Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications

On the first edition "...[the authors] present the basic concepts of analyzing the dynamics of rotating machinery.... Their intention is to provide a solid foundation from which students can understand the more complex methods..." (SciTech Book News) "...carefully written...a welcome addition to current reference texts now available in the area of rotordynamics." (Applied Mechanics Review) Authors Ishida and Yamamoto present a broad coverage of rotordynamics with an emphasis on understandign and solving vibration problems. This 2nd edition is completed by three topics of high importance: – A new chapter on vibration Suppression presents various methods and is a helpful guidance for professional engineers. – A chapter on Magnetic Bearings discusses advanced technologies which support the rotor without contact. As they support the highest speeds of any kind of bearings, they are in service in such industrial applications as electric power generation, petroleum refining, machine tool operation and natural gas pipelines. – 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.

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

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.