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Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications, 2nd Edition

ISBN: 978-3-527-40942-6
474 pages
December 2012
Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications, 2nd Edition (3527409424) cover image
A wide-ranging treatment of fundamental rotordynamics in order to serve engineers with the necessary knowledge to eliminate various vibration problems.
New to this edition are three chapters on highly significant topics:
Vibration Suppression - The chapter presents various methods and is a helpful guidance for professional engineers.
Magnetic Bearings - The chapter provides fundamental knowledge and enables the reader to realize simple magnetic bearings in the laboratory.
Some Practical Rotor Systems - The chapter explains various vibration characteristics of steam turbines and wind turbines.
The contents of other chapters on Balancing, Vibrations due to Mechanical Elements, and Cracked Rotors are added to and revised extensively.

The authors provide a classification of rotating shaft systems and general coverage of key ideas common to all branches of rotordynamics. They offers a unique analysis of dynamical problems, such as nonlinear rotordynamics, self-excited vibration, nonstationary vibration, and flow-induced oscillations. Nonlinear resonances are discussed in detail, as well as methods for shaft stability and various theoretical derivations and computational methods for analyzing rotors to determine and correct vibrations.
This edition also includes case studies and problems.
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Foreword to the First Edition XVII

Preface to the First Edition XIX

Preface to the Second Edition XXIII

Acknowledgements XXV

1 Introduction 1

1.1 Classification of Rotor Systems 1

1.2 Historical Perspective 3

References 8

2 Vibrations of Massless Shafts with Rigid Disks 11

2.1 General Considerations 11

2.2 Rotor Unbalance 11

2.3 Lateral Vibrations of an Elastic Shaft with a Disk at Its Center 13

2.3.1 Derivation of Equations of Motion 13

2.3.2 Free Vibrations of an Undamped System and Whirling Modes 14

2.3.3 Synchronous Whirl of an Undamped System 16

2.3.4 Synchronous Whirl of a Damped System 20

2.3.5 Energy Balance 22

2.4 Inclination Vibrations of an Elastic Shaft with a Disk at Its Center 23

2.4.1 Rotational Equations of Motion for Single Axis Rotation 23

2.4.2 Equations of Motion 23

2.4.3 Free Vibrations and Natural Angular Frequency 27

2.4.4 Gyroscopic Moment 29

2.4.5 Synchronous Whirl 33

2.5 Vibrations of a 4 DOF System 34

2.5.1 Equations of Motion 34

2.5.1.1 Derivation by Using the Results of 2 DOF System 35

2.5.1.2 Derivation by Lagrange’s Equations 37

2.5.2 Free Vibrations and a Natural Frequency Diagram 40

2.5.3 Synchronous Whirling Response 42

2.6 Vibrations of a Rigid Rotor 43

2.6.1 Equations of Motion 43

2.6.2 Free Whirling Motion and Whirling Modes 45

2.7 Approximate Formulas for Critical Speeds of a Shaft with Several Disks 46

2.7.1 Rayleigh’s Method 47

2.7.2 Dunkerley’s Formula 48

References 48

3 Vibrations of a Continuous Rotor 49

3.1 General Considerations 49

3.2 Equations of Motion 50

3.3 Free Whirling Motions and Critical Speeds 55

3.3.1 Analysis Considering Only Transverse Motion 56

3.3.2 Analysis Considering the Gyroscopic Moment and Rotary Inertia 58

3.3.3 Major Critical Speeds 59

3.4 Synchronous Whirl 60

References 65

4 Balancing 67

4.1 Introduction 67

