1st Edition

Vibration Analysis and Structural Dynamics for Civil Engineers
Essentials and Group-Theoretic Formulations

ISBN 9780415522564
Published November 30, 2014 by CRC Press
276 Pages 77 B/W Illustrations

USD $71.95

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Book Description

Appeals to the Student and the Seasoned Professional

While the analysis of a civil-engineering structure typically seeks to quantify static effects (stresses and strains), there are some aspects that require considerations of vibration and dynamic behavior. Vibration Analysis and Structural Dynamics for Civil Engineers: Essentials and Group-Theoretic Formulations is relevant to instances that involve significant time-varying effects, including impact and sudden movement. It explains the basic theory to undergraduate and graduate students taking courses on vibration and dynamics, and also presents an original approach for the vibration analysis of symmetric systems, for both researchers and practicing engineers. Divided into two parts, it first covers the fundamentals of the vibration of engineering systems, and later addresses how symmetry affects vibration behavior.

Part I treats the modeling of discrete single and multi-degree-of-freedom systems, as well as mathematical formulations for continuous systems, both analytical and numerical. It also features some worked examples and tutorial problems. Part II introduces the mathematical concepts of group theory and symmetry groups, and applies these to the vibration of a diverse range of problems in structural mechanics. It reveals the computational benefits of the group-theoretic approach, and sheds new insights on complex vibration phenomena.

The book consists of 11 chapters with topics that include:

  • The vibration of discrete systems or lumped parameter models
  • The free and forced response of single degree-of-freedom systems
  • The vibration of systems with multiple degrees of freedom
  • The vibration of continuous systems (strings, rods and beams)
  • The essentials of finite-element vibration modelling
  • Symmetry considerations and an outline of group and representation theories
  • Applications of group theory to the vibration of linear mechanical systems
  • Applications of group theory to the vibration of structural grids and cable nets
  • Group-theoretic finite-element and finite-difference formulations

Vibration Analysis and Structural Dynamics for Civil Engineers: Essentials and Group-Theoretic Formulations acquaints students with the fundamentals of vibration theory, informs experienced structural practitioners on simple and effective techniques for vibration modelling, and provides researchers with new directions for the development of computational vibration procedures.

Table of Contents



Definitions, aims and general concepts

Basic features of a vibrating system, and further concepts

Tutorial questions

Single degree-of-freedom systems

Basic equation of motion

Free vibration response

Equivalent spring stiffnesses for various structural and mechanical systems

Response to harmonic excitation

Tutorial questions

Systems with more than one degree of freedom

Introductory remark

Equations of motion

Techniques for assembling the stiffness matrix

The flexibility formulation of the equations of motion and assembly of the flexibility matrix

Determination of natural frequencies and mode shapes

The flexibility formulation of the eigenvalue problem

Worked examples

The modal matrix

Orthogonality of eigenvectors

Generalized mass and stiffness matrices

Worked examples

Modal analysis

Worked example

Tutorial questions

Continuous systems


Transverse vibration of strings

Axial vibration of rods

Flexural vibration of beams

Orthogonality of natural modes of vibration

Dynamic response by the method of modal analysis

Finite-element vibration analysis

The finite-element formulation

Stiffness and consistent mass matrices for some common finite elements

Assembly of the system equations of motion



Basic concepts of symmetry groups and representation theory

Symmetry groups

Group tables and classes

Representations of symmetry groups

Character tables

Group algebra




Rectilinear models


A Shaft-disc torsional system

A Spring-mass extensional system


Plane structural grids


Rectangular configurations

Square configurations


High-tension cable nets

Basic assumptions and geometric formulation

Outline of computational scheme

Illustrative examples

Symmetry-adapted functions

Symmetry-adapted flexibility matrices

Subspace mass matrices

Eigenvalues, eigenvectors and mode shapes

Summary and concluding remarks


Finite-difference formulation for plates

General finite-difference formulation for plate vibration

Group-theoretic implementation

Application to rectangular and square plates

Finite-difference equations for generator nodes of the basis vectors

Symmetry-adapted finite-difference equations and system eigenvalues

Concluding remarks


Finite-element formulations for symmetric elements

Group-theoretic formulation for finite elements

Coordinate system, node numbering and positive directions

Symmetry-adapted nodal freedoms

Displacement field decomposition

Subspace shape functions

Subspace element matrices

Final element matrices

Concluding remarks


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Alphose Zingoni is professor of structural engineering and mechanics in the Department of Civil Engineering at the University of Cape Town. He holds an M.Sc in structural engineering and a Ph.D in shell structures, both earned at Imperial College London. Dr. Zingoni has research interests encompassing shell structures, space structures, vibration analysis, and applications of group theory to problems in computational structural mechanics. He has written numerous scientific papers on these topics, which have been published in leading international journals and presented at various international conferences worldwide.


"… a valuable addition to the structural dynamics literature. In particular, the final six chapters provide a clear, concise and readable account of the group theoretical basis for simplifying the analysis of symmetric structural dynamical systems. … the book’s author presents an application of some of his own research work on group-theoretical formulations aiming to simplify the modelling of structures with symmetry."
—Computers and Structures, 2015

"… a novel approach to the vibration analysis of symmetric systems. …the book provides comprehensive guidance for students, practitioners and researchers interested in the essentials and group-theoretic formulations of vibration analysis and structural dynamics."
—ICE Proceedings-Structures-Buildings Journal, 2015

"Strengths of the book are the simplicity and clarity of explaining the basics of structural dynamics, including some worked examples and tutorial questions."
—Guido De Roeck, KU Leuven, Belgium

"This is a fabulous book, written by a true expert in the field. It is rigorous, but accessible, and it helps to simplify some of the most important but complex dynamics phenomena through an innovative link to the mathematics of group theory. This is a book I must have on my book shelf."
—Tim Ibell, President of the Institution of Structural Engineers, Bath, UK

"This book is well written and looks at an important topic in civil engineering education. It progresses from a fundamental treatment at undergraduate level to advanced topics at postgraduate coursework level and postgraduate research studies."
—Mark Bradford, UNSW Australia