rolling element bearing fault diagnostics using the blind ... · bearing condition monitoring has...

Rolling Element Bearing Fault Diagnostics using the

Blind Deconvolution Technique

Mahdi Karimi

BSc (Mech Engineering) (Isfahan University of Technology)

Master of Science (Toosi University of Technology)

Thesis submitted in total fulfilment of the requirements of the degree of

Doctor of Philosophy

School of Engineering Systems

Faculty of Built Environmental Engineering

Queensland University of Technology

September 2006

Bearing failure is one of the foremost causes of breakdown in rotating machinery. Such failure

can be catastrophic and can result in costly downtime. Bearing condition monitoring has thus

played an important role in machine maintenance. In condition monitoring, the observed signal

at a measurement point is often corrupted by extraneous noise during the transmission process.

It is important to detect incipient faults in advance before catastrophic failure occurs. In

condition monitoring, the early detection of incipient bearing signal is often made difficult due

to its corruption by background vibration (noise). Numerous advanced signal processing

techniques have been developed to detect defective bearing signals but with varying degree of

success because they require a high Signal to Noise Ratio (SNR), and the fault components

need to be larger than the background noise. Vibration analyses in the time and frequency

domains are commonly used to detect machinery failure, but these methods require a relatively

high SNR. Hence, it is essential to minimize the noise component in the observed signal before

post processing is conducted.

In this research, detection of failure in rolling element bearing faults by vibration analysis is

investigated. The expected time intervals between the impacts of faulty bearing components

signals are analysed using the blind deconvolution technique as a feature extraction technique to

recover the source signal. Blind deconvolution refers to the process of learning the inverse of an

unknown channel and applying it to the observed signal to recover the source signal of a

damaged bearing. The estimation time period between the impacts is improved by using the

technique and consequently provides a better approach to identify a damaged bearing. The

procedure to obtain the optimum inverse equalizer filter is addressed to provide the filter

parameters for the blind deconvolution process. The efficiency and robustness of the proposed

algorithm is assessed initially using different kinds of corrupting noises. The result show that

the proposed algorithm works well with simulated corrupting periodic noises. This research also

shows that blind deconvolution behaves as a notch filter to remove the noise components.

This research involves the application of blind deconvolution technique with optimum equalizer

design for improving the SNR for the detection of damaged rolling element bearings. The filter

length of the blind equalizer needs to be adjusted continuously due to different operating

conditions, size and structure of the machines. To determine the optimum filter length a

simulation test was conducted with a pre-recorded bearing signal (source) and corrupted with

varying magnitude noise. From the output, the modified Crest Factor (CF) and Arithmetic Mean

(AM) of the recovered signal can be plotted versus the filter length. The optimum filter length

can be selected by observation when the plot converges close to the pre-determined source

feature value. The filter length is selected based on the CF and AM plots, and these values are

stored in a data training set for optimum determination of filter length using neural network. A

pre-trained neural network is designed to train the behaviour of the system to target the

optimum filter length. The performance of the blind deconvolution technique was assessed

based on kurtosis values.

The capability of blind deconvolution with optimum filter length developed from the simulation

studies was further applied in a life bearing test rig. In this research, life time testing is also

conducted to gauge the performance of the blind deconvolution technique in detecting a

growing potential failure of a new bearing which is eventually run to failure. Results from un-

seeded new bearing tests are different, because seeded defects have certain defect characteristic

frequencies which can be used to track a specific damaged frequency component. In this test,

the test bearing was set to operate continuously until failures occurred. The proposed technique

was then applied to monitor the condition of the test bearing and a trend of the bearing life was

established. The results revealed the superiority of the technique in identifying the periodic

components of the bearing before final break-down of the test bearing.

The results show that the proposed technique with optimum filter length does improve the SNR

of the deconvolved signal and can be used for automatic feature extraction and fault

classification. This technique has potential for use in machine diagnostics.

ROLLING ELEMENT BEARING FAULTS DIAGNOSTICS USING THE BLIND DECONVOLUTION TECHNIQUE

ABSTRACT .................................................................................................. i ... TABLE OF CONTENTS ................................................................................. 111

LIST OF TABLES ........................................................................................ vi

. . LIST OF FIGURES ...................................................................................... vii

. . ACKNOWLEDGMENTS ............................................................................... xi1

... STATEMENT OF ORIGINAL AUTHORSHIP ................................................. xiii

TABLE OF CONTENT

CHAPTER 1: INTRODUCTION ........................................................................ 1 1.1 Research Problem and the Need for Fault Diagnostics ................................ 1

1.2 Motivation and Significance for This Research ........................................ 1

1.3 Existing Methods for Signal Enhancing ................................................. 2

1.4 Research Questions and Hypotheses ..................................................... 3

1.5 Aim and Objectives ........................................................................ 4

1.6 Research Method and Approach ......................................................... 4

1.7 Contribution of This Research ............................................................ 5

1.8 Organizational Overview of This Thesis ................................................. 5 . .

1.9 Author Publications ......................................................................... 7

CHAPTER 2: PRELIMINARY LITERATURE REVIEW ........................................ 8 . .

2.1 Condition Monitoring ....................................................................... 8

2.2 Time Domain Analysis ................................................................... 12

2.3 Frequency Domain Analysis ............................................................. 15

2.4 Signal Enhancing ......................................................................... 1 8

CHAPTER 3: ONTOLOGY: MACHINE FAULTS and CURRENT FAULT

DETECTION TECHNIQUES

3.1 Fault Occurrences ........................................................................ 25

3.1.1 Fatigue ............................................................................ 25

3.1.2 Lack of lubrication .............................................................. 26

............................................................................... 3.1.3 Wear 26

............................................ 3.1.4 Mechanism of Vibration generation 27 .................................................. 3.1.5 Bearing Defect Classifications 31

............................................ 3.2 Detection of Bearing Failure in Machines 31 ........................................................... 3.2.1 Maintenance Strategy 33

..................................................... 3.2.2 High Frequency Resonance 36 ..................................................... 3.2.3 Adaptive Noise Cancelling 37

............................................................... 3.2.4 Acoustic Emission 38

........................................................ 3.2.5 Time Domain Averaging 39

............................................................ 3.2.6 Blind Deconvolution 41

CHAPTER 4: BLIND DECONVOLUTION THEORY APPLIED TO BEARING

DIAGNOSTICS .................................................... 4.1 Applications of Blind Deconvolution 42

.......................................................... 4.2 Model of Blind Deconvolution 43

....................... 4.2.1 Known Probability Density Function of Input Signal 44

........................................................... 4.3 Higher-Order Statistics (HOS) 46

................................................. 4.4 Equalization Criterion and Algorithms 47

................................................... 4.4.1 Gradient Descent Algorithms 48

................................................ 4.4.2 Explicit HOS based Algorithms 50 ............................................................ 4.5 Maximum Kurtosis Criterion 51 .......................................................... 4.6 Eigenvector Algorithm (EVA) -52

4.7 Development of Theory in This Research: Determination of Equalizer ................................................................................... Parameters 54

...................................... 4.7.1 Redevelopment of Blind Deconvolution 54

...................... 4.7.2 Determining Optimum Filter Length of the Equalizer 57

... 4.7.3 Incorporating of Neural network technique with blind deconvolution 58

........................................................ CHAPTER 5: EXPERIMENTAL METHOD 61 ................................................... 5.1 Instrumentation for Data Acquisition 61

............................................................... 5.1 Experimental test rig 61 .................................................................... 5.2 Accelerometer 62

................................................................. 5.3 Charge Amplifier 63 . . . .................................................................. 5.4 Data Acqu~s~t~on 64

5.2 Benchmarking Blind Deconvolution through Computer Simulation Tests

............................................ 5.2.1 Simulation of Bearing Fault Signal 67 .................................................................... 5.2.2 Periodic Noise 68

............................................. 5.2.2.1 Time Interval Averaging 72

.................................................. 5.2.3 Summation of Periodic Noises 74

................................................. 5.2.4 Random Noise and Notch Filter 75 ......................................................... 5.2.5 Results and Discussions 78

5.3 Determining Blind Deconvolution Effectiveness and optimum Filter Length

5.3.1 Plan for bearing damage ......................................................... 79 ................................................... 5.3.2 Bearing Damaged Technique .80

........................................... 5.3.3 Measured Bearing Damage Signals -82 ....................................... 5.3.3.1 Outer Race Defect Experiments 82

...................................... 5.3.3.2 Inner Race Defect Experiments 84

.............................................. 5.3.3.3 Ball Defect Experiments 85

............................ 5.3.4 Fault Detection Using the Optimum Filter Length 87

5.3.5 Data Training Set for Optimization Based on a General Condition ...... 89

................... 5.3.6 Removing the High Resonance Frequency Components 92

.......................................................... 5.3.7 Results and Discussions 97

CHAPTER 6: EXPERIMENTAL METHOD: VALIDATION OF BLIND DECONVOLUTION THROUGH LIFE EXPERIMENTAL TESTS

................................................ 6.1 Bearing Test Rig for Life Time Testing 99 .................................................................. 6.2 Life Testing Procedure 10 1

.................................................................................... 6.3 Results -102

............................................................... . 6.3.1 Test bearing No 1 103

............................................ 6.3.1.1 Statistical feature analysis 103

.................................. 6.3.1.2 Bearing signal analysis using BD 111

.................................... 6.3.1.3 Description of damaged bearing 118

. 6.3.2 Test bearing No 2 ................................................................ 118

........................................... 6.3.2.1 Statistical feature analysis 118



................................................................ . 6.3.3 Test bearing No 3 128




................................................................ . 6.3.4 Test bearing No 4 132



................................................................. 6.4 Results and Discussions 136

.......................................... CHAPTER 7: CONCLUSIONS AND FUTURE WORK 138

7.1 Optimization of Equalizer Parameters Using Modified Crest Factor and ......................................................................... Arithmetic Mean 13 8

7.2 Simulation and Experimental Benchmarking of Blind Deconvolution ......... 139

7.3 Life Time Testing ........................................................................ 139

7.4 Future Work .............................................................................. 140

APPENDIX ............................................................................................... .IS%

LIST OF TABLES

Table 3.1 Bearing failure classification due to Wear and description ......................................... 27

Table 4.1 Estimated values for cumulants at zero-lag of random variables ................................ 47

Table 5.1 Average Frequency (Time inverse) of all Time Intervals ........................................... 72

Table 5.2 TSD of the observed and recovered signals ................................................................ 74

Table 5.3 Defect specifications of bearings ................................................................................. 79

Table 5.4 Different speeds of experiments .................................................................................. 79

Table 5.5 Data files for the simulation experiments .................................................................... 80

Table 5.6 Data training set for the neural network ...................................................................... 91

Table 6.1 Life Test Summary .................................................................................................... 103

Table 6.2 Deterministic characteristic defect frequencies ....................................................... 103

Table 6.3 Statistical features summary at each day - Test Bearing No . 1 ................................ 104

Table 6.4 Maximum magnitude of frequency components with and without ........................... 108

blind deconvolution (BD) over the last 40 minutes to failure for test bearing No . 1 ................ 108

Table 6.5 Kurtosis of the signal with and without .............. .. .................................................. 117


Table 6.6 Statistical features summary at each day -Test bearing No . 2 ................................... 119

Table 6.7 Maximum magnitude of frequency component with and without ............................ 121


Table 6.8 Kurtosis of the signal with and without ..................................................................... 127


Table 6.9 Statistical features summary -Test bearing No . 3 ...................................................... 129

Table 6.10 Statistical features summary -Test bearing No . 4 .................................................... 133

LIST OF FIGURES

Figure 3.1 Different mode shapes of bearing components .......................................................... 30

Figure 3.2 Basic dimensions of a general ball bearing with outer raceway waviness ................ 30

Figure 3.3 The time waveform due to a crack on the outer race of a rolling element bearing .... 32

Figure 3.4 The Weibull Curve for equipment reliability ................................................ 33

Figure 3.5 A schematic block diagram of the amplitude demodulation process ......................... 36

Figure 3.6 Adaptive noise cancelling principle ..................................................................... 37

Figure 3.7 Typical transient and continuous AE signals ............................................................. 38

Figure 3.8 AE features of transient signal ................................................................................... 39

Figure 4.1 Linear time-invariant system ...................................................................................... 42

Figure 4.2 Basic Diagram of Blind Deconvolution ..................................................................... 4.4 Figure 4.3 a) A random variable with Gaussian p.d.f. distribution [I661 ................................... 45

b) The output signal of an arbitrary channel c) to e) The recovered signal using an inverse filter

(equalizer) with different phases using second-order statistics ................................................... 45

Figure 4.4 a) A random spiky signal with Non-Gaussian p.d.f. distribution [I661 ..................... 45


(equalizer) with different phases using second-order statistics ................................................... 45

Figure 4.5 Cascade of transveral FIR blind equalizer length of L .............................................. 48

Figure 4.6 Block diagram of adaptive non-blind equalizer ......................................................... B Figure 4.7 Schematic illustration of the LMS algorithm ............................................... 49

Figure 4.8 Block diagram of adaptive blind equalizer updated with gradient descent

algorithm ..................................................................................................... 50

Figure 4.9 Block diagram of general HOS-based equalizer .......................................... 51

Figure 4.10 Schematic Block Diagram of Eigenvector Algorithm EVA .................................... 52

Figure 4.1 1 Schematic flowchart of A: The redeveloped algorithm of blind deconvolution for

optimization of specific conditions; B: Optimization of filter length for general conditions using

a neural network ............................................................................................. 56

Figure 4.12 An example architecture of a neural network .......................................................... 59

Figure 4.13 Schematic diagram of providing data training set .................................................... 60

Figure 5.1 Configuration of experimental set up ....................................................................... 6 1

Figure 5.2 Test Rig Assembly ................................................................................................... 6 1

Figure 5.3 Schematic drawing of a piezoelectronic accelerometer ............................................. 62

Figure 5.4 Charge per unit accelerometer (left), volt per unit accelerometer (right) ................. 63

Figure 5.5 Frequency response of the Bruel & Kjare 4332 accelerometer ................................. 63

Figure 5.6 Charge amplifier(left), Signal conditioning amplifier (right) .................................... 64

Figure 5.7 Second configuration of the simulation test rig ......................................................... 64

Fig 5.8 Connector NI (left). external KROHN-HITE filter (right) ............................................ 65

Figure 5.9 LabView Data Analysis Interface .............................................................................. 66

Figure 5.10 Redeveloped Blind Deconvolution Diagram .............................................. 67

Figure 5.1 1 Schematic of test rig ............................................................................................... 67

Figure 5.12 Koyo 6201 deep groove damaged ball bearing ........................................................ 67

Figure 5.13 Measured signals from different positions ............................................................... 68

Figure 5.14 Defective bearing signals recorded at different speeds (a) o=SOORPM;(b)

o =1000RPM; (c) o =1 5OORPM; (d) o =20000RPM .............................................................. 6 9

Figure 5.1 5 (a) Observed signal with 500 RPM (b) Corrupted signal with sinusoidal noise with

500 Hz and SNR=-8 dB (c) Recovered signal using blind deconvolution .................................. 69

Figure 5.16 (a) Observed signal with 500 RPM (b) Corrupted signal with sinusoidal noise with

500 Hz and SNR=-20 dB (c) Recovered signal using blind deconvolution ................................ 70


500 Hz and SNR=-43.9304 dB (c) Recovered signal using blind deconvolution ....................... 71

Figure 5.18 Observed signal before Blind Deconvolution (Left) and recovered signal after Blind

Deconvolution (Right) . A quarter of the signal is shown . Rotational frequency of the shaft

Figure 5.19 (a) Observed signal at 50 Hz; (b) Corrupted signal with combination of 2 sinusoidal

noise at 500, 1000 Hz; (c) Recovered signal after blind deconvolution ..................................... 74

Figure 5.20 (a) Observed signal at 500 Hz; (b) Corrupted signal with summation of 5 sinusoidal

noise frequencies at 25, 500, 1000, 2000 and 4000 Hz; (c) Recovered signal after BD ............ 75

Figure 5.21 Corrupting the Observed signal at 5OORPM with random noise and the result after

blind deconvolution ..................................................................................................................... 76

Figure 5.22 Gain response and phase plot of an equalizer for a sinusoidal noise with 500 Hz

frequency .................................................................................................................................. 7 7

Figure 5.23 Gain response and phase plot of an equalizer for a summation of sinusoidal noise

frequencies at 500, 1000, 2000, and 4000 Hz ............................................................................. 77

Figure 5.24 (a) Outer race defect with 0.1 mm width (left), (b) Inner race defect with 0.2 mm

width ............................................................................................................................................ 81

Figure 5.25 Ball defect which resembles a spot with width of 0.5 mm ...................................... 81

Figure 5.26 Cross section at view of ball bearing ....................................................................... 82

Figure 5.27 A typical observed signal for an outer race fault 0.1 mm rotating at (a) 600 RPM . 82

(b) 1200 RPM (c) 1800 RPM ...................................................................................................... 82


(b) 1200 RPM (c) 1800 RPM ...................................................................................................... 83


(b) 1200 RPM (c) 1800 RPM ................................................................................................ 83

Figure 5.30 A typical observed signal for an inner race fault 0.1 mm rotating at (a) 600 RPM . 84

(b) 1200 RPM (c) 1800 RPM .................................................................................................... 8 4

Figure 5.3 1 A typical observed signal for an inner race fault 0.2 mm rotating at (a) 600 RPM . 84

(b) 1200 RPM (c) 1800 RPM ...................................................................................................... 84

Figure 5.32 A typical observed signal for an inner race fault 0.5 mm rotating at (a) 600 RPM . 85

(b) 1200 RPM (c) 1800 RPM ...................................................................................................... 85

....... Figure 5.3 3 A typical observed signal for a ball fault with 0.1 mm in diameter rotating at 85

................................................................................ (a) 600 RPM (b) 1200 RPM (c) 1800 RPM 85

....... Figure 5.34 A typical observed signal for a ball fault with 0.2 mm in diameter rotating at 86

................................................................................ (a) 600 RPM (b) 1200 RPM (c) 1800 RPM 86

....... Figure 5.35 A typical observed signal for a ball fault with 0.5 mm in diameter rotating at 86

................................................................................ (a) 600 RPM (b) 1200 RPM (c) 1800 RPM 86

Figure 5.36 (a) Observed signal of an outer race defect at 600 RPM (b) Observed signal in

frequency domain sampled at 40 kHz (c) Modified Crest Factor graph versus filter length of the

equalizer (d) Arithmetic Mean graph versus filter length ........................................................... 87

Figure 5.37 Top- Observed signal with an outer race defect, Kurtosis=2.78, Bottom- Recovered

signal with the optimum filter length L=186, Kurtosis=9.06 ...................................................... 88

Figure 5.38 Demodulated recovered signal at 600 RPM with outer race defect ......................... 88

Figure 5.39 CF and AM plot for Outer Race Defect 0.1 mm width 600 RPM ........................... 89

Figure 5.40 CF and AM plot for Inner Race Defect 0.1 rnm width 600 RPM ............................ 90

Figure 5.41 CF and AM plot for Ball Defect 0.1 mm width 600 RPM ....................................... 91

Figure 5.42 (a) Observed signal at 600 RPM with an outer race defect (b) Corrupted signal with

sinusoid noise (c) Recovered signal after the blind deconvolution algorithm ............................ 93

.......... Figure 5.43 Gain response and phase response of the equalizer for 600 RPM outer race 93

Figure 5.44 (a) Observed Signal at 600 RPM Outer Race Defect (b) Spectrum of the observed

............. signal (c) Recovered signal with filter length 70 (d) Spectrum of the recovered signal 94

Figure 5.45 Gain response and phase response of the equalizer for processing of the observed

............................................................................. signal at 600 RPM with an Inner Race defect 95

Figure 5.46 (a) Observed Signal at 600 RPM with an Inner Race Defect (b) Spectrum of the

observed signal (c) Recovered signal with filter length 30 (d) Spectrum of the recovered

signal ......................................................................................................... 96


........................................................................................... signal at 600 RPM with a ball defect 96

Figure 5.48 (a) Observed Signal at 600 RPM with ball defect (b) Spectrum of the observed

............ signal (c) Recovered signal with filter length 98 (d) Spectrum of the recovered signal 97

Figure 6.1 Life Time Test Rig ................................................................................................... 100

Figure 6.2 Fixing of the bearing slave block to the loading table ............................................. 100

Figure 6.3 Test bearing housing with a mounted accelerometer ............................................. 101

Figure 6.4 Bearing 1 vibration signal features day l(first Day); (a) Temperature, (b) RMS, (c)

Peak to peak, (d) Kurtosis ......................................................................................................... 105

Figure 6.5 Bearing 1 vibration signal features day 4(Last Day); (a) Temperature, (b) RMS, (c)

Peak to peak, (d) kurtosis .......................................................................................................... 106

Figure 6.6 Temperature trend for test 1 until failure ................................................................. 106

Figure 6.7 Test bearing No. 1 vibration signal features day 4(Last Day), period of last 40

minutes up to failure; (a) Temperature, (b) RMS,(c) Peak to peak, (d) kurtosis ....................... 107

Figure 6.8(a) Observed Signal at 211 RPM at 1:25 PM ten minute before start point (b)

Spectrum of the observed signal (c) Recovered signal with filter length 32 (d) Spectrum of the

recovered signal ......................................................................................................................... 109


.................................................................................................. signal at 2 1 1 RPM at 1 :25 PM 1 10

Figure 6.10 Demodulated recovered signal at 21 1 RPM, 1:25 PM Band Passed Between (a)

1000 to 2500 Hz (b) Between 3000 to 3700 Hz (c) Between 3700 to 4500 Hz (d) Between 4700

to 5000 Hz ................................................................................................................................. 11 1

........ Figure 6.1 1 CF and AM plot for signal at 1 :40 PM with 21 1 RPM when crack initiated 1 12

Figure 6.12 (a) Observed Signal at 21 1 RPM at 1 :40 PM crack grows point (b) Spectrum of

the observed signal (c) Recovered signal with filter length 32 (d) Spectrum of the recovered

signal.. ........................................................................................................................................ 1 12


signal at 2 1 1 RPM at 1 :40 PM .................................................................................................. 1 14

Figure 6.14 Demodulated recovered signal rotating at 21 1 RPM at 1:40 PM when crack grows

.................................................................................... Band Passed Between 3700 to 4500 Hz 114



signal. .................................................................................................................................. 1 15


signal at 2 1 1 RPM at 1 : 50 PM .................................................................................................. 1 16

Figure 6.17 Demodulated recovered signal rotating at 21 1 RPM at 1:50 PM when the highest

peak point Band Passed Between 3700 to 4500 Hz .................................................................. 116

Figure 6.1 8 Test bearing No. 1 after failure (a)Bearing failure, inner race is split up (b) Fatigue

on outer race .............................................................................................................................. 1 18

Figure 6.19 Bearing test No. 2 vibration signal features day 3 (Last Day) ; (a) Temperature, (b)

........................................................................................... RMS,(c) Peak to peak, (d) Kurtosis 120

Figure 6.20 (a) Observed Signal at 21 1 RPM at 5 5 8 PM Start Point Test NO. 2 (b) Spectrum of


signal.. .................................................................................................................................... 122


.................................................................................................. signal at 21 1 RPM at 5:58 PM 122

Figure 6.22 Demodulated recovered signal at 21 1 RPM, 5:58 PM Band Passed Between (a)

1000 to 2500 Hz (b) Between 3000 to 3700 Hz (c) Between 3700 to 4500 Hz (d) Between 4700

to 5000 ESz .......................................................................................................................... 123

Figure 6.23 CF and AM plot for signal at 6:03 PM with 21 1 RPM when crack initiated ........ 124

Figure 6.24 (a) Observed Signal at 21 1 RPM at 6:03 PM middle point (b) Spectrum of the


signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . , , . . . . . . . . . . . . . . . . . . . .I25


signal at 2 1 1 RPM at 6:03 PM test bearing No. 2 .... . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Figure 6.26 Demodulated recovered signal rotating at 21 1 RPM at 6:03 Ph4 middle point test

bearing 2, Band Passed Between 3700 to 4500 Hz ................................................................ 127

Figure 6.27 Bearing 2 after failure ........................................................................................ 128

Figure 6.28 Bearing 3 vibration signal features, period of last 35 minutes up to failure; (a)

Temperature, (b) Kurtosis .................................................................................................. 129

Figure 6.29 (a) Observed Signal at 21 1 RPM at 9:38 AM (b) Spectrum of the observed signal

(c) Recovered signal with filter length 32 (d) Spectrum of the recovered signal ...................... 13 1

Figure 6.30 Demodulated recovered signal rotating at 21 1 RPM at 9:38 AM when craclc grows

Band Passed Between 3700 to 4500 Hz .................................................................................... 132

Figure 6.3 1 Test bearing No. 3 catastrophic failure ................................ ............................. 132

Figure 6.32 Bearing 4 vibration signal features over 35 Minutes; (a) Temperature, (b) RMS, (c)

Peak to peak, (d) Kurtosis ...................................................................................................... 133

Figure 6.33 (a) Observed Signal at 21 1 RPM at 1:30 PM test bearing 4 (b) Spectrum of the


signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . , . . . . . . . . . . . . . . . . . . . . . 1 3 5


peal< point Band Passed Between 3700 to 4500 tIz ................................................................ 136

OWLEDGEMENTS

This research work could not have been realised but for the support and encouragement of

many people. At the outset, I am very indebted to my Government of Islamic Republic of Iran

and Cooperative Research Centre for Integrated Engineering Asset Management for awarding

me a scholarship to study at Queensland University of Technology, Brisbane, Australia. I

convey my profound sense of gratitude to the Government and CIEAM.

I wish to express my appreciation to my Principal Supervisor, Associate Professor Andy Tan

for his pertinent guidance and encouragement during my candidature at QUT. I would also like

to thank my other Associate Supervisors, Professor Joseph Mathew and Dr Boshra Senadji for

their comments have been incorporated into this research work.

My personal thanks to Dr Fred Stapelberg for his help during the drafting phase of my thesis

and his valuable comments.

I thank my wife, Mrs Azam Haghgouyan, and express my gratitude to her for all her care and

encouragement during my studies. In the end, I thank my parents for providing me with this

care opportunity to purse a PhD program overseas and love to all those people out there for

helping me to achieve this objective.

I wish to thank Dr Sheng Zhang for his assistance

xii

STATEMENT OF ORIGINAL AUTHORSHIP

I declare that the work presented in this thesis is, to the best of my knowledge and belief, original, except as acknowledged in the text, and that the material therein has not been submitted, either in whole or in part, for a degree at this or any other university.

Signed: , ,

Chapter 1

Introduction

1.1 Research Problem and the Need for Fault Diagnostics

Condition monitoring of rolling element bearings for early detection of faults to prevent

catastrophic failure is important in industry. Vibration measurements are widely used for

detection of defects in bearings. Although the vibration signal may contain information of

defective parts, the main problem related to how a particular defect can be detected if the faulty

element is corrupted by noise during the transmission process. The transmission path which is

determined by the placement of a transducer as well as machine geometry relative to the fault

location will influence the final signal at the measurement point. Transmission path effects

(phase & amplitude) from the source of vibration to the transducer location are often unknown

or neglected. In practice, the vibration signal cannot always be measured directly at the source

of vibration. Vibration analysis is thus mainly concerned with the extraction of vibration data

from an observed signal.

In condition monitoring, the observed signal at a measurement point is often corrupted by

extraneous noise during the transmission process through the bearing housing. It is important to

detect incipient faults in advance before catastrophic failure occurs to enable precautionary

measures to be taken. Various signal processing techniques involving time, frequency and

statistical methods have been used to detect incipient faults. These techniques require a high

signal-to-noise-ratio (SNR), where the faulty component vibrations are higher than the

background noise. For a small defect the background noise is often higher and it is difficult to

detect an incipient fault. Hence, there is a need to minimize the noise component in the

observed signal in order to improve the SNR.

1.2 Motivation and Significance for This Research

Industrial machines are complex and have numerous components that could potentially fail.

There has been an increased interest in machine condition monitoring because of the potential

advantages to be gained from reduced maintenance costs, improved productivity and increased

plant availability. Today, the most fundamental issue of condition monitoring in most industrial

plant is in fault diagnostics and prognostics. One of the most effective approaches to investigate

this issue is condition monitoring based on vibration signal analysis. Hence, the motivation of

this research is to investigate further use of vibration analysis in solving efficiently some key

problems in condition monitoring. In order to avoid failure and to maintain product quality in a

highly automated factory, it is essential to monitor equipment condition continuously. This

research is motivated toward this goal.

The significance of this research is the development of an advanced signal processing technique

to recover the source signal corrupted during the transmission process, where detection of a

small failure is made difficult due to the heavy background noise. Current techniques are

inefficient to detect defective bearing signals in the presence of extraneous noise. The blind

deconvolution technique enhances the detectability of faults in rolling element bearings.

Blind deconvolution was the selected technique to be investigated in this research because the

technique does not need additional measurements such as applications in the adaptive noise

cancellation technique and multi-sensors in the blind source separation method. Blind

deconvolution thus has the potential to minimize the cost of multi sensors. Furthermore, in

some applications the contributions of sources are dependant. In this regard, the blind source

separation method may fail. Since the blind equalizer acts as a filter, if the knowledge of

background noise is known an analogue filter can be used to filter out the background noise.

However, blind deconvolution has the ability to remove periodic noise without prior knowledge

about noise characteristics.

1.3 Existing Methods for Signal Enhancing

There are several methods currently available for improving the signal to noise ratio. The first

method is based on the principle of coherent filtering and relies on a time domain averaging

signal [2]. The technique decomposes the observed signal into a periodic signal plus additive

noise. The periodic repetitive part of the signal is extracted by coherent averaging. In [3],

experimental examples were given for signals obtained from a combustion engine to detect

defects in the gear box using time averaging. In [4], a new application of time averaging to

machinery health monitoring was proposed. In [5, 6], an alternative method was used, namely

Adaptive Noise Cancelling (ANC), to suppress background noise of a diagnostic signal from a

complex machine. Its application has been successfully applied to acoustic signals in

identifying faults in a rotary machine [7 ] , speech signals [8], electrocardiography [9] and

adaptive antenna arrays [10]. In [l 1], the power cepstrum method was presented as an effective

diagnostic technique for rolling element bearings. In [12] bearing defect detection using

rotational frequencies was observed.

The problem of multi-path transmission has been a challenge in telecommunications. For

example mobile phone signals are subject to sever distortion, due to the reflection and

diffraction of the radio wave carrier. To combat the effects of multi-path environment the

communication channel must be estimated using an equalizer that reveals the effect of the

distortion. The received signal is processed to estimate the unknown channel and the

transmitted signal known as the desired signal. Adjusting adaptive filter taps to cancel

reverberation of the speech signals in a noisy environment is a blind deconvolution problem.

Another application of blind deconvolution is reflection seismology. Seismic exploration is

performed by generating an acoustic wave field that is reflected by geological layers with

different impedance. Blind deconvolution is used to remove the source waveform and other

undesirable influences from the seismogram. Yet another application is found in image

processing where blind deconvolution is used for the purpose of de-blurring and image

restoration, for example in astronomical imaging.

1.4 Research Questions and Hypotheses

From the identified research problem and the preliminary literature review, the following

questions can be formulated as input to this research.

Is it possible to minimize the background noise in the observed vibration signal of an

impending failure bearing and improve the signal-to-noise-ratio?

Would it be possible to detect an incipient fault in advance for all known types of

faults in rolling element bearings before a catastrophic failure occurs?

Is it possible to employ an advanced signal processing technique which requires

only one measurement and does not need to be trained by a sequence of data?

Would it be possible to apply such a technique in a real life application to detect a

bearing fault in the early stages of damage?

These research questions need to be addressed in selected set hypotheses which can be tested

through the appropriate research methodology:

1. Blind deconvolution has inherent abilities and possible limitations to enhance the

observed vibration signal of incipient faults in rolling element bearings. This

hypothesis will be tested through benchmarking the vibration signal corrupted by

simulated periodic and random noises.

2. The filter length of the blind equalizer in the blind deconvolution algorithm can be

optimized for general vibration monitoring applications. This hypothesis will be

tested through simulation of the type of fault, speed, and size of fault in bearing

components

3. Blind deconvolution is able to enhance the vibration signal of incipient faults of

rolling element bearings in real life-time applications. This hypothesis will be

trialled through life-time testing of a rolling element bearing running to failure,

whereby the enhancement characteristic of the blind deconvolution technique can be

validated.

1.5 Aim and Objectives

This research focuses on the development of a reliable method for enhancing of the observed

signal corrupted by noise through the transmission path. The aim of this project is to redevelop

and modify an iterative method of blind deconvolution based on the Generalized Eigenvector

Algorithm (EVA) developed by Jelonnek and Kamrneyer [13] for recovering the source

vibration signal. The main objectives of this project are as follow:

1. To redevelop the blind deconvolution algorithm and to determine the critical input EVA,

filter length and iteration number, suitable in rolling element bearing fault detection.

2. To incorporate a neural-network with blind deconvolution and link the output of blind

deconvolution to the input of a neural-network model for optimization of filter length of

the equalizer.

3. To automatically determine the optimum filter length and iteration number, thus

eliminating current trail-and-error methods.

4. To show that the blind deconvolution technique is an affective means for enhancing the

observed vibration signal in real life testing of the rolling element bearings.

1.6 Research Method and Approach

In order to address the various research hypotheses, the following quantitative research method

has been adopted:

1. Theory redevelopment of the blind deconvolution technique through incorporating a

neural network to optimize a general application of the technique as indicated in Section

4.7

2. Applying the redeveloped technique to detect bearing fault signals corrupted by noise

during the transmission process through various experimental methods including the

following:

a) A series of simulation tests with artificially damaged bearings for

benchmarking as indicated in Sections 5.2 and 5.3

b) Validation of experimental results using a life-time test rig with a healthy

bearing running to failure, as indicated in Chapter 6

1.7 Contribution of This Research

The main contribution of this research is the ability to effectively enhance incipient fault signals

of rolling element bearings through the blind deconvolution technique. Several other significant

contributions of the use of the blind deconvolution technique are as follows:

1. The ability to remove high frequency components around the excited resonant frequency

for outer race and inner race defects of rolling element bearings.

2. The effective use of crest factor as a parameter to indicate how much impacting occurs

in the recovered signal after blind deconvolution, which serves as a good criterion for

optimizing the filter length.

3. The capability to optimize the filter length of the blind equalizer for specific condition

of incipient fault in rolling element bearings using modified crest factor and estimating

the recovered signal with higher detectability.

4. The use of blind deconvolution as a signal enhancing technique for life-time testing.

1.8 Organizational Overview of This Thesis

Much of the detail obtained from the various experimental methods would normally be included

in an Appendix. However in order to substantiate the results particular to each experimental

method, inclusion of the detail in the text of the chapters describing each method was preferred.

Based on the aforementioned objectives in section 1.5, the thesis is structured as follows:

Chapter 1 explains the research problem and motivation behind this research, as well as the

research questions and hypotheses to be adopted through the appropriate research method. It

also describes the main objectives and contributions of this research and outlines an overview of

the dissertation.

