Table Of ContentStructural Engineering Structural
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“Martin Williams has come up with a novel way of presenting the theory and underlying principles L
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of structural dynamics. Mathematical derivations are kept as simple as possible and are fully I
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supported with a wealth of self-explanatory problems and solutions. This is an essential read for Dynamics
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undergraduate and graduate students of civil engineering.”
—Dr. Theodore Karavasilis, University of Warwick, UK S
“This book provides students with an in-depth knowledge of structural dynamics without boring
them! Each chapter sets out clearly the learning objectives and the theory is illustrated with many
clear and diverse examples. The commentaries are interesting to read and provide further insight
into the topics covered.”
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—Professor Tiziana Rossetto, University College London
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“I have no doubt that the book will be of use not only to advanced university level students but
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also to practitioners (including myself!) tackling engineering problems in the realms of structural
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dynamics, who will use the book not only to refresh and confirm their memory and understanding
of this complex subject, but also to stretch them into a few corners of it that they might not have C
previously considered.”
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—Edmund Booth, Consulting Engineer, UK
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“ … I feel that this book gives [a] clear explanation of the fundamental principles and theory, which
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is achieved from the author’s many years of teaching and research on structural dynamics.”
—Tianjian Ji, University of Manchester, UK A
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“I really appreciated the clear and simple way arguments are introduced, which is crucial for
undergraduate students who are not acquainted with structural dynamics and face the topic for D
the first time.”
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—Giuseppe Abbiati, ETH Zurich
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“ … this book provides an excellent introduction using a simple yet rigorous approach.”
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—Andrei M. Reinhorn, University at Buffalo, New York
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This short, accessible introduction to structural dynamics is aimed at upper level civil engineering
undergraduates and those starting an M.Sc. course, making extensive use of worked examples. It I
begins by describing dynamic systems and their representation for analytical purposes, then linear C
analysis of single- (SDOF) and multi-degree-of-freedom (MDOF) systems, under free vibration and S
in response to a variety of forcing functions. Hand analysis of continuous systems, methods of
calculation of non-linear dynamic response, and the key principles of random vibration analysis
are discussed.
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Structural
Dynamics
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Structural
Dynamics
MARTIN WILLIAMS
University of Oxford, UK
A SPON PRESS BOOK
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2016 by Martin Williams
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Version Date: 20160302
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Contents
Preface xi
About the front cover xiii
Symbols and abbreviations xv
1 Introduction to dynamic systems 1
1.1 What is dynamics and why is it important? 1
1.1.1 Why dynamics matters 2
1.1.1.1 The Tacoma Narrows Bridge 2
1.1.1.2 The Northridge and Kobe Earthquakes 3
1.1.1.3 The London Millennium Bridge 5
1.2 Basic definitions 6
1.3 Describing oscillatory motion 7
1.3.1 Time and frequency domains 7
1.3.2 The basics of harmonic motion 8
1.3.3 Phasors 12
1.3.4 Some everyday examples of harmonic motion 13
1.3.4.1 Pendulum 13
1.3.4.2 Bobbing in water 13
1.4 Dynamic properties of physical systems 14
1.4.1 Mass and stiffness 14
1.4.2 Damping 15
1.4.3 Modelling physical systems for dynamic analysis 17
1.5 Dynamic loads 19
1.5.1 Impulsive loads 19
1.5.2 Periodic loads 20
1.5.3 Random loads 22
Key points 24
Tutorial problems 24
vii
viii Contents
2 Single-degree-of-freedom systems 27
2.1 Introduction 27
2.2 Equation of motion 27
2.2.1 With an applied force 27
2.2.2 With a ground motion 29
2.3 Free vibration 30
2.3.1 Undamped free vibration 31
2.3.2 Damped free vibration 33
2.3.2.1 Critically damped: ξ = 1 (or c2 = 4 km) 34
2.3.2.2 Underdamped: ξ < 1 (or c2 < 4 km) 35
2.3.2.3 Overdamped: ξ > 1 (or c2 > 4 km) 36
2.3.2.4 Logarithmic decrement 36
2.4 Response to dynamic loads 37
2.4.1 Step load 38
2.4.2 Harmonic load 40
2.4.3 Harmonic base motion 45
2.4.4 Response to periodic loads 49
2.4.5 Impulse response 52
2.4.6 Response to irregular dynamic loads 54
2.4.7 Earthquake response spectrum 59
2.5 Transfer functions and block diagrams 64
2.5.1 Laplace transforms 65
2.5.2 Transfer function of a linear SDOF system 66
2.5.3 Relationship to impulse response 68
Key points 69
Tutorial problems 69
3 Multi-degree-of-freedom systems 73
3.1 Introduction 73
3.2 Equations of motion 74
3.2.1 MDOF systems with dynamic forces 74
3.2.2 MDOF systems subjected to base motions 77
3.2.3 Some comments on MDOF equations of motion 79
3.3 Free vibration of MDOF systems 80
3.3.1 Natural frequencies and mode shapes 80
3.3.2 Properties of mode shapes 86
3.3.3 Empirical methods 88
3.4 Vibration analysis of 2DOF systems 89
3.4.1 Equivalence between 2DOF and two SDOF systems 89
3.4.2 Undamped response to a harmonic load 92
3.4.3 Tuned mass dampers 94
3.5 Response of MDOF systems to dynamic loads 96
3.5.1 Mode superposition method: theory 96
3.5.2 Modelling damping in MDOF systems 97
Contents ix
3.5.2.1 Modal damping 98
3.5.2.2 Rayleigh damping 98
3.5.2.3 Other damping formulations 99
3.5.3 Implementing the mode superposition method 100
3.5.4 Mode superposition for MDOF systems with base motions 104
3.5.5 Truncation of modes 106
3.5.6 Mode superposition and response spectrum analysis 111
Key points 113
Tutorial problems 113
4 Continuous systems 117
4.1 Introduction 117
4.2 Vibrations of beams 117
4.2.1 Undamped equation of motion 117
4.2.2 Undamped free vibration of uniform beams 119
4.2.3 Forced vibration response 124
4.3 Equivalent single-degree-of-freedom systems 128
4.3.1 Generalised properties 128
4.3.2 Response to applied loads 133
4.3.3 Response to a base motion 135
Key points 138
Tutorial problems 139
5 Non-linear dynamics 141
5.1 Introduction 141
5.2 Non-linearity in structural properties 142
5.2.1 Variable mass 142
5.2.2 Non-proportional damping 142
5.2.3 Non-linear stiffness 147
5.2.3.1 Elastic-perfectly plastic hysteresis 149
5.2.3.2 A bilinear stiffness model 150
5.2.4 Geometric non-linearity 152
5.2.5 Dynamic characteristics of non-linear systems 156
5.3 Time-stepping algorithms 158
5.3.1 The need for a different approach 158
5.3.2 A simple algorithm: the central difference method 159
5.3.3 Stability, accuracy and time step 165
5.3.3.1 Accuracy 165
5.3.3.2 Stability 167
5.3.4 A more complex algorithm: the Newmark method 169
5.3.5 Extension to MDOF systems 171
5.3.5.1 Damping 171
5.3.5.2 Stability, accuracy and time step 172
5.3.6 How many non-linear analyses? 172
