Intended for introductory vibrations courses, Meirovitch offers a masterfully crafted textbook that covers all basic concepts at a level appropriate for undergraduate students. The book contains a chapter on the use of Finite Element Methods in vibrational analysis. Meirovitch uses selective worked examples to show the application of MATLAB software in this course. The author's approach challenges students with a precise and thoughtful explanations and motivates them through use of physical explanations, plentiful problems, worked-out examples, and illustrations.

This book presents material fundamental to a modern treatment of vibrations, placing the emphasis on analytical developments and computational solutions. It is intended as a textbook for a number of courses on vibrations ranging from the junior level to the second-year graduate level; the book can also serve as a reference for practicing engineers. Certain material from pertinent disciplines was included to render the book self-contained, and hence suitable for self-study. Consistent with this, the book begins with very elementary material and raises the level gradually. A large number of exam- ples and homework problems, as well as computer programs written in MATLAB, are provided.

  Preface

  Introduction

1 Concepts from Vibrations

  1.1 Newton's Laws

  1.2 Moment of a Force and Angular Momentum

  1.3 Work and Energy

  1.4 Dynamics of Systems of Particles

  1.5 Dynamics of Rigid Bodies

1.5.1 Pure translation relative to the inertial space

1.5.2 Pure rotation about a fixed point

1.5.3 General planar motion referred to the mass center

  1.6 Kinetic Energy of Rigid Bodies in Planar Motion

1.6.1 Pure translation relative to the inertial space

1.6.2 Pure rotation about a fixed point

1.6.3 General planar motion referred to the mass center

  1.7 Characteristics of Discrete System Components

  1.8 Equivalent Springs, Dampers and Masses

  1.9 Modeling of Mechanical Systems

  1.10 System Differential Equations of Motion

  1.11 Nature of Excitations

  1.12 System and Response Characteristics. The Superposition Principle

  1.13 Vibration about Equilibrium Points

  1.14 Summary

  Problems

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

  Excitations

  2.1 Undamped Single-Degree-of-Freedom Systems. Harmonic Oscillator

  2.2 Viscously Damped Single-Degree-of-Freedom Systems

  2.3 Measurement of Damping

  2.4 Coulomb Damping. Dry Friction

  2.5 Plotting the Response to Initial Excitations by MATLAB

  2.6 Summary

  Problems

3 Response of Single-Degree-of-Freedom Systems to Harmonic

  and Periodic Excitations

  3.1 Response of Single-Degree-of-Freedom Systems to Harmonic Excitations.

  3.2 Frequency Response Plots

  3.3 Systems with Rotating Unbalanced Masses

  3.4 Whirling of Rotating Shafts

  3.5 Harmonic Motion of the Base

  3.6 Vibration Isolation

  3.7 Vibration Measuring Instruments

3.7.1 Accelerometers--high frequency instruments

3.7.2 Seismometers--low frequency instruments

  3.8 Energy Dissipation. Structural Damping

  3.9 Response to Periodic Excitations. Fourier Series

  3.10 Frequency Response Plots by MATLAB

  3.11 Summary

  Problems

4 Response of Single-Degree-of-Freedom Systems to

  Nonperiodic Excitations

  4.1 The Unit hnpulse. Impulse Response

  4.2 The Unit Step Function. Step Response

  4.3 The Unit Ramp Function. Ramp Response

  4.4 Response to Arbitrary Excitations. The Convolution Integral

  4.5 Shock Spectrum

  4.6 System Response by the Laplace Transformation Method. Transfer Function

  4.7 General System Response

  4.8 Response by the State Transition Matrix

