Dynamics of Structures

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.

1 Introduction 1.1 Objectives of the study of structural dynamics 1.2 Importance of vibration analysis 1.3 Nature of exciting forces 1.3.1 Dynamic forces caused by rotating machinery 1.3.2 Wind loads 1.3.3 Blast loads1.3.4 Dynamic forces caused by earthquakes 1.3.5 Periodic and nonperiodic loads 1.3.6 Deterministic and nondeterministic loads 1.4 Mathematical modeling of dynamic systems 1.5 Systems of units 1.6 Organization of the text PART 12 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 2.7.1 Systems with localized mass and localized stiffness 2.7.2 Systems with localized mass but distributed stiffness 2.7.3 Systems with distributed mass but localized stiffness2.7.4 Systems with distributed stiffness and distributed mass 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 Selected readings Problems 3 Formulation of the equations of motion: Multi-degree-of-freedom systems3.1 Introduction3.2 Principal forces in multi-degree-of-freedom dynamic system 3.2.1 Inertia forces 3.2.2 Forces arising due to elasticity 3.2.3 Damping forces 3.2.4 Axial force effects 3.3 Formulation of the equations of motion 3.3.1 Systems with localized mass and localized stiffness 3.3.2 Systems with localized mass but distributed stiffness 3.3.3 Systems with distributed mass but localized stiffness 3.3.4 Systems with distributed mass and distributed stiffness 3.4 Transformation of coordinates 3.5 Static condensation of stiffness matrix 3.6 Application of Ritz method to discrete systems Selected readings Problems 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 multi-degree-of-freedom systems 4.14 Rayleigh’s dissipation function Selected readings Problems PART 25 Free vibration response: Single-degree-of-freedom system 5.1 Introduction 5.2 Undamped free vibration 5.2.1 Phase plane diagram 5.3 Free vibrations with viscous damping 5.3.1 Critically damped system 5.3.2 Overdamped system 5.3.3 Underdamped system 5.3.4 Phase plane diagram 5.3.5 Logarithmic decrement 5.4 Damped free vibration with hysteretic damping 5.5 Damped free vibration with coulomb damping 5.5.1 Phase plane representation of vibrations under Coulomb damping Selected readings Problems 6 Forced harmonic vibrations: Single-degree-of-freedom system 6.1 Introduction 6.2 Procedures for the solution of the 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 6.11.1 Measurement of support acceleration 6.11.2 Measurement of support displacement 6.12 Energy dissipated in viscous damping 6.13 Hysteretic damping 6.14 Complex stiffness 6.15 Coulomb damping 6.16 Measurement of damping 6.16.1 Free vibration decay 6.16.2 Forced-vibration response Selected readings Problems 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 7.7.1 Rectangular pulse 7.7.2 Triangular pulse 7.7.3 Sinusoidal pulse 7.7.4 Effect of viscous damping 7.7.5 Approximate response analysis for short-duration pulses 7.8 Response to ground motion 7.8.1 Response to a short-duration ground motion pulse 7.9 Analysis of response by the phase plane diagram Selected readings Problems 8 Analysis of single-degree-of-freedom systems: Approximate and numerical methods8.1 Introduction 8.2 Conservation of energy 8.3 Application of Rayleigh method to multi-degree-of-freedom systems 8.3.1 Flexural vibrations of a beam 8.4 Improved Rayleigh method 8.5 Selection of an appropriate vibration shape8.6 Systems with distributed mass and stiffness: analysis of internal forces 8.7 Numerical evaluation of Duhamel’s integral 8.7.1 Rectangular summation 8.7.2 Trapezoidal method 8.7.3 Simpson’s method 8.8 Direct integration of the equations of motion8.9 Integration based on piece-wise linear representation of the excitation 8.10 Derivation of general formulas 8.11 Constant-acceleration method 8.12 Newmark’s ß method 8.12.1 Average acceleration method 8.12.2 Linear acceleration method 8.13 Wilson-? method 8.14 Methods based on difference expressions8.14.1 Central difference method 8.14.2 Houbolt’s method 8.15 Errors involved in numerical integration8.16 Stability of the integration method 8.16.1 Newmark’s ß method 8.16.2 Wilson-? method 8.16.3 Central difference method8.16.4 Houbolt’s method 8.17 Selection of a numerical integration method8.18 Selection of time step Selected readings Problems 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 function9.6 Fourier integral representation of a nonperiodic load9.7 Response to a nonperiodic load 9.8 Convolution integral and convolution theorem9.9 Discrete Fourier transform9.10 Discrete convolution and discrete convolution theorem 9.11 Comparison of continuous and discrete fourier transforms9.12 Application of discrete inverse transform 9.13 Comparison between continuous and discrete convolution 9.14 Discrete convolution of an infinite- and a finite-duration waveform 9.15 Corrective response superposition methods 9.15.1 Corrective transient response based on initial conditions9.15.2 Corrective periodic response based on initial conditions 9.15.3 Corrective responses obtained from a pair of force pulses9.16 Exponential window method 9.17 The fast Fourier transform9.18 Theoretical background to fast Fourier transform 9.19 Computing speed of FFT convolution Selected readings Problems PART 310 Free vibration response: Multi-degree-of-freedom system 10.1 Introduction10.2 Standard eigenvalue problem10.3 Linearized eigenvalue problem and its properties10.4 Expansion theorem 10.5 Rayleigh quotient 10.6 Solution of the undamped free vibration problem10.7 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 Selected readings Problems 11 Numerical solution of the eigenproblem11.1 Introduction 11.2 Properties of standard eigenvalues and eigenvectors11.3 Transformation of a linearized eigenvalue problem to the standard form 11.4 Transformation methods 11.4.1 Jacobi diagonalization 11.4.2 Householder’s transformation 11.4.3 QR transformation 11.5 Iteration methods 11.5.1 Vector iteration 11.5.2 Inverse vector iteration 11.5.3 Vector iteration with shifts11.5.4 Subspace iteration 11.5.5 Lanczos iteration 11.6 Determinant search method 11.7 Numerical solution of complex eigenvalue problem 11.7.1 Eigenvalue problem and the orthogonality relationship11.7.2 Matrix iteration for determining the complex eigenvalues 11.8 Semidefinite or unrestrained systems 11.8.1 Characteristics of an unrestrained system11.8.2 Eigenvalue solution of a semidefinite system 11.9 Selection of a method for the determination of eigenvaluesS

