Fundamentals of the Theory of Mechanical Vibrations

Description

This book presents the fundamental concepts of modeling and analysis of vibrations in mechanical systems with one or more degrees of freedom. The presentation of classic topics is enriched by discussions on equilibrium, stability, and the linearization of the equations of motion. Practical examples throughout the text illustrate the applicability of the theory and explore the physics behind the equations. This book includes various Matlab codes, which allow readers to modify parameters and investigate the behavior of a wide range of mechanical systems. Furthermore, it is demonstrated how some of the mechanical systems studied can be constructed using ordinary materials, enabling readers to compare the theoretical results predicted by the mathematical models with the actual observed behavior. Preface Acknowledgments Contents Commonly Used Symbols and Abbreviations Part I Vibration of Systems with One Degree of Freedom 1 The Equation of Motion of Systems with One Degree of Freedom 1.1 Why Study Vibrations? 1.2 The Classical Oscillator 1.2.1 A Demonstration 1.2.2 Building a Model 1.2.2.1 Kinematic Analysis 1.2.2.2 Equation of Motion 1.2.3 Equilibrium Position 1.2.4 Energy Considerations 1.2.4.1 Potential Energy and Stability 1.2.5 Damping 1.3 The Simple Pendulum 1.3.1 A Model for the Simple Pendulum 1.3.1.1 Kinematic Analysis 1.3.1.2 Equation of Motion 1.3.1.3 Energy Conservation 1.3.1.4 Equilibrium Position and Stability 1.3.2 Simulating the Motion 1.3.2.1 Phase Portrait 1.3.3 Small Oscillation Approximation 1.3.3.1 Alternative Derivation of the Equation of Motion 1.3.3.2 Integral of Motion 1.3.4 The Damped Pendulum 1.3.4.1 Equation of Motion 1.3.4.2 Numerical Integration 1.3.4.3 Energy Considerations 1.3.4.4 Phase Portrait 1.3.4.5 Small Oscillations 1.4 Form of the Equation of Motion 1.5 Illustrative Examples 1.5.1 A Rolling Cylinder 1.5.1.1 Kinematic Analysis 1.5.1.2 Equation of Motion 1.5.1.3 Numerical Integration 1.5.1.4 Equilibrium Positions 1.5.1.5 Small Oscillations About the Equilibrium 1.5.1.6 Energy Approximation 1.5.1.7 Alternative Derivation of the Equation of Motion 1.5.2 Motion of a Metronome 1.5.2.1 Kinematic Analysis 1.5.2.2 Equation of Motion 1.5.2.3 Simulating the Motion 1.5.2.4 Equilibrium Positions 1.5.2.5 Linearization 1.5.2.6 Energy Approximation 1.5.2.7 Alternative Derivation of the Equation of Motion 1.6 Summary 1.7 Problems 2 Free Vibration of Systems with One Degree of Freedom 2.1 What Is Free Vibration? 2.2 Free Vibration of Undamped Systems 2.2.1 The Harmonic Oscillator 2.2.1.1 Computing the Response 2.2.1.2 Analysis of the Response 2.2.2 General Case of Undamped Free Vibration 2.2.3 Illustrative Examples 2.2.3.1 Small Oscillations of a Simple Pendulum 2.2.3.2 The Rolling Cylinder 2.2.3.3 Stable and Unstable Motions of a Metronome 2.2.3.4 A Rocking Semicylinder 2.2.3.5 A Bobbing Buoy 2.2.3.6 The Bifiliar Pendulum 2.3 Free Vibration in the Presence of Viscous Damping 2.3.1 The Mass-Spring-Damper System 2.3.1.1 Case 1: Undamped (ζ=0) 2.3.1.2 Case 2: Underdamped (0