Dynamic Analysis and Crack Detection in Stationary and

Dynamic Analysis and Crack Detection in

Stationary and Rotating Shafts

A thesis submitted to the University of Manchester for the degree of

Doctor of Philosophy in the Faculty of Engineering and Physical Sciences

2015

Zyad Nawaf Haji

School of Mechanical, Aerospace and Civil Engineering

The University of Manchester

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Contents

Contents ..................................................................................................................... 2

List of Figures ............................................................................................................ 8

List of Tables ........................................................................................................... 17

Nomenclature ........................................................................................................... 18

Abstract .................................................................................................................... 22

Declaration ............................................................................................................... 23

Copyright Statement ................................................................................................ 24

List of Publications .................................................................................................. 25

Acknowledgments ................................................................................................... 26

Introduction ......................................................................................... 27 CHAPTER 1

Background ......................................................................................................... 27 1.1

Types of Cracks and Importance of Crack Identification ................................... 28 1.2

Methods for Crack Identification ........................................................................ 29 1.3

Model-Based Methods ................................................................................. 30 1.3.1

Vibration-Based Methods ............................................................................ 30 1.3.2

Study Objectives and Methodology .................................................................... 31 1.4

Contributions to Knowledge ............................................................................... 33 1.5

Structure of the Thesis ......................................................................................... 34 1.6

Literature Review ................................................................................ 35 CHAPTER 2

Overview ............................................................................................................. 35 2.1

Types of Cracked Shafts ..................................................................................... 35 2.2

Transverse Cracks ........................................................................................ 35 2.2.1

Longitudinal Cracks ..................................................................................... 36 2.2.2

Slant Cracks.................................................................................................. 36 2.2.3

Behaviour of Cracked Shafts ............................................................................... 37 2.3

Non-Destructive Damage Detection Methods .................................................... 39 2.4

Identification of Cracks Based on Vibration Methods ........................................ 40 2.5

The Forward Problem................................................................................... 41 2.5.1

The Inverse Problem .................................................................................... 43 2.5.2

3

Applied Approaches to Investigate Cracked rotors ............................................. 43 2.6

Approach of Wavelet Transform and Wavelet Finite Element .................... 43 2.6.1

Hilbert –Haung Transform (HHT) ............................................................... 45 2.6.2

Analyses through Finite Element Technique ............................................... 46 2.6.3

Analysis of Crack through Numerical Simulations and Experiments.......... 48 2.6.4

Investigation through Nonlinear Dynamics of Cracked Rotors ................... 49 2.6.5

Analysis of Cracked Rotor using other Techniques ..................................... 51 2.6.6

Summary ............................................................................................................. 52 2.7

Mathematical Modelling of Rotor Systems and Theoretical Analysis CHAPTER 3

Tools 54

Introduction ......................................................................................................... 54 3.1

Coordinate Systems ............................................................................................. 54 3.2

Rigid Disc Elements ............................................................................................ 56 3.3

Shaft Elements ..................................................................................................... 58 3.4

Bernoulli-Euler Beam Element Theory........................................................ 58 3.4.1

Mass and Stiffness Matrices for Shaft Elements in Two bending Planes .... 61 3.4.2

Timoshenko Beam Element Theory ............................................................. 62 3.4.3

Gyroscopic Effects ....................................................................................... 67 3.4.4

Bearings ............................................................................................................... 69 3.5

Assembly Process ................................................................................................ 69 3.6

Boundary Conditions ........................................................................................... 71 3.7

System Equations of Motion ............................................................................... 72 3.8

Finite Element Model of Cracked Rotor Systems ............................................... 72 3.9

Cracked Rotor with an Open Crack ............................................................. 72 3.9.1

Cracked Rotor with Breathing Crack ........................................................... 77 3.9.2

Dynamic Analysis of the System ........................................................................ 81 3.10

Whirl Speed Analysis (Free Response System) ........................................... 81 3.10.1

Response of Rotors to Unbalance Forces and Moments .............................. 82 3.10.2

Theoretical Analysis Tool ................................................................................... 82 3.11

Description of Matlab Scripts ...................................................................... 83 3.11.1

Crack Definitions ......................................................................................... 84 3.11.2

Element Type Used in Ansys Numerical Model ................................................. 85 3.12

4

Matlab Script Verification ................................................................................... 86 3.13

Verification Models...................................................................................... 89 3.14.1

Summary ............................................................................................................. 91 3.15

Experimental Test Rig and Vibration Measuring Instruments ...... 92 CHAPTER 4

Introduction ......................................................................................................... 92 4.1

Experimental Test Rigs ....................................................................................... 92 4.2

Test Rig Used in Stationary Case ................................................................. 92 4.2.1

Test Rig used in Rotating Case .................................................................... 96 4.2.2

Rotor Alignment .................................................................................................. 98 4.3

Vibration Measuring Instruments ...................................................................... 100 4.4

Response Measurement .............................................................................. 100 4.4.1

4.4.1.1 Accelerometers versus PZTs ............................................................... 100

4.4.1.2 Strain Gauges versus PZTs ................................................................. 103

Data Acquisition Card ................................................................................ 107 4.4.2

Experimental Test Methodology ....................................................................... 108 4.5

Modal Analysis and Frequency Resolution Problems ............................... 112 4.5.1

Summary ........................................................................................................... 114 4.6

Detection and Localisation of a Rotor Crack Using a Roving Disc CHAPTER 5

and Normalised Natural Frequency Approach ........................................................ 115

Introduction ....................................................................................................... 117 5.1

Equation of Motion of a Cracked Rotor ............................................................ 120 5.2

Element Matrices of Rotor Systems in the Fixed Frame ........................... 120 5.2.1

Modelling of the Cracked Element ................................................................... 125 5.3

Numerical Solution ........................................................................................... 127 5.4

Crack Identification Technique ......................................................................... 129 5.5

Numerical Results and Analyses ....................................................................... 130 5.6

Validity of the NNF Curves Technique for Few Disc Positions ....................... 135 5.7

Experimental Testing and Validation ................................................................ 137 5.8

Experimental Test Rig and Instrumentation............................................... 137 5.8.1

Comparisons of Theoretical and Experimental Characteristics ................. 139 5.8.2

5

5.8.2.1 Case 1: Crack Parameters [μ, Γ] = [0.5, 0.3] ...................................... 139

5.8.2.2 Case 2: Crack Parameters [μ, Γ] = [0.3, 0.3] ...................................... 140

5.8.2.3 Case 3: crack parameters [μ, Γ] = [0.3, 0.5] ........................................ 142

5.8.2.4 Case 4: crack parameters [μ, Γ] = [0.3, 0.7] ........................................ 143

Summary of the Experimental Cases ......................................................... 145 5.8.3

Conclusions ....................................................................................................... 145 5.9

Vibration-based Crack Identification and Location in Rotors Using CHAPTER 6

a Roving Disc and Products of Natural Frequency Curves: Analytical Simulation

and Experimental Validation ..................................................................................... 146

Motivation and Background .............................................................................. 147 6.1

Modelling of the Uncracked Rotor .................................................................... 151 6.2

Equations of Motion ................................................................................... 151 6.2.1

Model of the Rotor with an Open Crack ........................................................... 153 6.3

Crack Modelling ................................................................................................ 153 6.4

Numerical Model ............................................................................................... 155 6.5

Investigation Procedures ............................................................................ 156 6.5.1

Methodology of Crack Identification ................................................................ 157 6.6

Numerical Simulations and Results .................................................................. 158 6.7

Effect of Crack Location and Size ............................................................. 159 6.7.1

Effect of Symmetrical Crack Location....................................................... 164 6.7.2

Spatial Interval Influence .................................................................................. 166 6.8

Experimental Testing and Results ..................................................................... 167 6.9

Experimental Rig and Instrumentation ............................................................. 167 6.10

Experimental Results and Analyses ........................................................... 169 6.10.1

Sensitivity of Including or Excluding the First Mode on the Accuracy of the 6.11

Crack Location .......................................................................................................... 174

Conclusions ....................................................................................................... 178 6.12

The Use of Roving Discs and Orthogonal Natural Frequencies for CHAPTER 7

Crack Identification and Location in Rotors ........................................................... 179

Introduction ....................................................................................................... 181 7.1

System Modelling ............................................................................................. 185 7.2

6

Finite Element Model ................................................................................. 185 7.2.1

Rotor System Equations of Motion ............................................................ 186 7.2.2

Crack Modelling ......................................................................................... 187 7.2.3

Simulation Model and Algorithm ..................................................................... 190 7.3

Finite Element Model ................................................................................. 190 7.3.1

Crack Identification Algorithm .................................................................. 190 7.3.2

Simulation Results and Discussions .................................................................. 192 7.4

Effects of Crack Depth and Locations .............................................................. 192 7.5

Case 1: μ = [0.3, 0.5, 0.7 1], Γ = 0.2, mass ratio = 20.4% ......................... 192 7.5.1






Sensitivity of the Proposed Technique to the Mass of the Roving Disc ........... 200 7.6

Crack Identification using a Lighter Roving Disc ..................................... 201 7.6.1

7.6.1.1 Case 7: μ = [0.3, 0.5, 0.7 1], Γ = 0.3, mass ratio = 8.2% .................... 202

7.6.1.2 Case 8: μ = [0.3, 0.5, 0.7 1], Γ = 0.5, mass ratio = 8.2% .................... 204

Crack Identification using a Heavier Roving Disc..................................... 205 7.6.2


Case 10: μ = [0.3, 0.5, 0.7 1], Γ = 0.5, mass ratio = 40.8% ....................... 208 7.6.4

Feasibility of the Proposed Technique based on Few Roving Disc Positions .. 208 7.7

Conclusions ....................................................................................................... 211 7.8

Crack Identification in Rotating Rotors ......................................... 212 CHAPTER 8

Introduction ....................................................................................................... 212 8.1

Characteristics of Rotating Rotor ...................................................................... 212 8.2

Natural Frequency Map (Campbell Diagram) .................................................. 213 8.3

Numerical Simulation ....................................................................................... 214 8.4

Case 1: Crack Parameters [μ, Γ] = [0.3, 0.33] ............................................ 216 8.4.1



Experimental Test ............................................................................................. 221 8.5

7

Experimental Results.................................................................................. 222 8.5.1

Experimental results of Case 1: Crack Parameters [μ, Γ] = [0.3, 0.33] ..... 223 8.5.2



Summary ........................................................................................................... 226 8.6

Discussions, Summary, Conclusions and Prospective Studies ...... 227 CHAPTER 9

Discussion of the Roving Disc Effect ............................................................... 227 9.1

Summary of the Thesis ...................................................................................... 229 9.2

Limitations of the Proposed Techniques ........................................................... 232 9.3

Conclusions ....................................................................................................... 233 9.4

Scope for Prospective Studies ........................................................................... 233 9.5

Design and Dimensions of Rotating Rig ........................................ 235 APPENDIX A

Specifications of Vibration Measuring Instruments ....................... 238 APPENDIX B

References .............................................................................................................. 240

8

List of Figures

Figure ‎1.1 Typical applications of rotors: (a) turbo-machinery (different stages); (b)

ship propeller shaft; (c) backward curved fan; (d) gas recirculation fan. .... 27

Figure ‎1.2: Effect of fatigue cracks. ................................................................................ 28

Figure ‎1.3: A transverse crack type. ............................................................................... 29

Figure ‎2.1: (a) A transverse crack; (b) A longitudinal Crack. ........................................ 36

Figure ‎2.2: A slant crack (or torsion crack) type. ........................................................... 37

Figure ‎2.3: Vibration-based methodologies for crack investigations. ............................ 42

Figure ‎3.1: Typical finite rotor element and coordinates. ............................................... 55

Figure ‎3.2: Coordinates are used in the analysis of the rotor-bearing system [102]. ...... 55

Figure ‎3.3: Typical local coordinates in the two bending planes: (a) x-z plane, (b) y-z

plane [102].................................................................................................... 58

Figure ‎3.4: Shear in a small section of the beam [102]. .................................................. 63

Figure ‎3.5: Assembly of beam element matrices to form the global matrix. .................. 70

Figure ‎3.6: Adding disc influence ................................................................................... 70

Figure ‎3.7: Adding bearing influence ............................................................................. 71

Figure ‎3.8: Effect of boundary conditions on the system global matrix. ........................ 71

Figure ‎3.9: Modelling diagrams of the cracked element cross-section. (a) Before

rotation. (b) After shaft rotation. The hatched part defines the area of the

crack segment [18, 21, 75]. .......................................................................... 73

Figure ‎3.10: Breathing crack states and centroidal positions of the cross-section of

the cracked element at various rotational angles [21]. ................................. 78

Figure ‎3.11: (a) Cracked and uncracked elements have equal width. (b) Cracked and

uncracked elements of different width. ........................................................ 84

Figure ‎3.12: (a) Geometry of beam189. (b) Geometry of the element type combine

14. ................................................................................................................. 86

Figure ‎3.13: Verification model. ..................................................................................... 89

Figure ‎3.14: Cross-section of the crack with 0.1 depth ratio in Ansys ........................... 90

Figure ‎3.15: Modelling of the cracked rotor with disc in Ansys. ................................... 91

Figure ‎4.1: Experimental test rig of the stationary case: 1. Left bearing 2. Right

bearing. 3. PZT sensors. 4. Terminal conector of the PZTs‘ wires. 5.

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Impact hammer. 6. Signal conditioner for impact hammer. 7. NI-data

aquestion card. 8. PC. ................................................................................... 93

Figure ‎4.2: Dimensions of the experimntal test rig (of the stationary case). .................. 94

Figure ‎4.3: Circumferential grooves at 90 degrees interval around the disc bore. ......... 95

Figure ‎4.4: shafts of experimental tests........................................................................... 96

Figure ‎4.5: Experimental test rig for the rotating case: 1. AC-motor 2. Flexible

coupling. 3. Invertor. 4 Left-bearing. 5. Accelerometers. 6 Tachometer. 7.

Zoomed local part of the shaft. 8. crack slot. 9. PZT sensors and wires.

10. Right-bearing. 11. Collar with four holes. 12. Accelerometers. 13.

Slip ring (24-channel) 14. Aluminuime disc with four groves at its bore.

15. 16-channel data-aquisition boxes (Data physics-Abacus). 16. PC......... 97

Figure ‎4.6: Assembly of the slip ring and the wires of the PZT sensors. ....................... 98

Figure ‎4.7: Alignment instrument ................................................................................... 99

Figure ‎4.8: Comparisons between rotating shaft responses measured using

accelerometers and PZTs during runup tests. Ver. and Hor. stand for the

vertical and horizontal planes, respectively. .............................................. 101

Figure ‎4.9: Operating principle of: (a) accelerometer, and (b) PZT sensor. 1.

accelerometer case 2. seismic mass 3. Piezoelectric crystals 4. Micro-

circuit 5. test structur. ................................................................................. 102

Figure ‎4.10: The Test rig for using the longitudinal wave propagation method to

compare PZTs with starin gauges for on-shaft vibration mesurment.1.

Stainless steel shaft. 2. Ball for striking the shaft. 3. Zoomed local shaft

area. 4 Strain gauge. 5 PZT. 6 Strain gauge‘s power amplifier (quarter-

bridge). 7 Strain gauge connecters. 8. Data acquisition system (Data

Physics-Abacus). 9. PC .............................................................................. 104

Figure ‎4.11: Output of both the PZT and strain gauge at the same axial location on

the shaft (both in volts)............................................................................... 105

Figure ‎4.12: Output of both the PZT and strain gauge at the same axial location on

the shaft. PZT in volts and strain gauge in strain (microstrain). ............... 106

Figure ‎4.13: Sensitivity of the PZT sensors in comparison with the strain gauges. ..... 106

Figure ‎4.14: Data acquistion systems: (a) NI-DAQ hardware and NI-Signal Express

software. (b). Data Physics (Abacus) system ............................................. 108

Figure ‎4.15: Frequency response functions at the same location of the disc at both the

intact and cracked rotor. Crack depth ratio μ = 0.3 at location Γ = 0.3.

Frequency resolution = 0.315 Hz ............................................................... 109

10

Figure ‎4.16: Time waveform (PZT sensor) of the rotating rotor at a point of the

roving disc for both the intact and cracked rotor. Crack depth ratio μ =

0.3 at location Γ = 0.3................................................................................. 110

Figure ‎4.17: A screenshot of waterfall plot of spectra of a PZT sensor on the shaft.

The analysis was done by DataPhysics hardware using built-in

SignalCalc software.................................................................................... 111

Figure ‎5.1: Finite element model of the rotor with a cracked cross-section. ................ 121

Figure ‎5.2: Typical finite shaft element and coordinates. ............................................. 121

Figure ‎5.3: Definition of the degrees of freedom for the shaft element. ....................... 123

Figure ‎5.4: A cracked element cross-section: (a) rotating, (b) non-rotating; the

hatched partdefines the area of the crack segment [18, 21]. ...................... 126

Figure ‎5.5: First four direct natural frequencies of the stationary cracked rotor with a

crack of different depth ratios at = 0.3: (a) 1st mode, (b) 2

nd mode, (c)

3rd

mode, (d) 4th

mode. ............................................................................... 129

Figure ‎5.6: First four theoretical NNF curves of the stationary cracked rotor with a


nd mode, (c)

3rd

mode, (d) 4th

mode. ............................................................................... 131



nd mode, (c)

3rd

mode, (d) 4th

mode. ............................................................................... 132



nd mode, (c)

3rd

mode, (d) 4th

mode. ............................................................................... 134



nd mode, (c)

3rd

mode, (d) 4th

mode. ............................................................................... 135

Figure ‎5.10: Variation of the first and second NNFCs against the locations of the

roving disc at only 5 points along the shaft length. Cracks at location Γ =

0.3. (a) 1st mode, (b) 2nd mode. ................................................................ 136

Figure ‎5.11: Variation of the first and second NNF curves agnaist the locations of the

roving disc at 5 points in the windowed sections shwon in Fig. 10. Cracks

at location Γ = 0.3. (a) 1st mode, (b) 2nd mode. ........................................ 137

Figure ‎5.12: The experimental rig, PZT sensors and the transverse crack. .................. 138

11

Figure ‎5.13: Comparison of the theoretical and experimental NNF curves for a

cracked shaft with μ = 0.5 at location Γ = 0.3: (a) 1st mode, (b) 2nd

mode, (c) 3rd mode, (d) 4th mode. ............................................................. 139









mode, (c) 3rd mode, (d) 4th mode .............................................................. 144

Figure ‎6.1: The finite element model of the intact and cracked rotor ........................... 151

Figure ‎6.2: Schematic view of a finite rotor element and coordinates for an intact and

cracked. ...................................................................................................... 152

Figure ‎6.3: A cracked element cross-section; (a) Rotating, (b) Non-rotating; the

hatched part defines the area of the crack segment [18, 21, 75]. ............... 153

Figure ‎6.4: Frequency curve products (FCP) of the cracked rotor with a crack of

various depth ratios μ at the location = 0.2: (a), (b) and (c) use

Equation (‎6.12); (d) uses Equation (‎6.13). ................................................. 159













12




Figure ‎6.10: Experimental rig: (a) Assembly setup; (b) Disc groves. .......................... 167

Figure ‎6.11: The transverse crack and bonded PZTs .................................................... 168

Figure ‎6.12: Experimental frequency curves product (FCP) of the cracked rotor with

a crack of μ = 0.3 and 0.5 at the location = 0.3: (a), (b) and (c) use



a crack of μ = 0.3 at the location = 0.5: (a), (b) and (c) use Equation

(‎6.12); (d) uses Equation (‎6.13). ................................................................ 171


a crack of μ = 0.3 at the location = 0.7: (a) use Equation (‎6.12); (b) use

Equation (‎6.13). .......................................................................................... 173

Figure ‎6.15: Theoretical and experimental NNF curves of the cracked shaft with μ =

0.3 at location Γ = 0.5: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th

mode. .......................................................................................................... 175

Figure ‎6.16: The NNF curves of the FCP method based on the modes 2, 3 and 4 of

the cracked rotor with a crack of μ = 0.3 and 0.5 at location = 0.3: (a)

use Equation (‎6.12); (b) use Equation (‎6.13). ............................................ 176

Figure ‎6.17: The NNF curves of the FCP method based on the modes 2, 3 and 4 of

the cracked rotor with a crack of μ = 0.3 at location = 0.5: (a) use

Equation (‎6.12); (b) use Equation (‎6.13).................................................... 177

Figure ‎7.1: Finite element model of an intact and cracked rotor. ................................. 185

Figure ‎7.2: Typical finite rotor element and coordinates for an intact and cracked

rotor. ........................................................................................................... 186

Figure ‎7.3: A cracked element cross-section; (a) Non-rotating, (b) Rotating; the

hatched part defines the area of the crack segment [18, 21]. ..................... 188

Figure ‎7.4: Second moments of area of the cracked segment in this study against

crack depth ratios. ...................................................................................... 191

Figure ‎7.5: Vertical normalised natural frequency curves of the non-rotating cracked

rotor for cracks of different depth ratios located at = 0.2 and a roving

disc of mass 0.5 kg. Based on natural frequencies of non-rotating intact

13

and cracked rotor in the vertical y-z plane: (a) 1st mode, (b) 2

nd mode, (c)

3rd

mode, (d) 4th

mode. ............................................................................... 193

Figure ‎7.6: Normalised orthogonal natural frequency curves of the non-rotating

cracked rotor cracks of different depth ratios located at = 0.2 and a

roving disc of mass 0.5 kg. Based on natural frequencies of non-rotating

cracked rotor in both the horizontal and vertical planes: (a) 1st mode, (b)

2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 194





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 196





2nd

mode, (c) 3rd

mode, (d) 4th

mode .......................................................... 197





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 198





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 199





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 201



14



2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 202





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 203

Figure ‎7.14: Vertical normalised natural frequency curves of the non-rotating

cracked rotor for cracks of different depth ratios located at = 0.5 and a


intact and cracked rotor in the vertical y-z plane: (a) 1st mode, (b) 2

nd

mode, (c) 3rd

mode, (d) 4th

mode. ............................................................... 205





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 206





2nd

mode, (c) 3rd

mode, (d) 4th

mode. ......................................................... 207

Figure ‎7.17: Variation of the first and second NONF curves agnaist the locations of

the roving disc at 5 points only along the shaft length. Cracks at location

ξ = 0.3. (a) 1st mode, (b) 2

nd mode. ............................................................ 210

Figure ‎7.18: Variation of the first and second NONF curves agnaist the locations of

the roving disc at 5 points in the windowed sections shwon in Figure 17.

Cracks at location ξ = 0.3. (a) 1st mode, (b) 2

nd mode. ............................... 210

Figure ‎8.1: Schematic of Campbell diagrame. Black square and star markers indictate

Forward (FW) and Backward (BW) whirling natural frequencies,

respectively................................................................................................. 214

Figure ‎8.2: Simulation model of the cracked rotating rotor .......................................... 215

15

Figure ‎8.3: Campbell diagram of the intact rotating rotor in case1 at three locations

of the roving disc. (a), (b) and (c) for the disc close to the left-bearing,

mid-shaft and close to right–bearing, respectively. Black square and star

marks indictate Forward (FW) and Backward (BW) whirling natural

frequencies, respectively. ........................................................................... 217

Figure ‎8.4: First mode normalised natural frequency curve of the craked rotating

shaf. Crack depth ratio μ = 0.3 at Γ = 0.33. Rotating speed range 0-3000

rpm: (a) and (b) are Forward and Backward whirling critical speeds,

respectively................................................................................................. 218

Figure ‎8.5: First mode normalised natural frequency curve of the craked rotating

shaft. Crack depth ratio μ = 0.3 at Γ = 0.53. Rotating speed range 0-3000

rpm: (a) and (b) are Forward and Backward whirling critical speeds,

respectively................................................................................................. 220

Figure ‎8.6: First mode normalised natural curve of the craked rotating shaft. Crack

depth ratio μ = 0.3 at Γ = 0.79. Rotating speed range 0-3000 rpm: (a) and

(b) are Forward and Backward whirling critical speeds, respectively ....... 221

Figure ‎8.7: Rotating test righ. (For details see Chapter 4 section ‎4.2.1 ........................ 222

Figure ‎8.8: Schematic of the PZT locations on the rotating shaft. ................................ 223

Figure ‎8.9: First mode normalised natural frequency of the experimental results of

the cracked rotating rotor with a crack of μ = 0.3 at location Γ = 0.33.

Rotational speed range 10-300 rpm: (a) Bottom PZT‘s response, (b)

Right PZT‘s response. ................................................................................ 224



Rotational speed range 10-300 rpm. (a) Bottom PZT‘s response. (b)

Right PZT‘s respons. .................................................................................. 225



Rotational speed range 10-300 rpm. (a) Bottom PZT‘s response. (b)

Right PZT‘s response. ................................................................................ 226

Figure ‎9.1: Adding a roving disc mass as a point mass-at a node of the beam............. 228

Figure ‎9.2: Schematic of a cracked beam with an auxiliary roving disc. ..................... 228

Figure ‎9.3: Dimensions of the rotating rig base (Dimensions in mm). ......................... 235

Figure ‎9.4: Bearing supports: (Dimension in mm) ....................................................... 235

Figure ‎9.5: Motor support (Dimensions in mm) .......................................................... 236

16

Figure ‎9.6: Dimensions of disc (mm). .......................................................................... 236

Figure ‎9.7: Bearing-collar dimensions (mm). ............................................................... 237

Figure ‎9.8: Specifications of the strain gauges used in Chapter 4. ............................... 239

17

List of Tables

Table ‎3.1: States of the breathing crack for full rotational angle ( ) ........................... 79

Table 3.2: Global stiffness matrix computed manually for two elements with a crack .. 87

Table ‎3.3: Results of using the exact solution and the developed Matlab scripts. .......... 88

Table ‎3.4: Comparison between Matlab scripts and Ansys solution of case 1 ............... 89

Table ‎3.5: Comparison between Matlab scripts and Ansys solution for case 2 .............. 90

Table ‎4.1: Dimensions and materials of the test rig ........................................................ 93

Table ‎4.2: Maximum acceptable misalignment limits (www.gearboxalignment.co.uk). 99

Table ‎5.1: Physical parameters of the rotor model ....................................................... 127

Table ‎5.2: Physical properties of cracks for Numerical simulations ............................ 128

Table ‎6.1: Numerical model parameters ....................................................................... 151

Table ‎6.2 : Values of the numerical and experimental cases ........................................ 156

Table ‎7.1: Values of parameters in the numerical model.............................................. 190

Table ‎7.2: Values for Numerical Cases......................................................................... 192

Table ‎8.1: Parameters of numerical cases ..................................................................... 215

Table ‎8.2: Computed FW and BW critical speeds of case 1 from the Campbell

diagrams of both the intact and cracked rotating shaft for each disc location.216

Table ‎8.3: Computed FW and BW crtical speeds of case 2 from the Campbell


Table ‎8.4: Computed FW and BW crtical speeds of Case 3 from the Campbell


Table ‎9.1: Specifications of the accelerometer sensors ................................................ 238

18

Nomenclature

u Displacement in x-axis direction relative to the fixed reference in space

u Displacement in y-axis direction relative to the fixed reference in space

Rotation about x-axis relative to the fixed reference in space

Rotation about y-axis relative to the fixed reference in space

Td Total Kinetic Energy of disc

Linear velocities in x direction

Linear velocities in y direction

Instantaneous angular velocities about the - axis

Instantaneous angular velocities about the - axis

Instantaneous angular velocities about the axis

md Mass of the disc

, - Transform matrix

* + Displacement vector

Med Element mass matrix

Ged Element gyroscopic matrix

( ) Lateral displacement of the beam neutral plane

Ee, Element Young‘s modulus

Local element displacement

( ) Shape functions

Ie Second moment of area of the cross section about the neutral axis

Element strain energy

Second derivatives of the shape function

Te Kinetic energy

Mass density

Beam cross-section area

Beam cross section angle

Ge Shear modulus

Poisson‘s ratio

19

Shear constant

Ratio between the inner and outer shaft radius

Constant shear angle

Constant dimensionless value

Matrix of element inertia force

Matrix of element elastic force

Second derivative of the element local coordinate

Ip Polar moment of inertia

Bearing horizontal stiffness

Bearing vertical stiffness

Bearing cross stiffness

Stiffness matrix of the bearings

Damping matrix of bearings

Bearing horizontal damping

Bearing vertical damping

Bearing cross damping

( ) Nodal displacement vector

Global mass matrix

Global damping matrix

Global stiffness matrix

Global gyroscopic matrix

Combination of unbalance forces and moments vector

ϕ Crack angle

Ω Rotor speed

Horizontal centroidal axis

Vertical centroidal axis

Second moment of area centroidal

Second moment of area centroidal

Horizontal stationary axis

Vertical stationary axis

Stiffness matrix of cracked element

Cracked element length

( ) Time-varying second moment of area of the cracked element about

( ) Time-varying second moment of area of the cracked element about

20

( ) Cracked element second moment of area about X

( ) Cracked element second moment of area about Y

Left uncracked area

Y-axis centroidal location

R Shaft radius

Overall cross-sectional area of the cracked element

Area of the crack segment (at fully open crack)

Constant value depends on the crack depth ratio

Crack depth ratio ⁄

Horizontal rotational second moment of area of the cracked cross-section

Vertical rotational second moment of area of the cracked cross-section

Second moment of area of the intact shaft

Stiffness matrix contains cracked and intact element

Crack angle commences to close

Fully closed crack angle

The second moment of area about X as the crack begins to close

The second moment of area about Y as the crack begins to close

Overall second moment of area of ( ) in horizontal direction

Overall second moment of area of ( ) in vertical direction

( ) Centroidal coordinates of ( ) relative to the stationary x axis

( ) Centroidal coordinates of ( ) relative to the stationary y axis

G.F Gauge factor of strain gauges

Vo Output voltage of strain gauges

Eex Excited voltage of strain gauges

Natural frequency of cracked rotors

Natural frequency of intact rotors

Natural frequency ratio

Maximum natural frequency ratio

Normalised natural frequency (NNF) curves of a mode

1st mode normalised natural frequency (NNF) curves

2nd

mode normalised natural frequency (NNF) curves

3rd


4th


21

ζ Non-dimensional roving disc location

Non-dimensional crack location

Shifted normalised natural frequency (NNF) curves

Normalised natural frequency at the pivot point

Ld Disc location

L Total shaft length

Lc Crack location

Lb Bearing span

( ) Non-dimensional frequency curve products

2nd

mode non-dimensional frequency curve products

3rd mode non-dimensional frequency curve products

4th

mode non-dimensional frequency curve products

Unified non-dimensional frequency curve products

Cracked shaft natural frequencies in vertical plane

Cracked shaft natural frequencies in horizontal plane

Orthogonal natural frequency ratios

Maximum orthogonal natural frequency ratio

i Normalised orthogonal natural frequency (NONF) curves

1 1st normalised orthogonal natural frequency (NONF) curves

2 2nd

normalised orthogonal natural frequency (NONF) curves

3 3rd


4 4th


22

Abstract

The University of Manchester

Zyad Nawaf Haji - PhD Mechanical Engineering - 2015

Dynamic Analysis and Crack Detection in Stationary and Rotating Shafts

The sustainability, smooth operation and operational life of rotating machinery

significantly rely on the techniques that detect the symptoms of incipient faults. Among

the faults in rotating systems, the presence of a crack is one of the most dangerous faults

that dramatically decreases the safety and operational life of the rotating systems,

thereby leading to catastrophic failure and potential injury to personnel if it is

undetected.

Although many valuable techniques and models have been developed to identify a crack

(or cracks) in stationary and rotating systems, finding an efficient technique (or model)

that can identify a unique vibration signature of the cracked rotor is still a great

challenge in this field. This is because of the unceasing necessity to develop high

performance rotating machines and driving towards significant reduction of the time

and cost of maintenance.

Most of the crack identification techniques and models in the available literature are

based on vibration-based methods. The main idea of the vibration-based method is that

the presence of a crack in a rotor induces a change in the mass, damping, and stiffness

of the rotor, and consequently detectable changes appear in the modal properties

(natural frequencies, modal damping, and mode shapes). Among all these modal

properties, the choice of the modal natural frequency change is more attractive as a tool

for crack identification. The changes in natural frequencies due to a crack can be

conveniently measured from just a few accessible points on the cracked rotor.

Furthermore, measuring the natural frequencies does not require expensive measuring

instruments, and the natural frequency data is normally less contaminated by

experimental noise. However, the change that a crack induces in the natural frequencies

is usually very small and can be buried in the ambient noise. Moreover, the natural

frequencies are not affected if the crack is located at the nodes of modes or far from the

location of inertia force and out-of-unbalance force that the disc generates in the shaft.

To overcome these problems (or limitations), therefore, this study is conducted using

the idea of the roving mass (roving disc in rotor case). The modal natural frequencies

are used for the identification and location of cracks of various severities at different

locations in both stationary and rotating shafts. The fundamental idea of the roving disc

is that an extra inertia force is traversed along the cracked rotor to significantly excite

the dynamics of the rotor near the crack locations. In other words, the location of a

crack can be anywhere on the shaft which is contrary to the developed techniques in the

available literature in which the location of a crack should be close to the disc.

Along with the roving disc idea, three crack identification techniques are developed in

this study using the natural frequencies of the cracked and intact shafts. Each of these

techniques has its merits and limitations for crack identification. These techniques are

implemented using data that are numerically generated by the finite element method

based on the Bernoulli-Euler shaft elements and experimentally validated in the

laboratory environment.

The numerical and experimental results clearly demonstrate the capability of the

suggested approach for the identification and location of cracks in stationary and

rotating shafts.

23

Declaration

I declare that no portion of the work referred to in the thesis has been submitted in

support of an application for another degree or qualification of this or any other

university or other institute of learning.

24

Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this thesis)

owns certain copyright or related rights in it (the ―Copyright‖) and he has given The

University of Manchester certain rights to use such Copyright, including for

administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic

copy, may be made only in accordance with the Copyright, Designs and Patents Act

1988 (as amended) and regulations issued under it or, where appropriate, in

accordance with licensing agreements which the University has from time to time.

This page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trademarks and other

intellectual property (the ―Intellectual Property‖) and any reproductions of copyright

works in the thesis, for example graphs and tables (―Reproductions‖), which may be

described in this thesis, may not be owned by the author and may be owned by third

parties. Such Intellectual Property and Reproductions cannot and must not be made

available for use without the prior written permission of the owner(s) of the relevant

Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and

commercialisation of this thesis, the Copyright and any Intellectual Property and/or

Reproductions described in it may take place is available in the University IP Policy

(see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant

Thesis restriction declarations deposited in the University Library, The University

Library‘s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations)

and in the University‘s policy on Presentation of Theses.

http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487

http://www.manchester.ac.uk/library/aboutus/regulations

25

List of Publications

Journal Papers

1. Z. N. Haji, S. Olutunde Oyadiji, ‗‗The use of roving discs and orthogonal natural

frequencies for crack identification and location in rotors‘‘, Journal of Sound and

Vibration, Vol. 333, Issue 23, November 2014, pp. 6237-6257.

2. Z. N. Haji, S. Olutunde Oyadiji, ‗‗Vibration-based Crack Identification and

Location in Rotors Using a Roving Disc and Products of Natural Frequency

Curves: Analytical Simulation and Experimental Validation‘‘, Journal of Sound and

Vibration (under review).

3. Z. N. Haji, S. Olutunde Oyadiji, ‗‗Detection and Localisation of a Rotor Crack

Using a Roving Disc and Normalised Natural Frequency Approach‘‘, Journal of

Finite Elements in Analysis and Design (under review)

Conference Papers

1. Z. N. Haji, S. O. Oyadiji, ‗‗Detection of Cracks in Stationary Rotors via the Modal

Frequency Changes Induced by a Roving Disc‘‘, Proceedings of the ASME 2014

12th Biennial Conference on Engineering Systems Design and Analysis

(ESDA2014), Copenhagen, Denmark, June 2014.

2. Z. N. Haji, S. O. Oyadiji, ‗‗Using Roving Disc and Natural Frequency Curves for

Crack Detection in Rotors‘‘, Conference on Vibration Engineering and Technology

of Machinery (VETOMACX 2014), the University of Manchester, UK, September

2014.

26

Acknowledgments

I would like to express special gratitude to my supervisor Dr S. O. Oyadiji for his

assistance and valuable guidance, strong motivation and encouragement during the

research period. I thank him for his unlimited support and invaluable knowledge that

enabled me to develop my own expertise in this area and gain insight into practical

experience.

I wish to thank all the workshop staff of the Pariser Building at The University of

Manchester, particularly Bill Storey, David Johns and Philip Oakes for their constant

support and assistance in preparation of the experimental testing rigs of this research.

I thank my sponsor, the Ministry of Higher Education and Scientific Research-

Kurdistan, for giving me this precious opportunity to learn.

Last but not the least, a special word of gratitude and appreciation goes to my family for

their sincere support and constant encouragement during my study. Also, I would like to

thank my nephew Maher and his wife Basima, who are also PhD students at

Birmingham University, for their moral and material support during the study period.

CHAPTER 1: Introduction

27

CHAPTER 1

Introduction

Background 1.1

Over the years, rotating machinery has been considerably used in modern industry

through various applications ranging from power plants to aerospace equipment and

marine propulsion systems. Gas turbines, compressors, centrifugal fans and ship

propellers are examples of rotating machinery that are driven by rotating shafts which

constitute the main component (or heart) of these high performance rotating systems as

shown in Figure ‎1.1.

(a) (b)

(c) (d)

Figure 1.1 Typical applications of rotors: (a) turbo-machinery (different stages);

(b) ship propeller shaft; (c) backward curved fan; (d) gas recirculation fan.


28

The unceasing necessity to develop high performance rotating machines has made

improving efficiency and increasing power and safety the main factors for the design of

high performance rotating machines. This has led to design machines with more flexible

part, specifically, shafts. Therefore the dynamic analysis of shafts has become more

important than ever.

Although shafts are carefully designed for fatigue loading and high level of safety by

using high quality materials and precise manufacturing techniques, disastrous failures of

rotors as a consequence of cracks still exists. This is, particularly the case in high speed

rotating machines, in which the shaft carries discs, blades, gears, etc. which are sources

of generating mechanical stresses such as flexural, torsional, axial radial and shear

forces during shaft rotation. As a result, the local stresses due to fatigue cracks will

increase and become more than the yield strength of the shaft material. With the passage

of time, the depth of the crack propagates until it reaches a limiting value beyond which

the shaft cannot withstand the static and dynamic load anymore and a sudden fracture of

the shaft occurs as shown in Figure ‎1.2 [1-6]. Therefore, an early detection of cracks in

shafts can extend the integrity of rotor systems and improve their safety, reliability and

operational life.

Types of Cracks and Importance of Crack Identification 1.2

Cracks in different configurations and severity can be developed on the shaft during the

operation of rotating machines. These cracks are classified according to their orientation

with respect to the shaft axis and are known as transverse cracks, longitudinal cracks

Figure 1.2: Effect of fatigue cracks.


29

and slant cracks. Of these crack types, the transverse crack has remained the most

dangerous and important kind of cracks, as the safety and the dynamic behaviour of

machines are significantly affected by its occurrence. This type of crack has been

intensively investigated because, it is perpendicular to the rotational axis of the shaft

(see Figure ‎1.3) and reduces the second moment of area of the shaft cross-section which

leads to considerable changes in the dynamic behaviour of the system [7, 8].

The presence of cracks in rotor-shafts is one of the most dangerous and critical defects

of rotating machinery. They are caused by cyclic fatigue loads (unavoidable in rotating

systems) and mechanical defects or high stress concentrations due to defects in the

manufacturing process. If a crack propagates continuously and is not detected, abrupt

failure, may occur and finally lead to a catastrophic failure with enormous costs in down

time, consequential damage to equipment and potential injury to personnel.

