S E C O N D E D I T I O N

ROTATING MACHINERY V I B R A T I O N

From Analysis to Troubleshooting

MAURICE L. ADAMS, JR Case Western Reserve University

Cleveland, Ohio

C\ CRC Press W ^ У Taylor &. Francis Group

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an inform a business

Contents

Preface xvii Acknowledgments xxi Author xxv

Part I Primer on Rotor Vibration

1 Vibration Concepts and Methods 3 1.1 One-Degree-of-Freedom Model 3

1.1.1 Assumption of Linearity 3 1.1.2 Unforced System 4 1.1.3 Self-Excited Dynamic-Instability Vibrations 6 1.1.4 Steady-State Sinusoidally Forced Systems 7 1.1.5 Damping 8 1.1.6 Undamped Natural Frequency: An Accurate

Approximation 10 1.1.7 1-DOF Model as an Approximation 11

1.2 Multi-DOF Models 13 1.2.1 Two-DOF Models 13 1.2.2 Matrix Bandwidth and Zeros 16 1.2.3 Standard Rotor Vibration Analyses 18

1.3 Modes, Excitation, and Stability of Multi-DOF Models 19 1.3.1 Modal Decomposition 19 1.3.2 Modal Damping 24 1.3.3 Forced Systems Decoupled in Modal

Coordinates 27 1.3.4 Harmonic Excitation of Linear Multi-DOF

Models 27 1.3.5 Dynamic Instability: The Complex

Eigenvalue Problem 28 1.4 Summary 31

2 Lateral Rotor Vibration Analysis Models 35 2.1 Introduction 35 2.2 Simple Linear Models 37

2.2.1 Point-Mass 2-DOF Model 37

viii Contents

2.2.2 Jeffcott Rotor Model 39 2.2.3 Simple Nontrivial 8-DOF Model 41

2.2.3.1 Lagrange Approach (i) 44 2.2.3.2 Lagrange Approach (ii) 49 2.2.3.3 Direct F — ma Approach 52

2.3 Formulations for RDA Software 55 2.3.1 Basic Rotor Finite Element 55 2.3.2 Shaft Element Lumped Mass Matrix 57 2.3.3 Shaft Element Distributed Mass Matrix 58 2.3.4 Shaft Element Consistent Mass Matrix 59 2.3.5 Shaft Element Stiffness Matrix 61 2.3.6 Shaft Element Gyroscopic Matrix 62 2.3.7 Addition of Nonstructural Mass and Inertia to

Rotor Element 62 2.3.8 Matrices for Complete Free-Free Rotor 63 2.3.9 Radial-Bearing and Bearing-Support Models 64

2.3.9.1 Bearing Coefficients Connect Rotor Directly to Ground 67

2.3.9.2 Bearing Coefficients Connect to an Intermediate Mass 68

2.3.10 Completed RDA Model Equations of Motion 70 2.4 Insights into Linear LRVs 70

2.4.1 Systems with Nonsymmetric Matrices 71 2.4.2 Explanation of Gyroscopic Effect 77 2.4.3 Isotropic Model 79 2.4.4 Physically Consistent Models 82 2.4.5 Combined Radial and Misalignment Motions 82

2.5 Nonlinear Effects in Rotor Dynamical Systems 83 2.5.1 Large Amplitude Vibration Sources that Yield

Nonlinear Effects 84 2.5.2 Journal-Bearing Nonlinearity with Large Rotor

Unbalance 85 2.5.3 Unloaded Tilting-Pad Self-Excited Vibration in

Journal Bearings 94 2.5.4 Journal-Bearing Hysteresis Loop 96 2.5.5 Shaft-on-Bearing Impacting 97 2.5.6 Chaos in Rotor Dynamical Systems 99 2.5.7 Nonlinear Damping Masks Oil Whip and

Steam Whirl 100 2.5.7.1 Oil Whip Masked 100 2.5.7.2 Steam Whirl Masked 101

2.5.8 Nonlinear Bearing Dynamics Explains Compressor Bearing Failure 101

2.6 Summary 104

Contents ix

Bibliography 104 Textbooks 104 Selected Papers Concerning Rotor Dynamics Insights 105 Selected Papers on Nonlinear Rotor Dynamics 105