Chapter 2 outlines the preliminary literature review of fault detection techniques, particularly

in the time and frequency domains. It explains the concept of energy of the vibration signal

which can be used for fault diagnosis. The applications of time and frequency domain

approaches in condition monitoring are also reviewed in this chapter. Literature on current

signal enhancing techniques is also reviewed especially the Power Cesptrum Technique and

Noise Cancellation Methods. The Adaptive Line Enhancer and Autoregressive Modelling

techniques are reviewed.

Chapter 3 considers some common machine faults and how they occur, including the

mechanism of vibration generation in rolling element bearings. The chapter also describes what

current techniques are applied in fault diagnostics. Instrument and software technology used in

routine monitoring and detection of machinery faults prior to failure is presented. High

frequency resonant techniques, adaptive noise cancelling methods and acoustic emission of

transient elastic wave generation in materials under stress are addressed. Furthermore, energy

based techniques such as RMS level which introduces time interval averaging to enhance the

SNR of the observed signal, is described in detail. An explanation of the application of the blind

deconvolution technique to recover the source of vibration is in rolling element bearing is also

addressed.

Chapter 4 proposes blind deconvolution as a signal processing technique to enhance the

monitored (observed) signal and to reconstruct the desired signal. Such a process is often

modelled as an input signal convolved with a filter with an unknown impulse response. In this

chapter the theory of blind deconvolution is discussed. Two forms of blind deconvolution,

considering the observed signal and the method of extraction, are explained. The eigenvector

approach (EVA) by Kammyar and Jelonnek [13] is chosen for its simplicity and ease. This

algorithm uses cross kurtosis of two higher order cumulants. Determination of equalizer

parameters is reviewed in Section 4.7. The redevelopment of the blind deconvolution technique

is presented in Section 4.7.1, and in Section 4.7.2 it is explained how filter length of the

equalizer can be optimized for bearing signals. A neural network is incorporated with blind

deconvolution and this is presented in Section 4.7.3. This chapter addresses objectives 1 and 2

given in Section 1.5.

Chapter 5 addresses the set objectives 3 of Section 1.5. Section 5.1 describes the experimental

method and data acquisition procedures. Different test apparatuses to simulate common

machine faults were used in obtaining the diagnostics signals. The first test rig incorporated a

damaged bearing and a coupling disk system to create shaft misalignment together with gear

meshing. Section 5.2 describes the benchmarking of blind deconvolution through computer

simulation tests to recover the original bearing signal corrupted by various types of noises. The

efficiency and robustness of blind deconvolution was assessed by corrupting the observed

signal with different simulated noise. Section 5.2 focuses on the use of blind deconvolution as a

notch filter to filter out noise components without any prior knowledge about the noise. In

section 5.3 a plan is designed to damage different components of a bearing for the simulation

study. In Section 5.3 a neural-network was designed to train input parameters of general

conditions for diagnostic signals. Training data sets for the neural network for different inputs is

provided using different types of faults, sizes and speeds which is explained in Section 5.3.1.

Fault detection using blind deconvolution is presented in Section 5.3.4. Some results such as

removing the high resonant frequency is presented in Section 5.3.6. Important notes and

discussions are presented in Section 5.3.7.

Chapter 6 presents the validation of blind deconvolution through life experimental tests. This

chapter addresses objective 4 of Section 1.5. A bearing test rig with a healthy bearing running to

failure is presented in Section 6.1. Life testing procedures is explained in Section 6.2. Four

different scenarios for life time testing are reviewed in Section 6.3. For every scenario the

Chapter 2

Preliminary Literature Review

2.1 Condition Monitoring

Numerous methods have been developed to monitor the condition of machines. Machine

condition monitoring has been performed by knowledgeable experts using sight and sound.

Although a variety of approaches may be used in condition monitoring applications the

vibration monitoring and analysis is the most widely used technique. Vibration monitoring is a

routine collection of vibration data on specific machines. The intent is to capture the pertinent

vibration frequencies and amplitudes in order to create a trend for each machine over an

extended period of usage. A base-line level need to be established when the machine is in good

health. If the level of vibration exceeds the base-line value, it is an indication of that the

machine has deteriorated and requires attention. Triggering an alarm does not identify the fault,

but merely warns an indication of a change is being occurred that require further attention [14].

There has been an increased interest in machine condition monitoring because of the potential

advantages to be gained from reduced maintenance costs, improved productivity and increased

plant availability [14]. The number of commercially available instruments to monitor the

machines has increased. The more sophisticated of these are the microprocessor based

instrumentation and allow continuous monitoring of items of equipments as well as orderly

keeping the records. The most fundamental application of condition monitoring in industrial

plants is fault diagnostics and prognostics [15-19]. There are only a few basic condition

monitoring approaches which are serious contenders for application in industry. The first of

these depends on the analysis of wear particles in lubrication oils. It was shown that ferrography

is particularly attractive in this category although other techniques such as the use of magnetic

plugs, spectrographic oil analysis and scanning electron microscopes have merit [20]. The value

of wear particle analysis is acknowledged although it is not considered here.

Several studies have been conducted to investigate the Acoustic Emission (AE) response of

defective bearings. AE is the phenomenon of transient elastic wave generation in materials

under stress. In [21] the application of acoustic emission as a measure of the condition of the

low speed antifriction bearing of was suggested. In [22, 23] it was shown that AE parameters,

such as ringdown counts, event and peak amplitude of the signal, can detect defects before they

appear in the vibration acceleration range and can also detect the possible source of AE

generation during a fatigue life test of thrust loaded ball bearings. In [24, 25] a system is

proposed to locate the source of vibration, was later improved by introducing two AE sensors

into the system for measuring the differences of arrival times of acoustic emission signal at the

sensors. Some researchers have studied the AE parameters with simulated local defects on

various bearing elements. Acoustic emission signals have been shown to detect defects in the

form of a fine scratch on the inner race of an axially loaded angular contact ball bearing at low

speeds [26, 27]. In [28] it was demonstrated that acoustic emission parameters, such as peak

amplitude and count, for detection of defects in radially loaded ball bearings at low and normal

speeds. In [29] it was suggested that measurement of the area under the amplitude time curve is

a preferred method for detection of defects in rolling element bearings. Distribution of events

by counts and peak amplitude have also been used for quality inspection of bearings [30]. In

[3 1] the inter-relationship among the AE parameters for various size of defect was studied. The

statistical distributions of event versus ringdown counts and peak amplitude of the AE signal

under different load and speeds were presented.

Vibration analysis is the art of using vibration information (waveform, spectral, phase, etc.) to

aid in the diagnosis of machinery. When the machine can not be taken out of service for close

inspection, the efforts of diagnosing the machine condition can be quite challenging. The efforts

require an understanding of the machinery behaviour operating conditions, signal conditioning,

and diagnostic techniques. In this analysis, the health of the machine based on collected data

should be analysed. This allows the changes within the machine to be determined precisely and

appropriate corrective action can be initiated. Although there are several methods of condition

monitoring, vibration analysis was chosen for several reasons. First, it is easy to implemented

and reliable [32]. Second, different defects produce different vibration patterns; and can relate

to a specific bearing defect [33]. Finally vibration monitoring is relatively inexpensive. The

assessment of machine's condition varies from machine to machine. The successful operation

of a machine relies on the performance of each machine component, and requires a detailed

understanding of the behaviour of these elements and their interaction. Bearing failure is one of

the foremost causes of breakdown in rotating machinery. Such failures can be catastrophic and

can result in costly downtime. Therefore, bearing monitoring introduces important challenges to

machine maintenance.

In [34] it was reported that a bearing in good condition tends to have a large amplitude ratio of

low to high frequency signals. A defect in a machine such as spalling on a bearing race or a

propagating crack can cause the amplitude ratio of low to high frequency signals to decrease

significantly. Therefore, in the earlier stages of bearing deterioration, the high frequency

information tends to provide better indication of the bearing condition. The abrupt changes in

the contact stresses at the interface between the rolling elements and the races when a rolling

element passes over a local defect generate impulsive forces at the defect. This impulsive force

produces vibrations which can be monitored to detect the presence of a defect in the bearings.

Impulses or impacts may produce oscillating forces and can damage the components of the

machine which could lead to catastrophic failure. The amplitudes of impacts are dependent on

the size of the faults and impact rates.

In [35] it was reported that defects in the rolling element bearings cause an increase in vibration

levels in the high frequency range of the spectrum. This is because the natural frequencies of

the bearing elements and the housing structure are excited by the impacts caused by fault in the

rolling elements. Vibration signals of defective bearings usually lie in the lower frequency

range, usually up to 500 Hz. However, their resonance frequency can be in the medium to high

frequency range (around 10 kHz). Monitoring the increase in vibration levels at these

frequencies which are usually higher than 5 kHz, can be used to detect the defects in the rolling

element bearings. The type of bearing loading such as axially loaded bearings and with effect of

varying defect size may affect the high frequency components.

The defect on a particular bearing element is expected to result in higher vibration level at the

element rotational frequency in the low frequency range of vibration spectrum. These

characteristic defect frequencies of rolling element bearings can be theoretically calculated [36].

The harmonics of these frequencies are also be present in the vibration spectrum. It was

observed that an inner race defect frequency may have sidebands at the shaft rotational

frequency due to the modulation of defect frequency at the resonant frequency. It was reported

that bearing defect detection can be achieved by using these rotational frequencies in 137] and

[38]. However the measurement results in [39] show that it was not possible to detect the defect

because of the absence of significant peak at the defect frequency in the spectrum in the early

stage of damage.

In [32] it was mentioned that it is difficult to detect a defect at bearing characteristic rotational

frequencies in direct spectrum of a faulty bearing signal because the fault is very small. An

important requirement for effective condition monitoring is the ability to detect an incipient

fault. At this stage the source signal is generally small and its detection would require the noise

component in the overall vibration signal to be severely attenuated with a consequential

improvement in the signal-to-noise-ratio (SNR). Detecting an incipient failure in the rolling

element bearings is difficult due to the low SNR. In situations where the vibration is

contaminated by either background noise or unwanted components, normal fault detection

techniques may fail to detect a growing defect at an early stage, due to the relatively low SNR.

Incipient failures do not produce large oscillating forces because the compliance is not varied

much [40, 41]. However, these forces can excite high frequency vibration at a known defect

frequency but attenuate relatively quickly with distance from the defect source. Incipient defect

frequency usually shows up in the high frequency area, normally above 20 kHz and can

occasionally extend to 400 kHz.

The high frequency resonance technique (HFRT) also known as demodulated resonance

analysis or Envelope Detector (ED) has been used for fault detection. This technique is the best

suited to vibration monitoring of gearboxes and turbo machinery due to its ability to separate

the vibration generated by a defective bearing and other machine cpmponents [42]. In the

HFRT the vibration signal of the defective bearing is processed to extract the characteristic

defect frequencies which may not be identifiable in the direct spectrum especially in the

presence of vibration from gears and other machine elements. In chapter 3 this technique is

explained in details.

Since the impact vibration generated by the collision of rotating elements, a bearing fault has

relatively low energy, and is overwhelmed by noise with higher energy and the vibration

generated from other macrostructural components. Therefore it is difficult to identify a small

bearing fault in the spectra at Bearing Characteristic Frequencies (BCF) using the conventional

Fast Fourier Transform (FFT). In order to improve detectability of a fault, the Envelope

Detection (ED) technique has been used in conjunction with FFT [43]. Although various

techniques have been proposed to use ED in bearing fault diagnosis, the most common practice

is to apply ED at the bearing resonance frequency located in the high frequency range [44].

Since the range of bearing resonance has to be known before ED method can be applied, several

runs of impact tests are needed to determine the bearing resonance frequency. The structural

resonances excited by the impacts due to a defect in the bearing elements are amplitude-

modulated at the characteristic defect frequency corresponding to the location of the defect on

the bearing elements. In [45] it was reported that a signal indicative of the bearing condition can

be recovered by demodulating one of these resonances. The spectrum of the enveloped signal

can be obtained to determine the characteristic defect frequency of the bearing element.

The interaction between a local defect on a bearing element and its mating components

produces abrupt changes at constant stresses in the interface and generates a pulse with a very

short duration. This pulse produces vibration which can be monitored to detect the presence of a

defect in the bearing. It was reported in [46] that there are two approaches to investigate the

bearing failure mechanism. The first step is to mount a healthy bearing on a rotating shaft in a

working machine and then run the bearing until failure. The vibration signals were monitored

and plotted against time [46, 47]. Any sudden change in the waveform was reported as an

indication of a possible defect [48]. The failure is accelerated by either overloading, over

speeding or starving the bearings of lubricants [49, 50]. The second approach is to seed or

introduce defects in a healthy bearing component intentionally by some techniques such as acid

etching, spark erosion, scratching, mechanical indentation or laser techniques. The vibration

response can be measured and compared to the result of a good bearing [33, 39]. In some of

these studies the size of simulated defects has been quantified and varied [51]. The former

approach of life tests is time-consuming. On the other hand, testing of bearings with simulated

defects is much quicker but preparation of the defective bearing requires special tools and this

procedure does not represent real life failure.

2.2 Time domain analysis

The Measurement of signal energy can be a good indicator of a bearing's health. The overall

root-mean-square (RMS) of a signal is a representative of the energy. This method has been

applied with limited success for the detection of localized defects [52]. However it is expected

that high value of RMS corresponds to an overall deterioration of the machine. However in

some cases this criterion had limited success [53]. The crest factor is a modified quantity of

RMS and is defined as a ratio of the maximum peak of the signal to its RMS value. The value

of the crest factor can be regarded as a feature for condition monitoring or fault diagnosis. In

[53], the overall RMS acceleration level and the frequency spectrum of the vibration signal in

the 0-25 kHz range for the good and the defective bearings were obtained. It was found that

there were two major peaks related to the resonant frequencies in the spectrum. The vibration of

the overall acceleration levels for different loads, speeds and defect sizes were plotted. It was

shown in the plots that RMS level measurements with defects are always above than without

defects and the defect can be detected effectively. It was found that defect is detected best in the

outer race followed by in the inner race and the ball in that order. Probably the detection of a

defect in the outer race is easiest, because (i) the vibration transmission path to the transducer is

the shortest compared with other cases; (ii) the vibration in the case of inner race defect may be

damped when transmitted through the balls; the outer race defect always remains in the zone of

maximum load whereas the inner race or ball defect move in and out of this zone during

rotation; (iii) the ball defect is the most difficult to detect because the defect may not be in touch

with either of the races for some time, as it is free to spin in any direction. The effectiveness of

these parameters has been investigated in terms of size of defect [54].

In [54] it was shown that higher levels for outer race defect larger than about 75 can be

detected by RMS and power measurements. The vibration parameters have been compared in

the form of defect detectability, which has been defined as the ratio of the level of the defective

bearing to the maximum level of the good bearings. The detectability plots for the outer race

and inner race indicated that the defect detectability of overall power is the best, followed by

RMS measurements. In [55] it was shown that crest factor can be used as an alternative

measurement instead of RMS level of vibration. It was found that crest factor can be used in

fault detection of rolling element bearing with limited access. With increasing bearing damage

it has been observed that RMS value increases [56]. Bearing rolling frequencies and associated

changes in their amplitude with particular defects have been identified by using RMS parameter

[12, 57]. In [58] it was found that changes in peak level over the frequency ranges up to 10 kHz

is a good indication of incipient damage using crest factor. It was found that the results were

partially insensitive to changes in bearing load and speed. In [32] similarly it was acknowledged

that incipient damage is hard to detect by observing changes in peak acceleration. Bearing

condition was assessed by comparison of peak counts for the measured signal with a Gaussian

amplitude distribution in [32]. The measured values of RMS and peak are dependent upon

bearing load, speed, housing tightness, quantity of lubricant and bearing clearance. It is

therefore difficult to define the condition of a bearing from only RMS or peak value

measurement, except in instances where substantial background information is available. In

[58] both crest factor and the statistical peak counting method in [32] permit a more direct

assessment of condition with minimal resource to previous history, as they are less sensitive to

operational changes.

In [48] ball bearings testing which involves a run-to-failure condition, it was observed that once

a crack appeared in the bearing component, the propagation occurred rapidly and the RMS level

increased significantly. The subsequent drop in the RMS level was attributed to a phenomenon

known as "healing". The term "healing" was defined as smoothing of the sharp edges of a crack

or small damages zone by continued rolling contact. In [48] it was reported that as the damage

spread over a broader area, the signal RMS showed an increase again. In the run-to-failure

experiment an acoustic emission sensor was used and the AE signal exhibited a single increase,

beginning about 10 minutes after the increase in accelerometer RMS. It was found that the

results in crest factor parameter were similar to the RMS level. The results revealed a

decreasing trend in crest factor versus speed for the accelerometer signal, and the reverse trend

for the AE signal.

In time domain analysis, statistical parameters are normally used for fault detection. Treating

the monitored signal as a random variable, statistical parameters such as probability density

function and the moments of data are often used. In [59] it was reported that the amplitude

characteristics of a vibration signal which is assumed to be a stationary random process and can

be expressed in terms of an instantaneous probability density function. In [59], it was shown

that in the early stage of the test, when the bearing is undamaged, the density function is an

inverted parabola which is an indicative of a Normal/Gaussian distribution. With an incipient

damage in bearing component changes can occur in the tails of the distribution curve. With

increasing time and advancing damage the tail of distribution curve is initially broadens. This

characteristic can be enhanced further by taking the integral of the probability density curve. It

was shown that observing changes in the probability at particular amplitude levels provides

significant information to fault diagnosis. In [55] near-Gaussian distribution for some damage

bearing was obtained.

It was reported in [60] that a series of statistical moments can be used to indicate the shape of

the probability density distribution. The first and the second moments are the mean value and

the variance respectively. Odd moments, r =I, 3, 5,. . . etc. where r is the order of moment,

relate the information about the position of the peak density relative to the median value. Even

moments, r =2, 4, 6, . . . etc., indicate the spread in distribution [60]. Among these moments, all

the odd moments close to zero, where indicative of a symmetrical acceleration amplitude

distribution, whereas higher even moments are sensitive to the impulsive signal associated with

the bearing damage [33].

The fourth moment, normalized with respect to the fourth power of standard deviation is quite

useful in fault diagnosis. This quantity is called kurtosis. Kurtosis is a compromise measure

between the insensitive lower moments and the over-sensitive higher moments. It was reported

that the kurtosis can be a good criterion to distinguish between a damaged and a healthy bearing

[60]. It was reported in [48] that a healthy bearing with Gaussian distribution will have a

kurtosis value about 3. When the bearing deteriorates this value goes up to indicate a damaged

condition. The value reduces again when the defect is well advanced. Therefore, this is most

effective in identifying impending failure, when the kurtosis significantly exceeds a value of 3.

In [33] it was suggested to apply kurtosis values at multiple frequencies in order to understand

the sensitivity of kurtosis to the bearing failure. One of the advantages of this method is that

there is no need to know time history of the signal and the bearing condition can be monitored

directly by observing the kurtosis. Although some researchers have shown the effectiveness of

kurtosis in bearing fault detection, this method was not able to detect an incipient damage

effectively [55]. The effectiveness of kurtosis versus shaft speed has been considered [60]. It was observed that the trend of statistical parameters such as kurtosis is diverted in a particular

speed range. However in an ideal situation the statistical variable such as kurtosis should not

change very much with the change in shaft speed. A detailed study showed one theoretical

explanation for the observed discrepancy [60]. In [61] the effectiveness of this method under a

simulated condition was studied. Several other studies [61-65] have also shown the

effectiveness of kurtosis in bearing defect detection. In [32, 55, 66] the method could not detect

an incipient damage effectively.

The shock pulse method is another approach for bearing fault detection in the time domain [67].

The principle of the technique is based on the structural resonance which is excited at high

frequency by the impulsive loading on the faults. Special transducers (piezoelectric) with the

resonant frequency tuned around the structural bearing resonant frequency (centred at 32 kHz

normally) are used. When the shock pulses are produced in the bearing, vibration including the

shock pulses is transmitted to the transducer and resulted in damped oscillations at its resonant

frequency. In a transient state, the maximum value of the damped vibration indicates the

condition of the bearing. As a threshold, initial value of a transient damped vibration for a

healthy bearing is measured first and then it is subtracted from the shock value of test bearing to

obtain a net shock pulse value. The final value corresponds to the condition of the bearing can

be used to classify the condition of the bearings. In [55] it was reported that shock pulse meters

are simple to use and a semi-skilled personnel can operate them. The shock pulse meter gives a

single value indicating the condition of the bearing, without the need for elaborate data

interpretation as required in some other methods. This technique has attracted wide industrial

attention [55, 64, 66, 68, 69]. In [27] and [70] it was reported that the shock pulse method could

not effectively detect defects at low speeds. However, in [68] it was reported that the shock

pulse method is effective in the detection of defects in low-speed spherical roller bearings in a

paper production line. An on-line bearing condition monitoring technique based on the shock

pulse method has been reported in [71].

2.3 Frequency Domain Analysis

Spectral analysis of vibration signal is widely used in bearing diagnostics. It was found that

frequency domain methods are generally more sensitive and reliable than time domain methods.

The advent of modern Fast Fourier Transform (FFT) analysers has made the job of obtaining

narrowband spectra easier and more efficient. The extent of changes in the spectrum is heavily

related to the nature of the faults. The source of some spikes can not be explained in the

spectrum due to the some micro-structural components in the machine; however in most cases

the peaks can be identified [36]. Both the low- and high-frequency ranges of the vibration

spectrum are of interest in assessing the condition of the bearing. In [72] it was demonstrated

that the spectrum of the monitored signal changes when faults occur. Exciting the resonant

frequency of the bearing by an impulsive loading can be the basis of a new approach in fault

detection [42]. It was mentioned that resonant frequencies normally occur above 5 kHz. The

resonant frequencies can be calculated theoretically based on the physical interactions between

bearing components [73-75]. It was indicated in [76] that it is difficult to estimate dimensions of

these resonances and are affected when assembled into a full bearing and mounting in housing.

However, it was mentioned that resonances are not altered significantly. In [37, 50, 55, 76] it was reported that monitoring the increase in the level of vibrations in the high-frequency range

of the spectrum is an effective method of predicting the condition of rolling element bearings

and has been used successfully.

In [77] a bearing mathematical model incorporating: the effect of the bearing geometry, shaft

speed, bearing load distribution, types of loads (both radial and axial), the shape of the

generated pulses, transfer function of the path and the exponential decay of vibration due to the

damping property of the bearing was designed. It was reported that the severity, extent and age

of damage can be better represented by pulses [38,61]. Thus, the shapes of pulse can influence

the amplitude response. The model, which was presented in this work [77], predicted a

frequency spectrum having peaks at characteristic defect frequencies in the both cases of

defective inner race and defective ball. A comparison between the analytical values of velocity

amplitudes and the experimental values obtained from reference [53] was conducted [77]. It

was shown that there was a fair agreement between analytical and experimental values. The

amplitude at these frequencies was also predicted by this model for the various defect locations.

The amplitude for the outer race defect was found to be high in comparison to those for the

inner race defect and rolling element defect. In [78] an alternative model was presented for the

vibration produced by a single point defect in a rolling element bearing. A demodulation

vibration spectrum was performed to calculate envelope function. It was shown that this model

can predict the frequencies and relative amplitudes of the components of the spectrum correctly.

Basically, three dimensional plots of spectra, i.e. amplitude in terms of frequency and time can

be used for fault detection. The time axis is important for condition monitoring. Although these

three dimensional plots contain valuable information, it was difficult to obtain an exact

indicator to determine the healthy or the faulty bearing. In [72] a number of discriminating

features is extracted from the spectrum to determine the change in the spectrum. These

extracted features are then compared with the references to detect a fault.

There are two methods to look at the discriminating features. The first method is based on a

single parameter by comparing narrow band spectra. Several parameters such as arithmetic

mean, geometric mean and correlation have been suggested [72, 78] to quantify the differences

in spectra for good bearings and damaged one. The value of Arithmetic mean, Amn, or

Geometric mean, Gmn, or correlation is related to the amplitude of the monitored signal in the

frequency domain. The other option is to look at both spectrums of the test bearing (damaged)

with reference to an undamaged bearing. A single parameter such as matched filter root mean

square or simply Mfrms was proposed [72]. To evaluate the performance of these single

parameters, time graphs of these parameters were plotted and it was found that in the damaged

bearing spectrum when a fault occurs there is a sudden change in the trend of the graph

announcing an alarm. Based on the amount of change in the plot, the necessary timing of any

action can be deduced [72].

The second method based on the frequency approach is based on some specific frequencies

depending on the kind of faults. These features may include defect frequency, principle and

harmonics of the rotational speed, line frequency and slip frequency. In these features, a

prominent spike in a narrow range of the nominated frequency with an increase in the energy

indicates that a failure is imminent. Basic characteristic defect frequencies might be employed

for bearing fault detection based on the knowledge of the bearing such as geometry of the

bearing and its rotational speed. In [75] these frequencies were calculated to represent the

number of impacts per unit time. In [38] it was mentioned that the frequency generated at the

outer race is roughly equal to 40 percent of the number of balls times the revolution per second

of the shaft (RPS); for the inner race, the defect frequency approximates 60 percent of the

product of the number of balls and RPS of the shaft. For a bearing with a stationary outer race,

the characteristic defect frequency equations were presented in [35, 37, 38, 791. In [27, 35, 38,

43, 79] it was reported that matching the theoretical frequency and the obtained spike in the

spectrum can be a good basis for fault detection. In [80] it was found that the computed

frequency does not always equal to the spectral frequency. The reason could be due to the some

slippage or skidding occurs in the rolling element bearings.

In [79] it was mentioned that defective raceways can be identified by a narrow band spike at the

mentioned frequencies of the race on which the defect exists. It was observed that a single

spectral spike was generated at frequency domain, i.e., ball pass frequency output (BPFO) when

the defect was not large enough. As the size of defect increased and it became larger, the ball

pass frequency modulated with the speed of rotating unit. At this point the modulation

generated narrow side band spikes at the BPFO plus or minus RPS [38]. The difference

frequency between the BPFO and the side lope was almost equal to the speed of rotating unit.

As the defect increased in size more side lobes were generated and spaced at RPS, consequently

the spectrum was a series of spikes; sometimes the BPFO disappeared. Similarly, in [35] it was

found that this phenomenon seems to occur when the defect length is greater than the length

requires generating one or two ball pass frequencies, and applies to both radial and axial loads.

In [37] it was reported that defects on the inner race tend to behave in a similar manner to the

outer race except when the amplitude of the spectrum of an inner race defect is less than that of

the outer race for a given size defect. Two reasons were suggested; (i) a defect on the inner race

is in the load zone only once per revolution; (ii) the signal is transmitted through a structural

interface path to the transducer position. It was observed that in the case of a defect on a

moving element such as the inner race, the spectrum has sidebands about the components at the

characteristic defect frequencies. In [53] a typical spectrum due to an inner race defect was

shown. The sidebands could be attributed to the time-related change in the defect position

relative to the accelerometer position. In [35] sidebands was also observed about the

characteristic defect frequencies.

In [43] it was reported that defects on rolling elements can generate a ball spin frequency (BSF)

or some multiple of it. It was shown that the spectrum can be either a narrow band single spike

or a series of narrow band spikes spaced at BSF or FTF. In [38] it was shown that when more

than one ball defects was present, sums of BSF were generated. The number of sums was equal

to the number of defective balls. It was mentioned that the appearance of BSF is not always a

representative of a defective ball. The BSF could be generated if the cage is broken at rivet.

Defects on the balls are often accompanied by a defective inner race and/or outer race defect. In

[27] it was reported that spectral analysis of bearings with multiple defects on different

components is usually complex. Frequencies generated in different defective components will

add and subtract, therefore some spectrum will contain more than one of the basic frequencies

i.e., BPFO, BPFI, BPFB, FTF. In some cases the harmonics of basic frequencies i.e., lx, 2x,

3x, etc., can be identified in the spectrum.

The characteristic defect frequencies may not appear in the direct spectrum in some cases due to

the presence of 'noise' or vibration from other sources which masks the vibration signal from

the bearing unless the defect is sufficiently large. In [53] it was found that the direct spectral

method can only detect defects of relatively large sizes. In [70] the conditions for bearing defect

detection becomes difficult were listed. In [1 1], one reason for the absence of defect frequencies

in the direct spectrum was found to be due to the averaging and shift effect produced by the

variation of the impact period and intermodulation effect. In [1 1] the time history measurement

data was used in the incipient stage of defective bearing. It was shown that it is difficult to

obtain a significant peak at the fundamental frequencies in the monitored signal spectrum due to

the multiple defects.

2.4 Signal enhancing

When the signal is masked by 'noise' or vibration from other sources, advanced signal

processing techniques have to be employed to detect the faults. If the fault is large enough the

defect may be detected by spectrum. In [79], the effectiveness of different vibration

measurements for the detection of defects in the bearing was investigated. Sensitivity of these

techniques has been assessed using different faults. A single parameter to determine the

contribution of the original vibration signal and the noise is the SNR. Frequency and time

domain analysis are usually not effective in the study of vibration signals when the SNR is quite

poor. In practice the vibration signal is generally corrupted by background noise. In order to

improve the signal-to-noise ratio, some signal processing techniques have to be employed.

Signal Decomposition; It was shown that the time domain signals collected from rotating

machines consist of periodic signal and additive noise [8 1]. Such periodic signatures were

extracted using time domain averaging. The procedure is based on applying coherent windows

to the observed signal. In conjunction with various windows, a new procedure for the

suppression of strong interferences was presented and simple design formula for this technique

was developed. There are two methods which have been used successfully to decompose the

signal. The first method is based on the principle of coherent filtering and relies on

synchronizing the signal [82, 83]. This method is particularly useful in situations where the

motion is phase locked, like a gearbox [3]. The second method is termed the minimum phase

reconstruction method [4], which makes use of Kolmogoroff method of spectral factorization.

In this technique the tooth meshing components and their harmonics are eliminated from the

spectrum of the time domain average and the remaining time signal is reconstructed to produce

the "residual" signal. It was demonstrated that this signal often shows evidence of a defect

before it can be seen in the time domain average. The time domain averaging technique is

reviewed in chapter 3 in details.

Power Cepstrum Technique; In [84] the Power Cepstrum technique was shown to be an

effective diagnostic technique. Power Cepstrum is defined as the Fourier transform of the

logarithmic power spectrum of the vibration signal. It was observed in [ l 1] that the logarithmic

power spectrum of the monitored signal was characterized by double pealts with a frequency

separation. This frequency separation was equal to the fundamental train frequency (FTF).

These enhanced double peaks, called doublets, started at shaft (spindle) frequency and were

spaced harmonically at one characteristic defect frequency i.e., BPFB. A Fourier transform of

the log power spectrum caused these peaks to result a large peak at the cepstrum diagram. This

diagram showed that individual impacts are deviated about 8-10 percent from the averaged

impact repetition rate; thus the impacts due to the defect are quasi-periodic. It was also found

that the enhanced doublet peaks is the result of two separate modulations. The first peak was the

result of the amplitude modulation of the impact source signal by the cage signal. The second

peak was the result of the modulation of the resonant carrier signal by the impact excitation.

The enhanced doublet structures are therefore the results of two separate amplitude modulations

and consequently are defect related. In [85] it was suggested that the fundamental impact

frequency occurs in a frequency band centered at characteristic defect frequency using power

cepstrum. In [54] it was shown that this technique is capable of diagnosing the fault on outer

race effectively but it failed to detect inner race defects.

The attractiveness of this technique in the diagnosis of faults in rotating machines is that a

significant amplitude change at the rotational frequencies indicates the presence of specific

malfunctions. On the other hand, in [79, 86, 87] it was reported that bearing defect detection is

not possible using the rotational frequencies because of the absence of a significant pealt at the

fundamental rotational frequencies in the power spectrum. In [87], the absence of significant

pealt at the fundamental impact frequency was found to be due to two causes which may

operate simultaneously. The first, a rapid variation on the impact rate in the waveform may

produce a shift effect. The second, an inter-modulation effect which translates defect related

information to frequency locations unrelated to the fundamental impact frequency. In the power

cepstrum approach it was found that fault detection and diagnosis were possible despite the

absence of a significant peak at the fundamental rotational frequency.

Adaptive Noise Cancellation; Adaptive Noise Cancellation (ANC) was used to suppress the

background noise [6]. In this technique the noise canceller output is fed back to the adaptive

filter and the filter weights are adjusted through an LMS adaptive algorithm to minimize the

total output power of the system [88]. In [89] it was mentioned that its application especially in

acoustic signals and in identifying faults in rotary machinery has been very successful. In [89],

a strong bearing signal together with background noise vibration was collected to form the

primary input for the ANC canceller. The reference input was obtained at the surface of the

housing where the bearing signal was severely attenuated on reaching the surface of the

housing. In [5] it was reported that ANC can be applied to improve the SNR of the monitored

signal from a complex machine. It was shown that the ANC technique works very well in

situations where the noise in the two inputs are mutually correlated and the reference input

contains no signal or a very weak signal. A good SNR was not achieved in some cases when a

delay introduced in the primary input to de-correlate the primary in particular when the

reference input noise varied too high. In [90] the application of ANC and the blind

deconvolution techniques to detect a damage in rolling element bearings especially when the

signals are contaminated severely by noise, was presented. The classical stochastic gradient

method based on Least Mean Square (LMS) algorithm was applied to both techniques to

compare their effectiveness in detecting the desired signal. It was observed that the ANC

technique requires a minimum of two inputs and the second input reference signal is greatly

dependant to the location of the probe. This method does need a second, reference,

measurement, which is correlated only with either the background noise or the impulsive signal.

There are applications where such a reference signal is not readily available.

Adaptive Line Enhancer; A scheme is developed, referred to as a two-stage Adaptive Line

Enhancer (ALE), specifically for situations where no synchronous or reference signals is

available. In [91] the first stage of the scheme is presented to remove the tonal signal

components of the background noise (harmonic components of the background noise), whilst

the second stage is aimed at enhancing the impulsive signal relative to the broadband random

components. It was found that a long filter is required to increase the attenuation of the filter at

the harmonic frequencies and also to remove the interaction between the various spectral lines

to achieve the objective of the first stage ALE. It was mentioned that the objective of the second

stage ALE is to reduce the broadband noise. It was reported in [92] that the second adaptive

filter algorithm must have fast transient characteristic to track any non-stationary nature of the

signal. This is because the nature of the bearing signal, excluding the noise, is impulsive and the

impulsive signals are cyclic with a short duration. In [91, 93] the adaptive algorithm was

successfully applied to data obtained from an industrial gearbox through careful choice of input

conditions. This technique was shown to perform effectively in the situations where the phased

are locked like a gearbox vibrations.