4.2 Classification of Rotors 67

4.3 Balancing of a Rigid Rotor 69

4.3.1 Principle of Balancing 69

4.3.1.1 Two-Plane Balancing 69

4.3.1.2 Single-Plane Balancing 70

4.3.2 Balancing Machine 71

4.3.2.1 Static Balancing Machine 71

4.3.2.2 Dynamic Balancing Machine 71

4.3.3 Field Balancing 75

4.3.4 Various Expressions of Unbalance 77

4.3.4.1 Resultant Unbalance U and Resultant Unbalance MomentV 77

4.3.4.2 Dynamic Unbalance (U1,U2) 79

4.3.4.3 Static Unbalance U and Couple Unbalance [Uc,−Uc] 80

4.3.5 Balance Quality Grade of a Rigid Rotor 82

4.3.5.1 Balance Quality Grade 82

4.3.5.2 How to Use the Standards 84

4.4 Balancing of a Flexible Rotor 86

4.4.1 Effect of the Elastic Deformation of a Rotor 86

4.4.2 Modal Balancing Method 87

4.4.2.1 N-Plane Modal Balancing 88

4.4.2.2 (N + 2)-Plane Modal Balancing 90

4.4.3 Influence Coefficient Method 90

References 92

5 Vibrations of an Asymmetrical Shaft and an Asymmetrical Rotor 93

5.1 General Considerations 93

5.2 Asymmetrical Shaft with a Disk at Midspan 94

5.2.1 Equations of Motion 94

5.2.2 Free Vibrations and Natural Frequency Diagrams 95

5.2.2.1 Solutions in the Rangesω > ωc1 andω < ωc2 98

5.2.2.2 Solutions in the Range ωc1 > ω > ωc2 99

5.2.3 Synchronous Whirl in the Vicinity of the Major Critical Speed 100

5.3 Inclination Motion of an Asymmetrical Rotor Mounted on a Symmetrical Shaft 102

5.3.1 Equations of Motion 103

5.3.2 Free Vibrations and a Natural Frequency Diagram 108

5.3.3 Synchronous Whirl in the Vicinity of the Major Critical Speed 109

5.4 Double-Frequency Vibrations of an Asymmetrical Horizontal Shaft 110

References 113

6 Nonlinear Vibrations 115

6.1 General Considerations 115

6.2 Causes and Expressions of Nonlinear Spring Characteristics: Weak Nonlinearity 115

6.3 Expressions of Equations of Motion Using Physical and Normal Coordinates 121

6.4 Various Types of Nonlinear Resonances 123

6.4.1 Harmonic Resonance 124

6.4.1.1 Solution by the Harmonic Balance Method 124

6.4.1.2 Solution Using Normal Coordinates 128

6.4.2 Subharmonic Resonance of Order 1/2 of a Forward Whirling Mode 130

6.4.3 Subharmonic Resonance of Order 1/3 of a Forward Whirling Mode 132

6.4.4 Combination Resonance 133

6.4.5 Summary of Nonlinear Resonances 136

6.5 Nonlinear Resonances in a System with Radial Clearance: Strong Nonlinearity 139

6.5.1 Equations of Motion 141

6.5.2 Harmonic Resonance and Subharmonic Resonances 142

6.5.3 Chaotic Vibrations 144

6.6 Nonlinear Resonances of a Continuous Rotor 145

6.6.1 Representations of Nonlinear Spring Characteristics and Equations of Motion 146

6.6.2 Transformation to Ordinary Differential Equations 149

6.6.3 Harmonic Resonance 150

6.6.4 Summary of Nonlinear Resonances 151

6.7 Internal Resonance Phenomenon 152

6.7.1 Examples of the Internal Resonance Phenomenon 152

6.7.2 Subharmonic Resonance of Order 1/2 153

6.7.3 Chaotic Vibrations in the Vicinity of the Major Critical Speed 156

References 158

7 Self-Excited Vibrations due to Internal Damping 161

7.1 General Considerations 161

7.2 Friction in Rotor Systems and Its Expressions 161

7.2.1 External Damping 162

7.2.2 Hysteretic Internal Damping 162

7.2.3 Structural Internal Damping 167

7.3 Self-Excited Vibrations due to Hysteretic Damping 168

7.3.1 System with Linear Internal Damping Force 169

7.3.2 System with Nonlinear Internal Damping Force 171

7.4 Self-Excited Vibrations due to Structural Damping 173

References 176

8 Nonstationary Vibrations during Passage through Critical Speeds 177