  4.9 Discrete-Time Systems. The Convolution Sum

  4.10 Discrete-Time Response Using the Transition Matrix

  4.11 Response by the Convolution Sum Using MATLAB

  4.12 Response by the Discrete-Time Transition Matrix Using MATLAB

  4.13 Summary

  Problems

5 Two-Degree-of-Freedom Systems

  5.1 System Configuration

  5.2 The Equations of Motion of Two-Degree-of-Freedom Systems

  5.3 Free Vibration of Undamped Systems. Natural Modes

  5.4 Response to Initial Excitations

  5.5 Coordinate Transformations. Coupling

  5.6 0rthogonality of Modes. Natural Coordinates

  5.7 Beat Phenomenon

  5.8 Response of Two-Degree-of-Freedom Systems to Harmonic Excitations.

  5.9 Undamped Vibration Absorbers

  5.10 Response of Two-Degree-of-Freedom Systems to Nonperiodic

  Excitations

  5.11 Response to Nonperiodic Excitations by the Convolution Sum

  5.12 Response to Initial Excitations by MATLAB

  5.13 Frequency Response Plots for Two-Degree-of-Freedom Systems by

  MATLAB

  5.14 Response to a Rectangular Pulse by the Convolution Sum Using

  MATLAB

  5.15 Summary

  Problems

6 Elements of Analytical Dynamics

  6.1 Degrees of Freedom and Generalized Coordinates

  6.2 The Principle of Virtual Work

  6.3 The Principle of D'Alembert

  6.4 The Extended Hamilton's Principle

  6.5 Lagrange's Equations

  6.6 Summary

  Problems

7 Multi-Degree-of-Freedom Systems

  7.1 Equations of Motion for Linear Systems

  7.2 Flexibility and Stiffness Influence Coefficients

  7.3 Properties of the Stiffness and Mass Coefficients

  7.4 Lagrange's Equations Linearized about Equilibrium

  7.5 Linear Transformations. Coupling

  7.6 Undamped Free Vibration. The Eigenvalue Problem

  7.70rthogonality of Modal Vectors

  7.8 Systems Admitting Rigid-Body Motions

  7.9 Decomposition of the Response in Terms of Modal Vectors

  7.10 Response to Initial Excitations by Modal Analysis

  7.11 Eigenvalue Problem in Terms of a Single Symmetric Matrix

  7.12 Geometric Interpretation of the Eigenvalue Problem

  7.13 Rayleigh's Quotient and Its Properties

  7.14 Response to Harmonic External Excitations

  7.15 Response to External Excitations by Modal Analysis

7.15.1 Undamped systems

7.15.2 Systems with proportional damping

  7.16 Systems with Arbitrary Viscous Damping

  7.17 Discrete-Time Systems

  7.18 Solution of the Eigenvalue Problem. MATLAB Programs

  7.19 Response to Initial Excitations by Modal Analysis Using MATLAB

  7.20 Response by the Discrete-Time Transition Matrix Using MATLAB

  7.21 Summary

  Problems

8 Distributed-Parameter Systems: Exact Solutions

  8.1 Relation between Discrete and Distributed Systems. Transverse

  Vibration of Strings

  8.2 Derivation of the String Vibration Problem by the Extended Hamilton

  Principle

  8.3 Bending Vibration of Beams

  8.4 Free Vibration. The Differential Eigenvalue Problem

  8.5 0rthogonality of Modes. Expansion Theorem

  8.6 Systems with Lumped Masses at the Boundaries

  8.7 Eigenvalue Problem and Expansion Theorem for Problems with

   Lumped Masses at the Boundaries

  8.8 Rayleigh's Quotient. The Variational Approach to the Differential

   Eigenvalue Problem

  8.9 Response to Initial Excitations

  8.10 Response to External Excitations

  8.11 Systems with External Forces at Boundaries

  8.12 The Wave Equation

  8.13 Traveling Waves in Rods of Finite Length

  8.14 Summary

   Problems

9 Distributed-Parameter Systems: Approximate Methods

9.1 Discretization of Distributed-Parameter Systems by Lumping

9.2 Lumped-Parameter Method Using Influence Coefficients

9.3 Holzer's Method for Torsional Vibration

