Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications, 2nd EditionISBN: 9783527409426
474 pages
December 2012

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, selfexcited vibration, nonstationary vibration, and flowinduced 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.
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 TwoPlane Balancing 69
4.3.1.2 SinglePlane 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 NPlane 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 DoubleFrequency 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 SelfExcited 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 SelfExcited 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 SelfExcited 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 SteadyState 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 SteadyState 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 RollingElement 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 RollingElement 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 SelfExcited 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 SelfExcited 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 SelfExcited Oscillation 229
References 232
10 FlowInduced Vibrations 235
10.1 General Considerations 235
10.2 Oil Whip and Oil Whirl 235
10.2.1 Journal Bearings and SelfExcited 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 SelfExcited 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 SqueezeFilm 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: SteadyState Oscillations of a Teetered TwoBladed 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 LowSpeed 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 SuperSubharmonic 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 SteadyState 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 SteadyState 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 PDControl 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 PIDControl 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 PDControl 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 HighSpeed 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
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 longtime editor of the Journal of Vibration and Control. Professor Ishida has authored about 120 research papers and several books.