Advanced Structural Dynamics

Contents
Preface
Notation and Symbols
Unit Conversions
1 Fundamental Principles
	1.1 Classification of Problems in Structural Dynamics
	1.2 Stress–Strain Relationships
		1.2.1 Three-Dimensional State of Stress–Strain
		1.2.2 Plane Strain
		1.2.3 Plane Stress
		1.2.4 Plane Stress versus Plane Strain: Equivalent Poisson’s Ratio
	1.3 Stiffnesses of Some Typical Linear Systems
	1.4 Rigid Body Condition of Stiffness Matrix
	1.5 Mass Properties of Rigid, Homogeneous Bodies
	1.6 Estimation of Miscellaneous Masses
		1.6.1 Estimating the Weight (or Mass) of a Building
		1.6.2 Added Mass of Fluid for Fully Submerged Tubular Sections
		1.6.3 Added Fluid Mass and Damping for Bodies Floating in Deep Water
	1.7 Degrees of Freedom
		1.7.1 Static Degrees of Freedom
		1.7.2 Dynamic Degrees of Freedom
	1.8 Modeling Structural Systems
		1.8.1 Levels of Abstraction
		1.8.2 Transforming Continuous Systems into Discrete Ones
			Heuristic Method
		1.8.3 Direct Superposition Method
		1.8.4 Direct Stiffness Approach
		1.8.5 Flexibility Approach
		1.8.6 Viscous Damping Matrix
	1.9 Fundamental Dynamic Principles for a Rigid Body
		1.9.1 Inertial Reference Frames
		1.9.2 Kinematics of Motion
			Cardanian Rotation
			Eulerian Rotation
		1.9.3 Rotational Inertia Forces
		1.9.4 Newton’s Laws
			(a) Rectilinear Motion
			(b) Rotational Motion
		1.9.5 Kinetic Energy
		1.9.6 Conservation of Linear and Angular Momentum
			(a) Rectilinear Motion
			(b) Rotational Motion
		1.9.7 D’Alembert’s Principle
		1.9.8 Extension of Principles to System of Particles and Deformable Bodies
		1.9.9 Conservation of Momentum versus Conservation of Energy
		1.9.10 Instability of Rigid Body Spinning Freely in Space
	1.10 Elements of Analytical Mechanics
		1.10.1 Generalized Coordinates and Its Derivatives
		1.10.2 Lagrange’s Equations
			(a) Elastic Forces
			(b) Damping Forces
			(c) External Loads
			(d) Inertia Forces
			(e) Combined Virtual Work
2 Single Degree of Freedom Systems
	2.1 The Damped SDOF Oscillator
		2.1.1 Free Vibration: Homogeneous Solution
			Underdamped Case (ξ < 1)
			Critically Damped Case (ξ = 1)
			Overdamped Case (ξ > 1)
		2.1.2 Response Parameters
		2.1.3 Homogeneous Solution via Complex Frequencies: System Poles
		2.1.4 Free Vibration of an SDOF System with Time-Varying Mass
		2.1.5 Free Vibration of SDOF System with Frictional Damping
		(a) System Subjected to Initial Displacement
		(b) Arbitrary Initial Conditions
	2.2 Phase Portrait: Another Way to View Systems
		2.2.1 Preliminaries
		2.2.3 Examples of Application
			Phase Lines of a Linear SDOF System
			Ball Rolling on a Smooth Slope
		2.2.2 Fundamental Properties of Phase Lines
			Trajectory Arrows
			Intersection of Phase Lines with Horizontal Axis
			Asymptotic Behavior at Singular Points and Separatrix
			Period of Oscillation
	2.3 Measures of Damping
		2.3.1 Logarithmic Decrement
		2.3.2 Number of Cycles to 50% Amplitude
		2.3.3 Other Forms of Damping
	2.4 Forced Vibrations
		2.4.1 Forced Vibrations: Particular Solution
			(a) Heuristic Method
			(b) Variation of Parameters Method
		2.4.2 Forced Vibrations: General Solution
		2.4.3 Step Load of Infinite Duration
			2.4.4 Step Load of Finite Duration (Rectangular Load, or Box Load)
		2.4.5 Impulse Response Function
		2.4.6 Arbitrary Forcing Function: Convolution
			Convolution Integral
			Time Derivatives of the Convolution Integral
			Convolution as a Particular Solution
	2.5 Support Motion in SDOF Systems
		2.5.1 General Considerations
		2.5.2 Response Spectrum
			Tripartite Spectrum
		2.5.3 Ship on Rough Seas, or Car on Bumpy Road
	2.6 Harmonic Excitation: Steady-State Response
		2.6.1 Transfer Function Due to Harmonic Force
		2.6.2 Transfer Function Due to Harmonic Support Motion
		2.6.3 Eccentric Mass Vibrator
			Experimental Observation
		2.6.4 Response to Suddenly Applied Sinusoidal Load