9.4 Myklestad's Method for Bending Vibration

9.5 Rayleigh's Principle

9.6 The Rayleigh-Ritz Method

9.7 An Enhanced Rayleigh-Ritz Method

9.8 The Assumed-Modes Method. System Response

9.9 The Galerkin Method

9.10 The Collocation Method

9.11 MATLAB Program for the Solution of the Eigenvalue Problem by the

   Rayleigh-Ritz Method

9.12 Summary

   Problems

10 The Finite Element Method

  10.1 The Finite Element Method as a Rayleigh-Ritz Method

  10.2 Strings, Rods and Shafts

  10.3 Higher-Degree Interpolation Functions

  10.4 Beams in Bending Vibration

  10.5 Errors in the Eigenvalues

  10.6 Finite Element Modeling of Trusses

  10.7 Finite Element Modeling of Frames

  10.8 System Response by the Finite Element Method

  10.9 MATLAB Program for the Solution of the Eigenvalue Problem by the

   Finite Element Method

  10.10 Summary

   Problems

11 Nonlinear Oscillations

  11.1 Fundamental Concepts in Stability. Equilibrium Points

  11.2 Small Motions of Single-Degree-of-Freedom Systems from Equilibrium..

  11.3 Conservative Systems. Motions in the Large

  11.4 Limit Cycles. The van der Pol Oscillator

  11.5 The Fundamental Perturbation Technique

  11.6 Secular Terms

  11.7 Lindstedt's Method

  11.8 Forced Oscillation of Quasi-Harmonic Systems. Jump Phenomenon

  11.9 Subharmonics and Combination Harmonics

  11.10 Systems with Time-Dependent Coefficients. Mathieu's Equation

  11.11 Numerical Integration of the Equations of Motion. The

   Runge-Kutta Methods

  11.12 Trajectories for the van der Pol Oscillator by MATLAB

  11.13 Summary

   Problems

12 Random Vibrations

  12.1 Ensemble Averages. Stationary Random Processes

  12.2 Time Averages. Ergodic Random Processes

  12.3 Mean Square Values and Standard Deviation

  12.4 Probability Density Functions

  12.5 Description of Random Data in Terms of Probability Density Functions...

  12.6 Properties of Autocorrelation Functions

  12.7 Response to Arbitrary Excitations by Fourier Transforms

  12.8 Power Spectral Density Functions

  12.9 Narrowband and Wideband Random Processes

  12.10 Response of Linear Systems to Stationary Random Excitations

  12.11 Response of Single-Degree-of-Freedom Systems to Random Excitations..

  12.12 Joint Probability Distribution of Two Random Variables

  12.13 Joint Properties of Stationary Random Processes

  12.14 Joint Properties of Ergodic Random Processes

  12.15 Response Cross-Correlation Functions for Linear Systems

  12.16 Response of Multi-Degree-of-Freedom Systems to Random Excitations...

  12.17 Response of Distributed-Parameter Systems to Random Excitations

  12.18 Summary

   Problems

Appendix A. Fourier Series

A.1 0rthogonal Sets of Functions

A.2 Trigonometric Series

A.3 Complex Form of Fourier Series

Appendix B. Laplace Transformation

B.1 Definition of the Laplace Transformation

B.2 Transformation of Derivatives

B.3 Transformation of Ordinary Differential Equations

B.4 The Inverse Laplace Transformation

B.5 Shifting Theorems

B.6 Method of Partial Fractions

B.7 The Convolution Integral. Borel's Theorem

B.8 Table of Laplace Transform Pairs

Appendix C. Linear Algebra

C.1 Matrices

 C.1.1 Definitions

 C.1.2 Matrix algebra

 C.1.3 Determinant of a square matrix

 C.1.4 Inverse of a matrix

 C.1.5 Transpose, inverse and determinant of a product of matrices.

 C.1.6 Partitioned matrices

C.2 Vector Spaces

 C.2.1 Definitions

 C.2.2 Linear dependence

 C.2.3 Bases and dimension of vector spaces

C.3 Linear Transformations

 C.3.1 The concept of linear transformations

 C.3.2 Solution of algebraic equations. Matrix inversion

Bibliography

Index