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 loads
1.3.4 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 1
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
2.7.1 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 stiffness
2.7.4 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 systems
3.1 Introduction
3.2 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 2
5 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 methods
8.1 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 shape
8.6 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 motion
8.9 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 β method
8.12.1 Average acceleration method
8.12.2 Linear acceleration method
8.13 Wilson-θ method
8.14 Methods based on difference expressions
8.14.1 Central difference method
8.14.2 Houbolt’s method
8.15 Errors involved in numerical integration
8.16 Stability of the integration method
8.16.1 Newmark’s β method
8.16.2 Wilson-θ method
8.16.3 Central difference method
8.16.4 Houbolt’s method
8.17 Selection of a numerical integration method
8.18 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 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 infinite- and a finite-duration waveform
9.15 Corrective response superposition methods
9.15.1 Corrective transient response based on initial conditions
9.15.2 Corrective periodic response based on initial conditions
9.15.3 Corrective responses obtained from a pair of force pulses
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
Selected readings
Problems
PART 3
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
Selected readings
Problems
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
11.4.1 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 shifts
11.5.4 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 relationship
11.7.2 Matrix iteration for determining the complex eigenvalues
11.8 Semidefinite or unrestrained systems
11.8.1 Characteristics of an unrestrained system
11.8.2 Eigenvalue solution of a semidefinite system
11.9 Selection of a method for the determination of eigenvalues
Selected readings
Problems
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
Selected readings
Problems
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
13.3.1 Mode superposition method
13.3.2 Mode acceleration method
13.3.3 Static condensation and Guyan’s reduction
13.3.4 Load-dependent Ritz vectors
13.3.5 Application of lanczos vectors in the transformation of the equations of motion
13.4 Direct integration of the equations of motion
13.4.1 Explicit integration schemes
13.4.2 Implicit integration schemes
13.4.3 Mixed methods in direct integration
13.5 Analysis in the frequency domain
13.5.1 Frequency analysis of systems with classical mode shapes
13.5.2 Frequency analysis of systems without classical mode shapes
Selected readings
Problems
PART 4
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 vibrations 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
Selected readings
Problems
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
15.3.1 Definition of the eigenvalue problem
15.3.2 Self-adjointness of operators in the eigenvalue problem
15.3.3 Orthogonality of eigenfunctions
15.3.4 Positive and positive definite operators
15.4 Expansion theorem
15.5 Frequencies and mode shapes for lateral vibrations of a beam
15.5.1 Simply supported beam
15.5.2 Uniform cantilever beam
15.5.3 Uniform beam clamped at both ends
15.5.4 Uniform beam with both ends free
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
15.7.1 Axial vibrations of a clamped–free bar
15.7.2 Axial vibrations of a free–free bar
15.8 Frequencies and mode shapes for the transverse vibration of a string
15.8.1 Vibrations of a string tied at both ends
15.9 Boundary conditions containing the eigenvalue
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
Selected readings
Problems
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
Selected readings
Problems
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 Reflection and refraction of waves at a discontinuity in the system properties
17.7 Characteristics of the wave equation
17.8 Wave dispersion
Selected readings
Problems
PART 5
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
18.5.1 One-dimensional elements
18.5.2 Two-dimensional elements
18.6 One-dimensional bar element
18.7 Flexural vibrations of a beam
18.7.1 Stiffness matrix of a beam element
18.7.2 Mass matrix of a beam element 884
18.7.3 Nodal applied force vector for a beam element
18.7.4 Geometric stiffness matrix for a beam element
18.7.5 Simultaneous axial and lateral vibrations
18.8 Stress-strain relationships for a continuum
18.8.1 Plane stress
18.8.2 Plane strain
18.9 Triangular element in plane stress and plane strain
18.10 Natural coordinates
18.10.1 Natural coordinate formulation for a uniaxial bar element
18.10.2 Natural coordinate formulation for a constant strain triangle
18.10.3 Natural coordinate formulation for a linear strain triangle
Selected readings
Problems
19 Component mode synthesis
19.1 Introduction
19.2 Fixed interface methods
19.2.1 Fixed interface normal modes
19.2.2 Constraint modes
19.2.3 Transformation of coordinates
19.2.4 Illustrative example
19.3 Free interface method
19.3.1 Free interface normal modes
19.3.2 Attachment modes
19.3.3 Inertia relief attachment modes
19.3.4 Residual flexibility attachment modes
19.3.5 Transformation of coordinates
19.3.6 Illustrative example
19.4 Hybrid method
19.4.1 Experimental determination of modal parameters
19.4.2 Experimental determination of the static constraint modes
19.4.3 Component modes and transformation of component matrices
19.4.4 Illustrative example
Selected readings
Problems
20 Analysis of nonlinear response
20.1 Introduction
20.2 Single-degree-of freedom system
20.2.1 Central difference method
20.2.2 Newmark’s β Method
20.3 Errors involved in numerical integration of nonlinear systems
20.4 Multiple degree-of-freedom system
20.4.1 Explicit integration
20.4.2 Implicit integration
20.4.3 Iterations within a time step
Selected readings
Problems
Answers to selected problems
Index
Biography
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 and became 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.