		2.6.5 Half-Power Bandwidth Method
			Application of Half-Power Bandwidth Method
	2.7 Response to Periodic Loading
		2.7.1 Periodic Load Cast in Terms of Fourier Series
		2.7.2 Nonperiodic Load as Limit of Load with Infinite Period
		2.7.3 System Subjected to Periodic Loading: Solution in the Time Domain
		2.7.4 Transfer Function versus Impulse Response Function
		2.7.5 Fourier Inversion of Transfer Function by Contour Integration
			Location of Poles, Fourier Transforms, and Causality
		2.7.6 Response Computation in the Frequency Domain
			(1) Trailing Zeros
			(2) Exponential Window Method: The Preferred Strategy
	2.8 Dynamic Stiffness or Impedance
		2.8.1 Connection of Impedances in Series and/or Parallel
			Standard Solid
	2.9 Energy Dissipation through Damping
		2.9.1 Viscous Damping
			Instantaneous Power and Power Dissipation
			Human Power
			Average Power Dissipated in Harmonic Support Motion
			Ratio of Energy Dissipated to Energy Stored
			Hysteresis Loop for Spring–Dashpot System
		2.9.2 Hysteretic Damping
			Ratio of Energy Dissipated to Energy Stored
			Instantaneous Power and Power Dissipation via the Hilbert Transform
		2.9.3 Power Dissipation during Broadband Base Excitation
			Best Match between Viscous and Hysteretic Oscillator
		2.9.4 Comparing the Transfer Functions for Viscous and Hysteretic Damping
		2.9.5 Locus of Viscous and Hysteretic Transfer Function
3 Multiple Degree of Freedom Systems
	3.1 Multidegree of Freedom Systems
		3.1.1 Free Vibration Modes of Undamped MDOF Systems
			Orthogonality Conditions
			Normalized Eigenvectors
		3.1.2 Expansion Theorem
		3.1.3 Free Vibration of Undamped System Subjected to Initial Conditions
		3.1.4 Modal Partition of Energy in an Undamped MDOF System
		3.1.5 What If the Stiffness and Mass Matrices Are Not Symmetric?
		3.1.6 Physically Homogeneous Variables and Dimensionless Coordinates
	3.2 Effect of Static Loads on Structural Frequencies: Π-Δ Effects
		3.2.1 Effective Lateral Stiffness
		3.2.2 Vibration of Cantilever Column under Gravity Loads
		3.2.3 Buckling of Column with Rotations Prevented
		3.2.4 Vibration of Cantilever Shear Beam
	3.3 Estimation of Frequencies
		3.3.1 Rayleigh Quotient
			Rayleigh–Schwarz Quotients
		3.3.2 Dunkerley–Mikhlin Method
			Dunkerley’s Method for Systems with Rigid-Body Modes
		3.3.3 Effect on Frequencies of a Perturbation in the Structural Properties
			Perturbation of Mass Matrix
			Perturbation of Stiffness Matrix
			Qualitative Implications of Perturbation Formulas
	3.4 Spacing Properties of Natural Frequencies
		3.4.1 The Minimax Property of Rayleigh’s Quotient
		3.4.2 Interlacing of Eigenvalues for Systems with Single External Constraint
		Single Elastic External Support
		3.4.3 Interlacing of Eigenvalues for Systems with Single Internal Constraint
			Single Elastic Internal Constraint
		3.4.4 Number of Eigenvalues in Some Frequency Interval
			Sturm Sequence Property
			The Sign Count of the Shifted Stiffness Matrix
			Root Count for Dynamically Condensed Systems
			Generalization to Continuous Systems
	3.5 Vibrations of Damped MDOF Systems
		3.5.1 Vibrations of Proportionally Damped MDOF Systems
		3.5.2 Proportional versus Nonproportional Damping Matrices
		3.5.3 Conditions under Which a Damping Matrix Is Proportional
		3.5.4 Bounds to Coupling Terms in Modal Transformation
		3.5.5 Rayleigh Damping
		3.5.6 Caughey Damping
		3.5.7 Damping Matrix Satisfying Prescribed Modal Damping Ratios
		3.5.8 Construction of Nonproportional Damping Matrices
		3.5.9 Weighted Modal Damping: The Biggs–Roësset Equation
	3.6 Support Motions in MDOF Systems
		3.6.1 Structure with Single Translational DOF at Each Mass Point
			Solution by Modal Superposition (Proportional Damping)
		3.6.2 MDOF System Subjected to Multicomponent Support Motion
		3.6.3 Number of Modes in Modal Summation
		3.6.4 Static Correction