There are several non-destructive monitoring techniques such as vibration testing,

thermography, visual inspection, ultrasonic and process monitoring employed to

diagnose and monitor the critical behaviour of rotating machines during maintenance

spells. Among those predictive maintenance techniques, vibration testing is the

powerful non-destructive maintenance technique used with maintenance programs [9].

Methods for Crack Identification 1.3

Identification of cracks in rotating shafts has been a major topic for both engineers and

dynamic analysts for a long time. A variety of models and techniques have been

Figure 1.3: A transverse crack type.


30

developed to identify cracks in shafts at an early stage of the crack initiation. In the light

of the vibration dynamic behaviour of cracks, these techniques can be broadly classified

(or grouped) into two methods as follows [7, 10]:

Model-Based Methods 1.3.1

Analytical or numerical models consider the base of the model-based methods for

simulating and investigating the vibration dynamic behaviour of cracked shafts during

rotation. In these methods, the changes that the induced cracks create in the

configuration of rotor systems are represented as equivalent loads in the developed

analytical or numerical models. Thus, these equivalent loads, which are as virtual forces

or moments, are applied on the intact structures or rotor systems to create an identical

dynamic behaviour to that determined in the cracked systems. However, identification

of cracks and dynamics analysis of cracked systems based on the model-based methods

can be unreliable or have big errors. This is because of the numerical errors and

assumptions that the model-based techniques are based on. For example, stiffness

consideration of cracked shafts produces significant errors, since some of the developed

models do not really represent periodic changes at different angle of rotations according

to the stiffness parameters that were used for modelling these models.

Vibration-Based Methods 1.3.2

It is known that the presence of a crack in structures and rotor systems increases the

flexibility of the system which tends to change its dynamic characteristics such as the

natural frequencies and mode shapes. On this principle, vibration-based methods have

been developed to identify cracks in rotating systems. Literally, these methods are used

to measure (or monitor) the changes in vibration signals (amplitude and phase) of the

system response with and without a crack which normally occurs at 1X and 2X

frequency responses (1X and 2X stand for the 1st and 2

nd order of the rotor‘s running

speed, respectively) [3, 11]. These indicators, unfortunately, cannot convincingly

differentiate the presence of a crack in shafts from other inevitable faults such as

imbalance and misalignment which generate vibration spectra and waveforms similar to

that that the crack creates. Therefore, more study and investigation are still required to

develop more robust models and techniques to overcome these serious drawbacks in the

field of identification and localisation of cracks in rotor systems.


31

The above discussion constitutes the motivation for this study with the main aim being

to add new knowledge in the field of identification and localisation of cracks in rotor

systems through introducing a new technique and methodology.

Study Objectives and Methodology 1.4

Over the decades, many researchers have intensively investigated the dynamic

behaviour of cracked shafts. During this period, numerous models and techniques for

crack detection have been derived and developed. Most (if not all) of these techniques

are based on the theories of fracture mechanics and rotor dynamics. Specifically,

monitoring the changes in the dynamic behaviour due to the presence of cracks has

helped to identify a crack in rotating systems. In spite of these valuable efforts, an

accurate and more reliable technique is still required to identify and locate a crack in

rotors. This is because the limitations and assumptions that these crack identification

models or techniques are based on could not be considered as a universal technique to

identify any type of crack in shafts. For example,

Stiffness parameters of cracked shafts which are considered and implemented in

some of these models do not really match the periodic change in stiffness at different

angles of rotation.

The crack in these models is induced at a location very close to the disc location in

order that the crack be more affected (or excited) by the inertia force of the disc. That

is, these models and techniques are not suitable if the crack is located beyond the

zone of these forces.

Most of these models and techniques are based on using rotor unbalance as a

harmonic excitation force or weight dominance as a static deflection force to

investigate the dynamic behaviour of the cracked rotor. That is, if the cracked shaft

works under the normal operating conditions, these approaches are not recommended

or applicable to use in this field.

A change in 1X, 2X and 3X of a frequency response (1X, 2X and 3X stand for the

1st, 2

nd and 3

rd order of the rotor‘s running speed, respectively) of a rotating shaft has

been used as typical symptoms of the presence of a crack in rotating shafts (extensive

surveys are presented in [7, 10, 12, 13]). In fact, these well-known symptoms can be

caused by many other faults such as rotor unbalance, shaft bows, coupling

misalignments which happen at 1X of frequency components. Similarly, 2X


32

frequency components are generated by asymmetries of polar stiffness as in

generators, and also, by non-linear effects in oil film journal bearings and errors in

surface geometry of journals. These last two faults can also excite 3X of frequency

components [14].

Now, an important question to consider is ‗‗how much is the reliability, accuracy and

applicability of the techniques or models that are based on this approach?‘‘ This

question, which has not been addressed in the literature, is addressed indirectly in this

study.

The main aim of this study is to analyse the dynamic behaviour of both the stationary

and rotating shafts to identify and localise a crack through using a roving auxiliary disc.

The changes exhibited in the natural frequencies due to the presence of cracks in

systems may be used not only for crack detection but also for quantifying the depth and

location of the crack. That is, introducing an accurate and reliable transverse crack

model or developing a technique will increase the knowledge about the dynamics of

cracked rotors which is the key point to develop devices to identify and localise

incipient cracks prior to failure occurrence.

In principle, the changes in the natural frequencies of structures and rotating systems

due to the presence of a crack are typically very small and hardly perceptible. Therefore,

most of the investigations in the available literatures that are based on natural frequency

changes have been carried out on cracked shafts which have cracks that are close to the

concentrated inertia of the rotor, which the mounted disc generates. This is because to

enhance or magnify the small changes in the natural frequencies due to the presence of

the crack. To take into account this problem, the idea of an auxiliary roving mass as an

extra movable dynamic inertia force is implemented in this study to identify and locate a

crack in the stationary and rotating shafts. Therefore, this work is carried out to

investigate the following objectives:

1. Conduct the idea of the roving mass on both the stationary and rotating shafts

2. Use change in natural frequencies for the identification and location of cracks in the

stationary and rotating shafts.

3. Investigate the identification and location of cracks of various depths at different

location along the shaft through simulations and experimental analyses.


33

4. Study the spatial interval effect of the roving disc on the crack identification and

location.

5. Investigate how the roving mass approach can affect the natural frequency during

shaft rotation.

6. Study the effect of measuring the dynamic response of the cracked shaft directly

from the shaft (using piezoelectric ceramic sensors) and indirectly from the bearings

(using accelerometers and strain gauges).

Contributions to Knowledge 1.5

The following are the contributions that have been achieved in this study.

1. It is known that the changes in the natural frequencies of a shaft due to the presence

of a crack are very small. Therefore, natural frequency changes cannot be used for

crack identification. However, the roving mass (herein roving disc) method

presented in this study overcomes this problem. The method produces a natural

frequency curve which is processed to identify and locate a crack.

2. The idea of roving mass or disc has been implemented on both non-rotating and

rotating shafts. The numerical simulations and experimental results show that using

a roving mass as a traversing inertia force has a significant impact on the dynamics

of a cracked shaft and enables identification of cracks in shafts using the developed

natural frequency curves methods

3. Three methods were developed for the identification and location of cracks using

the natural frequency curves. These are designated as (i) normalised natural

frequency curves (NNFCs), (ii) natural frequency curve product (FCP) method, and

(iii) normalised orthogonal natural frequency curves (NONFCs) method.

4. The severity and locations of a crack can be identified anywhere along the length of

non-rotating and rotating shafts, including critical locations such as modal nodes,

mid-shafts and shaft ends. Also, the roving disc method will enable crack

identification in rotating shafts even if misalignment and out-of-unbalance forces

are presented.

5. In practice, the vibration of rotating shafts is mainly measured at the bearing

pedestal. Such on-bearing measurement is greatly affected by noise as the bearing

pedestal is required to be very rigid. In this research, piezo ceramic (PZT) sensors


34

were used for on-shaft measurements. It is shown that they are more effective, have

higher performance, are more cost-effective and are more practicable than strain

gauges and accelerometers for on-shaft vibration measurements.

Structure of the Thesis 1.6

Following this general background on rotating machinery and crack problems with

details of the objectives of the present study, Chapter 2 presents a critical review of the

literature that have been devoted to the presence and causes of cracks in structural and

rotating systems, and the methods and methodology of the identification of cracks using

vibration-based methods. Chapter 3 introduces the derivations of the equations of

motion of the intact rotor and cracked rotor using the finite element (FE) method. Also

the mathematical models of the open crack and breathing crack are presented. This is

followed by the verification of the Matlab scripts that developed from the models. The

design of both the stationary and rotating testing rigs and the vibration measuring

instruments are described in Chapter 4. In this chapter, also the procedures of

conducting the experiments and the reason for choosing the PZT sensors rather than

accelerometers and strain gauges are described.

The normalised natural frequency (NNF), normalised orthogonal natural frequency

(NONF) and frequency curve product (FCP) curve techniques, which are used for crack

identification and location, are presented, respectively in Chapters 5, 6 and 7. The

techniques are theoretically simulated and experimentally validated using stationary

rotors with cracks of various severity and different locations.

Chapter 8 is devoted to the application of the developed technique to cracked rotating

rotors using the mathematical model and experimental testing rig for a rotating shaft.

Finally, Chapter 9 provides summary and general conclusions along with future

recommended studies.

CHAPTER 2: Literature Review

35

CHAPTER 2

Literature Review

Overview 2.1

Since 1970s, the idea of changes in the dynamic behaviour of rotors due to the presence

of general faults has become an effective tool for monitoring and diagnosing faults in

rotating machines. For all faults in rotors, cracks have been classified as the most

significant fault that affects the safety and the vibration/dynamic behaviour of rotors.

[15]. The presence of a crack in a shaft is a process of growing fracture slowly under

cyclic loads. If the crack continues to propagate in an operating machine and not

detected early, it grows continuously. Thus, the reduced rotor cross sectional area due to

the crack growth is unable to withstand the dynamic loads that are applied on it.

Eventually unpredictable rapid failure will occur as a brittle fracture mode once the

crack approaches the critical size. This sudden failure generates a considerable amount

of energy that is stored in the rotating shaft. This can lead to huge damages in the

operating system [9]. Obviously, finding an efficient model of a crack in rotor systems

may help in identifying a unique vibration signature of the cracked rotor. Also, this

model will assist in the early detection of the crack before damage occurs due to further

crack propagation.

Types of Cracked Shafts 2.2

The geometry of cracks has an impact on the characteristics and dynamics of the

cracked rotor. Therefore, cracks are categorised into three groups as follows.

Transverse Cracks 2.2.1

These kinds of cracks are the most serious and most common defects in rotating

systems. They are perpendicular to the axis of a shaft as shown in Figure ‎2.1 (a). They

reduce the cross sectional area of the shaft and produce serious damages to the shaft.


36

Therefore, they have been investigated intensively by past and present researchers [1,

16-18].

Longitudinal Cracks 2.2.2

This type of cracks is relatively uncommon and less serious, and it is parallel to the axis

of the shaft as shown in Figure ‎2.1 (b).

Slant Cracks 2.2.3

This kind of cracks appears at an angle to the axis of the shaft (see Figure ‎2.2). They

occur less frequently compared to transverse cracks. Slant cracks are similar to the

helicoidal cracks [19], and have a significant impact on the rotors‘ behaviour when

torsional stresses are dominant. Their behaviour is similar to the influence of transverse

cracks on lateral bending behaviour [10]. In terms of severity, the effect of transverse

cracks on the lateral vibration are more than that of slant cracks [20].

Maximum

tensile stress Crack

(a)

Crack

(b)

Figure 2.1: (a) A transverse crack; (b) A longitudinal Crack.


37

In addition to these crack nomenclature, cracks have been defined by other

nomenclatures by earlier scholars. The ‗‗Gaping Cracks‘‘ are cracks that are opened

permanently during the shaft rotation; they are more properly called ‗‗notches‘‘ [15].

The crack model presented by Papadopoulos [7] was based on the principle of a gap

crack. The effect of a crack on a shaft was investigated experimentally by some

researches through making a notch on a shaft. The results were discussed when these

models did not reflect the accurate shaft behaviour.

Another nomenclature frequently mentioned in the literature is ―breathing crack‖, in

which the generated stresses around the cracked surface during rotation of the shaft

depend on the circumferential location of the crack. The crack is open when the surface

of the shaft is under tensile stresses and closed when the surface stresses reverse to

compressive stresses and so on [5, 21]. The reduction of the component stiffness

increases when the shaft is under tensile stress. Small crack sizes, low rotational speeds

and large radial forces cause the cracks to breathe [22]. Recently many theoretical

studies have been carried out and focused extensively on breathing cracks among other

crack types due to their direct effect on the safety of rotors.

Behaviour of Cracked Shafts 2.3

The dynamics of a cracked rotor have been studied by many researchers based on the

‗‗breathing crack‘‘, which is one of the most common approaches for investigating the

dynamic behaviour of a cracked rotor [17]. During shaft rotation, the process of

breathing occurs by opening and closing gradually. For instance, in large turbine-

Crack Maximum

tensile stress

Figure 2.2: A slant crack (or torsion crack) type.


38

generator rotors, the weight is dominant, in other words the static deflection of the rotor

is much greater than the vibration due to unbalance forces which causes the crack to

breathe during each revolution. If any cracks occur in a rotor in this manner, the crack

will open and close in agreement with the shaft rotation [14].

Crack breathing has a crucial impact on the dynamic behaviour of rotors; it changes the

stiffness of the rotor [23]. Normally the stiffness of an intact rotor at various angles of

rotation remains the same value. However, when a crack commences in a shaft, the

stiffness of the shaft will change periodically at each angle of shaft rotation. At a

specific angle, the crack will close when the stresses on the surface of the shaft are

compressive stresses, and remains open when the stresses reversed to compressive

stresses which have intensive impact on the shaft stiffness reduction. Papadopoulos [7]

also defined the situation of a breathing crack between completely open crack and

completely closed crack states. It is also an intermediate state called partially open or

partially closed.

Although many studies have been carried out by researchers on the breathing crack, in

order to derive the dynamic characteristics of a cracked shaft, the concept of interaction

between the changes in the stiffness of a shaft and the closure of a partial crack has not

yet been found correctly. Therefore, due to the vital role of this phenomenon in the

dynamics of rotors, it must be studied accurately to model a crack correctly [13].

The simulation of the breathing crack in a rotor is often carried out using two familiar

models by Patel and Darpe. [24]. One of the models is a switching or hinge model, in

which the rotor stiffness switches between the stiffness in the closed crack state to the

stiffness of the fully open crack state. The other model was presented by Jun et al.[25],

is the response-dependent breathing crack model.

In spite of the fact that researchers and engineers have intensively considered and

investigated the problem of cracks in rotors and developed a variety of models and

methodologies for investigating the dynamics of cracked rotors, the challenge in this

area is still continuing. From time to time, survey (or review) papers are presented on

the investigation of cracked rotors. Surveys on cracked rotors have been presented by

Wauer [26]; Gasch [16], Dimarogonas [12], Sabnavis et al.[5]. The latter, have

presented a review in which focused on crack detection in shafts by using vibration-


39

based methods (signal measurements) and modal testing (measuring mode shape or

natural frequency changes). They used non-traditional methods such as fuzzy logic,

genetic algorithms, etc. to process the data

More recently Kumar and Rastogi [13] have presented an extensive review on the

dynamics of a cracked rotor. The concentration was on modelling approaches that have

been applied to investigate cracked rotors such as the finite element method, wavelet

transform, Hilbert-Huang transform, nonlinear dynamics techniques, and so forth. In

this survey, extended Lagrangian mechanics, genetic algorithm analysis, and a least

squares identification method were included as new analytical techniques for crack

detection in rotors. In the conclusions of the survey they stated the following; (1) the

breathing mechanism of a crack is a key point for modelling a crack and must be

modelled more accurately to detect cracks in rotors, (2) the finite element method

represents the local compliance matrix more effectively, therefore, the crack element

must be discretised accurately to reflect the real behaviour of a crack in a rotor. That is,

during this period, a crucial advancement has been achieved in the knowledge of the

dynamic behaviour of cracked rotors which has helped the identification of a crack in a

rotor. However, the development of an accurate and reliable model or technique to

identify and localise any cracks in rotating shafts is still required. Studies in this area are

still open and continuous, owing to the unceasing necessity for efficient and more

powerful rotors.

Non-Destructive Damage Detection Methods 2.4

There are a variety of non-destructive methods that are used to identify cracks in

structural and rotating machinery systems. The most commonly used non-destructive

crack identification (NDCI) methods can be classified into two groups as follows

1. Local crack identification techniques

Ultrasonic (transmission/reflection echo)

X-Ray

Magnetic

Liquid Penetrant

Thermography

Visual inspection


40

2. Global crack identification techniques

Vibration

Ultrasonic wave propagation

In principle, the local crack identification techniques (particularly Ultrasonic and X-Ray

methods) require that the area in which a crack may occur is known and easily

reachable for conducting a test. These conditions, however, may not (or cannot) be

guaranteed for most applications in civil or mechanical engineering. To overcome these

problems and limitations, therefore, the global crack identification technique using the

vibration-based method is developed [27].

The intrinsic principle of the idea of the vibration-based damage identification

technique is that the occurrence of damage in a structure induces changes in the physical

properties (mass, stiffness and damping) of a structure. Hence, these changes give rise

to notable changes in natural (modal or resonant) frequencies, modal damping, and

mode shapes. For example the presence of cracks in a structure results in reductions in

the stiffness of the structure. That is to say, damage (or crack) in structures and rotating

machinery can be identified by studying and analysing the changes in the vibration

signature of the damaged structure [27, 28].

Identification of Cracks Based on Vibration Methods 2.5

According to the explanation of the previous section, the vibration testing method is a

more powerful and preferable non-destructive method than the other non-destructive

crack identification methods to study the features of structural systems with cracks. In

the light of this importance, the investigation methodology of crack identification in

rotor systems through using the vibration technique can be categorised as shown in

Figure ‎2.3. To better understand, the figure is further classified into four levels

according to the classification for damage-identification methods that was defined by

Rattyr [29] [28]as follows:

Level 1: Investigation indicates that damage is present in the structure. (Detection

method)

Level 2: Level 1 plus giving information about the location of the damage.

(Detection and Localisation Method


41

Level 3: Level 2 plus giving information about the size of the damage.(Size

assessment)

Level 4: Level 3 plus giving information of the remaining service life of the damaged

structure. ( Consequence/Residual life)

An extensive survey of crack identification methods, which are based on the changes in

modal properties (i.e. natural frequencies, modal damping factors and mode shapes) was

presented by Doebling et al. [30]. The literature concentrated primarily on the crack

identification methods within Levels 1 to 3 without attempting to predict the remaining

service life of a structure. Broadly speaking, Level 4 prediction is associated with the

fields of fracture mechanics or fatigue-life analysis; therefore it was not addressed in the

literature review.

Among the crack identification methods that are categorised in Figure 2.3, the natural

(or modal) frequency based methods are very attractive methods for the identification of

cracks. This is because the changes in the natural frequencies due to the stiffness and

mass changes can be simply measured from a point on the structure using only a single

sensor. Also, the natural frequencies are usually less contaminated by experimental

noise. Literally, the natural (or modal) frequency based methods for crack identification

are categorised into the forward problem and the inverse problem [27, 28]. These two

problems are explained in the subsequent sections.

The Forward Problem 2.5.1

The forward problem usually falls under the crack vibration-based methods in Level 1.

In these methods the frequency changes (or shifts) are determined from a known type of

damage in a structure. Basically, the damage in a structure is modelled mathematically,

and then the determined frequency changes due to the location and severity of the

damage are used as a theoretical base for natural frequency-based methods to detect the

damage in the structure [27, 28, 31]. On this concept, many researchers have

investigated the damage in structures. For instance, Gudmundson [32] derived an

explicit expression for the resonance frequencies of a wide range of damaged structure

using an energy-based perturbation approach. A loss of the mass and stiffness of the

damaged structure can be accounted by this method. An analytical relationship between

the first-order changes in the eigenvalues and the location and severity of the crack was


42

Vibration-Based Diagnostics Methods for Crack Identification

Model Based Methods

(Theoretical Analysis)

Finite difference

Runge-Kutta

Newmark

Wilson

Houbolt

Finite element

analysis

Formulation of Equations of

Motions

Solutions of Equations of Motion

Analytical Methods Numerical Methods

Holzer Method

Jacobi Method

Iteration Method

Non-Model Based Methods

(Experimental Analysis)

Frequency Domain

Methods

Time Domain

Methods

Frequency Response

Functions (FRFs)

Modal Frequency

Mode Shapes

Modal Damping

Stationary

time response

Wave

propagation

Non-stationary

time response

Operational

Deflection

Shapes

Matrix Algebra Continuum

Mechanics Ralyeigh-Ritz

Hamilton‘s

Principle

Matrix

Methods Closed-form

Analytical

Solutions

Vibration Data Processing Techniques

Fast Fourier

Transform (FFT)

Genetic

Algorithm

Wavelet

Transform

Hilbert –Haung

Transform (HHT)

Neural

Networks

Short Time Fourier

Transform (STFT)

Prognostics Methods

Data-Driven

Prognostics

Model-Based

Prognostics

Lev

el 1

L

evel

2&

3

Lev

el 4

Figure 2.3: Vibration-based methodologies for crack investigations.


43

developed by Liang et al [33] to solve the problem of determining the frequency

sensitivity of a simply-supported (or cantilevered) beam with a single crack. Friswell et

al. [34] presented an alternative statistical method of crack identification based on the

theory of generalised least squares. In this method, the ratio of natural frequencies is

determined from both the measured and analytical data, and used to identify the

locations of cracks of different severity.

The Inverse Problem 2.5.2

The inverse problem, which typically falls under Level 2 or Level 3 of damage

identification, consists of determining the physical properties of cracks (i.e. location and

size) in a given structure form the natural frequency shifts [27, 28, 31]. The

investigations on the principle of the inverse problems was started in 1978 when Adams

et al. [35] introduced a method based on the natural frequencies of longitudinal

vibrations for detection of damage in a 1D component. In 1997, Salawu [36] introduced

a state of the art review of a variety of publications that had been published before 1997,

which deal with damage detection in structural systems through natural frequency

changes. Recently, an overview of inverse problems, which was formulated as an

optimisation problem was presented by Friswell [34]. The author outlined the

similarities and differences with design optimisation, and discussed the application of

model updating within a design optimisation environment.

Applied Approaches to Investigate Cracked rotors 2.6

During the last four decades, investigators have paid attention to the study of the

presence of cracks in rotating shafts due to the importance of the machines safety in

practical applications. Various techniques and models are employed by different

scholars. These approaches are classified into the following groups

Approach of Wavelet Transform and Wavelet Finite Element 2.6.1

Wavelet transform is another signal based technique that has been employed broadly by

investigators to detect the depth and location of a crack (or damage by considering the

natural frequencies, mode shapes and modal damping) and so forth. Zheng at al. [13]

employed the technique of wavelet transform for the study of bifurcation and chaos. The


44

local property in both time and frequency domains can be determined by the wavelet

transform, and they introduced a way to analyse domains of different motion types in

the parametric space of a nonlinear system. Darpe [37] has introduced a method to find

a transverse crack on the surface of a rotating shaft. The study also depended on the

technique of the wavelet transform. Xiang at al. [38] have identified a crack in a rotor

using a wavelet finite element method, and suggested a method to analyse the dynamic

behaviour of a Rayleigh beam. The characterization and detection of the effect of a

crack on the rotary instability was studied by Yang and Chan [39]. They developed and

introduced a wavelet-based algorithm to describe periodic, period doubling, fractal-like,

and chaotic motions as consequences of the inherent nonlinearity due to the opening and

closing of a crack during shaft rotation. The most significant advantage of this algorithm

is its capability to identify the transition state continuously that marks the initiation and

propagation of rotor dynamical and mechanical stability. Zhong and Oyadiji [40] have

proposed a new technique based on the stationary wavelet transform (SWT) for crack

detection in beam-like structures. They showed that the SWT is much better than the

conventional discrete wavelet transform (DWT) for the identification of a crack in

simply-supported beams. More recently a new approach for detecting small cracks (less

than 5% ) in beam-like structures without baseline modal parameters has been

developed by Zhong and Oyadiji [41]. The approach is based on the continuous wavelet

transforms (CWTs).

Using the traditional finite element method (FEM) to study cracked rotors has some

limitations, such as low efficiency, inadequate accuracy, slow convergence to correct

solutions, and so forth, in the case of complex rotor systems with high nonlinearity [13].

For these reasons, wavelet spaces have been used as approximate spaces to overcome

these difficulties which led Ma et al [42] to derive wavelet finite element methods

(WFEM). This technique was used by Chen et al. [43, 44] and applied to a dynamic

multi-scale lifting computation method using Daubechies wavelet. The desirable

benefits of WFEM are multi-resolution properties and diverse fundamental functions for

the analysis of structures. Li et al. [45] presented a methodology to specify the size and

location of a crack, employing the advantage of WFEM in the modal analysis of a

cracked beam. Reasonable results with small error were obtained by Xiang et al. [46]

when they applied WFEM and experimental testing on cantilever beams to detect and

locate cracks.


45

Hilbert –Haung Transform (HHT) 2.6.2

In the recent years, Hilbert-Haung transform (HHT) has become more powerful and

popular time–frequency analysis approach for monitoring machinery components and

crack detection, especially with the analysis of transient signals. The HHT works

through three steps: (i) perform a time adaptive decomposition operation which is called

empirical mode decomposition (EMD) on the signal. (ii) Then decompose this signal

into a set of complete components so-called intrinsic mode function (IMF) which is

almost orthogonal and mono-component. (iii) Finally, apply Hilbert transform on the

obtained IMFs to obtain a full energy time-frequency distribution of the signal, named

the Hilbert–Huang spectrum [47, 48].

The HHT approach has a good capability for analysing non-linear and non-stationary

time signals which are generated during the start-up or run-down of rotating machinery.

Also, the main merit of the HHT is that it is capable of dealing with signals of large size

without consuming time for computing and analysing these signals. This is because the

EMD operation, which is the most computation consuming step in the HHT, does not

involve convolution and other time consuming operations. Additionally, the HHT is a

powerful tool for detecting a small crack (such as real fatigue cracks) and rolling

bearing faults because it depends on the concept of the instantaneous frequency rather

than the concept of the frequency resolution and time resolution. These advantages

make HHT more attractive and popular than Fourier spectral and wavelet transforms

[48-51].

During these years, many studies have been presented for detection and monitoring of

cracks in transient response of cracked rotors using the Hilbert-Haung transforms

(HHT). Guo and Peng [52] have used the Hilbert-Huang transforms to identify and

monitor the presence of a small transverse in cracked rotors.This method is useful for

detecting a crack with very small depths and also has a good capability for analysing

non-linear and non-stationary data. Ramesh et al [49] have studied crack detection of a

rotor using HHT and wavelet transforms. The results revealed that HHT is a better

signal processing tool compared to wavelet transforms for crack identification. In 2011,

Feldman introduced an extensive review on the application of the Hilbert-transform in

mechanical vibration analysis. In order to show the concepts of the HHT approach on

actual mechanical signals, the reviewer demonstrated the HHT through many


46

applications. This demonstration also helps how HHT can be exploited in machine

monitoring, identification of faults in mechanical systems and decomposition of signal

components. Peng [47] has made a comparison between the application of HHT and

wavelet transforms on detection of rolling bearing faults. The comparison results

showed that the HHT has potentially diagnosed the faults in rolling bearing and has

better frequency resolution and time resolution than wavelet transforms.

Analyses through Finite Element Technique 2.6.3

The finite element method (FEM) is another popular investigative modal-based

technique that has enabled researchers to study and analyse the dynamic behaviour of a

breathing crack in a rotor. Due to the capability of this tool in three-dimension (3D)

modelling and ease in simulation, many researchers have adopted and applied it for

analysis of cracks in rotating shafts. Papadopoulos and Dimarogonas [53] [54] have

studied the dynamics of a rotating shaft with an open transverse surface crack. They

assumed that the open crack leads to a system with behaviour similar to that of a rotor

with dissimilar second area moments of inertia along the two perpendicular directions.

Then, they represented the local flexibility due to the presence of a crack by a matrix of

size 6×6 for six-DOFs in the cracked element. This matrix has off-diagonal terms,

which causes the coupling of the bending in two transverse directions; extension along

the longitudinal direction and torsion about the longitudinal direction. Sekhar and

Prabhu [55] have studied the behaviour of a transverse crack during the vibration and

stress fluctuation of a simply supported shaft with a crack. The free and forced vibration

has been carried out using finite element analysis (FEA). Dirr et al [1] have worked on

the existence of small transverse cracks in rotating shafts. In their work, they simulated

small transverse cracks in shafts and developed a spatial finite element model and

employed it for numerical simulation to detect the size and location of the small cracks

and their dynamic behaviour, in another work. Mohuiddin and Khulief [56] have studied

the model characteristics of a crack using the finite element analysis of a cracked

conical shaft and deduced the frequency of a shaft in multi cracked states.

Nelson and Nataraj et al. [57] and Nelson [58] have presented a theoretical analysis of

the dynamics of rotor-bearing system with a transversely cracked rotor. The rotating

assembly was modelled using finite rotating shaft elements and the presence of a crack

was taken into account by a rotating stiffness variation. This stiffness variation is a


47

function of the rotor‘s bending curvature at the crack location and represented by a

truncated Fourier series expansion. The static condensation method was employed to

reduce the degrees of freedom, and the resulting nonlinear parametrically excited

system was analysed using a disturbance method. The dynamic characteristics of a rotor

system regarding a slant crack in a shaft was studied by Sekhar and Prasad [59] using a

FE model of a rotor bearing system for bending vibration. They stated that a general

reduction occurs in all modal frequencies with an increase in the depth of a crack.

Wauer [60] has formulated the equations of motion for cracked rotating shafts. A

rotating Timoshenko finite shaft element with six degrees of freedom was considered.

The open crack was simulated by a local spring element, with reduced stiffness and

damping, that connects two uniform fields. Mei, and Moody et al [61] have developed a

wave approach to analyse the free and forced vibrations of Timoshenko beams with

various structural discontinuities. They derived the transmission and reflection matrices

for various discontinuities (such as cracks, boundary and change in section) in the

Timoshenko beam.

Darpe and Gupta [62] have studied the coupled longitudinal, bending, and torsional

vibrations for a rotating cracked shaft using a response dependent nonlinear breathing

crack model. Bending natural frequencies and the sum and difference frequencies were

observed in the lateral vibration spectrum due to the interaction of the torsional

excitation frequency and rotational frequency, when torsional excitation with a

frequency equal to the bending natural frequency is applied to the cracked rotor. Darpe

[37] has proposed a crack identification procedure in a rotating shaft by analysing

transient features of the resonant bending vibrations by wavelet transforms. The author

utilized both the nonlinear breathing phenomenon of the crack and the coupling of

bending–torsional vibrations due to the presence of a crack. Hamidi et al [63] have

presented a finite element model for the study of modal parameters of cracked rotors. In

this study, the open crack was modelled by additional local flexibility, and then the

stiffness matrix of the finite element model was found from inverting the flexibility

matrix. The finite element method (FEM) was used in modelling the equations of

motion of the cracked rotor in refs [20, 37, 64, 65], where the flexibility matrix was also

used in modelling the stiffness matrix of the cracked element. The finite element

stiffness matrix of a rod in space found in ref. [66] was used to represent the cracked

element stiffness matrix in refs [67-70], where the time-varying element stiffness matrix


48

of the cracked element was considered. The classical breathing function proposed in

ref.[71], was used to express the time change in the stiffness of the cracked element

during rotation. This resulted in a time-varying element stiffness matrix due to the

breathing mechanism of the crack. The finite element equations of motion were solved

using the harmonic balance (HB) method. The shapes of the orbits in the neighbourhood

of subcritical speeds and the emerged resonance peaks at these speeds can be used for

crack detection in rotor systems.

Sekhar [72] has presented a review work on multi-crack identification techniques in

structures such as beams, rotors, pipes. Cahsalevris and Papadopoulos [73] have studied

the dynamic behaviour of a cracked beam with two transverse surface cracks. Each

crack was characterised by its depth, position, and relative angle. The compliance

matrix was calculated for all angles of rotation. Singh and Tiwari [74] have developed a

two-step multi-crack identification algorithm which was based on forced responses from

a non-rotating shaft, the Timoshenko beam theory is used to model the shaft by using

the finite element method. The methodology identifies very well the presence of cracks

and also estimates quite accurately the location and the size of cracks on the shaft. Most

recently, Al-Shudeifat and Butcher [18, 21, 75] have proposed an accurate mechanism

for breathing crack model. A new time-varying function of the breathing crack model

was introduced. They applied this new model using the FEM, and formed an actual

periodically time-varying stiffness matrix for a breathing crack and then merged it into

the stiffness matrix of the global system matrix. This model drew on the principle of

reducing second moment of area locally, which Mayes and Davis proposed in 1984

[76].

Analysis of Crack through Numerical Simulations and Experiments 2.6.4

With numerical simulations and experiments, Mancilla and his research group have

proposed the analysis of three aspects to facilitate cracked shaft detection; vibratory

response at local resonances (vibration peaks occurring at rational fractions of the

fundamental rotating critical speed) using the discrete Fourier transform and Bode plots;

orbital evolution around 1/2, 1/3 and 1/4 of the critical speed [77] and the variation in

the threshold of vibratory stability for various crack-imbalance orientations [78]. They

also found in [79] that different mathematical models of the crack breathing mechanism

affect the vibratory responses of the system.


49

Machorro, Adams et al [80] have identified damaged shafts by using active sensing-

simulation and experimentation, which were based on Timoshenko beam theory.

Various kinds of defects such as transverse cracks, imbalance, misalignment, bent shafts

and a combination of them were considered. The numerical and experimental results

demonstrated that there are only slight changes in the passive vibration natural

frequencies and mode shapes due to cracks in the shaft. Therefore, active sensing is

necessary to detect damage. It was also shown that torsional and axial responses that

were measured using active vibration sensing are highly sensitive to cracks in the shaft.

Also, Sawicki and Frsiwell [81] have used an active magnetic bearing (AMB) as an

auxiliary harmonic excitation to detect cracks in shafts. They assumed that the crack is a

breathing crack type, which opens and closes due to the self-weight of the rotor and

produces a parametric excitation. The combinational frequencies of the AMB and the

rotational speed were used to detect the crack. However, the authors commented that

further studies are required to develop a robust technique for condition monitoring.

Recently, Cheng, Li et al [82] have studied a Jeffcott rotor with a transverse crack to

show the limitations of using weight dominance for analysing cracked rotors. The

breathing behaviour of the crack was demonstrated by using the angle between the

crack direction and the shaft deformation direction instead of using the rotor weight

dominance to study the dynamic response of the cracked rotor. The study showed that

the derived equations of motion of a cracked rotor with the assumption of weight

dominance are not recommended to investigate the dynamic behaviour of a cracked

rotor close to its critical speeds.

Investigation through Nonlinear Dynamics of Cracked Rotors 2.6.5

The process of a breathing crack in a rotating shaft generates nonlinear behaviour on the

dynamics of rotating shafts. During run-up or coast down of a rotating machine, this

behaviour could be monitored by measuring the vibration amplitude as a function of the

angular velocity of the rotating machine. In this context, many researchers have studied

the nonlinear dynamic behaviour of shafts due to the presence of a crack.

Qin et al. [83] have modelled a Jeffcott cracked rotor using a piecewise linear system

due to the effect of opening and closing (i.e. breathing) the crack in a rotating shaft. In

this study nonlinear dynamic behaviour of the cracked rotor due to bifurcation was

illustrated. Several phenomena such as jump between two periodic orbits, intermittent


50

chaos, and transition from periodic motion to quasi-periodic motion were observed due

to the presence of a grazing bifurcation in the response. Muller et al. [84] have studied

the nonlinear dynamics of a cracked shaft through changing the stiffness coefficients of

the cracked shaft. The crack was assumed as an external excitation force considering

that the presence of a crack causes a change in stiffness coefficients, hence, the system

becomes parametrically excited and nonlinear.

The influence of whirling on the nonlinear dynamics behaviour of a cracked rotor has

been investigated by Feng et al. [85]. They found distinct differences in the bifurcation

amplitude of orbits when the results of the cracked rotor with whirling compared were

compared with the cracked rotor without whirling speed effect. The authors stated that

this work may be useful to detect and diagnose a crack in the early stages. Zhu et al.

[86] have studied the dynamics of a cracked rotor with an active magnetic bearing. They

noted from the theoretical analysis that the crack cannot be detected by using the

traditional method with the 2X and 3X revolution super harmonic frequency

components in the supercritical speed region but it is possible in the case of a cracked

rotor with an active magnetic bearing. A numerical and experimental study on a

cantilever beam with a breathing crack has been conducted by Bovsunovskii [87, 88].

The forced and transient vibrations of the cracked beam were studied numerically by

simulating the beam with a closing crack under the action of various modes of a

nonlinear restoring force and linear viscous friction. In this study an analytical

approached was presented to determine the relative change of the vibration frequency of

cracked beams. Baschshmid et al. [89] have employed 3D non-linear models to

accurately investigate the breathing mechanism of cracks in rotating shafts. All the

effects of nonlinearity due to a rather deep transverse shaft that was developed in a full

size shaft were studied and evaluated. In this study, the numerical time integration

method was used for the nonlinear analysis of a heavy, horizontal axis, and well-

damped steam turbine rotor. The authors concluded that the overall deviations from

linear behaviour are rather small, and neither instabilities nor sub-harmonic components

appear.

The nonlinear resonances of rotating shafts due to a transverse crack have been

investigated by Ishida et al [90], through applying a harmonic excitation force to the

cracked rotor and its excitation frequency was swept. In this study various types of

nonlinear resonances that occur due to crack were illustrated, and numerically and


51

experimentally clarified the types of these resonances, their resonance points, and

dominant frequency component of these resonances. Furthermore, they clarified that the

amplitudes of these nonlinear resonances rely on the non-linear parametric

characteristics of the crack in rotors.

Al-Shudfiet [91] has recently presented an approach for identifying the dynamic

stability of cracked rotors with time-periodic stiffness. The time-periodic finite element

stiffness matrix was formulated using time-varying second area moments of inertia at

the cracked element cross-section of a cracked rotor. In this study, the harmonic balance

(HB) solution was applied to the finite element (FE) equations of motion to obtain the

semi-infinite coefficient matrix which was used to study the dynamic stability of the

cracked rotor. The author observed that the sign of the determinant of a scaled version

of a sub-matrix of this semi-infinite coefficient matrix at a finite number of harmonics

in the HB solution is sufficient for the identification of the major unstable zones of the

cracked rotor. Furthermore, the unstable zones of the FE cracked rotor appeared only at

the backward whirl-speeds.

