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Elements Of Structural Dynamics - A New Perspective
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ISBN13：9781118339626
出版社：John Wiley & Sons Inc
作者：Roy
出版日：2012/09/14
裝訂／頁數：精裝／438頁
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商品簡介
作者簡介
目次
商品簡介
Elements of Structural Dynamics: A New Perspective takes a novel ‘top-down’ approach to the subject of structural dynamics. It starts with the basic principles of linear momentum balance of isotropic and linearly elastic systems, which in turn yield the governing equations of motion in structural dynamics. It attempts to mathematically model the ‘real-life’ structural systems, whilst not sacrificing the important feature of a suitably graded exposition with gradually increasing complexity.
Elements of Structural Dynamics: A New Perspective is devoted to covering dynamics of linear structural systems and covers the computational aspects that enable the adapting of the basic theory into computer programs. This book also provides some information on the important mathematical concepts to allow a more insightful understanding of the subject.
Elements of Structural Dynamics: A New Perspective is devoted to covering dynamics of linear structural systems and covers the computational aspects that enable the adapting of the basic theory into computer programs. This book also provides some information on the important mathematical concepts to allow a more insightful understanding of the subject.
作者簡介
Debasish Roy, Indian Institute of Science, Bangalore, India
Debasish Roy is a professor in the Department of Civil Engineering at the Indian Institute of Science. He is a member of the editorial board for seven journals and has published circa 80 journal articles. His current research interests include Nonlinear and Stochastic Structural Dynamics, Linearization Techniques in Non-linear Dynamics, and Mesh-free and Finite Element Methods.
G. V. Rao, Vasthu Shilpa Associates, Bangalore, India
Dr. Rao obtained his Ph. D in civil engineering from the Indian Institute of Science and has since been employed in industry. He has many years experience in experimental testing and research in structural engineering and also in research and software development in the area of structural dynamics using FEM. He is currently employed as an engineering consultant at Vasthu Shilpa Associates in India.
目次
Prologue - An introduction to structural dynamics
Chapter 1 Dynamic equations of motion – structural systems
1.0 Introduction
1.1 System of rigid bodies and dynamic equations of motion
Principle of virtual work
Hamilton’s principle
Lagrangian equations of motion
1.2 Continuous dynamical systems and equations of motion from Hamilton’s
principle
Stain and stress tensors and strain energy
1.3 Dynamic equilibrium equations from principles of Newton’s force balance
Displacement- strain relationships
Stress-strain relationships
1.4. Equations of motion by Reynolds transport theorm
Mass conservation
Linear momentum conservation
Chapter 2 Continuous systems – PDEs and solution
2.0 Introduction
2.1 Some continuous systems and PDE-s
A taut string – the one dimensional wave equation
An Euler-Bernoulli beam – the one-dimensional bi-harmonic wave equation
Beam equation with rotary inertia and shear deformation effects
Equations of motion for 2 D plate by classical plate theory (Kirchhoff theory)
Strain-displacement relationships
Displacements due to bending
Stress-strain relationships
Energy expressions
Rectangular plate
Plate with non-smooth boundaries (sharp edges)
2.2 PDEs and general solution
PDE-s and canonical transformations
General solution to the wave equation
Particular solution (D’Alembert’s solution) to the wave equation
2.3 Method of separation of variables and solution to linear homogeneous PDE-s
Homogeneous PDE with homogeneous boundary conditions
Sturm-Liouville boundary value problem for the wave equation
Adjoint operator and self-adjoint property
Eigenvalues and eigenfunctions of the wave equation
Series solution to the wave equation
Mixed boundary conditions and wave equation
Sturm-Liouville boundary value problem for the biharmonic wave equation
Timshenko Beam PDE - Free vibration solution
Simply supported beam
Thin rectangular plates – free vibration solution
2.4 Orthonormal basis and eigenfunction expansion
Best approximation to f(x)
2.5 Solution of inhomogeneous PDE-s by eigenfunction-expansion method
2.6 Solutions of inhomogeneous PDE-s by Green’s function method
2.7 Solution of PDEs with inhomogeneous boundary conditions
2.8 Solution to non-self adjoint continuous systems
Eigen solution of non-self adjoint system
Bi-orthogonality relationship between L and L^*.
