Vibrations of Rotating Machinery: Volume 2. Advanced Rotordynamics: Applications of Analysis, Troubleshooting and Diagnosis

Preface
Contents
1 An Overview of Vibration Problems in Rotating Machinery
	1.1 Structure and Feature of Rotating Machinery
	1.2 Vibration Problems in Rotating Machinery and Countermeasures
	1.3 Guidelines for Health Evaluation
		1.3.1 International Standards on Vibrations of Rotating Machinery
		1.3.2 Allowable Vibration Criteria
		1.3.3 Condition Monitoring of Machines and Certification System of Vibration Diagnostics Engineers
	1.4 Vibrations in Rotating Machinery and Diagnosis
		1.4.1 Causal Relationship Matrix
		1.4.2 Discrimination Between Forced Vibrations and Self-excited Vibrations
2 Basics of Plain Bearings
	2.1 Operating Principles of Plain Bearings
		2.1.1 Features of Plain Bearings
		2.1.2 Oil Film Formation and Generation of Oil Film Pressure
	2.2 Hydrodynamic Lubrication Theory for an Oil Film Bearing
		2.2.1 Coordinates and Differential Equation of Oil Film Pressure
		2.2.2 Derivation of Reynolds Equation
	2.3 Steady-State Characteristics of a Plain Bearing Oil Film
		2.3.1 Infinitely Short Bearing Approximation Solution of Reynolds Equation (Cylindrical Bearing)
		2.3.2 Circumferential Boundary Condition of Pressure
		2.3.3 Equilibrium of Journal Center and Journal Center Locus
		2.3.4 Steady-State Oil Film Reaction Force and Sommerfeld Number
	2.4 Dynamic Characteristics of a Plain Bearing Oil Film
		2.4.1 Oil Film Force of a Cylindrical Bearing
		2.4.2 Linear Stiffness Coefficients and Linear Viscous Damping Coefficients of a Cylindrical Bearing
	2.5 Turbulent Oil Film
	2.6 Actual Plain Bearings
		2.6.1 Various Plain Bearings
		2.6.2 Details of Plain Bearing Specifications
3 Unbalance Vibration of a Rotor in Plain Bearings
	3.1 Feature of Unbalance Vibration
	3.2 Mathematical Expression for Unbalance Vibration
		3.2.1 Expression with X–Y Coordinates
		3.2.2 Complex Displacement Expression z = x + jy
	3.3 Simplified Expressions for the Dynamic Properties of an Oil Film with Unbalance Vibration
	3.4 Effects of Bearing Type and Design Variables
	3.5 Effects of Bearing Pedestal Stiffness
4 Stability of a Rotor in Plain Bearings
	4.1 Cause and Phenomena of Oil Whip
		4.1.1 Cause of Oil Whip
		4.1.2 Oil Whip Phenomena
	4.2 Stability Chart of Circular Bearing Based on Linear Vibration Analysis
		4.2.1 Coordinates and Equation of Motion
		4.2.2 Characteristic Equation and Stability Criterion
		4.2.3 Stability Chart
	4.3 Countermeasures Against Oil Whip in a Cylindrical Bearing
	4.4 Stability Charts of Non-circular Bearings and Oil Whip Countermeasures
	4.5 Explanation of Mechanism of Oil Whip
	4.6 Countermeasures for Actual Journal Bearings and Points of Attention
	4.7 Journal Bearing Specification to Suppress Flow-Excited Vibration
5 Vibration of Rolling Element Bearings
	5.1 Stiffness of Rolling Element Bearings [58, 59, 60]
		5.1.1 Stiffness in the Radial Direction: kr
		5.1.2 Thrust Direction (Axial) Stiffness: kz
	5.2 Excitation Frequency of a Rolling Element Bearing [60, B46]
		5.2.1 Whirling and Rotational Frequencies of Rolling Elements
		5.2.2 Exciting Frequency Caused by a Traveling Ball
		5.2.3 Race Vibration (Ringing) of Rolling Element Bearings
	5.3 Diagnosis of Rolling Element Bearing Vibrations and Signal Processing
		5.3.1 Acceleration Vibration Waveform of Bearing Box
		5.3.2 Amplitude Modulation
		5.3.3 Frequency Band
		5.3.4 Envelope Processing
		5.3.5 Concepts of Bearing Vibration Diagnosis
	5.4 Case Study of Rolling Element-Bearing Vibration Diagnosis [62]
		5.4.1 Degree of Degradation of a Rolling Element Bearing
		5.4.2 Vibration Waveform and FFT Analysis of a Damaged Rolling Element Bearing
6 Vibration in Magnetic Bearing Rotor Systems
	6.1 Functions and Characteristics of Magnetic Bearings
		6.1.1 Magnetic Circuit [91]
		6.1.2 Size of an Active Magnetic Bearing (AMB) [B48, 92]
		6.1.3 Attractive Force of a Magnetic Bearing
		6.1.4 Linearization of a Magnetic Force in Magnetic Bearing
	6.2 ISO Standards Related to Magnetic Bearings [93]
		6.2.1 Terms Related to Active Magnetic Bearings (ISO 14839-1)
		6.2.2 Vibration Evaluation Criteria (ISO 14839-2)
		6.2.3 Evaluation Criteria of Stability Margin (ISO 14839-3)
		6.2.4 Case Study
	6.3 Straight Control of an Active Magnetic Bearing Rotor
		6.3.1 Controller Transfer Function
		6.3.2 Dynamic Characteristics of AMB
		6.3.3 Modeling and Control of an Active Magnetic Bearing Rotor (One AMB)
		6.3.4 Modeling and Control of an AMB Rotor (Two AMBs + Symmetrical Rotor) [97–99, VB550]
	6.4 Cross-Control of an Active Magnetic Bearing
		6.4.1 Cross-Control [100]
		6.4.2 Whirl Vibration and Stability of Cross-Control
		6.4.3 Cross-Control of Unbalance Vibration
7 Case Studies of Forced Vibration Problems of a Rotor
	7.1 Approaches to Resonance Problems in Rotating Machinery
		7.1.1 Natural Frequencies and Damping Ratios Varying with Rotational Speed or Load
		7.1.2 Constant Speed Machine and Variable Speed Machine
		7.1.3 Vibration Amplitude at Critical Speed During Acceleration/Deceleration
		7.1.4 Bending Vibrations and Torsional Vibrations
	7.2 Criteria for Acceptable Vibration Levels of Rotating Machinery
		7.2.1 Transition of the Vibration Standards
		7.2.2 ISO 10816: Mechanical Vibration—Evaluation of Machine Vibration by Measurements on Non-rotating Parts [142]
		7.2.3 ISO 7919: Mechanical Vibration of Non-reciprocating Machines—Measurements on Rotating Shafts and Evaluation Criteria [143]