Autoregressive Modelling Technique; Sophisticated vibration analysis techniques have also

been devised for monitoring of complex rotating machinery. In [94] it was shown that

autoregressive of time series modelling was found to be useful in vibration analysis of rotating

machinery and it is able to provide some use in generating parametric spectra. In [95] it was

mentioned that parametric spectrum analysis methods were defined and developed in order to

determine signal power spectrum density. In [96] the resolution power of signal modelling

based parametric technique was used and conventional spectrum analysis technique such as

correlogram was compared. In [97] it was found that parametric methods were particularly

useful in early detection of faults, especially when two typical frequencies are close to each

other. In [98] it was found that the spectral analysis reduces to the identification of the filter

model where the number of the parameters is obtained by minimizing the error between the

measured signal and the output of the model according to an optimality criterion. The

autoregressive model, as part of parametric spectrum analysis, provides a 'one-step-ahead'

prediction of the vibration signal, as the function uses the previous outputs regressed on to itself

to provide an estimate of the current output. In [99] it was proposed that autoregressive time

series modelling of the vibration signal provides an alternative means of diagnosing faults in the

machinery operating under transient conditions when only short lengths of data is available

such as low speed. In [99] three distinct autoregressive models were evaluated, namely, linear

Box-Jenkiss models, non-linear back propagation neural networks and non-linear radial basis

functions. Performance of each autoregressive model in the observer bank was quantified and

compared based on the SNR of the output signal from each model. After increasing the SNR as

an enhancement tool, the classification stage of the diagnostic system was performed to identify

the most likely class of fault present based upon the performance of each autoregressive model.

It was found that if reliability of the diagnosis is of major concern, especially for very short data

length, then the back propagation neural network should be selected. However for situations

where data is freely available and a simple system is desired, then linear autoregressive models

may be considered. In [96, 97] higher order statistical signal processing procedures have been

used to detect knocks or faults in applications such as internal combustion engines and

industrial gearboxes. It was found that although these techniques are able to detect faults early,

the complexity of these techniques causes them to be used as supplementary procedures.

High Frequency resonance Techniques; The resonances of bearing components are amplitude

modulated at characteristic defect frequencies. In [75] it was mentioned that by demodulation at

one of these frequencies the signal containing information of the fault can be obtained, although

these characteristic frequencies could not be observed in the direct spectrum of the bearing

signal in some cases [75]. In [42] the spectrum of the signal after demodulation was shown to

contain prominent spikes as characteristic defect frequency related to the type of fault [42]. It

was found that the appearance of various spectral lines in the demodulated spectrum is related

to different fault locations [75]. This technique has been used extensively and its success has

been demonstrated by several researchers [39, 52, 54, 99-101]. In 178, 102] a single-mode

vibration model was developed to explain the appearance of various spectral lines owing to

different defect locations in the demodulated spectrum. It was suggested that the sidebands

around the defect frequency are a result of the modulation of carrier frequency due to loading

and transmission path. This model was further extended in [103] to characterize the vibrations

of bearings subjected to various loadings. In [104] the 'normalization' of the envelope-detected

frequency spectra of the faulty bearing with respect to the healthy bearing was proposed to give

greater sensitivity to the detection of defect frequencies. In [42] it was found that having

advanced damage on the bearing components, the defect frequencies may become submerged in

the rising background level of spectrum. This was attributed to the fact that in severe damage,

the impacts are generated more frequently and the leading edge of the impact is buried in the

decay of the previous damage ringing wave. However, in [105] it was argued that this is

because of the reduced difference in amplitudes of the random noise and the defect peak

heights, which become random as the defect progresses.

Wavelet Transform; Wavelet transform analysis has been developed as an alternative time-

frequency analysis of stationary and non-stationary signals and was suggested by some

researchers in fault diagnostics field [106]. In [I07] it was reported that wavelet transform

provides a variable-resolution time-frequency distribution from which periodic structural

ringing due to repetitive force impulses, generated upon the passing of each rolling element

over the defect, are detected. It was found that the existence of bearing faults can be revealed by

an increase of vibration energy in low frequency range using a fine resolution in frequency, and

appearance of the impact in frequency range. Moreover, in [108] it was mentioned that the

cause of bearing faults can be identified in this high frequency range with fine resolution in

time. In [109] the use of wavelet analysis in processing non-stationary signals was described for

fault diagnostics of industrial machine. In [109] it was stated that FFT is inadequate for fault

diagnosis and the wavelet analysis was suggested for the analysis of non-stationary signals. It

was proposed that wavelet is an ideal technique in detection of signal changes that usually

indicate failure in mechanical devices. However, no result was given on the detection of

localized defects occurring in rotary machines. In [110, 1 1 1] a method was proposed to measure

the changing spectral composition of non-stationary signals using wavelet maps. In [112]

wavelet analysis was used to monitor the mechanical process. It was proposed that the defects

of a process could be identified according to the changes of various forms of wavelet

coefficients. Several fault detection indices, such as the energy of the spectrum and the extreme

autocorrelation, were obtained using wavelet analysis. A case study was presented on the

detection of bearing failure through the significant changes of total energy and band energy

using wavelet analysis. In [108] several failure indices were defined using the discrete type of

wavelet analysis for the prediction of spalling in ball bearings from vibration signals. In [107] continuous wavelet analysis was used to detect localized bearing defects based on vibration

signals. They used an index to estimate the interval of bearing impacts, and identified the type

of bearing faults based on calculation of the autocorrelation of wavelet coefficients. However,

this method is too complicated for use in industry.

Blind Source Separation; Most recently the Blind Source Separation (BSS) tool has been used

to recover unknown original vibration signals (called sources) from a finite set of observations

recorded by sensors. BSS is a general signal processing method, which determines the

contributions of different physical sources independently based on a finite set of observations

signals recorded by sensors. In [113] the BSS technique was applied to the signals from various

machines which operate simultaneously to diagnose each element. It was intended to remove

the influence of the other machines without stopping them. It was indicated that BSS can be

applied to the signals from rotating machines to separate the mechanical signatures of several

sources of vibrations. In [114] it was shown that BSS provides a good estimation of sources. It

was demonstrated that each temporal approach can be effective even when there are few

exciting signals from rotating machines. It was stated that perfect recovery of mixing coefficient

filters is not necessary to obtain independent output signals. In [I151 Blind separation of wide

band sources was studies on rotating machine signals. It was assumed particular input sources

and observation signals have particular interaction. By minimizing a contrast function of a

probability distribution the source separation was achieved. In [I161 BSS was set as a pre-signal

processing stage to a fault diagnosis plan. It was shown that BSS allows the vibration generated

from a single rotating element machine to be recovered by the sensor signal from the

contribution of other working machines. It was reported that by adapting an unknown

separating function, the BSS method estimates a new vector, which is supposed to be as close as

possible to the source vector. Two BSS approaches, namely temporal and frequential methods

which were suited to solving this problem, were presented in [116]. It was shown that both

temporal and frequential BSS approaches give rise to similar results.

Blind deconvolution; In this work it is intended to use Blind Deconvolution as a principal

technique to extract fault features in noisy vibration signals. The original transmitted signal is

distorted by the channel as the input signal to an adaptive filter, whereas the desired signal is a

delayed version of the original signal. This method has been improved and has found

applications such as ultrasonic signals in non-destructive testing and seismic signals [117- 1 191.

This technique was also developed for fault detection in rolling element bearings with corrupted

vibration signals in specific, controlled laboratory conditions [120, 12 11. Fortunately, unlike the

other signal processing methods, blind deconvolution method does not need training. In fact

most of the other signal processing methods have to be trained by some sequences to recognize

the differences between the features. However there is an important requirement which

stipulates that optimum linear or non-linear equalization must be obtained from a few selected

samples of the received signals.

In [I221 it was proposed a monitoring scheme based on pattern recognition. The technique

employs short-time-signal processing techniques to extract useful features from bearing

vibration to be used by a pattern classifier to detect and diagnose the defect. To overcome the

shortcomings of this method, bispectral analysis of vibration has been applied to detect the

presence of the characteristic defect frequency and its harmonics [123]. The pattern recognition

technique is then applied to classify the condition of a bearing based on its bicoherence. An

automatic fault diagnosis system for ball bearings, based on processing of time-domain

signatures and a pattern recognition technique, has also been reported [124]. In recent times,

artificial neural networks have emerged as a popular tool for signal processing and pattern

classification tasks, and are suitable for condition monitoring programs. An artificial neural

network can be defined as a mathematical model of the human brain and has the ability to learn

to solve a problem, rather than having to be pre-programmed with a precise algorithm. In [125]

it was proposed a model based on a neural network for fault detection in rolling element

bearings. The system consists of a collection of parametric time-series models, one for each

class of bearing fault to be identified, based on a back-propagation neural network. This time-

domain-based model has the advantage that the diagnosis can be performed using very short

data lengths and is suitable for application in slow and variable-speed machinery. However, the

main drawback is that the model cannot handle very large data without misclassifying the fault.

To overcome this shortcoming, it was proposed that to present the vibration data to the neural

network in the frequency domain [126].

In [I271 it was suggested that characteristic defect frequencies can be used for classification of

faults. In separate research, a 'neural bearing analyser' model has been developed taking only

certain areas of the vibration spectrum and using a back-propagation network. A fuzzy set is the

cornerstone of a non-additive uncertainty theory, namely possibility theory and it is a versatile

tool for both linguistic and numerical modelling [128]. Fuzzy logic can signify the nature of the

machine diagnosis and can express the status of the severity of the machine damage. The

knowledge of operators can be incorporated into fuzzy logic systems. These systems lack the

ability of self-learning which is very important in intelligent fault diagnosis. Neural network as

a pattern recognition method can also be usefully applied in machine diagnosis [129, 1301. It

can diagnose and record a new class of fault, providing that the particular network has been

trained well. The neural network is usually viewed as a fault classifier but it can also output

local optimisation. Recently the neuro-fuzzy technique has emerged as an alternative solution to

diagnose rotating machinery faults and avoids some of the above mentioned disadvantages

[130]. A fuzzy logic is incorporated with the neural network to deal with unclear information

and this combination is called the neuro-fuzzy technique.

Chapter 3

Ontology: Machine Faults and Current Techniques

3.1 Fault Occurrences

Vibration is a vectorial parameter with three dimensions, namely time, frequency and

amplitude, and requires to be measured at carefully selected points. Bearings are the best

locations for measuring machinery vibration since this is where the basic dynamic loads and

forces of the machine are applied and they are a critical component of machinery. Rolling

element bearings can affect the vibration of machines, as a result of either inherent design

characteristics or imperfections and deviation from ideal running geometry within the bearing.

The imperfection and geometry deviations can occur during bearing component manufacturing,

during assembly of a bearing into a machine, or from bearing deterioration during operation.

Each can have profound effect on machine vibration, either by altering compliance properties or

by acting as a source of forces to directly generate vibration. It is shown that it is possible to

detect a bearing fatigue failure or initial rolling component surface spall using vibration

analysis. This chapter describes the cause of machine faults and then reviews current techniques

to detect bearing failure.

3.1.1 Fatigue

No rotating bearing can give unlimited service, because of fatigue in rolling contacts. The

stresses repeatedly acting on these surfaces can be extremely high as compared to other stresses

acting on engineering structures. If reversing stresses, which are applied cyclically, exceed the

endurance limit, a fatigue failure will occur in the structure. Rolling contact fatigue is

demonstrated by flaking off of metallic particles from the surface of raceways and rolling

elements. For a properly manufactured bearing, this flaking usually commences as a crack at

specified depth below the rolling contact surface, known as weak points, and is propagated to

the surface [13 1, 1321. In [I321 a yield stress criterion was given for initiating the crack. In

113 11 it was reported that fatigue is assumed to occur when the first crack or spall is observed

on a load-carrying surface. If fatigue failure is a function of the number of known weak points

in a highly stressed region, then as the region increases in volume, the number of weak points

increases and the probability failure increases although the specific loading is not altered. This

phenomenon is further explained in [133]. As a result the reliability, this is synonymous with

probability of survival, approach was taken to calculate fatigue life of the rolling element

bearings.

In [I341 the operation conditions of rolling element bearing were listed. These conditions were

shown to have the most profound effect on extending fatigue life compared to any other. In

many applications these conditions are exceeded due to overloading, variation in contacts

angles or over speeding. If the maximum rolling element load is significantly increased, fatigue

life is significantly decreased. The standard methods of fatigue life calculation do not account

for changes in ball bearing contact angle. In [I351 the variation of life with load and speed for a

number of rolling element bearings were studied. It was found that the fatigue life of the

bearing decreases under relatively heavy loading. In [I361 it was shown that misalignment in

non-aligning rolling elements bearings distorts the internal load distribution and thus alters

fatigue life. It was reported that the load distribution is altered by misalignments. Misalignment

was found a cause edge loading in roller-raceway contacts; edge loading of even small

magnitude can rapidly diminish fatigue life.

3.1.2 Lack of lubrication

If a rolling element bearing is adequately designed and lubricated, the rolling surfaces can be

separated by a lubricant film. In [I371 the effect of lubricant film thickness on fatigue life was

demonstrated. A method for estimating this lubricant film thickness was given. It was also

shown that lubricant film thickness is sensitive to bearing speed of operation and lubricant

viscous properties and the film thickness is virtually insensitive to load. It was stated that a

considerable improvement in the fatigue life occurs by using a sufficiently viscous lubricant at

slow speed. It was found that a bearing with very smooth raceway and rolling element surfaces

require less of a lubricant film than a bearing with relatively rough surfaces. In [I381 the

relationship of lubricant film thickness to surface roughness in rolling element bearing was

established by introducing the use of asperity slops as well as height of asperity peaks. In [I391

the effect of lubrication on bearing fatigue life was indicated. It was proved that if the height of

asperity peaks exceeds a certain value, the fatigue life is expected to be increased and if this

value is less than unity, the bearing will probably not attain the estimated life and it can lead to

a rapid fatigue failure. It was shown that the edge stress in a line contact is substantially reduced

if an adequate lubricant film separates the contacting rolling bodies. The lubricant film tends to

permit an increase in fatigue life by reducing the magnitude of normal stress and lack of proper

lubricant causes the bearing to fail.

3.1.3 Wear

Wear is the loss or displacement of material from a surface. Material loss may be loose debris.

Material displacement may occur by local plastic deformation or the transform of the material

from one location to another by adhesion. When wear has progressed to the degree that it

Table 3.1 Bearing failure classification due to Wear and descri~tion r 14 11

threatens the essential function of the bearing, the bearing is considered to have failed. In [140]

distinct classes of bearing failure were recognized through experiences and detailed failure

analysis. They are listed in table 3.1. It was reported that these bearing failure modes are

defined based on description of observations. Wear prevention is accomplished by forming

lubrication filrns by hydrodynamic lubricants. During the surface life of a bearing, the

lubrication and wear processes are interactive.

It is essential that both lubrications and wear processes within a bearing be considered as

tropological system. A tropological interaction system was described in [142]. It was reported

that there are numerous technical options for improving wear performance through material,

chemical properties of lubricants, finishing process and surface modification technologies.

3.1.4 Mechanism of Vibration Generation in Bearings

Rolling element bearings will generate vibrations during the operation even if they are

geometrically and elastically perfect. Machine vibration is affected by bearings in three ways:

halla

This table is not available online. Please consult the hardcopy thesis available from the QUT Library

as a structural element defining in part a machine's stiffness; as a generator of vibration by the

mean that load distribution within the bearing varies cyclically; as a vibration generator because

of geometrical imperfections from manufacturing, installation or wear and damage after

continued use.

e The first effect is as a structural element that acts as spring and also adds masses to a

system. In [41] it was shown that number of rolling elements under load varies with the

angular position of the cage. This gives rise to a periodical vibration of the total stiffness

of the bearing assembly and consequently generates vibrations. These are excited

vibrations that occur irrespective of the equality and accuracy of the bearing. In [40] it

was demonstrated that varying compliance of the bearing assembly can give rise to both

radial and axial displacement of a shaft supported by rolling bearings. A detail

mechanism of this vibration motion generation and a realistic method to calculate the

amplitudes and frequencies of these vibrations as functions of bearing geometry and

operating load and speed was presented in [40]. In [I431 it was stated that the shaft

centre undergoes a cyclic displacement with a time period equal to the ball (or roller)

passage time due to the variation of assembly stiffness. A shaft loci was derived in a

plane perpendicular to the axis of rotation for both ball and roller bearings. It was also

confirmed that the varying compliance vibration is valid only for very slowly rotating

bearings. In a bearing operating at normal speed, the inertia forces of the rotor have to

be taken into account. The equation of motion for the rotor-bearing system becomes

non-linear and time varying coefficients. If inertias of rotors are included in the equation

of motion the vertical and horizontal movements are strongly coupled.

The second effect of bearing on machine vibration occurs because bearings carry load

with discrete elements whose angular position, with respect to the line of action of the

load, continually changes with the time. In [36] it was reported that this change of

position causes the inner and outer raceways to undergo periodic relative motion even if

the bearing is geometrically perfect. Analysis of this motion was described in [36]. This

change of position also causes the load distribution changes. Hence, the ball deflection

will be altered in both radial and axial direction during shaft spinning. Both vertical and

horizontal vibration amplitudes are non-sinusoidal as the result of nonlinear deflection

characteristic.

* The third effect that bearings have on machine vibration arises from geometrical

imperfections. A set of component characteristic defect frequencies, which produce

energy, were determined considering the geometry of the bearing in Chapter 2. This

energy may be generated by a cyclic stress or by a periodic impact at a defect. In [36]

additional frequency components generated by errors such as lobbing, ovality (non-

roundness) and ball diameter differences, which interact with the rolling frequencies

were discussed. Each time a defect in a rolling element bearing makes contact under

load with another surface in the bearing, an impulse of vibration is generated in the time

domain signal corresponding to the time interval of the impacts. This impulse is of

extremely short duration compared with the interval between impulses, and so its energy

is distributed at a very low level over a wide range of frequencies. These impacts usually

excite the resonant frequency in the system (bearing) at a much higher frequency than

the vibration generated by the other machine components. More than one resonance may

be excited. These resonances may change and additional one can be produced when the

components are assembled and loaded in the presence of a lubricant. Such resonances

are exited by load changes due to rolling frequency harmonics or by irregular contact in

damaged regions. In [73, 75, 1441 these resonant frequencies were calculated

theoretically based on the physical interactions between bearing components as follows:

Resonant frequency of a ball

Ellipsoidal mode

Radial mode is called f,, and satisfies following frequency equation

where t ana = 4 ~ @ 4 p - a2 (A + 2a)

Resonant frequency of races

In-plane flexural modes

N 2 + 1 2 m

Transverse flexural modes

Radial mode

f R , =

where E = Young's modulus of the bearing

p = Density of the bearing material,

y = Poisson's ratio and m = mass = pA , c = Wave speed = ,E p = Lame's constants, N = Number associated with mode order

a = radius of neutral axis and A = Cross section area = bh

bh I = Area moment inertia = - , b = lateral dimension of the bearing

12 1

h = height of the cross section area = -(O.D. - I.D.) 2

Figure 3.1 shows mode shapes for the mentioned resonant frequencies of the ball as well

as the races respectively. In [145] three mechanisms by which imperfections in bearing

cause vibration were discussed. Waviness and other form of errors causing radial or

axial motion of raceway were presented. Figure 3.2 shows a basic dimension of a

general ball bearing with outer race waviness.

Figure 3.1 Different mode shapes of

Figure 3.2 Basic dimensions of a general ball bearing with outer raceway waviness

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This figure is not available online. Please consult the hardcopy thesis available from the QUT Library

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3.1.5 Bearing Defect Classifications

The bearing defects may be classified as "local" or "distributed". The beginning of progressive

bearing damage which can be called incipient failures often characterized by a sizable local

defect on the components. The local defects include cracks, pits and spall on the rolling

surfaces. The dominant mode of failure of rolling element bearing is spalling of the races or the

rolling elements, caused when a fatigue crack begins below the surface of the metal and

propagates towards the surface until the metal breaks away to leave a small pit or spall. In El461

it was reported that fatigue failure may be hastened by overloading or shock loading of the

bearing during operation and installations. Electric pitting or cracks due to the extensive shock

loading are also among the different types of bearing damage described in [147].

The distributed defects, on the other hand, include surface roughness, waviness, misaligned

races and off size rolling elements [77, 1481. The lobed surface of Figure 3.2 is a causative

factor of bearing vibration. The important feature is called waviness which is the number of

lobes per circumference. Some of distributed defects on the bearing such as waviness can be

traced after manufacturing process using a stylus transducer. In [I491 the correspondence

between waviness and the resulting vibration spectrum was examined. The relations between

waviness of order of the bearing in terms of circular frequency such as the inner ring, cage and

roller angular velocity were studied. Component inspection for waviness was used to assess the

degree of radial deviation from a true circle on the circumferences of raceways. In 11491 the

characteristic defect frequencies were used to establish the relations between the component

waviness testing and filter bands that coincide with the vibration signals. Three different

vibration frequency bands were introduced to calculate the number of waves on a component

that influence a particular band. It was proposed to divide the filter band frequencies by related

characteristic defect frequency to determine the number of waves per circumference of the

raceways. In [146, 1501 it was reported that distributed defects are caused by manufacturing

error, improper installation or abrasive wear. These are a measure of the quality of the bearing.

The variation in contact forces between rolling elements and raceways due to the distributed

defects results in an increased vibration level.

3.2 Detection of Bearing Failure in Machines

As previously shown, such a measurement may be used for machines with bearing in new

condition as well as for machines with defects on the bearing components. Some of defects

other than waviness can be difficult to detect with the conventional three bands inspection

methods. Such defects include local defects on raceways or ball, dirt, greases with improper

properties and cages with incorrect geometry. Some of these defect types may have only a small

effect on the average measured vibration in the inspection bands and vibration analysis needs to

be carried out. General methods for evaluating data include one or more of the following:

1. Comparison of vibration data with guidelines developed empirically on similar type of

equipment [ 1 5 1 1.

2. Evaluation of vibration data in an absolute sense with no prior history such as evaluation

the time or frequency signals to associate vibration with specific machine components.

3. Trending of vibration data from one machine over time and this method id discussed in

detail in Chapter 6.

When a local defect occurs, subsequent rolling over of the damage will produce repetitive

shocks or short-time duration impulses. Figure 3.3 shows the effect of successive rolling

element impacting a damaged area on the outer raceway. This impulsive time waveform

corresponds to lightly damped oscillation of some system natural frequency greater than the

repetition frequency of the train of impacts. This high resonant frequency could be a resonance

of the bearing outer ring or of the housing itself. Impulsive occurrences in bearings can cause

system vibration at many frequencies that can be harmonically related. In [38] it was found that

in the early stage of failure the impulses might have little effect on the amplitude of vibration at

the characteristic defect frequencies. In addition, significant normal machine vibration could

occur at these low frequencies, so small changes in vibration amplitude initially may be difficult

to detect. However higher order harmonics, with spacing related to specific components

frequencies might be detectable.

Figure 3.3 The time waveform due to a crack on the outer race of a rolling element bearing[l52]

Apart from evaluation of vibration spectrum to identify specific bearing component frequencies,

vibration signal can be obtained or analysed by other means to trend the onset of failure. In [55]

a comprehensive evaluation of vibration parameters over the life of bearings run to failure was

presented. Various conditions were applied to the tested bearings. Some vibration parameters

were computed by performing arithmetic operations on two frequency spectra, one of which

was usually the initial spectrum obtained when tests were begun (healthy bearing signal). The

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\

computed parameters were then trended. Statistical functions such as probability density,

skewness and kurtosis were also evaluated. The results indicated that several vibration

parameters evaluated from frequency spectra were successful trend indicators. It was

demonstrated that monitoring bearing vibrations and comparing the vibration signals against a

baseline for satisfactory bearing operation may be used to detect an impending bearing failure.

3.2.1 Maintenance Strategy

Mechanical mechanisms that are in motion such as bearings are particularly prone to problems

from wear, corrosion, erosion, fatigue contamination, abuse, etc. Many applications have relied

on preventive maintenance to minimize unscheduled downtime due to the bearing failure. In

[I531 it was reported that a real machine is complex and has numerous components which could

potentially fail. As a general rule, the failure rate of a machine is depicted by a "bathtub curve".

There are some exceptions to this distribution, but the "bathtub curve" is quite representative. In

[I531 it was stated that the behaviour of a new machine follows the pattern as follows: 0 Zone 1 : Infant mortality is an unexpected failure due to an error in assembly or part

manufacturer. There is no excuse for infant mortality, but it happens. The result is

that the probability of failure is higher when a machine is brand new than after a bit

of run time. See Figure 3.4.

Zone 2: After correcting any infant mortality issues, the machine enters a period of

time considered the normal service life time when the probability failure is relatively

low. Through good maintenance, this portion of the curve can be extended.

Zone 3: The machine has operated many hours at this point and unusual

complications are being to occur, increasing the probability of failures. Reliability

will continue to decrease unless the machine is totally refurbished. Obviously,

machinery can be kept operating through this period, but reliability is poor and the

cost of repairs increases rapidly. It probably would male good economic sense to

consider replacing the machine or rebuilding it.

Figure 3.4 The Weibull Curve for equipment reliability shows the

behaviour of the machine with respect to time [I]

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The "bathtube" curve also applies to a machine that has undergone a serious problem for

maintenance. In this case, machine was removed for operation for an extended period of time,

dismantled, cleaned calibrated, adjusted, and all impending or actual defects corrected. Due to

the dynamic nature of mechanisms, machines require appropriate maintenance and attention to

perform reliably and at peak performance. In [I531 it was suggested that there are four

fundamental approaches to maintenance:

e Run-to-failure or "breakdown".

e Routine superficial inspection.

Preventive maintenance (PM).

Predictive maintenance (PdM).

3.2.1.1 Run-to-failure or "breakdown"

In this approach, the machine is simply ignored and operated until it fails. This methodology

requires no effort or knowledge. The machine is fixed as an emergency basis. The loss of

production, unexpected coast and a number of other issues are disadvantages of this method.

The advantage is that there is no up front cost or concern. As a result "breakdown" maintenance

is not attractive.

3.2.1.2 Routine superficial inspection

Since "breakdown" maintenance has not proven to be very effective, a new approach of

performing simple task or visually inspecting for obvious issues such as loose fasteners, bad

belts, cracks, lubricant leakage, etc is adopted. This technique is effective and combined with

the sense of listening and feeling there is no a decent return or investment. An individual who

has developed a rapport with the machine can make a relatively quick assessment. For instant,

unusual vibrations as sensed by touch of an expert can be processed in hislher mind. The result

of processing can portend an impending failure on a bearing or warn an electrical problem. The

advantage is that there is no up front cost. The disadvantage is that superficial inspections may

not reveal developing problems with sufficient warning to avoid an unexpected failure. In

addition, previous experiences can not be much helpful to precisely define the nature of a

failure deeply when a new problem is involved.

3.2.1.3 Preventive Maintenance (PM)

PM is the time-based application of routine activities to lubricate, clean, inspect adjust, and

replace components that have a history of failing. The meaning of PM in this context is that of

routine minor but very important tasks. The advantage is that known potential problem areas are

addressed before failure occurs, minimizing unexpected downtime. PM makes a big difference

in reliability and reduces the cost of operation of the equipment over the long term. In this

approach based on calculation of bearing endurance, either from fatigue of rolling contact

surfaces or other wear phenomena or based on past experience of bearing failures, periodical

stoppage of machinery are scheduled to either inspect or replace the bearings. The disadvantage

is that PM is not free, requires effort, and may contribute to some unnecessary replacement of

parts such as the bearing which had been in operation where most liltely not pron to failure.

3.2.1.4 Predictive Maintenance (PdM)

Predictive Maintenance is the most intensive approach in the maintenance strategy. PdM is the

application of latest instrument and software technology to routine monitoring and detection of

machinery faults prior failure. The combination of PM and PdM is very powerful and can

assure the ultimate in equipment reliability and cost reduction. It is a non-invasive technique for

monitoring the health of the machine in an attempt to track any changes and pinpoint the source

well in advance of a failure. Based on bearing condition monitoring and prognostic ltnowledge

of the duration of effective bearing performance, failure-prevention can be taken after receiving

of the first signal of impending bearing failure. Having said that condition monitoring

techniques must be proven reliable and life prognostication methods must be proven sufficiently

accurate.

Some advantages of PdM are as follow:

PdM provides the ultimate machinery health assessment and reduces the cost of

unnecessary PM part replacements. 0 Since parts are not changed on a predetermined time based interval, the potential for

problems associated with less than perfect workmanship are minimized.

The potential for infant mortality (new parts that fail prematurely) is minimized. 0 Since unnecessary tasks can be avoided, maintenance manpower can be carefully

planned to achieve optimum efficiency. e The cost of repair a machine versus total replacement can be addressed quantitatively. 0 The quality of a new machine can be assessed in advance of purchase to assure that the

equipment meets the requirements.

PdM consists of myriad techniques with more being introduced to the market. Typical

categories include vibration monitoring/analysis, lubricant, conditionlwear particle analysis,

infrared monitoring/analysis, motor current analysis, specialized electrical system

monitoring/analysis, etc [I 541. Condition based maintenance (CBM), which originates from

PdM, is a relatively new concept and additional information is required for its effective

implementation. In [I551 it was reported that in CBM approach the ability to predict a

remaining bearing life based on actual accumulated condition of operation, namely load-speed-

temperature, has become important. Some micro-sized pressure, temperature, ultrasound

sensors were embedded in close proximity of the rolling elements to sense incipient fatigue

failure. In [I551 an algorithm was developed to determine the time available for effective

operation after incipient defect occurrences. It was determined that an effective lubricant film

was generated even in the presence of grossing spalling.

3.2.2 High Frequency resonance Techniques(WFRT); As mentioned earlier each time a

defect strikes its mating element, a pulse of short duration is generated that excites the

resonances periodically at the characteristic frequency related to the fault location. The

resonances are thus amplitude modulated at these frequencies. By demodulation at one of these

frequencies the signal containing information of the fault can be obtained. In [42] an enveloping

procedure was taken to demodulate the bearing signal. The sequence of operations is illustrated

in Figure 3.5 known as HFRT or demodulated resonance analysis.

The signal was band-pass filtered around one of the resonant frequencies to eliminate

frequencies generated by shaft imbalance, gear meshing, background noise and other

resonances or most unwanted vibration signal from the other machine components. The

remaining signal consisted of a narrowband carrier at the resonant frequency, amplitude

modulated at the characteristic defect frequency.

It can be extracted by the simple technique of envelope detection, in which the band-

passed signal is then rectified to convert bipolar input signal to a unipolar signal. In this

stage the rectifier must be designed with care that negative part of the signal is avoided

and a meaningful output signal is to be obtained.

The demodulated signal is smoothed by low-pass filtering to eliminate the carrier

resonant frequency. The smoothing circuit in the envelop detector is commonly figured

out as a peak-hold circuit.

LABVIEW Rectified Envelope Pass

INTERFACE Signal Filtered

Signal Signal

Figure 3.5 A schematic block diagram of the amplitude demodulation process

The spectrum of the low-passed signal showed prominent spikes as characteristic defect

frequency related to the kind of fault [42]. It was found that the envelope spectrum of a

defective inner race or a damaged roller element bearing are more complicated than the envelop

spectrum of an outer race defect. It was shown that the characteristic defect frequency for an

inner race defect is clearly visible in the spectrum but it is accompanied by a number of

modulation sideband spaced at multiples of shaft rotation. A satisfactory explanation for the

appearance of the modulation sidebands around the characteristic defect frequency was

presented.

3.2.3 Adaptive Noise Cancellation; In [5, 6, 9, 88, 891 the principle of this technique was

explained. The general concept of ANC is shown in Figure 3.6. This technique involves the monitoring of vibration signal v corrupted by a noise no and is received at the primary sensor.

A reference noise n, , which is related to the noise no in some unknown way but uncorrelated

with the vibration signal s, is received at the reference sensor. The reference input then is updated adaptively to match no as close a possible, and is then subtracted from the primary

input s +no to produce the system output c= v + no - y. Assuming v, no, n, and y are

statistically stationary and have zero means, an expression for the system output can be

obtained as follows:

.....................................

Figure 3.6 Adaptive noise cancelling principle

c=(v+ no) - y = c = v + ( n o - y ) (3.6)

The noise canceller output contains the signal plus residual interference. The noise canceller filter acts to minimize the average power, known as expectation of the squared error (no - y),

this residual interference at the noise canceller output (error signal). The noise canceller output

is fed back to the adaptive filter and the filter weights are adjusted through a LMS adaptive

algorithm to minimize the total output power of the system [9, 88, 1241. The concept of LMS

algorithm is well known in the literature and with the minium error signal, the filter output y then gives a best lest square estimate of the primary noise no. Hence, it causes c to be best least

square estimate of vibration signal v

3.2.4 Acoustic Emission

Acoustic Emission (AE) is a high frequency listening technique which was initially developed

as a Non-Destructive Testing (NDT) technique to detect crack growth in materials and the

structures. When a material is subjected to stress at certain level, a rapid release of strain energy

takes place in the form of elastic waves which can be detected by AE sensor placed on it.

Plastic deformation and growth of cracks are among the main source of acoustic emission in

metals. AE technique is therefore, widely used in non-destructive testing for the detection of

crack propagation and failure detection in rotating machinery. The signal is generated and

measured in the frequency range which is greater than 100 kHz. In [21] it was reported that AE

monitoring has an added advantage that it can even detect the growth of subsurface cracks

whereas vibration monitoring can normally detect a defect when it appears on the surface. In

El561 it was stated that there are two types of AE signals as shown in Figure 3.7. Figure 3.7a

represents a transient signal, known as the bursts, start and end points deviate clearly from

background noise. Unlike the transient signal, the continuos AE signal never ends (Figure 3.7b).

Figure 3.7 Typical transient and continuous AE signals [I 561

A detection threshold is used to eliminate background noise. The threshold is a subjective value

and is determined by the user because if the AE signal exceeds the threshold in either the

positive or negative direction a burst is assumed to be detected. Statistical evaluation of a AE

waveforms is based on certain features which are shown in Figure 3.8 [156]. These features are:

Arrival time (absolute time of first threshold crossing)

Peak amplitude

Rise time (time interval between the first threshold crossing and peak amplitude)

Signal duration ( time interval between first and last threshold crossing)

Ringdown counts ( number of threshold crossing )

Energy ( area under the diagram or square amplitude over time of duration )

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Figure 3.8 AE features of transient signal [I 561

An even consists of a group of ringdown counts and signifies a transient wave [21]. In 1311 it

was stated that the distribution of the events by ringdown counts and peak amplitudes were

found to be good indicators of bearing defect detection. With a defect on a bearing component,

the distributions of events tend to be over a wider range of peak amplitudes and counts.

3.2.5 Time Domain Averaging

An alternative technique of vibration analysis which was used for early detection of failure in

rolling element bearings is called time domain averaging. The changes in the vibration of a

bearing may be small when a defect is small, and may not be readily detected against the

normal vibration of the bearing. In [81] it was shown that it is possible to enhance the clarity of

the changes in the time domain average by digital signal processing, using techniques which

remove the normal vibration from the time domain average so that the changes in the vibration

are more readily apparent. The time domain averaging technique was used to extract the

periodic signals from noisy waveforms. It was a coherent detection process, and the period of

this signal has to be known. In [82] the averaging processing was explained by assuming a signal x(t) to consist of the sum of periodic signal f (t)and additive noise from the other

components s(t) . x(t> = f (t) + s(t)

In [83] it was shown that when summing up subsequent x, the repetitive periodic part f will add

coherently, and the noise incoherently. After N summing we obtain:

x(t,) = Nf (t, ) + f i s ( t , ) (3.8)

It was demonstrated that signal-to-noise-ratio was enhanced by a factor o f f i (Equation 3.8)).

