Figure
4.1 shows the experimental set-up of the real rotor system. Figure 4.2 shows vibratory
signal flow diagram for identifying Rub. It is composed of a rotor and a stator, a driving
motor, bearings and couplings. Vibration signals were collected, for both horizontal and
vertical direction, from the rotor system using non-contact eddy-current transducers at a
sampling rate of 2 kHz. The experiment was done at three different rotational speed.
4.1.2 Experimental Setup




                                                                                   Casing




                                                                              Rotor Disc




                             Figure 4.1 Rotor-casing setup




                                           16
                                                          Rotor-stator
                                                          arrangement
             Motor




                Figure 4.2(a) Experimental Setup




Rotor Disc
                                             Casing (Stator)




                Figure 4.2( b) Experimental Setup




                               17
4.1.3 Experimental Results
Experiment for rub between rotor and stator was conducted at different rotational speeds.
Rub between rotor and casing was introduced by lowing down the upper part of the
casing till it touched the rotor disc and then tightened it up at its new position with help of
nut arrangement. The shaft vibrations in both horizontal and vertical direction, was
measured with the help proximity probe fixed at the probe holder. The vibration signals
was then transferred to oscilloscope through proximitor and vibration monitor and the
data was recorded in the oscilloscope. The following figures shows the time-domain
waveform , FFT spectrum and the wavelet scalogram (using Morlet wavelet) of the
vibration signal at different rotational speeds with both rubbing and without rubbing .




 Figure 4.3 (a)Time domain signal at 588 rpm in        vertical direction(a) without-rub (b) with-rub


To get a better information of high frequencies and their occurring time scalogram with
low scale values is shown in the following figures and indicate


                                                  18
                                            (a)




                                                                                High Frequencies
                                                                                present




   Figure 4.4 (b) Scalogram with lower range of scale “a”, at 588 rpm (a) without rub
                                       (b)with-rub
absence of higher frequencies. But 4.4 (b) shows higher frequencies at the scale values
of a=9-13 , and these are occurring after every 200 time sample, i.e after one cycle . Thus
the rubbing is occurring once every cycle. Their approximate frequency can be calculated
by using equation 3.9.


                                            19
Figure 4.5 is the 3D plot of scalogram showing the values of wavelet coefficient on the z-
axis. In the Figure the main peak indicate the rotational frequency, and the lower Peaks
shows presence of higher frequencies.




      (a)                                                               (b)
 Figure 4.5 Scalogram 3-D at lower scale, at 588 rpm (a) without rub (b) with rub.
Figure 4.5 (b) shows distinct periodic peaks at higher frequencies(lower scale) due to
rubbing which is not visible in      Figure4.5 (a). Thus scalogram provides a better
information in both time and frequency domain.
Three set vibrations signals along the horizontal direction were recorded separately at
three different rpm i.e. 588 rpm, 912 rpm and 960 rpm. The following figures present
horizontal vibration due to rubbing at 588 rpm using time domain waveform along with
their FFT spectra and scalogram.




                                           20
                                          (a)




                                                                                         No rubbing
                                                                                         at the end of
                                                                                         the signal




                                          (b)
  Figure 4.6 Time domain signal at 588 rpm in horizontal direction (a) without-rub (b)
                                       with-rub




             Figure 4.7 Scalogram 3-D for signal shown in Figure 4.6b.


In Figure 4.7 that represents 3D scalogram for signal shown in Figure 4.6b, a periodic
bright peaks can be seen which indicate high frequency impacts due to rubbing. In the


                                          21
same scalogram we can observe the absence of these high frequencies at latter stage of
the signal. Thus this provides the confirmation that the rubbing is not present for a very
small time span at end of the recorded signal. Thus the wavelet scalogram is able to pick
up information about the frequency content as well as its temporal representation with
better clarity and detail.
          Experimental signals had been collected at different rotational speeds. It can be
concluded that the Scalogram (CWT) have distinct advantage for the rub-impact signal
analysis and they can characterize the dynamical behaviour of the rub-impact rotor better
and can extract the rub impact feature well.




