Vibration Control Engineering : Passive and Feedback Systems

"This book applies vibration engineering to turbomachinery, covering installation, maintenance and operation. With a practical approach based on clear theoretical principles and formulas, the book is an essential how-to guide for all professional engineers dealing with vibration issues within turbomachinery. Vibration problems in turbines, large fans, blowers, and other rotating machines are common issues within turbomachinery. Applicable to industries such as oil and gas mining, cement, pharmaceutical and naval engineering, the ability to predict vibration based on frequency spectrum patterns is essential for many professional engineers. In this book, the theory behind vibration is clearly detailed, providing an easy to follow methodology through which to calculate vibration propagation. Describing lateral and torsional vibration and how this impacts turbine shaft integrity, the book uses mechanics of materials theory and formulas alongside the matrix method to provide clear solutions to vibration problems. Additionally, it describes how to carry out a risk assessment of vibration fatigue. Other topics covered include vibration control techniques, the design of passive and active absorbers and rigid, non-rigid and Z foundations. The book will be of interest to professionals working with turbomachinery, naval engineering corps and those working on ISO standards 10816 and 13374. It will also aid mechanical engineering students working on vibration and machine design."--Back cover Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface Acknowledgements About the Author Abbreviations Part I: Vibration Theory of Sdof, Mdof and Continuous Dynamic Systems Chapter 1: Dynamics of Linear SDOF Systems 1.1 Introduction to Machine’s Vibration 1.2 The Basics of Vibrating Systems 1.2.1 Time Response 1.2.1.1 Transient Response Classification 1.2.2 Frequency Response 1.2.3 Vibration Graphical Representation 1.2.3.1 Resonance 1.2.4 Friction Damping 1.2.5 Vibration Causes and Consequences 1.3 Linear Mechanical System Description 1.4 Equation of Motion of Dynamic Systems 1.4.1 Vector Interpretation of the Equation of Motion 1.5 Natural frequency 1.5.1 The Natural Frequency of Linear Systems 1.5.2 The Natural Frequency of Rotating Systems 1.6 Natural Response of Second-Order Systems 1.7 Derivation of the Time Natural Response 1.7.1 Damping Ratio and Damped Frequency 1.7.2 Natural Transient Response Formula 1.7.3 Vector Interpretation of the Natural Time Response 1.7.4 Concepts to Remember Regarding Second-Order Systems 1.7.5 Natural Response and Decay Curves 1.7.5.1 Settling Time and Number of Cycles 1.7.5.2 Decay Ratio 1.7.5.3 The First Peak Time 1.7.5.4 Practical Assessment of Time Parameters 1.8 Transient Response to a Step Force Input 1.8.1 Conceptual Description 1.8.2 Transient Response Formula 1.8.2.1 Equation of Motion for a Step Input Force 1.8.2.2 Natural Response to a Step Input 1.8.3 Transient Response Overshoots to a Unit Step Input 1.9 Transient Response to a Harmonic Force Input 1.9.1 Conservative Vibrating System 1.9.1.1 Resonance of the Forced Response 1.9.2 Non-Conservative Vibrating System 1.9.2.1 Permanent Forced Response 1.9.2.2 Total Vibration 1.9.3 Practical Assessment of a Transient Response 1.9.3.1 Technical Assessment Summary 1.10 Frequency Response 1.10.1 Frequency Response of Second-Order Systems 1.10.2 Frequency Response Charts of Second-Order Systems 1.10.3 Resonance Parameters 1.11 Fundamental vibration forms 1.11.1 Externally Excited Mode 1.11.2 Self-Excited Mode 1.11.2.1 Note About the Recommended Velocities Range 1.11.3 Base-Excited Form 1.11.4 Transmitted Force Mode 1.11.5 Comparison of the Four Fundamental Vibration Forms Notes Chapter 2: Dynamics of Rotating SDOF Systems 2.1 Introduction to Torsional Vibration 2.2 Torsional Vibration of SDOF Systems 2.2.1 Torsional System Response 2.2.1.1 Natural Frequency of Rotating Systems 2.2.1.1.1 Natural Frequency of a Rotor-Shaft Assembly 2.2.1.2 Damping Ratio ζ 2.2.2 Transient Response With a Step Torque Input 2.2.2.1 Torsional Natural Response 2.2.2.2 Transient Response to a Step Torque 2.2.3 Velocity Transient of a Turbine-Generator Set 2.2.4 Frequency Response 2.2.5 Torsional Stress Under Vibration 2.2.6 Cumulative Fatigue Generated by Turbomachines Startup 2.2.7 Multidisciplinary Assessment of Torsional Vibration 2.2.7.1 Technical Scenario 2.2.7.2 Calculation Model 2.2.7.3 Technical Summary Notes Chapter 3: Dynamics of Linear and Rotating MDOF and Continuous Systems 3.1 Introduction to MDOF and Continuous Systems 3.1.1 Discrete Multi-Degree of Freedom Systems 3.1.2 Continuous Systems 3.1.2.1 Stress Waves and Propagation Velocity 3.2 Linear Multi-Degree of Freedom Systems 3.2.1 Matrix Model of Multi-Degree Systems 3.2.2 Natural Frequencies of a System with