3 Torsional Rotor Vibration Analysis Models I l l 3.1 Introduction I l l 3.2 Rotor-Based Spinning Reference Frames 113 3.3 Single Uncoupled Rotor 113

3.3.1 Lumped and Distributed Mass Matrices 115 3.3.1.1 Lumped Mass Matrix 115 3.3.1.2 Distributed Mass Matrix 116

3.3.2 Stiffness Matrix 117 3.4 Coupled Rotors 119

3.4.1 Coaxial Same-Speed Coupled Rotors 120 3.4.2 Unbranched Systems with Rigid and Flexible

Connections 121 3.4.2.1 Rigid Connections 122 3.4.2.2 Flexible Connections 124 3.4.2.3 Complete Equations of Motion 124

3.4.3 Branched Systems with Rigid and Flexible Connections 126 3.4.3.1 Rigid Connections 127 3.4.3.2 Flexible Connections 129 3.4.3.3 Complete Equations of Motion 129

3.5 Semidefinite Systems 130 3.6 Examples 130

3.6.1 High-Capacity Fan for Large Altitude Wind Tunnel 130

3.6.2 Four-Square Gear Tester 132 3.6.3 Large Steam Turbo-Generator Sets 134

3.7 Summary 135 Bibliography 137

Part II Rotor Dynamic Analyses

4 RDA Code for Lateral Rotor Vibration Analyses 141 4.1 Introduction 141 4.2 Unbalance Steady-State Response Computations 142

4.2.1 3-Mass Rotor Model + 2 Bearings and 1 Disk 145 4.2.2 Phase Angle Explanation and Direction

of Rotation 149

X Contents

4.2.3 3-Mass Rotor Model + 2 Bearings/Pedestals and 1 Disk 152

4.2.4 Anisotropie Model: 3-Mass Rotor + 2 Bearings/ Pedestals and 1 Disk 155

4.2.5 Elliptical Orbits 158 4.2.6 Campbell Diagrams 163

4.3 Instability Self-Excited-Vibration Threshold Computations 165 4.3.1 Symmetric 3-Mass Rotor + 2 Anisotropic Bearings

(Same) and Disk 166 4.3.2 Symmetric 3-Mass Rotor + 2 Anisotropic Bearings

(Different) and Disk 172 4.4 Additional Sample Problems 173

4.4.1 Symmetric 3-Mass Rotor + 2 Anisotropic Bearings and 2 Pedestals 174

4.4.2 Nine-Stage Centrifugal Pump Model, 17-Mass Stations, 2 Bearings 175 4.4.2.1 Unbalance Response 175 4.4.2.2 Instability Threshold Speed 178

4.4.3 Nine-Stage Centrifugal Pump Model, 5-Mass Stations, 2 Bearings 179

4.5 Summary 180 Bibliography 180

5 Bearing and Seal Rotor Dynamics 183 5.1 Introduction 183 5.2 Liquid-Lubricated Fluid-Film Journal Bearings 184

5.2.1 Reynolds Lubrication Equation 184 5.2.1.1 For a Single RLE Solution Point 187

5.2.2 Journal Bearing Stiffness and Damping Formulations 187 5.2.2.1 Perturbation Sizes 189 5.2.2.2 Coordinate Transformation Properties 190 5.2.2.3 Symmetry of Damping Array 192

5.2.3 Tilting-Pad Journal Bearing Mechanics 192 5.2.4 Journal Bearing Stiffness and Damping Data and

Resources 196 5.2.4.1 Tables of Dimensionless Stiffness and

Damping Coefficients 198 5.2.5 Journal Bearing Computer Codes 199 5.2.6 Fundamental Caveat of LRV Analyses 199

