Mechanical Vibrations: Theory and Applications takes an applications-based approach at teaching students to apply previously learned engineering principles while laying a foundation for engineering design. This text provides a brief review of the principles of dynamics so that terminology and notation are consistent and applies these principles to derive mathematical models of dynamic mechanical systems. The methods of application of these principles are consistent with popular Dynamics texts. Numerous pedagogical features have been included in the text in order to aid the student with comprehension and retention. These include the development of three benchmark problems which are revisited in each chapter, creating a coherent chain linking all chapters in the book. Also included are learning outcomes, summaries of key concepts including important equations and formulae, fully solved examples with an emphasis on real world examples, as well as an extensive exercise set including objective-type questions.

### Table of Contents

1. INTRODUCTION.

The Study of Vibrations. Mathematical Modeling. Generalized Coordinates. Classification of Vibration. Dimensional Analysis. Simple Harmonic Motion. Review of Dynamics.

2. MODELING OF SDOF SYSTEMS.

Introduction. Springs. Springs in Combination. Other Sources of Potential Energy. Viscous Damping. Energy Dissipated By Viscous Damping. Inertia Elements. External Sources. Free-Body Diagram Method. Static Deflections and Gravity. Small Angle or Displacement Assumption. Equivalent Systems Method.

3. FREE VIBRATIONS OF SDOF SYSTEMS.

Introduction. Standard Form of Differential Equation. Free Vibrations of an Undamped System. Underdamped Free Vibrations. Critically Damped Free Vibrations. Overdamped Free Vibrations. Coulomb Damping. Hysteretic Damping. Other Forms of Damping.

4. HARMONIC EXCITATION OF SDOF SYSTEMS.

Introduction. Forced Response of an Undamped System Due to a Single-Frequency Excitation. Forced Response of a Viscously Damped System Subject to a Single-Frequency Harmonic Excitation. Frequency-squared Excitations. Response Sue to Harmonic Excitation of Support. Vibration Isolation. Vibration Isolation from Frequency-Squared Excitations. Practical Aspects of Vibration Isolation. Multifrequency Excitations. General Periodic Excitations. Seismic Vibration Measuring Instruments. Complex Representations. Systems with Coulomb Damping. Systems with Hysteretic Damping. Energy Harvesting.

5. TRANSIENT VIBRATIONS OF ONE-DEGREE-OF-FREEDOM SYSTEMS.

Introduction. Derivation of Convolution Integral. Response Due to a General Excitation. Excitations Whose Forms Change at Discrete Times. Transient Motion Due to Base Excitation. Laplace Transform Solutions. Transfer Functions. Numerical Methods. Shock Spectrum. Shock Isolation.

6. TWO-DEGREE-OF-FREEDOM SYSTEMS.

Introduction. Derivation of the Equations of Motion. Natural Frequencies and Made Shapes. Free Response of Undamped Systems. Free Vibrations of a System with Viscous Damping. Principal Coordinates. Harmonic Response of Two-Degree-of Freedom Systems. Transfer Functions. Sinusoidal Transfer Function. Frequency Response. Dynamic Vibration Absorbers. Damped Vibration Absorbers. Vibration Dampers.

7. MODELING OF MDOF SYSTEMS.

Introduction. Derivation of Differential Equations Using the Free-Body-Diagram Method. Lagrange’s Equations. Matrix Formulation of Differential Equations for Linear Systems. Stiffness Influence Coefficients. Flexibility Influence Coefficients. Inertia Influence Coefficients. Lumped-Mass Modeling of Continuous Systems.

8. FREE VIBRATIONS OF MDOF SYSTEMS.

Introduction. Normal-Mode Solution. Natural Frequencies and Mode Shapes. General Solution. Special Cases. Energy Scalar Products. Properties of Natural Frequencies and Mode Shapes. Normalized Mode Shapes. Rayleigh’s Quotient. Principal Coordinates. Determination of Natural Frequencies and Mode Shapes. Proportional Damping. General Viscous Damping.

9. FORCED VIBRATIONS OF MDOF SYSTEMS.

Introduction. Harmonic Excitations. Sinusoidal Transfer Functions. Modal Analysis for Undamped Systems and Systems With Proportional Damping. Modal Analysis for Systems With General Damping. Numerical Solutions.

10. VIBRATIONS OF CONTINUOUS SYSTEMS.

Introduction. General Method. Second-Order Systems: Torsional Oscillations of a Circular Shaft. Transverse Beam Vibrations. Energy Methods.

11. FINITE ELEMENT METHOD.

Introduction. Assumed Modes Method. General Method. The Bar Element. Beam Element. Global Matrices.

12. NONLINEAR VIBRATIONS.

Introduction. Sources of Nonlinearity. Qualitative Analysis of Nonlinear Systems. Quantitative Method of Analysis. Free Vibrations of SDOF Systems. Forced Vibrations of SDOF Systems with Cubic Nonlinearities. MDOF Systems. Continuous Systems. Chaos.

13. RANDOM VIBRATIONS.

Introduction. Probability Density Functions. Standard Deviation. Autocorrelation. Fourier Transforms. Power Spectral Density. Wide Band and Narrow Band Processes. Response of SODF Systems. Response of MDOF Systems.

APPENDIX A: UNIT IMPULSE FUNCTION AND UNIT STEP FUNCTION.

APPENDIX B: LAPLACE TRANSFORMS.

APPENDIX C: LINEAR ALGEBRA.

APPENDIX D: DEFLECTIONS OF BEAMS DUE TO CONCENTRATED LOADS.

APPENDIX E: MATLAB PROGRAMS.