	7.3 Case Studies of Unbalance Vibration Problems
		7.3.1 Balancing of a Gas Turbine Rotor [VB017]
		7.3.2 Structural Resonance Problems of a Vertical Pump
		7.3.3 Vibration Caused by Low Stiffness of a Fan Framework [VB058]
		7.3.4 Thermal Bending Vibration
	7.4 Case Studies of Forced Vibration in an Asymmetric Rotor
		7.4.1 Secondary Critical Speed of an Asymmetric Rotor
		7.4.2 Elliptical Deformation of Circular Nyquist Plots Due to Rotor Asymmetry
	7.5 Vibrations Induced by Gears
	7.6 Vibration Generated by a Cross Joint
	7.7 Case Studies of Other Forced Vibration
		7.7.1 Electromagnetic Vibration
		7.7.2 Rotating Stall
		7.7.3 Rotor Blade
		7.7.4 Interaction Between the Rotor Blade and the Stator Vane
		7.7.5 Four-Cycle Engine
		7.7.6 Belt Drive Machines
		7.7.7 Reducing Torsional Vibration
8 Case Studies of Self-excited Vibration of Rotor Stability Problems
	8.1 Approaches to Self-excited Vibration Problems in Rotating Machinery
		8.1.1 How to Identify Self-excited Vibrations and Apply Countermeasures
		8.1.2 Examples of Destabilizing Force
	8.2 Self-excited Vibration Problems Caused by Oil Film Bearings or Seals
		8.2.1 Self-excited Vibration Caused by Oil Film Bearing (Oil Whip)
		8.2.2 Self-excited Vibration Caused by Seal (Clearance Flow)
	8.3 Self-excited Vibration Due to Fluid Force of an Impeller
		8.3.1 Self-excited Vibration Caused by Turbine Blades (Axial Flow)
		8.3.2 Self-excited Vibration Caused by Centrifugal Impellers
	8.4 Self-excited Vibration Due to Internal Damping (Hysteresis Whip)
	8.5 Fluid-Containing Rotor
	8.6 Friction Whip
	8.7 Bently–Muszynska Model for Clearance Flow
	8.8 Stabilization with Squeeze Film Damper
		8.8.1 Differences with and Without a Centering Spring
		8.8.2 Feeding/Draining of Oil and End Seals
		8.8.3 Dynamic Characteristics of a Simple Model of a  Squeeze Film Damper
		8.8.4 General Expression for Dynamic Characteristics of Squeeze Film Dampers
9 Torsional Vibration and Related Coupled Vibration
	9.1 Analysis and Measurement of Torsional Vibration
		9.1.1 Single-Shaft System
		9.1.2 Measurement of Torsional Vibration in an Engine Shaft System
		9.1.3 Geared Shaft System
	9.2 Torsional Vibration of a Turbine Generator Set
		9.2.1 Torsional Vibration of a Turbine Generator Shaft System
		9.2.2 Measurement of Torsional Vibration [246]
		9.2.3 Calculation of Torsional Natural Frequencies
	9.3 Blade–Shaft-Coupled Torsional Vibration [246, 247]
		9.3.1 Outset of Blade–Shaft-Coupled Torsional Vibration Problems
		9.3.2 Equivalent Mass Model of Blade Vibration (Nodal Diameter Number κ = 0) [251, 252]
		9.3.3 Example of Blade–Shaft-Coupled Model
		9.3.4 Example of Calculation of Blade–Shaft-Coupled Natural Frequencies
		9.3.5 Effect of Blade–Shaft Coupling
		9.3.6 Verification of Accuracy of Measurement of Blade–Shaft-Coupled Vibration [253, 254]
	9.4 Bending–Torsional Coupled Vibration
		9.4.1 Equation of Motion for Bending–Torsional Coupled Vibration [262, 263]
		9.4.2 Stability of Bending–Torsional Coupled Vibration
10 Signal Processing for Rotor Vibration Diagnosis
	10.1 Vector Monitor (Balance Analyzer)
		10.1.1 What is a Vector Monitor?
		10.1.2 PLL (Phase-Locked Loop)
		10.1.3 Bode Diagram and Nyquist Diagram
	10.2 Signal Processing for Unbalance Vibrations
		10.2.1 Waveform and Orbit of Shaft Center of Rotation for Unbalance Vibration
		10.2.2 Vibration Measurement
		10.2.3 Block Diagram of Unbalance Vibration
