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
2
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
Case 2: μ = [0.3, 0.5, 0.7 1], Γ = 0.3, mass ratio = 20.4% ......................... 195 7.5.2
Case 3: μ = [0.3, 0.5, 0.7 1], Γ = 0.4, mass ratio = 20.4% ......................... 196 7.5.3
Case 4: μ = [0.3, 0.5, 0.7 1], Γ = 0.5, mass ratio = 20.4% ......................... 198 7.5.4
Case 5: μ = [0.3, 0.5, 0.7 1], Γ = 0.6, mass ratio = 20.4% ......................... 199 7.5.5
Case 6: μ = [0.3, 0.5, 0.7 1], Γ = 0.7, mass ratio = 20.4% ......................... 200 7.5.6
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 9: μ = [0.3, 0.5, 0.7 1], Γ = 0.3, mass ratio = 40.8% ......................... 206 7.6.3
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
Case 2: Crack Parameters [μ, Γ] = [0.3, 0.53] ............................................ 218 8.4.2
Case 3: Crack Parameters [μ, Γ] = [0.3, 0.79] ............................................ 220 8.4.3
Experimental Test ............................................................................................. 221 8.5
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Experimental Results.................................................................................. 222 8.5.1
Experimental results of Case 1: Crack Parameters [μ, Γ] = [0.3, 0.33] ..... 223 8.5.2
Experimental results of Case 2: Crack Parameters [μ, Γ] = [0.3, 0.53] ..... 224 8.5.3
Experimental results of Case 3: Crack Parameters [μ, Γ] = [0.3, 0.79] ..... 225 8.5.4
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.
9
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
crack of different depth ratios at = 0.3: (a) 1st mode, (b) 2
nd mode, (c)
3rd
mode, (d) 4th
mode. ............................................................................... 131
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)
3rd
mode, (d) 4th
mode. ............................................................................... 132
Figure 5.8: First four theoretical NNF curves of the stationary cracked rotor with a
crack of different depth ratios at = 0.7: (a) 1st mode, (b) 2
nd mode, (c)
3rd
mode, (d) 4th
mode. ............................................................................... 134
Figure 5.9: First four theoretical NNF curves of the stationary cracked rotor with a
crack of different depth ratios at = 0.4: (a) 1st mode, (b) 2
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
Figure 5.14: Comparison of the theoretical and experimental NNF curves for a
cracked shaft with μ = 0.3 at location Γ = 0.3: (a) 1st mode, (b) 2nd
mode, (c) 3rd mode, (d) 4th mode. ............................................................. 141
Figure 5.15: Comparison of the theoretical and experimental NNF curves for a
cracked shaft with μ = 0.3 at location Γ = 0.5: (a) 1st mode, (b) 2nd
mode, (c) 3rd mode, (d) 4th mode. ............................................................. 143
Figure 5.16: Comparison of the theoretical and experimental NNF curves for a
cracked shaft with μ = 0.3 at location Γ = 0.7: (a) 1st mode, (b) 2nd
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
Figure 6.5: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.3: (a), (b) and (c) use
Equation (6.12); (d) uses Equation (6.13). ................................................. 160
Figure 6.6: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.4: (a), (b) and (c) use
Equation (6.12); (d) uses Equation (6.13). ................................................. 162
Figure 6.7: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.5: (a), (b) and (c) use
Equation (6.12); (d) uses Equation (6.13). ................................................. 163
Figure 6.8: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.6: (a), (b) and (c) use
Equation (6.12); (d) uses Equation (6.13). ................................................. 164
12
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). ................................................. 165
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
Equation (6.12); (d) uses Equation (6.13). ................................................. 170
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); (d) uses Equation (6.13). ................................................................ 171
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). .......................................................................................... 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
Figure 7.7: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.3 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. ......................................................... 196
Figure 7.8: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.4 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 .......................................................... 197
Figure 7.9: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.5 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. ......................................................... 198
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)
2nd
mode, (c) 3rd
mode, (d) 4th
mode. ......................................................... 199
Figure 7.11: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.7 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. ......................................................... 201
Figure 7.12: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.3 and a
14
roving disc of mass 0.2 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. ......................................................... 202
Figure 7.13: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor 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
cracked rotor in both the horizontal and vertical planes: (a) 1st mode, (b)
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
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) 3rd
mode, (d) 4th
mode. ............................................................... 205
Figure 7.15: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.3 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)
2nd
mode, (c) 3rd
mode, (d) 4th
mode. ......................................................... 206
Figure 7.16: Normalised orthogonal natural frequency curves of the non-rotating
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)
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
Figure 8.10: First mode normalised natural frequency of the experimental results of
the cracked rotating rotor with a crack of μ = 0.3 at location Γ = 0.53.
Rotational speed range 10-300 rpm. (a) Bottom PZT‘s response. (b)
Right PZT‘s respons. .................................................................................. 225
Figure 8.11: First mode normalised natural frequency of the experimental results of
the cracked rotating rotor with a crack of μ = 0.3 at location Γ = 0.79.
