# Tag: solutions to fundamentals of vibrations

**Fundamentals of Vibrations PDF**

**Textbook by Leonard Meirovitch**

*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.

## Fundamentals of Vibrations

**TABLE OF CONTENTS**:

1. Concepts from Vibrations 2. Response of Single-Degree-of-Freedom Systems to Initial Excitations 3. Response of Single-Degree-of-Freedom Systems to Harmonic and Periodic Excitations 4. Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations 5. Two-Degree-of-Freedom Systems 6. Elements of Analytical Dynamics 7. Multi-Degree-of-Freedom Systems 8. Distributed-Parameter Systems: Exact Solutions 9. Distributed-Parameter Systems: Approximate Methods 10. The Finite Element Method 11. Nonlinear Oscillations 12. Random Vibrations

Appendixes: Fourier Series / Laplace Transformation / Linear Algebra

**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