Analysis of Cracked Rotor using other Techniques 2.6.6

In addition to the techniques mentioned previously, researchers have adopted other tools

and methodologies to define and analyse dynamic behaviour of a crack in rotating

shafts. The behaviour of the cracked rotor in the neighbourhood of the subcritical speeds

was also studied in refs. [37, 92-99]. The transfer matrix method was employed in

studying the behaviour of the cracked rotor system where the second harmonic

characteristics are used in detecting the crack in the system [92]. Moreover, the transfer

matrix method was utilized to find the cracked rotor response of a simple rotor model

by Jun and Gadala [93]. The nonlinear behaviour of the cracked rotor was studied by

Xiao et al [94] where new peaks of vibration have appeared at 1/2 and 1/3 of the critical

speeds. A theoretical cracked beam model was used for detecting cracks in power plant

rotating machines by Stoisser and Audebert [95]. The vibration amplitude in the

neighbourhood of the first subcritical speed (1/2 first critical speed) were used in

detecting the crack while a good match was found between the numerical and

experimental results. Patel and Darpe [96] also studied the nonlinear dynamic behaviour

of the cracked Jeffcott rotor with switching and breathing crack models. Chaos and

bifurcation were observed only in the case of a switching crack. Zhou et al [97]


52

performed an experimental analysis of a cracked rotor in the neighbourhood of the

subcritical speeds. The effects of the crack depth and the additional eccentricity were

verified experimentally via the shapes of the orbits, response and waterfall plots for the

shaft with an open crack. Correlation between the cracked rotor response and subcritical

speeds during the passage through it was studied in refs. [98, 99] in which the two loops

orbit appear in the neighbourhood of the 1/2 the critical speed. This behaviour of the

orbit before and after the critical speed can be utilized as an indication of a propagating

crack in rotor systems.

Other signal-based approaches have been developed for crack detection and condition

monitoring. Sinha [100] has presented Higher Order Spectra (HOS) as another signal

processing tool for identifying a breathing transvers crack in rotors. The HOS tool is

based on using the higher harmonics in a signal that the crack breath produces during

shaft rotation and exhibits as nonlinear behaviour. However, the author stated that

further work is needed to enhance and increase the confidence level of the HOS tool in

machine condition monitoring.

Summary 2.7

An overview on risks of cracks, types of crack, the dynamic behaviour of structural and

rotating machinery when cracks occur are presented in this chapter. The types of non-

destructive crack identification methods are classified and presented, particularly the

vibration-based methods which are categorised and illustrated clearly due to the crucial

role of these methods in the investigations of cracks in structural systems and rotating

machinery. In the light of this, a comprehensive review on the approaches, techniques

and models that researchers have developed for the identification of cracks in systems

whether these systems are non-rotating or rotating systems is presented.

In spite of great advancements that researchers have made in crack identification, all

researches indicate that the area of crack identification in shafts is active and in

desperate need of more knowledge to develop reliable techniques because there is no

approved (or reliable) model or technique, that can be used to identify all different types

of cracks in rotating equipment.

The location of cracks with respect to inertia and out-of-balance forces is crucial to

make most of the developed techniques that are presented in the literature to work


53

properly. Also, the review shows that the amplitude indications at 1X, 2X and 3X

revolution of the rotation speed, which indicate the presence of a crack, is still a

problematic issue because these indications appear also for other faults such as

misalignment, out-of-balance and asymmetric stiffness. Furthermore, the literature

shows that more studies are required for modelling aspects of crack propagation and

residual life estimation. Several investigations have been presented using only

numerical simulations which require to be validated experimentally. Despite so much

advancement in signal processing techniques, new and reliable fault diagnostics need to

be developed. Investigations on the factors of crack sensitivity for condition monitoring

are required. Studies on modelling of a crack under the influence of impact forces and

surfaces friction interaction due to close and open of cracks during rotation are required.

Some of the aforementioned gaps in the literature on crack identification constitute the

objectives of this study. For instance, (i) enhancement of the role of modal frequencies

as a tool for crack identification, (ii) the study of the identification and location of

cracks irrespective of the locations of the inertia and out-of-balance forces.

Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools

54

CHAPTER 3

Mathematical Modelling of Rotor Systems and Theoretical

Analysis Tools

Introduction 3.1

The finite element (FE) method is the most convenient tool for analysing and modelling

structural systems numerically. This method has a crucial impact on the static and

dynamic analysis of structures and rotating machinery since it was developed in 1972.

This is because of its capability for handling small and large (complex) structural and

rotor systems and conducting a variety of analyses on these systems [101]. This

advanced and popular method is used as a numerical tool for simulating the rotor-

bearing system with and without cracks in this study.

In this chapter, we derive the mathematical formulation of the equations of motion of a

rotor system and its components, in addition to the effect of cracks. These equations will

be based on the FE method for modelling and analysing a cracked rotor-bearing system

supported on linear stiffness bearings. The equations will include the effects of rotary

inertia, gyroscopic moments, unbalance, transverse shear, transverse crack (open and

breathing cracks), and bearing‘s stiffness and damping.

Coordinate Systems 3.2

A flexible rotor-bearing system to be analysed consists of a rotor composed of discrete

discs, rotor- shaft segments with distributed mass and elasticity and discrete bearings. It

is essential first to divide the rotor into simple elements, and then assemble the

mathematical models of these elements to make a set of equations that represent the

system to an acceptable accuracy.

A rotor-bearing system in a shaft-line model is broken down into a number of shaft

elements with nodes at the ends. Each shaft element consists of two nodes, and other

components such as discs and bearings are assumed to be assembled to the element at

these nodes where required. In this study, only lateral or transverse vibrations are


55

considered. In this way each node has four degrees of freedom; transverse displacement

in the x- and y- axes directions and rotations about the x- and y-axes directions as shown

in Figure 3.1.

𝜓

e 𝜉

𝜓 𝜓

𝜃

𝜐 𝑢

𝜃

𝜐 𝑢

𝑢

𝑣

𝜃 Ω

Z

Y

X

Figure 3.1: Typical finite rotor element and coordinates.

Figure 3.2: Coordinates are used in the analysis of the rotor-bearing system [102].

𝑣

𝑦

𝜓

𝜃

O 𝑧

𝜙

Ox-axis into paper

𝑥

𝜓 𝑣

𝑦

𝜃

𝑢

𝜙

Oz-axis out of paper Oy-axis out of paper

𝑥

𝑢 𝜃

𝜓 𝑧

ϕ

O


56

The translations of the shaft from the equilibrium axis are u and v in the x- and y-axes

directions, respectively, and the rotations are defined as about the x-axis which is

positive and a positive rotation about the y-axis according to the assumed sign

convention in Figure ‎3.2 that will be used in this study [58, 102, 103]. The rotor-bearing

system is developed to include a set of associated components consisting of shaft

segments with distributed mass and elasticity, rigid discs and linear bearings.

Rigid Disc Elements 3.3

The element equation of motion of a typical rigid disc with mass centre coincident with

the elastic rotor centreline is derived by using energy method as follows:

The total Kinetic Energy (Td) due to the translation and rotation of the disc is

(

)

.

/

(‎3.1)

In matrix form Equation (‎3.1) is given by the following expression

. /

[

] . /

(

,

*

+(

, (‎3.2)

where md is the mass of the disc and and are the linear velocities in x and y

directions respectively and , and are the instantaneous angular velocities about the

, and axes, which are axes fixed in the disc and rotate with it. The displacements

(u, v, θ, ѱ) of a typical cross section relative to the fixed reference in space are

transformed to corresponding displacements ( ) relative to the rotating reference

and vice versa by using the orthogonal transformation

* + , -* + (‎3.3)

with

* + (

, * + (

, , - [

] (‎3.4)

To derive a valid expression for the Td term, the rotation must be related to coordinate

axes of the rotating disc by using Equation (‎3.3). The angular velocities are


57

(

, ( + [

](

+

[

] [

] (

+

(‎3.5)

Suppose that the rotations are applied in the following order: about the new y-axis, θ

about the new x-axis and then about the new z-axis (thus is the angle of rotation

about the shaft). The instantaneous angular velocity about the z-axis is , where

is the disc rotational speed.

Simplification of Equation (‎3.5) produces

(

, [

](

+ (‎3.6)

Substituting Equation ( 3.6) in ( 3.2), the total Td is then

(

)

(

)

(

) (‎3.7)

The element matrices are obtained by applying Lagrange‘s Equation to Equation (‎3.6)

considering no strain energy for rigid disc then

(

( *

(

)

)

[

](

, [

](

, (‎3.8)

[

], [

]

where Med and Ge

d are the element mass matrix and gyroscopic matrix for a rigid disc,

respectively.


58

Shaft Elements 3.4

The shaft contributes both mass and stiffness to the whole rotor model. It also may

possess gyroscopic effects, internal damping and shear effects. If the gyroscopic effects

are ignored, then, for a symmetric shaft, the two bending planes will often be uncoupled

so that the displacements and rotations that are caused by the forces and moments in one

plane will be only in the same plane. Subsequently, the element mass and stiffness

matrices in a single plane for beam bending may be derived, which are used to create

the element matrix for a flexible shaft in two bending planes.

The element mass and stiffness matrices are derived using both the Bernoulli-Euler and

Timoshenko beam theories, which are used in this study to investigate the effect of a

slender and thick beam on the dynamic characteristics of the rotor-bearing system.

Bernoulli-Euler Beam Element Theory 3.4.1

Beam element matrices are obtained first by ignoring shear effects and rotary inertia,

using the so-called Bernoulli-Euler beam theory which has precise approximation for

the slender beam. The element matrices are calculated on the basis of the kinetic and

strain energies within the beam element according to the lateral displacement ( ),

of the beam‘s neutral plane. The beam elements have two nodes per element and two

degrees of freedom per node. The transverse displacement and slope at the node for the

single beam bending x-z plane are shown in Figure ‎3.3a. The element material is

assumed to be linear and obey Hooke‘s law with Young‘s modulus Ee, and cross

sections perpendicular to the beam neutral axis.

Figure 3.3: Typical local coordinates in the two bending planes: (a) x-z plane, (b) y-z plane [102].

Z

𝑢𝑒

𝜓𝑒 𝜓𝑒

𝑢𝑒 𝑢𝑒(𝜉 𝑡)

𝜉

ℓ𝑒

(a)

Z

𝑣𝑒

𝜃𝑒 𝜃𝑒

𝑣𝑒 𝑣𝑒(𝜉 𝑡)

𝜉

ℓ𝑒

(b)


59

Supposing the element translation is a cubic polynomial in and satisfies the conditions

at the nodes as,

( ) ,

( ) , (ℓ ) ,

(ℓ ) ( 3.9)

Then the element deflection can be approximated by

( ) , ( ) ( ) ( ) ( )-

(

( )

( )

( )

( ))

(‎3.10)

where the shape functions, ( ), are

( ) .

ℓ

ℓ /, ( ) ℓ .

ℓ

ℓ

ℓ /,

( 3.11)

( ) .

ℓ

ℓ /, ( ) ℓ .

ℓ

ℓ /.

The strain energy, Ue, of the beam element [102] is

∫ (

( )

)

ℓ

( 3.12)

where Ie is the second moment of area of the cross section about the neutral axis.

Substituting equation (‎3.10) into (‎3.12), gives the beam element strain energy due to the

lateral displacement as

(

( )

( )

( )

( ))

[

]

(

( )

( )

( )

( ))

( 3.13)

where the stiffness matrix elements are

∫

ℓ

( ) ( ) ( 3.14)

where and

are the second derivatives of the ith shape function with respect to ξ,

which are

ℓ .

ℓ /,

ℓ .

ℓ /, ( 3.15)


60

ℓ .

ℓ /,

ℓ .

ℓ /.

Having Equation (‎3.15) substituted in Equation (‎3.14), the stiffness matrix elements are

developed, where only one term is generated here as an example. Thus,

∫

ℓ

( ) ( )

∫

ℓ (

ℓ *

ℓ (

ℓ *

ℓ

ℓ ∫ (

ℓ

ℓ )

ℓ

ℓ *

ℓ

ℓ +

ℓ

ℓ

[

]

ℓ

( 3.16)

Hence, computing the other terms gives the stiffness matrix elements for the single

bending x-z plane as

[

]

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ]

( 3.17)

In a similar way, the mass matrix is generated but by using the kinetic energy. Ignoring

the rotational effects, in this way the kinetic energy Te of the beam is

∫

ℓ

( ) ( 3.18)

where, is the mass density of the material, is the beam cross-section area, and

indicates the first derivative of the beam translation with respect to time. Substituting

Equation (‎3.10) into (‎3.18) gives the kinetic energy of the single bending x-z plane as

(

( )

( )

( )

( ))

[

]

(

( )

( )

( )

( ))

( 3.19)

where the mass matrix elements, for a uniform cross-section beam, are


61

∫

ℓ

( ) ( ) ( 3.20)

The element is calculated, as an example, as

∫

ℓ

( ) ( )

∫ .

ℓ

ℓ /

ℓ

.

ℓ

ℓ /

∫ .

ℓ

ℓ

ℓ

ℓ

ℓ /

ℓ

( 3.21)

0

ℓ

ℓ

ℓ

ℓ

ℓ 1

ℓ

ℓ 0

1

ℓ

Solving the other integrals gives the element mass matrix as

[

] ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ

ℓ ℓ ℓ ℓ

]

( 3.22)

Mass and Stiffness Matrices for Shaft Elements in Two bending Planes 3.4.2

According to the described local coordinates in Section 3.2 and Figure 3.2, the

definition of the bending coordinates of the x-y and y-z planes are shown in Figure 3.3.

The figure shows, that in the y-z plane the angles and have the opposite sense to

the angles and for the beam bending in the x-z plane, relative to both the positive

transverse translation and z-axis direction. Therefore, the elements of both stiffness and

mass matrices for the Bernoulli-Euler beam can be directly generated from Equations

( 3.17) and ( 3.22), based on the local coordinates vector

, - in Figure ‎3.3. Assuming the two bending

planes do not couple, then the element stiffness and mass matrices for the two planes are

merely entered into the proper location in the 8×8 beam element matrices (considering

the change in sign for matrix elements that associate with the angles and and


62

corresponding moments). Then, the stiffness and mass matrices for the two bending

planes are defined as

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

( 3.23)

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

( 3.24)

Timoshenko Beam Element Theory 3.4.3

The Timoshenko beam model corrects the Bernoulli-Euler beam model with shear

deformation and rotary inertia which are two significant effects for beams and shafts

that are relatively stubby with a low aspect ratio. In this theory, the assumption is that

the cross sections remain plane and rotate about the same neutral axis as for the

Bernoulli-Euler beam but do not remain normal to the deformed longitudinal axis [102,

104, 105]. Figure ‎3.4 illustrates the shear effect through an angle βe, which is the

difference between the plane of the beam cross-section and the normal to the beam

centreline. So the beam cross section angle, , is

( )

( ) (‎3.25)

Thus, the lateral displacement, ue, and the angle of the beam cross section, are used

as the degrees of freedom at the beam nodes. Therefore, based on the variable

assignment of the bending x-z plane of Figure ‎3.3,

|

( )

| ℓ

(ℓ ) (‎3.26)


63

To generate a relation between the shear angle βe, and the lateral displacement, ue, the

latter is assumed to be cubic and, therefore,

where are constants that depend on the nodal displacement conditions of Equation

(‎3.9). The correlation between the lateral displacement and shear angle is obtained by

regarding the beam moment equilibrium. Ignoring the inertia terms, the relationship

(Inman, 2008) may be written as

. ( )

( )

/ ( ) (‎3.28)

where Ge is the shear modulus, with ( ), where is Poisson‘s ratio,

and is the shear constant whose value depends on the beam cross section shape,

which is expressed [106] as

( )(

)

( )( ) ( )

(‎3.29)

where is the ratio between the inner and outer shaft radius. For a solid shaft

and ( ) ( ).

Having Equation (‎3.25) substituted in Equation (‎3.28) for an element with constant

cross section, the shear angle and lateral displacement relationship gives

( ) ( ) ( ) ( ) ( )

(‎3.27)

𝛽𝑒(𝜉)

𝑢 𝑒(𝜉)

𝑢𝑒(𝜉)

𝜉 𝜉 𝛿𝜉

Figure 3.4: Shear in a small section of the beam [102].


64

( )

( )

( ) (‎3.30)

When ue, which is given by Equation (‎3.27) is substituted in Equation (‎3.30), the

constant shear angle becomes

( ) ℓ

( )

(‎3.31)

where

ℓ is a constant dimensionless value [102].

Applying the lateral nodal conditions of Equation ( 3.9) and the rotational nodal

conditions including shear effects of Equation (‎3.26) into Equation (‎3.27), and grouping

terms in matrix form to give the new shape function with shear effects for the single

bending x-z plane as

( ) , ( ) ( ) ( ) ( )-

(

( )

( )

( )

( ))

(‎3.32)

where,

( )

.

ℓ

ℓ

ℓ /,

( ) ℓ

.

ℓ

ℓ

ℓ /,

( )

.

ℓ

ℓ

ℓ /, (‎3.33)

( ) ℓ

.

ℓ

ℓ

ℓ /.

The element strain energy, containing the shear effect, is

∫ .

( )

/ ℓ

∫

ℓ

( ) (‎3.34)

For a uniform element cross section

( )

( )

( )

( )

(‎3.35)

Because is constant along the length of the element, then


65

(

( )

( )

( )

( ))

[

]

(

( )

( )

( )

( ))

(‎3.36)

where the stiffness matrix elements, including shear are

∫ ℓ

( )

( ) ℓ

∫

ℓ

( )

( ) (‎3.37)

Hence, the integration results generate the stiffness matrix of the beam elements

including shear effect for the single bending x-z plane as,

( )ℓ

[ ℓ ℓ ℓ ℓ

( ) ℓ ℓ ( )

ℓ ℓ ℓ ℓ

( ) ℓ ℓ ( )]

(

, (‎3.38)

Similarly, the kinetic energy is computed to find the elements of the mass matrix, using

the same shape functions, which includes the shear effect, and then extended to

calculate the rotary inertia. The shaft element kinetic energy with the mentioned effects

is

∫ (

)

ℓ

(‎3.39)

∫

ℓ

.

/

(‎3.40)

Then the kinetic energy of terms of the mass matrix for the single bending x-z plane is

(

( )

( )

( )

( ))

[

]

(

( )

( )

( )

( ))

(‎3.41)

where, the mij terms for a uniform cross-sectional beam are

∫

ℓ

( ) ( )

∫ ( ℓ

( )

( ))( ℓ

( )

( ))ℓ

(‎3.42)

The process of integrating is similar to before. The result is the following elements of

mass matrix of single bending x-z plane:


66

ℓ

( ) [

]

( ) ℓ [

] (‎3.43)

where,

, (

)ℓ

( )ℓ ,

, ( )ℓ

( )ℓ , (

)ℓ

( )ℓ

, ( )ℓ

According to the description in Section ‎3.4.2, the element stiffness and mass matrices

for the two bending planes including the shear effect are given by

( )ℓ

[ ℓ ℓ ℓ ℓ

ℓ ( )ℓ ℓ ( )ℓ

ℓ ( )ℓ ℓ ( )ℓ

ℓ ℓ ℓ ℓ

ℓ ( )ℓ ℓ ( )ℓ

ℓ ( )ℓ ℓ ( )ℓ

]

(‎3.44)

ℓ

( )

[

]

( ) ℓ

[

]

(‎3.45)


67

The inertia and elastic forces within an element due to element mass and elastic strain,

respectively, are

(‎3.46)

(‎3.47)

where is the element inertia force, is the element elastic force and

, - is the second derivative of the element local

coordinates.

Gyroscopic Effects 3.4.4

Gyroscopic effects are generated in the beam in a similar way to the rigid discs.

Generally, these effects are insignificant unless the shaft has a large polar moment of

inertia Ip, and rotates at a high speed. The gyroscopic effects occur because of the

kinetic energy which is defined for a uniform shaft as [102],

∫ ( ) ( )ℓ

(‎3.48)

The equation of the kinetic energy shows that the gyroscopic effects couple the two

bending planes, so the shape functions must include both rotations and in terms of

the local node coordinates [58, 103]. Therefore,

( )

, ( )

(‎3.49)

and, thus,

( ( )

( )* [

]

[

] (‎3.50)

where , - is the vector local node

coordinates of the two bending beam planes.

Applying Equation (‎3.50) into Equation (‎3.48) gives

(‎3.51)

where , - is the first derivative of the local

coordinates, and


68

∫ ( ) ℓ

( ) (‎3.52)

Then, from Lagrange‘s equations

(

.

/

.

/

)

, - (‎3.53)

where is the element gyroscopic matrix. This matrix is a skew-symmetric matrix.

Thus, the elements of the gyroscopic matrix are computed from Equation (‎3.52) as

∫ ( ( ) ℓ

( ) ( ) ( )) (‎3.54)

Therefore, all the elements of the gyroscopic matrix of the two bending beam x-z and y-

z planes, neglecting the shear effect are generated by integrating Equation (‎3.54), to

give,

ℓ

[ ℓ ℓ ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎3.55)

If shear effects were implemented, the kinetic energy given by Equation (‎3.48)

including the angles of rotation would represent rotations of cross sections containing

the shear effects, given by Equation (‎3.26). The shape function would also represent the

shear effects, as demonstrated in Equation (‎3.32). Thus, the computing integrations are

the same as those for the rotary inertia inclusion, Equation (‎3.42), and the result would

be a skew-symmetric matrix including shear effects in the two bending beam planes as

follows:


69

( ) ℓ

[ ]

(‎3.56)

where

, ( )ℓ

,

( )ℓ , ( )ℓ

Bearings 3.5

The proposed bearings in this work are assumed to be linear and obey the following

governing equation which correlates the forces acting on the shaft because of the

bearings with resultant displacements and velocities of the shaft as [102, 103]

( * [

] . / 0

1 . / ( 3.57)

In vector notation, with ( * and .

/

( 3.58)

where, [

] is the stiffness matrix of the bearings, and 0

1 is

the damping matrix of bearings.

Assembly Process 3.6

The following steps describe the assembly procedures for the shaft–element matrices

(8×8), disc matrices (4×4), and bearing matrices (4×4) to the global matrix of dimension

4 (N×N), where N is the number of beam elements.

1. Global matrix: this matrix is built by overlapping the element matrices of the beam

as shown in Figure 3.5 which is the same for mass, gyroscopic and stiffness matrices.

Similarly, the element global forces are built.


70

2. Adding disc effects: An additional inertia matrix of a disc is included in rotor

dynamics by considering the mass and inertia of the disc as nodal properties. These

nodal properties can be included at one node (4×4 matrices) which gives a point

property or at two nodes (8×8 matrices) which are used when the thickness is

considered [107].

Figure 3.6 shows the element matrices and the two possibilities of the disc location

relative to the shaft elements.

Figure 3.5: Assembly of beam element matrices to form the global matrix.

Figure 3.6: Adding disc influence


71

3. Addition of bearings effect: the two positions of the bearing that may be placed

relative to the element matrices are described in Figure 3.7.

Boundary Conditions 3.7

The boundary conditions (BCs) are applied after completing the assembly of beam

matrices which have a potential influence on the results. The BCs, which are also named

equation constraints, are specified according to the bearing flexibility. If the supports

are considered flexible, additional terms are added to the degrees of freedom of the left

and right bearing. On the other hand, if the supports are regarded to be very stiff, then

Figure 3.7: Adding bearing influence

Figure 3.8: Effect of boundary conditions on the system global matrix.


72

the rotor is not able to displace in the vertical and horizontal directions at the 1st node

and the (N+1) the node. Therefore, the rows and columns corresponding to those nodes

in the vertical and horizontal displacement within the global matrix must be removed

[102, 103] as shown in Figure ‎3.8.

System Equations of Motion 3.8

The full equations of motion of the system is obtained by assembling the component

equations as

( ) ( ) ( ) ( ) ( 3.59)

where, ( ) ,

- is the nodal displacement vector with

dimension 4(N+1) × 1, is the global mass matrix, is the global damping matrix,

is the global stiffness matrix and is the global gyroscopic matrix of the rotor system

without any cracks, each of dimension 4(N+1) × 4(N+1). ( is the combination of

unbalance forces and moments vector matrix 4(N+1) × 1)

For a specified node k, if this is the node at which a disc is placed and if the node is

displaced by ε due to unbalance and by an angle λ due to bearing tilt, the vector of

unbalance forces associated with the element of disc [102] is

(

( )

( )

( ) ( )

( ) ( ) )

(

( )

( ) )

(

) ( 3.60)

where, is the force vector that acts at node k due to the offset and tilt of a disc,

and are the initial angles of the unbalance force and moment vectors with respect to

the OXY axes, when t = 0.

Finite Element Model of Cracked Rotor Systems 3.9

Cracked Rotor with an Open Crack 3.9.1

The crack status in a rotor remains either opened or closed according to the location and

amount of the out-of-balance forces arising in the cracked rotor. This means that the

vibration amplitude due to any unbalanced force acting on a rotor is greater than the

static deflection due to the rotor weight. The open transverse crack geometry is

modelled as illustrated in Figure ‎3.9, where the hatched section defines the segment of


73

open transverse crack. The crack is assumed to be at an initial angle ϕ with respect to

the fixed negative Y-axis at t = 0 as shown in Figure ‎3.9a. So the angle of crack relative

to the negative Y-axis changes with time to ϕ + Ωt as shown in Figure ‎3.9b, when the

shaft rotates [21].

The second moments of area and about centroidal and axes of the cracked

element are time-varying values during the shaft rotation. The centroidal and axes

remain parallel to the stationary and axes during the rotation of the shaft. Therefore,

the stiffness matrix of the cracked element can be written in a form similar to that of the

non-axisymmetric beam in space in [66] as

[ ( ) ℓ ( ) ( ) ℓ ( )

( ) ℓ ( ) ( ) ℓ ( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

( ) ℓ ( ) ( ) ℓ ( )

( ) ℓ ( ) ( ) ℓ ( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( ) ]

(‎3.61)

(a) (b)

Figure 3.9: Modelling diagrams of the cracked element cross-section. (a) Before

rotation. (b) After shaft rotation. The hatched part defines the area of the crack

segment [18, 21, 75].


74

where, is the stiffness matrix of the cracked element, is the cracked element

length, and ( ) and ( ) are the time-varying second moment of area of the cracked

element about the centroidal axes and , respectively. Equation (‎3.61) is used here to

define the cracked element itself with time-varying stiffness, while it was used by

Sekhar [65], Pilkey [66] and Sinou [68] to represent the crack segment only. If the shear

effects are considered, Equation (‎3.61) becomes,

ℓ

[ ]

(‎3.62)

where,

( )

,

ℓ ( )

,

( )

,

ℓ ( )

,

(‎3.63)

( )ℓ ( )

,

( )ℓ ( )

,

( )ℓ ( )

,

( )ℓ

( )

,

From Figure ‎3.9b, the centroidal coordinates relative to the stationary and axes are

( ) ( ) (‎3.64)

( ) ( ) (‎3.65)

Hence, the centroidal time-varying second moment of area and are computed as

( ) ( ) ( )

(‎3.66) ( ) ( )

( )

. ( ( ))/

( ) ( ) ( )

(‎3.67) ( )

( )

( )

. ( ( ))/


75

where ( ) and ( ) are the cracked element‘s second moment of area about X and Y

axes, is the left uncracked area, and is the y-axis centroidal location. Thus, and

are defined as

(‎3.68) ( ) ( )

. ( ) ( )√ ( )/

( ( ))

(‎3.69)

where R is the shaft radius, is the overall cross-sectional area of the cracked element

( = in the case of an open crack), is the area of the crack segment (at fully

open crack), √ ( ) is constant value that depends on the crack depth ratio, and

⁄ is the crack depth ratio and h is the depth of the crack in the transverse

direction of the shaft.

The cracked element cross-section second moment of area and about the rotational

x- and y-axes respectively, are derived for as [18]

(‎3.70)

(

(( )( ) ( )))

(( )( ) ( ))

(‎3.71)

(( )( ) ( ))

where ⁄ is the second moment of area of the shaft, when the crack is fully

closed.

As the shaft rotates, the varying time second moment of area ( ) and ( ) about the

stationary X and Y axes are defined in terms of rotational second moments of area in

Equations (‎3.70) and (‎3.71) as follows [66]:


76

( )

( ( )) ( ( )) (‎3.72)

( )

( ( )) ( ( )) (‎3.73)

For a shaft with a symmetrical cross-sectional area of the cracked element .

Hence, substituting Equations (‎3.72) and (‎3.73)) into Equations (‎3.66) and (‎3.67) yields

the centroidal time varying second moment of area and of the cracked element as

( ( )) (‎3.74)

( ( )) (‎3.75)

Thus ( ) ⁄ , (

) ⁄ and are constant

values during the shaft rotation. As a consequence, the finite element stiffness matrix of

the cracked element given in Equation (‎3.61) can be rewritten in the following form

( ( )) (‎3.76)

where,

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎3.77)

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎3.78)

As a result, the FE equations of motion of the uncracked rotor bearing-system given in

Equation (‎3.59) are rewritten including the effect of the cracked element of an open

crack model as


77

( ) ( ) ( ) . ( ( ))/ ( ) (‎3.79)

where is the 4(N+1) × 4(N+1) stiffness matrix of which the elements of the cracked

element i in the cracked stiffness matrix

are inserted instead of the elements of the

uncracked element i in , is the another 4(N+1) × 4(N+1) stiffness matrix which has

zero elements at all locations except at the cracked element location where the elements

are equal to .

Cracked Rotor with Breathing Crack 3.9.2

The mechanism of the breathing crack (opens and closes when the shaft rotates) in the

cracked rotor happens because of the shaft weight under gravity. The diameter and

length of the shaft in comparison with the maximum static deflection due to the shaft

weight itself are too big. Thus, the difference between the centroid and the neutral axes

of the cross-sectional of the cracked element is insignificant and negligible. As the shaft

begins to rotate, the locations of the axes of the cracked element change with respect to

time during rotation. Consequently, tensile and compressive stresses are generated

below and above the neutral axis, which tend to maintain the crack opens and closes,

respectively. Similarly, the breathing crack is modelled as shown in Figure ‎3.10, the

hatched segment defines the crack section [18].

The angle of crack with the negative Y-axis is changed by time to as illustrated

in Figure ‎3.9b, as the shaft begins to rotate. While the higher end of the crack segment

edge approaches the compression stress zone, the crack angle commences to close at

angle . Therefore, the crack becomes fully closed at angle as

shown in Figure ‎3.10 and Table ‎3.1, where and are computed with respect to the

negative Y-axis as

(

( )

√ ( )*,

( ) (‎3.80)


78

For the cross-sectional area of the cracked element is . Thus, the

overall cross sectional area of the cracked element ( ) during rotation is expressed

by

where, ( ) is the time varying quantity which represents the area of the closed part of

the crack segment, when the crack commences to close at .

The second moment of area of in Equations (‎3.70) and (‎3.71) about stationary X and

Y axes for t = 0 or about rotating x and y axes for , where these second moments of

area and

about X and Y axes become time-varying values when the shaft starts

( ) ( ) (‎3.81)

Figure 3.10: Breathing crack states and centroidal positions of the cross-section of the

cracked element at various rotational angles [21].


79

Table ‎3.1: States of the breathing crack for full rotational angle ( )

Rotational angle Breathing crack states

, Fully open

( ) ,

⁄ Partially open

( ) ( ) Fully closed

( ) ( )

to rotate and they are defined in Equations (‎3.72) and (‎3.73). As the crack begins to

close the second moment of area and

of ( ) about X and Y axes starts to

appear. As a result, the overall second moment of area and

of ( )

corresponding to each time step and the new values of the ( ) are calculated about

stationary X and Y axes as

(‎3.82)

(‎3.83)

Thus, the second moment of area of ( ) about centroidal and axes which remain

parallel to the stationary X and Y axes during the breathing crack rotation are calculated

as

( ) ( ( ))

( ) (‎3.84)

( ) ( ( ))

( ) (‎3.85)

where, ( ) and

( ) are the ( ) centroidal coordinates relative to the

stationary X and Y axes. A precise functional relationship for ( ) and

( ) is derived

[18] as

( )

( ) ( ) (‎3.86)

where


80

. (

)/

0. ⁄ / ∑ .

/ (( )

*

( ⁄ ) 1, m is a positive

even number , and is given in Equation (‎3.70).

Similarly, an equation for ( ) is derived as

( )

( ) ( ) ( ) (‎3.87)

where,

(

( )∑

( ) ( )

( ))

( ) , , and is given in Equation (‎3.71).

Thus, the FE stiffness matrix with a breathing crack model of the cracked element

according to the approximate time-varying second moment of area ( ) and

( ) in

Equations (‎3.86) and (‎3.87) respectively, is given as

ℓ

[

( ) ℓ ( )

( ) ℓ ( )

( ) ℓ

( ) ( ) ℓ

( )

ℓ ( ) ℓ

( ) ℓ

( ) ℓ

( )

ℓ ( ) ℓ

( ) ℓ

( ) ℓ

( )

( ) ℓ

( ) ( ) ℓ

( )

( ) ℓ

( ) ( ) ℓ

( )

ℓ ( ) ℓ

( ) ℓ

( ) ℓ

( )

ℓ ( ) ℓ

( ) ℓ

( ) ℓ

( ) ]

(‎3.88)

Equation (‎3.88), can be rewritten in vector form as follows

( ) ( ) (‎3.89)

where, is the cracked element i stiffness matrix when the crack is in the fully closed

state which is equal to the stiffness matrix of the uncracked element in Equation ( 3.23).

The and

are the secondary stiffness matrices that arise due to the existence of the

breathing crack effect. They are generated via Equations (‎3.86) and (‎3.87) as

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎3.90)


81

ℓ

[ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎3.91)

The finite element equations of motion of the cracked rotor in a rotor-bearing system

with a breathing crack effect, are given in matrix form as

( ) ( ) ( ) ( ( ) ( )) ( ) (‎3.92)

where, and are two additional global 4(N+1) × 4(N+1) stiffness matrices of zero

elements apart from the cracked elements where the elements equal to in and

in

.

Dynamic Analysis of the System 3.10

Whirl Speed Analysis (Free Response System) 3.10.1

In order to calculate the eigenvalues and eigenvectors of the rotor system, the equation

of motion in Equation (‎3.59) can be rewritten into 2n first-order differential equations,

in terms of and [102] as

0( )

1

. / 0

1 . / .

/ (‎3.93)

For an undamped system

Substituting . / and

. /, into Equation (‎3.93), gives

(‎3.94)

where, 0

1 and 0

1

Assuming a solution of the form ( ) , then ( )

; hence Equation

(‎3.94) becomes


82

(‎3.95)

This is 4(N+1) × 4(N+1) eigenvalue problem which can be solved numerically by

MATLAB code, to find eigenvalues for each shaft speed of interest, and eigenvectors

corresponding to each eigenvalue.

Response of Rotors to Unbalance Forces and Moments 3.10.2

The forced rotor equation of motion is of the form

( ) ( ) ( ) ( ) ( ) ( 3.96)

where, is the global force vector due to the offset and tilt of the disc.

The steady-state solution is of the same form:

( ) ( ) ( ) ( is complex) ( 3.97)

Substituting Equation (‎3.97) into Equation (‎3.96), gives the steady-state response to the

unbalance forces and moments as

, ( ) - , ( )-

( 3.98)

where the matrix ( ) , ( ) - is the dynamic stiffness matrix

and its inverse ( ) , ( )- is the matrix of the Frequency-Response Function.

Therefore, Equation (‎3.98) may be written as [102]

( ) ( 3.99)

For a system with n degrees of freedom, Equation (‎3.99) expands as

(

, [

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

](

, ( 3.100)

Theoretical Analysis Tool 3.11

The numerical analyses of the present study are based on the finite element method. The

finite element equations of the components of a cracked and non-cracked rotor-bearing

system, which were derived in Chapter 3, are programmed in Matlab (version 2010a).

The commercial FE software Ansys® (version 13) also has been used in this study as a


83

reference to verify the Matlab scripts that were developed from the theoretical

equations.

Description of Matlab Scripts 3.11.1

The rotor-dynamics software, which is a set of scripts written in Matlab by Friswell

[102], describes the dynamics of lateral motion of a rotor-disc-bearing system. In the

current study we have developed and extended this software to compute the dynamic

behaviour of a cracked rotor-disc-bearing system. The Matlab scripts of the time-

varying finite element stiffness matrix of an open crack and a breathing crack are

written and merged with global matrices of the cracked rotor system. These

developments have been carried out in the software in order to study the influence of the

following physical parameters on the dynamic characteristics of cracked rotors:

Position of crack.

Depth of crack.

Orientation of crack.

Number of cracks (with different depths and orientations).

Moreover, the thicknesses of the cracked and uncracked elements were modified and

controlled in order to calculate the effect of crack thickness on vibration behaviour as

shown in Figure ‎3.11. Thus, at the basic level, the software consists of three parts:

Defining the model, boundary conditions, forcing and operating conditions;

Analysing the system and generating the results; and

Graphical means for interpreting the model and results.

The system definition (model, boundary conditions, forcing and operating conditions) is

all merged into a single Matlab structured array. Having defined the system, the model

is passed easily to analysis and plotting functions. All definitions of the system are

explained in detail on Friswell‘s website except for the crack model that we have

developed and incorporated into the Matlab structured array which will be illustrated

here.


84

Crack Definitions 3.11.2

The model of a cracked rotor-discs-bearing system is defined using definitions of the

nodes, shafts, cracks, discs and bearings. The model is defined as a formed array in

Matlab. The definition of cracks is in two cases; one for open cracks and another for

breathing cracks. According to physical parameters in section 4.2 and Figures 3.9 and

3.10, cracks are defined as

model.crack = [ Crack_Type Node_1 Node_2 ...... properties ....; .......

Crack type 1 is an open crack, given by the following information

model.crack = [1 Nod_1 Nod_2 Depth_ratio initial_crack_angle Rotation_angle

ℓ𝑒𝑐 ℓ𝑒

ℓ

(a)

ℓ𝑒𝑐 ℓ𝑒

ℓ

(b)

Figure 3.11: (a) Cracked and uncracked elements have equal width. (b)

Cracked and uncracked elements of different width.


85

Rotor_Spd_rpm; ...

Crack type 2 is a breathing crack, given by the following information

model.crack = [2 Nod_1 Nod_2 Depth_ratio Rotation_angle Rotor_Spd_rpm m_1

P_1; .....

Both crack types are modelled and programmed for both the Bernoulli-Euler and

Timoshenko beam theories.

Element Type Used in Ansys Numerical Model 3.12

The beam element type beam189 has been used in the developed Ansys model to

represent the shaft and discs, whereas bearings and foundation are represented by using

element type combine14. Beam189 element is a quadratic element that has three nodes

in three dimensions (3-D); each node has six to seven degrees of freedom (6-7 DOFs) as

shown in Figure ‎3.12a. The DOFs include translations at each node in three dimension

x-, y- and z- axes and rotations about the three axes are contained in the model; a

seventh DOF (warping magnitude) can also be included. Beam189 is convenient for

modelling linear slender structures to moderately thick linear structures. The mechanical

characteristics of this element are based on the theory of Timoshenko beam, which

incorporates shear deformation effects. By including stress-stiffening terms, beam189

enables to study and analyse the dynamics and stability of axial, flexural, and torsional

vibration problems. In addition, the effect of creep, elasticity, plasticity and other non-

linear material models can be included. Beam189 can be employed with any beam

cross-section which can be modelled with more than one material.

Combine14 is an element that is used to represent springs and dampers. The element has

two nodes with three DOFs per node, which can represent longitudinal or torsional

motion in 1-D, 2-D, or 3-D as shown in Figure ‎3.12b. The longitudinal spring-damper is

an element that is used to represent a uniaxial tension-compression effect with up to

3DOFs per node: translations in the x, y, and z directions. The torsional spring damper

is a purely rotational element with 3DOFs at each node: rotations about the nodal x-, y-,

and z-axes. This element has no mass, and the characteristics of the spring or damper

may be easily removed from the element [108].


86

Matlab Script Verification 3.13

Verification of the extended Matlab script was conducted manually, in order to confirm

the validity of the scripts. A simple case with two elements was analysed using the

global stiffness and mass matrices which were derived in Chapter 3 for a cracked and

non-cracked element. These two elements with four degrees of freedom per node have

12-dgrees of freedom (DOFs) in the global matrix. The derived stiffness matrix was

compared manually with that computed by the Matlab software for the same case.

