Advanced Vibration Analysis

by Kelly; S. Graham

• ISBN13: 9780849334191
• ISBN10: 0849334195
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-12-19
• Publisher: CRC Press

Summary

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.

Table of Contents

Introduction and Vibration of Single-Degree-of-Freedom Systems	p. 1
Introduction	p. 1
Degrees of Freedom and Generalized Coordinates	p. 1
Scope of Study	p. 7
Newton's Second Law, Angular Momentum, and Kinetic Energy	p. 8
Particles	p. 8
Systems of Particles	p. 9
Rigid Bodies	p. 13
Components of Vibrating Systems	p. 17
Inertia Elements	p. 17
Stiffness Elements	p. 22
Energy Dissipation	p. 30
External Energy Sources	p. 34
Modeling of One-Degree-of-Freedom Systems	p. 38
Introduction and Assumptions	p. 38
Static Spring Forces	p. 39
Derivation of Differential Equations	p. 42
Model Systems	p. 48
One-Degree-of-Freedom Models of Continuous Systems	p. 49
Qualitative Aspects of One-Degree-of-Freedom Systems	p. 56
Free Vibrations of Linear Single-Degree-of-Freedom Systems	p. 63
Response of a Single-Degree-of-Freedom System Due to Harmonic Excitation	p. 70
General Theory	p. 70
Frequency-Squared Excitation	p. 73
Motion Input	p. 75
General Periodic Input	p. 80
Transient Response of a Single-Degree-of-Freedom System	p. 82
Derivation of Differential Equations Using Variational Methods	p. 87
Functionals	p. 87
Variations	p. 91
Euler-Lagrange Equation	p. 93
Hamilton's Principle	p. 100
Lagrange's Equations for Conservative Discrete Systems	p. 104
Lagrange's Equations for Non-Conservative Discrete Systems	p. 112
Linear Discrete Systems	p. 122
Quadratic Forms	p. 122
Differential Equations for Linear Systems	p. 125
Linearization of Differential Equations	p. 127
Gyroscopic Systems	p. 130
Continuous Systems	p. 136
Bars, Strings, and Shafts	p. 138
Euler-Bernoulli Beams	p. 150
Timoshenko Beams	p. 166
Membranes	p. 170
Linear Algebra	p. 173
Introduction	p. 173
Three-Dimensional Space	p. 174
Vector Spaces	p. 177
Linear Independence	p. 182
Basis and Dimension	p. 185
Inner Products	p. 189
Norms	p. 193
Gram-Schmidt Orthonormalization Method	p. 197
Orthogonal Expansions	p. 202
Linear Operators	p. 206
Adjoint Operators	p. 212
Positive Definite Operators	p. 219
Energy Inner Products	p. 222
Operators Used in Vibration Problems	p. 225
Summary of Basic Theory	p. 225
Differential Equations for Discrete Systems	p. 227
Stiffness Matrix	p. 227
Mass Matrix	p. 233
Flexibility Matrix	p. 234
M[superscript -1]K and AM	p. 240
Formulation of Partial Differential Equations for Continuous Systems	p. 242
Second-Order Problems	p. 245
Euler-Bernoulli Beam	p. 253
Timoshenko Beams	p. 262
Systems with Multiple Deformable Bodies	p. 266
Continuous Systems with Attached Inertia Elements	p. 272
Combined Continuous and Discrete Systems	p. 278
Membranes	p. 283
Free Vibrations of Conservative Systems	p. 287
Normal Mode Solution	p. 287
Properties of Eigenvalues and Eigenvectors	p. 292
Eigenvalues of Self-Adjoint Operators	p. 292
Positive Definite Operators	p. 297
Expansion Theorem	p. 298
Summary	p. 302
Rayleigh's Quotient	p. 303
Solvability Conditions	p. 306
Free Response Using the Normal Mode Solution	p. 309
General Free Response	p. 309
Principal Coordinates	p. 314
Discrete Systems	p. 316
The Matrix Eigenvalue Problem	p. 317
Natural Frequency Calculations Using Flexibility Matrix	p. 326
Matrix Iteration	p. 330
Continuous Systems	p. 341
Second-Order Problems (Wave Equation)	p. 342
Euler-Bernoulli Beams	p. 360
Repeated Structures	p. 375
Timoshenko Beams	p. 398
Combined Continuous and Discrete Systems	p. 409
Membranes	p. 414
Green's Functions	p. 430
Non-Self-Adjoint Systems	p. 437
Non-Self-Adjoint Operators	p. 437
Discrete Systems with Proportional Damping	p. 441
Discrete Systems with General Damping	p. 446
Discrete Gyroscopic Systems	p. 452
Continuous Systems with Viscous Damping	p. 458
Forced Response	p. 465
Response of Discrete Systems for Harmonic Excitations	p. 465
General Theory	p. 465
Vibration Absorbers	p. 470
Harmonic Excitation of Continuous Systems	p. 480
Laplace Transform Solutions	p. 490
Discrete Systems	p. 491
Continuous Systems	p. 497
Modal Analysis for Undamped Discrete Systems	p. 501
Modal Analysis for Undamped Continuous Systems	p. 504
Discrete Systems with Damping	p. 516
Proportional Damping	p. 516
General Viscous Damping	p. 517
Rayleigh-Ritz and Finite-Element Methods	p. 525
Fourier Best Approximation Theorem	p. 525
Rayleigh-Ritz Method	p. 528
Galerkin Method	p. 531
Rayleigh-Ritz Method for Natural Frequencies and Mode Shapes	p. 532
Rayleigh-Ritz Methods for Forced Response	p. 551
Admissible Functions	p. 556
Assumed Modes Method	p. 560
Finite-Element Method	p. 570
Assumed Modes Development of Finite-Element Method	p. 575
Bar Element	p. 577
Beam Element	p. 584
Exercises	p. 595
Chapter 1	p. 595
Chapter 2	p. 602
Chapter 3	p. 611
Chapter 4	p. 614
Chapter 5	p. 617
Chapter 6	p. 620
Chapter 7	p. 622
Chapter 8	p. 625
References	p. 627
Index	p. 629
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