The transfer function of the ideal averager was found assuming x(t) as a vibration signal before averaging process and y(t) as the signal after the process. Denoting x(nT) as the sampled

vibration signal to be averaged at a sampling interval nl'and averaged period mT , the signal after averaging, y(nT) was obtained as:

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By taking the discrete z transfer of the process and substituting by frequency response, the

amplitude and phase responses of the averaging were found as: W

sin nN --

1 a, I W o ) l = N

sin n -

W By taking - = K as an integer K the amplitude response function represents a gain response of

0 1

a comb filter with centre frequencies ofKu1. In [83] it was reported that if sufficient averages

are taken, the comb filter can be approximated by a train of ideal impulses located at the

multiples of the repetition frequency. It was noted that if the Fourier transform of the original

signal is multiplied by the comb filter it causes only components appear at multiples of the

repetition frequency in the frequency domain. Time domain coherent averaging was also shown

to be an especial case of sampled low pass filter. The averaging processing was found to be

considered as filtering process to enhance the vibration signal.

The time domain average of a signal is calculated by the convolution of the original signal in

the time domain with a train of ideal impulses, a process which is equivalent to the

multiplication of in the frequency domain of thc Fourier transform of the original signal by a

comb filter. If sufficient averages are taken, the comb filter can be approximated by a train of

ideal impulses located at the multiples of the repetition frequency. If the Fourier transform of

the original signal is multiplied by the comb filter it causes only components appear at multiples

of the repetition frequency in the frequency domain. Time domain coherent averaging was also

shown to be an especial case of sampled low pass filter. The averaging processing was found to

be considered as filtering process. The general approach used showed that time domain

extraction of periodic signals can be performed by a filter having a digital continuos or hybrid

realization.

If each impulsive portion of the monitored signal is taken with the start of each portion (frame)

being determined by the synchronising the next impulse, the ensemble average of all impulsive

portions can be calculated. It is found that after many averages all of the frames which are

synchronised with the start of next impulse, the time or frequency associated with the defect can

be estimated. This technique is particularly useful for complex systems such as gearboxes or

rolling elements bearings as it eliminates the vibration from other system components, thus

reducing the problem to the study of a simpler system. If sufficient averages are taken then the

time domain average closely approximates a truly periodic signal, with the very important result

that the Fourier transform of the time domain average is a pure line spectrum at the defect

characteristic frequency. This enables the manipulation of the time domain average in the

frequency domain representing the behaviour of a filter. These operations are not possible with

conventional spectral analysis because there are many frequency components present in the

incoming signal which are not in general exactly periodic within the block of data which is

sampled.

3.2.6 Blind Deconvolution

Many techniques have been used to enhance the source signal using only one single input. Blind

deconvolution (equalization) is one of those techniques and has been used to recover the desired

signal from a single received channel without any advance knowledge of the channel. In

conventional deconvolution, the system characteristics are known and the task is to estimate the

input signal. In blind deconvolution both the system and the system's input are not known. In

such a case, it is generally assumed that the system and the deconvolution filter must be a unit

impulse up to a scale factor. The technique has been turn keyed in a variety of applications

[119, 1571. In this research a closed form solution to blind deconvolution using the classical

stochastic gradient method based on the Least Mean Square (LMS) algorithm adopted from

Jelonnek and Kammyar [13] was applied to test the feasibility of using the technique to detect

the bearing signal. The LMS algorithm may not be as fast as in its convergence rate but it is

relatively easy to implement. It was shown that the results clearly reproduced the original input

signal [158]. This algorithm works very well with periodic noise and the equalizer behaves like

a notch filter in removing the noise at the corresponding studied frequencies. For random noise

this technique appears incapable of removing such noise. The coefficients of the equalizer

(inverse filter) were computed within the algorithm and the gain response of the equalizer

showed the notch at a certain frequency band.

Chapter 4

Blind Deconvolution Theory Applied to Bearing Diagnostics Convolution is referred to as the computation of the output signal y[n], given the knowledge of

both the input signal x[n] and the impulse response of the system h[n]. Deconvolution is

referred to the determination either the impulse response of the system or the input signal given

the output signal y[n]. The output of the systems is often known or accessible by signal

measurement. There are two deconvolution formats as follows: o The system is unknown, but the input signal is known and available. This is a problem

of system identification. In some cases a model the of system h[n] is available and it is

required to estimate the unknown input signal x(n). Since convolution is commutative,

both deconvolution problems are mathematically equivalent.

The system is unknown, the input signal is not available and only some statistical

properties of the input signal are assumed. Solution to such a problem is referred to as

blind deconvolution and it is more difficult task to solve compared to the normal

deconvolution problem.

Figure 4.1 depicts a linear time-invariant system with and input signal (source) passing through

an unknown channel and the output signal is measured by a sensor. It is assumed the operation

is performed in the discrete-time domain. The output signal y[n] is defined as the convolution of

the input signal x[n] and the impulse response h[n] of the system and can be expressed as:

where k is an index and n is the sampling discrete variable. Convolution is demonstrated as

follows:

Y [nl =h[nl *x[nl (4.2)

(unknown channel) 1

Input signal

(source) x[n] b

Figure 4.1 Linear time-invariant system

4.1 Applications of Blind Deconvolution

Blind deconvolution is being used in a number of areas such as data transmission, seismic

deconvolution, image restoration and recovering the vibration signal. Some examples of these

applications are as follows:

* In high-speed telecommunication systems, data is transmitted through a

communication channel with known observed signal is received by the receiver at

Impulse response of

system h[n]

Output signal

(measured) y[n]

the other end of the communication link. It is aimed to recover the source of an

unobserved transmitted signal without any prior knowledge of the channel and the

source. Mobile communication is one of the examples which employ blind

deconvolution.

In seismic deconvolution, the geological layers of earth are modelled with different

acoustic impedance. The sequence of reflection coefficients is estimated

corresponding to the various types of layer models. The received signal is itself

made up of echoes produced at different layers of the model in response to the

excitation which is in the form of short-duration pulse. The extraction of the

excitation wave form associated with the received signal is usually unknown and

blind deconvolution is applied to the observed or measured signal to recover the

source of excitation. The impulse response of the layered earth model is viewed as

equally spaced time sequence of reflection coefficients [159] and [I 601.

In image restoration, the unknown channel is regarded as blurring effects caused by

photographic or electronic imperfections. The channel output is a blurred version of

the original image. Given the blurred image as observed signal, it is required to

restore the original imagelsignal using blind deconvolution.

4.2 Model of Blind Deconvolution

In vibration signal, it is assumed that an original unknown input signal (desired signal) is

transmitted through a path between the source of vibration and the sensor pick-up point. The

task is to recover the original source of the vibration signal using blind deconvolution. In

vibration signal processing, the analogue waveform is converted to a series of discrete-time

signal by an analogue to digital convector. The noise signal has significant impact on

measurements and corrupts the observed signal and resulted in the observed signal different

from the original source of the vibration signal. Figure 4.2 shows a schematic diagram of blind

deconvolution process. It shows that an original input signal x[n], which represents the

unobserved source of vibration, is distorted by an unknown channel and then is corrupted by the

background noise. The unknown channel h[n] denotes the composite channel impulse response

which included physical medium. The observed signal y[n] can be measured easily using a

sensor. However, the original input signal x[n] is unknown. In this Figure s[n] represents the

noise signal which can be white Gaussian noise or deterministic noise. The objective is to

deconvolve the observed signal y[n] and to estimate the unknown channel. This is achieved by

passing y[n] through a blind equalizer with finite impulse response (FIR) e[n] of length L. The

process is called blind deconvolution because the performance is blind and there is no training

for the known sequence. The observed signal can be written as:

Noise s[n]

Estimation

of Original Signal

Input FIR Inverse Filter (L) Input

Signal Figure 4.2 Basic Diagram of Blind Deconvolution

Signal

A

The output of the blind equalizer is denoted by x[n] and is an estimation of original input

signal x[n] and can be expressed as:

The original input vibration signal x[n] is assumed to be a sequence with a zero mean,

independent and identically distributed (i.i.d). This signal is also assumed to be a random

variable. The vibration signal can be regarded as a random variable since the outcome of each

experiment is different.

4.2.1 Known Probability Density Function of Input Signal

The first mathematical characteristic of such random variables is the "Expectation". The

mathematical expectation of the random variable X is E[X] which may be read as "the expected

value of the X" or "the mean value of X" is defined as: +cO

E [XI = 5 xf, (x)dx (4.7) -m

Wheref,(x) is the probability density function (p.d.f.) of X happening to be discrete with N

possible values ofx, . The original input signal is also assumed to have a non-Gaussian p.d.f..

However, in other applications of blind deconvolution such as in geophysics, it was reported

that the original input x[n] has a continuous p.d.f.. In [161] a general Gaussian p.d.f. was

assumed as follows:

with values of a from about 0.6 to 1.5, and is a useful model for the reflection of coefficients of earth layers. In the Equation (4.8), p is a scale parameter that defines the variance of x and is

given as follows:

It is not preferred to use a known p.d.f. of the data such as Gaussian assumption because we are

compelled to choose some source models whereas the original input signal x[n] distribution is

uncertain and non-Gaussian. Furthermore, if the p.d.f. of input signal is Gaussian the recovered

signal after equalization gives the same second-order statistics using different equalizers

(inverse filters). Thus, second-order statistics alone are not able to identify the correct inverse

filter if the p.d.f. of input signal is Gaussian. Figure 4.3 from [162] shows that when a random

variable with Gaussian p.d.f. distribution is passed through a channel and is recovered by an

equalizer ( inverse filter) with different phases using second-order statistics then any qualitative

differences between the results can not be seen.

Figure 4.3 a) A random variable with Gaussian p.d.f. distribution [162].


(equalizer) with different phases using second-order statistics.

The same experiment with spiky time series was performed in [162] and the results in Figure

4 . 4 ~ to Figure 4.4e confirmed that spiky signal is not Gaussian. The assumption of Non-

Gaussianity of vibration signal is satisfied because an ideal vibration signal from a localized

defect on the rolling element bearing components is spiky as indicated in Figure 3.3. It was

demonstrated that the convergence to a unique inverse filter (equalizer) can be achieved if the

input signal is Non-Gaussian and the channel is stable. This was done by maximizing an

appropriate functional (cost objective function) namely, the kurtosis of the convolved sequence.

~+t~,-td,-e-~i-i

Figure 4.4 a) A random spiky signal with Non-Gaussian p.d.f. distribution 11621.


(equalizer) with different phases using second-order statistics.

halla


halla


The channel is assumed to be a composite channel with Equivalent Discrete-Time White Noise

Filter model. This implies that perfect equalization can be obtained by assuming infinite number

of taps. Although the channel is unknown but it is assumed to be time-invariant at least over

certain period of time. This implies that even if the real medium between the source and

observed vibration signals is time-variant the channel characteristics do not change over certain

period of time, hence quasi time- invariant. The channel is assumed to be casual; h[n] is zero for

n < 0, and possibly mixed-phase impulse response. If some of the zeros and poles of channel

impulse response h[n] shown in Figure 4.1, are inside the unit circle in z-plane whilst others lie

outside the unit circle the system is mix phased and described as a non-minimum phase (NMP) system. In [161] it was reported that any non-minimum phase system can be converted to a

minimum phase system by cascading with an appropriate allpass (AP) providing that a system

never has any poles or zeroes precisely on the unit circle. For a non-minimum pahse channel

h[n], the magnitude response of an unknown system can be identified from second order

statistics like autocorelation function or the power spectrum providing that the training

sequence (desired signal is known). On the other hand, phase information of the inverse system

is only preserved in statistics of order higher than two. Thus, equalization algorithms employ

higher-order statistics (HOS).

4.3 Higher-Order Statistics (HOS)

When the signals are Non-Gaussian the first two moments such as autocorrealtion do not define

their p.d.f. and only HOS can reveal other information about the signals such phase information

compared to the second-order statistics (SOS). Cumulants and momeents are statistics

paramaters which describe the distrubtion of the signal. For a zero-mean distrubitions, the first

three central momeents and the corresponding cummulants are identical and they differs from the fourth order. Needless to say for a Gaussian distributed signal C, =0, C, = 0, , C, = C, = 0

where C stands for the cumulant function and the index indicates the order of cummulant. For a

zero mean real signal x[n] the second-order and third order momements are difined as:

M , [k] = C, [k] = E[x(i)x(i + k)] (4.10)

M3 [k, m] = C3 [k, m] = E[x(i)x(i + k)x(i + m)] (4.1 1)

where i,k,m are the lags. The forth-order moment sequence is defined as:

M4 [k, m, n] = E[x(i)x(i + k)x(i + m)x(i + n)] (4.12)

The forth-order cumulant is different than forth-order moments as follows:

C4[k,m,n] = M4[k,m,n] - C2[k]C2[m - n] - C,[m]C,[k - n] - C2[n]C2[m - k] (4.13)

The higher order complex cepstrum is one of the HOS technique being used in blind deconvolution. Let Cn, (a,, a,,... .an-,) be the n~ order cumulant spectrum of the random sequence

x[n]. Then the nth order cepstrum is defined as:

~ ~ ~ ( k , , k , ,..., kn-,) = ~ - ~ ( l n ~ ~ ~ ( a ~ , a ~ ,.... an-l)) (4.14)

where F-' is the inverse (n-1)-dimensional Fourier transform. The tricepstrum (forth-order

cepstrum) of the observed signal is given by K n y ( k ~ , k , , k 3 ) = F-1{1nc4y(mI,m22m3)) (4.15)

In practice, only a finite number of data samples are available. An estimation of moments and

cumulants are used in the algorithm. In this estimation, expectation function E(.) is computed as

a summation function due to limited number of sample data. Any error in the values of

cumulants estimated from finite segments of time series will result in large variance in other

higher-order estimates. It is necessary to employ a greater number of samples for HOS

calculation compared to SOS. To determine the significant of HOS, a simulation study was

performed. Estimated cumulants were used to distinguish between three different independent

and identically distributed (i.i.d.) random variable signals with zero-mean and unit variance,

namely Exponential, Gaussian and Uniform. Results in table 4.1 shows that only the estimated

forth-order cumulant at zero-lag is able to distinguish between the random variable signals.

Table 4.1 Estimated values for cumulants at zero-lag of

I Estimated Cumulants I Exponential I Gaussian

mdom variables

Unlforml

4.4 Equalizatio~l Criterion and Algorithms

Since the equalizer is regarded as the inverse of the channel, perfect equalization can be

achieved when the convolution of the channel and the equalizer represents a unit impulse

response function as expressed as follows: s[n] = h[n]*e[n] = 6(n - k)ceJO

where k is the delay, c is the gain factor, 8 the linear phase shift and k is a constant delay. In

other words, it is required the power of the overall impulse response s[n] to be concentrated in

one tap as a relative large amplitude spike as follows:

s [n] = s[k] 6 (n-k) (4.17)

However the equalizer impulse response e[n] can not be determined from this criterion directly

because the channel h[n] is still unknown. This criterion provides only a measure of the inter

symbol interference (ISI) that can be useful to asses the performance of the blind deconvolution

algorithm. It is obvious from Equation 4.16 that perfect equalization gives zero IS1 as follows:

Xls[nl.Y - lsmm 12

where s,,, is the maximum magnitude of overall impulse response s[n] concentrated on one tap.

It is obvious that complete removal of IS1 can not be guaranteed since an ideal infinite channel

impulse response is convolved with a truncated length equalizer (FIR). The convolution error is

the difference between the output of estimated equalizer with a finite length and the output of an

ideal infinite equalizer and is given as follows:

g[n] = x[n] + q[n]

where the convolution error q[n] represents the residua1 inter symbol interference induced by

channel distortion. A detailed schematic diagram of blind equalizer or FIR filter is shown in

Figure 4.5.

Figure 4.5 Cascade of transveral FIR blind equalizer length of L

In order to obtain the FIR blind equalizer coefficients, there are three types of algorithms to

equalize the channel as follows:

Gradient Descent Algorithm; reminds everyone the conventional adaptive non-blind

equalizers. The conventional approach is based on a training sequence, which is

replaced by a non-linear estimator of the channel input. The estimation is designed to

minimize a cost function which is implicitly based oh HOS. Furthermore, an adaptive

filter is updated through the gradient descent algorithm. Bussgang algorithm is one of

the most commonly used procedures in this category. 0 Explicit HOS based Algorithm; employs higher-order cumulants or polyspectra. An

empirical estimation of statistics is used to equalize the channel. This requires

nonlinear computation and making use of equalizer input and output signals. A faster

convergence of the equalization can be achieved through this category.

Cyclostationary statistic based algorithm; is based on non-stationary signals have such

property that the samples separated by a period have the same p.d.f.. An opportunity

can be taken to exploit sample separated by the cycle-period to create an ensemble of

points and lead to better estimation of signal characteristics.

4.4.1 Gradient Descent Algorithm A

This is to minimize a cost function g(x[n]) with respect to the equalizer parameters to perform

a perfect blind equalization. The cost function is based on implicit HOS and characterises inter

symbol interference (Equation 4.18) and reduces the convolutional error (Equation 4.19). The

inverse filter taps (Figure 4.6) are adapted with gradient descent operator as follows: A

ek Ln1 = ek rn - + ~ . l ( - ~ , , g ( ~ [ ~ l ) (4.20)

where - ~ ~ , ~ ( x [ n ] ) represents the negative gradient and e,[n] denotes the kth inverse filter tap at

the arrival of equalizer input y[n]. The parameter p determines the step-size of the gradient

function and affects on the stability and convergence of the algorithm. This algorithm uses

similar technique to the non-blind adaptive equalization algorithm where a cost function is

minimized through the mean square error (MSE). The structure of non-blind equalizer is shown

in Figure 4.6. In Figure 4.6 the desired signal d[n] is a delay version of channel input and is

regarded as training sequence. The LMS algorithm is used to minimize the instanious error

function z[n] which is defined as follows: A

z[n] = x[n] - d[n]

where the cost function is defined as expectation of square error function as follows: A

g(x[nl) = ~ [ l z , [n1I2 I (4.22)

Traditionally an adaptive filter (FIR) is used to equalize the process. A training sequence is used

to provide the "desired signal" for the adjustment of the tap weights of a linear transversal filter

and to minimize the mean square values of error signal between the desired signal and the filter

output. The most popular algorithm used to perform this adjustment is the least mean square

(LMS) algorithm which is shown in Figure 4.7. The tap-weight vector of the FIR filter is

updated as illustrated in Figure 4.7. The blind deconvolution technique uses the same approach

and it does not require the use of the training sequence for the adjustment of tap weights [163].

Mobile communication is one of the examples which employ blind deconvolution.

Adaptive 1 I algorithm I Figure 4.6 Block diagram of adaptive non-blind equalizer

Weight factor Parameter

Error

Signal

Tap-input

Vector u Figure 4.7 Schematic illustration of the LMS algorithm (tap adjustment)

According to [92] the solution for the non-blind Minimum Mean Square Error equalizer, denoted as, ems,(ko, can be calculated based on the following equation:

emscck,, = R;:. * ryd with ryd = E[yk * d[k - ko]] and R, = E[yk * y;] (4.23)

where r,, denotes the cross-correlation of the training sequence and the channel output

(obseved signal), R,, represnts the autocorrealtion matrix of the observed signal y[n], and the

vector y and y *, are defined as:

yk = {Y* [kl, Y* lk - I],..., Y* - LI) (4.24)

Y: = { ~ [ k l , y[k - lI,...y[k - Ll) (4.25)

The gradient descent blind deconvolution algorithm is a logical extension of LMS-type non-

blind equalization technique. The training sequence d[n] is absent here because the process is A A

blind. A non linear estimator based on equalizer output x[n] to x[n - M I , where M denotes the

memory of the non-linear estimator and is employed to perform blind equalization. The idea of

replacing a non-linear estimator based on equalizer output with the training sequence d[n] was

first introduced by Sato [164]. This replacement is shown in Figure 4.8.

1 algorithm (LMS) I zfnl

Figure 4.8 Block diagram of adaptive blind equalizer updated with gradient descent algorithm

This algorithm uses memory-less nonlinearities whose output assumes Bussgang statistics [165]

and is the first series of Bussgang algorithms which use different types of cost functions. The

Sato's cost function is given as follows:

where the Sato parameter y is a cosnstant defined as:

The adaptive filter (LMS) in Figure 4.8 is used to update the blind equalizer taps using Equation

4.20. An improved series of cost functions were suggested by Godrad [166]. These cost A

functions involve only with magnitude ofx[n]. Although a number of different Bussgang

algorithms have been proposed, most are just variations of the Sato and Godrad algorithms. The

main disadvantage of these algorithms is the slow rate of the convergence.

4.4.2 Explicit NOS based Aigorithrn

The gradient decent algorithm does not use any explicit HOS. Explicit HOS based algorithm

employs higher-order cumulant, cepstra and their Fourier transforms (polyspectra) for

equalization of the channel. The algorithm correlates the HOS of the observed signal to reveal

non-minimum phase characteristics of the channel and outperform the Bussgang algorithm on

convergence rate. The tricepstrum equalization algorithm (TEA) is an early approach suggested

by Hatzinakos and Nikias [167], using cepstrum of the forth order cumulant which is defined in

Equation 4.1 5.

An alternative approach adopted by Shalvi and Weinstein [I681 to equalize the channel by

using a few statistics, was used for maximum kurtosis criterion. Furthermore, the super-

exponential algorithm (SEA) emerged and it was simpler to use compared to the TEA [169].

Jelonnek and Kammeyer [13] introduced an approach based on eigenvector algorithm (EVA)

which is very robust and fast in terms of performance. It is noted that TEA is basically a blind

identification procedure. This implies that this technique determines the channel before

obtaining the equalizer filter taps. On the other contrary, SEA and EVA are pure equalization

schemes that estimate the inverse of channel directly from the statistics. Figure 4.9 models a

basic blind equalization configuration for explicit HOS based algorithm.

4.5 Maximum Kurtosis Criterion

From Equation 4.13, Shalvi [168] proved a theorem that suggests; a necessary and sufficient

condition for a perfect equalization is that E[ . It is required to equalize

few statistical probability distributions of the channel input and the equalizer output. The

fundamental idea behind is to exploit the known Schwartz inequality:

where s, in Equation 4.17 represents the overall impulse response of the channel and equalizer.

This inequality is satisfied for only one non zero element of vectors,. Since the input signal to

the channel was assumed to be i.i.d., the application of this theorem to the HOS (second and

forth-order) cumulant is shown as:

Figure 4.9 Block diagram of general HOS-based equalizer

Equations 4.28 and 4.29 are derived from the elementary cumulant properties which suggest

that kurtosis and variance of input and output process must have the same sign. The cumulants

of the input and output process are substituted in the Schwartz inequality as follows: / \ 2

subject to C: (o,o,o) = a4c2 (o,o,o)

Maximize /Ci (O,O,O)/ subject to E[ x ] = e[lx12] I A/:

where a i s an arbitrary power gain through the system. This implies that the maximization of

the absolute value of the output forth-order cumulant c:(o,o,o) leads to the desired perfect

equalization as long as the right hand side of the inequality remains constant. In order to

equalize the channel, Shalvi [I691 obtained an appropriate Lagrangian equation based on forth-

order cumulant of equalizer output signal. The minimum of the fundamental cost function was

then obtained by setting the gradient of Lagrangian parameter to zero. This leads to the super-

exponential algorithm (SEA equilibrium) of Shalvi and Weinstein [169]. An appropriate

cumulant based method was developed to solve for the equalizer filter vector.

4.6 Eigenvector Algorithm (EVA)

Figure 4.10 is an extension of Figure 4.3 which schematically shows the model of EVA for

blind deconvolution [13]. The independent stationary zero-mean additive noise, s[n], corrupts

the channel output to produce the channel output y[n]. At this stage a Finite Impulse Response

(FIR), known as the blind equalizer, is introduced to the channel output y[n] to produce the out

put signal i [ n ] where the impulse response of FIR(L) is e[n]. The second filter with the same

order is introduced parallel to the blind equalizer as a reference system filter. The task of the

second filter is to generate an implicit sequence of reference data for the iteration process u[n].

The impulse response of the second filter is denoted by an]. The objective is to determine the

equalizer coefficient without access to the transmitted data x[n] from the observed (received)

data only. Similar to the SEA equilibrium produced by Shalvi/ Weinstein, the EVA solution is

based on maximum cross kurtosis criterion which defines:

:[.I

Equalizer

FIR Inverse Filter (L) Output Channel (Recovered)

I Reference I

Figure 4.10 Schematic block diagram of Eigenvector Algorithm EVA

t -+ Filter 6.nl uEn1

A

The cross kurtosis between x[n] and u[n] can be used as a measure of equalization quality:

subject to rxx (0) = c;(o) = oj (4.34)

where rxx (0) = C; (0) = oj represents the autocorrelation of the equalizer output. Since the

equalizer output is still unknown, it is preferred to express the quality function (equalization A

criterion) in terms of equalizer input (observed signal) by replacing x[n] in Equation 4.6 as:

Maximize le'c;~el subject to e * ~ , , e = o: (4.35)

where c:~ stands for cross cumulant matrix of the reference output signal u[n] and observed signal y[n], R,, represents a Toeplitz autocorrelation matrix of the observed signal y[n] and e

is the equalizer impulse response. In this equation, 0: denotes the variance of the signal. From A A

a block L of L received data samples y[O], ...y[ L-11, consistent estimates of Ryy and ~4 in

place of R, and C y can be calculated based on Equation 4.33 as follows: A 1 Ryy L - & =-ZL-lyky; k=! with Y; = {y[k],y[k -l],...y[k - V ] )

and

A A "y where represents the equalizer filter length. The values of Ryy and c4 are estimated by

unbiased sample averaging. The Hermitian cross-cumulant matrix can be calculated from the following equation and the Ciy (.) can be rewritten as scalar cross-cumulants as follows:

where the * signs denotes the conjugate complex of the element. The quality function in

Equation 4.35 is quadratic in the equalizer coefficient and can be expressed as follows: -

CYY EVA - it Ryy EVA (4.39)

Equation 4.39 is called the EVA equation and the coefficient vector eEVA is obtained by choosing the eigenvector of R;; ciY associated with the maximum magnitude eigenvector i t . It

is noted that EVA equalizes the channel very well if the magnitude of the combined impulse response sin] adopts its maximum value s,,, only once. However this condition can not be

guaranteed since the channel is unknown. It is recommended to load the reference filter with

impulse and iterative adjustment of the reference filter's coefficients. The procedure to

implement EVA can be divided into five steps: a) Take a block length of observed data (samples) called L. b) Set the reference filter with an impulse f(')(k) = 6(k-[t/2]) at the centre and the

iteration counter i = 0. From y(O),. . .,y(L-1), then estimate the autocorrelation matrix R,, based on Equation 4.36.

c) Compute the convolution of u[k] = y[k] * f(')(k) and then calculate the cumulant matrix ciY based on Equations 4.36 and 4.3 8.

d) Solve Equation 4.39 for the obtained~,, , c:Y and calculate the eigenvector e,,, by

choosing the maximum iZ in the R;; ciY . e) Load eg$,(k) into the both reference system as well as the equalizer and let

f(l+l) (k) = e$$, (k) and increase the iteration counter i + i + 1 . If i < r as a maximum iteration limit which is given in input parameters whole process is repeated.

The reason for loading an initial impulse function into the reference filter coefficient is to

achieve optimum convergence [13]. The input parameters to the EVA programs are: y[n] as the

sequence of observed vibration data, number of received data samples L, filter length and

iteration number i. An iterative procedure is taken to calculate the blind equalizer coefficient as

well as the outputs of the filters. The program outputs the final coefficient equalizer. By using

the convolution in the observed signal to the equalizer impulse response, the recovered signal

can be obtained and analysed for fault detection.

4.7 Development of Theory in This Research: Determination of Equalizer Parameters

The development of theory in this research has two features. The first feature is a bridging

functionality from the existing theory on blind deconvolution as gleaned from the literature, and

the developed theory proper, which is the second feature. The bridging functionality has been

developed as the inclusion of the computation of crest factor and arithmetic mean versus filter

length in Figure 4.1 1 part-B. The developed theory proper includes selecting optimum filter

length of the equalizer from crest factor and arithmetic mean plots and incorporating a neural

network for optimization of a general condition as indicated in the next sections.

4.7.1 Redevelopment of Blind deconvolution Algorithm

The redeveloped algorithm was used to test the feasibility of using blind deconvolution in

recovering the source signal of a defective bearing. The defective bearing used was a seeded

defect on the outer race and the signal was generated with a simulated defect on the bearing

outer ring. This signal represents the source signal (desired signal) x[k] propagating through an

unknown system (channel) h[k]. The source signal is corrupted by a periodic noise and

instrumentation and cable noise to form a sequence of noise signal n[k]. The convolution of the

desired signal with impulse response of the channel and the simulated noise are assumed

together to form the input signal y[k] for the EVA process. The objective of the EVA program

A

is to obtain an output signal x[n] as close as possible to the original input signal x[lc]. The EVA

approach, developed in [13], was chosen in this work for its simplicity and excellent

performance. A proper choice of filter length is critical in the computation of the eigenvectors.

In [13] it was shown that the convergence rate depends on the filter length of the equalizer. This

was confirmed experimentally through simulation studies and a trial and error method to

determine the optimum filter length [90, 1581. Figure 4.11 (part A) illustrates a practical

approach in determining optimum filter length using a computer simulation program. The EVA

approach with crest factor as criteria for determining the filter length value is also shown in

Figure 4.1 1. After initiating the observed signal y[k] and setting the crest factor to zero, the

input parameters to the EVA program are initialized. The input parameters are as follows: 0 v-vec: A vector (row or column) with the received sequence y[k] sampled at symbol

rate, where y[k] is the sum of the steady-state output of a possibly mixed-phase LTI

system excited by an i.i.d. non-Gaussian input sequence and stationary (coloured or

white) Gaussian noise. 0 i v ec : A vector (row or column) with the number of iterations executed for each

equalizer order. e Lk: Number of samples of a selected block of a reduced quality of received data y[l<l to

be used in EVA. It is noted that Lk must be less or equal in length of vector v-vec.

ell-vec: A length N (row or column) vector with the ascending orders of the symbol-rate

FIR equalizer to be adjusted. It is noted that the maximum number of ell-vec should be

smaller than L1c to ensure regularity of the estimate of the autocorrelation matrix. In order to initiate EVA, the reference filter with an impulse f(O)(k) = 6(k-[[/2]) is set at the

centre and the iteration counter i = 0. From y(O),. . .,y(L-1). The autocorrelation matrix R,, is

computed according to Equation 4.36. The convolution of u[k] = y[k] * fil)(k) is computed and then the cumulant matrixCyY is calculated based on Equations 4.36 and 4.38. The obtained R,,

CyY is substituted in the Equation 4.39. The coefficients of equalizer vector e$$,(k) is

calculated by choosing the maximum A in the^;; cYY . The e(E?,(k) is loaded into both the

reference system as well as the equalizer, with fii+')(k) =e$$,(lc) and the iteration counter

i --+ i + 1 increased. If i < r as a maximum iteration limit which is given in the input parameters,

the whole process is repeated. After executing the flow chart, the EVA algorithm the

coefficients of the equalizer are evaluated and used to update the reference filter's coefficients.

The formulation of this algorithm was programmed using MTLAB [170]. At the end of the

EVA process, the output result (recovered signal) for a particular filter length from the equalizer

is stored and compared with known simulated source data. The modified crest factor of the

recovered signal is calculated and the EVA process is repeated for a greater filter length value.

The graphs of modified crest factor and arithmetic mean of the recovered signal versus filter

length value are plotted and used as a feature for determining the optimum filter length.

+ Compute the autocorrelation matrix R,

I According to Equation 4.36

* Determine u[k] = y[k] * f (" (k) and C,"Y Cross-cumulant matrix

According to Equation 4.3 8 I

Substitute R, and Cty into Equation 4.39

CF = A R ,,,, e,?,,,

I Determine equalizer coefficients ej;iA (k)

From R;: C,"Y (most significant vector)

1 A

Let f "'"(k) = e$iA(k) andi -+ i + 1, x[n] = y[k] * e$L (k)

Plot the graphs of modified crest factor and arithmetic

Of the recovered signal versus filter length i

Using a compromised program to select the optimum filter length

value from CF and AM graphs and store this value in a data training set I

General Conditions: Optimum Filter Length

crest factor

kurtosis, AM B

Observed Signal optimum filter length for general conditions

Figure 4.1 1 Schematic flowchart of A: The redeveloped algorithm of blind deconvolution for optimization

of specific conditions; B: Optimization of filter length for general conditions using a neural network

4.7.2 Determining Optimum Filter Length of the Equalizer

A proper choice of filter length is critical in the calculation of the eigenvector and the

convergence rate of the equalizer. It is necessary to optimize the filter length of the blind

equalizer due to different operating conditions, size and structure of the machines. It can be

assumed that a pure defective bearing signal without other background noise is the source of

vibration. The characteristic feature of a defective bearing can be determined from the

dimensions and speed of the bearing. To determine the optimum filter length, two approaches

are developed.

A simulation test conducted with a pre-recorded bearing signal (source) and corrupted with

varying magnitude noise. The source signal can be corrupted by simulated noise to represent

machine noise. By varying the amplitude of the corrupting noise, different SNRs can be

generated to determine the optimum filter length for the equalizer. It can be assumed that the

corrupted bearing signal is the observed signal y[n] and the damaged bearing signal is the

source x[n], all shown Figure 4.3. Given the observed signal, it is required to recover the source

signal using the EVA method. From the output, certain features of the recovered signal can be

plotted versus the filter length to target the predetermined value of source features. The

optimum filter length can be selected when the plot converges close to a pre-determined source

feature value.

The optimum filter length is selected using a compromised program between modified Crest

Factor (CF) and the Arithmetic Mean (AM) plots, and is stored in a data training set. A pre-

trained neural network is designed to train the behaviour of the system and target the optimum

filter length in any general operational conditions. The input parameters for this neural network

are: crest factor, kurtosis, arithmetic mean and Mfrms. The target for this neural-network is the

obtained value of optimum filter length. At the end of the process in any general condition, the

optimum filter length is transferred to the blind deconvolution program to recover the original

source of vibration signal. The neural network is used to select the optimum filter length for a

general application where the operating conditions and types of bearing faults are not known.