The results given by the two matrices are exactly the same, as shown in Table 3.2.

Moreover, the exact natural frequency of this case, which represents a simply-supported

beam, was determined by Equation (‎3.101) and compared with the computed natural

(a)

(b)

Figure 3.12: (a) Geometry of beam189. (b) Geometry of the element type combine 14.


87

frequency by using Matlab scripts (see Table ‎3.3) . The percentage error between the

exact natural frequency and theoretical Matlab natural frequency was less than 1.5%.

This error percentage decreased dramatically to 0% for the uncracked case, but

increased to 2% for cracked case, when the number of elements was increased to 100.

This increase occurred due to the length of the cracked element in the two cases.

Therefore, the Matlab script is reliable and can be employed.

The beam natural frequency was calculated analytically by using Equation (‎3.101)

[102]. This equation is for the first lateral natural frequency of a beam with pinned ends

(simply-support beam).

√

ℓ (‎3.101)

Table 3.2: Global stiffness matrix computed manually for two elements with a crack

a

0 0 b -a 0 0 b 0 0 0 0

0 a a 0 0 -a -b 0 0 0 0 0

0 -b c 0 0 b d 0 0 0 0 0

b 0 0 e - b 0 0 d 0 0 0 0

-a 0 0 -b

a + a =

f 0 0

-b + b

= 0 -a 0 0 b

0 -a b 0 0

a + a =

f

b - b =

0

0 + 0 =

0 0 -a -b 0

0 -b d 0 0

b - b =

0

c + c =

g

0 + 0 =

0 0 b d 0

b 0 0 d

-b + b

= 0 0 0

c + c =

g -b 0 0 d

0 0 0 0

0 - a= -

a 0 0

0 - b= -

b a 0 0

-

b

0 0 0 0 0

0 - a =

-a

0 + b =

b 0 0 a b 0

0 0 0 0 0 0 - b = 0 + d = 0 0 b a 0


88

- b d

0 0 0 0

0 + b =

b 0 0

0 + d =

d -b 0 0 a

where a = 1.5080 E05 , b = 3.7699 E04, c = 1.2566 E04, d = 6.2832 E03, e = 1.2566

E04, f=3.0159 E05, g = 2.5133 E04

Table ‎3.3: Results of using the exact solution and the developed Matlab scripts.

Two elements with a crack depth ratio 0.1

Exact Using FE by Matlab Error

%

uncracked

(Hz)

cracked

(Hz)

uncracked

(Hz)

cracked

(Hz) uncracked cracked

39.770 38.808 39.927 39.242 0.4 1.1

100 elements with a crack depth ratio 0.1

Exact Using FE by Matlab Error

%

uncracked

(Hz)

cracked

(Hz)

uncracked

(Hz)

cracked

(Hz) uncracked cracked

39.770 38.8082 39.770 39.742 0 2

| |

Verification of Matlab Script by Using ANSYS 3.14

In the previous section, verification of the Matlab script has been conducted manually

on a simple case. The computations gave desirable results and proved the authenticity of

the developed Matlab scripts for simulating crack effects in rotor systems. However, the

Matlab script validation has been performed, also, by using a well-known commercial

finite element (FE) solver so-called Ansys. The Ansys package has been used for a long

time for numerous applications using the FE technique with various element types. On

this basis, the validity of the Matlab scripts has been verified by solving two cases using

the developed Matlab scripts and the ANSYS package. The procedures for conducting

these two cases are as follows:


89

Verification Models 3.14.1

A simply-supported rotor model shown in Figure ‎3.13 has been used as a verification

model for computing natural frequencies by both the developed Matlab scripts and the

Ansys package. The shaft diameter is 0.02 m, 1 m length and a crack with 0.1 depth

ratio is located at the centre of the shaft. The aluminium disc at the mid-span has mass

density 2700 kg/m3 and Poisson‘s ratio 0.33. The shaft element was divided into 250

elements, and the crack element is 0.5 mm long. The shaft mass density and Young‘s

modulus are 7800 kg/m3 and 200 GPa respectively.

Case 1: Simply-support rotor without crack and disc effects;

The developed Matlab scripts and the Ansys package have been used as FE solvers for

analysing the rotor in this case. The results of the first natural frequencies of this case in

Table ‎3.4 show that the two solvers predict the same frequencies of the rotor in both the

horizontal and vertical planes. This good agreement proves the validity and reliability of

the developed Matlab scripts as a FE solver.

Table ‎3.4: Comparison between Matlab scripts and Ansys solution of case 1

Mode Matlab scripts Ansys

1 fh = 39.77 Hz fh = 39.75 Hz

fv = 39.77 Hz fv = 39.75 Hz

Figure 3.13: Verification model.


90

Case 2: Simply-support rotor with crack and disc effects;

The effects of the crack and disc have been implemented in this case and solved using

both the developed Matlab scripts and Ansys package. The cross-sections of cracks with

various depth ratios were modelled in AutoCAD and then exported to Ansys in order to

obtain the exact cracked section coordinates. In this case the crack with 0.1 depth ratio

is used in Ansys with an aluminium disc (see Figure ‎3.14 and Figure ‎3.15 ). Hence the

results of this case have been compared with that obtained from the Matlab scripts. The

results of the two solvers in Table ‎3.5 show that the results of the developed Matlab

scripts are quite close to the results generated by the Ansys package.

Table ‎3.5: Comparison between Matlab scripts and Ansys solution for case 2

Mode Matlab scripts Ansys

1 fh = 20.47 Hz fh = 21.90 Hz

fv = 20.48 Hz fv = 21.92 Hz

Figure 3.14: Cross-section of the crack with 0.1 depth ratio in Ansys


91

Summary 3.15

In this chapter, the models of the shaft, disk and bearing are developed using the finite

element method. The stiffness matrices of the finite element models of both the open

and breathing cracks are represented as time-varying matrices. The interaction of these

models and the crack effects are developed and illustrated to analyse the dynamics of

intact and cracked rotor-disc-bearing system. Mathematical formulation of the equations

of motion of the cracked rotor-disc-bearing systems and its components are developed.

The effects of rotary inertia, gyroscopic moments, unbalance, transverse shear,

transverse crack (open and breathing cracks), and bearing stiffness and damping are

included. These models are presented as scripts in the Matlab environment and verified

using the commercial finite element analysis software Ansys.

Figure 3.15: Modelling of the cracked rotor with disc in Ansys.

CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments

92

CHAPTER 4

Experimental Test Rig and Vibration Measuring Instruments

Introduction 4.1

The experimental work is another module of this study, which is conducted to validate

theoretical results that are obtained from finite element simulations. In the literature,

there are numerous reports on investigations that have been carried out on a cracked

rotor-bearing system using different methods. Most of these studies are either

theoretical investigations without proving the accuracy and applicability of the results in

practice, or experimental investigations without studying the theoretical background of

a crack so as to improve crack models. A few scholars have studied cracked rotor-

bearing systems theoretically and experimentally, and their studies were performed on

simple cracked rotor-bearing systems. In this study, a rotor-disc-bearing system with an

open crack type of different severity and locations has been investigated theoretically

and experimentally. This study was conducted in two different states: stationary state

and rotating state. Therefore, two new test rigs were designed to validate the theoretical

results of the cracked rotor system in both these two states.

Experimental Test Rigs 4.2

Test Rig Used in Stationary Case 4.2.1

Figure ‎4.1, shows a photograph of the test rig that was used in the case that the cracked

rotor system is stationary, that is there no rotation, and, hence, there are no gyroscopic

effects. The rig consists of a uniform shaft and a rigid disc supported by two ball

bearings mounted on stiff pedestals (see Figure ‎4.1) as a simply-supported rotor. The

disc is used as a roving mass and it is traversed at 40 mm spatial interval along the

length of the shaft. The dimensions and materials of the rig are presented and illustrated

in Table ‎4.1 and Figure ‎4.2.

Four circumferential groves, at 90 degrees interval, were made around the disc bore in

order to traverse the disc over sensors and wires that are bonded to the surface of the


93

shaft as shown in Figure ‎4.3. The wires were insulated copper wires (for winding

transformers, coils, etc.) of 0.315 mm diameter. This type of wire was used to avoid

electrical contact between the wires and the shaft.

Table ‎4.1: Dimensions and materials of the test rig

Parameters Shaft Disc Bearings

Material Stainless steel Aluminium n/a

Young‘s Modulus, Es (GPa) 200 70 n/a

Poisson‘s ratio, vs 0.3 0.33 n/a

Density, ρ (Kg/m3) 7800 2700 n/a

Total length, L (m) 1.16 n/a n/a

2

5

1

6

7

4

8

3

Figure 4.1: Experimental test rig of the stationary case: 1. Left bearing 2. Right

bearing. 3. PZT sensors. 4. Terminal conector of the PZTs‘ wires. 5. Impact

hammer. 6. Signal conditioner for impact hammer. 7. NI-data aquestion card. 8.

PC.


94

Disc locations, Ld (m) n/a 0.01, 0.02, 0.3, …, 1

Bearing span, Lb (m) 1 n/a n/a

Crack locations, Lc (m) 0.38, 0.58, 0.78 n/a n/a

Outer diameter, Do (m) 0.02 0.2 n/a

Inner Diameter, Di (m) - 0.02 n/a

Thickness, td (m) - 0.04 n/a

Mass, m (Kg) 2.45 1.0 n/a

Bearing stiffness, ( kxx , kyy ) N/A n/a 7 × 107 N/m

Bearing damping, ( cxx , cyy ) N/A n/a 200 N.s/m

Transverse crack of depths 3 and 5mm (i.e. μ = h/R = 0.3 and 0.5) with 0.5 mm width at

locations Γ = 0.3, 0.5 and 0.7 were machined on the shafts by an Electrical Discharge

Machine (EDM) as shown in Figure ‎4.4.

Lead-Titanate-Zirconate (PZT) piezoelectric ceramic sensors with dimensions 5 mm

length, 3 mm width and 0.7 mm thickness were bonded to the surface of the shaft to

Dis

c

Lt

Lb

Ld

Lc

Crack

PZT sensors

Coated wires Left Bearing Right Bearing

td

Figure 4.2: Dimensions of the experimntal test rig (of the stationary case).


95

acquire the dynamic strain response of the shaft for each disc location. At each axial

location along the length of the shaft, four PZT sensors were mounted circumferentially

(Top, Bottom, Right, and Left) in 90-degree angular positions around the shaft. For each

of the four angular orientations (0o, 90

o, 180

o and 270

o), there are 12 PZT sensors which

were bonded to the shaft at 80 mm interval along the axial direction of the shaft using

conductive epoxy (see Figure ‎4.1 and 4.2). The reason for using PZT sensors is to avoid

the weight of the sensors and their wires. In addition, PZT sensors do not require any

amplifiers to amplify their output signals and operate with good accuracy which means

that the noise effect is very low.

Grooves

Figure 4.3: Circumferential grooves at 90 degrees interval around the disc

bore.


96

Test Rig used in Rotating Case 4.2.2

The test rig in Figure ‎4.5 was designed to investigate the dynamics of the cracked rotor

in the rotating case. The rig consists of a shaft and a rigid disc which are supported by

two ball bearings. The rotor is connected to a variable speed motor (max speed 3000

rpm) by a flexible coupling. The material and dimensions of the shafts and the disc and

the size of the PZTs are the same as in Figure ‎4.2 and Table ‎4.1 except for the bearing

span Lb which is 0.9 m for this case. The rotating test rig is bigger and more

sophisticated than the stationary test rig in Section ‎4.2.1 (see Figure ‎4.1). The rig was

Crack

Coated

wires

PZTs

Figure 4.4: shafts of experimental tests.


97

designed in this size in order to reflect the characteristics of a real rig, and also to be

able to excite the rotor at first order (1×rev.). The method of on-shaft vibration

measurement was used instead of the on-bearing measurement method for measuring

the lateral vibration of the rotor. This method is very difficult because all the PZT

sensors and wires must be adapted to rotate simultaneously with the shaft. For this

purpose, a slip ring was used and the bearings were modified, using a collar with four

holes in order to pass the wires through it to the slip ring as shown in Figure ‎4.5 and 4.6.

The dimensions and design of each part of the rig are presented in Appendix A.

1 2

13

8 6

15

3

4

7

9 11

10 12

5

16

14

Figure 4.5: Experimental test rig for the rotating case: 1. AC-motor 2. Flexible

coupling. 3. Invertor. 4 Left-bearing. 5. Accelerometers. 6 Tachometer. 7. Zoomed

local part of the shaft. 8. crack slot. 9. PZT sensors and wires. 10. Right-bearing.

11. Collar with four holes. 12. Accelerometers. 13. Slip ring (24-channel) 14.

Aluminuime disc with four groves at its bore. 15. 16-channel data-aquisition boxes

(Data physics-Abacus). 16. PC.


98

Rotor Alignment 4.3

Unlike the stationary test the rotating test requires the shaft to be aligned to avoid (or

reduce) the vibrations resulting from misalignment of the rotor. The centre line of the

shaft and bearings must be adjusted to be in line with the centre-line of the motor‘s

shaft. For this purpose, the alignment instrument in Figure ‎4.7a, which consists of two

clamps (one for the shaft and the other for the motor‘s shaft), dial gauge, and a spirit

level (for indicating the horizontal and vertical planes of the rotor for misalignment

measurements), was designed and manufactured. After the alignment instrument became

ready, the alignment of the rotor was performed by mounting the instrument firmly on the

shaft of the rig to be rotated as shown in Figure ‎4.7b. Then two locations at the

Figure 4.6: Assembly of the slip ring and the wires of the

PZT sensors.


99

circumference of the shaft or coupling of the stationary shaft were measured to calculate the

alignment and to make the rotor to be within an acceptable misalignment range according to

the misalignment limits in Table ‎4.2. The alignment of the rig was performed at 3000 rpm

which is the maximum speed that was used in this study.

(a) (b)

Figure 4.7: Alignment instrument

Table 4.2: Maximum acceptable misalignment limits

(www.gearboxalignment.co.uk).


100

Vibration Measuring Instruments 4.4

Response Measurement 4.4.1

In general, the response of excited rotors whether rotating (self-excitation) or stationary

(Impact hammer excitation) is measured using accelerometers, proximity probes or

strain gauges, Alternatively, Lead-Titanate-Zirconate (PZT) piezoelectric ceramic

patches can be used as a transducer to acquire the response of excited rotors, particular

for on-shaft vibration measurements. The following are the reasons for using PZTs in

this study rather than both accelerometers and strain gauges as transducers to capture

the rotor response.

4.4.1.1 Accelerometers versus PZTs

Accelerometers are normally mounted on the rotor‘s bearings which excite the base of

the mounted accelerometers. That is, the value of output vibration from an

accelerometer depends on the magnitude of the motion of the bearing housing or

pedestal on which the accelerometer is mounted. In this case, if the bearings are rigid

(such as ball bearings on stiff pedestals) or a rotor rotates at low rotating speeds the

accelerometers will not function well, and consequently, the signal-to-noise ratio (SNR)

will be very low. The literature on investigating the dynamics of rotors (whether intact

or cracked rotors) has shown that the method of on-shaft vibration measurement is more

accurate than the method of bearing vibration measurement. This is because the former

considers only the shaft‘s vibrations whereas the latter considers interaction vibrations

of the bearings and shaft. However, applying the former method by using

accelerometers is difficult (or impossible), particularly, in rotating shafts due to the

dimensions, configuration, weight, and mounting method of accelerometers. In contrast,

PZT sensors enable on-shaft vibration measurements, they have high output voltage for

small input excitation, no weight effect and no amplifier is required and low noise

effect.

This was proved by conducting a comparison between accelerometers and PZTs for

acquiring the response of a rotating shaft during run-up tests as shown in Figure ‎4.8.

These characteristics of PZTs gave rise to the use of PZT sensors in this study rather

than accelerometers. The vibration responses of shafts were measured directly using

PZT patches that were bonded on the surface of the shafts. Thus, the measured shaft

responses were not influenced to the same extent by bearing and structural faults,


101

journal bearing damping, structural noise, etc. The specifications of the accelerometers

are given in Appendix B.1.

Indeed, the basic principle of operation of accelerometers and PZT sensors for

generating electric charges according to the piezoelectric effect is identical, namely: The

application of a dynamic load or mechanical strain causes the piezoelectric element to

produce electric charges. To measure the charge, it is necessary to use a charge

amplifier. For accelerometers, the use of a charge amplifier is essential. However, for a

PZT strain sensor, the output signal is much greater to the extent that it can be measured

directly without the need for a charge amplifier. This is because the piezoelectric

0 100 200 300 400 500 600-2

0

2

Time (sec)

Vo

ltag

e (

V) PZT Sensor

0 100 200 300 400 500 600-1

0

1

Time (sec)

Accele

rati

on

(g

)

Accelerometer (Ver.)

0 100 200 300 400 500 600-2

0

2

Time (sec)

Accele

rati

on

(g

)

Accelerometer (Hor.)

Figure 4.8: Comparisons between rotating shaft responses measured

using accelerometers and PZTs during runup tests. Ver. and Hor.

stand for the vertical and horizontal planes, respectively.


102

element in an accelerometer is subjected to less strain than a PZT sensor bonded directly

on a structure.

The piezoelectric element in an accelerometer is strained by a force (F) when the

accelerometer is vibrated. The force, which results from the product of the acceleration

(As) of a seismic mass and its mass (Ms), acts on the piezoelectric element and causes

the piezoelectric element to produce an electric charge, which is proportional to the

applied force [109, 110]. Conversely, the PZT sensor is strained by the deflection of the

structure on which the PZT is mounted. Of course, the magnitude deflection of the

structure during vibration is much higher than the force that results from the product of

the acceleration of a seismic mass and the magnitude of the mass. Accordingly, the

magnitude of the released electric charges in PZTs will be much higher than that the

seismic force produced in accelerometers under the same vibration condition. The

principle of operation of the piezoelectric elements in both accelerometers and PZTs is

depicted in Figure ‎4.9.

F = Ms As

(a)

2

3

1

5 4

(b)

PZT

Beam before vibrating

Surface of beam

when vibrating in

the first mode

Figure 4.9: Operating principle of: (a) accelerometer, and (b) PZT sensor. 1.

accelerometer case 2. seismic mass 3. Piezoelectric crystals 4. Micro-circuit 5. test

structur.


103

4.4.1.2 Strain Gauges versus PZTs

Strain gauges are resistive transducers which are widely used in structural systems to

experimentally measure loads or strains. Strain gauges consist of conductive resistance

wires in which the gauge resistance is directly proportional to the change in length per

unit length of the wires. The characteristics of strain gauges such as small dimensions,

high accuracy, high sensitivity and negligible weight have made these transducers are

preferable to study the dynamics of rotors by measuring vibrations on-shaft directly.

Typically, the resistance change in a strain gauge is quite small and difficult to measure.

Therefore, a voltage change due do resistance change is always preferred which is done

by using a Wheatstone-bridge circuit. That is, each strain gauge needs a Wheatstone-

bridge circuit (i.e. voltage amplifier). A dummy gauge (or lead-wire) is also essential to

be used to compensate temperature variations in wires during tests. Normally, each

strain gauge requires three lead-wires (for more accurate results, 4 lead-wires are

required) to be connected to avoid temperature variations in the wires. In this case, if 4

strain gauges are mounted on each side of a shaft to be tested, there will be 24 wires (for

8 strain gauges with 3 wires each) and 8-Wheatstone-bridge circuit (i.e. 8-voltage power

supply) which need to be balanced at each time during the test. Also, skill is required to

bond stain gauges on a surface and balance the strain gauge bridge, particularly, for

rotating shafts. This is because using a slip ring generates high noise as a result of high

contact resistance between the slip ring‘s brushes. Consequently the strain gauge bridge

cannot be balanced. Due to these feature limitations, PZT transducers were preferred to

strain gauges in this study. In addition to the advantages of PZTs that are mentioned in

the previous section, the installation (or sticking) method of PZTs on the structure of

interest that is being tested is very easy and does not need skills. The cutting process of

PZT patches to the required dimensions for the test of interest is only the disadvantage

of PZTs because the PZTs patches are very brittle and easy to break. These combined

characteristics have a crucial impact on the sensor‘s configuration and accuracy of the

measurements.

In order to assess the capability and sensitivity (in physical units) of the PZT sensors to

capture the shaft vibration in comparison with metal strain gauges, the rig in Figure ‎4.10

was designed. The rig uses the longitudinal wave propagation method. The test rig

consists of a shaft with the same dimensions and material as the shafts that were used in

Table ‎4.1. The shaft is suspended by two piano wires as a free-free beam system. The

strain gauges, which have resistances of 120 ohm and a gauge factor, G.F = 2.01 (see


104

Appendix B.3 for more information about the specifications), were bonded on the shaft

surface. At the diametrically opposite locations to each strain gauge, a PZT sensor was

bonded in order to capture the vibration response at the same axial location as the strain

gauge captures. Each strain gauge was connected to a quarter-bridge power supply and

balanced at excited voltage, Eex. = 5volts. Afterwards, the shaft was aligned horizontally

and one end of the shaft was centrally struck by a steel ball. The longitudinal wave

propagation was then captured by both the PZT sensor and strain gauge and processed

and analysed using Data Physics a data acquisition module as shown Figure ‎4.11. The

figure shows that the output voltage of the PZT is nearly 20 times greater than the

voltages recorded by the strain gauges. It should be noted that the PZT response signal

was not amplified, whereas the strain gauge signal was amplified. In the case of a

rotating shaft, it is necessary to use a slip-ring to transfer the strain resistances.

However, the output voltage value of the strain gauge would be seriously affected by

inherent noise due to the instruments and the slip-ring. Additionally, the PZT is

3 2 4

6

8 1

7

5

9

Figure 4.10: The Test rig for using the longitudinal wave propagation method to

compare PZTs with starin gauges for on-shaft vibration mesurment.1. Stainless steel

shaft. 2. Ball for striking the shaft. 3. Zoomed local shaft area. 4 Strain gauge. 5 PZT. 6

Strain gauge‘s power amplifier (quarter-bridge). 7 Strain gauge connecters. 8. Data

acquisition system (DataPhysics-Abacus). 9. PC


105

connected by only one wire whereas each strain gauge at least must be connected to

three wires to avoid temperature variations. This merit in PZTs has definitely a crucial

impact on reducing the noise effect, space, and installation time, particularly in rotating

shafts. In principle, the output voltages of strain gauges are converted to units of strain

(microstrain) as a non-dimensional physical unit. The output voltage of the strain

gauges were transformed to strain (mm/mm) is given in Figure ‎4.12 by using Equation

(4.1). Then the sensitivity of the PZT sensor in comparisons with the strains of the strain

gauges was evaluated by plotting the output voltage of the PZT sensors against strains

as given in Figure ‎4.13.

(‎4.1)

where, is strain, is strain gauge‘s output voltage, is gauge factor, excitation

voltage and Gain = 1000.

Figure 4.11: Output of both the PZT and strain gauge at the same axial

location on the shaft (both in volts).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2

4

Time (ms)

Ou

tpu

tVo

ltag

e (

V)

PZT sensor

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

-0.1

0

0.1

0.2

Time (ms)

Ou

tpu

t V

olt

ag

e (

V)

Strain gauge


106

Figure 4.12: Output of both the PZT and strain gauge at the same axial

location on the shaft. PZT in volts and strain gauge in strain (microstrain).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2

4

Time (ms)

Ou

tpu

t V

olt

ag

e (

V)

PZT sensor

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-50

0

50

Time (ms)

Str

ain

(m

icro

stra

in)

Strain gauge

Figure 4.13: Sensitivity of the PZT sensors in comparison with the strain gauges.

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5

3

3.5

4

Strain (microstrain)

PZ

T O

utp

ut

Vo

ltag

e (

V)

Slope = 51.1 mV/microstrain


107

Data Acquisition Card 4.4.2

A process of measuring output electrical signals such as voltage, resistance or current,

with a computer is called Data acquisition (DAQ). In order to collect these output

electrical signals from sensors, a DAQ measurement hardware as well as a computer

with programmable software is required, collectively named DAQ system. The accuracy

of processing signals depends mainly on the specifications of the DAQ measurement

hardware that are chosen for the required tests. Therefore, the DAQ measurement

hardware of type PCI-6123 S series shown in Figure ‎4.14a, has been chosen for this

research. This DAQ hardware falls under the NI S Series product family The ―S‖

denotes simultaneous sampling which is the most prominent benefit of the dedicated

analogue-to-digital (A/D) converter per channel architecture. The acquired vibration

responses are processed and analysed by the software NI-Signal Express 2013 which is

based on the LabView platform. Detail specifications of the DAQ type NI-PCI-6123 S

series and BNC-2110 are presented in Appendix B.2.

An Abacus DAQ module made by Data Physics was also used as an advanced data

acquisition system for acquiring and processing vibration in rotating systems. Two Data

Physics (DP) modules with 8-channels each, as shown in Figure ‎4.14b, were used for

acquiring and processing the vibration signal in this study. The advanced specifications

and the advanced built-in software of the Data Physics modules made them more

preferable to the NI-PCI-6123 card for acquiring and processing the vibration signals of

the rotating systems. For more information about the Data Physics system see

(www.dataphysics.com).

http://www.dataphysics.com/


108

Experimental Test Methodology 4.5

The theoretical simulation results of the developed crack identification techniques,

which are presented in the subsequent chapters, were validated experimentally by using

the stationary and rotating test rigs in Figure ‎4.1 and 4.5. Tests were performed on

stationary and rotating rotors in both the intact and cracked states by inducing cracks

with the same depth ratios and locations that had been used in the theoretical

simulations which are presented in later chapters. In these tests, a disc, which represents

a roving mass that imparts an extra inertia force to the corresponding location, was

traversed along the intact and cracked rotors at a 40 mm spatial interval. Afterwards,

natural frequencies corresponding to each roving disc location were determined and

(a)

(b)

Figure 4.14: Data acquistion systems: (a) NI-DAQ hardware

and NI-Signal Express software. (b). Data Physics (Abacus)

system


109

used as a tool to identify and localise cracks in rotor systems through applying the

developed crack identification techniques that are presented in the next chapters.

For the stationary rotor tests, the modal analysis method was used to find modal

frequencies of the rotor using an impact hammer as the source of excitation. At each

0 100 200 300 400 500 600 700 800 900 100010

-5

10-4

10-3

10-2

10-1

100

Freq(Hz)

Am

pli

tud

e (V

)

0 100 200 300 400 500 600 700 800 900 100010

-6

10-4

10-2

100

Freq(Hz)

Am

pli

tud

e (V

)

Intact rotor

Cracked rotor

Figure 4.15: Frequency response functions at the same location of the disc at

both the intact and cracked rotor. Crack depth ratio μ = 0.3 at location Γ = 0.3.

Frequency resolution = 0.315 Hz


110

disc location, the frequency response functions (FRFs) were determined for both the

intact and cracked rotor. The first four modes were extracted from the highest peaks in

the FRFs as shown in Figure ‎4.15. This step was repeated and applied at all the disc

locations. Then, the extracted natural frequencies of both the intact and cracked rotor at

each disc location were used for the identification and location of the crack in the rotor

according to the explained scenario of the crack identification that are explained in the

subsequent chapters.

As for the rotating rotor tests, real-time waterfall analysis was used for capturing the

vibration response of the rotor during the rotor run up. The test rig at each disc location

0 100 200 300 400 500 600-4

-2

0

2

4

6

time (sec)

Am

pli

tud

e (

Vo

lts)

0 100 200 300 400 500 600-2

-1

0

1

2

time (sec)

Am

pli

tud

e (

Vo

lts)

Intact rotor

Cracked rotor

Figure 4.16: Time waveform (PZT sensor) of the rotating rotor at a point of

the roving disc for both the intact and cracked rotor. Crack depth ratio μ =

0.3 at location Γ = 0.3


111

was run up from 0 to 2900 rpm at a linear rate of 5 rpm/sec. The Mini-VLS series

Optical Speed Sensors, which are primarily designed for high speed monitoring, was

used as a tachometer for monitoring and capturing the rotor speeds during run-up tests

(see Figure ‎4.5). The vibration response of the shaft was collected by the PZT sensors

on the shaft and passed through the slip ring to the Data Physics modules for processing

and analysing. Figure ‎4.16 shows the time waveform history of two sensors during the

rotor run up. The presence of the highest amplitudes in the time waveform at different

speeds during run-up indicates the excitation of the rotor at the critical speeds which is

called rotor resonance frequencies. To understand the time waveform clearly, the

waterfall analysis, which is based on the Short Time Fourier Transform (STFT)

analysis, was conducted on the time waveform of each PZT sensor as shown in

Figure ‎4.17.

Figure 4.17: A screenshot of waterfall plot of spectra of a PZT sensor on

the shaft. The analysis was done by DataPhysics hardware using built-in

SignalCalc software.


112

Modal Analysis and Frequency Resolution Problems 4.5.1

The techniques, so-called NNF curves, FCP products and NONF curves that are

developed during this study are based on the vibration-based method using modal

frequency as a tool for crack identification in rotor systems. Typically, the modal

parameters, namely natural frequencies, damping ratios and mode shapes of a structure

are determined in two different methods called (1) theoretical modal analysis (2)

experimental modal analysis. The former assumes the knowledge of the stiffness, mass

and damping matrices of a structural system, and uses these matrices in solving an

eigenvalue problem; whereas the latter exploits the responses of the structural system

and applies modal analysis identification techniques for the determination of the modal

parameters. Sometimes the mathematical model of an existing structure is not available

or a structure is very complex and difficult to develop an exact mathematical model

without errors so the experimental modal analysis method has received more attention

than the theoretical modal analysis for the determination of modal parameters [111].

The responses of structural systems are measured in the time domain and presented in

the frequency domain. Experimental modal parameters are determined from a set of

frequency response functions (FRFs), which describe the input-output relationships

between two points on a structure and these responses are presented in the frequency

domain. Basically, some inevitable issues such as leakage or resolution of natural

frequencies affect the FRFs, which should be taken into account during the application

of the Fast Fourier Transform (FFT) to the responses in the time domain in order to gain

the FRFs in the frequency domain.

The resolution (or closeness) of discrete frequencies in FRFs results is governed by

some factors such as number of spectral lines (lines), the time duration of a capture

window (Tspan), the maximum frequency (Fspan) of interest and number of time

samples in each capture window (BlockSize). Accordingly, four relationships can be

generated to interrelate these factors and show how the frequency resolution (dF) can be

controlled. These four relationships are [112],

(‎4.2)

(‎4.3)


113

(‎4.4)

( ) (‎4.5)

Generally, in experimental modal testing, FRF measurements are conducted and

obtained by exciting the test structure artificially through using either an impact hammer

or one (or more) shakers. While both these excitation techniques have advantages and

disadvantages, the impact hammer method is a transient excitation technique that is very

efficient and portable compared to the need to move and align shakers for steady-state

vibration measurements. In transient testing, noise can relatively be reduced in the

computed FRFs by a small number of averages (three to six averages) and very high

quality FRFs can be obtained for lightly damped and linear test structures. The shaker

technique generates higher quality frequency response functions (FRF) over greater

bandwidths, much better control of the frequency ranges excited as well as the level of

force applied to the structure. However, greater caution must be taken during the setup

of a shaker test to avoid the shaker‘s mass and stinger‘s stiffness, which generate

undesirable FRFs [113, 114].

In principle, impact testing is conducted on a structure by striking several points on the

structure through using an instrumented impact hammer, which impart energy to the

structure as impulse force in a short time duration. Simultaneously, the response

acceleration (transient signals) of the structure is measured at a fixed reference site. That

is, the decay rate of the transient impact response will clearly be influenced by the

damping value of the structure being excited by an impact hammer. From the signal

processing standpoint, this is a key problem that affects the accuracy of the FFT because

of the leakage phenomenon, which is related to the sampling of the signal. Leakage is

mainly minimised by applying a windowing function such as Hanning, Hamming or

Flat windowing functions[112].

Ideally, the time window (Tspan) is defined to be adequate to allow the response signal

to decay back to zero value within the observation. In this case, if the defined Tspan is

too short (i.e. Tspan is shorter than the required time to allow the response signal to

completely die out to zero) to allow this to occur, the response measurement will be

truncated. This causes an error because a part of the response data is not be included in

the computation of the FFT. This problem can be overcome by the use of a response

window (Exponential Window type is recommended).


114

In this study, the rotor systems in the stationary case were artificially excited by using

the impact hammer method to conduct experimental modal testing, and to identify the

modal frequencies from the FRFs measurements. In fact, the applications of the

developed crack identification techniques in this work depend clearly on choosing

proper spectral lines (i.e. frequency resolution, dF,) that govern the closeness of natural

frequencies in the FRFs measurements. The frequency resolution of dF = 0.315 Hz,

which requires a time response duration of 3.17 seconds (dF = 1/Tspan)), was used in

the determinations of the modal frequencies, and used in the application of the crack

identification techniques, namely NNF curves, FCP products and NONF curves. That is,

the higher the spectral resolution, the greater is the accuracy in the application of the

crack identification techniques. Accordingly, it is not advisable to use the frequency

resolution (dF) greater than the 0.315 Hz. (Herein, frequency resolution dF <0.315

means high resolution and vice versa).

Summary 4.6

In this chapter, the design, dimensions and materials of the stationary and rotating test

rigs are presented. Also, the dimensions, locations and materials of the shafts and the

disc and how to use the disc as a roving mass are clearly explained. The type of the

sensors for acquiring the vibration of the rotor is provided. Using these test rigs, the

simulation results that have been obtained from the finite element analyses are

validated. The procedures for conducting the experimental tests of both the stationary

and rotating rotor are described.

CHAPTER 5

115

CHAPTER 5

Detection and Localisation of a Rotor Crack Using a Roving

Disc and Normalised Natural Frequency Approach

Zyad N Haji and S Olutunde Oyadiji

Journal of Finite Elements in Analysis and Design (under review)

CHAPTER 5

116

Detection and Localisation of a Rotor Crack Using a Roving Disc and Normalised

Natural Frequency Approach


School of Mechanical, Aerospace and Civil Engineering, University of Manchester,

Manchester M13 9PL, UK

Abstract

A modal frequency technique for the identification and localisation of cracks in

stationary rotor-bearing systems is addressed in this study. The proposed technique,

which is based on the natural frequencies of intact and cracked stationary rotor-bearing

system, is numerically investigated and validated experimentally. The rotor with an

open crack and carrying an auxiliary roving disc has been modelled using Bernoulli-

Euler finite elements. In order to identify and locate the crack in the stationary rotor-

disc-bearing system, the proposed approach utilises the variation of the normalized

natural frequency curves versus the non-dimensional location of a roving disc which

traverses along the rotor span. The merit of the proposed technique is that it uses the

roving disc as an extra dynamic mass to enhance the dynamics of the cracked rotor and,

thereby, to facilitate crack identification and localisation. The technique identifies,

locates and assesses the severity of a crack by using natural frequencies of the rotor

system. The presence of the crack is identified, and its location is determined, from the

appearance of sharp discontinuities in the plots of the normalised natural frequency

(NNF) curves versus the non-dimensional locations of the roving disc. In this work, the

first four natural frequencies are used for the identification and location of a crack in a

stationary rotor-bearing system. Both the numerical and experimental results prove

reasonably the feasibility and capability of the proposed technique for the identification

and localisation of cracks under the environment being investigated. Furthermore, this

approach is not only efficient and practicable for high crack depth ratios but also for

small crack depth ratios and for a crack close to or at the node of mode shapes, where

natural frequencies are unaffected.

Keywords: rotor dynamics; crack identification or detection; cracked rotor; roving disc

or mass; natural or modal frequency; non-destructive testing.

CHAPTER 5

117

Introduction 5.1

For the last three decades, many researchers have focused on the dynamic behaviour of

structures and rotors with cracks in order to develop techniques, theoretical models,

methodologies and technical tools to identify and localise cracks in their incipient

propagation stages [115, 116]. The presence of a crack in a structure reduces its local

stiffness at the location of the crack. Consequently, the dynamic vibration behaviour of

the structure is altered; this is manifested by changes in natural frequencies and mode

shapes. If a crack propagates continuously and is not detected early abrupt failure may

occur .This may lead to a catastrophic failure with enormous costs in down time,

consequential damage to equipment and potential injury to personnel. Therefore,

monitoring the integrity of structural components is very essential, to improve their

safety, reliability and operational life. Thus, identification of the depth and position of

cracks through non-destructive testing is important to ensure the integrity of structural

systems [117-119]. Many devoted work on the dynamics of cracked rotors and

structures can be found in [5, 26, 120, 121].

The change in the natural frequencies of structures due to a crack has been used by

many researchers as a tool to quantify the size and location of a crack at the early stages.

Lee and Chung [122] employed the natural frequency data of a one-dimensional beam-

type structure to identify and localise the crack in the beam. Chondros et al [123]

studied cracked Bernoulli-Euler continuous beams with single or double-edge open

cracks. They developed a theory to determine the crack size and location from measured

information of the cracked beam system by carrying out an inverse problem. Narkis

[124]developed a method for crack detection in a simply-supported beam. The method

was based on the variation of the first two natural frequencies of the simply-supported

beam.

Lin [125] derived a method based on a mathematical model in which the crack is

modelled as a rotational spring connecting two separate beams. The method was used to

identify a crack, its depth and location in a Timoshenko simply-supported beam. The

author indicated that the size and location of a crack can be determined by measuring

any two natural frequencies in this cracked system. A new technique for the

identification of the physical properties of a crack in structural systems was proposed by

Zhong and Oyadiji [126]. The technique was based on auxiliary mass spatial probing by

stationary wavelet transform. They indicated that it is difficult to locate the crack

CHAPTER 5

118

directly from the graphical plot of the natural frequency versus axial location of the

auxiliary mass. This curve of the natural frequencies can be decomposed by stationary

wavelet transform into a smooth, low order curve, called approximation coefficient, and

a wavy, high order curve called the detail coefficient, which includes crack information

that is useful for damage detection.

An approach based on the combination of wave propagation, genetic algorithm and the

gradient technique was presented by Krawczuk [116] for crack identification in beam-

like structures. The author concluded that the proposed method is better than methods

based on changes in modal parameters. Shen and Pierre [127] employed an approximate

Galerkin solution to identify the location and size of a crack in simply-supported

cracked beams in free bending vibrations. An analytical method to determine the

fundamental frequency of cracked Bernoulli-Euler beams in bending vibrations was

developed by Fernandez-Saez and Navarro [128] . The influence of the crack was based

on the mathematical model in which the cracked beam was represented by an elastic

rotational spring connecting the two segments of the beam at the cracked section.

Closed-form expressions for the approximate values of the fundamental frequency of

cracked Bernoulli-Euler beams in bending vibration are given. The results obtained

agree with those numerically obtained by the finite-element (FE) method. Cheng et al.

[129] applied the p-version finite element method to investigate the vibration

characteristics of cracked rotating tapered beam. They proposed an approach based on

spatial wavelet transform to detect cracks from the slight perturbation of the mode

shapes at the crack position.

Zhong and Oyadiji [130] also studied theoretically the natural frequencies of a damaged

simply-supported beam with a stationary roving mass based on the approximate

approach used by Fernandez-Saez and Navarro. They added a polynomial function,

which represents the effects of a crack, to the polynomial function which represents the

response of the intact beam in order to represent the transverse deflection of the cracked

beam. They showed that natural frequencies change due to the roving mass along the

cracked beam, therefore the roving mass can provide additional spatial information for

damage detection of the beam. Salawu [36] presented a review of various approaches

proposed for identifying damage using natural frequencies that are of crucial importance

in the integrity assessment of structures. Also, structural behaviour and condition can be

monitored by using natural frequency values obtained from the periodic vibration

testing of structural systems.