Eigensolutions of L and L^*
Chapter 3 Classical methods for solving the equations of motion
3.0 Introduction
3.1 Rayleigh-Ritz Method
Rayleigh’s principle
3.2 Weighted residuals method
3.3 Galerkin method
3.4 Collocation method
3.5 Sub-domain method
3.6 Least squares method
Chapter 4 Finite element method and structural dynamics
4.0 Introduction
4.1 Weak formulation of PDEs
Well-posedness of the weak form
Uniqueness and stability of solution to weak form
Numerical integration by Gauss quadrature
4.2 Element-wise representation of the weak form and the FEM
4.3 Application of the FEM to 2D problems
Membrane vibrations and FEM
Plane (2D) elasticity problems – plane stress and plane strain
4.4 Higher order polynomial basis functions
Beam vibrations and FEM
Plate vibrations and FEM
4.5 Some computational issues in FEM
Element shape functions in natural coordinates
Beam elements
Quadrilateral elements
4.6 FEM and error estimates
A-priori error estimate
Conclusions
Chapter 5 MDOF systems and eigenvalue problems
5.0 Introduction
5.1 Discrete systems through a lumped parameter approach
Positive definite and semi-definite systems
5.2 Coupled linear ODE-s and the linear differential operator
5.3 Coupled linear ODEs and eigensolution
5.4 First order equations and uncoupling
5.5 First order vs second order ODEs and eigensolutions
5.6 MDOF systems and modal dynamics
SDOF oscillator and modal solution
Rayleigh quotient
Rayleigh-Ritz method for MDOF systems
5.7 Damped MDOF systems
Dmped system and quadratic eigenvalue problem
Damped system and unsymmetric eigenvalue problem
Proportional damping and uncoupling MDOF systems
Damped systems and impulse response
Response under general loading
Response under harmonic input
Complex frequency response
Force transmissibility
System response and measurement of damping
Logarithmic decrement method
Half power method
CHAPTER 6 Structures under support excitations
6.0 Introduction
6.1 Continuous systems and base excitation
6.2 MDOF systems and base excitation
6.3 SDOF system and base excitation
Frequency response of SDOF system under base motion
6.4 Support excitation and response spectra
Peak response estimates of an MDOF system using response spectra
CHAPTER 7 Eigensolution procedures
7.0 Introduction
7.1 Power and inverse iteration methods and eigensolution
Power iteration
Inverse iteration
Order and rate of convergence – distinct eigenvalues
Shifting and convergence
Multiple eigenvalues
Eigenvalues within an interval – shifting scheme with Gram-Schmidt
orthogonalization and Sturm sequence property
7.2 Jacobi, Householder, QR transformation methods and eigensolutions
Jacobi method
Convergence of the Jacobi method
Jacobi method for the generalized eigenvalue problem
Householder and QR transformation methods
Householder transformation method
QR transformation method
Convergence of the QR method
Implementation issues with the QR method
7.3 Subspace iteration
Convergence in subspace iteration
7.4 Lanczos transformation method
Lanczos method and error analysis
7.5 Systems with unsymmetric matrices
Skew-symmetric matrics and eigensolution
Unsymmetric systems and eigensolutions
Unsymmetric eigenvalue problems and solution methods
Two-sided Lanczos transformation method
QR transformation method and unsymmetric matrices
7.6 Dynamic condensation and eigensolution
Symmetric systems and dynamic condensation
Unsymmetric systems and dynamic condensation
CHAPTER 8 Direct integration methods
8.0 Introduction
8.1 Forward and backward Euler methods
Forward Euler method
Backward (implicit) Euler method
8.2 Central difference method
8.3 Newmark-β method – a single step implicit method
Some degenerate cases of Newmark-β method and stability
Undamped case – amplitude and periodicity errors
Amplitude and periodicity errors
