Vibration Control Engineering Passive and Feedback Systems 1st Edition by Ernesto Novillo 1000482197 9781000482195

PART I Vibration Theory of SDOF, MDOF and Continuous Dynamic Systems

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

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

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

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

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

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

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

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

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

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

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