This approach is discussed in detail Section 5.3. Crest factor can be used as an optimization

criterion. It has been found that the crest factor can indicate the severity of bearing vibration

and this feature has been used to identify bearing fault [69]. The Crest Factor (CF) is defined as

a ratio of the maximum peak of the signal over the RMS value. This research makes use of a

modified Crest Factor which is the average of maximum peaks over the RMS value. The

modified Crest Factor can be expressed as follows:

Modified Crest Factor Average of Maximum Peaks RMS

where xp corresponds to the amplitude of vibration signal exceeding a certain threshold; M is j

the number of sequences exceeding the threshold; xi stands for the signal amplitude, and N is

the total number of sequences in a signal. The overall vibration level of a faulty bearing

increases with an increase in the size of fault. A maintenance decision can then be made if the

vibration level exceeds a predetermined value [69]. The simplest parameter which quantifies the

difference between the spectra is the Arithmetic Mean (AM) which is defined as:

where ~i are Fourier coefficients of the vibration spectrum. The change in the value of AM

with succeeding spectra gives a measure of the change in spectra. The blind deconvolution

technique with different filter length values was applied to the observed vibration signal to

recover the source signal. The modified Crest Factor (CF) and Arithmetic Mean (AM) of the

recovered signal were plotted with varying filter lengths starting from 2 to 240 (FIR). Both CF

and AM graphs can be used to determine the optimum filter length. It is expected that the values

of CF increases as the filter length increases. The CF value stabilizes at certain filter length and

remains fairly constant up to 240 filter length. In the AM plot the AM amplitudes fluctuate

between certain values of filter length. Beyond that value, the AM amplitude remains fairly

constant at pre-determined arithmetic mean values of the source signal. There is a good

correlation between both graphs in determining the optimum filter length where the CF values

increase, the AM values decrease and both trends remain fairly constant at a certain value as

indicated later in Section 5.3. The optimum filter length is selected where the CF and AM

values remain fairly constant close to the pre-determined values of the recovered signal features

(CF and AM). The optimum filter length is then compromised between the obtained results of

two graphs and is stored in a data training set for application of a neural-network.

4.7.3 Incorporating Neural-network Technique with blind deconvolution

In the general operating condition of a machine, different parameters may influence the

optimum filter length. These parameters are rotational speed, type of fault, size of fault, shape

of fault of the rolling element bearings and load. Since some of these parameters can not be

determined without inspection during the operation of a machine, it is required to train a neural

network to assimilate the behaviour of the system and target the optimum filter length in any

general condition. Neural networks are composed of simple elements operating in parallel,

originally inspired by biological nervous systems. As in nature, the network function is

determined largely by the connections between elements. A neural network can be trained to

perform a particular function by adjusting the values of the connections (weights) between

elements. Commonly neural networks are adjusted, or trained, so that a particular input leads to

a specific target output. Figure 4.12 depicts the architecture of a neural network which is typically organized in layers [171]. Layers are made up of a number of interconnected "nodes"

which contain an "activation function". The network is adjusted, based on a comparison of the

output and the target, until the network output matches the target. Typically many such

input/target pairs are used, in this supervised learning, to train a network. The architecture of a

multilayer network (called a Multi Layer Perceptron; MLP) is shown in Figure 4.12. The MLP

contains of Input Layers, Hidden Layers, and Output Layers. The number of inputs to the

network was constrained to four which are the variables of the crest factor, kurtosis, arithmetic

mean and Mfrsm of the observed signal. The hidden layers in the MLP are the constraints used

in back propagation in the neural network. The number of neurons in the output layer or target

of the neural network is constrained to the one which is the obtained optimum filter length.

Figure 4.12 An example architecture of a neural network [171]

The reason why kurtosis was selected as one input parameter to the neural network can be

stated as because the kurtosis value indicates the distribution of the observed signal. A data training set of the neural network was obtained with different defect sizes and by varying the

speed of the shaft (refer to data in the Table 5.4). For each case the recovered signal is plotted versus the filter length. The procedure mentioned in Section 4.7.2 is repeated, and the optimum

filter length then selected for each case. The obtained filter length for each case is stored in the

data training set for training process. Figure 4.13 depicts a data training set schematic. It shows

that under any general operational conditions such as speed, kind of fault, size of fault, shape of

fault and load in each case the crest factor, kurtosis, AM and M h s of the observed signal is

computed as the input nodes of the neural network. The redeveloped algorithm of blind

deconvolution is applied to the observed signal and the optimum filter length is selected and

stored in the data training set as the target of the neural network. One of the well known

supervised learning algorithms is backpropagation. It can train multilayer feed-forward

halla


networks with differentiable transfer functions to perform function approximation, pattern

association, and pattern classification. The term backpropagation refers to the process by which

derivatives of network error, with respect to network weights and biases, can be computed. This

process can be used with a number of different optimization strategies.

Data Training Set Generalcondition :

Speed crestfactor crestfactor TypeOfFault kurtosis

3 [Optimum filter lengthIl kurtosis SizeOfFault

3 [Optimum filter lengthI2 AM AM

ShapeOfFault Mfrms , Load

Mfrms

Figure 4.13 Schematic diagram of providing data training set

The block diagram of blind deconvolution with filter optimization using the neural network

technique is shown in Figure 4.14. It shows that an observed signal is used for a well trained

neural network to choose the optimum filter length in any general conditions. The observed

signal and the determined optimum filter length from the neural network are used as the inputs

to the blind deconvolution algorithm to recover the original source of vibration.

Optimum filter length

with optimum filter length for

eneral conditions

Figure 4.14 Schematic diagram of blind deconvolution with optimum I

Chapter 5

Experimental Method

5.1 Instrumentation for Data Acquisition The whole experimental set up to collect bearing fault signal is shown in Figure 5.1

Configuration of experimental set up. The set up consists of a test rig with a faulty bearing, an

accelerometer, a charge amplifier, an external filter, Labview interface and a data acquisition

software which will be explained in the following sections.

Figure 5.1 Configuration of experimental set up

5.1.1 Experimental test rig

The experimental test to evaluate the capability of blind deconvolution is a test rig capable of

simulating common machine faults, namely, gear damage, shaft misalignment, a defective

rolling element bearing and a combination of these faults. The test rig with the first

configuration of instrumentation is shown in Figure 5.2. The rig design incorporates a damaged

bearing, a coupling disk system to impose shaft misalignment, and a gear meshing set

consisting of a damaged gear. A damaged ball bearing was placed in position 3. The bearing

fault signals were measured from position 2 and 3 with different speeds. The ability of driving

the system using two separate motors allowed the damaged and undamaged bearing signal to be

observed simultaneously. Vibration responses of bearings with no defect and with the largest

defect size were measured at different positions and speeds. An external load can not be applied

on the test bearing and the rig runs the bearing in no load condition. The overall vibration levels

do not change significantly with load. Variation of load was not studied in this experiment.

Figure 5.2 Test Rig Assembly

5.1.2 Accelerometer

The basic construction of a piezoelectric accelerometer is shown schematically in Figure 5.3

Schematic drawing of a piezoelectronic accelerometer. The transuding element consists of

piezoelectric discs which are hold between a mass and a base. When the accelerometer is

subjected to vibration, the mass will exert a variable force on the piezoelectric discs and a

variable potential will be developed across the element. The potential will be directly

proportional to the acceleration over a particular frequency range. A piezoelectric material is

one which generates a charge when subjected to a stress. The important properties of

piezoelectric material of an accelerometer are, piezoelectric constant, stiffness, dielectric

constant, resistance and curie point. The sensitivity of an accelerometer is defined as the ratio of

its electrical output to the mechanical input. Accelerometers are calibrated for charge per unit

acceleration (pC/g= pC RMS/g, RMS=pC peaklg peak) and voltage per unit acceleration

(mV/g= mV RMS/g, RMS= mV pealdg peak). In these experiments both kinds of

accelerometers were used to collect vibration signal with different configurations. The natural

frequency of the of an accelerometer is not fixed and it depends on the mass and stiffness of the

accelerometer itself and the mass and stiffness of the ball bearing housing as well as the

stiffness of the mounting method.

output terminals

Figure 5.3 Schematic drawing of a piezoelectronic accelerometer

In these experiments an accelerometer was used to measure the vibration of the test bearing.

The transducer is designed for stud mounting, and its case ground is isolated from the mounting

surface. Two different accelerometers were used in this experiment namely Bruel & Kjare 4332

and IMI 621B51 named as two configurations. Figure 5.4 (left) shows the charge per unit

accelerometer and Figure 5.4 (right) shows the volt per unit accelerometer. The first one was

used in the first configuration with a charge amplifier and it has flat frequency response up to

10 kHz. It has resonant frequency 25 kHz. The second accelerometer was used with an

amplifier and an external filter. It has flat frequency response up to 10 kHz with a 35 kHz

resonant frequency. Both can detect tiny and small size of surface fault more accurately. The

frequency response curve given on the calibration chart is for the most rigid possible mounting

of the accelerometer. A typical plot of voltage sensitivity versus frequency for the

accelerometer Bruel & Kjare 4332 is shown in Figure 5.5. It can be seen from the Figure 5.5

that the useful frequency range of an accelerometer is limited by the natural resonance

frequency of the accelerometer. Figure 5.5 shows that this accelerometer can be used for the

frequency range up to 10 kHz and the frequency is not flat beyond this point.

Figure 5.4 Charge per unit accelerometer (left), volt per unit accelerometer (right)

- I)&*: ...-.......... ~oboiiom$tlrr; , . . ..-.....- bra h v d ; YC. . ...*. . -

Figure 5.5 Frequency response of the Bruel & Kjare 4332 accelerometer

5.1.3 Charge Amplifier

In order to transform the high output impedance of the accelerometer into a lower value and to

amplify the relatively weak output signal from the accelerometer, a pre-amplifier is required for

data acquisition. The charge amplifier is used to get rid of variable accelerometer cable length.

The only necessary information is the charge sensitivity of the accelerometer. The conditioning

amplifier Type 2626 (Figure 5.6 (left)) was used as a charge amplifier. It offers conditioning

possibilities to different accelerometers and measuring requirements. The charge has a 3 digit

sensitivity adjustment network which enables its sensitivity to be adjusted to the particular

transducer used. The network is calibrated in pC1g. The charge has rated output adjustable

between 1 mV/g and 10 Vlg dependant to a certain extent on the sensitivity of the

accelerometer. Apart from amplifying the signal based the sensitivity adjustment; the charge

amplifier passes the signal through a low pass filter and a high pass filter where the upper and

lower frequency is determined by the user. In order to avoid the aliasing phenomena the cut off

frequency was s the aliasing phenomena.

Figure 5.6 Charge amplifier(left), Signal conditioning amplifier (right)

In the first configuration of experimental set up shown in Figure 5.1 a charge per unit

accelerometer was used with the charge amplifier. In the second configuration shown in Figure

5.7 the experimental set up involves a volt per unit charge accelerometer with a BK

Conditioning Amplifier Type 2626 to amplify the signal (see Figure 5.6 (right)). An external

KROHN-HITE, model 3202 filter shown in Figure 6:8(right) was used to cut off frequency at

10 kHz.

Figure 5.7 Second configuration of the simulation test rig

5.1.4 Data Acquisition

A NI connector, model NI BNC 2120 shown in Figure 5.8 (left) and a NI DAQ6062E data

acquisition card were used to convert an analog signal to the digital signal in both experimental

set up. The NI data acquisition card and Driver s i ~

shown in Figure 5.1 and 5.7.

npli f ies configuration and measurements as

FIIJIIB 1. IItlG.212U ICOlll l$lK4

Fig 5.8 Connector NI (left), external KROHN-IHITE filter (right)

Labview was used to record and analyse data. The data acquisition software was used to convert

the analog data into a digital sequence for off-line analysis. This software is a graphical

programming language that uses icons instead of text to create applications. In contrast to text

based programming languages, where instructions determine program execution, LabVIEW

uses dataflow programming, where flow of data determines execution. In LabVIEW a user

interface known as a front panel with a set of tools and objects is built. Different codes are

added to the panel using graphical representation of functions to control the front panel objects.

The block diagram which resembles a flowchart contains these codes. Figure 5.9 shows a

typical LabView Data Acquisition interface. It can be observed from Figure 5.9 that the

sampling frequency and time duration of the signal can be adjusted in this interface. The data

was sampled at 40 kHz with a cut-off frequency at 10 kHz. It is also noted that different

channels can be selected in this software to collect different signals simultaneously. It can be

observed that this software is able to analyze the signal in frequency domain.

Figure 5.9 LabView Data Analysis Interface

5.2 Benchmarking Blind Decanvolution through Computer Simulation Tests Bearing failure is one of the foremost causes of breakdown in rotating machinery. Whenever a defect on a rolling element of a bearing interacts with its mating element such as the inner race or outer race, abrupt changes in the contact stresses at the interface of the element generates a pulse or impact of very short duration. This produces vibration and resulting noise which can be monitored. Unwanted noise generated from other related structural components has higher energy than the impact vibration generated by bearing faults, and often overwhelms the vibration generated by bearing faults.

Noise s[n]

x[nl i.i.d. Unknown Y [nl Redeveloped - Channel ' b Blind

Estimation hrnl Observed Deconvolution

original of Original Signal

Input Input Signal

Figure 5.10 Redeveloped Blind Deconvolution Diagram Signal

Figure 5.10 depicts a bearing fault signal x[n] which is transmitted through an unknown channel h[n] and is increases by the contaminating noise signal s[n], which is a by-product of related structural components of many industrial systems, becoming the observed signal y[n]. The most fundamental cause of noise and unsteady running of rolling element bearings is called varying compliance vibration. An early detection of incipient bearing signal is often difficult due to its

corruption by background vibration (noise). It is essential to minimize the noise component in

the observed signal before any diagnostics can be conducted. The redeveloped blind

deconvolution algorithm is used to recover the original source signal. Simulation studies for

fault detection in rolling element bearings with corrupted vibratory signals by different kinds of

noise in the specific controlled conditions of the laboratory, was carried out and the results are

discussed in the following sections.

5.2.1 Simulation of Bearing Fault Signal

The experimental test rig shown in Figure 5.1 1 is capable of simulating common machine

faults, namely, gear damage, shaft misalignment and rolling element bearing defect. The test

bearing consists of a Deep Groove Ball Bearing 6201 and was damaged using a 1 mm cut to a

section of the outer race. The damaged ball bearing shown in Figure 5.12 was placed in position

3 of the test rig shown in Figure 5.1 1. The DaqEZ professional data acquisition software

package was used to collect and analyze the data in real time for off-line analysis. UNDAMAGED DAMAGED AC MOTOR2 BEARING BEIRING 1

DC MOTOR1 COUPLING DISC DAMACED Figure 5.12 Koyo 6201

SYSTEM GEAR SET deep groove damaged

ball bearing Figure 5.1 1 Schematic of test rig

Since the defect is located on the outer race of the bearing, the Ball Pass Frequency (BPFO) can

be calculated from the following equation [75]:

n Bd BPFO=- (1 - - cos 8 )s (5.1) 2 Pd

where n is the number of rolling elements; Bd is given by ( 4 Da- 4 da)/2 and Da and da are the

outer and inner race diameters of the ball bearing; Pd is the ball pitch diameter; 8 is the contact

angle and s is the revolutions per second. The experimental signals were measured from the test

rig at positions 2 and 3 shown in Figure 5.1 1 Position 3 in the test rig is where the damaged

bearing is placed and position 2 is a place distance from the damaged bearing position.

Although the signals were collected at different positions 2 and 3 under the same operating

conditions such as speed and sampling frequency, it can be shown in Figure 5.13 that the

signals from a further distance relative to the position of the faulty bearing have lower vibration

amplitudes compared to the signals from the position where the faulty bearing is placed. All the

signals contain information of the faults and can be used for fault detection. Figures 5.13(a) to

5.13(d) show that if the signals are collected from a far distance relative to the fault position, the

amplitudes and impacting responses have different patterns depending on the corrupted noise

and position of the accelerometer. The function of the test rig is to generate background noise

and the aim is to recover the original signal of the damaged bearing using the blind

deconvolution technique. In time domain plot, the damaged bearing signal produces a series of

impulsive vibration responses at the characteristic bearing defect frequency. Four types of

damaged bearing signals with different driving speeds can be seen in Figure 5.14. It can be

observed that when the speed of rotation increases, the time interval between the impulses

decreases based on Equation 5.1.

(a) Damaged Beanng Siynel from Pasihon 3 Volt 0.5 1 I 1 I

' 1

*'.' '(d ~amoged ~ e i r i n g Signsl from ~ds i t ion 3 pius Noise of damaged gear I

-0 L m*--m.*-me 1 m-m-w-*M- A ------------ I (dl Datnagsd Bsanng Siy nal from Position 2 plus Noise of damaged gear

0 5

-0.5 85 0 1042 0.2083 0 3t25 0.418"r'ls)

Figure 5.13 Measured signals from different positions

5.2.2 Periodic Noise

The technique of blind deconvolution is assessed based on its ability to recover the input signal

when the channel is unknown. The robustness of blind deconvolution can be tested by

corrupting the observed signal with the simulated noises. These simulated noises can be divided

into two categories; deterministic and random noises. To simulate a real case, a periodic noise

was generated in the laboratory using a deterministic noise generator. The amplitude and

frequency of the generated noise was varied to produce different signal-to-noise-ratios (SNR).

Three different SNR were chosen starting from -8 dB to -43.9 dB. Figure 5.15(a) depicts a

typical impacting bearing fault signal which is also shown in Figure 5.15(a) at rotating speed of

500 RPM, using the faulty bearing shown in Figure 5.12. The observed signal was further

corrupted by 500 Hz sinusoidal noise with -8 dB SNR which is shown in Figure 5.15(b). Figure

5.15(c) shows that the blind deconvolution technique has the capability of removing sinusoidal

noise with a -8 dB SNR, recovering a clean signal as close as possible to the source signal

(bearing fault signal). At this stage the filter length of equalizer was selected by trail and error.

Tlme Domaln Observed Signal 500 RPM

Tlme Domaln Observed Slgnal 1000 RPM

1500 RPM 2000 RPM 0.15 1 I 0 4 , I

0.1 0.3 - 0 0 5 0 2

> - 0 - 0.1 EZ e U

-0.05 0 P > -0 1 -0.1

-0.15 -0 2

-0 2 -0.3 0 0.2 0 4 0.6 0 8 1 0 0 2 0 4 0.6 0.8 1

Time (Second) Time (Second)

Figure 5.14 Defective bearing signals recorded at different speeds (a) o =5OORPM;

(b) o =1000RPM; (c) o =1500RPM; (d) o =20000RPM

(a) Observed Signal 5OORPM - 0.5

(b) Corrupted Signal with SIN noise SNR=-8 dB - 0.5 t 1 I . . +- - B - S 0 .- +- C!

9 -0.5

(c) Recovered Signal After Blind Deconvolution ! I I I

Time ( Second )

Figure 5.1 5 (a) Observed signal with 500 RPM (b) Corrupted signal with sinusoidal noise with

500 Hz and SNR=-8 dB (c) Recovered signal using blind deconvolution

Figure 5.16(a) shows a typical impacting observed signal rotating at 500 RPM using the faulty

bearing shown in Figure 5.12. The observed signal was further corrupted by 500 Hz sinusoidal

noise with -20 dB SNR, shown in Figure 5.16(b). Figure 5.16(c) shows that the blind

deconvolution technique has the capability of removing sinusoidal noise with a -20 dB SNR and

recovering a clean signal as close as possible to the source signal. The lowest SNR reached was

-43.93 dB which is shown in Figure 5.17(b). Figure 5.17(c) shows that the blind deconvolution

technique has the capability of removing sinusoidal noise with a -43.93 dB SNR, recovering a

clean signal as close as possible to the source signal (bearing fault signal). For higher SNR, the

bearing fault signal impacts are more prominent and for lower SNR the bearing fault impacts

are masked by background noise. (a) Obsened Signal 5OORPM - 0.5 I I I I I I I

(b) Corrupted Signal with SIN noise SNR=-20 dB - 0.51 I I I I t I I 1

> -0.5 I I I 1 I I I I I I

(c) Recovet.ed Signal After Blind Deconvolution ,, 0.5 I

..- - - 3 - 6 0 . - +- C!

9 -0.5 0 0.085 0.17 0.255 0.33 0.415 0.50 0.85 0.66 0.745 0.83

Time ( Second )


500 Hz and SNR=-20 dB (c) Recovered signal using blind deconvolution

In practice the defect bearing signal is not as clear as the signals in Figure 5.14 and the observed

signals measured from an accelerometer are contaminated by unknown vibration (noise). In

order to understand the affects of blind deconvolution on the speed parameter, the blind

deconvolution technique was applied to observed signals to recover the original bearing signal

at four separate speeds. The input of the blind deconvolution algorithm is the observed signal

and the output is the source signal. Figure 5.1 8 compares the observed signal with contaminated

noise before blind deconvolution and the recovered signal after blind deconvolution in the time

domain. It can be seen from Figure 5.18 that the recovered signals are cleaner than the observed

signals in terms of identifying the impacts and the average time interval corresponds well to

BPFO. In order to minimize the error of the average between the impacts, and since the

outcome of each experiment can be different it is required to repeat the experiments for at least

ten different realizations and take the average of the time intervals between impacts.

5.2.2.1 Time Interval Averaging

In order to minimize the error in calculating the mean (Bias) and the variance of estimated time

interval between the impacts it is required to collect a long signal as indicated in preliminary

literature review [33, 63, 64, 66, 761. The length of the observed signal is N=20000 samples

(equal to 1.667 seconds using 12k sampling frequency), the length of the equalizer and iteration

number were set at 32 and 120 respectively to obtain the best results by trail and error. The time

period (intervals between the impacts) of the recovered signal after blind deconvolution was

calculated using a pre-set threshold to get rid of the minor fluctuation around the spikes, as

well as the unknown noise which comes from the other components. In order to get rid of

minor fluctuations and probable noises the threshold is set to five times the standard deviation

of the deconvolved signal (1721. Then the next largest peak is found and W samples around the

peak are set to zero. The value of W is usually taken to be less than half of the interval samples.

This procedure is repeated until no more peaks are found. The detected time interval between

the impacts can be regarded as a random variable since the outcome of each experiment can be

different. Table 5.1 shows the results of ten different realizations or observations in terms of the

four speeds. Ten tests were carried out in the same conditions and the blind deconvolution

technique was applied to these signals. Taking the average of all intervals for each speed, the

time between impacts and thus the characteristic defect frequency for outer race fault was

computed and compared with BPFO in Equation 5.1. It can be seen in Table 5.1 Average

Frequency (Time inverse) of all Time Intervals that the calculated average defect frequency is

almost close to the BPFO. The variation in these frequency values at each speed is due to the

fact that a constant speed was not maintained during data collection.

Table 5.1 Average Frequency (Time inverse) of all Time Intervals

4th Observation

5th Observation

6th Observation

7th Observation

gth Observation - --

9th Observation

2OOORPM

83.63 (Hz)

84.40

83.73

1 oth Observation

Average (Hz)

BPFO (Hz)

1500RPM

62.85 (Hz)

60.28

62.82

Shaft Speed

lSt Observation

2nd Observation

3rd Observation

23.50

21.82

21.62

19.80

20.35

22.86

24.1 1

22.27

21.21

5OORPM

22.88 (Hz)

21.62

24.16

41.17

43.71

42.32

41.84

41.77

42.39

1 OOORPM

42.37 (Hz)

40.58

42.36

43.53

42.20

42.42

61.61

62.89

62.41

63.15

62.56

62.52

83.89

83.67

83.71

83.71

83.93

83.68

62.39

62.35

63.63

83.7 1

83.81

84.84

The performance of blind deconvolution can be evaluated more effectively if the original input

signal of the channel is already known. Using the sum of the squared deviation (SSD) of each A

sample for the estimated signalx[n], and the known original input signal x[n] can be an

indication of the performance of blind deconvolution or perfect estimation of the inverse

channel [119]. The SSD can be expressed as follows:

N A

SSD = x [ x[n] - x[n] ]

A small value for SSD indicates a good estimation of the inverse channel, but it can only be

calculated when the original signal x[n], is known. In the present study x[n] is unknown thus

this method is not applicable. There are some other criteria which could be used instead of

SSD, such as the full width at the half maximum (FWHM) of the highest peak in the output of

the equalizer [173]. This criterion is independent of the input signal and it is useful when the

width of the impact signal can be measured in the time domain in particular when the ringing

waves are considerably high. Another approach to evaluate blind deconvolution performance is

to compare the recovered signalx[n] with the observed signal y[n] by employing a trimmed

standard deviation (TSD) of the observed signal and the recovered signal as follows [174]:

A

where N the is number of samples of the recovered sorted signal Xi and; A A A A

(xi 1 2, . . . N), where XI 5 x,... 5 xN (5.4)

The value of TSD is an indication of noise in the signal and is a measurement of energy. The

trimmed mean, TM, is defined as:

where T is the length of truncated samples with the nearest integer y~ where y is the percent of

the highest and lowest data. Small values of TSD indicate a good estimation of the inverse

filter. In this experiment the filter length was set at 32 and the number of block samples at L=N=12000. The value of TSD was computed with fixed y =0.05 to exclude the largest and the

smallest spikes. The calculated TSD for the observed and recovered signals are shown in Table

5.2 TSD of the observed and recovered signals. The results show that the values of TSD for the

recovered signals are reduced when compared to the values of TSD for the observed signals for

each speed. This is because the impulsive bearing fault signals are extracted by blind

deconvolution and the contaminating noise is suppressed. Hence a reduced TSD value for the

recovered signal is obtained.

Table 5.2 TSD of the observed and recovered signals

5.2.3 Summation of Periodic Noises

Shafi Speed

The most fundamental cause of noise in rolling element bearings are the so called varying

compliance vibration. These are parametrically excited vibrations that occur irrespective of the

manufactured quality and accuracy of the bearing. In real operating systems most noises from

Observed Signal

yrn1

faulty components of a machine are generally periodic. Therefore it is essential to simulate a

Recovered signal

ir.1

combination of periodic noise and determine the performance of blind deconvolution in this

case. (a) Obsetved Signal 5OORPM

I I ! I I I

(b) Corrupted Signal with Two SIN noise at 500, 1000 Hz SNR=-29 dB I I I I I I I I I

- .-

(c) Recovered Signal After Blind Deconvolution - 0.51 I t t I I

Time Second )

Figure 5.19 (a) Observed signal at 50 Hz; (b) Corrupted signal with combination of 2 sinusoidal

noise at 500, 1000 Hz; (c) Recovered signal after blind deconvolution

The bearing fault signal with a rotational speed of 500 RPM shown in Figure 5.19(a), was

corrupted with a summation of two periodic noises at 500 and 1000 Hz. The blind

deconvolution technique with a random filter length was applied to the corrupted signal to

obtain a recovered signal shown in Figure 5.19(c). Figure 5.19(c) shows that the blind

deconvolution technique has the capability of removing sinusoidal noise with a -29 dB SNR and

recovering a clean signal as close as possible to the source signal (bearing fault signal). The

bearing fault signal was further modified using a summation of five different levels of periodic

noise frequencies shown in Figure 5.20(b). The frequency of corrupting noises were 25, 500,

1000,2000,4000 Hz. It can be observed from Figure 5.20(b) that the original vibration signal is

severely corrupted by the simulated noise and the bearing fault impacts are invisible. Figure

5.20(c) shows that the blind deconvolution technique has the capability of removing sinusoidal

noise with a very low -49.64 dB SNR and recovering a clean signal as close as possible to the

source signal (bearing fault signal). It can be seen from Figure 5.20(b) that the bearing fault

signal was corrupted by a frequency component of 25 Hz being very close to the BPFO and

ended with a very high corrupting frequency component of 4000 Hz, covering a broad range of

frequency domain. Blind deconvolution successfully eliminated these components and the time

interval between impacts of the recovered signal shown in Figure 5.20(c) was calculated to be

about 21.41 Hz which is very close to the BPFO, 21 Hz. It is noted that the impacts in the

observed signal are buried in heavy background noise which are not easily detectable by other

methods; even to classify the type of fault under such a low SNR.

Observed Signal 503RPM 0.5

-0.5 (b) Corrupted Signal with SIN noise SNR=-49.6846

0.5

(c) Recovered Signal After Blind Deconvolution 0.2

-0.2 0 0.083 0.167 0.25 0.333 0.416

Time (second)

Figure 5.20 (a) Observed signal at 500 Hz; (b) Corrupted signal with summation of 5 sinusoidal

noise frequencies at 25, 500, 1000,2000 and 4000 Hz; (c) Recovered signal after BD

5.2.4 Random Noise and Notch Filter

Although the blind deconvolution technique was designed to work with white Gaussian noise,

the results show that if the observed signal is corrupted by either a random noise or a white

noise, blind deconvolution is not able to remove the noise. Figure 5.21(b) shows a corrupted

observed signal with a random noise. The SNR obtained was 3.8539 dB which is a low level of

corruption. It can be seen from Figure 5.21(c) that the corruption noise is still present after blind

deconvolution. The reason can be attributed to the fact that blind deconvolution only identifies a

prominent corrupting frequency component such as distinguished frequency in periodic noise.

In order to determine wether the noise components have been removed in the blind

deconvolution process, the gain response of the FIR filter or equalizer is plotted. It can be seen

that this equalizer acts as a notch filter and removes all corrupted sinusoidal noise. Figure 5.22

shows the gain response and phase plots of the equalizer that was used in Figure 5.17 and a

bearing fault signal subjected to 500 Hz sinusoidal noise with -43.93 dB.

Obsetved Signal 503RPM 0.5

0

-0.5 I I I I I I

jb) Corrupted Signal with Guassian noise SNR= 3.2140 0.5

0

-0.5

0.5

0 I I I , . I

-0.5 I I I I

0 0.17 0.33 0.50 0.67 0.83 Time (second)

Figure 5.21 Corrupting the Observed signal at 5OORPM with random noise and the result after

blind deconvolution

The gain response of the equalizer was plotted using the computed coefficient of the blind

equalizer e[n] (Figure 4.3). It can be observed from Figure 5.22 that there is a notch in the gain

response of the filter at a frequency of exactly 500 Hz. The technique was designed to

automatically remove the induced noise at the corrupting frequency. Although there was no

prior knowledge of the introduced noise (including the frequency) for the eigenvector

algorithm, it can be seen that the equalizer successfully eliminated the noise component at 500

Hz. The gain response of the equalizer was only plotted for the lowest SNR to analyse the worst

case and higher SNR was not considered. In general, all the noise that is likely to come from

other faulty components on a machine are periodic and can thus be treated the same as the

simulated noise. Figure 5.23 shows the gain response of an equalizer which was subjected to a

summation of four different frequencies sinusoidal noises specifically 500 Hz, 1000 Hz, 2000

Hz and 4000 Hz. The results confirm that blind deconvolution successfully removes sinusoidal

noise frequencies between 500 Hz and 4000 Hz. The blind equalizer filter shows that the signal

at these frequencies has been suppressed. Figure 5.23 shows four notches at the four studied

frequencies. It can also be observed that the phase plot versus the frequency is linear. The gain

response of the equalizer was only plotted for the lower SNR values to analyse the worst case. Gain Response of Eaualizer for

10 nois=0.l'"sin(2*pi*500*t) I I I I I

._-c

............. L.......... .. ..

.

............ 1 ..............#.............. >.............-

............ : .............. L ............. :

-40 I I I I I

0 1000 2000 3000 4000 5000 6000 Frequency (Hz)

Frequency (Hz)

Figure 5.22 Gain response and phase plot of an equalizer for a sinusoidal noise with 500 I-Iz

frequency Gain Response of Equalizer for

nois=0.1"sin(2"pi"500Xt~+0.1*sin(2"pi"l 000*tj+0.1*sin(2*pi*2000st~+0.1"sin(2"pi*4000*t~ 20 I I I I I

-60 I I I I I

0 1000 2000 3000 4000 5000 6000 Phase (Hz)

1000 I I I I I

0 ...........................

F m a, n -1000 - a, LO ; -2000 .......... i ..............

-3000 I I I I I

0 1000 2000 3000 4000 5000 6000 Frequency (Hz)

Figure 5.23 Gain response and phase plot of an equalizer for a summation of sinusoidal noise

frequencies at 500, 1000,2000, and 4000 Hz

5.2.5 Results and Discussions

This research has revealed the advantages of blind deconvolution to recover the original signal

of a typical faulty bearing corrupted by noise and distorted by the signal transmission path. The

principal reason for conducting the simulation study in this research was to asses the

effectiveness of the blind deconvolution algorithm as a technique for enhancing pre-signal

processing, in recovering bearing fault signals. It is difficult to detect defects in the direct

spectrum of bearing characteristic rotational frequencies because they are either absent or occur

without a fault being present [9]. Detecting incipient failure in rolling element bearings is

difficult. In a situation where the vibration is contaminated by either background noise or

unwanted components, other normal fault detection techniques may fail to detect a growing

defect at the incipient stage, due to the relatively low signal-to-noise-ratio (SNR). In [67] it was

shown that Power Cepstrum as an alternative enhancing enabled the diagnosis of a fault on

outer race effectively but it failed to detect inner race defects. In [4] it was reported that

adaptive noise cancellation (ANC), the another enhancer technique could be applied to improve

the SNR of the monitored signal from a complex machine. It was shown that the ANC

technique works very well in situations where the noise in the two inputs are mutually

correlated and the reference input contains no signal or a very weak signal. It was further

observed that the ANC technique requires a minimum of two inputs where the second input

reference signal is greatly dependant upon the location of the probe. This method thus needs a

second reference measurement, which is correlated only with either the background noise or the

impulsive signal. There are applications where such a reference signal is not readily available,

justifying blind deconvolution technique.

In the simulation study the expected value of time intervals between the impacts of a faulty

bearing signal, indicating defect frequency as a random variable, was considered using blind

deconvolution as a feature extraction technique. The time estimation between impacts was

improved and a better characteristic defect frequency achieved. It was further found that this

technique has the capability of removing sinusoidal noise with a very low SNR, recovering a

clean signal. The technique works very well with periodic noise whereby the equalizer behaves

like a notch filter in removing the noise at the corresponding frequencies. The observed signal

was also corrupted with a frequency component of 25Hz which is very close to BPFO, which is

misleading since the notch is very close to BPFO and the technique obtained the recovered

signal representing a typical outer race faulty bearing signal. The efficiency and robustness of

the proposed algorithm was assessed using different levels of corrupting noises. The result

showed that the proposed algorithm works very well with a range of periodic noise and the

technique is successful in removing the various noise components. It was found that the

recovered signal had in fact improved when compared with the observed signal. However, for

random noise this technique is incapable of removing the noise. It was also found that the

severity of the corrupting noise SNR has no effect on the created notch in the gain response

plot.

5.3 Determining Blind Deconvolution Effectiveness and Optimum Filter Length

5.3.1 Plan for Bearing Damage

In order to optimize filter length based on the general conditions of a vibration signal including

type of fault on the bearing components, nine Koyo 6201 ball bearings were damaged according

to the specifications presented in Table 5.3 Defect specifications of bearings. The tenth bearing

was used as a base for healthy bearing. The experiments were carried out in three different

speeds as presented in Table 5.3. It can be observed from Table 5.3 that the size of this

rectangular groove fault starts from 0.1 mm in width of the groove up to the 0.5 mm. The fault

is seeded in the three rolling element components; outer race, inner race and ball. The shape of

the faults on the outer and inner races simulate crack and the ball simulates spot with varying

diameters.

Table 5.3 Defect specifications of bearings

Table 5.4 Different speeds of experiments, indicates three different speeds, cut off frequency

No.