8.1 General Considerations 177

8.2 Equations of Motion for Lateral Motion 178

8.3 Transition with Constant Acceleration 179

8.4 Transition with Limited Driving Torque 183

8.4.1 Characteristics of Power Sources 183

8.4.2 Steady-State Vibration 184

8.4.3 Stability Analysis 187

8.4.4 Nonstationary Vibration 188

8.5 Analysis by the Asymptotic Method (Nonlinear System, Constant Acceleration) 189

8.5.1 Equations of Motion and Their Transformation to a Normal Coordinate Expression 190

8.5.2 Steady-State Solution 192

8.5.3 Nonstationary Vibration 194

References 196

9 Vibrations due to Mechanical Elements 199

9.1 General Considerations 199

9.2 Ball Bearings 199

9.2.1 Vibration and Noise in Rolling-Element Bearings 199

9.2.1.1 Vibrations due to the Passage of Rolling Elements 200

9.2.1.2 Natural Vibrations of Outer Rings 202

9.2.1.3 Geometrical Imperfection 204

9.2.1.4 Other Noises 205

9.2.2 Resonances of a Rotor Supported by Rolling-Element Bearings 205

9.2.2.1 Resonances due to Shaft Eccentricity 205

9.2.2.2 Resonances due to the Directional Difference in Stiffness 206

9.2.2.3 Vibrations of a Horizontal Rotor due to the Passage of Rolling Elements 208

9.2.2.4 Vibrations due to the Coexistence of the Passage of Rolling Elements and a Shaft Initial Bend 208

9.3 Bearing Pedestals with Directional Difference in Stiffness 209

9.4 Universal Joint 211

9.5 Rubbing 215

9.5.1 Equations of Motion 217

9.5.2 Numerical Simulation 218

9.5.3 Theoretical Analysis 220

9.5.3.1 Forward Rubbing 220

9.5.3.2 Backward Rubbing 221

9.6 Self-Excited Oscillation in a System with a Clearance between Bearing and Housing 222

9.6.1 Experimental Setup and Experimental Results 223

9.6.2 Analytical Model and Reduction of Equations of Motion 224

9.6.3 Numerical Simulation 226

9.6.4 Self-Excited Oscillations 227

9.6.4.1 Analytical Model and Equations of Motion 227

9.6.4.2 Stability of a Synchronous Whirl 228

9.6.4.3 Mechanism of a Self-Excited Oscillation 229

References 232

10 Flow-Induced Vibrations 235

10.1 General Considerations 235

10.2 Oil Whip and Oil Whirl 235

10.2.1 Journal Bearings and Self-Excited Vibrations 236

10.2.2 Reynolds Equation 239

10.2.3 Oil Film Force 240

10.2.3.1 Short Bearing Approximation 241

10.2.3.2 Long Bearing Approximation 243

10.2.4 Stability Analysis of an Elastic Rotor 243

10.2.5 Oil Whip Prevention 246

10.3 Seals 248

10.3.1 Plain Annular Seal 248

10.3.2 Labyrinth Seal 251

10.4 Tip Clearance Excitation 251

10.5 Hollow Rotor Partially Filled with Liquid 252

10.5.1 Equations Governing Fluid Motion and Fluid Force 254

10.5.2 Asynchronous Self-Excited Whirling Motion 256

10.5.3 Resonance Curves at the Major Critical Speed (Synchronous Oscillation) 257

References 261

11 Vibration Suppression 263

11.1 Introduction 263

11.2 Vibration Absorbing Rubber 263

11.3 Theory of Dynamic Vibration Absorber 263

11.4 Squeeze-Film Damper Bearing 264

11.5 Ball Balancer 266

11.5.1 Fundamental Characteristics and the Problems 266

11.5.2 Countermeasures to the Problems 268

11.6 Discontinuous Spring Characteristics 271

11.6.1 Fundamental Characteristics and the Problems 271

11.6.2 Countermeasures to the Problems 273

11.6.3 Suppression of Unstable Oscillations of an Asymmetrical Shaft 274

11.7 Leaf Spring 276

11.8 Viscous Damper 277

11.9 Suppression of Rubbing 278

References 280

12 Some Practical Rotor Systems 283

12.1 General Consideration 283

12.2 Steam Turbines 283