CHAPTER 5

119

In other studies, crack location and stiffness reductions of a beam, due to a crack, have

been studied by modelling a crack as a linear spring. Loya et al. [131] have applied

perturbation method to obtain the natural frequencies for bending vibrations of a

cracked Timoshenko beam with simple boundary conditions (BCs). The cracked beam

is modelled as two segments that are connected by two massless springs (one

extensional and the other rotational). They have shown that the method provides simple

expressions for the natural frequencies of cracked beams and it gives good results for

shallow cracks. Also, Sinou [132] presented a technique based on using frequency

contour lines method which use the changes of frequency ratios in cracked beams.

Sayyad and Kumar [133] studied a relationship between the natural frequencies and the

physical characteristics of the crack in a simply-supported beam. The crack was

included in the beam as an equivalent torsional spring connecting the two segments of

the beam. They showed that the variation of the first two natural frequencies is

sufficient for identification of the crack size in a cracked simply supported beam.

Sekhar and Balaji [59] used the finite element (FE) technique to detect a slant crack in a

rotor-bearing system based on the bending vibration. They stated that a general

reduction occurs in all modal frequencies with an increase in the depth of a crack.

Dharmaraju et al [134] developed a general technique based on the information of beam

force–response to identify and estimate crack flexibility coefficients and crack depth.

They employed the finite element method for modelling a Bernoulli-Euler beam

element, and the crack was modelled by a local compliance matrix.

The current work proposes a technique based on the natural frequencies of the intact and

cracked stationary rotor-bearing systems to identify, locate and determine the severity of

a crack in the rotor. The stationary rotor bearing system behaves as a simply–supported

beam and it carries a disc which acts as a roving mass which is traversed along the shaft

from one end to the other. The proposed technique has merits over the methods that

have been presented in the literature as the technique uses a roving disc to enhance the

dynamics of a crack in the rotor-bearing system, which facilitates the identification and

localisation of the crack in the shaft. Also, the experimental implementation of the

method requires the use of simple instrumentation and simple testing techniques. In this

proposed method, the natural frequencies of a cracked shaft are normalised by the

corresponding natural frequencies of an intact shaft. The presence of a crack is

identified, and its location is determined, from the appearance of sharp discontinuities in

the plots of the normalised natural frequency (NNF) curves versus the non-dimensional

CHAPTER 5

120

locations of the roving disc. The proposed technique is theoretically investigated and

experimentally validated. The results show clearly that the proposed technique is

feasible and capable to identify, locate and determine the severity of cracks in stationary

rotor-bearing systems.

Equation of Motion of a Cracked Rotor 5.2

The mathematical modelling of the cracked rotor with a single transverse surface crack

in Figure ‎5.1 is formulated using the finite element method (FEM). The constant cross

sectional of the shaft is divided into N Bernoulli-Euler beam finite elements with two

nodes for each element and four degrees of freedom per node, which are the transverse

displacements, u and v in the x- and y-axes directions, respectively, and the rotations θ

and ψ about the x- and y-axes directions, respectively as shown in Figure ‎5.2. The rotor

is assumed to be simply-supported by two ball bearings mounted in stiff pedestals. The

equations of motion of the rotor are obtained from Lagrange‘s equation which is given

by [135].

(

*

(‎5.1)

where, T, D and U are the kinetic, dissipation and strain energies, respectively, qi is the

generalised displacement and Qi is the external forcing. Formulation of the energy

expressions results in the element mass, stiffness and damping matrices which are

assembled to give the global matrices. Thus, the element matrices of each component of

the rotor will be firstly derived.

Element Matrices of Rotor Systems in the Fixed Frame 5.2.1

The elements of mass and stiffness matrices for Bernoulli-Euler shaft bending are

identical to the standard formulations of a beam which are determined by using the

conservation energy method (i.e. the kinetic and strain energy expressions). Thus, these

expressions are extended to the model of a shaft by considering the x-z plane and y-z

plane as two independent transverse bending planes. In order to apply the energy

method, the physical deformation within the element is approximated by using the

standard cubic shape functions which is based on the boundary conditions of the

simply-supported beam. The displacement within the element in the x-z plane is

interpolated by [58],

CHAPTER 5

121

Crack Segment

1 2 3 . . . . . . . . . . N-1 N .

L x

Lc wc

td

Dis

c

R

h

Y

X

Crack

Figure 5.1: Finite element model of the rotor with a cracked

cross-section.

𝜓

e 𝜉

𝜓𝑒(ξ ) 𝜓

𝜃

𝜐 𝑢

𝜃𝑒(ξ )

𝜐𝑒(ξ )

𝑢𝑒(ξ )

𝑢

𝑣

𝜃

Z

Y

X

Figure 5.2: Typical finite shaft element and coordinates.

CHAPTER 5

122

( ) , ( ) ( ) ( ) ( )-

(

( )

( )

( )

( ))

(‎5.2)

The strain energy within the shaft element can be approximated by

∫ (

( )

)

ℓ

(‎5.3)

This approximation together with the approximation to the lateral displacement of the

shaft that is given in Equation (‎5.2) can be used to obtain

, - (‎5.4)

where the elements of the stiffness matrix are

∫

ℓ

( ) ( ) (‎5.5)

where the double prime represents the second derivatives of the shape functions in

Equation (‎5.2), is the Young‘s modulus, and is the second moment of area of the

cross section about the neutral plane.

Similarly the elements of the mass matrix are obtained by using the kinetic energy

method. The kinetic energy of the Bernoulli-Euler shaft element is

∫

ℓ

( ) (‎5.6)

Substituting the shape functions, Equation (2), gives

, - (‎5.7)

where the elements of the mass matrix, eM are

∫

ℓ

( ) ( ) (‎5.8)

According to the local coordinates described in Figure ‎5.2, the definition of the bending

coordinates of the x-y and y-z planes are shown in Figure ‎5.3. Therefore, the elements of

both stiffness and mass matrices for the Bernoulli-Euler beam can be directly generated

from Equations (‎5.4) and (‎5.7), based on the local coordinate vector

, - in Figure ‎5.2. Assuming the two bending

CHAPTER 5

123

planes do not couple, then the elements of the mass and stiffness matrices for the two

perpendicular bending planes are

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎5.9)

and

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎5.10)

e

𝑢𝑒 𝑢𝑒

𝜓𝑒 𝜓𝑒

𝜉

z

𝑢𝑒(ξ )

x-z plane

z

e

𝜉

𝜃𝑒 𝜃𝑒

𝑣𝑒 𝑣𝑒

𝜐𝑒(ξ )

y-z plane

Figure 5.3: Definition of the degrees of freedom for the shaft element.

CHAPTER 5

124

The mass matrix of the disc is computed in a similar way assuming that the disc is rigid

(i.e. the strain energy within the disc element is neglected). The kinetic energy of a rigid

disc in the fixed frame is given by

(

)

(

)

(

) (‎5.11)

Based on the kinetic energy terms of Lagrange‘s equation, the element matrices of the

rigid disc can be derived from Equation (‎5.11) as

(

( *

(

)

)

(‎5.12)

Thus, the elements of the mass and gyroscopic matrices for the rigid disc are defined as

[

] (‎5.13)

and

[

] (‎5.14)

The ball bearings in this paper are assumed to be linear and obey the following

governing equations which correlate the forces acting on the shaft resulting from the

bearings with resultant displacements and velocities of the shaft as

(‎5.15)

where

[

] is the bearing stiffness matrix, 0

1 is the bearing

damping matrix, and . / is the bearing displacement vector.

Thus, assembling the aforementioned equations of the components of the rotor system,

the free vibration equation of motion of an intact rotor supported by rigid bearings at

both ends can be defined as

( ) ( ) ( ) ( ) (‎5.16)

CHAPTER 5

125

where, ( ) ,

- ( ) [

] is the nodal

displacement vector with dimension 4(N+1) × 1 corresponding to the local coordinate

vector for each element , - in Figure ‎5.2. M

is the global mass matrix which contains the mass matrices for each element of the

rotor and the mass matrix Md of the disc corresponding to degrees of freedom

, - , - . G and C are the global gyroscopic and damping

matrices, respectively.

Modelling of the Cracked Element 5.3

In this study, the crack model, which was derived by Al-Shudeifat and Butcher [24-26]

is used to define the stiffness properties of the cracked element in the considered rotor.

In this model, the open transverse crack geometry was modelled as illustrated in

Figure ‎5.4. The crack is assumed to be at an initial angle with respect to the fixed

negative Y-axis at t = 0. When the shaft rotates, the crack angle relative to the negative

Y-axis changes with time to + Ωt as shown in Figure ‎5.4a. The stiffness reductions of

the cracked element in a beam are considered as time-varying values during the shaft

rotation and defined through the reduction of the second moments of area and

about the centroidal and axes, respectively as

( ( )) (‎5.17)

and

( ( )) (‎5.18)

where ( ) ⁄ , (

) ⁄ and are constant

values during the shaft rotation. The second moments of area and about the

rotational x- and y-axes respectively, of the cracked element cross-section are derived

for as (see Al-Shudeifat [24-25])

(( )( ) ( )) (‎5.19)

and

.( )( ) ( )/ (‎5.20)

CHAPTER 5

126

Thus, A1 and e are defined, respectively, as

. ( ) ( )√ ( )/ (‎5.21)

and

( ( ))

(‎5.22)

where, √ ( ) is a constant which depends on the crack depth ratio, = h/R.

As a result, the finite element stiffness matrix of the cracked element can be written as

( ( )) (‎5.23)

As a consequence, the FE equations of motion of the uncracked rotor bearing-system

given in Equation (‎5.16) are rewritten to include the effect of the cracked element of an

open crack model as

( ) ( ) ( ) . ( ( ))/ ( ) (‎5.24)

where is the 4 (N+ 1) × 4 (N+ 1) global stiffness matrix derived from the global

stiffness matrix of the uncracked beams by replacing the uncracked element stiffness

matrix of element i by the cracked element stiffness matrix

. is another

(a) (b)

Figure 5.4: A cracked element cross-section: (a) rotating, (b) non-

rotating; the hatched partdefines the area of the crack segment [18, 21].

CHAPTER 5

127

4(N+1) × 4(N+1) global stiffness matrix; it has zero elements apart from those at the

cracked element location where the elements are equal to .

Numerical Solution 5.4

The finite element model of the simply-supported rotor in Figure ‎5.1, which represents

an intact and cracked rotor, has been coded in Matlab environment using the dimensions

and material properties stated in Table ‎5.1. The disc is treated as a roving concentrated

mass, which is traversed along the cracked rotor length at 10 mm spatial interval.

Hence, the natural frequencies of the rotor system in a cracked and intact status,

corresponding to the each spatial interval, are computed. In this study the assumptions

are that the rotor is a stationary rotor and the crack is always fully open (see

Figure ‎5.4b).

Table ‎5.1: Physical parameters of the rotor model

Parameters Shaft Disc Bearing

Material Stainless steel Aluminium n/a

Young‘s Modulus, Es (GPa) 200 70 n/a

Density, ρ (Kg/m3) 7800 2700 n/a

Length, (m) L=1 x = 0.01, 0.02, 0.03, ..., 1 n/a

Outer diameter, Do (m) 0.02 0.178 n/a

Inner Diameter, Di (m) n/a 0.02 n/a

Thickness, t (m) n/a 0.015 n/a

Mass, m (Kg) 2.45 1.0 n/a

Bearing stiffness, ( kxx , k yy ) n/a n/a 7 × 107 N/m

Bearing damping, ( cxx , cyy ) n/a n/a 100 N.s/m

In order to verify that the proposed approach is feasible to identify, locate and determine

the size of a crack in rotors, four numerical cases of different damage locations and

crack sizes are investigated. The location and size of a crack of 0.5 mm width, wc = 0.5

mm, for each case is defined in Table ‎5.2. It is known that if a crack is located at a node

of a mode (e.g. at the centre of a simply-supported beam for the second mode shape,

and at one-third or two-thirds of the rotor for the third mode shape), the natural

frequencies are not affected. For that reason, the arrangement of the crack location and

CHAPTER 5

128

sizes in Table ‎5.2 have been chosen to demonstrate the robustness of the proposed

approach when the crack is located close to the nodes of modes 2, 3 and 4. Cracks at

locations of = 0.3, 0.5 and 0.7 are investigated. Additionally, a crack at = 0.4, which

is close to all the nodes of the mode shapes of modes 1 to 4, is investigated. Therefore,

Case 1 studies a crack located near to the nodes of the third and fourth mode shapes,

which occur at one-third of the rotor length, Case 2 studies a crack located at the node

of the second and fourth modes, which occur at the centre of the rotor, and Case 3

studies a crack located near the vibration nodes of the third and fourth mode shapes. The

effects of the locations and sizes of a crack that is not too close to the nodes of modes

one to four are investigated in Case 4.

Table ‎5.2: Physical properties of cracks for Numerical simulations

Case No. Crack Location

= Lc /L

Crack depth ratio

µ

Relative Disc Location

ζ = x/L

1 0.3 0.3, 0.5, 0.7, 1.0

0.3, 0.5, 0.7, 1.0

0.3, 0.5, 0.7, 1.0

0.3, 0.5, 0.7, 1.0

0.01, 0.02,..., 1

0.01, 0.02,..., 1

0.01, 0.02,..., 1

0.01, 0.02,..., 1

2 0.5

3 0.7

4 0.4

Figure ‎5.5 shows the variations of the first four natural frequencies of the cracked rotor,

according to the geometrical and physical parameters stated in Table ‎5.1 and 5.2 for

Case 1. The figure shows that as the roving disc is traversed from one end of the shaft to

the other, the first natural frequency decreases and reaches a minimum value at the

centre of the shaft. Beyond the shaft centre, the first natural frequency increases until it

reaches the maximum value at the other end of the shaft. For modes two to four, the

figure shows that as the roving disc is traversed from one end of the shaft to the other,

the natural frequencies decrease to a minimum value and then increase to a maximum

value alternatively. This produces sinusoidal curves, which are referred to as natural

frequency curves [126] , for modes two, three and four that are double, triple and

quadruple of the natural frequency curve of mode one, respectively. Figure ‎5.5 also

indicates that the natural frequencies of the rotor hardly change as the non-dimensional

crack depth ratio (µ) is increased. However, the inserts in each figlet, which are the

enlarged views of segments of the curves, show that the natural frequencies slightly

decrease as the crack depth ratio is increased.

CHAPTER 5

129

Crack Identification Technique 5.5

The identification of a crack and its physical properties (i.e. location and size) cannot be

directly observed from the natural frequency curves shown in Figure ‎5.5. Therefore, to

enhance the clarity and, thereby, facilitate the identification and localisation of cracks

using the natural frequency curves, a method is proposed in this paper which is called

normalised natural frequency curves (NNFCs) method. This method, which is based on

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 128

30

32

34

36

38

40

x (m)

f c1 (

Hz)

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1120

125

130

135

140

145

150

155

160

x (m)f c2

(H

z)

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1290

300

310

320

330

340

350

x (m)

f c3 (

Hz)

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1530

540

550

560

570

580

590

600

x (m)

f c4 (

Hz)

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)

Figure 5.5: First four direct natural frequencies of the stationary cracked rotor with a


nd mode, (c) 3

rd mode, (d)

4th

mode.

CHAPTER 5

130

the natural frequency curves of an intact and cracked rotor, is implemented via three

steps, namely:

Step 1: The natural frequencies (fci) of a cracked beam for each crack depth ratio

corresponding to the roving disc locations are divided by the natural frequencies (foi) of

the intact rotor corresponding to the roving disc locations, in order to obtain the natural

frequency ratios (βi = fci/foi ) of each mode.

Step 2: The natural frequency ratios ( ) of each mode are divided by the maximum

natural frequency ratios ( ) of each mode to determine the normalised natural

frequency (NNF) curves (i) (i.e. i = ).

Step 3: The variation of the NNF curves (i) is plotted against the non-dimensional

roving disc locations ζ.

After applying the aforementioned steps, the crack can be identified and located from

the graphical results by two criteria: (i) the sharp notches in the graphs, and (ii) the

minimum points of the graphs.

To illustrate the application of the these steps for the identification, localisation and

sizing of a crack in rotors, the variation of the normalised natural frequencies (i)

against the normalised roving disc locations (ζ) of the four cases are presented in the

subsequent section.

Numerical Results and Analyses 5.6

The first set of the numerical results presented are for Case 1 for which the crack is

located at Γ = 0.3 and has depth ratios between μ =0.3 and 1.0. The normalised natural

frequency curves for the first four modes are shown in Figure ‎5.6. The figure shows that

the location and size of a crack are clearly indicated by all the non-dimensional

normalised natural frequency curves of all the modes. The cracks are identified and

located by means of the notch-shaped sections of the normalised natural frequency

(NNF) curves. The sections of the NNF curves that are neither near nor at the crack

location are smooth curves. For modes 1, 2 and 4, Figure ‎5.6a, b and c, respectively,

show sharp notches with minimum values of i at the correct crack location. For mode

3, the notched and pointed segments of the curves occur at the correct crack location but

the minimum values of 3 occur at smooth sections of the curves. Thus, for Case 1, this

CHAPTER 5

131

proposed technique is feasible to identify, locate and determine the size of a crack from

most of the first four modes of bending vibrations.

In Case 2, a crack is induced at = 0.5 from the left inner bearing, where a modal node

of the even modes (second and fourth) is located. In this case, the robustness of the

proposed method is demonstrated to identify and determine the location and sizes of the

crack in rotor systems. The sharp notches and the minimum points of the curves in

Figure ‎5.7 indicate that the first and third normalized natural frequency curves λ1, and

λ3, respectively, have determined precisely the location and depth of the crack in the

rotor, whereas the second and fourth normalized natural frequency curves λ2, and λ4,

respectively, have not. This is because of the location of the crack at a modal node of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9975

0.998

0.9985

0.999

0.9995

1

1

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9975

0.998

0.9985

0.999

0.9995

1

2

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9985

0.9988

0.9991

0.9994

0.9997

1

3

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9992

0.9994

0.9996

0.9998

1

4

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)

Figure 5.6: First four theoretical NNF curves of the stationary cracked rotor with a


nd mode, (c) 3

rd mode, (d)

4th

mode.

CHAPTER 5

132

the even modes. Although, the λ2 curves have not clearly identified and located the

crack, they obviously indicate the sizes of the cracks, in comparison with the λ2 curves

shown in Figure ‎5.5b in which the identification of the sizes of a crack are barely

perceptible. For the fourth mode λ4, the sharp notch in Fig.6d identifies and locates the

crack at the middle of the rotor even though the notch points are not the minimum

points of the curves. For all modes, comparing Figure ‎5.7a, b, c and d with Figure ‎5.5a,

b, c and d, respectively, it is seen that the crack sizes are identified by all sections of the

NNF curves whether they are at the notched or smooth sections of the curves.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9984

0.9988

0.9992

0.9996

1

1

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9996

0.9997

0.9998

0.9999

1

2

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9995

0.9997

0.9999

1

3

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9988

0.9992

0.9996

1

4

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)

Figure 5.7: First four theoretical NNF curves of the stationary cracked rotor with a crack

of different depth ratios at = 0.5: (a) 1st mode, (b) 2

nd mode, (c) 3

rd mode, (d) 4

th mode.

CHAPTER 5

133

For Case 3, the location of the crack is moved to about two-thirds of the rotor length

(Γ= 0.7) from the left bearing, which is close to a node of the third and fourth mode

shapes. The results in Figure ‎5.8 show the suitability of the proposed approach to

identify and determine the location and sizes of a crack in a rotor. The figure also

indicates that the reduction in the normalised natural frequency ratios is more

pronounced when the crack occurs near or at the node of the associated mode shape.

Cracks with various depth ratios μ have been identified and localised at the exact crack

locations by sharp discontinuities in all the λ1, λ2, λ3 and λ4 curves. In addition, the

minimum points of the λ1, λ2, and λ4 curves occur at their notch tips, which are the

correct crack locations. Even though the minimum values of the λ3 curves do not occur

at the correct crack location, the sharp-notch tips correctly identify and locate the crack

at Γ= 0.7.

As shown, the second and fourth normalized natural frequency curves λ2 and λ4 in Case

2 Figure ‎5.7b and d, respectively, as well as the third normalized natural frequency

curves λ3 in Case 1 Figure ‎5.6c and Case 3 Figure ‎5.8c have indicated that the minimum

values of the curves do not occur at the exact crack location. This is because the crack is

located at (or close to) the nodes of the mode shapes. To explore this further, Case 4 has

been studied, in which a crack is induced at Γ= 0.4 where the effect of the nodes is

almost negligible for modes one to four (i.e. at a location where the natural frequencies

are not significantly affected). The results of Case 4 in Figure ‎5.9 show that the location

and sizes of the crack are clearly observable in the first four normalized natural

frequency curves except for the second normalized natural frequency curves (see

Figure ‎5.9b) due to the closeness of a node of the second mode to the crack location.

CHAPTER 5

134

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9975

0.9985

0.9995

1

1

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9972

0.9979

0.9986

0.9993

1

2

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.9985

0.999

0.9995

1

3

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9994

0.9996

0.9998

1

4

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)



nd mode, (c) 3

rd mode, (d) 4

th mode.

CHAPTER 5

135

Validity of the NNF Curves Technique for Few Disc Positions 5.7

As can be seen, the NNF curves technique is based on the natural frequencies of intact

and cracked rotors at each interval of traversing the roving disc along the shaft. In other

words, not only natural frequencies govern this technique but also spatial interval of

traversing the roving disc along a shaft. The smaller the spatial interval of traversing the

roving disc, the more precise and sharper the NNF curve will be in identifying and

locating the crack. In the previous sections, a 10 mm spatial interval of traversing the

roving disc has been used to investigate the proposed technique theoretically. In

practice, this spatial interval value for testing long rotors, such as ship propeller shafts,

backward centrifugal fan shafts etc., will be tedious and time consuming. However, the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.998

0.9985

0.999

0.9995

1

1

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9975

0.998

0.9985

0.999

0.9995

1

2

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9995

0.9997

0.9999

1

3

= 0.3

= 0.5

= 0.7

= 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9993

0.9996

0.9999

1

4

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)



nd mode, (c) 3

rd mode, (d) 4

th mode.

CHAPTER 5

136

NNF curves technique can be applied through a large spatial interval with accuracy

close to the accuracy of using the small spatial interval. This can be accomplished by

using the coarse-fine mesh approach which is similar to that applied in finite element

analysis. Firstly, a coarse spatial interval of 5 points is used for the roving disc locations

along the rotor. The results of this coarse grid will give an approximate location for the

structural fault. Then, a fine grid of say 5 points is again used for the roving disc

locations around the approximate location that has been obtained using the coarse grid

of the first 5 points. The results of this coarse-fine mesh approach show that the

approach is quite feasible.

Figure ‎5.10 shows the variation of the first two NNF curves, which are the plots of λ1

and λ2 against the dimensionless locations of the roving disc ζ for 5 disc positions. The

results indicate that λ1 and λ2 can identify cracks located at ζ = 0.3 based on the coarse

grid results.

A more accurate location of the crack can be achieved by repeating the identification

process in the locality of the singular region that has been identified by the coarse grid.

Again, 5 roving disc positions are used in the windowed region in Figure ‎5.10. The

results for this region are shown in Figure ‎5.11. It is clearly seen from these Figures that

the crack is clearly identified and located in the rotor at ζ = 0.3.

0.1 0.3 0.5 0.7 0.90.9975

0.998

0.9985

0.999

0.9995

1

1

= 0.3

= 0.5

= 0.7

= 1.0

0.1 0.3 0.5 0.7 0.90.9975

0.998

0.9985

0.999

0.9995

1

2

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

Figure 5.10: Variation of the first and second NNFCs against the locations of the

roving disc at only 5 points along the shaft length. Cracks at location Γ = 0.3. (a) 1st

mode, (b) 2nd mode.

CHAPTER 5

137

Experimental Testing and Validation 5.8

Experimental Test Rig and Instrumentation 5.8.1

Figure ‎5.12 shows the photographic representation of the experimental test rig which is

located in the Dynamics Laboratory of the University of Manchester. The rig, which

represents a stationary rotor-disc-bearing system, consists of a uniform shaft and a disk

of 1 kg mass (see Table ‎5.1 for dimensions and materials). The shaft is simply-

supported by two ball bearings mounted in stiff pedestals (Figure ‎5.12a). The disc has

four grooves that are machined in its bore and are arranged in the bore at 90o interval as

shown in Figure ‎5.12b. These grooves enable the disc to be moved over sensors along

the shaft. The disc is used as a roving mass and it is traversed at a 40 mm spatial interval

along the length of the shaft. A crack with 0.5 mm width and of depths 3 and 5mm (i.e.

μ = h/R = 0.3 and 0.5) at locations Γ = 0.3, 0.5 and 0.7 from the bearing that is closer to

the motor has been cut in the shaft (see Figure ‎5.12c) using a laser cutting machine.

The vibration response of the rig at each disc location has been acquired by using

piezoelectric ceramic sensors made from Lead-Titanate-Zirconate (PZT). At each axial

location, four PZT sensors with dimensions 5 mm x 3 mm were mounted

circumferentially (Top, Bottom, Right, and Left) in 90-degree angular positions around

0.2 0.26 0.3 0.34 0.380.9975

0.998

0.9985

0.999

0.9995

1

1

= 0.3

= 0.5

= 0.7

= 1.0

0.2 0.26 0.3 0.34 0.380.9975

0.998

0.9985

0.999

0.9995

1

2

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

Figure 5.11: Variation of the first and second NNF curves agnaist the locations of the

roving disc at 5 points in the windowed sections shwon in Fig. 10. Cracks at location Γ =

0.3. (a) 1st mode, (b) 2nd mode.

CHAPTER 5

138

the shaft. Each row consists of 24 PZTs which are bonded along the shaft at 40 mm

intervals using conductive epoxy.

In this study, only three sensors in each row were used to determine the first four natural

frequencies in the vertical and horizontal planes of the stationary rotor-bearing system.

However, all the sensors will be used in subsequent rotating shaft tests for real-time

measurements of operational deflected shapes of the shaft. The use of PZT sensors has

led to a reduction of the effect of the weight of the sensors and their wires. PZT sensors

operate with good accuracy without amplifier and do not require any amplifiers to

amplify their output signals. The shaft-rotor system has been excited by using an Impact

Hammer (PCB Model: 086C04) and the corresponding responses were measured by the

PZT sensors. All vibration data were acquired via a 16-channel, 16-bit Data Acquisition

Card (NI-PCI6123) and recorded in the PC and processed by LabView Signal Express

Figure 5.12: The experimental rig, PZT sensors and the transverse crack.

CHAPTER 5

139

data acquisition software. The accuracy of these measurements was also verified by

using Data Physics (Abacus) data acquisition system. In each case, the frequency

response functions (FRFs) of the strain response with respect to the excitation forces

were determined.

Comparisons of Theoretical and Experimental Characteristics 5.8.2

5.8.2.1 Case 1: Crack Parameters [μ, Γ] = [0.5, 0.3]

This case relates to the identification of a crack of depth ratio μ = 0.5 which is located at

Γ = 0.3 in a stationary shaft. Figure ‎5.13 shows the comparisons of the first four

theoretical and experimental NNF curves which are the variations of λ1, λ2, λ3 and λ4

against the locations of the roving disc ζ. The results show that the experimental λ3 and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9996

0.9997

9998

0.9999

1

1

= 0.5 (Th.)

= 0.5 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99965

0.99975

0.99985

0.99995

1

2

= 0.5 (Th.)

= 0.5 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9997

0.9998

0.9999

1

3

= 0.5 (Th.)

= 0.5 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9988

0.9991

0.9994

0.9997

1

4

= 0.5 (Th.)

= 0.5 (Exp.)

(a) (b)

(c) (d)

Figure 5.13: Comparison of the theoretical and experimental NNF curves for a

cracked shaft with μ = 0.5 at location Γ = 0.3: (a) 1st mode, (b) 2nd mode, (c) 3rd

mode, (d) 4th mode.

CHAPTER 5

140

λ4 curves are relatively identical to the theoretical curves. Also, the experimental λ2

curve shows very close agreement with the theoretical λ2 curve. However, while the

experimental λ1 curve shows a similar trend to the theoretical λ1 curve, the effect of

measurement noise is more pronounced on the experimental λ1 curve than the effect of

numerical noise on the theoretical λ1 curve. This higher noise level in the experimental

λ1 curve is due to the more dominant effect of measurement noise, which comes from

the electrical measurement instrumentation including the impact hammer force

transducer and PZT sensors. Nevertheless, the results show that the crack can still be

reasonably identified and localised by using the first NNF curve λ1. In any case, the

other NNF curves clearly identify and locate the structural fault in the cracked rotor

without any ambiguity. Overall, the experimental results have proved that the proposed

technique is feasible and applicable to identify and localise structural faults such as

slots, which represent open cracks, in stationary rotor-bearing systems.

5.8.2.2 Case 2: Crack Parameters [μ, Γ] = [0.3, 0.3]

In the previous case, the experimental results of the NNF curves have shown reasonable

identity to the theoretical results when the crack depth ratio reaches μ = 0.5 which is

relatively high. Therefore, in this case, the proposed technique has been tested

experimentally by decreasing μ to 0.3 at the same location Γ = 0.3. This is to show the

validity and practicability of the NNF curves technique for crack identification in a

cracked stationary rotor with a small crack depth ratio. The comparisons of the

experimental and theoretical results of this case in Figure ‎5.14 show that the

experimental NNF curves have relatively similar trends to the theoretical NNF curves.

However, the figure shows that there are significant deviations between the theoretical

and experimental λ1 and λ3 curves. In the case of the λ1 curves, the discrepancies are due

to random numerical and experimental noise effects. In the case of the λ3 curves, the

discrepancies are due to the nodes of the third mode of vibration being close to the crack

location Γ = 0.3. Since the roving disc has very little (or no) effect at a nodal point, then

the responses in the λ3 curves when the crack is located at Γ = 0.3, which is close to the

nodal point Γ = 0.33, is not as strong as in the other modes. Nevertheless, the figure

shows that there is less deviation between the theoretical and experimental λ2 and λ4

curves. Furthermore, both the theoretical and experimental λ2 and λ4 curves show a

clearer and unambiguous crack identification and localisation than the λ1 and λ3 curves.

The minimum values and the sharp notches in the λ2 and λ4 curves occur at the correct

CHAPTER 5

141

crack location. In addition, the λ1 curves have a random and wavy behaviour which are

due to numerical noise (in the case of the theoretical curves) and experimental noise (in

the case of the experimental curves), and the minimum values of the experimental

curves deviate slightly from the correct crack location.

The figure also shows two minimum values of the theoretical λ3 curve are

approximately the same in value. One of these minimum values occurs at a section of

the curve which is a sharp notch whereas the other occurs at a smooth and rounded

section of the curve. The minimum value at the notched section of the theoretical λ3

curve correctly locates the crack. However, for the experimental λ3 curve, the minimum

value occurs at the wrong location of the crack. In fact, the section of the curve that has

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9998

0.99985

0.9999

0.99995

1

1

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9999

0.99992

0.99994

0.99996

0.99998

1

2

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.9999

0.99992

0.99994

0.99996

0.99998

1

3

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.99997

0.99998

0.99999

1

4

= 0.3 (Th.)

= 0.3 (Exp.)

(a) (b)

(c) (d)



mode, (d) 4th mode.

CHAPTER 5

142

the sharpest notch, which coincides with the correct crack location, does not have the

minimum value. This is because mode 3 has a nodal point close to the location Γ = 0.3

of the crack. Therefore, in both the theoretical and experimental cases, the λ3 curves do

not show the minimum values at Γ = 0.3.

Although, the minimum value of the experimental λ3 curve occurs at the wrong crack

location and is much less than the minimum value of the theoretical λ3 curves, sharp

troughs occur in both curves at the right crack location Γ = 0.3. Thus, sharp troughs of

the NNF curves can be used to identify the correct crack location; the rounded troughs

which have the minimum values of λ are not the crack locations. Overall, fairly close

agreement between the experimental and theoretical results proves the feasibility and

practicability of the NNF curve technique for the identification and localisation of

cracks in rotors.

5.8.2.3 Case 3: crack parameters [μ, Γ] = [0.3, 0.5]

In this case, the crack depth ratio has remained as μ = 0.3 but the crack location has

been moved to the middle of the shaft (Γ = 0.5) which is exactly at the location of a

node of the second and fourth modes. The first four experimental NNF curves in

Figure ‎5.15 show fairly similar trends to the first four theoretical NNF curves λ1, λ2, λ3,

and λ4. The theoretical and experimental λ1 curves (see Figure ‎5.15a) are very chaotic

due to numerical and experimental noise which is affected by the size and location of a

crack. It is obvious that the λ1 curves cannot be used to identify and locate the crack

unambiguously. Although these noise effects have less impact on λ2 and λ4, these curves

do not provide unambiguous identification of the crack. This is not surprising since the

crack is located at a node of the second and fourth modes. In this particular case,

Figure ‎5.15 shows clearly that it is only the λ3 curves that provide unambiguous

identification and localisation of the crack. This is then confirmed by the symmetry of

the λ2, λ3, and λ4 NNF curves around the crack location, which appears solely in the case

of a crack at middle of a shaft.

CHAPTER 5

143

5.8.2.4 Case 4: crack parameters [μ, Γ] = [0.3, 0.7]

This case is similar to Case 2 except that the crack is located at the symmetrically

opposite location of Γ = 0.7. Thus, Case 4 for [μ Γ] = [0.3 0.7] whereas Case 2 is for [μ

Γ] = [0.3 0.3]. This case has been performed in order to investigate the similarities and

differences that are possible when the NNF technique is used to identify and locate

cracks of identical severity but at symmetrically opposite locations in a shaft. Therefore,

it is interesting to compare the results of Case 4 presented in Figure ‎5.16 with the results

of Case 2 presented in Figure ‎5.14. For the λ1 curves, Figure ‎5.16a shows that when the

crack is located at the symmetrical location of Γ = 0.7, the numerical and experimental

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99985

0.9999

0.99995

1

1

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99998

0.99999

1

2

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9998

0.99985

0.9999

0.99995

1

3

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99994

0.99996

0.99998

1

4

= 0.3 (Th.)

= 0.3 (Exp.)

(a) (b)

(c) (d)



mode, (d) 4th mode.

CHAPTER 5

144

noise effects are greater than when it is located at the near symmetrical location of Γ =

0.3. In the theoretical case, this difference is probably due to the fact that the finite

element analysis (FEA) technique is an approximate iterative technique. In addition, the

boundary conditions used in the FEA are that the motion of the left end of the shaft is

constrained along x, y and z axis direction whereas the motion of the right end of the

shaft is constrained only in the y and z axis directions. However, the shaft is free to

move along the x-axis (axial) direction. Thus, the FEA can produce unsymmetrical

results for a system that is geometrically symmetric but which becomes unsymmetrical

with the imposition of the boundary conditions. In the experimental case, the difference

is due to the roving sensor impact test method used. In this method, the shaft is

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99985

0.9999

0.99995

1

1

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9999

0.99992

0.99994

0.99996

0.99998

1

2

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99992

0.99994

0.99996

0.99998

1

3

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99965

0.99975

0.99985

0.99995

1

4

= 0.3 (Th.)

= 0.3 (Exp.)

(a) (b)

(c) (d)



mode, (d) 4th mode

CHAPTER 5

145

impacted at a fixed point while measurements are made at different locations. The

impact (excitation) point is closer to the near symmetrical location. Thus, the impact

energy reaching the far symmetrical locations is less. Consequently, the measurement

noise is greater at the far symmetrical locations. However, in the λ2, λ3, and λ4 curves for

the higher modes of vibration, the effects of these numerical and measurement noises

are much less. As in Case 2, the sharp notches observed in the λ2, λ3, and λ4 curves

clearly identify and locate the crack at the correct location.

Summary of the Experimental Cases 5.8.3

The four cases investigated in this section have focused mainly on cracks of μ = 0.3

which is equivalent to 30% of the shaft radius. This is the minimum crack depth ratio

that has been found to give clear and unambiguous experimental crack identification

and localisation in the present study. Any crack of greater depth than this gives even

much clearer results as shown in Case 1 (see Figure ‎5.13) for which μ = 0.5. In addition,

the correlation between the theoretical and experimental characteristics for the NNF

curves λ1, λ2, λ3 and λ4 is very high when μ > 0.3 as demonstrated in Figure ‎5.13 for

Case 1. In general, the experimental results have shown that the NNF curves technique

can be used to identify and localise the crack in stationary rotor systems.

Conclusions 5.9

In this chapter, a technique for the identification, localisation and determination of the

severity of cracks in stationary rotor-bearing systems is proposed. The numerical

simulation results show that the variation of the first four normalised natural frequency

curves against non-dimensional location of a roving disc is quite sufficient to identify,

locate and determine the size of the crack in rotors. The unique characteristics of the

NNF curves, which are utilised for crack identification and localisation, are the sharp

notches in the NNF curves at the crack location and the rounded shapes at the intact

locations. In addition, the proposed technique is effective to identify and localise a crack

at any locations. The theoretical and experimental results have shown that the proposed

technique can only clearly identify and locate cracks of crack depth ratios greater than

5%. In general, the reasonable match between the results of the theoretical and

experimental investigation confirms clearly the applicability and practicability of the

proposed technique to identify and localise a crack in stationary rotor bearing systems.

CHAPTER 6

146

CHAPTER 6

Vibration-based Crack Identification and Location in Rotors

Using a Roving Disc and Products of Natural Frequency

Curves: Analytical Simulation and Experimental Validation


Journal of Sound and Vibration (under review)

CHAPTER 6

147

Vibration-based Crack Identification and Location in Rotors Using a Roving Disc

and Products of Natural Frequency Curves: Analytical Simulation and

Experimental Validation




Abstract

A new method for the identification and location of cracks in stationary-rotor systems is

proposed. The method, which is called frequency curve product (FCP), is based on the

normalised natural frequency curves (NNFCs) of cracked and intact rotors. The finite

element model of the rotor-disc-bearing system with an open crack and a roving disc is

developed using Bernoulli-Euler beam theory. The reduction of the natural frequencies

of a rotor due to a crack and a roving disc, which is traversed along the rotor to enhance

the dynamics of the rotor near the crack locations, is exploited to produce the NNFCs.

The FCP method uses the first four NNFCs of the rotor and products of the NNFCs to

identify and locate cracks clearly, irrespective of the crack location with respect to the

modal nodes. This method is numerically investigated and validated experimentally.

The numerical and experimental results clearly demonstrate the robustness of the

suggested approach for the identification and location of cracks in stationary rotor

systems.

Keywords: vibration analysis; crack identification; roving disc; open crack; finite

element modelling; rotor dynamics.

Motivation and Background 6.1

The derivation of a robust, accurate and feasible model for cracked shafts has attracted

considerable attention of many scholars for the past four decades. This is because the

presence of cracks in shafts, which may be induced during operation, poses a serious

risk to the performance of rotating machinery and can lead to mechanical failure. If the

crack is not detected early, abrupt failure may occur, and this may lead to catastrophic

accidents such as damage to equipment and potential injury to personnel. Therefore, an

CHAPTER 6

148

early detection of cracks in structures can extend the integrity of structural components

and improve their safety, reliability and operational life.

In the last three decades many approaches for modelling a crack in shafts have been

introduced by many researchers. These known approaches can be classified into three

categories [136, 137]: local stiffness reduction, discrete spring model and complex

models in two or three dimensions, which includes breathing crack models based on

finite element analysis. The simplest approaches assign a reduced stiffness to the shaft

near the location of the crack where the stiffness reduction is proportional to the crack

depth ratio to a certain extent. This reduction is normally represented by reducing the

second moment of area of the cracked shaft cross-section [23, 67, 138]. The stiffness

reduction in the fixed angular direction may remain constant, which gives rise to linear

equations of motion for the cracked shaft [15, 139]. This behaviour corresponds to the

well-known fully open cracks.