4.2 Case Study II: Crack Detection

Fatigue cracks are a potential source of catastrophic failures in rotors. Researchers have
put in considerable effort to develop a foolproof and reliable strategy to detect cracks in
rotors.




                         Figure 4.8. Breathing behaviour of crack


This case study presents a novel way to detect fatigue transverse cracks in rotating shafts.
The proposed detection methodology exploits both the typical nonlinear breathing
phenomenon of the crack and the coupling of bending-torsional vibrations due to the


                                               22
presence of crack for its diagnosis. A transient torsional excitation is applied for a very
short duration at specific angular orientation of the rotor and its effect in the lateral
vibrations is investigated. Wavelet transforms is used in revealing the transient features
of the resonant bending vibrations, which are set up for a short duration of time upon
transient torsional excitation. Variation of peak absolute value of wavelet coefficient (of
the transient lateral vibration response) with angle at which torsional excitation is applied
is investigated. The correlation of this variation with the breathing pattern of the crack is
studied. The sensitivity of the proposed methodology to the depth of crack is
investigated. The detection methodology gives a vibration response signature that closely
correlates with and is very specific to the behaviour of the transverse surface crack in a
horizontal rotor. The response features are not likely to be exhibited by other common
rotor faults under similar excitation conditions making the proposed detection
methodology unambiguous. The fact that the rotor is not required to stop and that the
detection process is applied for a rotating shaft with only a short period transient external
excitation makes the methodology more convenient.




             Figure 4.9 Finite element model of the rotor bearing system.




Consider a rotor segment containing a single transverse surface crack. A beam finite
element containing a transverse surface crack of depth a as shown in Figure 4.9. To study
the coupled bending-torsional vibrations of a cracked rotor, a simply supported rotor with



                                             23
a single centrally situated disc of mass 1kg is considered. A single transverse surface
crack is assumed just adjacent to the central disc. The total rotor span is divided into 14
elements of equal length (Figure 4.9). A crack finite element is used to represent the
crack. Rest of the rotor is modeled with Timoshenko beam elements with six degrees of
freedom per node.
The torsional excitation is applied to the cracked rotor with depth ratio of a  0.2
rotating at 22rad/sec. The torsional excitation Te= Mtsin(221t) at frequency of 221rad/sec
is applied for a very short duration of 0.02857sec (equal to period of one cycle). The
frequency 221rad/sec is the natural frequency of bending vibrations of the rotor. The
exact timing of the application of transient torsional excitation during rotation of the
cracked rotor can be varied. Figure 4.10 shows the response of the cracked rotor
( a  0.2 ) when the transient torsional excitation is applied when rotor is oriented at
=00.




    Figure 4.10. Unbalance response of a cracked rotor ( a  0.2 ) with transient
        torsional excitation applied at =00 during 5th revolution =22rad/sec.




