1st Edition
Vibration Analysis and Structural Dynamics for Civil Engineers Essentials and Group-Theoretic Formulations
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.
PART I: ESSENTIALS
Introduction
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
Introduction
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
References
PART II: GROUP-THEORETIC FORMULATIONS
Basic concepts of symmetry groups and representation theory
Symmetry groups
Group tables and classes
Representations of symmetry groups
Character tables
Group algebra
Idempotents
Applications
References
Rectilinear models
Introduction
A Shaft-disc torsional system
A Spring-mass extensional system
Conclusions
Plane structural grids
Introduction
Rectangular configurations
Square configurations
Conclusion
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
References
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
References
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
References
Biography
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