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
diagrams of both the intact and cracked rotating shaft for each disc location.219
Table 8.4: Computed FW and BW crtical speeds of Case 3 from the Campbell
diagrams of both the intact and cracked rotating shaft for each disc location.221
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
mode normalised natural frequency (NNF) curves
4th
mode normalised natural frequency (NNF) curves
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
normalised orthogonal natural frequency (NONF) curves
4 4th
normalised orthogonal natural frequency (NONF) curves
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.
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.
CHAPTER 1: Introduction
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.
CHAPTER 1: Introduction
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.
CHAPTER 1: Introduction
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.
CHAPTER 1: Introduction
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
CHAPTER 1: Introduction
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.
CHAPTER 1: Introduction
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
CHAPTER 1: Introduction
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.
CHAPTER 2: Literature Review
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.
CHAPTER 2: Literature Review
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.
CHAPTER 2: Literature Review
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-
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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.
CHAPTER 2: Literature Review
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
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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.
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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
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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.
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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]
CHAPTER 2: Literature Review
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
CHAPTER 2: Literature Review
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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)
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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)
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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)
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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].
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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:
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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)
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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:
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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].
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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) ( )
( )
( )
. ( ( ))/
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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]:
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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)
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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].
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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)
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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].
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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:
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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.
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
Chapter 3: Mathematical Modelling of Rotor Systems and Theoretical Analysis Tools
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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).
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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).
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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,
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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).
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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.
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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)
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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).
CHAPTER 4: Experimental Testing Rig and Vibration Measuring Instruments
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
Zyad N Haji and S Olutunde Oyadiji
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)
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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
crack of different depth ratios at = 0.3: (a) 1st mode, (b) 2
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
crack of different depth ratios at = 0.3: (a) 1st mode, (b) 2
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.
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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)
Figure 5.8: First four theoretical NNF curves of the stationary cracked rotor with a crack
of different depth ratios at = 0.7: (a) 1st mode, (b) 2
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)
Figure 5.9: First four theoretical NNF curves of the stationary cracked rotor with a crack
of different depth ratios at = 0.4: (a) 1st mode, (b) 2
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)
Figure 5.14: Comparison of the theoretical and experimental NNF curves for a
cracked shaft with μ = 0.3 at location Γ = 0.3: (a) 1st mode, (b) 2nd mode, (c) 3rd
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)
Figure 5.15: Comparison of the theoretical and experimental NNF curves for a
cracked shaft with μ = 0.3 at location Γ = 0.5: (a) 1st mode, (b) 2nd mode, (c) 3rd
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)
Figure 5.16: Comparison of the theoretical and experimental NNF curves for a
cracked shaft with μ = 0.3 at location Γ = 0.7: (a) 1st mode, (b) 2nd mode, (c) 3rd
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
Zyad N Haji and S Olutunde Oyadiji
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
Zyad N Haji and S Olutunde Oyadiji
School of Mechanical, Aerospace and Civil Engineering, University of Manchester,
Manchester M13 9PL, UK
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
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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
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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,
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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
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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
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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.
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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].
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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,
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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.
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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
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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
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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.
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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).
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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)
Figure 6.5: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.3: (a), (b) and (c) use Equation (6.12);
(d) uses Equation (6.13).
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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.
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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)
Figure 6.6: Frequency curve products (FCP) of the cracked rotor with a crack of
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)
Figure 6.7: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.5: (a), (b) and (c) use Equation (6.12); (d)
uses Equation (6.13).
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)
Figure 6.8: Frequency curve products (FCP) of the cracked rotor with a crack of
various depth ratios μ at the location = 0.6: (a), (b) and (c) use Equation (6.12); (d)
uses Equation (6.13).
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)
uses 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
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);
(d) uses Equation (6.13).
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
Zyad N Haji and S Olutunde Oyadiji
Journal of Sound and Vibration, Vol. 333, Issue 23, 2014, PP 6237–6257
doi:10.1016/j.jsv.2014.05.046
180
The Use of Roving Discs and Orthogonal Natural Frequencies for Crack
Identification and Location in Rotors
Zyad N Haji and S Olutunde Oyadiji
School of Mechanical, Aerospace and Civil Engineering, University of Manchester,
Manchester M13 9PL, UK
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.
Case 2: μ = [0.3, 0.5, 0.7 1], Γ = 0.3, mass ratio = 20.4% 7.5.2
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.
Case 3: μ = [0.3, 0.5, 0.7 1], Γ = 0.4, mass ratio = 20.4% 7.5.3
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
Figure 7.7: Normalised orthogonal natural frequency curves of the non-rotating cracked
rotor cracks of different depth ratios located at = 0.3 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.
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.