		3.6.5 Structures Subjected to Spatially Varying Support Motion
	3.7 Nonclassical, Complex Modes
		3.7.1 Quadratic Eigenvalue Problem
		3.7.2 Poles or Complex Frequencies
		3.7.3 Doubled-Up Form of Differential Equation
		3.7.4 Orthogonality Conditions
		3.7.5 Modal Superposition with Complex Modes
		3.7.6 Computation of Complex Modes
	3.8 Frequency Domain Analysis of MDOF Systems
		3.8.1 Steady-State Response of MDOF Systems to Structural Loads
		3.8.2 Steady-State Response of MDOF System Due to Support Motion
		3.8.3 In-Phase, Antiphase, and Opposite-Phase Motions
		3.8.4 Zeros of Transfer Functions at Point of Application of Load
		3.8.5 Steady-State Response of Structures with Hysteretic Damping
		3.8.6 Transient Response of MDOF Systems via Fourier Synthesis
		3.8.7 Decibel Scale
		3.8.8 Reciprocity Principle
	3.9 Harmonic Vibrations Due to Vortex Shedding
	3.10 Vibration Absorbers
		3.10.1 Tuned Mass Damper
		3.10.2 Lanchester Mass Damper
		3.10.3 Examples of Application of Vibration Absorbers
		3.10.4 Torsional Vibration Absorber
4 Continuous Systems
	4.1 Mathematical Characteristics of Continuous Systems
		4.1.1 Taut String
		4.1.2 Rods and Bars
		4.1.3 Bending Beam, Rotational Inertia Neglected
		4.1.4 Bending Beam, Rotational Inertia Included
		4.1.5 Timoshenko Beam
		4.1.6 Plate Bending
		4.1.7 Vibrations in Solids
		4.1.8 General Mathematical Form of Continuous Systems
		4.1.9 Orthogonality of Modes in Continuous Systems
	4.2 Exact Solutions for Simple Continuous Systems
		4.2.1 Homogeneous Rod
			Normal Modes of a Finite Rod
			Fixed–Fixed Rod
			Free–Free Rod
			Fixed–Free Rod
			Normal Modes of a Rod without Solving a Differential Equation
			Orthogonality of Rod Modes
		4.2.2 Euler–Bernoulli Beam (Bending Beam)
			Normal Modes of a Finite- Length Euler–Bernoulli Beam
			Simply Supported Beam
			Other Boundary Conditions
			Normal Modes of a Free– Free Beam
			Normal Modes of a Cantilever Beam
			Orthogonality Conditions of a Bending Beam
			Strain and Kinetic Energies of a Beam
		4.2.3 Bending Beam Subjected to Moving Harmonic Load
			Homogeneous Solution
			Particular Solution
		4.2.4 Nonuniform Bending Beam
		4.2.5 Nonclassical Modes of Uniform Shear Beam
			Dynamic Equations of Shear Beam
			Modes of Rotationally Unrestrained Shear Beam
			Concluding Observations
		4.2.6 Inhomogeneous Shear Beam
			Solution for Shear Modulus Growing Unboundedly with Depth
			Finite Layer of Inhomogeneous Soil
			Special Case: Shear Modulus Zero at Free Surface
			Special Case: Linearly Increasing Shear Wave Velocity
		4.2.7 Rectangular Prism Subjected to SH Waves
			Normal Modes
			Forced Vibration
		4.2.8 Cones, Frustums, and Horns
			(a) Exponential Horn
			(b) Frustum Growing as a Power of the Axial Distance
			(c) Cones of Infinite Depth with Bounded Growth of Cross Section
		4.2.9 Simply Supported, Homogeneous, Rectangular Plate
			Orthogonality Conditions of General Plate
			Simply Supported, Homogeneous Rectangular Plate
	4.3 Continuous, Wave-Based Elements (Spectral Elements)
		4.3.1 Impedance of a Finite Rod
		4.3.2 Impedance of a Semi-infinite Rod
		4.3.3 Viscoelastic Rod on a Viscous Foundation (Damped Rod)
			Stress and Velocity
			Power Flow
		4.3.4 Impedance of a Euler Beam
		4.3.5 Impedance of a Semi-infinite Beam
		4.3.6 Infinite Euler Beam with Springs at Regular Intervals
			Cutoff Frequencies
			Static Roots
		4.3.7 Semi-infinite Euler Beam Subjected to Bending Combined with Tension
			Power Transmission
			Power Transmission after Evanescent Wave Has Decayed
5 Wave Propagation
	5.1 Fundamentals of Wave Propagation
		5.1.1 Waves in Elastic Bodies
		5.1.2 Normal Modes and Dispersive Properties of Simple Systems
			An Infinite Rod
			Gravity Waves in a Deep Ocean
			An Infinite Bending Beam
			A Bending Beam on an Elastic Foundation
			A Bending Beam on an Elastic Half-Space
			Elastic Thick Plate (Mindlin Plate)