5.2.6.1 Example 200 5.3 Experiments to Measure Dynamic Coefficients 201

Contents xi

5.3.1 Mechanical Impedance Method with Harmonic Excitation 203

5.3.2 Mechanical Impedance Method with Impact Excitation 208

5.3.3 Instability Threshold-Based Approach 210 5.4 Annular Seals 212

5.4.1 Seal Dynamic Data and Resources 215 5.4.2 Ungrooved Annular Seals for Liquids 215

5.4.2.1 Lomakin Effect 216 5.4.2.2 Seal Flow Analysis Models 218 5.4.2.3 Bulk Flow Model Approach 219 5.4.2.4 Circumferential Momentum Equation 219 5.4.2.5 Axial Momentum Equation 220 5.4.2.6 Comparisons between Ungrooved Annular

Seals and Journal Bearings 222 5.4.3 Circumferentially Grooved Annular Seals

for Liquids 224 5.4.4 Annular Gas Seals 225

5.4.4.1 Steam Whirl Compared to Oil Whip 226 5.4.4.2 Typical Configurations for Annular

Gas Seals 227 5.4.4.3 Dealing with Seal LRV-Coefficient

Uncertainties 229 5.5 Rolling Contact Bearings 230 5.6 Squeeze-Film Dampers 235

5.6.1 Dampers with Centering Springs 236 5.6.2 Dampers without Centering Springs 237 5.6.3 Limitations of Reynolds Equation-Based

Solutions 238 5.7 Magnetic Bearings 239

5.7.1 Unique Operating Features of Active Magnetic Bearings 240

5.7.2 Short Comings of Magnetic Bearings 241 5.8 Compliance Surface Foil Gas Bearings 243 5.9 Summary 246 Bibliography 246

6 Turbo-Machinery Impeller and Blade Effects 251 6.1 Centrifugal Pumps 251

6.1.1 Static Radial Hydraulic Impeller Force 251 6.1.2 Dynamic Radial Hydraulic Impeller Forces 255

6.1.2.1 Unsteady Flow Dynamic Impeller Forces . . . . 255 6.1.2.2 Interaction Impeller Forces 257

xii Contents

6.2 Centrifugal Compressors 260 6.2.1 Overall Stability Criteria 260 6.2.2 Utilizing Interactive Force Modeling Similarities

with Pumps 262 6.3 High-Pressure Steam Turbines and Gas Turbines 263

6.3.1 Steam Whirl 263 6.3.1.1 Blade Tip Clearance Contribution 264 6.3.1.2 Blade Shroud Annular Seal Contribution . . . . 265

6.3.2 Partial Admission in Steam Turbine Impulse Stages 269

6.3.3 Combustion Gas Turbines 270 6.4 Axial Flow Compressors 270 6.5 Summary 272 Bibliography 272

Part III Monitoring and Diagnostics

7 Rotor Vibration Measurement and Acquisition 277 7.1 Introduction to Monitoring and Diagnostics 277 7.2 Measured Vibration Signals and Associated Sensors 281

7.2.1 Accelerometers 281 7.2.2 Velocity Transducers 283 7.2.3 Displacement Transducers 284

7.2.3.1 Background 284 7.2.3.2 Inductance (Eddy-Current) Noncontacting

Position Sensing Systems 285 7.3 Vibration Data Acquisition 289

7.3.1 Continuously Monitored Large Multibearing Machines 289

7.3.2 Monitoring Several Machines at Regular Intervals . . . 291 7.3.3 Research Laboratory and Shop Test Applications . . . . 292

7.4 Signal Conditioning 292 7.4.1 Filters 293 7.4.2 Amplitude Conventions 294

7.5 Summary 295 Bibliography 295

8 Vibration Severity Guidelines 297 8.1 Introduction 297 8.2 Casing and Bearing Cap Vibration Displacement

Guidelines 298 8.3 Standards, Guidelines, and Acceptance Criteria 300 8.4 Shaft Displacement Criteria 301