Figure 7.8: Normalised orthogonal natural frequency curves of the non-rotating cracked
rotor cracks of different depth ratios located at = 0.4 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
198
Case 4: μ = [0.3, 0.5, 0.7 1], Γ = 0.5, mass ratio = 20.4% 7.5.4
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
Figure 7.9: Normalised orthogonal natural frequency curves of the non-rotating cracked
rotor cracks of different depth ratios located at = 0.5 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.
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.
Case 5: μ = [0.3, 0.5, 0.7 1], Γ = 0.6, mass ratio = 20.4% 7.5.5
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.
Case 6: μ = [0.3, 0.5, 0.7 1], Γ = 0.7, mass ratio = 20.4% 7.5.6
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
Figure 7.11: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor cracks of different depth ratios located at = 0.7 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.
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
Figure 7.12: Normalised orthogonal natural frequency curves of the non-rotating cracked
rotor cracks of different depth ratios located at = 0.3 and a roving disc of mass 0.2 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.
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).
Figure 7.13: Normalised orthogonal natural frequency curves of the non-rotating
cracked rotor 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 cracked rotor in both the
horizontal and vertical planes: (a) 1st mode, (b) 2
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
Case 9: μ = [0.3, 0.5, 0.7 1], Γ = 0.3, mass ratio = 40.8% 7.6.3
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
Figure 7.15: Normalised orthogonal natural frequency curves of the non-rotating cracked
rotor cracks of different depth ratios located at = 0.3 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.
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.
Figure 7.16: Normalised orthogonal natural frequency curves of the non-rotating
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
Case 10: μ = [0.3, 0.5, 0.7 1], Γ = 0.5, mass ratio = 40.8% 7.6.4
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,
CHAPTER 8: Crack Identification in Rotating Rotors
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
CHAPTER 8: Crack Identification in Rotating Rotors
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.
CHAPTER 8: Crack Identification in Rotating Rotors
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
CHAPTER 8: Crack Identification in Rotating Rotors
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
BW critical speeds (Hz)
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
CHAPTER 8: Crack Identification in Rotating Rotors
217
0 500 1000 1500 2000 2500 30000
10
20
30
40
50
Rotor spin speed (rev/min)
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
Rotor spin speed (rev/min)
Und
ampe
d na
tura
l fr
eque
ncie
s (H
z)
Forward whirl is sequar; Backward whirl is star.
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)
Forward whirl is sequar; Backward whirl is star.
0 500 1000 1500 2000 2500 30000
10
20
30
40
50
Rotor spin speed (rev/min)
Und
ampe
d na
tura
l fr
eque
ncie
s (H
z)
Forward whirl is sequar; Backward whirl is star.
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.
CHAPTER 8: Crack Identification in Rotating Rotors
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.
Case 2: Crack Parameters [μ, Γ] = [0.3, 0.53] 8.4.2
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.
CHAPTER 8: Crack Identification in Rotating Rotors
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
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)
47.121 43.5348 39.696 36.9007 35.3769 35.9333 35.0628 38.0478 41.4066 45.4058
BW critical speeds (Hz)
45.308 42.374 39.1455 36.6986 35.328 35.0414 35.8311 37.7113 40.5931 43.8741
CHAPTER 8: Crack Identification in Rotating Rotors
220
Case 3: Crack Parameters [μ, Γ] = [0.3, 0.79] 8.4.3
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.
CHAPTER 8: Crack Identification in Rotating Rotors
221
Table 8.4: Computed FW and BW crtical speeds of Case 3 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)
47.1238 43.5373 39.6984 36.903 35.3793 35.0652 35.9356 38.0491 41.4077 45.4079
BW critical speeds (Hz)
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
CHAPTER 8: Crack Identification in Rotating Rotors
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
CHAPTER 8: Crack Identification in Rotating Rotors
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.
CHAPTER 8: Crack Identification in Rotating Rotors
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.
Experimental results of Case 2: Crack Parameters [μ, Γ] = [0.3, 0.53] 8.5.3
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.
CHAPTER 8: Crack Identification in Rotating Rotors
225
Experimental results of Case 3: Crack Parameters [μ, Γ] = [0.3, 0.79] 8.5.4
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)
Figure 8.10: First mode normalised natural frequency of the experimental results of the
cracked rotating rotor with a crack of μ = 0.3 at location Γ = 0.53. Rotational speed range
10-300 rpm. (a) Bottom PZT‘s response. (b) Right PZT‘s respons.
Bottom PZTs Right PZTs
CHAPTER 8: Crack Identification in Rotating Rotors
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.
Figure 8.11: First mode normalised natural frequency of the experimental results of the
cracked rotating rotor with a crack of μ = 0.3 at location Γ = 0.79. Rotational speed range
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
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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.
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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.
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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.
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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:
CHAPTER 9: Discussions, Summary, Conclusions and Prospective Studies
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)
APPENDIX A: Design and Dimensions of the Rotating Test Rig
236
Figure 9.5: Motor support (Dimensions in mm)
Figure 9.6: Dimensions of disc (mm).
APPENDIX A: Design and Dimensions of the Rotating Test Rig
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
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