8.4 HHT-α and generalized-α methods
Chapter 9 Stochastic structural dynamics
9.0 Introduction
9.1 Probability theory and basic concepts
9.2 Random variables
Joint random variables, distributions and density functions
Expected (average) values of a random variable
Characteristic and moment generating functions
9.3 Conditional probability, independence and conditional expectation
Conditional expectation
9.4 Some oft-used probability distributions
Binomial distribution
Poisson distribution
Normal distribution
Uniform distribution
Rayleigh distribution
9.5 Stochastic processes
Stationarity of a stochastic process
Properties of autocovariance / autocorrelation functions of stationary processes
Spectral representation of a stochastic process
S_XX (λ) as the mean energy density of X(t)
Some basic stochastic processes
Markov process
Poisson process
Wiener process
9.6 Stochastic dynamics of linear structural systems
Continuous systems under stochastic input
Response of a SDOF oscillator under a stochastic input
Response of an SDOF oscillator to a Stationary input
Response of an SDOF oscillator in frequency domain
Discrete systems under stochastic input - modal superposition method
9.7 An introduction to Ito calculus
Brownian filtration
Measurability
An adapted stochastic process
Ito integral
Martingale
Ito’s process
Ito’s formula
Quadratic variation (QV) of X(t)
Quadratic covariation
Integration by parts
Higher dimensional Ito’s formula
Quadratic covariation of Brownian components
Computing the response moments
Time integration of SDEs
Conclusions
Chapter 1 Dynamic equations of motion – structural systems
1.0 Introduction
1.1 System of rigid bodies and dynamic equations of motion
Principle of virtual work
Hamilton’s principle
Lagrangian equations of motion
1.2 Continuous dynamical systems and equations of motion from Hamilton’s
principle
Stain and stress tensors and strain energy
1.3 Dynamic equilibrium equations from principles of Newton’s force balance
Displacement- strain relationships
Stress-strain relationships
1.4. Equations of motion by Reynolds transport theorm
Mass conservation
Linear momentum conservation
Chapter 2 Continuous systems – PDEs and solution
2.0 Introduction
2.1 Some continuous systems and PDE-s
A taut string – the one dimensional wave equation
An Euler-Bernoulli beam – the one-dimensional bi-harmonic wave equation
Beam equation with rotary inertia and shear deformation effects
Equations of motion for 2 D plate by classical plate theory (Kirchhoff theory)
Strain-displacement relationships
Displacements due to bending
Stress-strain relationships
Energy expressions
Rectangular plate
Plate with non-smooth boundaries (sharp edges)
2.2 PDEs and general solution
PDE-s and canonical transformations
General solution to the wave equation
Particular solution (D’Alembert’s solution) to the wave equation
2.3 Method of separation of variables and solution to linear homogeneous PDE-s
Homogeneous PDE with homogeneous boundary conditions
Sturm-Liouville boundary value problem for the wave equation
Adjoint operator and self-adjoint property
Eigenvalues and eigenfunctions of the wave equation
Series solution to the wave equation
Mixed boundary conditions and wave equation
Sturm-Liouville boundary value problem for the biharmonic wave equation
Timshenko Beam PDE - Free vibration solution
Simply supported beam
Thin rectangular plates – free vibration solution
2.4 Orthonormal basis and eigenfunction expansion
Best approximation to f(x)
2.5 Solution of inhomogeneous PDE-s by eigenfunction-expansion method
2.6 Solutions of inhomogeneous PDE-s by Green’s function method
2.7 Solution of PDEs with inhomogeneous boundary conditions
2.8 Solution to non-self adjoint continuous systems
Eigen solution of non-self adjoint system
Bi-orthogonality relationship between L and L^*.