12.2.1 Construction of a Steam Turbine 283

12.2.2 Vibration Problems of a Steam Turbine 286

12.2.2.1 Poor Accuracy in the Manufacturing of Couplings 286

12.2.2.2 Thermal Bow 287

12.2.2.3 Vibrations of Turbine Blades 287

12.2.2.4 Oil Whip and Oil Whirl 290

12.2.2.5 Labylinth Seal 290

12.2.2.6 Steam Whirl 290

12.3 Wind Turbines 290

12.3.1 Structure of a Wind Turbine 290

12.3.2 Campbell Diagram of a Wind Turbine with Two Teetered Blades 292

12.3.3 Excitation Forces in Wind Turbines 294

12.3.4 Example: Steady-State Oscillations of a Teetered Two-Bladed Wind Turbine 295

12.3.4.1 Wind Velocity 296

12.3.4.2 Vibration of the Tower 296

12.3.4.3 Flapwise Bending Vibration of the Blade 297

12.3.4.4 Chordwise Bending Vibration of the Blade 297

12.3.4.5 Torque Variation of the Low-Speed Shaft 297

12.3.4.6 Variation of the Teeter Angle 297

12.3.4.7 Variation of the Pitch Angle 297

12.3.4.8 Gear 297

12.3.5 Balancing of a Rotor 298

12.3.6 Vibration Analysis of a Blade Rotating in a Vertical Plane 299

12.3.6.1 Derivation of Equations of Motion 299

12.3.6.2 Natural Frequencies 302

12.3.6.3 Forced Oscillation 302

12.3.6.4 Parametrically Excited Oscillation 303

References 305

13 Cracked Rotors 307

13.1 General Considerations 307

13.2 Modeling and Equations of Motion 309

13.2.1 Piecewise Linear Model (PWL Model) 309

13.2.2 Power Series Model (PS Model) 311

13.3 Numerical Simulation (PWL Model) 312

13.3.1 Horizontal Rotor 312

13.3.2 Vertical Rotor 313

13.4 Theoretical Analysis (PS Model) 313

13.4.1 Forward Harmonic Resonance [+ω] (Horizontal Rotor) 313

13.4.2 Forward Harmonic Resonance [+ω] (Vertical Rotor) 315

13.4.3 Forward Superharmonic Resonance [+2ω] (Horizontal Rotor) 315

13.4.4 Other Kinds of Resonance 317

13.4.4.1 Backward Harmonic Resonance [−ω] 317

13.4.4.2 Forward Superharmonic Resonance [+3ω] 317

13.4.4.3 Forward Subharmonic Resonance [+(1/2)ω] 318

13.4.4.4 Forward Super-Subharmonic Resonance [+(3/2)ω] 319

13.4.4.5 Combination Resonance 320

13.5 Case History in Industrial Machinery 321

References 324

14 Finite Element Method 327

14.1 General Considerations 327

14.2 Fundamental Procedure of the Finite Element Method 327

14.3 Discretization of a Rotor System 328

14.3.1 Rotor Model and Coordinate Systems 328

14.3.2 Equations of Motion of an Element 329

14.3.2.1 Rigid Disk 329

14.3.2.2 Finite Rotor Element 330

14.3.3 Equations of Motion for a Complete System 336

14.3.3.1 Model I: (Uniform Elastic Rotor) 336

14.3.3.2 Model II: Disk–Shaft System 340

14.3.3.3 Variation of Equations of Motion 343

14.4 Free Vibrations: Eigenvalue Problem 345

14.5 Forced Vibrations 347

14.6 Alternative Procedure 349

References 350

15 Transfer Matrix Method 351

15.1 General Considerations 351

15.2 Fundamental Procedure of the Transfer Matrix Method 351

15.2.1 Analysis of Free Vibration 351

15.2.2 Analysis of Forced Vibration 355

15.3 Free Vibrations of a Rotor 359

15.3.1 State Vector and Transfer Matrix 359

15.3.2 Frequency Equation and the Vibration Mode 364

15.3.3 Examples 365

15.3.3.1 Model I: Uniform Continuous Rotor 365

15.3.3.2 Model II: Disk–Shaft System 366

15.4 Forced Vibrations of a Rotor 367

15.4.1 External Force and Extended Transfer Matrix 367

15.4.2 Steady-State Solution 370

15.4.3 Example 371

References 371

16 Measurement and Signal Processing 373

16.1 General Considerations 373

16.2 Measurement and Sampling Problem 374

16.2.1 Measurement System and Digital Signal 374

