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ISBN 10: 0367389657
ISBN 13: 978-0367389659
Author: Graham Kelly
Delineating a comprehensive theory, Advanced Vibration Analysis provides the bedrock for building a general mathematical framework for the analysis of a model of a physical system undergoing vibration. The book illustrates how the physics of a problem is used to develop a more specific framework for the analysis of that problem. The author elucidates a general theory applicable to both discrete and continuous systems and includes proofs of important results, especially proofs that are themselves instructive for a thorough understanding of the result. The book begins with a discussion of the physics of dynamic systems comprised of particles, rigid bodies, and deformable bodies and the physics and mathematics for the analysis of a system with a single-degree-of-freedom. It develops mathematical models using energy methods and presents the mathematical foundation for the framework. The author illustrates the development and analysis of linear operators used in various problems and the formulation of the differential equations governing the response of a conservative linear system in terms of self-adjoint linear operators, the inertia operator, and the stiffness operator. The author focuses on the free response of linear conservative systems and the free response of non-self-adjoint systems. He explores three method for determining the forced response and approximate methods of solution for continuous systems. The use of the mathematical foundation and the application of the physics to build a framework for the modeling and development of the response is emphasized throughout the book. The presence of the framework becomes more important as the complexity of the system increases. The text builds the foundation, formalizes it, and uses it in a consistent fashion including application to contemporary research using linear vibrations.
Chapter 1 Introduction and Vibrations of Single- Degree-of-Freedom Systems
1.1 Introduction
1.1.1 Degrees of Freedom and Generalized Coordinates
1.1.2 Scope of Study
1.2 Newton’s Second Law, Angular Momentum, and Kinetic Energy
1.2.1 Particles
1.2.2 Systems of Particles
1.2.3 Rigid Bodies
1.3 Components of Vibrating Systems
1.3.1 Inertia Elements
1.3.2 Stiffness Elements
1.3.3 Energy Dissipation
1.3.4 External Energy Sources
1.4 Modeling of One-Degree-of-Freedom Systems
1.4.1 Introduction and Assumptions
1.4.2 Static Spring Forces
1.4.3 Derivation of Differential Equations
1.4.4 Model Systems
1.4.5 One-Degree-of-Freedom Models of Continuous Systems
1.5 Qualitative Aspects of One-Degree-of-Freedom Systems
1.6 Free Vibrations of Linear Single-Degree-of-Freedom Systems
1.7 Response of A Single-Degree-of-Freedom System Due To Harmonic Excitation
1.7.1 General Theory
1.7.2 Frequency-Squared Excitation
1.7.3 Motion Input
1.7.4 General Periodic Input
1.8 Transient Response of A Single-Degree-of-Freedom System
Chapter 2 Derivation of Differential Equations Using Variational Methods
2.1 Functionals
2.2 Variations
2.3 Euler–Lagrange Equation
2.4 Hamilton’S Principle
2.5 Lagrange’S Equations for Conservative Discrete Systems
2.6 Lagrange’S Equations for Non-Conservative Discrete Systems
2.7 Linear Discrete Systems
2.7.1 Quadratic Forms
2.7.2 Differential Equations for Linear Systems
2.7.3 Linearization of Differential Equations
2.8 Gyroscopic Systems
2.9 Continuous Systems
2.10 Bars, Strings, and Shafts
2.11 Euler–Bernoulli Beams
2.12 Timoshenko Beams
2.13 Membranes
Chapter 3 Linear Algebra
3.1 Introduction
3.2 Three-Dimensional Space
3.3 Vector Spaces
3.4 Linear Independence
3.5 Basis and Dimension
3.6 Inner Products
3.7 Norms
3.8 Gram-Schmidt Orthonormalization Method
3.9 Orthogonal Expansions
3.10 Linear Operators
3.11 Adjoint Operators
3.12 Positive Definite Operators
3.13 Energy Inner Products
Chapter 4 Operators Used in Vibration Problems
4.1 Summary of Basic Theory
4.2 Differential Equations for Discrete Systems
4.3 Stiffness Matrix
4.4 Mass Matrix
4.5 Flexibility Matrix
4.6 M−1K And Am
4.7 Formulation of Partial Differential Equations for Continuous Systems
4.8 Second-Order Problems
4.9 Euler–Bernoulli Beam
4.10 Timoshenko Beams
4.11 Systems with Multiple Deformable Bodies
4.12 Continuous Systems with Attached Inertia Elements
4.13 Combined Continuous and Discrete Systems
4.14 Membranes
Chapter 5 Free Vibrations of Conservative Systems
5.1 Normal Mode Solution
5.2 Properties of Eigenvalues and Eigenvectors
5.2.1 Eigenvalues of Self-Adjoint Operators
5.2.2 Positive Definite Operators
5.2.3 Expansion Theorem
5.2.4 Summary
5.3 Rayleigh’S Quotient
5.4 Solvability Conditions
5.5 Free Response Using the Normal Mode Solution
5.5.1 General Free Response
5.5.2 Principal Coordinates
5.6 Discrete Systems
5.6.1 The Matrix Eigenvalue Problem
5.7 Natural Frequency Calculations Using Flexibility Matrix
5.8 Matrix Iteration
5.9 Continuous Systems
5.10 Second-Order Problems (Wave Equation)
5.11 Euler–Bernoulli Beams
5.12 Repeated Structures
5.13 Timoshenko Beams
5.14 Combined Continuous and Discrete Systems
5.15 Membranes
5.16 Green’S Functions
Chapter 6 Non-Self-Adjoint Systems
6.1 Non-Self-Adjoint Operators
6.2 Discrete Systems with Proportional Damping
6.3 Discrete Systems with General Damping
6.4 Discrete Gyroscopic Systems
6.5 Continuous Systems with Viscous Damping
Chapter 7 Forced Response
7.1 Response of Discrete Systems for Harmonic Excitations
7.1.1 General Theory
7.1.2 Vibration Absorbers
7.2 Harmonic Excitation of Continuous Systems
7.3 Laplace Transform Solutions
7.3.1 Discrete Systems
7.3.2 Continuous Systems
7.4 Modal Analysis for Undamped Discrete Systems
7.5 Modal Analysis of Undamped Continuous Systems
7.6 Discrete Systems with Damping
7.6.1 Proportional Damping
7.6.2 General Viscous Damping
Chapter 8 Rayleigh–Ritz and Finite-Element Methods
8.1 Fourier Best Approximation Theorem
8.2 Rayleigh–Ritz Method
8.3 Galerkin Method
8.4 Rayleigh–Ritz Method for Natural Frequencies and Mode Shapes
8.5 Rayleigh–Ritz Methods for Forced Response
8.6 Admissible Functions
8.7 Assumed Modes Method
8.8 Finite-Element Method
8.9 Assumed Modes Development of Finite-Element Method
8.10 Bar Element
8.11 Beam Element
Chapter 9 Exercises
9.1 Chapter 1
9.2 Chapter 2
9.3 Chapter 3
9.4 Chapter 4
9.5 Chapter 5
9.6 Chapter 6
9.7 Chapter 7
9.8 Chapter 8
References
Index
Related Titles
advances in vibration
advanced vibration
advances in vibration engineering journal
advanced vibration analysis
vibration analysis book
Tags: Graham Kelly, Vibration Analysis