Chondros and Dimarogonas [123] have developed a theory for lateral vibration of

cracked Bernoulli-Euler continuous beams with single or double-edge open cracks.

They have determined the crack size and location from measured information of the

cracked beam system by carrying out an inverse problem. In other cases, crack location

and stiffness reductions of a beam have been studied by modelling a crack as a linear

spring. Loya and Rubio [131] have applied perturbation method to obtain the natural

frequencies for bending vibrations of a cracked Timoshenko beam with simple

boundary conditions (BCs). The cracked beam is modelled as two segments that are

connected by two massless springs (one extensional and the other rotational). They have

shown that the method provides simple expressions for the natural frequencies of

cracked beams and it gives good results for shallow cracks.

Lin [125] has proposed a method to identify crack depth and location in a simply

supported Timoshenko beam. This method has been derived according to a

mathematical model in which the crack is modelled as a rotational spring connecting

two separate beams. An analytical method to determine the natural frequency of cracked

Bernoulli-Euler beams in bending vibration has been derived by Fernandez-Saez and

Navarro [128]. The influence of the crack is based on the mathematical model in which

the cracked beam is represented by an elastic rotational spring connecting the two

segments of the beam at the cracked section. Closed-form expressions for the

approximate values of the natural frequency of cracked Bernoulli-Euler beams in

CHAPTER 6

149

bending vibration are given. The results obtained agree with those numerically obtained

by the finite element method.

However, the results of many experimental investigations have indicated that the

mechanism of opening and closing of a crack periodically in rotating shafts should be

taken into account during shaft spinning [136]. This mechanism is the so-called

breathing crack, which manifests itself in periodical stiffness changes with shaft

spinning. The breathing crack model has been introduced relatively in simple models by

Mayes and Davies [76], and in more sophisticated complex models, which correspond

to the third category, by Ostachowicz and Krawczu [140], and Darpe et al [62]. Mayes

and Davies [76] have investigated the vibration behaviour of a rotor by introducing a

model for breathing crack, in which the stiffness change of a cracked rotor is sinusoidal,

where the crack opens and closes gradually due to external loads. Recently, two new

time-varying functions of the breathing crack model have been developed by Al-

Shudeifat and Butcher [18, 21] who applied these functions for a more exact evaluation

of the stiffness changes of a cracked shaft.

The finite element method (FEM) is a popular investigative technique that has enabled

researchers to study the dynamic behaviour of a cracked rotor using complex models.

For instance, Papadopoulos and Dimaroganas [53, 54] have studied the dynamics of a

rotating shaft with an open transverse surface crack. They represented the local

flexibility due to the presence of a crack by a matrix of size 6×6 for six degree-of-

freedom in the cracked element. Sekhar and Balaji [59] have used the finite element

(FE) technique to detect a slant crack on a rotor-bearing system based on the bending

vibration. They stated that a general reduction occurs in all modal frequencies with an

increase in the depth of a crack. Hall and Potirniche [138] has developed a new three-

dimensional finite element with an embedded edge crack to model local stiffness in

cracked structures. The method used is based on the deviation of a modified stiffness

matrix considering that the presence of cracks in structures changes the element

flexibility. The analytical results showed that the new three-dimensional element is

applicable to represent cracks in three-dimensional structures.

Kulesza and Sawicki [118] have proposed a new rotor crack detection method based on

control theory. Indicators of the cracked rotor in the form of two auxiliary state

variables have been implemented into the FE model of the rotor-bearing system. Wang

et al [141] have used various order tracking techniques such as computed order tracking,

CHAPTER 6

150

Vold-Kalman filter order tracking and Gabor order tracking, in a finite element model

of a cracked rotor. They used this technique to calculate the response of a cracked rotor

under varying rotational speed conditions. An analytical expression for measuring

damage in beams with notch like non-propagating cracks has been presented by Dixit

and Hanagud [142]. The presented analytical expression combines the strain energy

with the depth and location of damage using modes and natural frequencies of damaged

beams to calculate the strain energy. Vaziri and Nayeb-Hashemi [143] have studied the

local and frictional energy losses at the crack location due to the plasticity at the crack

tip and the interaction between the crack surfaces, respectively, to evaluate the

vibrational characteristics of turbo-generator shafts with a circumferential crack of

various lengths. The study showed that the total energy loss in the circumferentially

cracked shaft may be less dominant than one of the energy losses associated with the

amplitude of the applied Mode III stress intensity factor.

Although many studies have been carried out by researchers on cracked rotors, in order

to derive the dynamic characteristic of a cracked shaft, the diagnosis of cracked rotors

remains problematic. Therefore, due to the vital importance of cracked rotor diagnosis,

the dynamics of cracked rotors must be accurately analysed in order to correctly

identified and locate cracks in rotors. All the investigations in the literature on the

dynamic vibration behaviour of cracked rotors have been carried out by using various

crack models. These crack models are based on the fact that the presence of a crack in

rotors reduces the cross sectional area at the crack location and changes the stiffness of

the rotor. These changes have a crucial effect in decreasing the natural frequencies of

the cracked rotor which have been used as a tool in various ways to detect a crack in a

rotor system. In addition, the reduction of the natural frequencies due to the traversing

of a roving auxiliary mass along cracked beams has been recently employed by Zhong

and Oyadiji [126, 130] for the identification and location of cracks.

In this study a new technique, which is called frequency curve product (FCP) method, is

presented to identify and locate a crack very clearly in stationary-rotor systems. The

proposed technique is based on the normalised natural frequency curves (NNFCs) of

cracked and intact rotors using the principle of roving masses and natural frequency

curves which was introduced by Zhong and Oyadiji [130]. These NNFCs are obtained

from the finite element modelling of the cracked rotor with an open crack through using

a roving disc. This technique is developed in order to firstly, solve the problem of the

disappearance of a crack effect when the crack is close to or exactly at a node of a mode

CHAPTER 6

151

shape. Where, this problem makes it difficult to decide the exact crack location in rotor

systems. Secondly, FCP curves combine the first four NNFCs in a single plot to identify

the crack location clearly. Different pairs of the normalised natural frequencies of

different modes of vibration are multiplied together in order to enhance the

identification and location of cracks. It is shown that this technique identifies the exact

crack location through unifying all the first four natural frequency curves at the

maximum positive value in the plot of both the numerical experimental results.

Modelling of the Uncracked Rotor 6.2

Equations of Motion 6.2.1

A rotor, which is supported by rigid bearings as a simply-supported beam, consisting of

a shaft and a movable disc of 1.0 kg has been used for the investigations of intact and

cracked rotors. The illustrative diagram and dimensions of the rotor are presented in

Figure ‎6.1 and Table ‎6.1. The rotor, which has a constant cross-section, is divided into

N Bernoulli-Euler beam finite elements, where each element has two nodes with four

degrees of freedom per node: transverse displacement in the x- and y-axes directions

and rotations about the x- and y-axes directions as shown in Figure ‎6.2. The rotor is

assumed to be a stationary rotor.

Table ‎6.1: Numerical model parameters

Parameters Shaft Disc Bearings

Material Stainless steel Aluminium N/A

Young‘s Modulus, Es (GPa) 200 70 N/A

Figure 6.1: The finite element model of the intact and cracked rotor

CHAPTER 6

152

Poisson‘s ratio, vs 0.3 0.33 N/A

Density, ρ (Kg/m3) 7800 2700 N/A

Length, L (m) 1 0.01, 0.02, 0.3, ..., 1 N/A

Outer diameter, Do (m) 0.02 0.2 N/A

Inner Diameter, Di (m) - 0.02 N/A

Thickness, t (m) - 0.04 N/A

Mass, m (Kg) 2.45 1.0 N/A

Bearing stiffness, ( kxx , kyy ) N/A N/A 7 × 107 N/m

Bearing damping, ( cxx , cyy ) N/A N/A 200 N.s/m

The free vibration equation of motion of a rotor which is assumed as simply-supported

at both ends can be defined as

( ) ( ) ( ) ( ) (‎6.1)

where, ( ) ,

- ( ) [

] is the nodal

displacement vector with dimension 4(N+1) × 1 corresponding to the local coordinate

vector for each element , -

, - in Figure ‎6.2. M is the global mass matrix

which contains the mass matrices Me for each element of the rotor and the mass

matrix Md of the disc corresponding to degrees of

freedom , - , - . G and C are global gyroscopic and damping

matrices, respectively. K is the global stiffness

.

u1

v1

X

Y

Z

Node 1

Node 2

θ1

ψ1

e

u2

v2

θ2

ψ2

Ω

Figure 6.2: Schematic view of a finite rotor element and coordinates for an intact and

cracked.

CHAPTER 6

153

matrix which includes the stiffness of intact elements and the stiffness reduction at the

location of a cracked element which are defined in detail in the next section.

Model of the Rotor with an Open Crack 6.3

A single transverse crack, which is perpendicular to the rotational axis of the cracked

rotor, is induced on the rotor surface as shown in Figure ‎6.3. The crack is assumed to be

a fully open crack. The status of the crack, which can be fully opened or closed, is

governed by the location and amount of the out-of-balance forces arising in the cracked

rotor. The open crack behaviour manifests when the static deflection due to the rotor

weight is less than the vibration amplitude due to any unbalanced force acting on a rotor

[21, 102]. The modelling of the cracked element and equations of motion of the cracked

rotor are presented in the following sections.

Crack Modelling 6.4

The crack model of Al-Shudeifat and Butcher [18, 21, 75] is used in this study. In this

model, the open transverse crack geometry is modelled as illustrated in Figure ‎6.3. The

crack is assumed to be at an initial angle ϕ with respect to the fixed negative Y-axis at t

= 0. When the shaft rotates; the angle of the crack relative to the negative Y-axis

changes with time to ϕ + Ωt (see Figure ‎6.3a.)

(b) (a)

Figure 6.3: A cracked element cross-section; (a) Rotating, (b) Non-rotating; the

hatched part defines the area of the crack segment [18, 21, 75].

CHAPTER 6

154

The stiffness matrix of the cracked element can be written in a form similar to that of a

asymmetric rod in space as [66].

[ ( ) ℓ ( ) ( ) ℓ ( )

( ) ℓ ( ) ( ) ℓ ( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

( ) ℓ ( ) ( ) ℓ ( )

( ) ℓ ( ) ( ) ℓ ( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( ) ]

(‎6.2)

where, is the stiffness matrix of the cracked element, ℓ is the cracked element

length. The second moments of area and about the centroidal and axes are

derived, respectively, as [18, 21, 75, 115]

( ( )) (‎6.3)

and

( ( )) (‎6.4)

where, 2)( 2

1eAIII yx , 2)( 2

12 eAIII yx and I3 = - I2 are constant values

during the shaft rotation.

. ( ) ( )√ ( )/ (‎6.5)

and

( ( ))

(‎6.6)

Where, A1 is cross-sectional area of the cracked element in the case of an open crack.

√ ( ) is a constant which depends on the crack depth ratio, = h/R is the

crack depth ratio and h is the depth of the crack in the transverse direction of the shaft

with radius R.

As a consequence, the finite element stiffness matrix of the cracked element given in

Equation (‎6.2) can be rewritten as

( ( )) (‎6.7)

where,

CHAPTER 6

155

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎6.8)

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎6.9)

Thus, the FEM equations of motion of the uncracked rotor bearing-system given in

Equation (‎6.1) are rewritten including the effect of the cracked element of an open crack

model as

( ) ( ) ( ) . ( ( ))/ ( ) (‎6.10)

where is the 4 (N+ 1) × 4 (N+ 1) global stiffness matrix derived from the global

stiffness matrix K of the uncracked beams by replacing the uncracked element stiffness

matrix of element i by the cracked element stiffness matrix

. K is another 4(N+1)

× 4(N+1) global stiffness matrix; it has zero elements apart from those at the cracked

element location where the elements are equal to . is the rotor speed, and ϕ is the

crack angle; both variables are equal to zero in this study, according to the assumptions

that the rotor is non-rotating and the crack is always fully open (see Figure ‎6.3b).

Numerical Model 6.5

Figure ‎6.1 shows the finite element model of a simply-supported rotor, which represents

intact and cracked rotors. The dimensions and material properties employed for the FE

simulations are stated in Table ‎6.1. The disc is a roving concentrated mass which is

traversed along the cracked rotor length at 10 mm spatial interval. Hence, the natural

frequencies of the rotor system in a cracked and intact status are computed for each 10

mm spatial interval of the roving disc along the cracked rotor. The equations of motion

of the cracked rotor are developed in the Matlab® programming environment.

CHAPTER 6

156

Investigation Procedures 6.5.1

The numerical investigations have been carried out on six numerical cases by using the

analytical cracked model shown in Figure ‎6.1. In each of these cases, cracks of 0.5 mm

width, wc = 0.5 mm, of various depth ratios µ at different locations = Lc/L along the

rotor as shown in Table ‎6.2 have been investigated. These six numerical cases have

been analysed in order to show the reliability and robustness of the suggested method to

identify and localise a crack in symmetrical locations along the rotor and close to or at

nodes of the rotor. This is because some NNFCs are unable to clearly identify and locate

the crack in some of these symmetrical configurations particularly when the crack

location coincides with the location of the nodes of some modes. The cases investigated

are as follows:

Case 1: Highlights the sensitivity of the suggested method for the identification of a

crack at Γ=0.2 where the vibration amplitude is very little and the locations of the nodes

of the first four bending modes of vibration of the rotor are insignificant.

Case 2: Studies a crack located near to the nodes of the third and fourth vertical and

horizontal modes (Γ=0.3).

Case 3: Demonstrates the effect a crack at 40% of the rotor length (Γ=0.4) where the

crack location is not too close to the nodes of modes one to four.

Case 4: Studies a crack located at the node of the second and fourth vertical and

horizontal modes at the centre of the rotor (Γ=0.5).

Cases 5 and 6: Study the behaviour of cracks which are located symmetrically to those

of Cases 2 and 3 beyond the mid-rotor at Γ=0.6 and 0.7 for Cases 5 and 6, respectively,

in order to demonstrate that the suggested method is responsive to symmetrical crack

locations.

Table ‎6.2 : Values of the numerical and experimental cases

Case

No.

Crack

Location

Γ=Lc/L

Disc

mass

(kg)

Ratio of disc

mass to shaft

md/ms

Crack depth ratios

μ

Roving Disc Location

ζ = Ld/L

Numerical Simulation Cases

1 0.2 1.0 40.8% 0.1 0.3, 0.5, 0.7, 1.0 0.01, 0.02,..., 1.0

2 0.3 1.0 40.8% 0.1 0.3, 0.5, 0.7, 1.0 0.01, 0.02,..., 1.0

3 0.4 1.0 40.8% 0.1 0.3, 0.5, 0.7, 1.0 0.01, 0.02,..., 1.0

4 0.5 1.0 40.8% 0.1 0.3, 0.5, 0.7, 1.0 0.01, 0.02,..., 1.0

5 0.6 1.0 40.8% 0.1 0.3, 0.5, 0.7, 1.0 0.01, 0.02,..., 1.0

CHAPTER 6

157

Methodology of Crack Identification 6.6

In this section, a new approach, which is called frequency curve product (FCP)

technique, is developed for the identification and location of a crack in rotor systems.

This technique is inspired by the orthogonality principle of the normalised natural

frequency (i) curves of the cracked rotor. This approach is implemented according to

the following steps:

Step 1: Implementation of Normalised Natural Frequency Approach

1. The natural frequencies fci of the cracked rotor are divided by the corresponding

natural frequencies foi of the intact rotor for the same roving disc locations in

order to obtain the natural frequency ratios (βi = fci/foi ) of each mode.

2. The natural frequency ratios βi of each mode are divided by the maximum

natural frequency ratio of each mode to determine the NNFCs λi (i.e.

⁄ ).

3. For each mode, and a specified crack depth ratio , let the normalised natural

frequency value at the zero location ( = 0) of the roving disc act as a pivot, and

subtract its value from the normalised natural frequencies corresponding to the

roving disc locations in order to obtain the shifted NNFCs Ψm,μ of a mode (m =

1, 2, 3 and 4) at an assigned crack depth ratio (μ) defined as,

(‎6.11)

where 0,μ is the normalised natural frequency at the pivot point at which ζ =

Ld/L= 0 for each mode, i,μ is the normalised natural frequency corresponding to

the roving disc locations, μ is the crack depth ratio and i= ζ = 0, 0.01, 0.02,

0.03, …, 1.

4. The next step is to form products of the NNFCs. This is carried out by

multiplying the shifted normalised natural frequency curve. Each result is

normalised by dividing by its maximum value in order to obtain non-

dimensional frequency curve products (ζi,m) which are defined as

6 0.7 1.0 40.8% 0.1 0.3, 0.5, 0.7, 1.0 0.01, 0.02,..., 1.0

Experimental Cases

7 0.3 1.0 40.8% 0.3, 0.5 0.04, 0.08,…, 1.0

8 0.5 1.0 40.8% 0.3 0.04, 0.08,…, 1.0

9 0.7 1.0 40.8% 0.3 0.04, 0.08,…, 1.0

CHAPTER 6

158

( ) (‎6.12)

where m = 2, 3, 4, i = 1, 2, 3 or 4

5. The threshold positive values of ζi,m (i.e. ζi,m > 0) are plotted for each pair of

modes against the non-dimensional roving disc locations for each crack depth

ratio.

6. This step unifies all the non-dimensional frequency curve products ζi,m (i.e. ζ1,2,

ζ1,3 and ζ1,4) or ζ2,m (i.e. ζ2,1, ζ2,3 and ζ2,4) or ζ3,m (i.e. ζ3,1, ζ3,2 and ζ3,4) or ζ4,m

(i.e. ζ4,1, ζ4,2 and ζ4,3) for each crack depth ratio in a single plot under the

designation ζ1(234). This is determined by multiplying the non-dimensional

frequency curve products ζ1,2, ζ1,3 and ζ1,4 together according to the following

equation.

( ) (‎6.13)

j,k,l = 1, 2, 3, 4

e.g. ( )

and

( ) ……and so on

Numerical Simulations and Results 6.7

The merits, feasibility and robustness of the frequency curves product (FCP) method for

identifying and locating cracks in rotors are herein investigated for cracked rotors of

various crack depths, symmetrical versus asymmetrical crack location and different

roving discs. Each figure of the FCP method contains four subplots, the first three

subplots (i.e. a, b and c) are the results of the implementation of Equation (‎6.12), whilst

the fourth subplot is the result of Equation (‎6.13). The natural frequencies for each

configuration are computed using a MATLAB programming script based on the

analytical equations which are presented in Section ‎6.3. The predictions are carried out

for all the three cases which are summarised Table ‎6.2. From the results, the products of

natural frequency curves are derived following the procedures in Section ‎6.6.

CHAPTER 6

159

Effect of Crack Location and Size 6.7.1

In this section, the efficiency and limitations of the implementation of the FCP method

have been tested for the identification of cracks of various locations and depth ratios by

using a roving disc of mass 1.0 kg.

Case 1

Figure ‎6.4 shows the variation of the FCP curves against the non-dimensional roving

disc locations for the rotors with cracks located at Γ=0.2. The method uses all four

modes in order to produce results for a particular crack depth ratio as shown by Figures

6.4a, b and c for μ = 0.3, 0.5 and 1.0, respectively, and for all crack depth ratios as

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 1

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

(23

4)

= 0.1

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)(d)(d)(d)(d)

Figure 6.4: Frequency curve products (FCP) of the cracked rotor with a crack of

various depth ratios μ at the location = 0.2: (a), (b) and (c) use Equation (6.12);

(d) uses Equation (6.13).

CHAPTER 6

160

shown in Figure ‎6.4d. The crack is identified at the location where the three frequency

curve products ζ1,2, ζ1,3 and ζ1,4 show peak positive values at the same location Γ = 0.2

which is the correct crack location. However, the presence of other peaks in the results

of the ζ1,2, ζ1,3 and ζ1,4 which are computed according to Equation (‎6.12) means that

extra effort is required to identify where the positive peaks of ζ1,2, ζ1,3 and ζ1,4 coincide

which identifies the exact crack location. This extra effort is avoided by using Equation

(‎6.13) which unifies ζ1,2, ζ1,3 and ζ1,4 of a particular μ in a single plot under the

designation ζ1(234) as shown in Figure ‎6.4d. The figure shows that ζ1(234) identifies the

crack location clearly without any ambiguity or uncertainty for all the crack depth ratios

except for μ = 0.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 1

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

1

(23

4)

= 0.1

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)(d)(d)(d)(d)


various depth ratios μ at the location = 0.3: (a), (b) and (c) use Equation (6.12);


CHAPTER 6

161

Case 2

In this case, numerical simulations have been carried out on the rotor with an open crack

of various depth ratios () at the location = 0.3 from the left bearing. The results of

this case in Figure ‎6.5 show that all the frequency curve products ζ1,2, ζ1,3 and ζ1,4 have

peaks at the same location Γ= 0.3 for the crack depth ratios μ = 0.3, 0.5 and 1.0 but with

non-coincidental peaks at other locations. The results of the composite frequency curve

product ζ1(234) in Figure ‎6.5d show that the problem of non-coincidental peaks has no

longer any effect on the correct identification and location of cracks. The effect of the

crack being located close to a node of a mode of bending vibration is considerably

suppressed.

Case 3

In this case the crack has been moved to = 0.4 from the left inner bearing. This

location is close to the nodal positions of modes 2, 3 and 4. The frequency curve

products (FCP) of this case presented in Figure ‎6.6, identifies and locates the crack

clearly at the maximum positive value of the three products of normalised frequency

curves ζ1,2, ζ1,3 and ζ1,4 (see Figures 6.6a, b and c). In the plot, the first four modes are

unified for each crack depth ratio. The correct crack location is identified from the

location where the peaks of the three products of normalised frequency curves coincide.

The figure also shows that the clarity and sensitivity of ζ1,m increases when increases

but the non-coincidental peaks still manifest in the curves of ζ1,2, ζ1,3 and ζ1,4. These

non-coincidental peaks disappear totally in the composite frequency curve ζ1(234) as

shown in Figure ‎6.6d which identifies and localises the crack convincingly without any

ambiguity. This is because of the merit of the FCP method of unifying the maximum

positive values of ζ1,2, ζ1,3 and ζ1,4 at the correct crack location in the composite

frequency curve ζ1(234). Figure ‎6.6d shows that the peaks of ζ1(234) manifest exactly and

explicitly only at the crack location = 0.4.

CHAPTER 6

162

Case 4

Now in this case, open cracks of various depth ratios, which are induced in the rotor at

= 0.5 from the left bearing, have been investigated. It is known that the system natural

frequencies are not affected, if a crack is located at modal nodes. That is, the

identification of the crack location in this case is difficult to achieve. However, the

application of the FCP method on the results of the simulation has identified and

localised the crack location with reasonable accuracy. These results in Figure ‎6.7 show

that the peaks of ζ1,2, ζ1,3 and ζ1,4 somewhat coincide at the correct crack location.

However, each of the frequency curves products ζ1,2, ζ1,3 and ζ1,4 has many peaks, the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 1

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

(23

4)

= 0.1

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)(d)(d)(d)(d)


various depth ratios μ at the location = 0.4: (a), (b) and (c) use Equation (6.12); (d)

uses Equation (6.13).

CHAPTER 6

163

overlay of these curves looks very chaotic, and identifying where the peaks of the three

frequency curves coincide is very challenging. This chaotic behaviour is due to the

location of the crack at the middle of the rotor, which coincides with the nodes of modes

2 and 4 of bending vibration of the rotor. This chaotic behaviour makes it very difficult

to identify the crack locations. In contrast, the results of the curves ζ1(234) in Figure ‎6.7d

identify and localize the cracks with high accuracy, clarity and without any chaotic

behaviour irrespective of how small the crack depth ratio μ is. This unique feature of the

composite frequency curve ζ1(234) facilitates the identification and location of cracks

along the rotor without any ambiguity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 1

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

(23

4)

= 0.1

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)(d)(d)(d)(d)




CHAPTER 6

164

Effect of Symmetrical Crack Location 6.7.2

This section deals with Cases 5 and 6 in which cracks are located at = 0.6 and 0.7

along the rotor respectively. The crack locations for these two cases are symmetrical

locations to crack locations at = 0.3 and 0.4 for Cases 2 and 3, respectively.

Case 5

Figure ‎6.8 shows the variation of the frequency curve products (FCP) with the non-

dimensional roving disc locations of a simply-supported rotor with cracks located at =

0.6. The results show that the peaks of ζ1,2, ζ1,3 and ζ1,4 are unified together clearly at

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 1

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

(23

4)

= 0.1

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)(d)(d)(d)(d)




CHAPTER 6

165

the crack location but with other peaks occurring at other locations. On the other hand,

the clarity and the crack identification accuracy are considerably increased and the non-

coincidental peaks totally vanish in the results of the composite frequency curves

products ζ1(234) in Figure ‎6.8d. The results confirm the accuracy and consistency of the

proposed method for the correct detection and location of cracks. It can be seen that

Figure ‎6.8 for Γ=0.6 is practically the mirror image of Figure ‎6.6 for Γ=0.4.

Case 6

In this case, the crack is located at = 0.7 from the left inner bearing and is in a

symmetrical location to the crack at = 0.3 in Case 2. Figure ‎6.9 shows the results of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 1

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1

(23

4)

= 0.1

= 0.3

= 0.5

= 0.7

= 1.0

(a) (b)

(c) (d)(d)(d)(d)(d)

Figure 6.9: Frequency curve products (FCP) of the cracked rotor with a crack of various

depth ratios μ at the location = 0.7: (a), (b) and (c) use Equation (6.12); (d) uses

Equation (6.13).

CHAPTER 6

166

the application of the FCP approach on this case. The figure indicates that the FCP

approach has maintained the same consistency for the identification of the crack at the

location where the peaks of the three frequency curve products ζ1,2, ζ1,3 and ζ1,4

coincide. Also, the composite frequency curves ζ1(234) in Figure ‎6.9d have maintained at

the same unique features of clarity and crack detection accuracy at all crack depth ratios

μ except for μ = 0.1 which shows peaks at other locations along the rotor. This lack of

uniqueness for μ = 0.1 is due to the fact that small crack depths have very little effect on

the natural frequencies of the rotor. Overall, the consistency and accuracy of the FCP

method, which is illustrated by all the cases, suggests that the developed approach can

be considered a unique solution for the identification and location of a crack in rotor

systems.

Spatial Interval Influence 6.8

Up to this point, investigations of the proposed method have focused on crack depth

ratios, μ, and locations, Γ, of a crack along the shaft. In fact, these crack physical

parameters are not only parameters that govern the accuracy and application of the FCP

method. The spatial interval of moving the disc along the shaft is another parameter that

has a crucial influence on the identification of a crack at the exact crack location. It has

been found that when the spatial interval decreases, the accuracy of the crack

identification increases.

In the section on numerical simulations, the disc was traversed at 10 mm spatial

intervals along the shaft which gives a total of 100 disc locations for a shaft of 1 m

length. Basically, 10 mm spatial interval and less is easy to apply theoretically, but in

practice, this spatial interval value will be tedious and time consuming, particularly, for

testing long rotors. However, the proposed method can be applied using a large spatial

interval with accuracy close to the accuracy of using the small spatial interval. This can

be accomplished by using the coarse-fine mesh approach with a coarse spatial interval

of say 10 points; like the application of the finite element method. From the results of

this coarse grid, an approximate location for the structural fault will be identified.

Thereafter, only the grids in the identified region are refined to 10 points again to

determine the exact crack location.

In the experimental part of this work, a 40 mm spatial interval (i.e. only 25 points along

the shaft) has been used instead of 10 mm spatial interval (i.e. 100 points along the

CHAPTER 6

167

shaft) which was implemented in the numerical simulation section. The experimental

results of using a 40 mm spatial interval give reasonable identification and location of

the crack at different locations. These results are demonstrated in the subsequent

section.

Experimental Testing and Results 6.9

Experimental Rig and Instrumentation 6.10

The experimental test rig shown in Figure ‎6.10 a, which is located in the Dynamics

Laboratory of the University of Manchester, is used to conduct the experimental tests of

this investigation. The rig consists of a uniform shaft and a disk whose dimensions and

materials are presented in Table ‎6.1. The shaft is supported by two ball bearings

mounted in stiff pedestals (see Figure ‎6.10a) as a simply-supported rotor. To enable the

traversing of the disc over sensors that are bonded to the surface of the shaft, four

grooves, at 90 degrees circumferential interval, were made around the disc bore as

shown in Figure ‎6.10b. The disc is used as a roving mass and it is traversed at 40 mm

spatial interval along the length of the shaft. Transverse crack of depths 3 and 5mm (i.e.

μ = h/R = 0.3 and 0.5) with 0.5 mm width at locations Γ = 0.3, 0.5 and 0.7 were

machined in the shafts as shown in Figure ‎6.11, using an Electrical Discharge Machine

(EDM).

Lead-Titanate-Zirconate piezoelectric (PZT) ceramic sensors were bonded to the

surface of the shaft and were used to acquire the dynamic strain response of the shaft for

(a) (b)

Grooves

Figure 6.10: Experimental rig: (a) Assembly setup; (b) Disc groves.

CHAPTER 6

168

each disc location when the shaft was subjected to transient free vibration. At each axial

location along the length of the shaft, four PZT sensors with dimensions 5 mm x 3 mm

were mounted circumferentially (Top, Bottom, Right, and Left) in 90-degree angular

positions around the shaft. For each of the four angular orientations (0o, 90

o, 180

o and

270o), there were 24 PZT sensors which were bonded to the shaft at 40 mm interval

along the axial direction of the shaft using conductive epoxy (see Figure ‎6.11).

In the current work, only three sensors in each row of PZTs were employed to

determine the frequency response functions (FRFs) of the shaft. From these FRFs the

first four natural frequencies in the vertical and horizontal planes of the stationary rotor-

bearing system were determined. However, all the sensors will be used in subsequent

rotating shaft tests for real-time measurements of operational deflected shapes of the

shaft.

The use of PZT sensors led to a reduction of the effect of the weight of the sensors and

their wires. In addition, PZT sensors do not require any amplifiers to amplify their

output signals and operate with good accuracy. An Impact Hammer (PCB Model:

086C04) was used to excite the rotor and the corresponding responses were measured

by the PZT sensors. A 16-channel-16-bit Data acquisition Card (NI-PCI6123) was used

to acquire and record vibration data in the PC. The vibration data was processed by

LabView Signal Express data acquisition software. Thereafter, the frequency response

functions (FRFs) of each case were computed.

Figure 6.11: The transverse crack and bonded PZTs

CHAPTER 6

169

Experimental Results and Analyses 6.10.1

Case 7

This case is the experimental validation of the proposed FCP method which has been

investigated theoretically in Case 2. In this case, only the crack of μ = 0.3 and 0.5 at the

same location Γ = 0.3 have been experimentally investigated through using 40 mm

spatial interval instead of 10 mm which was used in the numerical investigation in Case

2 (see Figure ‎8.6). Figure ‎6.12 shows the results of the application of the FCP method

on the experimental results of this case for both crack depth ratios μ = 0.3 and 0.5 at

location Γ = 0.3. This figure consists of 4 rows and 3 columns, each row represents a

mode which has been chosen as reference to form the FCP (i.e. if mode 1 is a reference,

the product is ζ1,2, ζ1,3 and ζ1,4, if mode 2 is a reference, the product result is ζ2,1, ζ2,3

and ζ1,4 and so on ). The first 2 columns show the results of the application of Equation

(‎6.12) for both μ = 0.3 and 0.5, and the third column is the result of Equation (‎6.13).

The figure shows that the experimental results of the application of both Equation (‎6.12)

and (13) for μ = 0.3 and 0.5 show reasonable agreement with the theoretical results of

Case 2 in Figure ‎6.5. The figure, also, shows that choosing a mode (i.e. mode 1 or 2 or 3

or 4) as a product reference, has considerable effect on the somewhat random behaviour

of the curves at intact locations, and the accuracy of the identification of the crack at the

correct location when using Equation (‎6.12). However, this randomness has no

remarkable influence on the results of Equation (‎6.13) except of a slight deviation from

the crack location Γ = 0.3 when μ = 0.3 (see Fig. 12c1). Nevertheless, the use of all

modes as product reference decreases the ambiguity about the exact crack location.

Generally, the experimental results of the FCP method show that this method is

applicable to identify and locate a crack in stationary rotors.

CHAPTER 6

170

Figure 6.12: Experimental frequency curves product (FCP) of the cracked rotor with a

crack of μ = 0.3 and 0.5 at the location = 0.3: (a), (b) and (c) use Equation (6.12); (d)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.3

4,1

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.5

4,1

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

(12

3)

= 0.3

= 0.5

(a4) (b4) (c4)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.5

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

(23

4)

= 0.3

= 0.5

(a1) (b1) (c1)(c1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.3

2,1

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.5

2,1

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

(13

4)

= 0.3

= 0.5

(a2) (b2) (c2)(c2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.3

3,1

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.5

3,1

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

3

(12

4)

= 0.3

= 0.5

(a3) (b3) (c3)

CHAPTER 6

171

Case 8

In this case, the experiment was performed on a shaft with a crack of μ = 0.3 located at

the middle of the shaft (i.e. Γ = 0.5) which is exactly at a node of the second and fourth

modes. The experimental results of the FCP application for this case (i.e. μ = 0.3 and Γ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

1

(23

4)

= 0.3

(a1) (b1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.3

2,1

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

2

(13

4)

= 0.3

(a2) (b2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.3

3,1

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

3

(12

4)

= 0.3

(a3) (b3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.3

4,1

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

4

(12

3)

= 0.3

(a4) (b4)

Figure 6.13: Experimental frequency curves product (FCP) of the cracked rotor

with a crack of μ = 0.3 at the location = 0.5: (a), (b) and (c) use Equation (6.12);


CHAPTER 6

172

= 0.5) are given in Figure ‎6.13. The figure shows that the experimental results of the

application of Equation (‎6.12) in Figures 6.13a1-a4 have the same behaviour as the

theoretical results demonstrated in Case 4 (see Figure ‎6.7a) for the crack identification

and location at Γ = 0.5. But the crack is difficult to identify and locate in both the

theoretical and experimental results of the application of Equation (‎6.12). This

uncertainty and confusion about the exact location of the crack has been reasonably

resolved by the application of Equation (‎6.13). Compared to the experimental results of

Equation (‎6.12) in Figures 6.13a1- a4, the experimental results of Equation (‎6.13) in

Figures 6.13b1-b4 manifest clearly the crack location except for a slight deviation of the

peak ζ1(234), ζ2(134), ζ3(124), and ζ4(123) curves from the exact location Γ = 0.5. This is

mainly due to the influence of using large spatial interval (40 mm) which has interacted

with the noise of the instrumentation. This problem can be solved by applying the local-

refinement technique which has been explained in Section ‎6.8. In this way, the

experimental peaks of ζ1(234), ζ2(134), ζ3(124), and ζ4(123) will occur at the exact location Γ

= 0.5 similar to that shown in Case 4 (see Figure ‎6.7d) in which the spatial interval is 10

mm.

Case 9

In this case, the shaft has a crack of μ = 0.3 at location Γ = 0.7. This case has been

investigated in order to determine the similarities and differences that are possible when

cracks of identical severity are located at symmetrically opposite positions in a shaft.

This case is similar to Case 7 (see Figure ‎6.12) in which the crack depth ratio μ = 0.3

and location Γ= 0.3. The results of the application of the FCP method on this case

presented in Figure ‎6.14 show that even if the results are smeared by the effects of the

numerical and measurement noises as well as location of the crack, the FCP method

reasonably identifies and locates the crack on the shaft. Also the figure shows good

agreement between numerical and experimental results except for a slight deviation of

the experimental FCP curves ζ1(234), ζ2(134), ζ3(124), and ζ4(123) which is due to the

interaction of the effects of both the experimental noises and spatial interval of the

roving disc.

CHAPTER 6

173

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.3

2,1

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1x 10

-3

2

(13

4)

= 0.3

(a2) (b2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1

,m

= 0.3

1,2

1,3

1,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

1

(23

4)

= 0.3

(a1) (b1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.3

3,1

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

3

(12

4)

= 0.3

(a3) (b3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.3

4,1

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

4

(12

3)

= 0.3

(a4) (b4)

Figure 6.14: Experimental frequency curves product (FCP) of the cracked rotor with a

crack of μ = 0.3 at the location = 0.7: (a) use Equation (6.12); (b) use Equation

(6.13).

CHAPTER 6

174

Sensitivity of Including or Excluding the First Mode on the Accuracy of the 6.11

Crack Location

In the previous section, the proposed FCP method identified reasonably (not exactly)

the crack location in all the experimental cases 7, 8 and 9 in Figures 6.12c1-c4, 6.13c1-

c4 and 6.14c1-c4, respectively. The results show that that the peaks of the FCP curves

deviated slightly from the exact locations Γ = 0.3, 0.5 and 0.7 in cases 7, 8 and 9,

respectively. This deviation and inaccuracy are not only due to experimental

instrumentation noise but also due to the use of the first mode λ1 in all the damage

indices ζ1(234), ζ2(134), ζ3(124), and ζ4(123). The first experimental NNF curve λ1 in all

experimental Cases 7 to 9 is greatly affected by experimental noise compared to the

second, third and fourth NNF curves λ2, λ3 and λ4. For example, the NNF curves of Case

8 (μ = 0.3 and Γ = 0.5) in Figure ‎6.15 show that the random noise and ambiguity of the

crack location is more dominant in the λ1 curves than it is in the second , third and

fourth NNF curves λ2, λ3 and λ4, respectively. Therefore, the first mode λ1 cannot be

relied on and used to locate the exact crack location. Hence, in this section, the use of

damage indices which exclude the first experimental NNF curve λ1 is investigated. The

other three experimental NNF curves λ2, λ3 and λ4 are used to determine ζ2(34), ζ3(24), and

ζ4(23).

Figure ‎6.16 shows the FCP experimental results of Case 7 (μ = 0.3, 0.5 and Γ = 0.3).

The figure shows the variation of the damage indices ζ2(34), ζ3(24), and ζ4(23) with

auxiliary mass location ratio ζ. In comparison with the results of the same case in

Figure ‎6.12, which is based on including and using the first NNF curves λ1 in the

damage indices, there is significant change in the FCP curves ζ2(34), ζ3(24), and ζ4(23).

Similarly, The FCP curves ζ1(234), ζ2(134), ζ3(124), and ζ4(123) of Cases 8 and 9 in Figures

6.13b1-4 and Figures 6.14b1-4, respectively, have shown a slight deviation of peaks

from the exact crack location Γ = 0.5 and Γ = 0.7 when the FCP curves are based on the

first NNF curve λ1. This problematic issue has been resolved substantially when λ1 of

each case is excluded from the damage indices as can be seen from the ζ2(34), ζ3(24), and

ζ4(23) curves of Cases 8 and 9 shown in Figures 6.16 and 6.17, respectively. The results

show that the ambiguity, deviation and inaccuracy are considerably eradicated and the

peaks of the ζ2(34), ζ3(24), and ζ4(23) curves are located exactly at the correct crack location

Γ = 0.3 and Γ 0.5 without any ambiguity (see Figures 6.16b2-4 for Case 8 and Figures

6.17b2-4 for Case 9).

CHAPTER 6

175

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99985

0.9999

0.99995

1

1

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99998

0.99999

1

2

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9998

0.99985

0.9999

0.99995

1

3

= 0.3 (Th.)

= 0.3 (Exp.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.99994

0.99996

0.99998

1

4

= 0.3 (Th.)

= 0.3 (Exp.)

