Dynamics of Structures: Second Edition

This major textbook provides comprehensive coverage of the analytical tools required to determine the dynamic response of structures. The topics covered include: formulation of the equations of motion for single- as well as multi-degree-of-freedom discrete systems using the principles of both vector mechanics and analytical mechanics; free vibration response; determination of frequencies and mode shapes; forced vibration response to harmonic and general forcing functions; dynamic analysis of continuous systems;and wave propagation analysis. The key assets of the book include comprehensive coverage of both the traditional and state-of-the-art numerical techniques of response analysis, such as the analysis by numerical integration of the equations of motion and analysis through frequency domain. The large number of illustrative examples and exercise problems are of great assistance in improving clarity and enhancing reader comprehension. The text aims to benefit students and engineers in the civil, mechanical and aerospace sectors.

Table of Contents:
TABLE OF CONTENTS PREFACE 1 INTRODUCTION 1.1 Objectives of the Study of Structural Dynamics 1.2 Importance of Vibration Analysis 1.3 Nature of Exciting Forces 1.4 Mathematical Modeling of Dynamic Systems 1.5 Systems of Units 1.6 Organization of the Text PART I 2 FORMULATION OF THE EQUATIONS OF MOTION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS 2.1 Introduction 2.2 Inertia Forces 2.3 Resultants of Inertia Forces on a Rigid Body 2.4 Spring Forces 2.5 Damping Forces 2.6 Principle of Virtual Displacement 2.7 Formulation of the Equations of Motion 2.8 Modeling of Multi Degree-of-Freedom Discrete Parameter System 2.9 Effect of Gravity Load 2.10 Axial Force Effect 2.11 Effect of Support Motion 3 FORMULATION OF THE EQUATIONS OF MOTION: MULTI-DEGREE-OF-FREEDOM SYSTEMS 3.1 Introduction 3.2 Principal Forces in Multi Degree-of-freedom Dynamic System 3.3 Formulation of the Equations of Motion 3.4 Transformation of Coordinates 3.5 Static Condensation of Stiffness matrix 3.6 Application of Ritz Method to Discrete Systems 4 PRINCIPLES OF ANALYTICAL MECHANICS 4.1 Introduction 4.2 Generalized coordinates 4.3 Constraints 4.4 Virtual Work 4.5 Generalized Forces 4.6 Conservative Forces and Potential Energy 4.7 Work Function 4.8 Lagrangian Multipliers 4.9 Virtual Work Equation For Dynamical Systems 4.10 Hamilton's Equation 4.11 Lagrange's Equation 4.12 Constraint Conditions and Lagrangian Multipliers 4.13 Lagrange's Equations for Discrete Multi-Degree-of-Freedom Systems 4.14 Rayleigh's Dissipation Function PART II 5 FREE VIBRATION RESPONSE: SINGLE-DEGREE-OF-FREEDOM SYSTEM 5.1 Introduction 5.2 Undamped Free Vibration 5.3 Free Vibrations with Viscous Damping 5.4 Damped Free vibration with Hysteretic Damping 5.5 Damped Free vibration with Coulomb Damping 6 FORCED HARMONIC VIBRATIONS: SINGLE-DEGREE-OF-FREEDOM SYSTEM 6.1 Introduction 6.2 Procedures for the Solution of Forced Vibration Equation 6.3 Undamped Harmonic Vibration 6.4 Resonant Response of an Undamped System 6.5 Damped Harmonic Vibration 6.6 Complex Frequency Response 6.7 Resonant Response of a Damped System 6.8 Rotating Unbalanced Force 6.9 Transmitted Motion due to Support Movement 6.10 Transmissibility and Vibration Isolation 6.11 Vibration Measuring Instruments 6.12 Energy Dissipated in Viscous Damping 6.13 Hysteretic Damping 6.14 Complex Stiffness 6.15 Coulomb Damping 6.16 Measurement of Damping 7 RESPONSE TO GENERAL DYNAMIC LOADING AND TRANSIENT RESPONSE 7.1 Introduction 7.2 Response to an Impulsive force 7.3 Response to General Dynamic Loading 7.4 Response to a Step Function Load 7.5 Response to a Ramp Function Load 7.6 Response to a Step Function Load With Rise Time 7.7 Response to Shock Loading 7.8 Response to a Ground Motion Pulse 7.9 Analysis of Response by the Phase Plane Diagram 8 ANALYSIS OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 8.1 Introduction 8.2 Conservation of Energy 8.3 Application of Rayleigh Method to Multi Degree of Freedom Systems 8.4 Improved Rayleigh Method 8.5 Selection of an Appropriate Vibration Shape 8.6 Systems with Distributed Mass and Stiffness: Analysis of Internal Forces 8.7 Numerical Evaluation of Duhamel's Integral 8.8 Direct Integration of the Equations of Motion 8.9 Integration Based on Piece-wise Linear Representation of the Excitation 8.10 Derivation of General Formulae 8.11 Constant Acceleration Method 8.12 Newmark's beta Method 8.13 Wilson-theta Method 8.14 Methods Based on Difference Expressions 8.15 Errors involved in Numerical Integration 8.16 Stability of the Integration Method 8.17 Selection of a Numerical Integration Method 8.18 Selection of Time Step 9 ANALYSIS OF RESPONSE IN THE FREQUENCY DOMAIN 9.1 Transform Methods of Analysis 9.2 Fourier Series Representation of a Periodic Function 9.3 Response to a Periodically