                                            24
   The transient torsional excitation is evident from the torsional vibration signal in
Figure 4.10 at the start of 5th rotation (c=14400). The torsional excitation causes a
broadband excitation as seen in the spectrum of the torsional vibration signal (Figure
4.10h). Although a transient torsional excitation was applied, the time domain signal in
bending and longitudinal vibrations do not show any sign of transient disturbance to the
rotor after c=14400. It may be mentioned that the excitation is applied when =00 (angle
turned during the 5th cycle of rotation) and when cumulative angle turned by the rotor
from the start is c=14400.
   Figure 4.11 shows response of the cracked rotor when the same transient torsional
excitation is applied at =1800 during the 5th rotation (i.e., c=16200). In this case,
Figures 4.11a and 4.11b show some noticeable changes in the response for a very short
duration for a couple of cycles or rotation (c=16200 - 25200) after the application of
excitation. The changes in the form of high frequency ripples are more perceptible in
horizontal vibrations. However, being for a very short duration, these vibrations are not
properly represented in the frequency domain signal of the vibrations shown in Figures
4.11e and 4.11f. The spectra in Figures 4.11e and 4.11f are almost identical to those in
Figures 4.10e and 4.10f although the time domain signals are slightly different as the
disturbance is noticed in Figures 4.11a and 4.11b.
   When the torsional excitation is applied at the two different angular orientation of the
rotor, Figures 4.10 and 4.11 show that there is small and barely noticeable change in time
domain signal. There is no change in the longitudinal vibration response in either case.
When the excitation is applied at =1800, the crack is on the underside (in the tensile
region) and hence is fully open. A fully open crack allows cross coupling between
torsional and bending vibration and torsional excitation generates resonant bending
vibrations that dies out quickly and are not sustained beyond couple of cycles of rotation.
In contrast, when the torsional excitation applied at =00, the crack is at the top side (in
the compressive region) and hence is fully closed. In this case the closed crack prevents
cross coupling of torsional and bending vibrations; the transient torsional excitation
therefore fails to generate any additional response in the bending vibrations. It is also
noticed from the bending stiffness variation of the cracked rotor (Figure 4.11i) that the
application of excitation does not alter the periodic variation of the stiffness indicating an


                                             25
unchanged opening and closing (breathing) of the crack. The observation is important as
it ensures that a different quality of rotor response is generated depending upon whether
the excitation is applied at =00 (crack closed) or at =1800 (crack closed).




    Figure 4.11. Unbalance response of a cracked rotor ( a  0.2 ) with transient
      torsional excitation applied at =1800 during 5th revolution =22rad/sec.


       Although a qualitatively different lateral vibration response is observed from
Figures 4.10 and 4.11, the response dies down quickly causing detection of such transient
phenomenon a difficult task using traditional signal analysis techniques such as
frequency domain analysis. This is evident from Figures 4.11e and 4.11f. Wavelet
transforms are better suited for such situations as they have time/space localization
feature. Wavelet analysis provides an excellent access to information that may be
obscured by the other techniques such as Fourier analysis.




                                             26
   Over the past 15 years, wavelet transforms (WT) have become one of the fast-
emerging and effective mathematical and signal processing tools for its distinct
advantage in analyzing signals, particularly the transient ones. Using different window
functions through dilation and translation of a prototype function called mother wavelet,
the wavelet transform can provide multi-resolution (multi-scale) analysis of a signal.
Hence it can extract time-frequency features of a signal effectively, which are obscured in
the traditional Fourier Transform analysis. The limitation of Fourier Transform stem from
the fact that the integral transform is applied on the signal globally and hence in the
process the time localization of the spectral component is not highlighted and is lost if the
transient decays quickly.
   To reveal the subtle but important information hidden in the transient vibration signal
generated due to transient torsional excitation, the response estimated using a nonlinear
rotor dynamic analysis of the cracked rotor is analyzed using continuous wavelet
transforms. Matlab’s (version 7) standard functions are used in the analysis with Morlet
mother wavelet. Figure 4.12 shows the 3D and 2D wavelet scalogram of torsional
vibration signal applied at =1800 (c=16200). It may be observed that the application of
excitation is observed in the form of large wavelet coefficient value after c=16200 for a
short duration thereafter. The large coefficients are observed for a wide scale range (from
scale value of 4 to 100 and beyond) indicating that wide band of frequencies are excited
at c=16200.




                                             27
                                                                             (a)




                                                                            (b)


Figure 4.12. Wavelet scalogram of the torsional vibration signal with excitation applied
                       at c=16200 i.e., =1800 a) 3D b)2D plot


                                          28
   The broadband torsional excitation also excites the torsional resonance in the
torsional vibration signal as observed in Figure 4.12b. The scale value of 8.2 maps to the
frequency of 125Hz, which is the torsional natural frequency of the rotor. At the scale
value of 8.2 and c=16200, a gradually decreasing coefficient value is observed from the
fading white patch between c=16200 and c=19000. This indicates a decaying torsional
vibration signal of torsional natural frequency.

