Vibration Control Engineering: Passive and Feedback Systems

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<0.707)
			8.4.3 Example of Spring Calculation to Isolate a Z Foundation
		Notes
	Chapter 9: Vibration Absorption
		9.1 Introduction to Vibration Absorption
		9.2 Vibration Absorbers for Rotating Machines
			9.2.1 Conceptual Description of Frahm’s Absorber
		9.3 Frahm’s Absorber Model
			9.3.1 Equations of Motion
				9.3.1.1 Machine and Absorber Vibration Amplitude
				9.3.1.2 Vibration Absorption Condition
			9.3.2 Example of Forces in a Machine-Absorber Assembly
				9.3.2.1 Forces With a Tuned Absorber
				9.3.2.2 Forces With an Untuned Absorber
			9.3.3 Frequency Response of Machine-Absorbers
				9.3.3.1 Definition of Non-Dimensional Variables and Parameters
				9.3.3.2 Vibration Amplitudes and Frequency Response
				9.3.3.3 Resonant Frequencies
			9.3.4 Frahm’s Absorber Design and Performance
				9.3.4.1 Frequency Difference to Resonance (FDTR)
				9.3.4.2 Absorber Design Procedure
				9.3.4.3 Mass Ratio Determination
				9.3.4.4 Tuning Error
				9.3.4.5 Tolerance to the Frequency Deviation
			9.3.5 Damped Absorption
				9.3.5.1 Conceptual Description
				9.3.5.2 Equations of Motion
				9.3.5.3 Derivation of Impedance Ratios z
				9.3.5.4 Den Hartog\'s Method
			9.3.6 Practical Assessment of a Fan Vibration Neutralization
				9.3.6.1 Scenario
				9.3.6.2 Undamped Absorber Design
				9.3.6.3 Damped Absorber Design
		9.4 Absorption of Overhead Lines Vibration
			9.4.1.1 Example of Force Produced by Karman Vortices
			9.4.2 Stockbridge Absorbers
		Notes
	Chapter 10: Vibration Control Techniques
		10.1 Introduction to Techniques to Reduce Vibration
		10.2 Control Vibration Philosophy
		10.3 Techniques General Procedure
			10.3.1 Scenario Description
			10.3.2 General Calculation Procedure
				10.3.2.1 Initial Scenario. Point 1 Calculation
				10.3.2.2 Final Scenario. Point 2 Calculation
				10.3.2.3 Design Ratios
				10.3.2.4 Example of the General Procedure Applied to the Four Fundamental Vibration Forms
			10.3.3 Description of the Seven Basic Techniques
				10.3.3.1 Technique 1. Externally-Excited Machine
				10.3.3.2 Technique 2. Self-Excited Machine
				10.3.3.3 Techniques 3 and 4. Base-Excited and Force Transmitted Machine
		10.4 Predicting and Preventing Harmful Vibrations
			10.4.1 Admissible Vibration Amplitude
				10.4.1.1 Turbomachine Rotor
				10.4.1.2 Machine Case and Bearings Cap
		Notes
	Chapter 11: Feedback Control Techniques
		11.1 Introduction to Feedback Control Techniques
			11.1.1 Main Definitions of Feedback Control Theory
		11.2 Control Systems Basics
			11.2.1 Closed-Loop Systems
		11.3 Time Response of Linear Systems
		11.4 Control Actions in Closed-Loop Systems
			11.4.1 Proportional Control Action
				11.4.1.1 Error with a P Controller
				11.4.1.2 Time Response With a P Controller
					What Changes Does the Adjustable Gain of the P cController Produce?
			11.4.2 Proportional Plus Integral Control Action
				11.4.2.1 Error With a PI Controller
			11.4.3 Proportional Plus Derivative Action
			11.4.4 PID Control Action
		11.5 Closed-Loop Stability
			11.5.1 Absolute Stability Determination
				11.5.1.1 Numerical Determination of the Absolute Stability
			11.5.2 Relative Stability Determination
		11.6 Controller Settings Calculation
			11.6.1 Ziegler-Nichols Tuning Methods
				11.6.1.1 Ziegler-Nichols Method Based on the S Reaction Curve
					Example of Controller’s Setting Calculation Based on the S Reaction Curve
				11.6.1.2 Ziegler-Nichols Method Based on Ultimate Dynamic Gain and Frequency
					Example of Controller’s Setting Calculation Based on Ultimate Gain and Period
		11.7 Active Vibration Control
			11.7.1 Design of Active Control for a Vibrating Structure
				11.7.1.1 Converting the Transfer Function to Obtain the Frequency Response
				11.7.1.2 Frequency Response
				11.7.1.3 Controller\'s Design
			11.7.2 Absorption of Ship\'s Roll
		Bibliography about Feedback Control Systems
			Books
			Classical Papers Based on the Frequency Response
		Notes
Index