Dynamics of Structures: Second Edition

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