		5.1.3 Standing Waves, Wave Groups, Group Velocity, and Wave Dispersion
			Standing Waves
			Groups and Group Velocity
			Wave Groups and the Beating Phenomenon
			Summary of Concepts
		5.1.4 Impedance of an Infinite Rod
	5.2 Waves in Layered Media via Spectral Elements
		5.2.1 SH Waves and Generalized Love Waves
			(A) Normal Modes
			(B) Source Problem
			(C) Wave Amplification Problem
		5.2.2 SV-P Waves and Generalized Rayleigh Waves
			Normal Modes
		5.2.3 Stiffness Matrix Method in Cylindrical Coordinates
		5.2.4 Accurate Integration of Wavenumber Integrals
			Maximum Wavenumber for Truncation and Layer Coupling
			Static Asymptotic Behavior: Tail of Integrals
			Wavenumber Step
6 Numerical Methods
	6.1 Normal Modes by Inverse Iteration
		6.1.1 Fundamental Mode
		6.1.2 Higher Modes: Gram–Schmidt Sweeping Technique
		6.1.3 Inverse Iteration with Shift by Rayleigh Quotient
		6.1.4 Improving Eigenvectors after Inverse Iteration
		6.1.5 Inverse Iteration for Continuous Systems
	6.2 Method of Weighted Residuals
		6.2.1 Point Collocation
		6.2.2 Sub-domain
		6.2.3 Least Squares
		6.2.4 Galerkin
	6.3 Rayleigh–Ritz Method
		6.3.1 Boundary Conditions and Continuity Requirements in Rayleigh–Ritz
		6.3.2 Rayleigh–Ritz versus Galerkin
		6.3.3 Rayleigh–Ritz versus Finite Elements
		6.3.4 Rayleigh–Ritz Method for Discrete Systems
		6.3.5 Trial Functions versus True Modes
	6.4 Discrete Systems via Lagrange’s Equations
		6.4.1 Assumed Modes Method
		6.4.2 Partial Derivatives
		6.4.3 Examples of Application
		6.4.4 What If Some of the Discrete Equations Remain Uncoupled?
	6.5 Numerical Integration in the Time Domain
		6.5.1 Physical Approximations to the Forcing Function
		6.5.2 Physical Approximations to the Response
			Constant Acceleration Method
			Linear Acceleration Method
			Newmark’s β Method
			Impulse Acceleration Method
		6.5.3 Methods Based on Mathematical Approximations
			Multistep Methods for First-Order Differential Equations
			Difference and Integration Formulas
			Multistep Methods for Second-Order Differential Equations
		6.5.4 Runge–Kutta Type Methods
			Euler’s Method
			Improved and Modified Euler Methods
			The Normal Runge–Kutta Method
		6.5.5 Stability and Convergence Conditions for Multistep Methods
			Conditional and Unconditional Stability of Linear Systems
		6.5.6 Stability Considerations for Implicit Integration Schemes
	6.6 Fundamentals of Fourier Methods
		6.6.1 Fourier Transform
		6.6.2 Fourier Series
		6.6.3 Discrete Fourier Transform
		6.6.4 Discrete Fourier Series
		6.6.5 The Fast Fourier Transform
		6.6.6 Orthogonality Properties of Fourier Expansions
			(a) Fourier Transform
			(b) Fourier Series
			(c) Discrete Fourier Series
		6.6.7 Fourier Series Representation of a Train of Periodic Impulses
		6.6.8 Wraparound, Folding, and Aliasing