Applied Load 9.4 Exponential Form of Fourier Series 9.5 Complex Frequency Response Function 9.6 Fourier Integral Representation of a Nonperiodic Load 9.7 Response to a Nonperiodic Load 9.8 Convolution Integral and Convolution Theorem 9.9 Discrete Fourier Transform 9.10 Discrete Convolution and Discrete Convolution Theorem 9.11 Comparison of Continuous and Discrete Fourier Transforms 9.12 Application of Discrete Inverse Transform 9.13 Comparison Between Continuous and Discrete Convolution 9.14 Discrete Convolution of an Infnite and a Finite duration Waveform 9.15 Corrective Response Superposition Methods 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution PART III 10 FREE VIBRATION RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEM 10.1 Introduction 10.2 Standard Eigenvalue Problem 10.3 Linearized Eigenvalue Problem and its Properties 10.4 Expansion Theorem 10.5 Rayleigh Quotient 10.6 Solution of the Undamped Free-Vibration Problem 10.7 Mode Superposition Analysis of Free-Vibration Response 10.8 Solution of the Damped Free-Vibration Problem 10.9 Additional Orthogonality Conditions 10.10 Damping Orthogonality 11 NUMERICAL SOLUTION OF THE EIGENPROBLEM 11.1 Introduction 11.2 Properties of Standard Eigenvalues and Eigenvectors 11.3 Transformation of a Linearized Eigenvalue Problem to the Standard Form 11.4 Transformation Methods 11.5 Iteration Methods 11.6 Determinant Search Method 11.7 Numerical Solution of Complex Eigenvalue Problem 11.8 Semi-definite or Unrestrained Systems 11.9 Selection of a Method for the Determination of Eigenvalues 12 FORCED DYNAMIC RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEMS 12.1 Introduction 12.2 Normal Coordinate Transformation 12.3 Summary of Mode Superposition Method 12.4 Complex Frequency Response 12.5 Vibration Absorbers 12.6 Effect of Support Excitation 12.7 Forced Vibration of Unrestrained System 13 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 13.1 Introduction 13.2 Rayleigh-Ritz Method 13.3 Application of Ritz Method to Forced Vibration Response 13.4 Direct Integration of the Equations of Motion 13.5 Analysis in the Frequency Domain PART IV 14 FORMULATION OF THE EQUATIONS OF MOTION: CONTINUOUS SYSTEMS 14.1 Introduction 14.2 Transverse Vibrations of a Beam 14.3 Transverse Vibrations of a Beam: Variational Formulation 14.4 Effect of Damping Resistance on Transverse Vibrations of a Beam 14.5 Effect of Shear Deformation and Rotatory Inertia on the Flexural Vibrations of a Beam 14.6 Axial Vibrations of a Bar 14.7 Torsional Vibrations of a Bar 14.8 Transverse Vibrations of a String 14.9 Transverse Vibration of a Shear Beam 14.10 Transverse Vibrations of a Beam Excited by Support Motion 14.11 Effect of Axial Force on Transverse Vibrations of a Beam 15 CONTINUOUS SYSTEMS: FREE VIBRATION RESPONSE 15.1 Introduction 15.2 Eigenvalue Problem for the Transverse Vibrations of a Beam 15.3 General Eigenvalue Problem for a Continuous System 15.4 Expansion Theorem 15.5 Frequencies and Mode Shapes for Lateral Vibrations of a Beam 15.6 Effect of Shear Deformation and Rotatory Inertia on the Frequencies of Flexural Vibrations 15.7 Frequencies and Mode Shapes for the Axial Vibrations of a Bar 15.8 Frequencies and Mode Shapes for the Transverse Vibration of a String 15.9 Boundary Conditions Containing the 15.10 Free-Vibration Response of a Continuous System 15.11 Undamped Free Transverse Vibrations of a Beam 15.12 Damped Free Transverse Vibrations of a Beam 16 CONTINUOUS SYSTEMS: FORCED-VIBRATION RESPONSE 16.1 Introduction 16.2 Normal Coordinate Transformation: General Case of an Undamped System 16.3 Forced Lateral Vibration of a Beam 16.4 Transverse Vibrations of a Beam Under Traveling Load 16.5 Forced Axial Vibrations of a Uniform Bar 16.6 Normal Coordinate Transformation, Damped Case 17 WAVE PROPAGATION ANALYSIS 17.1 Introduction 17.2 The Phenomenon of Wave Propagation 17.3 Harmonic Waves 17.4 One Dimensional Wave Equation and its Solution 17.5 Propagation of Waves in Systems of Finite Extent 17.6 Reection and Refraction of Waves at a Discontinuity in the System Properties 17.7 Characteristics of the Wave Equation 17.8 Wave Dispersion PART V 18 FINITE ELEMENT METHOD 18.1 Introduction 18.2 Formulation of the Finite Element Equations 18.3 Selection of Shape Functions 18.4 Advantages of the Finite Element Method 18.5 Element Shapes 18.6 One-dimensional Bar Element 18.7 Flexural Vibrations of a Beam 18.8 Stress-strain Relationship for a Continuum 18.9 Triangular Element in Plane Stress and Plane Strain 18.10 Natural Coordinates 19 COMPONENT MODE SYNTHESIS 19.1 Introduction 19.2 Fixed Interface Methods 19.3 Free Interface Method 19.4 Hybrid Method 20 ANALYSIS OF NONLINEAR RESPONSE 20.1 Introduction 20.2 Single-degree-of-freedom System 20.3 Errors involved in Numerical Integration of Nonlinear Systems 20.4 Multiple Degree-of-freedom System ANSWERS TO SELECTED PROBLEMS INDEX