16.2.2 Problems in Signal Processing 375

16.3 Fourier Series 376

16.3.1 Real Fourier Series 376

16.3.2 Complex Fourier Series 376

16.4 Fourier Transform 378

16.5 Discrete Fourier Transform 379

16.6 Fast Fourier Transform 383

16.7 Leakage Error and Countermeasures 383

16.7.1 Leakage Error 383

16.7.2 Countermeasures for Leakage Error 384

16.7.2.1 Window Function 384

16.7.2.2 Prevention of Leakage by Coinciding Periods 385

16.8 Applications of FFT to Rotor Vibrations 386

16.8.1 Spectra of Steady-State Vibration 386

16.8.1.1 Subharmonic Resonance of Order 1/2 of a Forward Whirling Mode 386

16.8.1.2 Combination Resonance 388

16.8.2 Nonstationary Vibration 388

References 391

17 Active Magnetic Bearing 393

17.1 General Considerations 393

17.2 Magnetic Levitation and Earnshaw’s Theorem 393

17.3 Active Magnetic Levitation 394

17.3.1 Levitation Model 394

17.3.2 Current Control with PD-Control 396

17.3.2.1 Physical Meanings of PD Control 397

17.3.2.2 Transfer Function and Stability Condition 397

17.3.2.3 Determination of Gains 398

17.3.2.4 Case with a Static Load 399

17.3.3 Current Control with PID-Control 399

17.3.3.1 Transfer Function and Stability Condition 399

17.3.3.2 Determination of Gains 400

17.3.3.3 Case with a Static Load 400

17.3.4 Practical Examples of Levitation 401

17.3.4.1 Identification of System Parameters 401

17.3.4.2 Digital PD-Control with DSP 402

17.3.5 Current Control with State Feedback Control 403

17.4 Active Magnetic Bearing 405

17.4.1 Principle of an Active Magnetic Bearing 405

17.4.2 Active Magnetic Bearings in a High-Speed Spindle System 405

17.4.3 Dynamics of a Rigid Rotor system 406

References 408

Appendix A Moment of Inertia and Equations of Motion 409

Appendix B Stability above the Major Critical Speed 413

Appendix C Derivation of Equations of Motion of a 4 DOF Rotor System by Using Euler Angles 415

Appendix D Asymmetrical Shaft and Asymmetrical Rotor with Four Degrees of Freedom 421

D.1 4 DOF Asymmetrical Shaft System 421

D.2 4 DOF Asymmetrical Rotor System 423

Reference 425

Appendix E Transformation of Equations of Motion to Normal Coordinates: 4 DOF Rotor System 427

E.1 Transformation of Equations of Motion to Normal Coordinates 427

E.2 Nonlinear Terms 428

References 429

Appendix F Routh–Hurwitz Criteria for Complex Expressions 431

References 432

Appendix G FFT Program 433

References 435

Index 437

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Toshio Yamamoto (1921-2007) was professor at Nagoya University. He was an internationally acknowledged expert for Rotor Dynamics and Nonlinear Vibrations. Professor Yamamoto published about 130 research papers and a number of books. For his academic achievements, he received one of the highests awards of the Japanese Society of Mechanical Engineering (JSME).

Yukio Ishida (born 1948) is professor at Nagoya University. His main research fields are Rotor Dynamics, Nonlinear Dynamics and Vibration Suppressions. During his academic career, he received the Pioneer Award of the Japanese Society of Mechanical Engineering JSME (2001), the Nagai Scientific Foundation Award (2003), the JSME Medal for Outstanding Paper (2006), and the JSME Education Award (2006). Yukio Ishida is editor of the JSME Journal of System Design and Dynamics, and a long-time editor of the Journal of Vibration and Control. Professor Ishida has authored about 120 research papers and several books.
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