1

2

3

Quantity

and sampling frequency for each experiment. Sets of data with different operational conditions

were collected to provide data training sets as discussed in the theory of determining filter

length in Chapter 5. Table 5.5 Data files for the simulation experiments, presents data files for

the simulation experiments. In Table 5.5, C is the configuration number and A is the amplifying

factor.

Fault location & type

Outer

race

(width)

Inner

race

(width)

Ball

(diameter)

Spot

0.1

0.2

0.5

3

Crack

0.1

0.2

0.5

3

0.1

0.2

0.5

3

Table 5.5 Data files for the simulation experiments

5.3.2 Bearing Damage Technique

A laser cutter was used to artificially damage the ball bearings. The required damage was

initially drawn as a 3 dimensional solid model and then transferred to the laser machine where

the laser software was able to generate the cutting program from the 3D model. The ball

bearing was then set up on the 'work table' of the machine and x, y and z positions were

established. Basically two laser beams are generated to intersect at a focal point, where cutting

is required. A sample of a damaged bearing with an outer race defect is shown in Figure

5.24(a). The defect is a crack with 0.1 mm width. An inner race crack with a width of 0.2 mm is

shown in Figure 5.24(b). A ball defect which resembles a spot with the width of 0.5 mm is

shown in Figure 5.25. The depth of defect was 0.5 mm from the bottom of the race for inner or

outer race and from the surface for the ball.

ize

(rnrn)

0.1

0.2

0.5

Ball Inner race Speed

(RPM)

600

1200

1800

600

1200

1800

600

1200

1800

Outer race

Name

IS1

IS2

IS3

IMl

IM2

IM3

ILl

IL2

IL3

Healthy

Name

OS1

OS2

OS3

OM1

OM2

OM3

OL1

OL2

OL3

A*

0.1

0.1

0.1

0.1

0.1

0.1

10

10

1

Name

BS1

BS2

BS3

BMl

BM2

BM3

BLl

BL2

BL3

C*

2

2

2

2

2

2

2

2

2

A*

0.1

0.1

0.1

C*

1

1

1

1

1

1

2

2

2

Name

HI

H2

H3

C*

1

1

1

1

1

1

2

2

2

A*

10

10

1

10

10

1

10

10

1

C*

1

I

1

A*

0.1

0.1

0.1

0.1

0.1

0.1

10

10

1

Figure 5.24 (a) Outer race defect with 0.1 rnm width (left), (b) Inner race defect with 0.2 mm width

Figure 5.25 Ball defect which resembles a spot with width of 0.5 rnm

Figure 5.26 illustrates a cross section view of the damaged bearing to show the depth of crack.

The crack starts from the bottom surface of the groove. In order to investigate the affect of size

of damage on the vibration signals, and provide a comprehensive data training set, the width of

the rectangular crack was varied according to Table 5.3 and the depth of fault was kept

constant. The characteristic defect frequency of the bearing for each case was tabulated for

comparison.

Figure 5.26 Cross section at view of ball bearing

5.3.3 Measured Bearing Damage Signals

5.3.3.1 Outer Race Defect Experiments

Figure 5.27 represents a typical observed signal from an outer race defect experiment. The size

of fault is 0.1 mm in width. In Figure 5.27 three different speeds are plotted and the

characteristic defect frequency is computed for each speed, matched with average time interval

between the impacts. Observed S~gnal Outer Race Fault 0 1 600 RPM

O l 1 I I 8 , I

Obsewed Slgnal Outer Race Fault 0.1 1200 RPM 0 5 , , , I I

- 0 5 l I I I I I I I 1 O b s e ~ e d Signal Outer Race Fault 0 1 1800 RPM

I, I I I I I I I I

-1 1 1 I I I I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4

Tlme ( Second )

Figure 5.27 A typical observed signal for an outer race fault 0.1 mm rotating at (a) 600 RPM

(b) 1200 RPM (c) 1800 RPM



characteristic defect frequency is computed for each speed, matched with the average time

interval between the impacts.

(a) Obsewed S~gnal Outer Race Fault 0 2 600 RPM

-0 2 1 I I I I I I I I

ibi Obse~ed Slanal Outer Race Fault 0 2 1200 RPM

(c) Obseived Slgnal Outer Race Fault 0 2 1800 RPM 0.4 1 I I I I % I I I

- 0 4 ~ I I I I I I I 1 0 0 05 0 1 0 15 0 2 0 25 0.3 0 35 0 4

Time ( Second )


(b) 1200 RPM (c) 1800 RPM





(a) Observed S~gnal Outer Race Fault 0 5 600 RPM I I I I I I

ib l Observed Sianal Outer Race Fault 0 5 1200 RPM

(c) Observed Stgnal Outer Race Fault 0 5 1800 RPM 0 2 , , 4 ,

-0 2 1 I I I I I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4

Tlme ( Second )


(b) 1200 RPM (c) 1800 RPM

5.3.3.2 Inner Race Defect Experiments

Figure 5.30 represents a typical observed signal from an inner race defect experiment. The size




(a) Observed S~gnal lnner Race Fault 0 1 600 RPM I 1 I I I

(b) Obsewed Signal lnner Race Fault 0 1 1200 RPM

- - lc i Observed S~anal lnner Race Fault 0 1 1800 RPM

-021 I I I I I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0.4

Tlme (Second )

Figure 5.30 A typical observed signal for an inner race fault 0.1 mm rotating at (a) 600 RPM

(b) 1200 RPM (c) 1800 RPM

Figure 5.3 1 represents a typical observed signal from an inner race defect experiment. The size




(a) Obsetved Signal inner Race Fault 0 2 600 RPM

- ,

(b) Observed Slgnal lnner Race Fault 0 2 1200 RPM

(c) Obsetved Signal lnner Race Fault 0 2 1800 RPM 0 2 , I I I I I I I

-021 I I , I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4

Time ( Second )

Figure 5.3 1 A typical observed signal for an inner race fault 0.2 mm rotating at (a) 600 RPM

(b) 1200 RPM (c) 1800 RPM

Figure 5.32 represents a typical observed signal from an inner race defect experiment. The size



interval between the impacts. (a) Observed Slgnal lnner Race Fault 0 5 600 RPM

n c

[b) Observed Slgnal lnner Race Fault 0.5 1200 RPM

(c ) Observed Slgnal lnner Race Fault 0 5 1800 RPM

-021 I I 1 I I 1 I I 0 0.05 0 1 0.15 0 2 0 25 0 3 0 35 0.4

Tlme (Second )

Figure 5.32 A typical observed signal for an inner race fault 0.5 mm rotating at (a) 600 RPM

(b) 1200 RPM (c) 1800 RPM

5.3.3.3 Ball Race Defect Experiments

Figure 5.33 represents a typical observed signal from a ball defect experiment. The size of fault

is 0.1 mm in diameter. In Figure 5.33 three different speeds are plotted and the characteristic

defect frequency is computed for each speed, matched with the average time interval between

the impacts.

(a) Observed Slgnal Ball Race Fault 0.1 600 RPM 0.05 1 I I I f I I I

(b) Observed Signal Ball Race Fault 0.1 1200 RPM 011 I I I I I I I

(cj Observed Signal Ball Race Fault 0 1 1800 RPM 0 2 I I I I I t I

-

-0 2 1 I , I I I I I I

0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 Ttme j Second )

Figure 5.33 A typical observed signal for a ball fault with 0.1 mm in diameter rotating at

(a) 600 RPM (b) 1200 RPM (c) 1800 RPM




the impacts. (a) Observed Signal Bail Race Fault 0 2 600 RPM

(b) Observed Slgnal Ball Race Fault 0 2 1200 RPM 0 1 ( I , I , , I I

(c) Obsewed Slgnal Ball Race Fault 0 2 1800 RPM

-021 1 I I I I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4

Time ( Second )


(a) 600 RPM (b) 1200 RPM (c) 1800 RPM




the impacts.

(a) Observed Slgnal Ball Race Fault 0 2 600 RPM 0 2 1 I I , I I

(b) Observed Signal Ball Race Fault 0 2 1200 RPM 011 I I I , , I I I - 1 I

t <? ' . . -. I I. . 6 0

. , o s l l l ' r l 1 ; ' 1 ',.. I ' I " " ' , "'I > >

(c) Observed Signal Ball Race Fault 0 2 1800 RPM 0 1 I I I I I I I

-0 11 I I I I I I I

0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 T~me ( Second )


(a) 600 RPM (b) 1200 RPM (c) 1800 RPM

5.3.4 Fault Detection Using Optimum Filter Length

In order to obtain the optimum filter length, the blind deconvolution technique with different

filter length values was applied to the observed vibration signals to recover the source signal. A

typical trace of the observed signal from a far distance of the fault position with 600 RPM and

0.1 outer race defect can be seen in Figure 5.36(a) and the frequency spectrum of the observed

signal in Figure 5.36(b). The modified Crest Factor (CF) and Arithmetic Mean (AM) of the

recovered signal were plotted with varying filter length starting from 2 to 240 FIR. The typical

graphs of CF and AM are shown in Figures 5.36(c) and 5.36(d) respectively. It can be seen

from Figures 5.36(c) and 5.36(d) that both the CF and AM graphs can be used to determine the

optimum filter length. In Figure 5.36(c) the value of CF increases as the filter length increases.

This is an indication that the average peak gets larger over the RMS value. The values stabilize

at around 186 filter length and remained fairly constant all the way to 240 filter length. In the

AM plot, Figure 5.36(d), it can be seen that the AM amplitudes fluctuate between -75 dB to -90

dB until 186 filter length. Beyond that value, the AM amplitude remained fairly constant at

about -90 dB. It can be seen that there is good correlation between both graphs in determining

the optimum filter length. Because where the CF values increase, the AM values decrease and

both trends remain fairly constant at a certain value.

Out0 1RPM600F1ltlOkHzSampl40kHzPosi2 (Modlfled) T~me Domaln Obsewed Signal q0-3 Frequency Domain Signal

1 .Q6 1 4 1

9 -70

- 8 - s U

a -90 - - TI

4 0 40 80 120 160 200 240

- 9 5 40 80 120 160 200 240

F~lter Length Fllter Length

Figure 5.36 (a) Observed signal of an outer race defect at 600 RPM

(b) Observed signal in frequency domain sampled at 40 kHz (c) Modified Crest Factor

graph versus filter length of the equalizer (d) Arithmetic Mean graph versus filter length

The top trace of Figure 5.37 illustrates the observed signal with an outer race defect rotating at

600 RPM and the bottom trace shows the recovered signal with the optimum filter length. In the

left top trace, it is not possible to see the damaged bearing signal and the impulses are masked

by background noise. The bottom trace shows a consistent impulsive signal of a damaged

bearing with an average time interval of 0.039 seconds very close to the characteristic defect

frequency. A healthy bearing with Gaussian distribution has a kurtosis value close to 3. The

kurtosis of the observed signal was found to be 2.78 because it indicates a Gaussian

distribution, while the kurtosis of the recovered signal was 9.06 and it can be seen that the

recovered signal was improved. Figure 5.38 shows the result of a demodulation process

performed on the recovered signal with the outer race defect shown in Figure 5.36. In the

demodulation process, the signal was filtered with band pass between 5000 and 7500 Hz which

was to be the correct bands pass to be the range that would result in the detection of the best

characteristic frequency (25.5 Hz). The reason for this is that the characteristic defect frequency

with a dominant spike at 25.6 Hz is clearly visible in the spectrum which is close to the

calculated frequency of 25.5 Hz. The spike is accompanied by a number of harmonics spaced at

multiples of the characteristic frequency. Time Doman Observed Slgnal Kurloas=2 78

--. ..-7----.-T

-----I

Tlme Damaln Recovered Signal Kuttosls=9 06

' g6r---T-- '

-1 9lj0 0 , 0 I , I # I

0.1 0 2 0 3 04 0 5 0 6 0.7 0 8 0 9 1 Time (Second)

Figure 5.37 Top- Observed signal with an outer race defect, Kurtosis=2.78, Bottom- Recovered

signal with the optimum filter length L=186, Kurtosis=9.06 Demodulated Recovered Signal F1lterLength=186

0 00147 r

Characteristic Defect Frequency

0 00123 -

Frequency (Hz)

Figure 5.38 Demodulated recovered signal at 600 RPM with outer race defect

5.3.5 Data Training Set for Optimization Based on a General Condition

A neural network was used to train the behaviour of the system and target the optimum filter

length. Nine different artificially damaged bearings with outer race defect, inner race defect and

ball defect with varying defect sizes were placed in the housing of the test rig shown in Figure

5.1, position 3, to collect the vibration signals. A data training set of the neural network was

obtained with different defect sizes and by varying the speed of the shaft from 600 RPM to

1800 RPM. For each case the recovered signal after blind deconvolution was plotted versus

filter length. The optimum filter length was selected based on modified crest factor and

arithmetic mean criterion. Figure 5.39 shows the plot of CF and AM versus filter length for an

outer race fault of 0.1 mm defect width with rotational speed of 600 RPM. An automatic

program selects the optimum filter length and stops the algorithm where the variation of CF and

AM is minimum and graphically is smooth. It can be observed from Figure 5.39 that the CF

values stabilized at around 70 filter length and remained fairly constant all the way to 240 filter

length. In the AM plot it can be seen that the AM amplitudes decrease from -35 dB to -50 dB

until 70 filter length. Beyond that value, the AM amplitude remained fairly constant at about - 50 dB. The optimum filter length of 70 for this particular case was then input to a data training

set for the training process. Figure 5.40 shows the plot of CF and AM versus filter length for an

inner race fault of 0.1 mm defect width with rotational speed of 600 RPM. It can be observed

from Figure 5.40 that the CF values stabilized at around 30 filter length and remained fairly

constant all the way to 240 filter length. In the AM plot it can be seen that the AM amplitudes

decrease from -95 dB to -1 15 dB until 30 filter length. Beyond that value, the AM amplitude

remained fairly constant at about -1 15 dB. The optimum filter length 30 was then input to a data

training set for the training process. Crest Factor Modified

OuterRaceO I GOORPM

Arithmet~cMean (dB) & Opt1mumFiiter=70 -35

-40

-45

-50

-55 0 50 100 150 200 250

Filter Length

Figure 5.39 CF and AM plot for Outer Race Defect 0.1 mm width 600 RPM

Crest Factor Modified InnerRaceO I GOORPM

I I I I i

I I I I I 50 100 150 200 250

Filter Length

Figure 5.40 CF and AM plot for Inner Race Defect 0.1 mm width 600 RPM

Although there was a good agreement between both CF and AM plots to select an optimum

filter length for outer race and inner race faults, it can be observed that for a ball defect these

plots would not be the same. I-Ience the program selected an optimum filter length which is a

compromise between these two criteria. Figure 5.41 shows the plot of CF and AM versus filter

length for a ball defect of 0.1 mm width with rotational speed of 600 RPM. It can be observed

from Figure 5.41 that the CF values stabilized at around 80 filter length and remained fairly

constant all the way to 240 filter length. In the AM plot it can be seen that the AM amplitudes

fluctuate from -35 dB to -50 dB until 110 filter length. Beyond that value, the AM amplitude

decreases to -55 dB. A compromised optimum filter length of 98 was selected and was then

input to a data training set for the training process.

Four vibration parameters were considered as the input parameters to the neural network

namely; crest factor, kurtosis, arithmetic mean and Mfrsm of the observed signal. The target of

the neural network was set at the obtained optimum filter length. Table 5.6 shows the provided

data training set for each case in terms of type of fault, speed and fault size. A well known

supervised learning algorithm for neural network is backpropagation. It can train multilayer

feed-forward networks with differentiable transfer functions to perform function approximation,

pattern association, and pattern classification. Based on Figure 4.14 Schematic diagram of blind

deconvolution with optimum filter, this data training set is used for a well trained neural

network to choose the optimum filter length in any general condition.

Figure 5.41 CF and AM plot for Ball Defect 0.1 mm width 600 RPM

Table 5.6 Data training set for the neural network OuterO.l 600RPM Outer0.11200RPM OuterO. I I800RPM

CF 1.41E+01 1.26E+O 1 1.03Et-01 Kurtosis 7.50E+00 2.06E+Ol 1.53E+01

AM 1.45E+O 1 4.04Et-01 5.57E+O 1 Mfrms 6.41E+01 6.84 E+Ol 9.29E+01

-.# -~

Optimum Filter Length 7.00E+Ol 4.60E+01 5.60E+Ol

Outer0.2 6OORPM Outer0.2 1200RPM OuterO.2 18OORPM CF 1.68Ei-0 1 9.1 8E+00 7.86Ei-00

Kurtosis 4.44E+0 1 1.79Et-0 1 1.37E-+O 1

AM 1.73E+O 1 1.70E+OI 3.40E+O 1 Mfrmr 7.36E+O1 6.829E+0 1 1.14E+02

Optimum Filter Length 3.60E+01 6.840E+01 5.40E+01

OuterO.5 600RPM Outer0.5 1200RPM OuterO. 5 1800RPM CF 1.26E+0 1 1.39E+O1 7.96E+00

Kurtosis 3.86E+O 1 2.92Et-0 1 1.68Ei-0 1

AM 2.47E+Ol 4.08E+Ol 1.34E+01

' - - J - - " -

Optimum Filter Length 6.80E+01 6.800E+01 5.20E+01

InnerO. I 600RPM ZnnerO.1 1200RPM InnerO.1 I800RPM CF 1.91E+01 1.17E+01 9.13Et-00

Kurtosis 6.47E+0 I 2.48E+01 1.58E+01

AM 9.1 1E+00 3.83E+01 1.02Ei-0 1

Znner0.2 600RPM Znner0.2 1200RPM Inner0.2 1800RPM CF 2.73E+01 1.4 1E+0 1 1.13E+01

Kurtosis 4.94E+0 1 3.40E+ 1 2.56E-t-0 1 AM 3.16E+01 5.00E+0 1 1.82E+O 1

Mfrms 6.872E+0 1 9.73E+01 7.35E+01 ., Optimum Filter Lengtlt 7.00E+01 7.60E+01 5.40E+01

Inner0.5 600RPM Znner0.5 1200RPM Inner0.5 1800RPM CF 2.07E-tOl 1.09E+0 1 7.96Ei-00

Kurtosis 6.61E+01 1.77E"rO 1 1.68E-tO 1 AM 2.3 1 E+O 1 5.48Ei-01 1.34E+OI

Mfrms 6.808Ei0 1 1.00E-i-02 5.10E+01

Optimum Filter Lengtlt 7.00E+01 3.20E+01 5.20E+01

Ball0.1600RPM Ball0.11200RPM Balf0.11800RPM CF 3.65E+00 4.88E+00 5.04E+00

Kurtosis 2.47Ei-00 2.88E+00 3.09E-t-00

AM 5.93E+00 2.52E+01 3.32EiOl

Mfrms 6.05E+O1 6.45Ei-01 6.37Et-01 Optimum Filter Le~tgtllr 9.80E+01 1.04E+02 1.66E+02

Ba110.2 600RPM Ba110.2 1200RPM Ba110.2 1800RPM CF 4.11E+00 4.64E+00 5.25E+00

Kurtosis 2.68E+00 3.00Ei-00 3.08E+00 AM 6.823E+00 2.65E+01 3.69E+O 1

-.8

Optimum Filter Lengtli 1.62E+02 6.88OE+Ol 6.80E+01

Ba110.5 600RPM Ba110.5 1200RPM Ba110.5 1800RPM CF 5.76E+00 1.04E+O 1 7.72E+00

Kurtosis 3.08E+00 6.855E+00 4.07E+00 AM 3.83E+Ol 5.91Ei01 1.78E+O 1

Optimum Filter Lengl11 2.063+02 7.40E+Ol 1.40E+02

5.3.6 Removing the High Resonance Frequency Components

As a result of optimum filter length, an observed signal with an outer race defect of 0.1 mm

width RPM was input to the blind deconvolution algorithm. The results are presented in Figure

5.42(a). It can be observed from Figure 5.42(b) that the observed signal of an outer race defect

of 0.1 mm and rotating speed of 600 RPM was further corrupted with 500 Hz sinusoid noise to

mask the impacts. It can be seen from Figure 5.42(c) that blind deconvolution has recovered the

source of vibration sufficiently to identify the impacts. It can be seen that blind deconvolution

with optimum filter length has enhanced the original bearing signal and has eliminated the

background noise. The impulses from the defect on the outer race can be clearly seen and the

SNR has been improved. In order to understand the nature of the equalizer in the blind

deconvolution algorithm, the gain response of the equalizer was plotted using the computed

coefficient of the equalizer e[n]. Figure 5.43 shows the gain response of the equalizer (filter)

which was used in this experiment. It can be observed from the gain response plot that there is a

notch at the corrupting frequency of 500 Hz. Although there was no prior knowledge of the

simulated noise it can be seen that the equalizer has successfully eliminated the sinusoidal noise

at 500 Hz. (a) Time Domain Observed Signal OuterRace 6OORPM

0.1 1 I I I I I I I I I I

(b) Corrupted Signal with SIN noise, noise=O.O25*s1n(2*pi"5OO*t) 0.1 1 I I I I I I I I I

(c) Recovered Signal After B l~nd Deconvolution with Opt~mum Filter Length L=70

Time (Second)

Figure 5.42 (a) Observed signal at 600 RPM with an outer race defect (b) Corrupted signal with

sinusoid noise (c) Recovered signal after the blind deconvolution algorithm

Gain Resoonse of Eaualizer with 500Hz corruotion noise L.70

Figure 5.43 Gain response and phase response of the equalizer for 600 RPM outer race

It can be observed from Figure 5.42(c) that the signal shows a consistent impulsive signal of a

damaged bearing. The signal-to-noise-ratio SNR has been improved as a result of blind

deconvolution. It was observed that the kurtosis can be a good criterion to distinguish between a

defective and undamaged bearing. A healthy bearing in good condition with Gaussian

distribution has a kurtosis value close to 3. In this situation, the value of kurtosis can be

misleading if the bearing signal is corrupted by Gaussian noise. When the bearing deteriorates

this value increases to indicate a damaged condition. The MATLAB [170] program calculation

of kurtosis of the observed signal in Figure 5.42(a) was found to be 2.78 while the kurtosis of

the recovered signal in Figure 5.42(c) was 9.06. Kurtosis has been found to be a good criterion

to distinguish between a defective and undamaged bearing.

The resonant frequencies of the defective roller bearing were found by performing an impact

test on the ball bearing. It was found that one of the excited frequencies of the bearing's

structure was around 8000 Hz. The resonant frequencies of the bearing when located in the

housing will alter and it is not practical to determine them by performing an impact test. The

resonant frequency could be higher due to the different structures of the bearing and the test rig.

It can be observed from the gain response plot that there is a wide notch between 8000 Hz and

10000 Hz centred at 9000 Hz. The centre point of this band could be the resonant frequency of

the combined bearing and test rig. This has the implication of the blind deconvolution method

which could simulate a notch filter effect corresponding to excited resonant frequency.

Time Domain Observed Signal , Frequency Domain Signal 0 1 , I 2 /

0.1 2 I O - ~

,.. 5 0.05 1.5 3 - C .- +d

P O -2 >

-0.05 - 0.5

-0.1 0 0 0.2 0 4 0.6 0.8 1 0 0.5 1 1.5 2

Time (Second) Frequency (Hz) l o 4

Figure 5.44 (a) Observed Signal at 600 RPM Outer Race Defect (b) Spectrum of the observed

signal (c) Recovered signal with filter length 70 (d) Spectrum of the recovered signal

The spectrum of the observed signal and recovered signal were plotted to confirm that the high

frequency resonance components were eliminated. Figure 5.44(a) shows time domain observed

signal at 600 RPM with an outer race defect. The lower spilte in the frequency spectrum in

Figure 5.44(b), shows the bearing defect frequency and two resonant frequencies at around

6000 and 9000 I-Iz. Figure 5.44(c) shows the time domain plot of the recovered signal after the

blind deconvolution process with the optimum filter length. The spectrum of the recovered

signal was plotted in Figure 5.44(d). It can be seen that the high frequency resonance

components were successfully eliminated around 6000 and 9000 Hz. This again indicates that

blind deconvolution has the capability of removing high frequency resonance components.

Figure 5.45 shows the gain response and phase plot of a 0.1 mm width inner race defect at 600

RPM. After the blind deconvolution process, the gain response of the equalizer was plotted

using the computed coefficient of the equalizer e[n]. It can be observed from the gain response

plot that the high frequency resonance around 6000 Hz was removed by a notch. The spectrum

of the observed signal and recovered signal for this particular test (inner race defect with 600

RPM) were plotted. Figure 5.46(b) shows the frequency spectrum of the observed signal with

the inner race defect. There is one major resonant frequency around 6000 Hz. The frequency

spectrum of the recovered signal in Figure 5.46(d) shows that this component was removed and

the spectrum confirms the location of the notch in the gain response plot.

Gain Response of Equalizer without corruption noise L=30

Frequency (Hz) x l o4

0 I I I I I I I I

-500 - .................................................... ............r...........- - UI

a, 1000 - & - g-1500 ........ i ............ 1 ............. ............ a,

....... ........... ............ m -2000 - 1 E

........... ............ ............. ...................................... ............. ............ -2500 - j j j j j j j j j j j j j j j j ;

-3000 I I I I I I I I I

0 0 2 0.4 0.6 0 8 1 1 2 1 4 1.6 1 8 2 Frequency (Hz) l o 4

Figure 5.45 Gain response and phase response of the equalizer for processing of

the observed signal at 600 RPM with an Inner Race defect

T~me Doma~n Observed S~gnal Frequency Domeln Signal 0 15 1 5

- 0 1 u -

0 0 5 1 C

i? A

m 0 il

-0 05 0 5 - m - - 0 1

0 0 0 5 1 1 5 2

Time Domain Recovered Signal 1 E

I I i 0 0 2 0.4 0 6 0.8 1

I m e (Second) 0 5 1 1 5 2

Frequency (Hz) l o 4

Figure 5.46 (a) Observed Signal at 600 RPM with an Inner Race Defect (b) Spectrum of the

observed signal (c) Recovered signal with filter length 30 (d) Spectrum of the recovered signal

Figure 5.47 is the result of the equalizer gain response of a ball defect at 600 RPM. It can be

observed that there are several notches from 6000 Hz up to 20000 Hz. A distinct notch related

to the resonant frequency was not observed in this plot probably due to the non-stationary

nature of rolling element faults. However it appears that the blind deconvolution algorithm was

able to generate a series of notches to remove the high frequency components.

Gain Response of Equalizer wlthout corruption noise L=70 20 I I I I I I I I I

-50 I I I I I I I I I

0 0.2 0.4 0.6 0 8 1 1.2 1 4 1.6 1.8 2 Frequency (Hz) l o 4

Frequency (Hz) x in4


the observed signal at 600 RPM with a ball defect

Figure 5.48 shows the observed and recovered signals accompanied by their frequency spectra

for the ball defect experiment. It can be observed that the high frequency components were

removed (Figures 25 (b) and 25 (d)).

T~me Domain Observed Signal Frequency Domain Signal 0.05 1 I 2 I

Time Domain Recovered Signal 0.1 2

1c3

- 0.05 1.5

3, - C

'n - P O 9 1 A2 5 ;?- -0.05 - 0.5

-0.1 0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1 5 2

T~me (Second) Frequency (Hz) l o 4

Figure 5.48 (a) Observed Signal at 600 RPM with ball defect (b) Spectrum of the observed

signal (c) Recovered signal with filter length 98 (d) Spectrum of the recovered signal

5.3.7 Results and Discussions

This study has revealed the advantages of blind deconvolution as a technique to recover the

original signal of a typical faulty bearing corrupted by noise and distorted by the transmission

path. Since characteristic frequencies contain very little energy, and usually are overwhelmed

by noise and higher levels of macro-structure vibration, it is difficult to identify them in the

frequency spectrum. The high frequency resonance technique (HFRT) was used to identify a

fault occurring at a characteristic frequency. The procedure to obtain the optimum inverse filter

is addressed, considering the influences of input parameters of blind deconvolution. The

modified Crest Factor (CF) and Arithmetic Mean (AM) graphs could be used to determine the

optimum filter length. It was found that there is a good correlation between both graphs to

determine the optimum filter length. Using the slope and finite difference programs, the

optimum filter length of the equalizer can be identified and applied to the blind deconvolution

algorithm automatically. This study has revealed the value of blind deconvolution as a

technique for recovering a damaged bearing signal. A defect on the outer race remains at the

same position relative to the loaded zone, unless the race is spinning in its housing. For rapidly

loaded bearings, defect on the inner race and rolling elements periodically can move into and

out of the loading zone with the rotation of the shaft and the cage. The result show that the

proposed algorithm works very well in regard to the detection of the characteristic frequency. It

was found that the recovered signal had been improved when compared with the observed

signal. The kurtosis factor of greater than 3 is an indication of spikiness of the signal and an

indication of a damaged bearing. It was found that the recovered signal has a higher kurtosis

compared to the observed signal. The results of optimum filter length can also be used to train a

neural network. Blind deconvolution behaves like a notch filter in eliminating the high

resonance frequency components. Although an analog filter can be used to filter out high

frequency components, this task requires prior knowledge of the unwanted noise. It was found

that the algorithm removes the high frequency components of the observed signal around the

excited resonance frequency without any prior knowledge of the resonant frequency

components. This technique was found to also work effectively in noise removal of signal

measured from faulty balls, despite the non-stationary nature of such faults. The results show

that this algorithm works well in eliminating related resonant frequencies of the outer and inner

race defects and acts like a notch filter.

Chapter 6

Experimental Method: Validation of Blind Deconvolution through Life Experimental Tests Bearing failure is one of the foremost causes of breakdown in rotating machinery. Such failure

can be catastrophic and can result in costly downtime. Bearing condition monitoring has thus

played an important role in machine maintenance. Many previous works have focused on crack

propagation failure resulting from an artificial or "seeded" damage [33, 63, 64, 66, 761. The

damage is normally seeded by inserting debris into the lubricant or by removing minute parts of

a bearing component using electric discharge or laser techniques. In this chapter, life time

testing is conducted to gauge the performance of the blind deconvolution technique in detecting

a growing potential failure of a new bearing which is eventually run to failure. Results from un-

seeded new bearing tests are different, because seeded defects have certain defect characteristic

frequencies which can be used to track a specific damaged frequency component. The difficulty

in detecting an early failure in life time testing is that it is not possible to predict which bearing

component will fail first. Furthermore it may consist of combinations of the characteristic defect

frequencies which makes detection difficult.

6.1 Bearing Test Rig for Life Time Testing

The final design of the life time bearing test rig is shown in Figures 6.1 to Figure 6.3. The test

rig consists of a TECO 5.5 kW 4 pole electric motor to drive the bearing shaft, a hand actuated

hydraulic load unit, a safety guard, a slave bearing block assembly to hold the bearing in

position and a load table. A detail drawing of the test trig is presented in appendix A. A

hydraulic ram provides the vertical force on the bearing housing as shown in Figure 6.1. The

slave bearing block contains two high precision bearings (7018CTDBLP6) to maintain the

alignment of the shafts. The slave bearing block is fixed onto the load table with two M10x20

bolts, as shown in Figure 6.4.2. The input shaft on the slave bearing block is driven by an

electric motor via a 630mm pulley to reduce the shaft motor speed from 1450 RPM to 21 1

RPM. The slave bearing block output shaft diameter for locating the bearings is 25 mm as

shown in Figure 6.2. Since the test bearing is to undergo heavy load (greater than the basic load

capacity of the bearing) the shaft tolerance has t28pm upper limit and t15pm lower limit.

Once the shaft position is fixed on the load table, the test bearing is mounted to the shaft. The

test bearing housing is manufactured to accommodate the applied load from the press and is

easy for mounting and dismounting the bearing as shown in Figure 6.3. It also has an

attachment for mounting an accelerometer on a bolt which can be screwed onto the bearing

ring. Three different bolt lengths were designed to give a variation in radial transmission path of

the signal from source to sensor. Bolt locations are situated at 90 degrees apart to allow for

multi directional measurement using different sensors such as AE sensors.

Figure 6.1 Life Time Test Rig

Figure 6.2 Fixing of the bearing slave block to the loading table

of a bearing's life still operating perfectly. Bearing fatigue life is a statistical probability;

therefore the estimated value was used only as a guide.

Experimental instrumentation for data acquisition as considered in detail in Section 5.1 and

consist of an accelerometer (IMI 621B51), a signal amplifier set to 10 amplification

(PCB482a20), an analogue filter with a cut off frequency of 3 kHz (KROHN3202), an analogue

to digital converter (NI BNC2120) and a laptop computer with LabView software to select a

certain sampling frequency. A voltage per unit IMI 621B51 accelerometer measured the

vibration and an optional AE sensor could be used in parallel to measure stress waves. The

focus of study was to collect only vibration signals. Type J thermocouples (iron and copper-

nikel) with a range of 0" to 750 "C were used to measure temperature. A thermocouple was

pressed against the outer race of the bearing to record a discrete sampled temperature signal.

Data from the thermocouple was collected continuously and data from the accelerometer was

acquired at intervals of 5 minutes. The sampling frequency for data acquisition was 12 kHz with

a cut off frequency of 3 kHz. A total of 217 data points were recorded for a time duration of

10.55 seconds.

The data was presented in the time domain. This allows trends to be plotted over time. In the

absence of a seeded bearing failure it is necessary to run the test until a bearing failure occurs.

Certain features were extracted from the time domain data such as Kurtosis, RMS and peak-to-

peak of the vibration signal. Definitions of these features are well known in the literature [42,

134, 1351 and demonstrate the statistical properties of the time based data. Trends of these

features were plotted to indicate the progressive failure period of the bearing over time. The test

results are presented in the following sections. The bearing temperatures attained steady state

for the tests run at constant speed. There was a clear trend in temperature relative to bearing

damage. Temperature trend for each case was plotted as a feature to determine the critical

failure period.

6.3 Results

The following results are taken from the bearing test rig under different conditions. The results

also demonstrate the working capability of the test rig. A description of the bearing condition

and photographs of the damaged bearing are presented at the end of each section. Based on the

bearing life equation, the load capacity was calculated and a suitable load on the bearing was

chosen. To effectively initiate bearing failure in a short period of time at a low rotational speed

required a high load. This is not an ideal scenario, as the elastic/plastic stability of the bearing is

reduced as the radial load approaches its load capacity. In some cases, attempts were made to

fail the bearing quickly through applying an excessive load, which resulted in catastrophic

failure. The signal features in terms of kurtosis, RMS, peak-to-peak and temperature of the test

bearing are recorded for comparison.

It is necessary to determine the estimated life of the bearing versus load. This allows radial load

to be determined for accelerated bearing failure. A spreadsheet with bearing life versus radial

load is shown in appendix B. Four tests with different operational conditions were carried out

and all tests resulted in damage to the bearing. The test conditions, load condition, test duration,

estimated life and the failure mode of the test are presented in Table 6.1. It can be observed

from Table 6.1 that for three experiments, failure occurred on the inner race of the roller

bearing. The test bearing runs continuously until bearing failure is reached. Based on the

theoretical life calculations of the bearing, the test length could be up to several days.

