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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
                          Monitoring"
              Introduction
► Signal Analysis of Vibration Data – KEY for
  Fault Detection & Monitoring
► Time Domain & Fourier Analysis has some
  inherent disadvantages
► Wavelet Transforms scores over traditional
  techniques for transient signals



                 "Diagnostic Maintenance and Machine   2
                        Condition Monitoring"
               HUMAN BODY




                                                        Thermometer
Glucometer
                  B P Apparatus




             Angiogram, Echocardiogram



                  "Diagnostic Maintenance and Machine                 3
                         Condition Monitoring"
                             MACHINE




                                                                  Thermocouple
Wear / Oil Analysis
                            Vibration Meter




                      New Signal Analysis Techniques
                                Wavelets



                            "Diagnostic Maintenance and Machine                  4
                                   Condition Monitoring"
    Different Techniques
to analyse raw vibration data




       Waterfall, Trend Plot, Acoustic Emission,
       Wavelet Transform


           "Diagnostic Maintenance and Machine     5
                  Condition Monitoring"
 Topics of Discussion

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


        "Diagnostic Maintenance and Machine   6
               Condition Monitoring"
                       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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                                     Condition Monitoring"
               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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                            Condition Monitoring"
                            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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                                                                       Condition Monitoring"
                                                                                                             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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                                                                              Condition Monitoring"
   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)


                                                      "Diagnostic Maintenance and Machine                                                           12
                                                             Condition Monitoring"
                            Original signal




Signal with disturbance


                                                        Wavelet Transform Locates the
                                                        disturbance in Time-Frequency
                                                        Representation




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                                 Condition Monitoring"
      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

                           "Diagnostic Maintenance and Machine       14
                                  Condition Monitoring"
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

         "Diagnostic Maintenance and Machine                           15
                Condition Monitoring"
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

                           "Diagnostic Maintenance and Machine       16
                                  Condition Monitoring"
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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           Condition Monitoring"
          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
                           "Diagnostic Maintenance and Machine               18
                                  Condition Monitoring"
                          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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                                    Condition Monitoring"
                              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
                                 "Diagnostic Maintenance and Machine                                             20
                                        Condition Monitoring"
              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




                "Diagnostic Maintenance and Machine   21
                       Condition Monitoring"
                  Scaling




Scaling a wavelet means stretching (or compressing) it




               "Diagnostic Maintenance and Machine       22
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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
                                 "Diagnostic Maintenance and Machine                          25
                                        Condition Monitoring"
 Continuous Wavelet Transform
                      290Hz

              120Hz



       50Hz

20Hz




                              "Diagnostic Maintenance and Machine   26
                                     Condition Monitoring"
Process of CWT
          Sweep over the
          entire span of the
          signal




                                                  Dilate the mother wavelet
                                                  Redo the above sweeping




            "Diagnostic Maintenance and Machine                           27
                   Condition Monitoring"
 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
                       "Diagnostic Maintenance and Machine        28
                              Condition Monitoring"
       Case Studies
    a) Rotor Stator Rub
►




         "Diagnostic Maintenance and Machine   29
                Condition Monitoring"
Rotor-Stator Rub Test Setup
             Rotor-stator
             arrangement




                            Rotor Disc         Casing (Stator)




         "Diagnostic Maintenance and Machine                     30
                Condition Monitoring"
Experimental Results

                                           NO RUB




                                           RUB



     "Diagnostic Maintenance and Machine            31
            Condition Monitoring"
CWT of the Signals
                                   NO RUB



                                            RUB




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PARTIAL/INTERMITTENT RUB

                                              NO RUB




                                              Partial
                                              RUB


        "Diagnostic Maintenance and Machine             33
               Condition Monitoring"
CWT of Partial Rub




    "Diagnostic Maintenance and Machine   34
           Condition Monitoring"
      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




                 "Diagnostic Maintenance and Machine   35
                        Condition Monitoring"
Case Studies - b) Rotor Crack




Breathing behaviour of crack
           "Diagnostic Maintenance and Machine   36
                  Condition Monitoring"
Finite Element Model




     "Diagnostic Maintenance and Machine   37
            Condition Monitoring"
Cross coupled Stiffness Variation




            "Diagnostic Maintenance and Machine   38
                   Condition Monitoring"
Response of Cracked Rotor w/o Torsional
              Excitation




              "Diagnostic Maintenance and Machine   39
                     Condition Monitoring"
 Response of Cracked Rotor with Transient
Torsional Excitation at =00 during 5th cycle




                "Diagnostic Maintenance and Machine   40
                       Condition Monitoring"
CWT of the Torsional Vibration




          "Diagnostic Maintenance and Machine   41
                 Condition Monitoring"
CWT of Lateral Response of Cracked Rotor
  with Transient Torsional Excitation



                                                    at =00
                                                    during
                                                    5th cycle



              "Diagnostic Maintenance and Machine         42
                     Condition Monitoring"
  Response of Cracked Rotor with Transient
Torsional Excitation at =1800 during 5th cycle




                  "Diagnostic Maintenance and Machine   43
                         Condition Monitoring"
CWT of Lateral Vibration Response




           "Diagnostic Maintenance and Machine   44
                  Condition Monitoring"
CWT of Lateral Vibration
      Response




       "Diagnostic Maintenance and Machine   45
              Condition Monitoring"
Sensitivity of CWT coefficients to
            crack depth




                                                  5% crack depth
            "Diagnostic Maintenance and Machine                    46
                   Condition Monitoring"
     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

                 "Diagnostic Maintenance and Machine   47
                        Condition Monitoring"
       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


                 "Diagnostic Maintenance and Machine   48
                        Condition Monitoring"
Thank You !!



 "Diagnostic Maintenance and Machine   49
        Condition Monitoring"

				
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