		10.2.4 Extraction of Forward Unbalance Vibration Component
		10.2.5 Balancing to Decrease the Forward Unbalance Vibration Amplitude
	10.3 Fourier Series Expansion [274, 275]
		10.3.1 Example: Estimation of an Inherent Cosine/Sine Wave’s Amplitude
		10.3.2 Principles of Fourier Series Expansion
	10.4 Discrete Fourier Transformation [276]
		10.4.1 Principles of the Discrete Fourier Transformation
		10.4.2 Mirror (Aliasing) Phenomenon of Complex Amplitudes
		10.4.3 Sampling Values
		10.4.4 Aliasing Error
		10.4.5 Suppression of Noise
	10.5 Practicalities of FFT Analysis [276, 60]
		10.5.1 Basic Specifications
		10.5.2 Signal Processing Procedure
		10.5.3 DFT of Synchronous and Asynchronous Waveforms
		10.5.4 Fourier Transforms
		10.5.5 Resolution
		10.5.6 Overall (OA)
		10.5.7 Application of FFT Analyzers [277, 278]
	10.6 Zooming of a FFT Analyzer
		10.6.1 FFT of the Original Signal
		10.6.2 Frequency Shifting and LPF
		10.6.3 DFT and Half-Spectrum Displays
		10.6.4 Spectrum Zoom Displays
	10.7 Full Spectrum
		10.7.1 Concept
		10.7.2 One-Channel FFT
		10.7.3 Two FFTs for Full Spectrum
		10.7.4 Example of Vibration Diagnosis Using a Full Spectrum
11 Our Latest Topics Relating to Simplified Modeling of Rotating Systems
	11.1 Practical Techniques Recommended for Model Order Reduction (MOR)
		11.1.1 Procedures for Guyan Reduction
		11.1.2 Mode Synthesis Technique
		11.1.3 Accuracy Comparison Between Guyan and Mode Synthesis Models
		11.1.4 Discrete Modeling for Continuous Medium
		11.1.5 Mode Separation
	11.2 Simplified Prediction of the Stability Limit for an Oil Film Bearing Supported Rotor
		11.2.1 Single-Degree-of-Freedom System and Simplified Stability Criterion [B37, 293]
		11.2.2 Two-DOF System and Undamped Natural Frequency [294]
		11.2.3 Simplified Stability Criterion for Two-Degree-of-Freedom System
		11.2.4 Calculation Example for a Rotor System with Three Disks [295]
		11.2.5 Example of Numerical Calculation for Muszynska’s Rotor System [293]
		11.2.6 Example of Numerical Calculation for Casing Whirl
		11.2.7 Anisotropic Properties of Oil Film Bearing Dynamic Force
	11.3 Vibration Analysis of Blade-and-Shaft Coupled Systems [296]
		11.3.1 Coupling Mass of Mode Synthesis Model
		11.3.2 Modeling of Mode Synthesis Method with 3D Finite Element Method (FEM)
		11.3.3 Mode Synthesis Model of Blade
12 Exercises of ISO Certification Examination for Vibration Experts
	12.1 The First 30 Questions and Multiple Choice Answers
	12.2 Additional 70 Questions
	12.3 Answers and Hints for the Additional Questions 31–100
Appendix A: Spring and Damping Coefficients of a Cylindrical Bearing Assuming Infinitely Short Width
Appendix B: Elliptical Coordinates
Appendix C: Fourier Transformation [291]
	C.1 Definition of Fourier Transformation
		C.2 Examples of Fourier Transformations
		C.3 Comparison of Amplitude Spectrum of Window Function
Appendix D: PLL Circuit and Synchronized Sinusoidal Wave Generation Circuit
Appendix E: Campbell Diagram
Appendix F: Details for Obtaining Eqs. (11.46) and (11.49)
Appendix G: Details for Obtaining Eq. (11.51)
Appendix H: Mode Synthesis Modeling for Rotational Blade Assemblies [299]
	H.1 Variables and Definition
	H.2 Equation of Motion for an Entire Blading System
		H.3 Mode Synthesis Model for Blading
References
R2 References (JSME v_BASE Data Book)
R2 References (Papers)
R2 References (Books)
Index