Table 6.1 Life Test Summary

Test Test Load Test Estimated Failure

Number Condition Condition Duration Life L, Mode

(hour) (hour)

1 Normal Heavy 5 1 116.16 Inner Race

2 No Lubricant Heavy 34 116.16 Inner Race

3 Normal Beyond Static Rate 3 116.6 Inner Race

4 Contaminant Lubricant Heavy 35 Min. 116.6 Ball Bearing

Table 6.2 presents the determinist characteristic defect frequencies for the test bearing 6805

rotating at 211 RPM. BPFB indicates Ball Pass Frequency Ball (roller defect); FTF

Fundamental Transfer Frequency (cage frequency); BPFO Ball Pass Frequency Output (outer

race defect) and BPFI Ball Pass Frequency Input (inner race defect). Since the equations used to

calculate the characteristic defect frequencies are based on bearing geometry and speed alone,

variations due to high loading and slipping were not considered. Furthermore during the test the

oil gets hot in the oil container, causing an increase in pressure beyond what was planned.

Table 6.2 Deterministic characteristic defect frequencies

Test Shaft Speed Shaft BPFB FTF BPFO BPFI

Number (RPM) Frequency ( f, ) ( f, ) ( f b ~ ~ ~ ) ( f b ~ ~ ~ r )

(1 x) Hz Hz Hz Hz Hz

1,2,3,4 21 1 3.52 28 1.6 24.95 30.3 1

6.3.1 Test bearing No. 1

6.3.1.1 Statistical feature analysis

A 6805 narrow test bearing was press-fitted onto a slave bearing block output shaft. A spacer

was used to position the test bearing on the shaft. The speed was set at 21 1 RPM and radial load

to 67% of the dynamic rated load as indicated in Table 6.1. An average of 67% of the dynamic

rated load is the result of load variations during the experiment because of the heat transfer from

the press to the oil container of the hydraulic jack. Test bearing No. 1 was operated for four

days under a heavy load. Test condition was normal and lubricant was not removed. The test

duration lasted 51 hours over four consecutive days whilst the estimated life was calculated to

the 1 16.16 hours. On each day the test was run for approximately 13 hours. From the obtained

signal features there was no indication of fault generation until day 4. The statistical features of

the data at each day are presented in Table 6.3 for comparison. It can be observed from Table

6.3 that the RMS level ranged from 1.1 to 1.3 with peak to peak ranging from 10 to 12 and

kurtosis ranging from 3 to 3.2 at day 1.

Table 6.3 Statistical features summary at each day - Test Bearing No. 1

Day Temperature RMS Peak to Kurtosis range peak

I 28°C-32°C 1.1-1.3 10-12 3 -3.2

2 28°C - 33°C 1.1 - 1.3 10 - 12.5 3 - 3.3

3 28°C-34°C 1.1-1.35 10-13 3 -3.4

4 (Before the last 40 Min) 28°C - 34°C 1.1 - 1.25 10 - 13 3 -3.4

4 ( last 40 Minutes) 37°C 1.21 14.5 3.6

It can be observed from Table 6.3 that day 2 data is similar to day 1. No clear trend in any of the

signal features is evident. Temperature increase exhibits the same trend as day 1. Day 3 data

again is similar to day 1 and 2. Day 4 data again is similar to day 1 and 2 and 3 except the last

40 minutes. No clear trend in any of the signal features is evident. Temperature increase

exhibits the same trend as day 1 and 2. This indicates that no fault has been initiated on any of

the bearing components. Bearing signal and feature data for the first three days of operation

provide an understanding of the baseline level of these features. The baseline level represents

the feature levels at the non-defect condition. The baseline level of this data can be generally

stated as the RMS range from 1.1 to 1.3, peak to peak range from 10 to 13 and kurtosis from 3

to 3.4. The failure appears to happen over a period of approximately 40 minutes. It can be

observed from Table 6.3 that the statistical features (RMS, P2P, and Kurtosis) for the last day of

testing (last 40 minutes) are different compared to the baseline level. The maximum values of

the signal features over the last 40 minutes are: temperature 37"C, RMS 1.21, peak to peak 14.5

and kurtosis 3.6. These values are higher than the no fault condition of the first three days.

Interestingly, the RMS value appears to be the least conclusive. The RMS initially trends

downwards before failure and then increases, whilst kurtosis and peak to peak show failure

more clearly. The temperature at failure also indicates bearing failure.

The features, RMS, Peak to peak, and kurtosis of the vibration signal at day 1 are presented in

Figures 6.4(b) to 6.4(d) respectively. Each data point represents the calculated time signal

feature values for 10 second duration of the signal. The time data is taken at 5 minutes intervals.

Thus for 13 hours, a total of 156 data points are taken. For each of the features there is no clear

trend of failure and the bearing operation is stable. The temperature data (Figure 6.4(b)) shows

the temperature rising to a steady state level at each day. This could be due to the rise in

ambient temperature throughout the day and also reaching the steady state operating

temperature of the bearing. A brief summary of each day of data is discussed below. It can be

observed from Figure 6.4 that the feature plots (RMS, P2P, and Kurtosis) for the first day of

testing are generally stable. There is no clear indication of the onset of a bearing failure.

Figure 6.4 Bearing 1 vibration signal features day 1 (first Day); (a) Temperature, (b) RMS,

(c) Peak to peak, (d) Kurtosis

Data captured on day 4 are shown in Figure 6.5 which shows the bearing failure at the end of

each plot. The vibration signal features, RMS, peak to peak, and kurtosis of day 4 are presented

in Figures 6.5(b), (c), and (d) respectively. It can be observed from Figure 6.5 that the feature

plots (RMS, P2P, and kurtosis) for the last day of testing is different compared to Figure 6.4.

The failure appears to happen over a period of approximately 40 minutes, with eight data points

showing a sharp rise starting at 1:35 PM to 2: 10 PM when failure occurred. It can be observed

from Figures 6.5(b), (c), and (d) that the feature values over the last 40 minutes are higher than

the no fault condition in Figure 6.4. In Figure 6.5(d), the kurtosis trend drops down at 1:50 PM

and rises again up to the failure point at 2: 10 PM. The last day of the temperature data from test

No. 1 is shown in Figure 6.6. There is a minor fluctuation in temperature plot 50 minutes before

failure point at 2:10 PM. During the last hour of this test, the temperature of the roller bearing

did increase as spa11 damage developed on a single rolling element. This was the only test in

which a change in temperature with bearing failure occurred. Bearing temperature point vales

are normally gauged against ambient temperature and viewed in isolation provide little insight

to bearing condition. However trends, showing sharp rising of temperature may indicate

deterioration in bearing conditions.

I I I ~ t a d t Point 1:35 PM/ Q:40 AM 10:45 AM 11:50 AM 12:55 PM 2 05 PM 3:10 PM

Figure 6.5 Bearing 1 vibration signal features day 4(Last Day); (a) Temperature, (b) RMS,

(c) Peak to peak, (d) kurtosis

Figure 6.6 Temperature trend for test 1 until failure

Features plots for the last 40 minutes period up to failure point is shown in Figure 6.7. The

RMS values in Figure 6.7(b) show an increase from 1.1 to 1.2 at 1 :25 PM and a decrease from

1.2 to 1.0 at time 1:35 PM. The peak to peak value in Figure 6.7(c) shows a smooth trend at

1 :35 PM and an increase from 10 to 12 at 1 :35 PM. It can be observed from Figure 6.7(b), (c)

that the increase points of the features are different and the start point of the failure can not be

specified exactly. The increase in RMS may be attributed to the result of impact which can

produce a large elastic stress wave due to the larger inherent flexibility of the bearing

components. It was therefore decided to look at the kurtosis plot in Figure 6.7(d) as the

alternative feature to determine the start point of the failure.

Figure 6.7 Test bearing No. 1 vibration signal features day 4(Last Day), period of last 40

minutes up to failure; (a) Temperature, (b) RMS,(c) Peak to peak, (d) kurtosis

. . 15 r I I I 1 I I I

& l o , ......... i ........ ...+ ' ,__<.: ......... .......... : ............ :... ...... - I

a

5 I I I I I I I I I

4 (dl

The kurtosis value of 3 indicates no failure condition. With crack initiation, the impact of

rolling elements generates impulses, leading to a spiky vibration signal and increasing the

kurtosis value. It is obvious from Figures 6.7(d) that there is a sign of bearing failure as the

kurtosis level increases to 3.5. The kurtosis value increased from 3 at 1:35 PM as the start point

of the failure, to a maximum level of 3.5 at 1 5 0 PM, then decreased to 3.3 at 1:55 PM and rose

again. This fluctuating trend can be explained by the nature of the damage. The initial increase

was caused by the appearance and initial propagation of the surface defectt. This was reviewed

in [48]. The subsequent drop in kurtosis level could be attributed to a healing process. The

Ln .- $ 3.5 3

healing process is the smoothing of the sharp edges of a crack or small damage zone by

continued rolling contact. There may have been a stall in the crack propagation during this

period, or decrease in vibration amplitude due to these smoothing effects. As the damage spread

over a broader area, the kurtosis rose again over the time. Once the damage area becomes larger

than the spacing of the rolling elements, continuous shock load (resulting from one decaying

~II& 1.15 & ?:35 I.& I I : 1 k 5 2:bo 2b5 ~:IOPM T~me

I I I I I I I I I

&hest ~ d a k ~ o i n t 1 5 0 PM i ~ a i l u i e Point - ........... i ............ : ............. : ...... i .......................

1 start Poy---- -c\

impulses merging into the beginning of the next impulse) would bring the signal to a normal

2:10 PM

distribution and return the kurtosis value to 3. This trend was very clear in the Figure 6.8(d)

with the bearing test run at constant speed.

Table 6.4 lists the maximum magnitude of frequency components present in the spectrum of the

time domain signal before and after blind deconvolution (BD) over the last 40 minutes to

failure. In Table 6.4 before start point began at 1:25 PM with 10 minutes laps up before the

kurtosis trend increased until failure point at 2: 10 PM. The characteristic defect frequency was

30.5 Hz at each point after the start point at 1 :35 PM, each within 0.063 % of the BPFI of 30.3 1

It can be observed from Table 6.4 that no defect frequency and peak was detected at 1:25 PM

and 1:35 PM (before failure). It can also be seen from Table 6.4 that maximum peak amplitude

in the spectrum reduced from 1.4 E-01 to 2.96 E-03 at 1:40 PM. The reduction in the maximum

amplitude before and after BD was also observed at 1 :50 PM and 2: 10 PM. The reduction of the

peak amplitude after blind deconvolution can be attributed to the nature of the blind equalizer

(filter). Although the individual peak is stronger without blind deconvolution, the reduction of

signal-to-noise-ratio makes the signal and the overall modulation pattern clearer after the

equalizer filter.

Table 6.4 Maximum magnitude of frequency components with and without

blind deconvolution (BD) over the last 40 minutes to failure for test bearing No. 1

Detected Defect Peak Peak

Time Status Shaft Frequency Amplitude Amplitude Failure

History Frequency Present without with Mode

(1 x) Hz BD BD

1:25 PM Before Start Point 3.7 Nil Nil Nil Nil

1:35 PM Start point 3.7 Nil Nil Nil Nil

1.40 PM Crack Grows 3.7 30.5 1.4 E-02 2.96 E -03 Inner Race

1 5 0 PM Highest Peak 3.7 30.5 2.10 E-02 2.96 E-04 Inner Race

2: 10 PM Failure Point 3.7 30.5 4.96 E-02 6.40 E-05 Inner Race

The spectrum of the observed signal and recovered signal after BD at 1 :25 PM (before failure)

in time and frequency domains are presented in Figure 6.8. Figure 6.8(a) shows the signal of the

observed signal at ten minutes before the commencement of failure (the start point in Table

6.4). The lower spike in the frequency spectrum, Figure 6.8(b), shows 4 frequency components

at around 200 Hz, 980 Hz, 1300 Hz and 1800 Hz. Figure 6.8(c) shows the time domain plot of

the recovered signal after the blind deconvolution process. The spectrum of the recovered signal

is shown in Figure 6.8(d) indicating 5 frequency peak components at around 200 Hz, 1300 Hz,

1800 Hz, 2700 Hz and 3600 Hz. The kurtosis of the observed signal, Figure 6.9(a), was found

to be 3.03 which shows the bearing is in a good condition, while the kurtosis of the recovered

signal was 3.1 1. It can be seen that the recovered signal is beginning to show sign of impending

failure. A typical spiky time domain trace of the inner race fault was not found in either of the

observed or recovered signals due to the heavy background noise while the crack is still very

small. Some inconsistent spikes might be seen in the recovered signal after the blind

deconvolution process. They do not correspond to any specific characteristic defect frequency.

It can be observed from Figure 6.8(d) that the recovered signal contains new resonant frequency

components at 2700 and 3600 Hz which could be useful for the high frequency resonance

technique (HFRT).

Bearing 1 Time @ 1:25 PM (a) Time Domain Obsewed Signal L=32 10 minutes before Start point jb) Frequency Domain Signal

0 4 1 I 0.03 I I

(c) T~me Domain Recovered Signal 1f4 ( 4 0.03 I I 3 I 1

d . 2

5 0.01 2 > - g 0 1.5 'n F e -0.01 >

1

-0.02 0.5

-0.03 0 0 0.2 0.4 0.6 0.8 0 1000 2000 3000 4000 5000 6000

Time ( Second ) Frequncy (Hz)

Figure 6.8(a) Observed Signal at 21 1 RPM at 1 :25 PM ten minute before start point (b)

Spectrum of the observed signal (c) Recovered signal with filter length 32 (d) Spectrum of the

recovered signal

Figure 6.9 shows the gain response and phase plot of the equalizer for processing of the

observed signal at 1:25 PM. After the blind deconvolution process, the gain response of the

equalizer was plotted using the computed coefficient of the equalizer e[n]. It can be seen from

Figure 6.8(b) that the frequency components at 980 Hz, 1300 Hz, and 1800 Hz were attenuated

as shown in the gain response plot in Figure 6.9, ranging between -25 dB to -50 dB over the

frequency range from 0 to 3000 Hz. It can be observed from the gain response plot that the high

frequency resonance around 980 Hz was removed by a wide notch with a value of -50 dB.

Figure 6.9 also shows some other notches at 1600 Hz with a value of -40 dB, 1800 Hz with a

value of -37 dB and at 2400 Hz with a value of -25 dB.

Gain Resoonse of Eaualizer L=32

Phase Response of Equalizer L=32 500 I 1 I I I

-2000 I I I I I 0 1000 2000 3000 4000 5000 6000

Frequency (Hz)


the observed signal at 21 1 RPM at 1 :25 PM

Figure 6.10 shows the result of a demodulation process performed on the recovered signal at

1 :25 PM. In the demodulation process, the signal was band pass filtered at four different bands

to obtain the best modulated resonant frequency region. The resonant frequencies of the bearing

when located in the housing will be altered due to different stiffness which is not practical to

determine by only performing an impact test. The best range for band passing is normally

determined by a trail and error process based on certain estimates. These four bands were

selected based on a possible excitation of the resonant frequencies within the bearing signal.

Figure 6.10(a) is the result of a demodulation process between 1000 to 2000 Hz; Figure 6.10(b)

between 3000 to 3700 Hz; Figure 6.10(c) between 3700 to 4500 Hz; and Figure 6.10(d)

between 4300 to 5000 Hz. It was assumed that these frequency bands would result in the

detection of the best characteristic frequency. It can be seen that at the higher bands (Figures

6.10(c) and 6.10(d)) there are no signs of predominant frequency components. In the lower

frequency bands (Figures 6.10(a) and Figure 6.10) there are indications of shaft frequencies at

3.7 Hz and 2 ~ 3 . 7 Hz. No characteristic bearing defect frequency was observed in the spectra of

the recovered signal as the detection signal is insignificant or non-existent. Only the harmonics

of the shaft speed were found due to the misalignment of the shafts. Figures 6,10(a) to 6.10(d)

also confirm the result of Figure 6.8(c), that no spikes correspond to the characteristic defect

frequency after blind deconvolution, indicating no bearing failure.

Demodulsted Recovered SIQnaI at I 25 PM Demodulated Recovered S l~na l at 1 25 PM Band Passed 1000M To 2500Hz Band Passed 3000 Hz To 3700 Hz

r--------

Frequency (Fz) Frequency l hz)

Demodulated Recovered Sgnal at I 25 PM

'0 5 10 15 20 25 30 35 40 45 50 ciequenw ( H z )

Figure 6.10 Demodulated recovered signal at 21 1 RPM, 1 :25 PM Band Passed Between (a) 1000 to 2500 Hz

(b) Between 3000 to 3700 Hz (c) Between 3700 to 4500 Hz (d) Between 4700 to 5000 Hz

6.3.1.2 Bearing signal analysis using BD

Figure 6.11 shows the plot of CF and AM versus filter length for the recovered signal at 1 :40

PM. At 1 :40 PM there is an increase in kurtosis level of 3.3 which indicated there is a bearing

failure. An automatic program selects the optimum filter length for the blind equalizer which

involves the CF and AM process as shown in Figure 6.12. It can be observed from Figure 6.12

that the CF values stabilized at around 18 while the AM values stabilized around 40 filter

length. It can also be observed from Figure 6.1 1 that an optimum point of filter length 32 is

achieved. The optimum filter length L=32 is used in this test. In the AM plot it can be seen that

the AM amplitudes decrease from -40 dB to 0 dB until 32 filter length. Beyond that value, the

AM amplitude remained fairly constant at about 0 dB.

Crest Factor Modified Recovered Signal @ 1:40 PM 14 1 v I I I I I I

ArithmeticMean (dB) & OptimumFilter=32 80 I I I I I 1

-20 1 I 1 I 1 I I 0 20 40 60 80 100 120

Filter Length

Figure 6.1 1 CF and AM plot for signal at 1 :40 PM with 2 1 1 RPM when crack initiated

Bearing 1 Time @ 1:40 PM (a) Tirrie Donlain Observed Signal L=32 Crack Grows Point (bj Frequency Doinain Signal

0.6 ll I 0.035

(c) Time Domain Recovered Signal ''03 Rotitional Freqhency 3.7 Hz 0.02

11 Detected 1dner Race Frequency 30.5 HZ// -0.03

0 0.2 0.4 0.6 0.8 Time ( Second )

"0 1000 2000 3000 4000 5000 6000 Frequncy (Hz)

Figure 6.12 (a) Observed Signal at 21 1 RPM at 1:40 PM crack grows point (b) Spectrum of


signal

Figure 6.12(a) shows time domain of the observed signal at 1:40 PM, which shows bearing

crack is progressive. The result of a recovered signal after blind deconvolution at 1 :40 PM is

shown in Figure 6.12(c). The spectrum of the observed signal and recovered signal at 1:40 PM

with optimum filter length in the frequency domain are presented in Figures 6.12(b) and

6.12(d). The lower spike in the frequency spectrum, Figure 6.12(b), shows 4 frequency

components at around 200 Hz, 800 Hz, 1200 Hz and 1700 Hz. Figure 6.12(c) shows the time

domain plot of the recovered signal after the blind deconvolution process with the optimum

filter length L=32. The observed signal in Figure 6.12(a) has a kurtosis level of 3.3 which

shows bearing damage. This value may not represent the severity of the defect due to

background noise. The kurtosis of the recovered signal was 10.07 which is a much clearer

indication of a damaged bearing. In the observed signal of Figure 6.12(a), it is not possible to

identify the spike of an inner race fault and the bearing fault signal was masked by background

noise. Figure 6.12(c) shows a typical trace of an inner race fault modulated on the shaft

frequency at 3.7 Hz after blind deconvolution in the recovered signal. It can be seen from

Figures 6.12(a) and Figure 6.12(c) that the recovered signal was improved compared to the

observed signal. The consistent time intervals of inner race fault impulses (30.5 Hz) match very

well with BPFI of 30.31 Hz. It can be observed from Figure 6.12(d) that the recovered signal

contains new resonant frequency components at 3500 and 4000 Hz which can be used in the

demodulation process.

Figure 6.13 shows the gain response and phase plot of the equalizer for the processing of the

observed signal. After the blind deconvolution process, the gain response of the equalizer was

plotted using the computed coefficient of the equalizer e[n]. It can be seen from Figure 6.14 that

the vibration amplitudes were attenuated as shown in the gain response plot of Figure 6.13.

These ranged between -10 dB to -50 dB over the frequency range from 0 to 4000 Hz. It can be

observed from the gain response plot that the frequency resonance around 1200 Hz was

removed by a notch with a value of -47 dB. Figure 6.13 also shows some other notches at 1700

Hz with a value of -50 dB, 2500 Hz with a value of -32 dB and at 2800 Hz with a value of -27

dB. Figure 6.14 shows the result of a demodulation process performed on the recovered signal

at 1:40 PM of Figure 6.12(c). In the demodulation process, the signal was band pass filtered

between 3700 to 4500 Hz as the best modulated resonant frequency region. Figure 6.14 is the

demodulation process between 3700 and 4500 Hz. From Figure 6.13(d) it was obvious that

band passing within this range would result in detection of the best characteristic frequency.

The characteristic defect frequency with a dominant spike at 30.5 Hz is clearly visible in the

spectrum, shown in Figure 6.14, which is close to the calculated frequency of 30.31 Hz. The

spike is accompanied by a number of harmonics spaced at multiples of the rotational speed

frequency which is a typical characteristic of an inner race mode failure. In order to confirm the

detected frequency matches with the actual excited resonant frequency region, the highest peak

point in the crack propagation range of test bearing No. 1, at 1 :50 PM was chosen for analysis.

Gain Response of Equalizer L=32 20 1 I I I I I I



Phase Response of Equalizer L=32

Demodulated Recovered Signal at 1.40 PM lo-4 Band Passed 3700 Hz To 4500 Hz

2 t t t I I 1 t I I 1

1000

0 - m a,

g, -1 000 a, U a, 2000 m - m E -3000

-4000

_ Inner Race Mode Failure BPFI = 30.5 Hz

I I I I I

-L .............

...............

- .................

................. -

I I I I I

14 - 3.7 Hz = 1*RPM

p 1.2 Side Band = 26.76 Hz - Side Band = 34 24 H +- -

g 0 1 i? r! - 7.4 Hz = 2"RPM

0.8

0.6

0.4

0.2

0 0 5 10 15 20 25 30 35 40 45 50

0 1000 2000 3000 4000 5000 6000 Frequency (Hz)

Frequency j Hz)

Figure 6.14 Demodulated recovered signal rotating at 21 1 RPM at 1 :40 PM when crack grows

Band Passed Between 3700 to 4500 Hz

The time and frequency spectra of the observed and recovered signals at 1 :50 PA4 with optimum

filter L=32 are presented in Figure 6.15. Figure 6.15(a) shows the time domain of the observed

signal at the highest peak point as shown in Table 6.3. The frequency spectrum, in Figure

6.15(b), shows 4 frequency components at around 200 Hz, 800 Hz, 1050 Hz and 1700 Hz.

Figure 6.15(c) shows the time domain plot of the recovered signal after the blind deconvolution

process with the optimum filter length L=32. The spectrum of the recovered signal is presented

in Figure 6.15(d).

Bearing 1 Time @ 1'50 PM (a) Time Domain Observed Signal L=32 The Highest Peak Point (bi Freauencv Domain Sianal

(c) Time Domain Recovered Signal

I O4 1 Inner Race Xreqnency 30.5 Hz 1




signal

With the crack occurring on the inner race, the kurtosis of the observed signal was found to be

3.50 which shows a small deviation from a good bearing in this specific case, while the kurtosis

of the recovered signal was 12.5. In the observed signal in Figure 6.15(a), it is not possible to

identify the impact of an inner race fault as the bearing fault signal was masked by background

noise. After the blind deconvolution, the recovered signal in Figure 6.15(c) shows a typical

trace of an inner race fault with a detected inner race fault frequency of 30.5 Hz modulated on

the shaft frequency at 3.7 Hz. It can be seen from Figures 6.15(a) and Figure 6.15(c) that the

recovered signal was improved compared to the observed signal. The consistent time intervals

of inner race fault impulses match very well with BPFI of 30.31 Hz. It can be seen that the

recovered signal was significantly improved. From the spectrum in Figure 6.15(d), the

recovered signal contains a new resonant frequency component at 4000 Hz which can be used

for the demodulation process.

Gain Response of Equalizer L=32 20 - I

-

0 -

-.20 - .................... ....... a 73

r" ! 2300 Hz .................... i...- ................. L ..................... i................

-80 I I I I I



Demodulated Recovered Signal at 150 PM x 1 6 ~ Band Passed 3700 Hz To 4500 Hz

I1 F

Frequency ( Hz J

Figure 6.17 Demodulated recovered signal rotating at 21 1 RPM at 1 :50 PM when the highest

peak point Band Passed Between 3700 to 4500 Hz




Figure 6.16 that the vibration amplitudes were attenuated at 1050 Hz and 2300 Hz ranging

between -60 dB to -20 dB from 0 to 4000 Hz frequency range. Figure 6.17 shows the result of

the demodulation process performed on the recovered signal at 1:50 PM as shown in Figure

6.1 5(c). In the demodulation process, the signal was band pass filtered between 3700 and 4500

Hz as the best modulated resonant frequency region. From Figure 6.15(d) it was obvious that

band passing within this range would result in the best detection of the characteristic frequency

because the new resonance component in the recovered signal matches with the excited

resonant frequency region. The characteristic defect frequency with a dominant spike at 30.5 Hz

is clearly visible in the spectrum, which is close to the calculated frequency of 30.31 Hz. The

spilte is accompanied by a number of harmonics spaced at multiples of the rotational speed

frequency which is a typical characteristic of an inner race mode failure.

The lcurtosis value appears to be more sensitive in assessing the performance of blind

deconvolution because crack initiation and propagation through bearing components would

generate impulses, thus changing the distribution of the vibration signal and increasing the

kurtosis.

Table 6.5 Kurtosis of the signal with and without


Detected Defect Kurtosis Kurtosis of

Time Status Shaft Frequency of the the Signal Failure

Frequency Present Signal without Mode

(1 x) Hz with BD BD

1:25 PM Before Start Point 3.7 Nil 3.03 3.11 Nil

1:35 PM Start point 3.7 Nil 3.07 3.50 Nil

1.40 PM Crack Grows 3.7 30.5 3.20 10.07 Inner Race

1:50 PM Highest Peak 3.7 30.5 3.50 12.5 Inner Race

2: 10 PM Failure Point 3.7 30.5 3.42 11.13 Inner Race

Table 6.5 lists the defect frequencies present in the observed signal and the kurtosis of the

signals before and after blind deconvolution (BD) with optimum filter length over the last 40

minutes to failure for test bearing No. 1. It can be observed from Table 6.5 that the kurtosis

value with and without BD vary significantly. The increase in the kurtosis value after blind

deconvolution can be regarded as an advantage of blind deconvolution as a sensitive technique

for diagnostic when a crack is initiated in the bearing component.

(a) Figure 6.18 Test bearing No. 1 after failure (a)Bearing failure, inner race is split up

(b) Fatigue on outer race

Figure 6.18 shows test bearing No. 1 after failure. In Figure 6.18(b) bearing fatigue can be seen

on the inner race. It can be observed from Figure 6. 18(b) that a crack was initiated on the inner

race causing the inner race to split up and the balls to be dropped between the shaft and the

inner race under the pressing force as shown in Figure 6.1 8(a).



A 6805 narrow test bearing was press fitted onto a slave bearing block output shaft. A spacer

was used to position the test bearing on the s M . The speed was set at 21 1 RPM. Bearing 2

experiments were undertaken on the bearing without lubricant, as indicated in Table 6.1. The

bearing lubricant was removed with kerosene. Test bearing No. 2 was operated for three days

under a heavy load. The test duration lasted 34 hours over three consecutive days whilst the

estimated life was calculated to the 1 16.16 hours as indicated in Table 6.1. On each day the test

was run for approximately 12 hours. Signal features of this bearing show the bearing operates in

a steady state manner until the catastrophic failure point. The statistical features of the data at

each day are presented in Table 6.6 for comparison. It can be observed fiom Table 6.6 that the

RMS level ranged h m 1.1 to 1.3 with peak to peak ranging from 10 to 12 and kurtosis ranging

from 3 to 3.15 at day 1. It can be observed fiom Table 6.6 that day 2 data is similar to day 1. No

clear trend in any of the signal features is evident. Temperature increase exhibits the same trend

as day 1. Day 3 data again is similar to day 1 and 2 except the last 10 minutes. No clear trend in

any of the signal features is evident. It can be seen that there is no initial trend upwards in the

data. Data at these points is similar to that for the first three days of running for test bearing No.

1. Temperature increase exhibits the same trend as day 1 and 2. Bearing signal and feature data

for the first day of operation provide an understanding of the baseline level of these features.

The baseline level represents the feature levels at the non-defect condition. The baseline level

from this data can be generally stated as RMS range from 1.1 to 1.35, peak to peak range from

10 to 13 and kurtosis from 3 to 3.2. The failure appears to happen over a period of

approximately 10 minutes. It can be observed from Table 6.6 that the statistical features (RMS,

P2P, and Kurtosis) for the last day of testing (last 10 minutes) are different compared to the

baseline level. The maximum values of the signal features over the last 40 minutes are:

temperature 42"C, RMS 3.5, peak to peak 41 and kurtosis 3.6.4. These values are higher than

the no fault condition of the first three days. All statistical features tend to indicate the failure

clearly. A sharp rise in temperature can also be seen at the bearing failure point. This can be

attributed to friction heat generation due to dry running.

Table 6.6 Statistical features summary at each day -Test bearing No. 2

Day Temperature RMS Peak to Kurtosis range peak

1 28°C - 33°C 1.1 - 1.3 10 - 12 3 - 3.15

2 28°C - 34°C 1.1 - 1.35 10 - 13 3 -3.2

3 (Before the last 10 Min) 28°C - 34°C 1.1 - 1.4 10 - 16 3 - 3.25

3 (Last 10 minutes) 42°C 3.5 4 1 3.8

Data captured on day 3 for test bearing No. 2 are shown in Figure 6.19 which shows the bearing

failure at the end of each plot. Each data point represents the calculated time signal feature

values for 10 second duration of the signal. The time data is taken at 5 minutes intervals. Thus

for 12 hours a total of 144 data points are taken. The vibration signal features, RMS, peak to

peak, and kurtosis of day 3 are presented in Figures 6.19(b), (c), and (d) respectively. The

failure appears to happen over a period of approximately 10 minutes, with three data points

showing a sharp rise starting at 5:58 PM to 6:08 PM when failure occurred. It can be observed

from Figures 6.19(b), (c), and (d) that the feature values over the last 10 minutes are higher than

the no fault condition baseline. In Figure 6.19(d), the kurtosis rises sharply at 5:58 PA4 up to the

failure point at 6:08 PM. Bearing operation is stable until the catastrophic failure point shown

at the end of the signal features plot in figure 6.19. This failure appears to be due to the

development of a crack due to dry running. This crack has propagated and separated the bearing

outer race from the bearing. Subsequently the rolling elements are removed.

Figure 6.19 Bearing test No. 2 vibration signal features day 3 (Last Day) ; (a) Temperature, (b)

RMS,(c) Peak to peak, (d) Kurtosis

0

0 20 40 60 80 100 120 60 I I I I I

40 - .......................................... % 0. 20 I ................

w - ~ . . . u c - - - - - - - ~ - - + 0 I I I I I

0 20 40 60 80 100 120 4 I I I I I

................... .:... .................. !. . .Highe&Peak~oint .@a %nreP~@f) ...

It can be observed from Figures 6.19(b), (c), and (d) that the increase points of the features are

almost the same and the start point of the failure can be specified as 5:58 PM. It was decided to

look at the kurtosis plot in Figure 6.19(d) to determine the start point of the failure. The kurtosis

value of 3 indicates no failure condition. W& crack initiation, the impact of rolling elements

generates impulses, leading to a spiky vibration signal and increasing the kurtosis value. It is

obvious from Figures 6.19(d) that there is a sign of bearing failure as the kurtosis level

increases to 3.9. The kurtosis value increased from 3.1 at 5:058 PM as the start point of the

failure to a maximum level of 3.9 at 6:08 PM.

2

Table 6.7 lists the maximum magnitude of frequency components present in the spectrum of the

time domain signal before and after blind deconvolution (BD) over the last 10 minutes to

failure. In Table 6.7, the start point began at 5:58 PM until failure point at 6:08 PM. The

characteristic defect frequency was 30.5 Hz at each point after the start point at 5:58 PM, each

within 0.063 % of the BPFI of 30.31. It can be observed from Table 6.7 that no defect

frequency and peak was detected at 5:58 PM (before failure). It can also be seen from Table 6.7

that maximum peak amplitude in the spectrum reduced from 2.03 E-02 to 3.45 E -03 at 6:08

PM. The reduction in the maximum amplitude before and after BD was also observed at 6:03

PM. The reduction of the peak amplitude after blind deconvolution can be attributed to the

nature of the blind equalizer (filter).

I I I I

100

3 -.--.--- ~ t a r t Point 5:58 PM

50 20 40 60 80 120

Table 6.7 Maximum magnitude of frequency component with and without


Detected Defect Peak Peak

Time Status Shaft Frequency Amplitude Amplitude Failure

History Frequency Present without with Mode

(1 x) Hz BD BD

558 PM Start Point 3.7 Nil Nil Nil Nil

6:03 PM Middle point 3.7 30.5 1.2 E-02 2.32 E -03 Inner Race

6:08 PM Highest Peak Point 3.7 30.5 2.03 E-02 3.45 E -03 Inner Race

The spectrum of the observed signal and recovered signal after BD at 5:58 PM (before failure)

in time and frequency domains are presented in Figure 6.20. Figure 6.20(a) shows the signal of

the observed signal at ten minutes before the failure (the start point in Table 6.7). The lower

spike in the frequency spectrum, Figure 6.20(b), shows 3 frequency components at around 200

Hz, 800 Hz, and 1800 Hz. Figure 6.20(c) shows the time domain plot of the recovered signal

after the blind deconvolution process. The spectrum of the recovered signal is shown in Figure

6.20(d) indicating 7 frequency peak components at around 200 Hz, 790 Hz, 1800 Hz, 2700 Hz,

3600 Hz, 4500 Hz and 5200 Hz. The kurtosis of the observed signal, Figure 6.20(a), was found

to be 3.1 which is misleading which shows the bearing is in a good condition, while the lurtosis

of the recovered signal was 6.37. It can be seen that the recovered signal is beginning to show

sign of impending failure. A typical spiky time domain trace of the inner race fault was not

found in either of the observed or recovered signals due to the heavy background noise while

the crack is still very small. Some consistent spikes might be seen in the recovered signal after

the blind deconvolution process correspond to the shaft rotational frequency 3.7 Hz. It can be

observed from Figure 6.20(d) that the recovered signal contains new resonant frequency

components at 2700,3600 Hz, 4500 Hz and 5200 Hz which could be useful for the HFRT.




Figure 6.21 that the frequency components at 250 Hz, 700 Hz, 1300 Hz, 1600 Hz, 2200 Hz, and

3750 Hz were attenuated as shown in the gain response plot in Figure 6.21, ranging between -20

dB to -55 dB over the frequency range from 0 to 4000 Hz. It can be observed from the gain

response plot that the high frequency resonance around 250 Hz was removed by a wide notch

with a value of -50 dB. Figure 6.21 also shows some other notches at 500 Hz with a value of - 55 dB, 1300 Hz with a value of -45 dB, 1600 Hz with a value of -47 dB, 2200 Hz with a value

of -54 dR and at 3700 Hz with a value of -20 dB.