		6.6.9 Trigonometric Interpolation and the Fundamental Sampling Theorem
		6.6.10 Smoothing, Filtering, Truncation, and Data Decimation
		6.6.11 Mean Value
		6.6.12 Parseval’s Theorem
		6.6.13 Summary of Important Points
		6.6.14 Frequency Domain Analysis of Lightly Damped or Undamped Systems
			Exponential Window Method: The Preferred Tool
	6.7 Fundamentals of Finite Elements
		6.7.1 Gaussian Quadrature
			Normalization
	6.7.2 Integration in the Plane
		(a) Integral over a Rectangular Area
		(b) Integral over a Triangular Area
		(c) Curvilinear Triangle
		(d) Quadrilateral
		(e) Curvilinear Quadrilateral
		Inadmissible Shapes
	6.7.3 Finite Elements via Principle of Virtual Displacements
		(a) Consistency
		(b) Conformity
		(c) Rigid Body Test
		(d) Convergence (Patch Test)
	6.7.4 Plate Stretching Elements (Plane Strain)
		(a) Triangular Element
		(b) Rectangular Element
	6.7.5 Isoparametric Elements
		Plane Strain Curvilinear Quadrilaterals
		Cylindrical Coordinates
7 Earthquake Engineering and Soil Dynamics
	7.1 Stochastic Processes in Soil Dynamics
		7.1.1 Expectations of a Random Process
		7.1.2 Functions of Random Variable
		7.1.3 Stationary Processes
		7.1.4 Ergodic Processes
		7.1.5 Spectral Density Functions
		7.1.6 Coherence Function
		7.1.7 Estimation of Spectral Properties
		7.1.8 Spatial Coherence of Seismic Motions
			Coherency Function Based on Statistical Analyses of Actual Earthquake Motions
			Wave Model for Random Field
			Simple Cross-Spectrum for SH Waves
			Stochastic Deconvolution
	7.2 Earthquakes, and Measures of Quake Strength
		7.2.1 Magnitude
			Seismic Moment
			Moment Magnitude
		7.2.2 Seismic Intensity
		7.2.3 Seismic Risk: Gutenberg–Richter Law
		7.2.4 Direction of Intense Shaking
	7.3 Ground Response Spectra
		7.3.1 Preliminary Concepts
		7.3.2 Tripartite Response Spectrum
		7.3.3 Design Spectra
		7.3.4 Design Spectrum in the style of ASCE/SEI-7-05
			Design Earthquake
			Transition Periods
			Implied Ground Motion Parameters
		7.3.5 MDOF Systems: Estimating Maximum Values from Response Spectra
			Common Error in Modal Combination
			General Case: Response Spectrum Estimation for Complete Seismic Environment
	7.4 Dynamic Soil–Structure Interaction
		7.4.1 General Considerations
			Seismic Excitation (Free-Field Problem)
			Kinematic Interaction
			Inertial Interaction
		7.4.2 Modeling Considerations
			Continuum Solutions versus Finite Elements
			Finite Element Discretization
			Boundary Conditions
		7.4.3 Solution Methods
			Direct Approach
			Superposition Theorem
			Three-Step Approach
			Approximate Stiffness Functions
		7.4.4 Direct Formulation of SSI Problems
			The Substructure Theorem
			SSI Equations for Structures with Rigid Foundation
		7.4.5 SSI via Modal Synthesis in the Frequency Domain
			Partial Modal Summation
			What If the Modes Occupy Only a Subspace?
