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Fundamentals of Vibrations provides a comprehensive coverage of mechanical vibrations theory and applications. Suitable as a textbook for courses ranging from introductory to graduate level, it can also serve as a reference for practicing engineers. Written by a leading authority in the field, this volume features a clear and precise presentation of the material and is supported by an abundance of physical explanations, many worked-out examples, and numerous homework problems.

The modern approach to vibrations emphasizes analytical and computational solutions that are enhanced by the use of MATLAB. The text covers single-degree-of-freedom systems, two-degree-of-freedom systems, elements of analytical dynamics, multi-degree-of-freedom systems, exact methods for distributed-parameter systems, approximate methods for distributed-parameter systems, including the finite element method, nonlinear oscillations, and random vibrations. Three appendices provide pertinent material from Fourier series, Laplace transformation, and linear algebra.

Table of Contents

1. Concepts from Vibrations

Newton’s Laws / Moment of a Force and Angular Momentum / Work and Energy / Dynamics of Systems of Particles / Dynamics of Rigid Bodies / Kinetic Energy of Rigid Bodies in Planar Motion / Characteristics of Discrete System Components / Equivalent Springs, Dampers and Masses / Modeling of Mechanical Systems / System Differential Equations of Motion / Nature of Excitations / System and Response Characteristics. The Superposition Principle / Vibration about Equilibrium Points

2. Response of Single-Degree-of-Freedom Systems to Initial Excitations

Undamped Single-Degree-of-Freedom Systems. Harmonic Oscillator / Viscously Damped Single-Degree-of-Freedom Systems

3. Response of Single-Degree-of-Freedom Systems to Harmonic and Periodic Excitations

Response of Single-Degree-of-Freedom Systems to Harmonic Excitations / Frequency Response Plots / Systems with Rotating Unbalanced Masses / Whirling of Rotating Shafts / Harmonic Motion of the Base / Vibration Isolation / Vibration Measuring Instruments / Energy Dissipation. Structural Damping / Response to Periodic Excitations. Fourier Series / Frequency Response Plots by MATLAB, b>4. Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations. The Unit Impulse. Impulse Response / The Unit Step Function. Step Response / The Unit Ramp Function. Ramp Response / Response to Arbitrary Excitations. The Convolution Integral / Shock Spectrum / System Response by the Laplace Transformation Method. Transfer Function / General System Response / Response by the State Transition Matrix / Discrete-Time Systems. The Convolution Sum / Discrete-Time Response Using the Transition Matrix / Response by the Convolution Sum Using MATLAB / Response by the Discrete-Time Transition Matrix Using MATLAB

5. Two-Degree-of-Freedom Systems

System Configuration / The Equations of Motion of Two-Degree-of-Freedom Systems / Free Vibration of Undamped Systems. Natural Modes / Response to Initial Excitations / Coordinate Transformations. Coupling / Orthogonality of Modes. Natural Coordinates / Beat Phenomenon / Response of Two-Degree-of-Freedom Systems to Harmonic Excitations / Undamped Vibration Absorbers / Response of Two-Degree-of-Freedom Systems to Nonperiodic Excitations / Response to Nonperiodic Excitations by the Convolution Sum / Response to Initial Excitations by MATLAB / Frequency Response Plots for Two-Degree-of-Freedom Systems by MATLAB / Response to a Rectangular Pulse by the Convolution Sum Using MATLAB

6. Elements of Analytical Dynamics

Degrees of Freedom and Generalized Coordinates / The Principle of Virtual Work / The Principle of D’Alembert / The Extended Hamilton’s Principle / Lagrange’s Equations