Dr. Jag Mohan Humar, is currently Distinguished Research Professor of Civil Engineering at Carleton University, Ottawa, Canada. Dr. Humar obtained his Ph.D. from Carleton University in 1974. He joined Carleton as a faculty member in the Department of Civil Engineering in 1975 andbecame a full professor in 1983, and served as the Chairman of the Department of Civil and Environmental Engineering from 1989 to 2000. Dr. Humar’s main research interest is in structural dynamics and earthquake engineering. He has published over 120 journal and conference papers in this and related areas. He is also the author of a book entitled "Dynamics of Structures," published by Prentice Hall, USA in 1990. The second edition of the book has been published by Balkema Publishers of Netherlands in 2002. In February 2000 Dr. Humar led a Canadian Scientific mission to Gujarat to study the damage caused by the Bhuj earthquake. Dr. Humar is actively involved in the development of seismic design provisions of the National Building Code of Canada. Over the last 15 years he has served as a member of the Standing Committee on Earthquake Design, an advisory body to National Building Code of Canada (NBCC) for its seismic design provisions. During these years the NBCC seismic provisions have undergone substantial revisions, and many of the changes and new requirements have been influenced by Dr. Humar’s work in the field. Along with teaching, academic administration, and research, Dr. Humar has also been active in engineering consulting He served as a special consultant for several outstanding civil engineering projects, including the National Aviation Museum in Ottawa and the SkyDome in Toronto. He was a seismic design consultant on several other projects, which include the Earthquake Response Study of the Alexandria Bridge across the Ottawa River, Seismic Rehabilitation of the Victoria Museum, Ottawa, Blast Load Analysis of the Mackenzie Tower, Parliamentary Precinct, Ottawa. He also served as a member and chair of the experts panel to review the seismic rehabilitation and upgrade of the West Block, Parliamentary Precinct, Ottawa. Dr. Humar has received several awards for his outstanding contributions to teaching, research, engineering practice, and the profession. Dr. Humar serves as a field referee for many international journals including the ASCE Journals of Structures and Engineering Mechanics, the Journal of Sound and Vibration, the Journal of Structural Dynamics and Earthquake Engineering, and the Canadian Journal of Civil Engineering. For 7 years he served as an Associate Editor for the Canadian Journal of Civil Engineering. Currently he is the Associate Editor of the International Journal of Earthquake Engineering and Structural Dynamics.