Contents xiii

8.5 Summary 302 Bibliography 303 Bibliography Supplement 303

9 Signal Analysis and Identification of Vibration Causes 307 9.1 Introduction 307 9.2 Vibration Trending and Baselines 307 9.3 FFT Spectrum 308 9.4 Rotor Orbit Trajectories 310 9.5 Bode, Polar, and Spectrum Cascade Plots 317 9.6 Wavelet Analysis Tools 321 9.7 Chaos Analysis Tools 325 9.8 Symptoms and Identification of Vibration Causes 330

9.8.1 Rotor Mass Unbalance Vibration 330 9.8.2 Self-Excited Instability Vibrations 331

9.8.2.1 Oil Whip 333 9.8.2.2 Steam Whirl 333 9.8.2.3 Instability Caused by Internal Damping

in the Rotor 334 9.8.2.4 Other Instability Mechanisms 336

9.8.3 Rotor-Stator Rub-Impacting 336 9.8.4 Misalignment 339 9.8.5 Resonance 340 9.8.6 Mechanically Loose Connections 341 9.8.7 Cracked Shafts 342 9.8.8 Rolling-Element Bearings, Gears, and

Vane/Blade-Passing Effects 342 9.9 Summary 343 Bibliography 344

Part IV Trouble-Shooting Case Studies

10 Forced Vibration and Critical Speed Case Studies 349 10.1 Introduction 349 10.2 HP Steam Turbine Passage through First

Critical Speed 350 10.3 HP-IP Turbine Second Critical Speed through

Power Cycling 352 10.4 Boiler Feed Pumps: Critical Speeds at Operating Speed . . . 354

10.4.1 Boiler Feed Pump Case Study 1 354 10.4.2 Boiler Feed Pump Case Study 2 358 10.4.3 Boiler Feed Pump Case Study 3 360

10.5 Nuclear Feed Water Pump Cyclic Thermal Rotor Bow . . . . 361

xiv Contents

10.6 Power Plant Boiler Circulating Pumps 364 10.7 Nuclear Plant Cooling Tower Circulating

Pump Resonance 367 10.8 Generator Exciter Collector Shaft Critical Speeds 367 10.9 Summary 369 Bibliography 370

11 Self-Excited Rotor Vibration Case Studies 371 11.1 Introduction 371 11.2 Swirl Brakes Cure Steam Whirl in a 1300 MW Unit 371 11.3 Bearing Unloaded by Nozzle Forces Allows

Steam Whirl 375 11.4 Misalignment Causes Oil Whip/Steam

Whirl "Duet" 377 11.5 Summary 378 Bibliography 379

12 Additional Rotor Vibration Cases and Topics 381 12.1 Introduction 381 12.2 Vertical Rotor Machines 381 12.3 Vector Turning from Synchronously Modulated Rubs . . . . 384 12.4 Air Preheater Drive Structural Resonances 391 12.5 Aircraft Auxiliary Power Unit Commutator

Vibration-Caused Uneven Wear 393 12.6 Impact Tests for Vibration Problem Diagnoses 397 12.7 Bearing Looseness Effects 398

12.7.1 350 MW Steam Turbine Generator 398 12.7.2 BFP 4000 hp Electric Motor 399 12.7.3 LP Turbine Bearing Looseness on a 750 MW

Steam Turbine Generator 400 12.8 Tilting-Pad versus Fixed-Surface Journal Bearings 401

12.8.1 A Return to the Machine of Section 11.4 of Chapter 11 Case Study 402

12.9 Base-Motion Excitations from Earthquake and Shock . . . . 403 12.10 Parametric Excitation: Nonaxisymmetric Shaft Stiffness . . 404 12.11 Rotor Balancing 406

12.11.1 Static Unbalance, Dynamic Unbalance, and Rigid Rotors 407

12.11.2 Flexible Rotors 408 12.11.3 Influence Coefficient Method 410 12.11.4 Balancing Computer Code Examples and the

Importance of Modeling 412 12.11.5 Case Study of 430 MW Turbine Generator 418

Contents xv

12.11.6 Continuous Automatic In-Service Rotor Balancing 419

12.11.7 In-Service Single-Plane Balance Shot 421 12.12 Summary 422 Bibliography 422

Index 425