Advanced Vibration Analysis (Mechanical Engineering)
By: S. Graham Kelly (author)Hardback
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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.
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
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