(a) (b)

(c) (d)

Figure 6.15: Theoretical and experimental NNF curves of the cracked shaft with μ =

0.3 at location Γ = 0.5: (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode.

CHAPTER 6

176

Figure 6.16: The NNF curves of the FCP method based on the modes 2, 3 and 4 of the

cracked rotor with a crack of μ = 0.3 and 0.5 at location = 0.3: (a) use Equation (6.12);

(b) use Equation (6.13).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.3

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.5

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

(34

)

= 0.3

= 0.5

(a2) (b2) (c2)(c2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.3

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.5

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

3

(24

)

= 0.3

= 0.5

(a3) (b3) (c3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.3

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.5

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

4

(23

)

= 0.3

= 0.5

(a4) (b4) (c4)

CHAPTER 6

177

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

2

,m

= 0.3

2,3

2,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

2

(34

)

= 0.3

(a2) (b2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

3

,m

= 0.3

3,2

3,4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

3

(24

)

= 0.3

(a3) (b3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

4

,m

= 0.3

4,2

4,3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

4

(23

)

= 0.3

(a4) (b4)

Figure 6.17: The NNF curves of the FCP method based on the modes 2, 3

and 4 of the cracked rotor with a crack of μ = 0.3 at location = 0.5: (a) use

Equation (6.12); (b) use Equation (6.13).

CHAPTER 6

178

Conclusions 6.12

This work has proposed the FCP method for the identification and location of cracks in

rotor systems through exploiting the influence of a roving disc mass. The results of the

investigations have shown that the application of the FCP approach enhances the

identification and location of a crack. The results also showed that the FCP method has

the merit of unifying the peaks of the products of frequency curves only at the crack

locations.

The first NNF curves λ1 are more susceptible to random noise in both experimental and

theoretical cases than the other NNF curves λ2, λ3, and λ4. As a consequence, the first

NNF curves λ1 are not recommend for computing the FCP curves.

Overall, the accuracy and consistency of the FCP method for the identification and

location of cracks under conditions of varying locations along a rotor show that the

method can be considered as a technique for the unique identification and location of

cracks in rotor systems.

179

CHAPTER 7

The Use of Roving Discs and Orthogonal Natural Frequencies

for Crack Identification and Location in Rotors


Journal of Sound and Vibration, Vol. 333, Issue 23, 2014, PP 6237–6257

doi:10.1016/j.jsv.2014.05.046

http://dx.doi.org/10.1016/j.jsv.2014.05.046

180

The Use of Roving Discs and Orthogonal Natural Frequencies for Crack

Identification and Location in Rotors




Abstract

A variety of approaches that have been developed for the identification and localisation

of cracks in a rotor system, which exploit natural frequencies, require a finite element

model to obtain the natural frequencies of the intact rotor as baseline data. In fact, such

approaches can give erroneous results about the location and depth of a crack if an

inaccurate finite element model is used to represent an uncracked model. A new

approach for the identification and localisation of cracks in rotor systems, which does

not require the use of the natural frequencies of an intact rotor as a baseline data, is

presented in this paper. The approach, named orthogonal natural frequencies (ONFs), is

based only on the natural frequencies of the non-rotating cracked rotor in the two lateral

bending vibration x-z and y-z planes. The approach uses the cracked natural frequencies

in the horizontal x-z plane as the reference data instead of the intact natural frequencies.

Also, a roving disc is traversed along the rotor in order to enhance the dynamics of the

rotor at the cracked locations. At each spatial location of the roving disc, the two ONFs

of the rotor-disc system are determined from which the corresponding ONF ratio is

computed. The ONF ratios are normalised by the maximum ONF ratio to obtain

normalised orthogonal natural frequency curves (NONFCs). The non-rotating cracked

rotor is simulated by the finite element method using the Bernoulli-Euler beam theory.

The unique characteristics of the proposed approach are the sharp, notched peaks at the

crack locations but rounded peaks at non-cracked locations. These features facilitate the

unambiguous identification and locations of cracks in rotors. The effects of crack depth,

crack location, and mass of a roving disc are investigated. The results show that the

proposed method has a great potential in the identification and localisation of cracks in a

non-rotating cracked rotor.

Keywords: vibration analysis; crack identification; finite element modelling; rotor,

cracks; roving disc; rotor dynamics.

181

Introduction 7.1

The early detection and diagnosis of incipient faults associated with rotating machines

have aroused considerable interest from researchers over the decades. It is known that

the presence of a crack in rotors gives rise to stress concentration in the vicinity of the

crack tip which introduces local flexibility at the crack location. This consequent

reduction in the rotor stiffness which is associated with decrease in natural frequencies

and mode shapes of the rotor lead to considerable changes in the dynamic behaviour of

the cracked rotor. This, in turn, causes dangerous and catastrophic problems on the

dynamics of rotating systems and result in serious damage to the rotating systems.

Therefore, the early detection of a crack has significant importance on the safety,

reliability, performance and efficiency of a cracked rotor system [40, 67, 141, 144-146].

A non-rotating rotor system with a transversal open crack can be considered as a

simply- support beam. As a result, the investigations of the identification of an open

crack in non-rotating structures such as beams, rods and columns is useful for

identifying and locating cracks in rotating machinery (see [127, 147, 148] ). Over the

past three decades, the aforementioned dynamic behaviour and non-destructive

techniques (vibration-based) have been utilised by many researchers for the

identification and location of cracks in rotating machines and structures. Mayes and

Davies [23, 76], Friswell and Lees [149], Doebling et al. [30], Tsai and Wang [150],

Lee and Chung [122] and Sekhar [151] conducted several investigations on the

dynamics of rotating cracked shafts. They have indicated that the change in natural

frequencies and mode shapes due to the presence of a crack may be useful for the

identification and localisation of cracks in rotors.

Narkis [124] studied the behaviour of cracked simply supported uniform beams and

uniform free-free vibrating rods in bending. He has shown that the variation of the first

two natural frequencies is adequate to identify the crack location. The identification of a

single crack in a vibrating rod which is based on the knowledge of the shifting of a pair

of natural frequencies due to the presence of a crack has been presented by Morassi

[152]. The analysis was based on an explicit expression of the sensitivity of frequencies

to crack location which can be used for non-uniform rods under general boundary

conditions. Messina and Williams [153] have introduced a new algorithm to improve

the computational efficiency of the multiple damage location assurance criterion

(MDLAC). This algorithm requires measurements of the changes in a few natural

frequencies of an intact and cracked structure in order to provide good prediction for the

182

location and size of single or multiple. Zhong and Oyadiji [126, 130, 154] have

investigated theoretically and experimentally the natural frequencies of a cracked

simply supported beam with a stationary roving mass. The results have shown that the

roving mass along the cracked beam increases the variation of natural frequencies. That

is, using the roving mass enables to provide additional spatial information for damage

detection of the beam. From their results; they produced natural frequency curves,

which show the variations of the natural frequencies with the location of the roving

mass. Fan and Qiao [27] have presented a more comprehensive literature survey on

using a variety of techniques which are based on the variation of natural frequencies for

the identification of cracks in structures.

Some researchers have investigated the use of changes in the flexibility matrix to

identify cracks. Zhao and Dewolf [155] have conducted a theoretical sensitivity

investigation comparing the application of natural frequencies, mode shapes, and model

flexibility for the identification of damage in structures. The results indicate that model

flexibility performs slightly better than the other two methods. Pandey and Biswas [156]

have presented a new method for damage localisation based on the changes in the modal

flexibility of a damaged beam. Lu and Zhao [157] have proposed the modal flexibility

curvature method with higher sensitivity than Pandey‘s method for multiple damage

localisations.

The identification of cracks through utilising mode shape measurements has been

conducted by some researchers. Pandey et al [156] have indicated that the damage

location in structures can be detected by using the changes in the curvature mode shapes

which are localized in the region of damage. These changes increase with increasing

damage size. Ratcliffe [158] has shown that the location of damage can be identified by

using the mode shapes associated with higher natural frequencies, but their sensitivity is

less than the sensitivity of the lower mode. Zhong and Oyadiji [159] have used the

derivatives of the mode shapes of simply supported beams for the identification and

localisation of small cracks in beams. The results have shown that using the first, second

and third derivatives of the displacement mode shape provide good indication of the

presence of a crack.

Some non-model based algorithms which are shown as damage index methods require

the characteristics of the intact structure as a baseline data. The baseline is used as a

reference to determine the changes in the modal parameters due to a crack. Normally,

the baseline data is determined by two ways: either from undamaged structure

183

measurements or by modelling the intact structure using the finite element method

(FEM). In the context of using this methodology, Al-Said [160] has proposed a new

algorithm to identify the location and depth of a crack in a stepped cantilevered

Bernoulli-Euler beam with two masses. The variation of the difference between the

natural frequencies of the cracked and intact systems was utilised for the proposed

algorithm. However, a finite element model and excremental responses were used to

obtain the natural frequencies of the intact beam as a baseline data. Pandey et al. [156]

have compared the mode shape curvatures of the intact and damaged structures to

identify the damage location. Cornwell et al. [161] have extended the one-dimensional

strain energy method which was presented by Stubbs and Kim [162] to two-dimensional

structures, but these two approach still requires the baseline data of the intact structure.

Most of the previous reviewed methods depend on the baseline data of intact structures

to compare with the cracked structures. In fact, this approach can give erroneous results

about the location and depth of a crack if an inaccurate finite element model is used to

represent an intact model. To overcome this problem, several approaches have been

presented to identify the crack location in cracked systems without resorting to the data

of the intact states. Law [163] has proposed an algorithms for the damage localisation in

two-dimensional plates. The algorithm based on changes in uniform load surface (ULS),

which requires only the first few of frequencies and mode shapes of the plate before and

after damage, or only the damage state eigenpairs if a gapped-smoothing technique is

implemented. Ratcliffe [158] has developed a method for the identification of damage

in one-dimensional beams. The method is based on a modified Laplacian operator

which operates only on the data of the damaged beam. The procedure is performed by

applying a cubic curve fit to the modal data and determines the variation between the

curve fit and the actual data to locate the damage. Recently, a new method for crack

detection in beam-like structures has been presented by Zhong and Oyadiji [40]. The

method is based on finding the difference between two sets of detail coefficients

obtained by the use of the stationary wavelet transform (SWT) of two sets of mode

shape data of a cracked beam. The difference of the detail coefficients of the two new

signal series, which are obtained, respectively, from the left half and reconstructed right

half of the modal displacement data of a damaged simply supported beam, was used for

crack detection. The method is implemented without prior knowledge about the modal

parameters of an uncracked beam as a baseline for crack detection.

184

Most of the aforementioned investigations have been focused on the identification of

cracks in stationary structures, such as beams, using approaches that require natural

frequencies or mode shapes of the intact state as a baseline date for crack detection.

Similarly, the approaches employed for the identification of cracks in rotating

machinery require the baseline modal data of the intact rotor. It is crucial to generate an

appropriate approach for the identification and localisation of cracks in rotors without

using baseline data for crack identification and locations. This is to avoid the drawbacks

of an inaccurate finite element modelling of intact rotors, as stated in the literature, and

the problem of losing the baseline data of intact rotors which were designed decades

ago. In this work, a new approach is presented for the identification and localisation of

cracks in rotor systems. The approach, which does not require the natural frequencies of

intact rotors as baseline data, is based on the cracked natural frequencies in both the x-z

and y-z planes of the non-rotating cracked rotor.

It is known that the changes that a crack causes to the natural frequencies are typically

very small and may be buried in the changes caused by operational and environmental

conditions [27, 36]. To account for this problem, the roving auxiliary mass method,

which was recently developed by Zhong and Oyadiji [40, 126, 130], is used. The roving

of the mass along a rotor enhances the effects of the crack on the dynamics of the rotor

which facilitates the crack identification and location in the cracked rotor. In this paper,

the roving auxiliary mass is a roving disc which is stationary (zero velocity) at each

location for which the natural frequencies of the cracked rotor are evaluated. The roving

disc of three different masses of magnitudes 8.0%, 20.4% and 40.8% of the shaft mass

are used in this study in order to show the sensitivity of the proposed approach to the

mass and location of the roving disc.

The cracked natural frequencies in the x-z and y-z planes are computed by using the

finite element technique for modelling the cracked rotor system. The cracked region is

modelled as varying-time stiffness and it is used for the vibration analysis and crack

identification and location of a non-rotating rotor with an open crack. The rotor is

considered as a simply-supported beam which satisfies the Bernoulli-Euler beam theory.

The orthogonal natural frequencies of the cracked rotor in the x-z and y-z planes, namely

fxci and fyci are computed. From these, the normalised orthogonal natural frequency ratios

(NONFRs) are determined and the corresponding normalised orthogonal natural

frequency curves (NONFCs) are obtained. Then the variations of the NONFCs, which

185

are NONFRs, against roving disc locations, clearly identify and locate the crack in the

non-rotating cracked rotor system.

System Modelling 7.2

The rotor, under this study, is composed of a rotor shaft with a single open crack and a

moveable disc of three different masses as illustrated in Figure ‎7.1. It is considered as a

simply-supported beam supported by two rigid bearings. The Bernoulli-Euler beam

theory is used in the modelling of the shaft for lateral bending vibrations in the x-z and

y-z planes. That is, the shear deformation and rotational inertia are neglected. It is

assumed that there is no damping in the rotor. The finite element method (FEM) is used

to implement the analytical model of the non-rotating rotor system.

Finite Element Model 7.2.1

In this investigation, the non-rotating rotor-shaft, which has a constant cross-section, has

been segmented into N elements using two noded Bernoulli-Euler beam elements. Each

element has four degrees-of-freedom per node: transverse displacement in the x- and y-

axes directions and rotations about the x- and y-axes directions as shown in Figure ‎7.2.

Consequently, the nodal displacement of a shaft element is defined by

, - (‎7.1)

Figure 7.1: Finite element model of an intact and cracked rotor.

186

where, the superscript ‘T ’‎denotes transpose of a matrix/vector. In this paper, matrices

and vectors are written in bold letters and are mentioned wherever they appear.

Rotor System Equations of Motion 7.2.2

After assembling the governing equations for all the shaft elements and the rigid disc,

the free vibration equations of motion of the uncracked rotor-bearing system are defined

as:

( ) ( ) (‎7.2)

where, ( ) ,

- is the nodal displacement vector with

dimension 4(N+1) × 1 corresponding to the local coordinate vector for each element in

Figure ‎7.2, M is the global mass matrix which contains the mass matrices Me for each

element of the rotor and the mass matrix Md of the disc corresponding to degrees of

freedom , - . K is the global stiffness matrix which includes the stiffness

of intact elements and the stiffness reduction at the location of a cracked element which

will be defined in detail in the next section. All the matrices are of size 4(N+1) ×

4(N+1). These matrices are given by

Figure 7.2: Typical finite rotor element and coordinates for an intact and cracked

rotor.

187

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎7.3)

where e is the beam element length, e is the beam mass density and Ae is the beam

cross-section area.

[

] (‎7.4)

where md is the mass of the disc and Id is the diametric mass moment of inertia of the

disc about any axis perpendicular to the rotating axis.

After using the rotor response vector as tj

oet qq )( and allocating the boundary

conditions on the system equations on assembly of elemental equations, the

uncracked rotor equations of motion reduce to

( ) (‎7.5)

Crack Modelling 7.2.3

The presence of a transverse crack in rotors generates local flexibility at the crack

location due to the concentration of strain energy in the vicinity of the crack tip under

load. Therefore, an appropriate crack model is essential to accurately predict the

dynamic response of the rotor system with an active crack. Many scholars have

investigated this serious problem and proposed and developed a variety of crack model

(see [23, 27, 67, 76, 155] ). In this paper, the model of an open crack in Figure ‎7.3 has

been proposed recently by Al-Shudeifat and Butcher [18, 21, 75, 115, 164, 165] and is

implemented in order to locally represent the stiffness properties of the crack cross-

section. In this model, the stiffness reductions of the cracked element in a beam are

considered as time-varying values during the shaft rotation and defined through the

reduction of the second moment of area and about the centroidal X and Y axes,

respectively, as

( ( )) (‎7.6)

and

188

( ( )) (‎7.7)

The crack is assumed to be at an initial angle with respect to the fixed negative Y-axis

at t = 0 (Figure ‎7.3a), and the angle of the crack relative to the negative Y-axis changes

with time to + Ωt when the shaft rotates (Figure ‎7.3b). Therefore, the stiffness matrix

of the cracked element, in [18, 21] can be defined in a form similar to that of an

asymmetric rod in space [66].

[ ( ) ℓ ( ) ( ) ℓ ( )

( ) ℓ ( ) ( ) ℓ ( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

( ) ℓ ( ) ( ) ℓ ( )

( ) ℓ ( ) ( ) ℓ ( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( )

ℓ ( ) ℓ ( ) ℓ ( ) ℓ

( ) ]

(‎7.8)

where, 2)( 2

1eAIII yx , 2)( 2

12 eAIII yx and I3 = - I2 are constant values

during the shaft rotation, [18, 21] and

(( )( ) ( )) (‎7.9)

(a) (b)

Figure 7.3: A cracked element cross-section; (a) Non-rotating, (b) Rotating; the hatched

part defines the area of the crack segment [18, 21].

189

(( )( ) ( )) (‎7.10)

√ ( ) is a constant, μ = h/R is the crack depth ratio and h is the depth of the

crack in the transverse direction of the shaft.

As a consequence, the finite element stiffness matrix of the cracked element given in

Equation (‎7.8) can be rewritten [18, 21] as

( ( )) (‎7.11)

where,

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎7.12)

ℓ

[ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

]

(‎7.13)

Thus, the FEM equations of motion of the uncracked rotor bearing-system given in

Equation (‎7.2) are rewritten including the effect of the cracked element of an open crack

model as

( ) . ( ( ))/ ( ) (‎7.14)

where is the global stiffness matrix derived from the global stiffness matrix K of the

uncracked beams by replacing the uncracked element stiffness matrix of element i

by the cracked element stiffness matrix . is another global stiffness matrix; it has

zero elements apart from those at the cracked element location where the elements are

equal to . is the rotor speed, and is the crack angle; both variables are equal to

zero in this study, according to the assumptions that the rotor is non-rotating and the

crack is always fully open (see Figure ‎7.3a).

190

Simulation Model and Algorithm 7.3

Finite Element Model 7.3.1

In this study, the finite element model in Figure ‎7.1, with the physical dimensions and

material properties in Table ‎7.1, has been used for the simulation of the intact and

cracked rotors. The rotor is assumed as a simply-supported beam with an open crack

carrying a moveable disc. The disc is a stationary roving mass which is traversed along

the cracked rotor length at 10 mm spatial interval to enhance the dynamics of the rotor

at the cracked locations. Consequently, the natural frequencies of the non-rotating rotor

in a cracked and intact status in the two planes x-z and y-z are determined for each 10

mm spatial interval of the roving disc along the cracked rotor. The proposed method has

been verified by conducting a number of simulations on the cracked rotor with an open

crack of different locations and sizes. All simulations have been performed by coding

the equations of motion of the cracked rotor in the MATLAB

(R2010a) environment.

Table ‎7.1: Values of parameters in the numerical model

Parameters Shaft Disc

Material Stainless steel Aluminium

Young‘s Modulus, Es (GPa) 200 70

Poisson‘s ratio, vs 0.3 0.33

Density, ρ (Kg/m3) 7800 2700

Length, (m) L = 1 Ld = 0.01, 0.02, 0.3, ..., 1

Outer diameter, Do (m) 0.02 0.2

Inner Diameter, Di (m) - 0.02

Thickness, t (m) - 0.04

Mass, m (Kg) 2.45 0.2, 0.5 and 1.0

Crack Identification Algorithm 7.3.2

It is acknowledged, that the second moments of area and of an uncracked beam

with circular cross sectional area are identical about the centroidal and axes,

respectively. According to Equations (‎7.6) and (‎7.7) and the crack coordinates in

Figure ‎7.2 and are not identical when a crack occurs and their values decrease

191

unequally as the crack depth ratio increases as seen in Figure ‎7.4. This unequal

reduction in the and has been exploited in this study to develop a new approach,

named orthogonal natural frequencies (ONFs) method, for the identification and

location of an open crack in non-rotating rotor systems. The approach is based only on

the cracked natural frequencies of the cracked rotor in the two lateral bending vibration

x-z and y-z planes. That is, the approach does not need the uncracked natural frequencies

as a baseline data; the cracked natural frequencies in the horizontal x-z plane are used as

the reference data instead of the intact natural frequencies. This approach is

implemented as follows:

1. The cracked natural frequencies in the vertical y-z plane (fyci) are divided by the

corresponding cracked natural frequencies in the horizontal x-z plane (fxci) for

the same roving disc locations in order to obtain the orthogonal natural

frequency ratios (ONFRs), i, ( i.e. i = fyci / fxci) of each mode.

2. The orthogonal natural frequency ratio i of each mode is divided by the

maximum orthogonal natural frequency ratio of each mode to determine the

normalised orthogonal natural frequency curves (NONFCs),i, (i.e. ⁄ ).

3. The variation of the NONFCs, i, is plotted against the non-dimensional roving

disc locations = Ld /L.

Sec

on

d m

om

ent

of

area

Figure 7.4: Second moments of area of the cracked segment in this study against

crack depth ratios.

192

Simulation Results and Discussions 7.4

In this section, a number of simulations with different scenarios have been conducted to

manifest the validity and reliability of the proposed ONF approach for the identification

and localisation of a crack in a non-rotating cracked rotor-bearing system. These

numerical simulations have been performed through using ten numerical cases

considering the effects of the depth ratios µ and locations = Lc/L of the crack with 0.5

mm width, that is wc = 0.5 mm, and the effects of the roving disc mass of 8.0%, 20.4%

and 40.8% of the 2.45 kg shaft mass. These cases are illustrated more clearly in

Table ‎7.2.

Table ‎7.2: Values for Numerical Cases

Effects of Crack Depth and Locations 7.5

In this part, the applicability of the proposed method has been investigated through six

numerical cases considering the effects of crack depth ratios and locations . In these

six cases the mass of the roving disc used is 0.5 kg (md/ms = 20.4%).

Case 1: μ = [0.3, 0.5, 0.7 1], Γ = 0.2, mass ratio = 20.4% 7.5.1

In this case, the cracks of various depths are located at =0.2. The ONF method does

not depend on the intact natural frequencies. It relies only on the orthogonal frequencies

fxci and fyci of the cracked rotor. But in order to establish the merits of the proposed ONF

method, a reference set of the first four normalised natural frequency curves (NNFCs)

Case

No.

Crack

Locations

Γ=Lc/L

Mass of Disc

(kg)

Ratio of

mass of

disc to

shaft

Crack Dept

Ratios

μ

Roving Disc

Location

= Ld / L

1 0.2

0.5 20.4%

0.3, 0.5, 0.7,

1.0 0.01, 0.02, ..., 1.0

2 0.3

3 0.4

4 0.5

5 0.6

6 0.7

7 0.3 0.2 8.2%

8 0.5

9 0.3 1.0 40.8%

10 0.5

193

1, 2, 3, and 4, which are based on the natural frequencies of the intact and cracked

non-rotating rotor in the y-z plane, is derived. These four NNFCs are compared with the

first four orthogonal normalised natural frequency curves (NONFCs) 1, 2, 3 and 4

using the ONFs method.

Figure ‎7.5 shows the variation of the NNFCs 1, 2, 3, and 4 of the non-rotating

cracked rotor in the vertical y-z plane against non-dimensional roving disc locations .

These NNFCs are based on dividing the cracked natural frequency in the y-z plane by

the intact natural frequencies in the y-z plane for each 10 mm spatial interval of the

roving disc along the cracked and intact non-rotating rotor. The results in the figure

Figure 7.5: Vertical normalised natural frequency curves of the non-rotating cracked

rotor for cracks of different depth ratios located at = 0.2 and a roving disc of mass 0.5

kg. Based on natural frequencies of non-rotating intact and cracked rotor in the vertical

y-z plane: (a) 1st mode, (b) 2

nd mode, (c) 3

rd mode, (d) 4

th mode.

194

identify very well the presence, location and depth of the crack by the very sharp peaks

in the NNFCs 1 and 2 at the crack locations; whereas the NNFCs 3, and 4 do not

introduce the same clarity for the crack identification and location. This is because of

the closeness of the crack location to the nodes of the third and fourth modes. But the

difference between the shapes of the peaks, which are very sharp and convergent at the

cracked location; whereas they are rounded and divergent at non cracked locations, is a

very helpful indication to overcome reasonably the problem of the closeness of the

crack location to nodes, which in turn makes it easy to identify and localise cracks.

Figure 7.6: Normalised orthogonal natural frequency curves of the non-rotating cracked

rotor cracks of different depth ratios located at = 0.2 and a roving disc of mass 0.5 kg.

Based on natural frequencies of non-rotating cracked rotor in both the horizontal and

vertical planes: (a) 1st mode, (b) 2

nd mode, (c) 3

rd mode, (d) 4

th mode.

195

Figure ‎7.6 shows the variation of the NONFCs 1, 2, 3 and 4 of the non-rotating

cracked rotor against non-dimensional roving disc locations . The figure shows, that

the first four NONFCs 1, 2, 3 and 4, which are based on the ONFs approach in this

paper, identify and localise the presence, location and depth of the crack similarly to the

first four NNFCs 1, 2, 3, and 4 in Figure ‎7.5. In addition to the fact that the ONF

method employs only the cracked natural frequencies in the x-z and y-z planes, the

results of the ONF method are clear and less noisy. For example, Figure ‎7.6a shows that

the mode 1 NONFCs 1 are smoother than the mode 1 NNFCs 1 shown in Figure ‎7.5a.

This is because the same level of numerical noise is associated with the orthogonal

frequencies fxci and fyci in the ONF approach. Therefore, in forming the ratios i = fyci /

fxci and ⁄ the numerical noise effect cancels out. However, in the NNFC

approach, different levels of numerical noise are associated with the intact and cracked

natural frequencies foi and fci, respectively. Thus, the ratios ⁄ (where,

⁄ and ) have a residual numerical noise effect on the NNFCs which

is manifested prominently in the 1 NNFCs shown in Figure ‎7.5a, particularly at small

crack depths ratios. Therefore, the proposed ONF approach is applicable and reliable for

the identification and location of cracks in non-rotating cracked rotors. In the interest of

brevity, therefore, in subsequent cases, only the NONFCs 1, 2, 3 and 4 are presented.


In this case, the crack has been moved to the location = 0.3 from the left bearing to be

nearly close to the nodes of the third mode shape. The variation of the first four

NONFCs 1 to 4 of modes 1 to 4, respectively, against non-dimensional roving disc

locations are shown in Figure ‎7.7. The results of the curves 1 to 4 of all the crack

depth ratios in the figure show acute convergences and very sharp turn in the curves at

the crack location, whereas the peaks of the curves are rounded at the locations where

there are no cracks. Thus, the crack location is recognised clearly by the very sharp

spiky points of discontinuity at the cracked position. Also, the figure highlights that in

all the first four NONFCs except for the 3 curves the location and size of the cracks are

identified clearly. The tips of the sharp notches of the NONF curves intersect at the

highest values for 1, 2 and 4 curves. But for 3 curves, the sharp notch tips of NONF

curves do not intersect at the highest values. This is because of the vicinity of the crack

location to a node of the third mode shape, which occurs at = 0.33. Moreover, all

NONF curves are smooth and there is no ambiguity in the crack identification and

196

location. Overall, the implementation of the proposed ONF method is more robust for

the identification and location of cracks than the NNFC method.


The non-rotating cracked rotor with an open crack of various at the location = 0.4

from the left bearing have been simulated numerically in this case. The crack location

= 0.4 is close to the nodal positions of modes 2, 3 and 4. The NONFCs 1, 2, 3 and 4

of this case, which are presented in Figure ‎7.8 , reveal that the location and depths of the





nd mode, (c) 3

rd mode, (d) 4

th mode.

197

crack are reasonably observable in the 1, 3 and 4 curves (see Figures 7.8 a, c, d). In

these curves, the maximum values of the sharp notches appear at the correct crack

location with acute convergence; whereas the highest value of 2 curves occur at the

wrong crack location with flattened shapes. However, the sharp notches of the 2 curves

(see Figure ‎7.8b), which are not the highest value of the 2 curves, occur at the correct

crack location. These unique features, that the peaks of the NONF curves become very

sharp and convergent at the crack location only, and flattened at the non-cracked

locations, enable the clear and unambiguous identification and location of cracks of

various sizes in rotors.





nd mode, (c) 3

rd mode, (d) 4

th mode

198


In the previous cases 1 to 3, the applicability and reliability of the ONF method have

been investigated for the identification and localisation of cracks located near to nodes

of mode shapes. In this case, these investigations have been extended to show the

consistency of the method for the crack identification and location of cracks of various

at location = 0.5 which coincides exactly with the nodes of the second and fourth

modes. It is known that, if a crack is located at modal nodes, the system natural

frequencies are not affected. The results of the simulation in Figure ‎7.9, using the ONF

method, reveal the location and sizes of the cracks with reasonable accuracy. The





nd mode, (c) 3

rd mode, (d) 4

th mode.

199

location of the crack is very clearly identified in the 1, and 3 NONFCs (see Figures

7.9a, c). Also, the crack location can be observed in the 4 curves for the cracks with μ

0.5 (see Figure ‎7.9d) by the sharp notches in these curves at the crack location, and

rounded peaks at the non-crack locations. However, Figure ‎7.9b shows that the crack

location in the 2 curves is still not clearly identified because the crack is located at the

node of the mode.


In this case, the crack is induced at the location Γ = 0.6, such that, it is symmetrical to

Figure 7.10: Normalised orthogonal natural frequency curves of the non-rotating

cracked rotor cracks of different depth ratios located at = 0.6 and a roving disc of

mass 0.5 kg. Based on natural frequencies of non-rotating cracked rotor in both the

horizontal and vertical planes: (a) 1st mode, (b) 2

nd mode, (c) 3

rd mode, (d) 4

th mode.

200

the location of the crack at Γ = 0.4 in Case 3. The variation of the NONFCs 1 to 4 in

Figure ‎7.10 shows that the highest values of 1 and 4 curves, which occur individually

at convergent sharp notches, clearly and correctly identify the location and sizes of the

cracks (see Figure ‎7.10a and d).Also the results in the figure indicate that despite the

maximum values of the 2 and 3 curves occurring at flattened peaks which are located

at the incorrect crack location, the 2 and 3 curves contain sharp notches at the exact

crack location. That is, the application of the ONF method has the same consistency and

characteristics for the identification and location of cracks which are located

everywhere along non-rotating rotors. The figure also shows the clarity and smoothness

of the curves for all the crack depth ratios including the small value of µ = 0.3.


This case is similar to the previous case which deals with the effect of symmetrical

crack location. Here, the crack is moved to the location = 0.7 from the left inner

bearing which corresponds to the location of the crack at = 0.3 in Case 2. The

simulation results in Figure ‎7.11 indicate that the location and depth of the crack are

clearly identified in all the first four NONFCs by the sharp converged peaks. Similar to

the results of the crack at = 0.3 in Case 2 (see Figure ‎7.7); the converged peaks in all

the first four NONFCs have the highest values at the correct crack location except for

the 3 curves. This confirms the applicability and consistency of the proposed method

for the identification and localisation of the crack wherever along the non-rotating

cracked rotor.

Sensitivity of the Proposed Technique to the Mass of the Roving Disc 7.6

Up to this point, the focus has been on the applicability of the proposed ONF method

for crack identification and location by investigating the effect of the crack depths and

locations. But there are other concerns about choosing an appropriate mass for the

roving disc. It has been stated that the roving of a mass along a beam enhances the

effects of the crack on the dynamics of the beam which facilitates the crack

identification and location in a cracked rotor (see Zhong and Oyadiji [40, 126, 130]).

This is does not mean using a heavy roving mass as big as possible because this will

damage the cracked beam and increase the crack depth further which is definitely not

desirable. In order to avoid this problem and to choose an appropriate mass for the

roving disc, the sensitivity of the proposed method to the mass of the roving disc has

201

been studied by using two different roving disc masses in four cases. These two mass

values have been chosen to be lower and higher than 0.5 kg (md/ms = 20.4%), which is

used as a datum of masses in this paper. The two masses chosen are 0.2 kg and 1 kg

which are 8.2% and 40.8% of the shaft mass, respectively, (see Table ‎7.1).

Crack Identification using a Lighter Roving Disc 7.6.1

The effect of using a lighter roving disc has been investigated in this sub-section. The

roving disc mass is decreased from 0.5 kg to 0.2 kg, that is 8.2% of the shaft mass





nd mode, (c) 3

rd mode, (d) 4

th mode.

202

(md/ms = 8.2%). The investigations are presented as cases 7 and 8 for cracks located at

= 0.3 and 0.5, respectively, along the non-rotating cracked rotor length.

7.6.1.1 Case 7: μ = [0.3, 0.5, 0.7 1], Γ = 0.3, mass ratio = 8.2%

The simulation results for the 1 , 2 , 3 and 4 curves of the non-rotating cracked rotor

with cracks of various depth ratios and located at = 0.3 using a roving disc of 0.2

kg (8.2%) are shown in Figure ‎7.12. In comparison with the results of Case 2 in

Figure ‎7.7, in which a roving disc of 0.5 kg (20.4%) is used, the results of Figure ‎7.12

show that the corresponding values of the 1, 2, 3 and 4 curves are higher. This is





nd mode, (c) 3

rd mode, (d) 4

th mode.

203

because the use of a lighter roving disc causes smaller shifts in the natural frequencies

of the non-rotating cracked rotor in both the x-z and y-z planes. Also, the sharpness of

the notched tips and smoothness of the i curves at the crack location is slightly

reduced, particularly in the 3 curves in which the sharpness of the notched tips at the

crack location decreased observably at the small μ values. This causes uncertainty on

the exact crack location in the 3 curves (see Figure ‎7.12c).





nd mode, (c) 3

rd mode, (d) 4

th mode.

204

7.6.1.2 Case 8: μ = [0.3, 0.5, 0.7 1], Γ = 0.5, mass ratio = 8.2%

The crack in this case is induced at the mid-shaft point, where the central nodes of the

second and fourth mode shapes are exactly located. Consequently, the natural

frequencies of the cracked rotor are barely affected. This simulation and investigation

have been conducted to estimate the perceptibility of the proposed method when using a

small roving disc to identify and locate a crack which is located exactly at a node.

Figure ‎7.13 shows that the accuracy of the first and fourth NONFCs 1 and 3 for the

identification and localisation of the crack has relatively decreased in comparison with

the first and fourth NONFCs 1 and 3 of Case 4 in Figure ‎7.9. Furthermore, the

numerical noise of the 1 curves has increased and the smoothness of the curves has

decreased, particularly when 0.7 (see Figure ‎7.13a). The sharp notched tips at the

location of the crack have almost totally been buried and disappeared from the 4 curves

(Figure ‎7.13d). Comparing to all the i NNFCs in Figure ‎7.14, which are based on the

intact natural frequencies of the non-rotating rotor as reference, for the same conditions,

Figure ‎7.13 shows that the proposed method maintains its consistency in providing

fairly smooth and accurate NONF curves with very small numerical noise. On the other

hand, there is a greater level of numerical noise associated with the NNFC method as

shown by the results in Figure ‎7.14, especially for modes 1 and 2.

205

Crack Identification using a Heavier Roving Disc 7.6.2

In this section, the roving disc mass is increased to 1 kg, that is 40.8% of the shaft mass

(md/ms = 40.8%). This effect has been investigated through two cases, that is cases 9

and 10, for cracks located at = 0.3 and 0.5, respectively, along the non-rotating

cracked rotor. The results of these two cases are as follows:

Figure 7.14: Vertical normalised natural frequency curves of the non-rotating cracked

rotor for cracks of different depth ratios located at = 0.5 and a roving disc of mass 0.2

kg. Based on natural frequencies of non-rotating intact and cracked rotor in the vertical y-

z plane: (a) 1st mode, (b) 2

nd mode, (c) 3

rd mode, (d) 4

th mode.

206


Figure ‎7.15 shows the first four NONFCs of the non-rotating cracked rotor with cracks

of various and located at = 0.3 using a roving disc of 1 kg (40.8%). In comparison

with the results of the 1 to 4 curves in Figure ‎7.7 (md = 0.5 kg) and Figure ‎7.12 (md =

0.2 kg), the simulation results in the figure show that increasing the roving disc mass

affects considerably the accuracy and clarity of the four NONFCs 1 , 2 , 3 and 4.

Also, the smoothness of the 1 curve is increased greatly, and the notched peaks become

sharper at the correct crack location. Similarly, the mass increasing of the roving disc





nd mode, (c) 3

rd mode, (d) 4

th mode.

207

results in sharp discontinuities in the 3 and 4 curves (see Figures 7.15 c, d) at the exact

crack location. Thus, the accuracy and clarity of the 1, 2, 3 and 4 curves for crack

identification and location depend substantially on choosing an appropriate mass for the

roving disc. Also, the results have highlighted that using a mass less than 8.0% of the

shaft mass is not recommended to use for enhancing the effects of the crack on the

dynamics of the rotor which facilitates the crack identification and location in the

cracked rotor.


cracked rotor cracks of different depth ratios located at = 0.5 and a roving disc of mass

1.0 kg. Based on natural frequencies of non-rotating cracked rotor in both the horizontal

and vertical planes: (a) 1st mode, (b) 2

nd mode, (c) 3

rd mode, (d) 4

th mode.

208


The importance and impact of using a roving disc of mass 40.8% of the shaft on the

accuracy of the crack detection and localisation by using the proposed method have

been illustrated in the previous case. For this case, the investigation has been extended

by moving the crack to at location = 0.5. The results of in Figure ‎7.16 show that the

clarity and accuracy of the 1, 2, 3 and 4 curves has been increased in comparison

with the results of using the roving disc of mass 0.5 kg in Case 4 and 0.2 kg in Case 8 as

shown in Figure ‎7.9 and Figure ‎7.13, respectively. In addition, the notches in the 4

curves have been interestingly increased at the crack location (see Figure ‎7.16 (d))

which is hardly perceptible in the previous cases. However, the 2 curves have

maintained the same behaviour as in the previous cases except for the curve of μ = 1, in

which a very small spiky tip has appeared at the correct crack location (see Figure ‎7.16

b). The 2 curves do not give any indication about the crack due to the location of the

crack at the node of the second and fourth modes where there is very small vibration

amplitude.

Feasibility of the Proposed Technique based on Few Roving Disc Positions 7.7

In the previous sections, the investigations have focused on the applicability of the

NONF curves technique for the identification and localisation of cracks of various crack

depth ratios and locations. The effect of the mass of the roving disc has also been

considered. In general, the technique has been shown to be capable to identify and

localise cracks reasonably under the stated varying conditions. In all these investigations

in the previous section, many roving disc positions are used in order to obtain smooth

NONF curves. In real applications, the desire is to obtain a quick indication of the

presence and location of cracks in rotors. One of the main drawbacks of the ultrasonic

technique is the need to scan over the entire domain of the structure under test. Thus, for

a rotor, an ultrasonic sensor will need to be traversed at small intervals along the length

of the rotor. Also, each axial position, the ultrasonic sensors may need to be traversed

circumferentially. This will result in hundreds of sensors locations and will be quite

tedious and time consuming. In the proposed technique, the number of roving disc

positions required is far less. Nevertheless, for quick practical application, it is

advantageous to reduce this further. This can be achieved by starting with a coarse grid

of say 5 points for the roving disc locations along the length of the rotor. The results of

this coarse grid will give an approximate location for the structural fault. A ‗‗mesh

209

refinement ‘‘, which is similar to that performed in finite element analysis, will then be

carried out around the approximated location using a fine grid of say 5 points for the

roving disc locations. The result of this coarse-fine mesh approach is shown in this

suction to be quite feasible. The approach ‗‗coarse-refine mesh‘‘ can be summarised by

the following steps:

a. Coarse identification: Use few locations of roving mass (e.g. 5 points) along the

whole length of the shaft.

b. Plot orthogonal normalised natural frequency curves for modes 1 to 4.

c. Identify crack location from the troughs/peaks in the curves.

d. Repeat Steps b and c for fine identification around troughs/peaks identified

(similar to re-meshing in FEA).

e. If necessary, repeat Step d.

