- Comprehensive coverage of numerical techniques, including frequency domain analysis methods, numerical integration techniques etc.
- Clarity of presentation, which will appeal to the reader, and particularly to students
- Large number of illustrative examples and end-of-chapter exercises
- Availability of a solution manual (152 pages), which will assist instructors
- Proven track record such that the previous editions have been acquired by engineering libraries around the world and the book has been adopted as a text book at many universities

*Solutions manual available upon qualifying course adoption*

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 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 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 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 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 310 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 414 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 518 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 *

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.