Cracked Rotors (eBook)

Foreword 5
Contents 7
Acronyms 12
Cracks in Rotating Shafts 14
Introduction 14
The Development of Transverse Cracks in Rotating Shafts 17
References 27
Typical Dynamic Behaviour of Cracked Shafts 29
Introduction 29
Analysis of the Vibrations at Rated Speed 29
Thermal Sensitivity of Cracked Rotors 34
Case A – Steam Turbine 34
Case B – Generator 36
Dynamic Behaviour during Speed Transients 38
Case A – Generator 39
Case B – Generator 43
Case C – Test Rig 45
Final Comments 47
References 47
Rotor Testing for Crack Detection 48
Dynamic and Static Tests for the Detection of Cracks in Rotors 48
Dynamic Tests 48
Static Tests 52
Non-destructive Testing for Power Generation Rotors 56
Basic Criteria for the Selection of Inspection Techniques 58
Methods Based on Visual Checks 60
Ultrasonic Inspections (UT) 71
Reliability of NDT Inspections 89
NDT as a Fundamental Aspect of Damage Tolerance Design 95
References 98
Laboratory Tests on Cracked Shafts 102
Introduction 102
Laboratory Test about Breathing Mechanism on a Specimen 102
Description of the Experimental Apparatus 105
Experimental Results 107
Factory Test Results about Thermal Sensitivity on a Steam Turbine 111
Tests at Low Rotating Speed 112
Tests during Speed Transients 115
References 118
Crack Modelling 119
A Review about Crack Modelling 119
Fracture Mechanics Approach and Propagation of Cracks 121
Strain Energy Release Rate Approach 121
Finite Element Modelling of Cracked Elements 127
Fatigue Crack Propagation 136
Modelling the Breathing Behaviour and ts Thermal Sensitivity 141
The Breathing Crack Simplified Model 145
Modelling the Crack 154
Approach Based on Fracture Mechanics (SERR Approach) 154
Approach Based on Energy Balance (EDF) 160
Approach Based on “Equivalent Beam” (FLEX) 163
The 3D Model 171
Comparison of Results Obtained with the Models 172
Basic Modelling of a Cracked Rotor 179
Equations of Motion and Linearization for a Horizontal Shaft with Breathing Crack 184
Stability 188
Vibration Forced by the Crack Influence 193
Crack and Unbalance Response 199
Deeper Cracks 200
Final Remarks 201
Modelling the Cracked Rotor Dynamical Behaviour 201
References 206
Results Obtained Using Simulations 209
Simulations Compared to Experimental Results 209
Results Obtained on EDF EURoPE Test-Rig 209
Results Obtained on PdM Test-Rig 211
Results Obtained on Real Machines 213
Sensitivity of Crack Induced Vibrations to Different Parameters 217
Sensitivity of Crack Excited Vibrations to Crack Depth and Position 217
Effect of Crack Shape 239
Effects of Shear Forces on Cracked Shaft Deflections and Vibrations 246
References 256
Some Special Effects Caused by Cracks 257
Effect of Transverse Cracks on Torsional and Axial Vibrations 257
Static Axial and Torsional Deflections due to Coupling with Bending and Torsional Loads 258
Shift of Natural Torsional Frequency 262
Excitation of Sideband Component in Torsional Vibrations of a Test-Rig Shaft 264
Axial Vibration Excitation of a Test-Rig Shaft 268
Torsional Excitation in a Vertical Axis Centrifugal Pump 269
Excitation of Torsional Vibrations in a 1300 MW Turbo Generator 275
Slant and Helical Cracks 283
Description of the Model 287
Breathing Mechanism 290
Deflections 291
Dynamic Tests 295
Final Remarks about Slant and Helical Cracks 296
Non-linear Behaviours in Cracked Rotors 296
Cracked Rotor Linear and Non-linear Modelling 298
Description of the Model and of the Method 299
Results Obtained with Shaft Loaded by Weight Only 302
Results Obtained with Shaft Loaded by Weight and Unbalance 302
Sub-harmonics 309
References 310
Crack Diagnosis in Rotating Shafts 312
Diagnosis 312
Qualitative Approach 312
Definition of the Fault Matrix 313
Inference 316
Knowledge Representation 320
Symptom Generation 320
Fault Description 328
Fault Matrix 333
Baseline 343
Results for the Qualitative Approach 346
Model Based Approach 349
Definition of Equivalent External Forces to Faults 350
Definition of Equivalent External Forces to Cracks 353
Crack Depth Identification 360
Results of Model Based Approach 366
Test-Rig of Politcnico di Milano 366
Test-Rig of Électricité de France 376
References 401
Index 404