   The 3D scalogram of lateral vibration signal is shown in Figure 4.13a. The x-axis
represents the cumulative angle turned by the rotor (c), the y-axis represents the scale
value and the z-axis represents the absolute values of wavelet coefficients. The figure
represents the wavelet scalogram of the time domain signal of lateral vibration shown in
Figure 4.11b; wherein the excitation is applied at =1800. The high energy concentration
is observed at scale values of 293 and 146, which represents 1x and 2x harmonic
frequency components for the rotational speed of 3.5Hz. The figure also shows a
gradually decreasing ridge (shown encircled) from angle of rotation c=16200 at the scale
value of 29 that corresponds to the bending natural frequency (35Hz) of the cracked
rotor. The same 3D plot for the scale value from 1 to 100 is shown in Figure 4.13b, that
clearly shows the bending natural frequency vibration initiated during the 5th cycle of
rotation after the application of excitation at c=16200.




                                             29
                                                                              (a)




                                                                              (b)




Figure 4.13. Wavelet scalogram of the bending vibration signal with excitation applied at
            c=16200 i.e., =1800 a) scale range 1-350 b) scale range 1-100




                                           30
                                                                              (c)




                                                                              (d)




                                         Peak Ca,b value at
                                         c=16930




Figure 4.13. Wavelet scalogram of the bending vibration signal with excitation applied at
c=16200 i.e., =1800 c) 3D scalogram; scale range 1-40 d) 2D scalogram; scale range 1-
                                           40


                                           31
   As the scale value of interest is 29, the 2D wavelet map with reduced scale range
(Figure 4.13) shows the generation of bending resonance vibration initiated after
c=16200 and gradually decays. It may also be observed that although the resonant
bending vibrations are seen immediately following the application of torsional excitation
at c=16200, the peak value of wavelet coefficient is observed at c=16930 (indicated by
dark brown patch) as it takes finite time to build up resonant vibration. The vibrations
decay after c=16930 as seen in Figure 4.13d. The excitation is applied in torsional mode
and resonant vibrations are observed in the bending mode, indicating coupling of
vibrations.

   Wavelet transform is now applied to the bending vibration signal shown in Figure
4.10b. All other conditions being the same, only timing of the excitation is different in
this case. Instead of applying at =1800 (c=16200), the excitation is applied at =00
(c=14400) when the crack is closed as observed from the stiffness variation in Figure
4.11i. Compared to Figures 4.13b and 4.13d, Figures 4.14a and 4.14b shows a complete
absence of the ridge representing transient resonant bending vibrations at scale value of
29 beyond c=14400. The wavelet map thus can pick up the transient part of the signal
and locate the presence of these transients in a time-frequency representation shown in
Figures 4.13 and 4.14. The exact timing of excitation and the frequency of the transient
disturbance can be judged easily from a single graph.

   The simulation is repeated for different depths of crack and it has been found that the
wavelet coefficients are highly sensitive to the depth of crack (Figure 4.15) and hence the
strategy can be adopted to detect crack at its early stage of progress. Figure 4.16 shows
that even for a very shallow depth crack (crack depth 5% of diameter of crack), the
presence of high frequencies can be detected using wavelet scalogram and the
frequencies would have been undetected using conventional signal analysis techniques.




                                            32
                                                                            (b)




 Figure 4.14. Wavelet scalogram of the bending vibration signal with excitation
applied at c=14400 i.e., =00 a) 3D scalogram; scale range 1-100 b) 3D scalogram;
                                 scale range 1-40




              Figure 4.15. Sensitivity of Cap,b to crack depth ratio a




                                         33
 Figure 4.16. Wavelet scalogram of the bending vibration signal with excitation
                  applied at =1800 for shallow crack ( a =0.05)




CONCLUDING REMARKS:

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 using wavelet transform. For short duration transient signals, as the
wavelet transform preserves the time and frequency information, the cause of such
signals can be easily correlated. The technique has been successfully applied for
detection of rotor rub and also tried for unique identification of rotor crack.




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