Advanced Vibration Analysis: 1st Edition (Hardback) book cover

Advanced Vibration Analysis

1st Edition

By S. Graham Kelly

CRC Press

664 pages | 220 B/W Illus.

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pub: 2006-12-19
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Description

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.

Reviews

"This book is a useful textbook on vibration analysis and may be very useful to students, post-graduate students and engineers who work in the area of dynamics of discrete and continuous systems."

– Yuri N. Sankin, in Zentralblatt Math, 2009

Table of Contents

INTRODUCTION AND VIBRATION OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS

Introduction

Newton's Second Law, Angular Momentum, and Kinetic Energy

Components of Vibrating Systems

Modeling of One-Degree-of-Freedom Systems

Qualitative Aspects of One-Degree-of-Freedom Systems

Free Vibrations of Linear Single-Degree-of-Freedom Systems

Response of a Single-Degree-of-Freedom System Due to Harmonic Excitation

Transient Response of a Single-Degree-of-Freedom System

DERIVATION OF DIFFERENTIAL EQUATIONS USING VARIATIONAL METHODS

Functionals

Variations

Euler-Lagrange Equation

Hamilton's Principle

Lagrange's Equations for Conservative Discrete Systems

Lagrange's Equations for Non-Conservative Discrete Systems

Linear Discrete Systems

Gyroscopic Systems

Continuous Systems

Bars, Strings, and Shafts

Euler-Bernoulli Beams

Timoshenko Beams

Membranes

LINEAR ALGEBRA

Introduction

Three-Dimensional Space

Vector Spaces

Linear Independence

Basis and Dimension

Inner Products

Norms

Gram-Schmidt Orthonormalization Method

Orthogonal Expansions

Linear Operators

Adjoint Operators

Positive Definite Operators

Energy Inner Products

OPERATORS USED IN VIBRATION PROBLEMS

Summary of Basic Theory

Differential Equations for Discrete Systems

Stiffness Matrix

Mass Matrix

Flexibility Matrix

M -1 K and AM

Formulation of Partial Differential Equations for Continuous Systems

Second-Order Problems

Euler-Bernoulli Beam

Timoshenko Beams

Systems with Multiple Deformable Bodies

Continuous Systems with Attached Inertia Elements

Combined Continuous and Discrete Systems

Membranes

FREE VIBRATIONS OF CONSERVATIVE SYSTEMS

Normal Mode Solution

Properties of Eigenvalues and Eigenvectors

Rayleigh's Quotient

Solvability Conditions

Free Response Using the Normal Mode Solution

Discrete Systems

Natural Frequency Calculations Using Flexibility Matrix

Matrix Iteration

Continuous Systems

Second-Order Problems (Wave Equation)

Euler-Bernoulli Beams

Repeated Structures

Timoshenko Beams

Combined Continuous and Discrete Systems

Membranes

Green's Functions

NON-SELF-ADJOINT SYSTEMS

Non-Self-Adjoint Operators

Discrete Systems with Proportional Damping

Discrete Systems with General Damping

Discrete Gyroscopic Systems

Continuous Systems with Viscous Damping

FORCED RESPONSE

Response of Discrete Systems for Harmonic Excitations

Harmonic Excitation of Continuous Systems

Laplace Transform Solutions

Modal Analysis for Undamped Discrete Systems

Modal Analysis for Undamped Continuous Systems

Discrete Systems with Damping

RAYLEIGH-RITZ AND FINITE ELEMENT METHODS

Fourier Best Approximation Theorem

Rayleigh-Ritz Method

Galerkin Method

Rayleigh-Ritz Method for Natural Frequencies and Mode Shapes

Rayleigh-Ritz Methods for Forced Response

Admissible Functions

Assumed Modes Method

Finite Element Method

Assumed Modes Development of Finite Element Method

Bar Element

Beam Element

Exercises

References

Index

About the Series

Mechanical Engineering

A Series of Textbooks and Reference Books

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
SCI041000
SCIENCE / Mechanics / General
TEC009070
TECHNOLOGY & ENGINEERING / Mechanical