			Member Forces
		7.4.6 The Free-Field Problem: Elements of 1-D Soil Amplification
			Effect of Location of Control Motion in 1-D Soil Amplification
		7.4.7 Kinematic Interaction of Rigid Foundations
			Iguchi’s Approximation, General Case
			Iguchi Approximation for Cylindrical Foundations Subjected to SH Waves
			Geometric Properties
			Free-Field Motion Components at Arbitrary Point, Zero Azimuth
			Surface Integrals
			Volume Integrals
			Effective Motions
	7.5 Simple Models for Time-Varying, Inelastic Soil Behavior
		7.5.1 Inelastic Material Subjected to Cyclic Loads
		7.5.2 Masing’s Rule
		7.5.3 Ivan’s Model: Set of Elastoplastic Springs in Parallel
		7.5.4 Hyperbolic Model
		7.5.5 Ramberg–Osgood Model
	7.6 Response of Soil Deposits to Blast Loads
		7.6.1 Effects of Ground-Borne Blast Vibrations on Structures
		Frequency Effects
		Distance Effects
		Structural Damage
8 Advanced Topics
	8.1 The Hilbert Transform
		8.1.1 Definition
		8.1.2 Fourier Transform of the Sign Function
		8.1.3 Properties of the Hilbert Transform
		8.1.4 Causal Functions
		8.1.5 Kramers–Kronig Dispersion Relations
			Minimum Phase Systems
			Time-Shifted Causality
	8.2 Transfer Functions, Normal Modes, and Residues
		8.2.1 Poles and Zeros
		8.2.2 Special Case: No Damping
		8.2.3 Amplitude and Phase of the Transfer Function
		8.2.4 Normal Modes versus Residues
	8.3 Correspondence Principle
	8.4 Numerical Correspondence of Damped and Undamped Solutions
		8.4.1 Numerical Quadrature Method
		8.4.2 Perturbation Method
	8.5 Gyroscopic Forces Due to Rotor Support Motions
	8.6 Rotationally Periodic Structures
		8.6.1 Structures Composed of Identical Units and with Polar Symmetry
		8.6.2 Basic Properties of Block-Circulant Matrices
		8.6.3 Dynamics of Rotationally Periodic Structures
	8.7 Spatially Periodic Structures
		8.7.1 Method 1: Solution in Terms of Transfer Matrices
		8.7.2 Method 2: Solution via Static Condensation and Cloning
			Example: Waves in a Thick Solid Rod Subjected to Dynamic Source
		8.7.3 Method 3: Solution via Wave Propagation Modes
			Example 1: Set of Identical Masses Hanging from a Taut String
			Example 2: Infinite Chain of Viscoelastically Supported Masses and Spring-Dashpots
	8.8 The Discrete Shear Beam
		8.8.1 Continuous Shear Beam
		8.8.2 Discrete Shear Beam
9 Mathematical Tools
	9.1 Dirac Delta and Related Singularity Functions
		9.1.1 Related Singularity Functions
			Doublet Function
			Dirac Delta Function
			Unit Step Function (Heaviside Function)
			Unit Ramp Function
	9.2 Functions of Complex Variables: A Brief Summary
	9.3 Wavelets
		9.3.1 Box Function
		9.3.2 Hanning Bell (or Window)
		9.3.3 Gaussian Bell
		9.3.4 Modulated Sine Pulse (Antisymmetric Bell)
		9.3.5 Ricker Wavelet
	9.4 Useful Integrals Involving Exponentials
		9.4.1 Special Cases
	9.5 Integration Theorems in Two and Three Dimensions
		9.5.1 Integration by Parts
		9.5.2 Integration Theorems
		9.5.3 Particular Cases: Gauss, Stokes, and Green
	9.6 Positive Definiteness of Arbitrary Square Matrix
	9.7 Derivative of Matrix Determinant: The Trace Theorem
	9.8 Circulant and Block-Circulant Matrices
		9.8.1 Circulant Matrices
		9.8.2 Block-Circulant Matrices
10 Problem Sets
Author Index
Subject Index