Three Degrees of Freedom 3.3 Rotating Multi-Degree of Freedom Systems 3.3.1 Natural Frequencies of Two Degrees of Freedom System 3.3.2 Practical Assessment of Natural Frequencies 3.4 The Euler-Bernoulli Equation 3.4.1 Deflections and Efforts at Beam’s Supports 3.4.1.1 Boundary Conditions at Beam Supports 3.4.2 Derivation of the Euler-Bernoulli Equation 3.4.3 Solution to the Euler-Bernoulli Equation 3.4.3.1 Solution to the Spatial Equation 3.4.3.2 Beam’s Vibration Shapes 3.4.3.3 Solution to the Temporal Equation 3.4.3.4 General Solution of the Euler-Bernoulli Equation 3.4.4 Natural Frequencies with the Euler-Bernoulli Equation 3.4.5 Practical Assessment. Turbogenerator Set Frequencies 3.5 The Wave Equation 3.5.1 Derivation of the Wave Equation 3.5.2 Solution to the Wave Equation 3.5.2.1 Solution to the Spatial Equation 3.5.2.2 Solution to the Temporal Equation 3.5.2.3 General solution of the wave equation 3.5.3 Torsional Natural Frequencies With the Wave Equation 3.5.4 Practical Assessment. Oil Drill Rig Notes Part II: Turbo Machines and Ship Vibrations Chapter 4: Critical Velocity of Turbomachines 4.1 Introduction to the Critical Velocity 4.1.1 Calculation and Measurement of the Resonant Frequency 4.1.2 Type of Rotors 4.2 Rayleigh-Ritz Method 4.2.1 Critical Velocity Versus Static Deflection 4.2.2 A Practical Determination of Critical Velocity 4.2.3 Stepped Shafts 4.3 Dunkerley Method 4.3.1 Turbomachines With More than One Wheel 4.4 Critical Velocity Assessment. Example 4.5 Rotor Balancing 4.5.1 Conceptual Introduction to Balancing 4.5.2 Causes of an Unbalanced Rotor 4.5.3 Static Balancing 4.5.4 Dynamic Balancing 4.5.4.1 Dynamically Unbalanced Rotor 4.5.4.2 Balancing Masses Calculation 4.5.5 Balancing Machine Notes Chapter 5: Lateral Vibration of Turbomachines 5.1 Introduction to Lateral Vibration 5.2 Lateral Vibration Formulas 5.3 Centrifugal Deflection 5.4 Gyration Radius Frequency Response 5.4.1 Deflections and Gyration Radius at Singular Angles θ 5.5 Natural Frequency Versus Deflection 5.5.1 Correction by the Rotor Mass 5.5.2 Calculation of Shaft Deflection 5.6 Natural Frequency Versus Stress Propagation Velocity 5.6.1 Shaft Lateral Resonance in Power Plants Notes Chapter 6: Vibratory Forces in Turbomachines 6.1 Introduction to Vibratory Forces 6.2 Forces on Blades and Bearings 6.3 Radial Vibratory Forces 6.3.1 Assessment of Radial Vibratory Forces 6.3.2 Technical Scenario and Assessment Request 6.4 Vertical and Horizontal Vibratory Forces 6.4.1 Horizontal Vibratory Force 6.4.1.1 Maximum Horizontal Force 6.4.2 Assessment of Vibratory Forces on Pedestals 6.5 Frequency Response of Vibratory Forces 6.5.1 Frequency Response of the Vertical Force 6.5.2 Frequency Response of the Horizontal Force 6.6 Blade Subject to Impulse Force 6.6.1 Example of Centrifugal Force on a Blade 6.6.2 Vibration Produced by the Flow Impact on Blades 6.6.3 Assessment of Blades Resonance Risk 6.7 Rotor-Shaft Subject to Pulsating Torque Notes Chapter 7: Ship’s Oscillation and Vibration 7.1 Introduction to Ships 7.2 Ship’s Propulsion System 7.3 Ship’s Motions and Oscillation 7.3.1 Ship’s Transversal Oscillation 7.3.1.1 Roll’s Natural Frequency 7.3.2 Ship’s Longitudinal Oscillation 7.3.3 Ship’s Equation of Motion 7.3.4 Absorption of Ship’s Oscillations 7.3.4.1 Anti-Roll Tanks 7.3.4.2 Bilge Keels and Stabilizer Fins 7.4 Ship’s Mechanical Vibration 7.4.1 Longitudinal Vibration Excited by the Propeller 7.4.2 Isolation of Longitudinal Vibration 7.4.3 Isolation of Shaft Torsional Vibration 7.4.4 Diesel Motors Excitation 7.5 Beam Ship Vibration 7.5.1 Beam-Ship Natural Frequencies 7.5.1.1 Natural Frequencies by Euler-Bernoulli Equation 7.5.1.2 Hull Girder’s Natural Frequencies 7.5.2 The Hull Resonance Diagram 7.5.3 Finite Element Method. Brief Description 7.5.3.1 Ship’s Deformation by Torsional Torques 7.5.4 Vibration Tolerance Standards Notes Part III: Vibration Control Systems Chapter 8: Vibration Isolation 8.1 Introduction to Transmissibility of Foundations 8.2 Transmissibility of Rigid Foundation 8.2.1 Mechanical Impedance Definition 8.2.2 Transmissibility Ratio 8.2.3 Spring-Damper Set Design 8.2.4 Practical Assessment of Transmissibility Attenuation. Perfectly Rigid Foundation 8.3 Transmissibility of a Non-Rigid Foundation of Known Mass 8.3.1 The Undamped Non-Rigid Foundation of Known Mass 8.3.1.1 Vibration Amplitude Ratios 8.3.2 Isolator design 8.3.2.1 Practical Assessment of Spring Rigidity for a Non-Rigid Foundation 8.4 Transmissibility of Off-Land Z Foundation 8.4.1 Frequency Response Test of a Z Foundation 8.4.1.1 Frequency Response With No Resonance Peak 8.4.1.2 Frequency Response With Resonance Peak 8.4.2 Impedances Calculation of a Z Foundation 8.4.2.1 Z Model of First-Order 8.4.2.2 Z model of Second-Order 8.4.2.3 Frequency Response Curve with No Peak (ζ>0.707) 8.4.2.4 Frequency Response Test with Peak (ζ