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Table of Contents
Intro
Preface
Contents
About the Author
Notations
1 Vibration Basics
1.1 Introduction
1.2 Single Degree of Freedom Systems
1.2.1 Undamped Free Vibrations
1.2.2 Damped Free Vibrations
1.2.3 Harmonic Forcing
1.2.4 Arbitrary Forcing
1.3 Multiple Degree of Freedom Systems
1.3.1 Equations of Motion
1.3.2 Undamped Free Vibrations
1.3.3 Orthogonality of Modes
1.3.4 Damped Free Vibrations
1.3.5 Harmonically Forced Vibrations, No Damping
1.3.6 Harmonically Forced Vibrations, Damping Included
1.3.7 General Periodic Forcing
1.3.8 Arbitrary Forcing, Transients
1.4 Continuous Systems
1.4.1 Equations of Motion
1.4.2 Undamped Free Vibrations
1.4.3 Orthogonality of Modes
1.4.4 Normal Coordinates
1.4.5 Forced Vibrations, No Damping
1.4.6 Forced Vibrations, Damping Included
1.4.7 Complex-Valued Eigenvalues and Mode Shapes
1.4.8 Rayleigh's Method
1.4.9 Ritz Method
1.5 Energy Methods for Setting up Equations of Motion
1.5.1 Lagrange's Equations
1.5.2 Hamilton's Principle
1.5.3 From PDEs to ODEs: Mode Shape Expansion
1.5.4 Using Lagrange's Equations with Continuous Systems
1.6 Nondimensionalized Equations of Motion
1.7 Damping: Types, Measures, Parameter Relations
1.7.1 Damping in Equations of Motion
1.7.2 Damping Models
1.7.3 Damping Measures and Their Relations
1.7.4 Damping Influence on Free Vibration Decay
1.7.5 Damping Influence on Resonance Buildup
1.7.6 Estimating Mass/Stiffness-Proportional Damping Constants
1.7.7 Mass/Stiffness Damping Proportionality Constants for Beams
1.8 The Stiffness and Flexibility Methods for Deriving Equations of Motion
1.8.1 Common Basis
1.8.2 The Flexibility Method
1.8.3 The Stiffness Method
1.8.4 Maxwell's Reciprocity Theorem
1.9 Classification of Forces and Systems
1.9.1 Force Classification
1.9.2 System Classification
1.9.3 Stability Assessment
1.10 Problems
2 Eigenvalue Problems of Vibrations and Stability
2.1 Introduction
2.2 The Algebraic EVP
2.2.1 Mathematical Form
2.2.2 Properties of Eigenvalues and Eigenvectors
2.2.3 Methods of Solution
2.3 The Differential EVP
2.3.1 Mathematical Form
2.4 Stability-Related EVPs
2.4.1 The Clamped-Hinged Euler Column
2.4.2 The Paradox of Follower-Loading
2.4.3 Buckling by Gravity
2.5 Vibration-Related EVPs
2.5.1 Axial Vibrations of Straight Rods
2.5.2 Flexural Vibrations of Beams
2.6 Concepts of Differential EVPs
2.6.1 Multiplicity
2.6.2 Boundary Conditions: Essential or Natural/Suppressible
2.6.3 Function Classes: Eigen-, Test-, and Admissible Functions
2.6.4 Adjointness
2.6.5 Definiteness
2.6.6 Orthogonality
2.6.7 Three Classes of EVPs
2.6.8 The Rayleigh Quotient
2.7 Properties of Eigenvalues and Eigenfunctions
2.7.1 Real-Valueness of Eigenvalues
2.7.2 Sign of the Eigenvalues
2.7.3 Orthogonality of Eigenfunctions
Preface
Contents
About the Author
Notations
1 Vibration Basics
1.1 Introduction
1.2 Single Degree of Freedom Systems
1.2.1 Undamped Free Vibrations
1.2.2 Damped Free Vibrations
1.2.3 Harmonic Forcing
1.2.4 Arbitrary Forcing
1.3 Multiple Degree of Freedom Systems
1.3.1 Equations of Motion
1.3.2 Undamped Free Vibrations
1.3.3 Orthogonality of Modes
1.3.4 Damped Free Vibrations
1.3.5 Harmonically Forced Vibrations, No Damping
1.3.6 Harmonically Forced Vibrations, Damping Included
1.3.7 General Periodic Forcing
1.3.8 Arbitrary Forcing, Transients
1.4 Continuous Systems
1.4.1 Equations of Motion
1.4.2 Undamped Free Vibrations
1.4.3 Orthogonality of Modes
1.4.4 Normal Coordinates
1.4.5 Forced Vibrations, No Damping
1.4.6 Forced Vibrations, Damping Included
1.4.7 Complex-Valued Eigenvalues and Mode Shapes
1.4.8 Rayleigh's Method
1.4.9 Ritz Method
1.5 Energy Methods for Setting up Equations of Motion
1.5.1 Lagrange's Equations
1.5.2 Hamilton's Principle
1.5.3 From PDEs to ODEs: Mode Shape Expansion
1.5.4 Using Lagrange's Equations with Continuous Systems
1.6 Nondimensionalized Equations of Motion
1.7 Damping: Types, Measures, Parameter Relations
1.7.1 Damping in Equations of Motion
1.7.2 Damping Models
1.7.3 Damping Measures and Their Relations
1.7.4 Damping Influence on Free Vibration Decay
1.7.5 Damping Influence on Resonance Buildup
1.7.6 Estimating Mass/Stiffness-Proportional Damping Constants
1.7.7 Mass/Stiffness Damping Proportionality Constants for Beams
1.8 The Stiffness and Flexibility Methods for Deriving Equations of Motion
1.8.1 Common Basis
1.8.2 The Flexibility Method
1.8.3 The Stiffness Method
1.8.4 Maxwell's Reciprocity Theorem
1.9 Classification of Forces and Systems
1.9.1 Force Classification
1.9.2 System Classification
1.9.3 Stability Assessment
1.10 Problems
2 Eigenvalue Problems of Vibrations and Stability
2.1 Introduction
2.2 The Algebraic EVP
2.2.1 Mathematical Form
2.2.2 Properties of Eigenvalues and Eigenvectors
2.2.3 Methods of Solution
2.3 The Differential EVP
2.3.1 Mathematical Form
2.4 Stability-Related EVPs
2.4.1 The Clamped-Hinged Euler Column
2.4.2 The Paradox of Follower-Loading
2.4.3 Buckling by Gravity
2.5 Vibration-Related EVPs
2.5.1 Axial Vibrations of Straight Rods
2.5.2 Flexural Vibrations of Beams
2.6 Concepts of Differential EVPs
2.6.1 Multiplicity
2.6.2 Boundary Conditions: Essential or Natural/Suppressible
2.6.3 Function Classes: Eigen-, Test-, and Admissible Functions
2.6.4 Adjointness
2.6.5 Definiteness
2.6.6 Orthogonality
2.6.7 Three Classes of EVPs
2.6.8 The Rayleigh Quotient
2.7 Properties of Eigenvalues and Eigenfunctions
2.7.1 Real-Valueness of Eigenvalues
2.7.2 Sign of the Eigenvalues
2.7.3 Orthogonality of Eigenfunctions