7. Multi-Degree-of-Freedom Systems

Equations of Motion for Linear Systems / Flexibility and Stiffness Influence Coefficients / Properties of the Stiffness and Mass Coefficients / Lagrange’s Equations Linearized about Equilibrium / Linear Transformations. Coupling / Undamped Free Vibration. The Eigenvalue Problem / Orthogonality of Modal Vectors / Systems Admitting Rigid-Body Motions / Decomposition of the Response in Terms of Modal Vectors / Response to Initial Excitations by Modal Analysis / Eigenvalue Problem in Terms of a Single Symmetric Matrix / Geometric Interpretation of the Eigenvalue Problem / Rayleigh’s Quotient and Its Properties / Response to Harmonic External Excitations / Response to External Excitations by Modal Analysis / Systems with Arbitrary Viscous Damping / Discrete-Time Systems / Solution of the Eigenvalue Problem. MATLAB Programs / Response to Initial Excitations by Modal Analysis Using MATLAB / Response by the Discrete-Time Transition Matrix Using MATLAB

8. Distributed-Parameter Systems: Exact Solutions

Relation between Discrete and Distributed Systems. Transverse Vibration of Strings / Derivation of the String Vibration Problem by the Extended Hamilton Principle / Bending Vibration of Beams / Free Vibration. The Differential Eigenvalue Problem / Orthogonality of Modes. Expansion Theorem / Systems with Lumped Masses at the Boundaries / Eigenvalue Problem and Expansion Theorem for Problems with Lumped Masses at the Boundaries / Rayleigh’s Quotient. The Variational Approch to the Differential Eigenvalue Problem / Response to Initial Excitation / Response to External Excitations / Systems with External Forces at Boundaries / The Wave Equation / Traveling Waves in Rods of Finite Length

9. Distributed-Parameter Systems: Approximate Methods

Discretization of Distributed-Parameter Systems by Lumping / Lumped-Parameter Method Using Influence Coefficients / Holzer’s Method for Torsional Vibration / Myklestad’s Method for Bending Vibration / Rayleigh’s Principle / The Rayleigh-Ritz Method / An Enhanced Rayleigh-Ritz Method / The Assumed-Modes Method. System Response / The Galerkin Method / The Collocation Method / MATLAB Program for the Solution of the Eigenvalue Problem by the Rayleigh-Ritz Method

10. The Finite Element Method

The Finite Element Method as a Rayleigh-Ritz Method / Strings, Rods and Shafts / Higher-Degree Interpolation Functions / Beams in Bending Vibration / Errors in the Eigenvalues / Finite Element Modeling of Trusses / Finite Element Modeling of Frames / System Response by the Finite Element Method / MATLAB Program for the Solution of the Eigenvalue Problem by the Finite Element Method

11. Nonlinear Oscillations

Fundamental Concepts in Stability. Equilibrium Points / Small Motions of Single-Degree-of-Freedom Systems from Equilibrium / Conservative Systems. Motions in the Large / Limit Cycles. The van der Pol Oscillator / The Fundamental Perturbation Technique / Secular Terms / Lindstedt’s Method / Forced Oscillation of Quasi-Harmonic Systems. Jump Phenomenon / Subharmonics and Combination Harmonics / Systems with Time-Dependent Coefficients. Mathieu’s Equation / Numerical Integration of the Equations of Motion. The Runge-Kutta Methods / Trajectories for the van der Pol Oscillator by MATLAB

12. Random Vibrations

Ensemble Averages. Stationary Random Processes / Time Averages. Ergodic Random Processes / Mean Square Values and Standard Deviation / Probability Density Functions / Description of Random Data in Terms of Probability Density Functions / Properties of Autocorrelation Functions / Response to Arbitrary Excitations by Fourier Transforms / Power Spectral Density Functions / Narrowband and Wideband Random Processes / Response of Linear Systems to Stationary Random Excitations / Response of Single-Degree-of-Freedom Systems to Random Excitations / Joint Probability of Distribution of Two Random Variables / Joint Properties of Stationary Random Properties / Joint Properties of Ergodic Random Processes / Response Cross-Correlation Functions for Linear Systems / Response of Multi-Degree-of-Freedom Systems to Random Excitations / Response of Distributed-Parameter Systems to Random Excitations.

Appendix A: Fourier Series
Appendix B: Laplace Transformation
Appendix C: Linear Algebra

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