Eigensolutions of L and L^*
Chapter 3 Classical methods for solving the equations of motion
3.0 Introduction
3.1 Rayleigh-Ritz Method
Rayleigh’s principle
3.2 Weighted residuals method
3.3 Galerkin method
3.4 Collocation method
3.5 Sub-domain method
3.6 Least squares method
Chapter 4 Finite element method and structural dynamics
4.0 Introduction
4.1 Weak formulation of PDEs
Well-posedness of the weak form
Uniqueness and stability of solution to weak form
Numerical integration by Gauss quadrature
4.2 Element-wise representation of the weak form and the FEM
4.3 Application of the FEM to 2D problems
Membrane vibrations and FEM
Plane (2D) elasticity problems – plane stress and plane strain
4.4 Higher order polynomial basis functions
Beam vibrations and FEM
Plate vibrations and FEM
4.5 Some computational issues in FEM
Element shape functions in natural coordinates
Beam elements
Quadrilateral elements
4.6 FEM and error estimates
A-priori error estimate
Conclusions
Chapter 5 MDOF systems and eigenvalue problems
5.0 Introduction
5.1 Discrete systems through a lumped parameter approach
Positive definite and semi-definite systems
5.2 Coupled linear ODE-s and the linear differential operator
5.3 Coupled linear ODEs and eigensolution
5.4 First order equations and uncoupling
5.5 First order vs second order ODEs and eigensolutions
5.6 MDOF systems and modal dynamics
SDOF oscillator and modal solution
Rayleigh quotient
Rayleigh-Ritz method for MDOF systems
5.7 Damped MDOF systems
Dmped system and quadratic eigenvalue problem
Damped system and unsymmetric eigenvalue problem
Proportional damping and uncoupling MDOF systems
Damped systems and impulse response
Response under general loading
Response under harmonic input
Complex frequency response
Force transmissibility
System response and measurement of damping
Logarithmic decrement method
Half power method
CHAPTER 6 Structures under support excitations
6.0 Introduction
6.1 Continuous systems and base excitation
6.2 MDOF systems and base excitation
6.3 SDOF system and base excitation
Frequency response of SDOF system under base motion
6.4 Support excitation and response spectra
Peak response estimates of an MDOF system using response spectra
CHAPTER 7 Eigensolution procedures
7.0 Introduction
7.1 Power and inverse iteration methods and eigensolution
Power iteration
Inverse iteration
Order and rate of convergence – distinct eigenvalues
Shifting and convergence
Multiple eigenvalues
Eigenvalues within an interval – shifting scheme with Gram-Schmidt
orthogonalization and Sturm sequence property
7.2 Jacobi, Householder, QR transformation methods and eigensolutions
Jacobi method
Convergence of the Jacobi method
Jacobi method for the generalized eigenvalue problem
Householder and QR transformation methods
Householder transformation method
QR transformation method
Convergence of the QR method
Implementation issues with the QR method
7.3 Subspace iteration
Convergence in subspace iteration
7.4 Lanczos transformation method
Lanczos method and error analysis
7.5 Systems with unsymmetric matrices
Skew-symmetric matrics and eigensolution
Unsymmetric systems and eigensolutions
Unsymmetric eigenvalue problems and solution methods
Two-sided Lanczos transformation method
QR transformation method and unsymmetric matrices
7.6 Dynamic condensation and eigensolution
Symmetric systems and dynamic condensation
Unsymmetric systems and dynamic condensation
CHAPTER 8 Direct integration methods
8.0 Introduction
8.1 Forward and backward Euler methods
Forward Euler method
Backward (implicit) Euler method
8.2 Central difference method
8.3 Newmark-β method – a single step implicit method
Some degenerate cases of Newmark-β method and stability
Undamped case – amplitude and periodicity errors
Amplitude and periodicity errors
8.4 HHT-α and generalized-α methods
Chapter 9 Stochastic structural dynamics
9.0 Introduction
9.1 Probability theory and basic concepts
9.2 Random variables
Joint random variables, distributions and density functions
Expected (average) values of a random variable
Characteristic and moment generating functions
9.3 Conditional probability, independence and conditional expectation
Conditional expectation
9.4 Some oft-used probability distributions
Binomial distribution
Poisson distribution
Normal distribution
Uniform distribution
Rayleigh distribution
9.5 Stochastic processes
Stationarity of a stochastic process
Properties of autocovariance / autocorrelation functions of stationary processes
Spectral representation of a stochastic process
S_XX (λ) as the mean energy density of X(t)
Some basic stochastic processes
Markov process
Poisson process
Wiener process
9.6 Stochastic dynamics of linear structural systems
Continuous systems under stochastic input
Response of a SDOF oscillator under a stochastic input
Response of an SDOF oscillator to a Stationary input
Response of an SDOF oscillator in frequency domain
Discrete systems under stochastic input - modal superposition method
9.7 An introduction to Ito calculus
Brownian filtration
Measurability
An adapted stochastic process
Ito integral
Martingale
Ito’s process
Ito’s formula
Quadratic variation (QV) of X(t)
Quadratic covariation
Integration by parts
Higher dimensional Ito’s formula
Quadratic covariation of Brownian components
Computing the response moments
Time integration of SDEs
Conclusions
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