Bearing 2 Time @ 5-58 PM (a) Time Domain Observed Signal 1.32 The Start Point (b) Frequency Domain Signal

0.6 I I 0.04 I I

-0.8 I I Time ( Second )

(c) Time Doma~n Recovered Signal

Oo6 0

I I -'.02 Rotational Frequency 3.7 Hz

id)

I200 Ilk

Figure 6.20 (a) Observed Signal at 21 1 RPM at 5:58 PM Start Point Test NO. 2

(b) Spectrum of the observed signal (c) Recovered signal with filter length 32 (d) Spectrum of

the recovered signal

Gain Response of Equalizer @ 5.58 PM L=32

Phase Response of Equalizer 500 I I I I I

.............. .i. ..................... -

a,

3 -1000 I ' m

.................... -1500 - a -2000

-2500 I I I I I

0 1000 2000 3000 4000 5000 6000 Frequency (Hz)


the observed signal at 21 1 RPM at 5:58 PM


5:58 PM. In the demodulation process, the signal was band pass filtered at four different regions

to obtain the best modulated resonant frequency region. The resonant frequencies of the bearing

when located in the housing will be altered due the different stiffness which is not practical to

determine by only performing an impact test. The best range for band passing is normally

determined by a trail and error process based on certain estimates. These four bands were

selected based on a possible excitation of the resonant frequencies within the bearing signal.

Figure 6.22(a) is the result of a demodulation process between 1000 to 2000 Hz; Figure 6.22(b)

between 3000 to 3700 Hz; Figure 6.22(c) between 3700 to 4500 Hz; and Figure 6.22(d)

between 4300 to 5000 Hz. It was assumed these frequency bands would result in the detection

of the best characteristic frequency. It can be seen that at the lower bands (Figures 6.22(a) and

6.22(b) there are only a sign of shaft frequencies at 3.7 Hz. In the higher frequency bands

(Figures 6.22(c) and Figure 6.22(d)) there are indications of shaft frequencies at 3.7 Hz and

4x3.7=14.8 Hz. An inner race characteristic defect frequency with a dominant spike was found

in Figures 6.22(a) and 6.22(b) at 30.5 Hz very close to the BPFI of 30.31 because the observed

signal was talcen when the crack initiated.

(a) Demodualed Recovsred Slgnal at 5 58 PM

(b) Demoduletsd Recovered slgnal a1 5 58 PM

Band Passed 1000Hi To 2500 tlr BandPassed 3000 Hz To 3700Hi

I i -i---7---

1

4

5 I 3 5

B 4 : 3 -

9 9 - ~ 2 5

$ 3 3 2

3 7 hz= 1"RPU

2 1 5 -74Hzn2"PPM

0 5

I i

'0 5 10 15 20 25 30 35 40 45 50 '0 5 10 15 20 25 30 35 40 45 50 Frequency ( I + j Frequency( H i 1

(c) Damoduiated Recoverod SlQnal a1 5 58 PM

(4 Demodulated Rscovorod SIgnBI at 5 58 PM

Band Parsed 3700 Hz lo 4500Hz Band Passed 4300 Hz To 5000 Hz

Figure 6.22 Demodulated recovered signal at 21 1 RPM, 5:58 PM Band Passed Between (a) 1000 to 2500 Hz

(b) Between 3000 to 3700 Hz (c) Between 3700 to 4500 Hz (d) Between 4700 to 5000 Hz


Figure 6.23 shows the plot of CF and AM versus filter length for the recovered signal at 6:03

PM. At 6:03 PM shown in Table 6.7, there is an increase in kurtosis level of 3.45 which

indicated there is a bearing failure. An automatic program selects the optimum filter length for

the blind equalizer which involves the CF and AM process as shown in Figure 6.23. It can be

observed from Figure 6.23 that the CF values stabilized at around 18 while the AM values

stabilized around 40 filter length. It can also be observed from Figure 6.23 that an optimum

point of filter length 32 is achieved. The optimum filter length L=32 is used in this test. In the

AM plot it can be seen that the AM amplitudes decrease down from -40 dB to 0 dB until 32

filter length. Beyond that value, the AM amplitude remained fairly constant at about 0 dB.

Crest Factor Modified Recovered Signal @ 1:40 PM 14 v I I I I I

ArithmeticMean (dB) & OptimumFilter=32 80 I I I I I

-20 1 1 I I I I I 0 20 40 60 80 100 120

Filter Length

Figure 6.23 CF and AM plot for signal at 6:03 PM with 21 1 RPM when crack initiated

Figure 6.24(a) shows time domain of the observed signal at 6:03 PM, which shows bearing

crack is progressive. The result of a recovered signal after blind deconvolution at 6:03 PM is

shorn in Figure 6.24(c). The spectrum of the observed signal and recovered signal at 6:03 PM

with optimum filter in the frequency domain are presented in Figures 6.24(b) and 6.24(d). The

lower spike in the frequency spectrum, Figure 6.24(b), shows 3 frequency components at

around 200 Hz, 800 Hz and 1800 Hz. Figure 6.24(c) shows the time domain plot of the

recovered signal after the blind deconvolution process with the optimum filter length L=32. The

observed signal in Figure 6.24(a) has a kurtosis level of 3.43 which shows bearing damage. This

value may not represent the severity of the defect due to background noise. The kurtosis of the

recovered signal was 6.83 which is a much clearer indication of a damaged bearing. In the

observed signal of Figure 6.24(a), there is little probability to identify the spike of an inner race

fault and the bearing fault signal was masked by background noise. In Figure 6.24(c), a typical

trace of spiky signal is obvious after blind deconvolution in the recovered signal and few

frequency components such as 3.7 Hz, 14.8 Hz and 25 Hz were detected. The 3.7 Hz frequency

component can be attributed to the shaft misalignment and the 25 Hz component is very close

value to the outer race defect frequency (BPFO) of 24.96 Hz shown in Table 6.2. Since the

observed signal was taken in the progressive failure period, a combination of some of the

deterministic characteristic frequencies shown in Table 6.2 may appear in the time and

frequency domain recovered signal. It can be observed from Figure 6.24(d) that the recovered

signal contains new resonant frequency components at 3600 Hz which can be used in the

demodulation process. Bearing 2 T~me @ 6:03 PM

(a) Time Domain Obseived Signal L=32 The Middle Point [b) Frequency Domain Signal

(c) Time Domain Recovered Signal - 4 . x 10 (dl

-" 8

0 0 2 0.4 0 6 0.8 "0 1000 2000 3000 4000 5000 6000 Time ( Second ) Frequency (Hz)

Figure 6.24 (a) Observed Signal at 21 1 RPM at 6:03 PM middle point (b) Spectrum of the


Figure 6.25 shows the gain response and phase plot of the equalizer for the processing of the



Figure 6.25 that the vibration amplitudes were attenuated as shown in the gain response plot of

Figure 6.25. These ranged between -44 dB to -58 dB over the frequency range from 0 to 2000

Hz. It can be observed from the gain response plot that the frequency resonance around 400 Hz

was removed by a notch with a value of -58 dB. Figure 6.25 also shows the other notch at 1650

Hz with a value of -44 dB.

Gain Response of Equalizer @ 6:03 PM L=32 20 I I I I I

.- K m

Phase Response of Equalizer 500 I I I I I

0 ?-.. U1 a! 500 g - g-1000 ............... ................... .................... / a!

.................. % -1500 - .................... ; ..................... ..................... ..................... 1 E -2000 .........

-2500 I I I I I

0 1000 2000 3000 4000 5000 6000 Frequency (Hz)


the observed signal at 21 1 RPM at 6:03 PM test bearing No. 2


6:03 PM. In the demodulation process, the signal was band pass filtered between 3700 to 4500

Hz as the best modulated resonant frequency region. From Figure 6.26(d) it was obvious that

band passing within this range would result in detection of the best characteristic frequency. In

Figure 6.26 shaft rotational frequency with a dominant spike at 3.7 Hz, four times of the shaft

frequency harmonic 14.8 Hz, outer race defect (BPFO) of 25.6 Hz and inner race defect (BPFI)

of 30.5 Hz are clearly visible in the spectrum very close to the characteristic defect frequencies

presented in Table 6.2. The spikes are accompanied by a number of harmonics spaced at

multiples of the rotational speed frequency. The results of the start point and the middle point in

test bearing No. 2 also showed that blind deconvolution is capable of recovering the bearing

fault signal to identify the characterise defect frequencies .

The kurtosis value appears to be more sensitive to asses the performance of blind deconvolution

because crack initiation and propagation through bearing components would generate impulses,

thus changing the distribution of the vibration signal and increased the kurtosis.

Demodulated Recovered Signal at 6:03 PM lo -4 Band Passed 4300 Hz To 5000 Hz

8 I I I I I I I I I

Frequency (Hz)

Figure 6.26 Demodulated recovered signal rotating at 21 1 RPM at 6:03 PM middle point

test bearing 2, Band Passed Between 3700 to 4500 Hz

Table 6.8 Kurtosis of the signal with and without


Detected Defect Kurtosis Kurtosis of

Time Status Shaft Frequency of the the Signal Failure

Frequency Present Signal without Mode

(1 x) Hz with BD BD

558 PM Start point 3.7 Nil 3.1 6.37 Inner Race

6:03 PM Middle Pint 3.7 30.5 3.42 6.83 Combination

6:08 PM Highest Point 3.7 30.5 3.85 4.48 Combination

Table 6.8 lists the defect frequencies present in the observed signal and the kurtosis of the

signals before and after blind deconvolution (BD) with optimum filter length over the last 15

minutes to failure for test bearing 2. The time started at 5:58 PM to 6:08 PM failure point. It can

be observed from Table 6.6 that the kurtosis value with and without BD have changed. The

increase in the kurtosis value after blind deconvolution can be regarded as an advantage of blind

deconvolution as a sensitive technique for diagnostic when a crack is initiated in the bearing

components.

6.3.2.3 Description of di-- - --' bearing

b Fatigue spalling

Vn lubrication

- w- Figure 6.27 Bearing 2 a r failure

It can be observed from Figure 6.27 that fatigue spalling was initiated on the inner race of the

test bearing No. 2. The bearing inner race in figure 6.4.27 shows the development of fatigue spalling due to high loading and no lubrication.



A 6805 narrow test bearing was pressed fit into the slave bearing block output shaft. A spacer

was used to position the test bearings on the shaft. The speed was set at 21 1 RPM. Test bearing

No. 3 was run under a constant heavy load beyond the static rating capacity. To effectively

initiate bearing fatigue in a short period of time at a low rotational speed, an average load of 4.4 kN was applied. Since oil in the hydraulic container got hot, the load was not maintained

constant and a total average of 4.4 kN beyond the dynamic load of 2.6 kN was achieved. This is

not an ideal scenario, as the elastic, plastic stability of the bearing is reduced as radial load

approaches load capacity. The test duration lasted 3 hours whilst the estimated life was

calculated as 1 16.16 hours as indicated in Table 6.1. Signal features of this bearing show the

bearing operates in a steady state manner until the catastrophic failure point. The statistical

features of the data are presented in Table 6.9 for comparison. It can be observed fiom Table

6.9 that the RMS level ranged fiom 1.5 to 2.4 with peak to peak ranging fiom 20 to 28 and

kurtosis ranging fiom 3 to 3.35 at first three data points. The failure appears to happen over a

period of approximately 10 minutes. It can be observed from Table 6.9 that the statistical

features (RMS, P2P, and Kurtosis) for the failure point are different compared to the first three

data pints. The maximum values of the signal features at failure pints are: temperature 43"C, RMS 3.5, peak to peak 41 and kurtosis 4.6. These values are higher than the no fault condition

of the fust three data points. All statistical features tend to indicate the failure clearly.

Table 6.9 Statistical features summary -Test bearing No. 3

Description Temperature EMS Peak to Kurtosis

range peak

The first 3 data points 28OC - 34OC 1.5 - 2.4 20 - 28 3 - 3.35

Failure point 43°C 3.5 4 1 4.6

Data captured for test bearing No. 3 are shown in Figure 6.28 which shows the bearing failure

at the end of kurtosis plot. Each data point represents the calculated time signal feature values

for 10 second duration of the signal. The time data is talcen at 5 minutes intervals. The vibration

signal feature, kurtosis value of data is presented in Figures 6.28(b). Figure 6.28(a) shows

temperature of the bearing versus time over the last 35 minutes up to the failure point at 9:43

AM while Figure 6.28(b) shows the kurtosis of the collected signal versus time over the same

period. The failure appears to happen over a period of approximately 10 minutes, with three

data points showing a sharp rise starting at 9:33 AM to 9:43 AM when failure occurred. It can

be observed from Figures 6.28(b) that the kurtosis value at the failure point are higher than the

no fault condition baseline. In Figure 6.28(b), the kurtosis rises sharply at 9:33 PM up to the

failure point at 9:43 PM. Bearing operation is stable until the catastrophic failure point shown

at the end of the signal features plot in figure 6.28. This failure appears to be due to the

development of a crack due to significant overloading. This crack has propagated and caused to

separate the bearing outer race from the bearing. Subsequently the rolling elements are

removed.

Figure 6.28 Bearing 3 vibration signal features, period of last 35 minutes

up to failure; (a) Temperature, (b) Kurtosis


Since the equations to calculate the characteristic defect frequencies are based on bearing

geometry and speed alone, variations due to high loading and slipping are not considered. The

detected characteristic defect frequencies were within 0.063 % of related failure mode

frequencies. Figure 6.29(a) shows time domain of the observed signal at 9:38 AM shown in

Figure 8:30, which shows the crack is progressive. The result of a recovered signal after blind

deconvolution at 9:38 AM is shown in Figure 6.29(c). The spectrum of the observed signal and

recovered signal at 9:38 AM with optimum filter in the frequency domain are presented in

Figures 6.29(b) and 6.29(d). The lower spike in the frequency spectrum, Figure 6.29(b), shows

3 frequency components at around 200 Hz, 800 Hz and 1800 Hz. Figure 6.29(c) shows the time

domain plot of the recovered signal after the blind deconvolution process with the optimum

filter length L=32. The observed signal in Figure 6.29(a) has a kurtosis level of 3.07 which

shows a bearing damage. The value may not represent the severity of the defect due to

background noise. The lturtosis of the recovered signal was 10.17 which is a much clear

indication of a damaged bearing. In the observed signal Figure 6.29(a), it is not possible to

identify the spike of an inner race fault and the bearing fault signal was masked by background

noise. After blind deconvolution in the recovered signal Figure 6.29(c), a typical trace of spiky

signal is obvious and few frequency components such as 3.7 Hz, 7 Hz, 9Hz and 14.8 Hz were

detected. The 3.7 Hz frequency component can be attributed to the shaft misalignment and the

14.8 Hz component is harmonic of the shaft speed. Since the observed signal was taken in the

progressive failure period, a combination of some of the deterministic characteristic frequencies

shown in Table 6.2 may appear in the time and frequency domain recovered signal. It can be

observed from Figure 6.30(d) that the recovered signal contains new resonant frequency

components at 5500 Hz. which can be used in the demodulation process.


9:38 AM shown in Figure 6.29. In the demodulation process, the signal was band pass filtered

between 3700 to 4500 Hz as the best modulated resonant frequency region. From Figure 6.29(d)

it was obvious that band passing within this range would result in the best detection of the

characteristic frequency. In Figure 6.26 shaft rotational frequency with a dominant spike at 3.7

Hz, harmonics of the shaft frequency at 7.35 Hz and 11 Hz can be detected. No characteristic

defect frequency was detected in this spectrum due to a significant overloading. The spikes are

accompanied by a number of harmonics spaced at multiples of the rotational speed frequency.

The results of the collected signal at 9:38 AM in the bearing No. 3 also showed that blind

deconvolution is capable of recovering the bearing fault signal to identify the characterise

defect frequencies . The kurtosis value appears to be more sensitive to asses the performance of

blind deconvolution because the crack initiation and propagation through a bearing components

would generate impulses, thus changing the distribution of the vibration signal and increasing

the kurtosis.

Test Bearing 3 T~me @ 9:38 PM (a) Time Domain Observed Signal L.32 The Start Point (b) Frequency Domain Signal

0 035 I I 1

(c) Time Domain Recovered Signal

0.3 1


Figure 6.29 (a) Observed Signal at 21 1 RPM at 9:38 AM (b) Spectrum of the observed signal

(c) Recovered signal with filter length 32 (d) Spectrum of the recovered signal

Shaft Rotational Frequenciy and harmonics 3.7 Hz, 7.35 Hz and I I Hz

Demodulated Recovered Signal at 9 3 8 AM lo5 Band Passed 4300 Hz To 5000 Hz

Frequency (Hz)

9

8

7

I I I ! ! I ! I I

-

-

-

-

Figure 6.30 Demodulated recovered signal rotating at 21 1 RPM at 9:38 AM when crack grows

Band Passed Between 3700 to 4500 Hz

6.3.3.3 Description of damaged bearing - Figure 6.3 1 Test bearing No. 3 catastrophic failure

The kurtosis values in figure 6.29@) shows an increase when a crack was initiated at 9:33 AM.

The crack initiation in to any contacting surface would generate impulses, this changing the

distribution of the vibration signal and increasing the kurtosis value. It can be observed fiom

Figures 6.29@) which resulted in catastrophic damage. There may have been a stall in the crack

propagation during this period, or decrease in vibration amplitude due to these smoothing

effects. As the damage spread over a broader area, the kurtosis roes again over time. From

figure 6.3 1, the catastrophic bearing failure of test bearing No. 3 can be seen. The bearing inner

ring is fractured along its circderence, and also shows a hoop stress fracture. This indicates

the bearing has been loaded beyond its ultimate tensile strength and has subsequently failed.

This is supported by the signal data that indicates that a bearing fault was present at the

beginning of the test.



A 6805 narrow test bearing was pressed fit into the slave bearing block output shaft. A spacer

was used to position the test bearings on the shaft. The speed was set at 21 1 RPM. Test bearing

No. 3 was run with hard particle contaminants placed inside the bearing under a constant heavy

load to initiate bearing fatigue in a short period of time at a low rotational speed. The test

duration lasted 35 minutes whilst the estimated life was calculated as 116.16 hours as indicated

in Table 6.1. Signal features of this bearing show the bearing operates in a steady state manner

until the catastrophic failure point except the kurtosis plot. The statistical features of the data are

presented in Table 6.10 for comparison. It can be observed from Table 6.10 that the RMS level

ranged fiom 1.2 to 1.65 with peak to peak ranging from 1 1 to 14 and kurtosis ranging from 3 to

3.1 at first three data points. The failure appears to happen over a period of approximately 5

minutes. It can be observed from Table 6.10 that the statistical features (RMS, P2P, and

Kurtosis) for the failure point are different compared to the first three data pints. The maximum

values of the signal features at failure pints are: temperature 39OC, RMS 2.75, peak to peak 32

and kurtosis 3.6.4. These values are higher than the no fault condition of the first three data

point. All statistical features tend to indicate the failure clearly.

Table 6.10 Statistical features summarv -Test bearing No. 4

Description Temperature RMS Peak to Kurtosis

range peak

The first 3 data points 28OC - 34OC 1.2 - 1.65 1 1 - 14 3 - 3.1

Failure point 39°C 2.75 32 3.8

Figure 6.32 Bearing 4 vibration signal features over 35 Minutes; (a) Temperature, (b) RMS,

(c) Peak to peak, (d) Kurtosis

40 ic)

! I I I I

........ .....................................,.. _J .............. * -

/

....... : .................. : .................. : ................. 10 I I I I I !

(dl

Data captured for test bearing No. 4 are shown in Figure 6.32 which shows the bearing failure

Ln .- Ln e 3 3

at the end each plot. Each data point represents the calculated time signal feature values for 10

second duration of the signal. The time data is taken at 5 minutes intervals. The vibration signal

features, RMS, peak to peak, and kurtosis of data are presented in Figures 6.32(b), (c), and (d)

respectively. The RMS value for test bearing No. 4 is slightly higher than the no fault condition,

peak to peak and kurtosis remain similar. The failure appears to happen over a period of

I I I I I !

& &. - ................. : ................. " ......... ........ .................. d.................

I I I I I I

1'15 PM 1:20 PM 1:25 PM 1:30 PM 1:35 PM 1:40 PM 1:45 PM 1:50 PM Time

approximately 5 minutes, with two data points showing a rise starting at 1:45 PM to 1:50 PM

when failure occurred. The inclusion of hard particles is similar to a distributed defect in that

the overall vibration level increases. The bearing was run until a large crack is developed, as

seen at 1:30 PM in Figure 6.32. It can be observed from Figures 6.32(d) that the kurtosis value

at the failure point are higher than the no fault condition baseline. In Figure 6.32(d), the kurtosis

rises at 1:33 PM up to the failure point at 1:55 PM. Bearing operation is stable until the

catastrophic failure point shown at the end of the signal features plot in Figure 6.32. This failure

appears to be due to the development of a crack due to contamination. This indicates a change

in the bearing; this is most likely to be the development of a large crack. This crack has

propagated and separated the bearing outer race from the bearing. Subsequently the rolling

elements are removed. Interestingly temperature remains relatively unchanged at this point, and

does not indicate any change in bearing.


Since the equations to calculate the characteristic defect frequencies are based on bearing

geometry and speed alone, variations due to contamination is not considered. Figure 6.33(a)

shows time domain of the observed signal at 1:45 PM shown in Figure 8:33, which shows the

crack is progressive. The result of recovered signal after blind deconvolution at 1:45 PM is

shown in Figure 6.33(c). The spectrum of the observed signal and recovered signal at 1:45 PM

with optimum filter in frequency domain are presented in Figures 6.33(b) and 6.33(d). The

lower spike in the frequency spectrum, Figure 6.33(b), shows 3 frequency components at

around 200 Hz, 800 Hz and 1800 Hz. Figure 6.33(c) shows the time domain plot of the

recovered signal after the blind deconvolution process with the optimum filter length L=32.

The observed signal in Figure 6.33(a) has a kurtosis level of 3.08 which shows a bearing

damage. The value may not represent the severity of the defect due to background noise. The

kurtosis of the recovered signal was 26.16 which is a much clear indication of a damaged

bearing. In the observed signal Figure 6.33(a), it is not possible to identify the spike of an inner

race fault and the bearing fault signal was masked by background noise. After blind

deconvolution in the recovered signal Figure 6.33(c), a typical trace of spiky signal is obvious

and few frequency components such as 3.7 Hz, 4.5 Hz and 1OHz were detected. The 3.7 Hz

frequency component can be attributed to the shaft misalignment. Since the observed signal was

taken in the progressive failure period, a combination of some of the deterministic characteristic

frequencies shown in Table 6.2 may appear in the time and frequency domain recovered signal.

It can be observed from Figure 6.33(d) that the recovered signal contains new resonant

frequency components at 4000 Hz. and it can be used in the demodulation process. Since the

observed signal was taken in the progressive failure period, a combination of some of the

deterministic characteristic fi-equencies shown in Table 6.2 may appear in the time and

frequency domain recovered signal. It can be observed from Figure 6.33(d) that the recovered

signal contains new resonant frequency components at 4000 Hz which can be used for

demodulation process.

Bearing 4 Time @ 1:45 PM (a) Time Domain Obseived Signal L=32 The Middle Point (b) Frequency Domain Signal

(c) Time Domain Recovered Signal 0.5 I I 2.5 I

10-3 (dl I

-0.5 J 0 0.2 0.4 0.6 0.8

Time ( Second ) " 0 1000 2000 3000 4000 5000 6000

Frequncy (Hz)

Figure 6.33 (a) Observed Signal at 21 1 RPM at 1:30 PM test bearing 4 (b) Spectrum of the



1:45 PM shown in Figure 6.33. In the demodulation process, the signal was band pass filtered

between 3700 to 4500 Hz as the best modulated resonant frequency region. From Figure 6.33(d)

it was obvious that band passing within this range would result in the best detection of the

characteristic frequency because the new resonance component in the recovered signal

(4000Hz) matches with the actual excited resonant frequency region. In Figure 6.26 shaft

rotational frequency with a dominant spike at 3.7 Hz can be detected. No prominent spike as the

characteristic defect frequency was detected in this spectrum due to the contamination. The

spikes are accompanied by a number of harmonics spaced at multiples of the rotational speed

frequency. The results of the collected signal at 9:38 AM in the bearing No. 3 also showed that

blind deconvolution is capable of recovering the bearing fault signal to identify the characterise

defect frequencies . The kurtosis value appears to be more sensitive to asses the performance of

blind deconvolution because the crack initiation and propagation through a bearing components

would generate impulses, thus changing the distribution of the vibration signal and increasing

the kurtosis.

Demodulated Recovered Slgnal at 1.45 P M 1 0 ' ~ Band Passed 3700 Hz To 4500 Hz

3 I I I I I I I I I

Frequency (Hz)


peak point Band Passed Between 3700 to 4500 Hz

6.4 Results and Discussions

This chapter involves the application of blind deconvolution with high resonant frequency

techniques for improving the SNR for the detection of a fault in life time testing of the bearing.

The contribution of this part of the research is the implementation of life tests, further

determining the capabilities of the blind deconvolution technique. The trending of time history

shows that the technique is capable of providing crack propagation information. This chapter

focuses on condition monitoring of the bearing without an artificially induced crack. This

approach of recording data through the bearing life is unique because it differs from other

studies that are limited to artificially induced defect. It is important to detect incipient faults in

advance before catastrophic failure occurs. The study reveals the advantages of blind

deconvolution as a technique to recover the original signal from a signal of a potentially

damaged bearing that has been distorted by the transmission path. Blind deconvolution and

HFRT have proven to be effective in monitoring bearing condition with incipient damage. The

results show that the proposed technique behaves like a notch filter around the band frequency

of the resonance and it improves the SNR of the deconvolved signal. It was found that the

recovered signal had been improved when compared with the observed signal. The constraining

feature of the blind deconvolution technique is the non-linear scaling peak magnitudes (due to

the nature of the equalizer filter used in blind deconvolution). The non-linear transformation of

the equalizer filtration process nullifies the relation between absolute signal magnitude and

physical darnage level. In practice relative changes between peaks are not absolute magnitudes.

Kurtosis values were used to compare the observed and recovered signals because kurtosis

factor of greater than 3 is an indication of spikiness of the signal and an indication of a damaged

bearing. It was found that the kurtosis value significantly increased after blind deconvolution

which improved the ability of fault detection. The kurtosis trend for the observed signal over

time increased, but then decreased due to the healing phenomena of smoothing of the sharp

edges of the bearing crack. High frequency resonance technique (HFRT) was used to identify a

fault occurring at a characteristic frequency. The results showed that inner race ways are more

prone to failure than outer raceways or rolling elements in these specific experiments. The inner

race defect frequency was heavily modulated by the shaft frequency producing a series if

sidebands as strong harmonics were present in the spectrum. Two possible explanations are: (1)

the rotational speed of the cage relative to the inner race is higher than that relative to the outer

race. (2) Contact forces in the inner race concentrate in smaller areas due to a smaller radius of

curvature and defect on the inner race, and rolling elements periodically can move into and out

of the loading zone with the rotation of the shaft and the cage; However a defect on the outer

race remains at the same position relative to the loading zone, unless the race is spinning in its

housing.

Chapter 7

Conclusion and Future Work

7.1 Optimization of Equalizer Parameters Using Modified Crest Factor and Arithmetic Mean

This research has revealed the advantages of blind deconvolution as a technique to recover the

original signal of a typical faulty bearing corrupted by noise and distorted by the transmission

path. Since characteristic frequencies contain very little energy when the defect size is small,

and are usually overwhelmed by noise and higher levels of macro-structure vibration, it is

difficult to identify them. The results show that the proposed technique behaves like a notch

filter in removing correlated noise and it improves the signal-to-noise ratio (SNR) of the

deconvolved signal. A procedure to obtain the optimum equalizer parameter was addressed, by

considering the influences of the input parameters of blind deconvolution. The modified Crest

Factor (CF) and Arithmetic Mean (AM) graphs were used to determine the optimum equalizer

filter length. Using the slope and finite difference programs, the optimum filter length of the

equalizer was identified and applied to the blind deconvolution algorithm. The optimum filter

length is selected using a compromised program between CF and the AM plots, and is stored in

a data training set for use in a neural network. A pre-trained neural network is designed to train

the behaviour of the system and target the optimum filter length in any general operational

conditions. The input parameters for this neural network are: crest factor, kurtosis, arithmetic

mean and Mfrms. The target for this neural-network is the value of optimum filter length. At the

end of the process, the optimum filter length is transferred to the blind deconvolution program

to recover the original source of vibration signal. The neural network is used to select the

optimum filter length for a general application where the operating conditions and types of

bearing faults are not known.

The results showed that the proposed algorithm works very well in recovering the characteristic

bearing fault frequencies. It was found that the recovered signal had been improved when

compared with the observed signal in terms of increment in kurtosis value. Blind deconvolution

was found to be a suitable technique for recovering a damaged bearing signal because the

recovered signal had a higher kurtosis compared to the observed signal. It was shown that the

kurtosis factor of greater than 3 is an indication of spikiness of the signal of a damaged bearing.

The results showed that the algorithm works well in eliminating related high resonance without

any prior knowledge of the resonant frequency components and the blind equalizer acts like a

notch filter. The technique was also found to be effective in noise removal of signals measured

from faulty bearing balls, despite the non-stationary nature of such faults.

7.2 Simulation and Experimental Benchmarking of Blind Deconvolution

The principal reason for conducting the simulation study in this research was to asses and to

verify the effectiveness of the blind deconvolution algorithm as a technique for enhancing pre-

signal processing, in recovering bearing fault signals. It is difficult to detect defects in the direct

spectrum of characteristic rotational frequencies of bearings because they are either absent or

occur or too small to be detected and corrupted by noise. Detecting incipient failure in rolling

element bearings is just as difficult. In a situation where the vibration is contaminated by either

background noise or unwanted components, other normal fault detection techniques may fail to

detect a growing defect at the incipient stage, due to the low SNR.

In the simulation study the expected value of time intervals between the impacts of rolling

elements within a faulty bearing signal was used as feature of a damaged bearing. The observed

time period between impacts was improved, and better identification of the defect frequency

was achieved. In simulation studies it was found that the technique has the capability of

removing sinusoidal noise as SNR as low as -49 dB. The studies also showed that the technique

behaves like a notch filter in removing periodic noise. The efficiency and robustness of the

proposed algorithm was assessed using different levels of corrupting noises. The results showed

that the proposed algorithm works very well with a range of periodic noise and the technique is

successful in recovering the damped bearing signal. The blind deconvolution technique already

showed that it is incapable of removing random noise. Current theory [I31 has shown that the

technique does show the capability of analysing Guassian noise, but remnants of the corruption

signal does remains, indicating retention of random noise. It was also found that the severity of

the corrupting noise SNR had no effect on the created notch in the gain response plot. A

proportional relationship between the severity of the corrupting noise and notch dept could not

be established, which could have provided a means for eliminating all the corrupting noise.

7.3 Life Time Testing

The contribution of this part of the research was the application of blind deconvolution in real

life testing of a rolling element bearing until failure occurs. The trending of failure time history

showed that it was capable of detecting bearing failure. This research focused on condition

monitoring of bearings operating continuously. In this operation it is essential to detect

symptoms of failure in advance and blind deconvolution was applied to monitor the

performance of the bearing continuously. In the life time testing a crack was initiated on the

inner race due to the development of fatigue spalling due to high loading causing the inner race

to split up and the balls to be dropped between the shaft and the inner race under the pressing

force. As the damage spread over a broader area, the bearing inner ring is fractured along its

circumference, and also a hoop stress. The catastrophic bearing failure of life time testing can

be seen at the end of the test. This indicates the bearing has been loaded beyond its ultimate

tensile strength and has subsequently failed.

It was found that the technique worked well and the recovered signal had been improved when

compared with the observed signal in terms of increment in kurtosis level. A constraint of the

blind deconvolution technique however, was the effect of non-linear scaling of the peak

magnitudes due to the nature of the equalizer filter used in blind deconvolution. The non-linear

transformation of the equalizer filtration process nullified the relation between absolute signal

magnitude and physical damage level and therefore it is not a useful parameter to asses the

magnitude of the defect. The kurtosis value was used to compare the observed and recovered

signals as the technique is based on relative peak values to the mean level. It was found that the

kurtosis value significantly increased after blind deconvolution and improved the ability of fault

identification.

Blind deconvolution was thus found to have a number of advantages in practical condition

monitoring applications. First, it can enhance pre-signal processing by eliminating unwanted

noise components. Observed signals from faulty bearings processed thorough the blind

deconvolution technique enhanced the fault signal. For example, the time interval related to the

defect impacts can be clearly identified in time domain. Also the signal to noise ratio was much

higher after blind deconvolution as the background noise had been suppressed. Second, blind

deconvolution requires only one input signal, hence overcoming the problem of coast associated

with sensors. Adaptive noise cancellation (ANC) has also been applied to improve the SNR of

the monitored signal from a complex machine [4]. The ANC technique proved to work very

well in situations where the noise in the two inputs are mutually correlated and the reference

input contains no signal or a very weak signal. However, ANC requires a minimum of two

inputs. When applied in bearing fault detection, two sensors must be employed to generate the

signals. When only one signal is available as with many practical cases, blind deconvolution

was found to be the preferred technique for signal enhancing.

7.4 Future Work

The work thus far in the development of the blind deconvolution technique has highlighted

certain areas for future research.

1. This research attempted to establish a proportional relationship between the severity of

corrupting noise and notch depth. However, the blind deconvolution technique was

found to be only within a SNR range whereby the minim SNR was -49 dB in specific

periodic noise. It was thus found that the severity of corrupting noise SNR had no effect

on the created notch in the gain response plot. Further research in this respect would be

to establish a means for eliminating all corrupting noise through the establishment of an

algorithmic relationship between the corrupting noise severity and notch depth.

2. The blind deconvolution technique was found to be effective in removing high

frequencies, which could be construed to be resonant; but this research has not evaluated

resonant frequencies as such. However, the removal of these higher frequencies was

considered to be a constraint in that the frequencies were preferred not to be removed

and rather retained to elicit bearing fault information through other techniques. Further

research in this respect would therefore be to either consider a combined analysis using

the blind deconvolution technique together with other methods or to establish a shift

effect in the equalizer filter so as not to remove the higher frequencies and thus retain

the higher SNR.

3. In this research modified Crest Factor and Arithmetic Mean were used for optimization

of the filter length. Other parameters such as kurtosis could potentially be used in future

work to select an optimum filter length of the blind equalizer.

4. The selection of blind equalizer parameters is based on a trail-and-error approach. This

is generally not effective in real life application. It is proposed that to automate the

whole process would be beneficial for industry and could be considered as research

application in future work. . 5. The application of the blind deconvolution in suppressing background noise in condition

monitoring applications for other rotational machine components such as, gears, pumps,

etc., could be investigated in future work.

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