For the sake of brevity, only the first two NONF curves will be used for illustrating

graphically the above steps. Figure ‎7.17 shows the variation of the first two NONF

curves δ1 and δ2 against the locations of the roving disc at 5 points only. The results

indicate that δ1 and δ2 can identify cracks located at ζ = 0.3 based on the coarse grid

results. A more accurate location of the crack can be achieved by repeating the

identification process in the locally of the singular region that identified by the coarse

grid. Again, 5 roving disc positions are used in the windowed region in Figure ‎7.17. The

results for this region are shown in Figure ‎7.18. It is clearly seen from these figures that

the crack is clearly identified and located in the rotor.

210

(a) (b)

Figure 7.17: Variation of the first and second NONF curves agnaist the locations of the

roving disc at 5 points only along the shaft length. Cracks at location ξ = 0.3. (a) 1st mode,

(b) 2nd

mode.

(a) (b)

Figure 7.18: Variation of the first and second NONF curves agnaist the locations of the

roving disc at 5 points in the windowed sections shwon in Figure 17. Cracks at location ξ =

0.3. (a) 1st mode, (b) 2

nd mode.

211

Conclusions 7.8

In this study, a new approach, which is based only on the orthogonal natural frequencies

in both the horizontal and vertical lateral bending vibration planes, is presented for the

identification and location of cracks in non-rotating rotors. A roving disc is traversed

along the length of the rotor in order to enhance the dynamics of the rotor and to,

thereby, amplify small crack effects. The proposed method has been verified through

various cases which include the effect of crack depth and location, and the effect of the

mass of the roving disc. The results of numerical simulations have shown that the

proposed method is efficient and has a high potential for applications in identifying and

localising cracks of different sizes and locations in non-rotating rotors. This is because

of the unique features of the normalised orthogonal natural frequency curves, which

converge at sharp notched peaks at the correct crack location, whereas the other peaks

of the curves are rounded. Moreover, the curves are very smooth and the numerical

noise is negligibly small. These capabilities are achieved without the need for the

baseline data of the intact state.

The effect of the mass of the roving disc significantly affects the accuracy of the crack

identification and location, as well as the smoothness of the curves. The results indicate

that using a roving disc with a mass ratio less than 8.0% (md/ms < 8.0%) of the shaft

mass is not recommended for the identification and localisation of cracks. Also,

although the use of a roving disc with a mass ratio of more than 20.4% (md/ms > 20.4%)

gives better and clear results with much less numerical noise, there is the danger of

increasing the severity of the crack. In this context, a roving disc with a heavy mass is

only advisable when a crack is located close to or exactly at the nodes of particular

modes.

CHAPTER 8: Crack Identification in Rotating Rotors

212

CHAPTER 8

Crack Identification in Rotating Rotors

Introduction 8.1

Up to this chapter, the validity and feasibility of the developed crack identification

techniques to identify and localise cracks in the rotor have been investigated by

traversing a roving disc along the cracked rotor in the stationary state. In general, real-

world rotating machinery, during its operating life, is susceptible to periodic

maintenance due to component life-limit or major overhaul caused by faults in some

components. In that case, the machine is at rest which means that the source excitation

forces such as out-of-balance, misalignment, bearing faults and so forth cannot induce

responses (in fact there is no response) in the rotor which can be measured and used to

identify faults in the rotor particularly for the crack fault. Therefore the developed crack

identification techniques in this study have been verified using the cracked rotor in both

the stationary case and rotating case to show the applicability of the crack identification

techniques under different operating conditions.

Characteristics of Rotating Rotor 8.2

In the previous chapters, the idea of using a roving disc as an extra inertia force to excite

the dynamics of the cracked rotor has given reasonable results for the identification and

localisation of cracks in the stationary rotor systems. Unlike the stationary rotors, the

rotating rotors are subjected to extra inherent disturbance forces which generate

synchronous vibrations with a dominant frequency that coincides with the shaft rotating

speed. These forces usually result from rotor unbalance, misalignment, and gyroscopic

moments and other forces. In the literature, a variety of crack identification techniques

have been presented in which the crack was induced close to the disc and deliberately

unbalance forces were created as extra excitation forces. However, all rotating

machinery is fully balanced and well-aligned when they are setup in order to function

properly and increase their operating lifetime. Therefore, the investigations in this study

are conducted on the rotor that is balanced and aligned to acceptable levels. That is,


213

only inherent unbalance forces, which are always generated due to limitations in

machining and assembling accuracy, are considered in this investigation.

Gyroscopic effects, on the other hand, are inevitable disturbance forces that affect the

lateral vibrations of rotor-dynamic systems, which are generated due to the tilt of a rotor

axis relative to the axis of rotation. The angle of tilt, polar moment of inertia of the rotor

and the rotor rational speed are principal factors that govern the magnitudes of

gyroscopic forces. These factors to some extent may be (or can be) manipulated in the

laboratory test rig to reduce (not to exclude totally) the gyroscopic effects but in real-

world rotors it is impossible to do this manipulation. This is also another reason to

investigate the roving disc idea in this study using both the stationary and rotating

cracked rotors. In principle, the gyroscopic moments affect the dynamics of rotor

systems in two ways. First, the gyroscopic moments couple the lateral dynamics of the

rotor in the horizontal and vertical directions of motion. That is, a change in the vertical

vibrations of the rotor affects the dynamics in the horizontal direction, and vice versa.

Second, the gyroscopic moments tend to drift the critical speeds of the rotor system

form their original predictions at zero speed [166]. This implies that the gyroscopic

moments tend to increase (or decrease) the rotor‘s natural frequencies even if there are

no cracks. Therefore, the implementation of the roving disc idea is extended in this

chapter to the rotating cracked rotors.

Natural Frequency Map (Campbell Diagram) 8.3

The existence of gyroscopic effects causes a variation in the roots of the characteristic

equation, so-called eigenvalues, with spinning speed, and this effect appears not only at

specific rotational speeds but also at other speeds. For that reason, it is convenient to

illustrate graphically the variation of the eigenvalues with spinning speed. In general,

the graph of the change of the eigenvalues with rotational speeds is plotted such that the

spinning speed axis is horizontal and the natural frequency axis is vertical. This plot is

called Natural frequency map or Campbell Diagram which provides a considerable

amount of information about the eigenvalues of the system in a single diagram. In

addition to this information, the Campbell Diagram is considered one of the most

general methods to determine the critical speeds of rotating systems. This is because the

Campbell Diagram enables the determination of the critical speeds of a machine over a

given range of speeds under any circumstances [102, 167]. The critical speeds are

defined by intersecting each natural frequency curve with the starting lines that pass


214

through the origin of the Campbell Diagram and represent the first order (1X), second

order (2X), etc. variation of the frequency with respect to rotational speed as shown in

Figure ‎8.1. Herein, these lines through the origin represent the harmonics of the running

speed (i.e. forcing line (ω) = n × rotating speed ( )). In this study, the critical speeds

corresponding to each location of the roving disc are defined by the Campbell Diagram

over the range of rotational speeds from 0 to 3000 rpm.

Numerical Simulation 8.4

In this section, simulations of a rotating rotor with an open crack are carried out

including gyroscopic effects. A roving disc, which is traversed along the rotating rotor

at 80 mm spatial interval (i.e. only ten points), is employed to enhance the dynamic

behaviour at the crack locations. The materials and dimensions of the rotating rotor are

the same as those that were used in the stationary rotor (see Table ‎4.1 in Chapter 4)

except for the bearings span which was changed to 0.9 m instead of 1 m as shown in

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300

350

400

450

Rotor spin speed (rev/min)

Natu

ral

freq

uen

cie

s (H

z)

1X

1st Mode

3rd Mode

4th Mode

2nd Mode

9X

8X

7X

6X

5X

4X

3X

2X

Figure 8.1: Schematic of Campbell diagrame. Black square and star markers

indictate Forward (FW) and Backward (BW) whirling natural frequencies,

respectively.


215

Figure ‎8.2. This is because of the slip ring‘s dimensions. The simulations were

performed through four cases as illustrated in Table ‎8.1

In the simulations, the critical speeds over the rotational speed range 0-3000 rpm were

determined for both the intact and cracked rotors at each location of the roving disc.

Then, the normalised natural frequency (NNF) curves method, which is demonstrated in

Chapter 3 Section Error! Reference source not found., was applied to identify and

locate the crack in the cracked rotating rotor.

Table ‎8.1: Parameters of numerical cases

At each location of the roving disc the the Campbell Diagram was computed to provide

Forward (FW) and Backward (BW) natural whirling frequencies, and, hence, to

evaluate the critical speeds. The simulation results of the three cases as follows.

Case

No.

Crack

Location

Γ=Lc/Lb

Mass of Disc

(kg)

Ratio of mass

of disc to shaft

Rotating

Speeds

(RPM)

Crack

Depth

Ratios μ

Non-dimensional

roving Disc

locations

= Ld/Lb

1 0.33 1.0 40.8% 0 - 3000 0.3 0.078, 0.167, ...,0.878

2 0.52 1.0 40.8% 0 - 3000 0.3 0.078, 0.167, ...,0.878

3 0.79 1.0 40.8% 0 - 3000 0.3 0.078, 0.167, ...,0.878

Couplin

g

Lt = 1.16 m

Lb = 0.9 m

Ld

Lc Crack

td

Slip

Ring

AC-

Motor

Left-Bearing Right-Bearing

Roving disc

Figure 8.2: Simulation model of the cracked rotating rotor


216

Table ‎8.2: Computed FW and BW critical speeds of case 1 from the Campbell diagrams

of both the intact and cracked rotating shaft for each disc location.

Non-dimensional Disc Locations (ζ) - 10 points- 0.078 0.167 0.256 0.344 0.433 0.522 0.611 0.700 0.789 0.878

Intact shaft FW critical speeds (Hz)

47.1264 43.5394 39.7000 36.9045 35.3808 35.0669 35.9378 38.0521 41.4112 45.4109

BW critical speeds (Hz)

45.313 42.3784 39.1495 36.7026 35.3325 35.047 35.8359 37.7158 40.5976 43.879

Cracked-shaft FW critical speeds (Hz)

47.1224 43.5353 39.6958 36.9003 35.3777 35.0644 35.9354 38.0496 41.4084 45.4074


45.3089 42.3743 39.1452 36.6985 35.3291 35.0447 35.8334 37.7133 40.5949 43.8758

Case 1: Crack Parameters [μ, Γ] = [0.3, 0.33] 8.4.1

A crack with μ = 0.3 at Γ = 0.33 form the left-bearing was investigated in this case. The

Campbell Diagram of each location of the roving disc was computed and on which the

critical speeds of the first mode were determined as shown in Figure ‎8.3. The figure

shows that all natural frequencies were affected by gyroscopic effects whether the

roving disc is located near the end or at the middle of the shaft. Some of the natural

frequencies increased (FW) with the rotational speeds and others decreased (BW),

although there is no crack. At each intersection of the forcing line (ω = 1x ) with the

forward and backward natural frequencies there is a critical speed. Similarly, The

Campbell Diagram of the cracked rotor was computed and the critical speeds at the

intersection of the forcing line with both the FW and BW whirling natural frequencies

were determined as shown in Table ‎8.2. The results show that, despite the separation of

the natural frequencies into FW and BW due to the gyroscopic effects, there are only

very slight reductions in FW and BW natural frequencies at each location of the roving

disc when the crack is induced in the shaft. Therefore, these very slight reductions in the

natural frequencies cannot be used to identify the crack. Hence, it is necessary to use a

technique to enhance the natural frequency data. Thus, applying the normalised natural

frequency (NNF) curves method on the results in Table ‎8.2, the severity and location of

the crack was determined as shown in Figure ‎8.4. The figure shows the first normalised


217

0 500 1000 1500 2000 2500 30000

10

20

30

40

50


Undam

ped

nat

ura

l fr

equen

cies

(H

z)

Forward whirl is sequar; Backward whirl is star.

BW Critical Speed

FW Critical Speed

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

30

35

40

45

50


Und

ampe

d na

tura

l fr

eque

ncie

s (H

z)


2100 2110 2120 2130

35.2

35.3

35.4

35.5

Rotor spin speed (rev/min)Undam

ped

nat

ura

l fr

equen

cies

(H

z)


0 500 1000 1500 2000 2500 30000

10

20

30

40

50


Und

ampe

d na

tura

l fr

eque

ncie

s (H

z)


BW Critical Speed

FW Critical Speed

ω = 1×

ω = 1×

ω =1×

(a)

(b)

(c)

Rotor spin speed (rpm)

Nat

ura

l fr

equen

cies

(H

z)

Figure 8.3: Campbell diagram of the intact rotating rotor in case1 at three locations of the

roving disc. (a), (b) and (c) for the disc close to the left-bearing, mid-shaft and close to right–

bearing, respectively. Black square and star marks indictate Forward (FW) and Backward

(BW) whirling natural frequencies, respectively.


218

natural frequency curves of the simulation of case 1 using both FW and BW critical

speeds that were determined from the Campbell diagram at each location of the roving

disc along the rotating shaft. The minimum value of λ1 occurs at a sharp notch which

indicated the correct crack location Γ = 0.33 in both the FW and BW whirling-critical

speeds (see Figures 8.4a and b, respectively). The figure also shows that the minimum

notch of the λ1 in FW critical speeds is sharper and more obvious than in BW critical

speeds at the crack location. However, the NNF curves remarkably identify the exact

location of the crack at location Γ = 0.33 in both the FW and BW whirling-critical

speeds.


In order to investigate the influence of the crack location on the sharpness of the NNF

curves at the cracked location, irrespective of using large spatial interval of the roving

disc, this case was investigated by using a crack of μ = 0.3 at the location Γ= 0.53 from

the left-bearing. Namely, the crack is located approximately in the middle of the shaft.

Similar to the investigation procedures of Case 1, the Campbell diagram of the intact

and cracked rotor at each location of the roving disc was evaluated and the FW and BW

critical speeds were determined. In the interest of brevity, therefore, only the FW and

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.99995

0.99996

0.99997

0.99998

0.99999

1

1

= 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.99995

0.99996

0.99997

0.99998

0.99999

1

1

= 0.3

(FW)

(a)

(BW)

(b)

Figure 8.4: First mode normalised natural frequency curve of the craked rotating shaf.

Crack depth ratio μ = 0.3 at Γ = 0.33. Rotating speed range 0-3000 rpm: (a) and (b) are

Forward and Backward whirling critical speeds, respectively.


219

BW critical speeds derived from the Campbell diagrams are tabulated as shown in

Table ‎8.3 instead of demonstrating the Campbell diagrams of each disc location. The

application of the NNF curves method on these results identified clearly the exact crack

location at Γ = 0.53 as shown in Figure ‎8.5. The figure shows that the sharp notch of the

λ1 curves of both FW and BW results indicated exactly the location of the crack at Γ =

0.53. Additionally, the trough sharpness of the NNF curves of both the FW and BW

curves was noticeably increased compared to that shown in case 1. That is, the trough

sharpness of the NNF curve at the crack location not only depends on the spatial

interval of the roving disc but it also depends on the crack location.

Table ‎8.3: Computed FW and BW crtical speeds of case 2 from the Campbell diagrams




47.1264 43.5394 39.7000 36.9045 35.3808 35.0669 35.9378 38.0521 41.4112 45.4109


45.313 42.3784 39.1495 36.7026 35.3325 35.047 35.8359 37.7158 40.5976 43.879

Cracked shaft FW critical speeds (Hz)

47.121 43.5348 39.696 36.9007 35.3769 35.9333 35.0628 38.0478 41.4066 45.4058


45.308 42.374 39.1455 36.6986 35.328 35.0414 35.8311 37.7113 40.5931 43.8741


220


This is another case that was also investigated to show the feasibility and robustness of

the roving disc idea and the NNF curves method to identify a crack at different location.

In this case the crack of μ = 0.3 was induced beyond the middle of the shafts at location

Γ = 0.79 from the left-bearing and closer to the right-bearing. The critical speeds at the

intersection of the forcing line, ω = 1× , with the FW and BW whirling natural

frequencies in the Campbell diagrams, which were computed for all the 10 points of the

roving disc, are shown in Table ‎8.4. The minimum value in the λ1 curve of this case in

Figure ‎8.6 indicates clearly the location of the crack at Γ = 0.79. The figure also shows

that the trough of the λ1 curve for the BW critical speeds is slightly drifted to the right of

the correct location whereas the trough of the FW curves is slightly drifted to the left of

the exact crack location. Overall, the location of the crack can be reasonably identified.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.999975

0.99998

0.999985

0.99999

0.999995

1

1

= 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.99994

0.99995

0.99996

0.99997

0.99998

0.99999

1

1

= 0.3

(FW) (BW)

(a) (b)

Figure 8.5: First mode normalised natural frequency curve of the craked rotating shaft.

Crack depth ratio μ = 0.3 at Γ = 0.53. Rotating speed range 0-3000 rpm: (a) and (b) are

Forward and Backward whirling critical speeds, respectively.


221

Table ‎8.4: Computed FW and BW crtical speeds of Case 3 from the Campbell diagrams




47.1264 43.5394 39.7000 36.9045 35.3808 35.0669 35.9378 38.0521 41.4112 45.4109


45.313 42.3784 39.1495 36.7026 35.3325 35.047 35.8359 37.7158 40.5976 43.879

Cracked-shaft FW critical speeds (Hz)

47.1238 43.5373 39.6984 36.903 35.3793 35.0652 35.9356 38.0491 41.4077 45.4079


45.3106 42.3765 39.1479 36.7011 35.3309 35.0451 35.8337 37.713 40.594 43.8758

Experimental Test 8.5

In the previous section, the results of the three cases have shown theoretically that the

roving disc idea and the method of processing the intact and cracked natural frequencies

by using the NNF curve method can be used to identify the location of a crack at any

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.99995

0.99996

0.99997

0.99998

0.99999

1

1

= 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.99995

0.99996

0.99997

0.99998

0.99999

1

1

= 0.3

(FW) (BW)

(b)(a)

Figure 8.6: First mode normalised natural curve of the craked rotating shaft. Crack

depth ratio μ = 0.3 at Γ = 0.79. Rotating speed range 0-3000 rpm: (a) and (b) are

Forward and Backward whirling critical speeds, respectively


222

location on the cracked rotating shaft. Although the theoretical results of the three cases

are reasonable, in practice these results may (or may not) be obtained. This is because of

inevitable disturbance forces (or factors) that exist in a real-life rotor which have not

been applied on the rotor in the theoretical state such as inherent unbalance and

misalignment forces, bearing noise, instrument noise, measurement errors, ambient

machine noise and so forth. Therefore, these three cases were also tested experimentally

to validate the theoretical results of the three cases that were demonstrated in the

previous section.

Experimental Results 8.5.1

In this section, the experimental results of the three cases, which were theoretically

investigated in the previous sections, are demonstrated. The procedures of the

experiments and computing results, and the details of the rotating test rig in Figure ‎8.7

are clearly discussed and illustrated in Chapter 4 Sections ‎4.2.1 and ‎4.5). In fact, the

location of cracks is anonymous (or unknown) in real-world rotors. Also, it is known

Figure 8.7: Rotating test righ. (For details see Chapter 4 section 4.2.1


223

that the deflection value of the plane in which a crack occurs is more dominant than the

deflection of a plane with no crack. These issues have a crucial impact on the

performance of the PZT sensors because PZTs work on the principle of the bending

tension and compression (in bending cases) on the surface that the PZTs are mounted.

That is, the greater the bending strain exerted on the PZT, the greater the output voltage

from the PZT. To take these issues into account, therefore, in these tests, the PZT

sensors on both the bottom and right of the rotating shaft were considered for measuring

the vibration response of the shaft during rotation. The bottom PZT sensors are located

exactly in the same plane that the crack is located whereas the right PZT sensors

(according to the view from the motor‘s side) are located at 90-degree apart from the

crack plane as shown in Figure ‎8.8. The following are the results of the three cases.

Experimental results of Case 1: Crack Parameters [μ, Γ] = [0.3, 0.33] 8.5.2

Figure ‎8.9 shows the first mode normalised natural frequency curve, λ1, of the

experimental results of Case 1 which is theoretically investigated in Section ‎8.4.1. The

minimum value of the λ1 curve locates fairly the location of the crack at Γ = 0.33 except

for a slight drift from the exact crack location in the λ1 curve derived from the

measurement by the right PZT sensors (see Figure ‎8.9b). The figure also shows that the

location of the crack and the arrangement scenario of the PZTs around the shaft have

crucial impact on the clarity of the λ1 curve but less influence on the accuracy in

determining the crack location. That is, The NNF curve method, which is based on the

Crack

PZT sensors

Left Bearing Right Bearing

Disc

Motor

Side

Crack

Right

Top

Bottom

Left

Motor Side View

Figure 8.8: Schematic of the PZT locations on the rotating shaft.


224

roving disc idea, can reasonably identify the crack location whether the PZT sensors are

mounted in the same or beyond the plane in which a crack occurs.


The first NNF curve, λ1, of the experimental investigation of this case is presented in

Figure ‎8.10. The location of the crack at Γ = 0.53 is clearly identified by the sharp notch

in the λ1 curve of both the bottom and right PZTs results as shown in Figure ‎8.10a and b,

respectively. The smoothness and accuracy of the λ1 curve of this case compared to the

features of the λ1 curve of the crack at Γ = 0.33 (in Case 1), confirm that the location of

a crack has more influence on the accuracy of the crack identification and localisation

than the location of PZT sensors. This is because the stiffness of the shaft with a crack

located beyond the middle of the shaft is higher than the shaft with a crack in the

middle. This also shows the merit and capability of both the roving disc idea and NNF

curve method for identifying a crack irrespective of the location of the crack.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.98

0.985

0.99

0.995

1

1

= 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.995

0.996

0.997

0.998

0.999

1

1

= 0.3(b)(a)

Bottom PZTs Right PZTs

Figure 8.9: First mode normalised natural frequency of the experimental results of the

cracked rotating rotor with a crack of μ = 0.3 at location Γ = 0.33. Rotational speed range

10-300 rpm: (a) Bottom PZT‘s response, (b) Right PZT‘s response.


225


In this case the location of the crack is moved to Γ= 0.79 from the left-bearing and it is

close to the right-bearing. The experimental results of this case in Figure ‎8.11 show the

consistency of the NNF curve to identify a crack at any location along the rotor. The

sharp tip of the λ1 curve is located exactly at the crack location Γ= 0.79. The figure also

shows the slight drift in the sharp tip of the λ1 curve from the crack location is still a

problematic issue for the results derived from the measurements using the right PZTs

(see Figure ‎8.11a). Even though the minimum values of the λ1 curves based on the right

PZT‘s results do not occur at the correct crack location, the sharp-notch tip identify and

locate the crack very close to Γ= 0.79. Broadly speaking, the location of the crack in

both the bottom and right PZTs results can be fairly identified by the NNF curve

method.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1

= 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.994

0.995

0.996

0.997

0.998

0.999

1

1

= 0.3(b)(a)



10-300 rpm. (a) Bottom PZT‘s response. (b) Right PZT‘s respons.

Bottom PZTs Right PZTs


226

Summary 8.6

The application of the roving disc idea and the normalised natural frequency (NNF)

method to identify and localise cracks in rotors is extended in this chapter to the rotating

cracked rotors. Important things about the characteristics of rotating rotors, particularly

gyroscopic effects have been discussed. The roving disc idea and the NNF curves

method are investigated theoretically and experimentally verified through three different

cases. It is worth noting that both the Forward (FW) and Backward (BW) whirling

critical speeds are possible to be used to identify and localise a crack in rotating rotor

systems. The theoretical and experimental results show that the roving disc idea and the

NNF curves method are applicable and practicable to be used to identify a crack in

rotating rotor systems. It is also important to note that the PZT sensors have

approximately the same sensitivity whether they are mounted or not mounted in the

plane of a crack.



10-300 rpm. (a) Bottom PZT‘s response. (b) Right PZT‘s response.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.965

0.97

0.975

0.98

0.985

0.99

0.995

1

1

= 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.975

0.98

0.985

0.99

0.995

1

1

= 0.3

(b)(a)

Right PZTs Bottom PZTs

CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies

227

CHAPTER 9

Discussions, Summary, Conclusions and Prospective Studies

Discussion of the Roving Disc Effect 9.1

The results of simulations and experiments of this study show that the roving disc

enhances the small changes in the natural frequencies resulting from a crack. To explain

physically why the roving disc enhances the dynamics of a system at the crack location,

first and foremost, it is necessary to know the physical meaning of modes, which are

used as a simple and effective tool to characterise resonant vibration. Modes are defined

as inherent material properties of a structure. In principle, the material properties (mass,

stiffness and damping properties), and boundary conditions govern (or determine) the

resonances of a structure during vibration. That is, a natural (or resonant) frequency,

modal damping, and a mode shape are the intrinsic factors that define each mode. Thus,

modes of a structure will change if either material properties or the boundary conditions

of the structure change. As an example, if a mass (whether uniformly distributed or

concentrated) is added to a structure, say a simply-supported beam as shown in

Figure ‎9.1, the beam will vibrate differently than before adding the mass because the

modes of the beam have changed.

Now, consider that the added mass to the simply-supported beam in Figure ‎9.1 is a

roving disc of constant mass that is traversed along the shaft (without cracks). This

shows that at each time of traversing the roving disc, the modes of the structure will

change due to the location of the roving disc, which gives rise to update (or change) of

the elements of both the mass and stiffness matrices. However, when a crack is

presented (see Figure ‎9.2), the mass matrix elements will not be affected by the severity

and location of the crack as much as the stiffness matrix elements will be affected at

each time the location of the roving disc is changed. That is, the mass matrix will

behave the same way whether there is a crack or no crack during the traversing of the

roving disc along the shaft, in contrast with the stiffness matrix that will be thoroughly

different in its value when there is a crack or no crack. In this case, dramatic increase

occurs in the flexibility of the shaft at the crack location when the roving disc


228

approaches the crack location; hence, remarkable decrease appears in the stiffness

matrix at the crack location, which results in prominent changes in the modes. These

characteristics appear in the normalised natural frequency curves as sharp troughs with

minimum value at the exact crack location and rounded curve tips at the wrong crack

location.

1 2 3 4

Node

L

Le

x

Added mass

(Disc)

Figure 9.1: Adding a roving disc mass as a point mass-at a node of the beam.

x

1 2 3 4

Nod

e

L

Le Crack

Roving Disc

(Circular contact with shaft)

Figure 9.2: Schematic of a cracked beam with an auxiliary roving disc.


229

Summary of the Thesis 9.2

In the context of the rotor dynamics analysis, the influence of a roving disc on the

dynamics of stationary and rotating shafts with a crack is investigated in this study. The

roving disc is used as an extra inertia force to enhance the slight change in the natural

frequencies of the shaft due to the presence of a crack. As a consequence, the location

and severity of the crack can be readily determined. Experimental and finite element

(FE) models have been built to understand and simulate the impact of the roving disc

idea on the crack identification in a rotor system. The application of the roving disc of

different masses on stationary and rotating cracked rotors have been studied through a

variety of cases using a crack of various severity and locations along the cracked rotor.

The influence of the mass of the roving disc has a crucial impact on the accuracy of the

crack identification and location, as well as the smoothness of the natural frequency

curves. The results indicate that a roving disc of a mass ratio less than 8.0% (md/ms <

8.0%) of the mass of the shaft is not recommended for the identification and localisation

of cracks. Also, although the roving disc of mass ratio more than 20.4% (md/ms >

20.4%) gives better and clear results with much less numerical noise, there is in

principle the possibility of increasing the severity of the crack. While the upper limit of

the roving mass ratio has not been investigated in this work, it is suggested that an upper

limit of 50% should be considered in order that the roving mass will not significantly

aggravate the severity of the crack. Suffice it to say that a roving disc of a heavy mass is

only advisable when a crack is very close to the ends of the shaft or located close to (or

exactly) at the nodes of particular modes. Of course the location of a crack is unknown

in real-world rotors but increasing the mass of the disc is only the alternative method to

identify clearly the presence of a crack that is located close to (or exactly) at the nodes

of particular modes.

Although the investigation shows that greater change in natural frequencies of a cracked

shaft results from the roving disc, the identification of a crack and its physical properties

(i.e. location and size) cannot be directly observed from the natural frequency curves.

Therefore, the normalised natural frequency (NNF) curves technique, which is based on

the natural frequencies of the intact and cracked rotor, has been developed. The

proposed approach utilises the variation of the normalized natural frequency curves

versus the non-dimensional location of a roving disc which traverses along the rotor

span. The proposed technique has merits over the methods that have been presented in


230

the literature as the technique uses a roving disc to enhance the dynamics of a crack in

the rotor-bearing system, which facilitates the identification and localisation of the crack

in the shaft. Also, the experimental implementation of the method requires the use of

simple instrumentation and simple testing techniques. The presence of the crack is

identified, and its location is determined, from the appearance of sharp discontinuities in

the plots of the normalised natural frequency (NNF) curves versus the non-dimensional

locations of the roving disc.

The theoretical and experimental results show that when using a roving disc of mass

ratio more than 20.4% (md/ms > 20.4%), the minimum crack depth ratio that the NNF

curves method can identify is μ = 0.3 which is equivalent to 15% of the shaft diameter.

Any crack of greater depth than this gives even much clearer results. In addition, the

correlation between the theoretical and experimental characteristics for the NNF curves

of the four modes is very high when μ > 0.3. Since the crack detectability increases as

the roving mass increases, then the NNF curves method can also be used to identify a

crack of depth ratio less than 15% of the shaft diameter if the mass of a roving disc is

increased. Overall, the numerical and experimental results prove reasonably the

feasibility and capability of the proposed technique for the identification and

localisation of cracks under the environment being investigated.

The frequency curve product (FCP) has been also developed in this study to identify and

locate a crack in rotor systems. The proposed technique is based on the normalised

natural frequency curves (NNFCs) of cracked and intact rotors using the principle of

roving masses and natural frequency curves. This technique has been developed in order

to solve the problem of the disappearance of a crack effect when the crack is close to or

exactly at a node of a mode shape. The FCP curves method uses the first four NNFCs of

the rotor and products of the NNFCs to produce a single plot to identify and locate

cracks clearly, irrespective of the crack location with respect to the modal nodes. It is

shown that this technique identifies the exact crack location through unifying all the

first four natural frequency curves at the maximum positive value in the plot of both the

numerical experimental results.

The results of the investigations show that the FCP method has the merit of unifying the

peaks of the products of frequency curves only at the crack locations. Also the results

show that the first mode NNF curves are more susceptible to random noise in both

experimental and theoretical cases than the first, third and fourth NNF curves. As a


231

consequence, the first NNF curves λ1 are not recommended for computing the FCP

curves. Broadly speaking, the accuracy and consistency of the FCP method for the

identification and location of cracks under conditions of varying locations along a rotor

show that the method can be considered as a technique for the unique identification and

location of cracks in rotor systems.

The normalised orthogonal natural frequency (NONF) curves approach, which is based

only on the natural frequencies of the non-rotating cracked rotor in the two lateral

bending vibration x-z and y-z planes, has been developed in this study. The approach

uses the natural frequencies of a cracked shaft in the horizontal x-z plane as the

reference data instead of the natural frequencies of the intact shaft. A roving disc is also

traversed along the rotor in order to enhance the dynamics of the rotor at the cracked

locations. The results show that the proposed method is efficient and has a high

potential for applications in identifying and localising cracks of different sizes and

locations in non-rotating rotors. This is because of the unique features of the normalised

orthogonal natural frequency curves, which converge at sharp notched peaks at the

correct crack location, whereas the other peaks of the curves are rounded. Moreover, the

curves are very smooth and the numerical noise is negligibly small. These capabilities

are achieved without the need for the baseline data of the intact state.

In addition to the influence of the physical properties of a crack and the mass of the

roving disc on the potential of the roving disc and the accuracy of the proposed

techniques for the crack identification, spatial interval of traversing the roving disc

along a shaft is also important. The smaller the spatial interval of traversing the roving

disc, the more precise and sharper the NNF, NONF and FCP curves will be in

identifying and locating the crack. In practice, a small spatial interval value, particularly

for testing long rotors, will be tedious and time consuming. However, the roving disc

idea and the proposed techniques can be applied through a combination of small and

large spatial intervals with accuracy close to the accuracy of using the small spatial

interval. This can be accomplished by using the coarse-fine mesh approach which starts

with a coarse mesh of 5 to 10 points as initial traversing points of the roving disc along

a shaft. Then, initial results are used to locate the region of the shaft with a defect.

Another mesh of 5 to 10 points can then be made around the defect region to obtain a

more accurate result.


232

The roving disc idea and the proposed crack identification technique also have been

applied on the cracked rotating shaft. This is to investigate the potential of the roving

disc idea for crack identification under the presence of gyroscopic effects and inherent

faults such as misalignment and out-of-balance. The investigation findings show that the

roving disc idea and NNF curves technique work very well to identify the location and

severity of the crack in the rotating shaft despite the presence of gyroscopic moments

and inherent misalignment and out-of-balance forces. Both Forward (FW) whirling

critical speeds and Backward (BW) whirling critical speeds have the same potential to

be used in the NNF curves technique to identify and localise the crack in the rotor. The

smoothness and accuracy of the NNF curves of a crack locates in the middle of the rotor

is better than that of a crack that is located near the ends of the rotor.

Like all types of transducers that are used to measure vibration responses such as

accelerometers, strain gauges, proximity probes, etc. the PZT sensors must be mounted

in the same (or opposite) plane of the excitation force in order to obtain high output

voltage. The PZT sensors location with respect to the location of the crack has no

crucial impact on the accuracy and smoothness of the curves of the proposed techniques

for the identification of the crack.

Unlike most of the models (or techniques) that are presented in the available literature

for the identification and localisation of cracks in structural systems and rotating

machinery, which work as model based methods, the Normalised Natural Frequency

(NNF) curves, Frequency Curve Products (FCP) and Normalised Orthogonal Natural

Frequency (NONF) curves work as non-model based methods. This is another distinct

advantage that makes these techniques are more tangible during implementation.

Limitations of the Proposed Techniques 9.3

Similar to all the developed techniques in the literature for the identification of cracks in

rotor systems, the roving disc idea and the proposed techniques in this study have some

limitations. These certain limitations can be summarised as:

1. A roving disc of mass less than half of the shaft‘s mass is suitable to theoretically

identify and localise a crack of any depth but in practice it is not applicable to be

used for the identification of cracks in shafts.


233

2. In the case of non-uniform cracked shafts, roving discs of different bore sizes are

required.

3. The roving disc idea is difficult to be used in complex rotor systems.

4. The NNF curves technique cannot simultaneously localise the crack location in all

the first four modes if the crack is located exactly on the nodes of modes.

5. The FCP curves method solves thoroughly the drawback of the NNF curve method

but at least four modes are required to be extracted in order to apply the FCP curves

method.

Conclusions 9.4

Based on the research findings of this study, the following points have been concluded:

1. A roving disc enhances very well the dynamics of the rotor near the crack locations.

2. The developed approaches are feasible to identify and localise a crack in rotor system

irrespective of the severity and location of the crack.

3. Sharp-pointed peaks of the natural frequency curves occur at the exact crack location

whereas round-shaped peaks occur at the wrong crack locations.

4. The techniques use only the natural frequencies of a rotor to locate and assess the

severity of a crack in the rotor.

5. The experimental implementation of the techniques requires using simple

instrumentation and simple vibration testing technique to determine the modal

frequencies.

6. The techniques perform reasonably even if gyroscopic moments, misalignment and

out-of-balance forces are generated.

Scope for Prospective Studies 9.5

According to the literature, the topic of this study is one of the most important research

studies that have been considered in the protective maintenance of rotor systems. This

research topic has a crucial impact on the factors that govern the sustainability of

rotating machinery. The approaches that have been presented in this research study can

be considered as a beginning of an idea that will provide further insight into the

dynamics of cracked rotors. These approaches need to be studied more and extended

further to meet diversity of use in stationary and rotating systems. Future works,

therefore, should focus on the following suggestions:


234

1. The application of the proposed techniques was conducted using a simple rotor

system. A rotor system with large discs and blades, which generate complicated

natural frequency maps, should be considered in future studies.

2. Use of Timoshenko beam elements for the finite element analysis of the rotor to

include shear and rotary inertia effects of the shaft.

3. Fluid film and active magnetic bearings, which are more commonly used in real-

world rotors, should be considered because they have a crucial impact on the stability

and linearity of rotor system.

4. The dynamics of breathing cracks is different from the dynamics of open cracks;

therefore, the former type also should be studied.

5. In this study the dynamics of a rotor with one crack was conducted. The rotor with

two or more cracks should also be considered

6. The imitation crack in this study was as a slot on the shaft. However, the

configuration and dynamics of slotted cracks do not exactly reflect the configuration

and dynamics of real fatigue cracks. In future, therefore, a rotor with a real fatigue

crack is also important to be investigated.

APPENDIX A: Design and Dimensions of the Rotating Test Rig

235

Design and Dimensions of Rotating Rig APPENDIX A

Figure 9.3: Dimensions of the rotating rig base (Dimensions in mm).

Figure 9.4: Bearing supports: (Dimension in mm)


236

Figure 9.5: Motor support (Dimensions in mm)

Figure 9.6: Dimensions of disc (mm).


237

Figure 9.7: Bearing-collar dimensions (mm).

APPENDIX B: Specifications of vibration measuring instruments

238

Specifications of Vibration Measuring APPENDIX B

Instruments

Accelerometers’ Specifications APPENDIX B.1

Table ‎9.1: Specifications of the accelerometer sensors

Specifications Dytran Model 3055B2 Dytran Model 3225F1

Weight (gm) 10 0.6

Housing titanium titanium

Sensing Element ceramic, shear quartz, shear

Sensitivity (mV/g) 100 10

Range (g) 50 500

Frequency Response, ±5% (Hz) 1 to 10000 1.6 to 10000

Electrical Isolation yes no

Maximum Shock (g) 2000 5000

Mode IEPE IEPE

Temperature Range (ºC) -51 to +121 -51 to +121

Thermal Coefficient of sensitivity

(%/ºC) 0.12 0.06

Specifications of DAQ type NI-PCI-6123 S series and BNC-2110 APPENDIX B.2

a- Specification of DAQ type PCI-6123 S series

Product Name: PCI-6123 S series

8 simultaneously sampled analogue inputs, 16 bits, 500 kS/s per channel

4 input ranges from ±1.25 to ±10 V

Deep on-board memory (16 or 32 MS)

8 hardware-timed digital I/O lines; two 24-bit counters; analogue and digital

triggering

APPENDIX B: Specifications of vibration measuring instruments

239

NI-DAQmx driver software and Lab VIEW Signal Express interactive data-logging

software

Optimized integration with NI Lab VIEW, Lab Windows™/CVI, and Measurement

Studio.

b- Specification of BNC-2110

BNC connectors for analogue I/O

Terminal block for digital and timing I/O connections

Interfaces to X Series

Specifications of Strain Gauges APPENDIX B.3

Figure 9.8: Specifications of the strain gauges used in Chapter 4.

REFERENCES

240

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[11] S.S. Ganeriwala, Detection of Shaft Cracks Using Vibration Signature Analysis, in:

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