Nonlinear Vibration with Control: For Flexible and Adaptive Structures

461 Pages·2015·11.95 MB·English

by  David Wagg

Solid Mechanics and Its Applications
David Wagg
Simon Neild
Nonlinear
Vibration
with Control
For Flexible and Adaptive Structures
Second Edition
Solid Mechanics and Its Applications
Volume 218
Founding Editor
G.M.L. Gladwell, Waterloo, ON, Canada
Series editors
J.R. Barber, Ann Arbor Michigan, USA
Anders Klarbring, Linköping, Sweden
Aims and Scope of the Series
The fundamental questions arising in mechanics are: Why?, How?, and How
much? The aim of this series is to provide lucid accounts written by authoritative
researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics;statics,kinematicsanddynamicsofrigidandelasticbodies:vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
Themedianlevelofpresentationistothefirstyeargraduatestudent.Sometexts
aremonographsdefiningthecurrentstateofthefield;othersareaccessibletofinal
year undergraduates; but essentially the emphasis is on readability and clarity.
More information about this series at http://www.springer.com/series/6557
David Wagg Simon Neild
(cid:129)
Nonlinear Vibration with
Control
For Flexible and Adaptive Structures
Second Edition
123
David Wagg SimonNeild
Department of Mechanical Engineering Department of Mechanical Engineering
Universityof Sheffield Universityof Bristol
Sheffield Bristol
UK UK
ISSN 0925-0042 ISSN 2214-7764 (electronic)
ISBN 978-3-319-10643-4 ISBN 978-3-319-10644-1 (eBook)
DOI 10.1007/978-3-319-10644-1
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Preface
Identifying, modelling and controlling nonlinear vibrations is becoming increas-
ingly important in a range of engineering applications. This is particularly true in
thedesignofflexiblestructuressuchasaircraft,satellites,bridges,sportsstadiaand
other tall/slender structures. There are also applications in the areas of robotics,
mechatronics, micro-electro-mechanical systems (MEMS) and non-destructive
testing (NDT) and related disciplines such as structural health monitoring (SHM).
In the majority of cases, the trend is towards lighter structures, increased flexi-
bility and other higher levels of performance requirements. It is increasingly
commonforstructurestohaveintegratedactuatorandsensornetworkstocarryout
tasks such as limiting unwanted vibrations, detecting damage and in some cases
changing theshape ofthe structure.Thesetypes ofstructures have becomeknown
as smart structures (sometimes called adaptive or intelligent structures). They are
often made of new composite materials and their ability to perform multiple tasks
means that these types of smart structures are multifunctional.
Nonlinear behaviour in structural dynamics arises naturally from a range of
commonmaterialandgeometricnon-linearities.Bytheirnature,thesestructuresare
typicallymadeupofhighlyflexiblecontinuouselementssuchasbeams,cablesand
plates.Theyarealsorequiredtooperateinadynamicenvironmentand,asaresult,
understanding the vibration behaviour of the structures is critically important.
The focus of this book is first to give a comprehensive treatment of nonlinear
multi-modal structuralvibration problems, and then to show how (alimitedset of)
control techniques can be applied to such systems. The emphasis is on continuous
structural elements with relatively simple geometry, which enables a range of
analytical and approximate techniques to be presented, without the need for
extensivenumericalsimulation.Itshouldbeemphasizedthatthereisnoattemptto
provide a comprehensive treatment of nonlinear control techniques in this book.
Instead, a limited set of control approaches which apply to problems of vibration
control are presented.
The aim was to make the book accessible to the reader with some background
knowledge in linear vibration. The book falls into two main parts. The first five
chapters have been developed from lecture notes taught at masters level, and
v
vi Preface
example problems are included at the ends of Chaps. 2–4. The second half of the
book,Chaps.5–8,hasmoreofaresearchemphasis,withcasestudiesandresearch
examples shown where appropriate.
Chapters 1–3 contain introductory material on nonlinear vibration phenomena
and controlmethodsfor nonlinear vibration. Chapter 4introduces the approximate
techniquessuchasharmonicbalance,andperturbationmethodswhichcanbeused
for analysis of nonlinear vibration problems. The topic of modal analysis for
nonlinear structures is discussed in detail in Chap. 5. In particular, normal
form analysis is used to model multi-modal vibration response for nonlinear
structures.
Then each of Chaps. 6–8 is dedicated to a particular type of structural element.
Chapter6isfocusedonbeams,Chap.7oncablesandChap.8onplatesandshells.
In these chapters, a selection of nonlinear vibration case studies is presented.
Discussions of control methods are also included where appropriate.
Since the first edition of this book was published, there has been continued
interest in modelling and controlling nonlinear vibrations. The main change
wehaveintroducedisthatthenormalformmethodsdescribedinChaps.4and5are
now based on the second order form of the governing equations. This is a
more natural approach for structural dynamics problems and has other advantages
that are explained in the relevant sections. In addition to this, many more minor
additions have been made to update the text.
This book has only been possible with the generous help and support of many
colleaguesandcollaborators.Inparticular,wewouldliketoacknowledgethework
of Andres Arrieta Diaz, Andrea Cammarano, Alicia Gonzalez-Buelga, Tom Hill,
Irina Lasar, Xuanang Liu, Julian Londono, Nihal Malik, Claire Massow, Jack
Potter, Alex Shaw and Zhengfan Xin who carried out some of the original work,
which is presented in this book. For informed discussion on the scope of the book
andfeedbackonthedraftmanuscript,wewouldliketothankNickAlexander,Alex
Carrella, Mike Davies, David Ewins, Peter Gawthrop, Peter Green, Dan Inman,
Irfan Khan, Bernd Krauskopf, Steve Shaw, Lawrie Virgin, PaulWeaverand Keith
Worden. We would also like to thank Series Editor, Graham Gladwell, for his
detailed technical comments on the draft manuscript. In addition, we are very
grateful to Paul Neild, who meticulously proofread the manuscript of the first
editionandtoTomHill,whokindlyproducedFigs.5.6–5.12and7.10–7.13forthe
second edition. Finally, we would like to thank Tom Spicer from Springer for his
help and support.
Contents
1 Introduction to Nonlinear Vibration and Control. . . . . . . . . . . . . . 1
1.1 Vibration of Flexible Structures. . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Causes of Nonlinear Vibration. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Geometric Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 External Forces and Constraints . . . . . . . . . . . . . . . . . . 7
1.2.4 Freeplay, Backlash, Impact and Friction. . . . . . . . . . . . . 9
1.2.5 Control and Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Mathematical Models for Vibration . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Linear Vibration Modelled Using Sine Waves. . . . . . . . . 12
1.3.2 Nonlinear Vibration Modelled Using Sine Waves . . . . . . 17
1.3.3 Multiple Degrees-of-Freedom . . . . . . . . . . . . . . . . . . . . 20
1.4 Control of Nonlinear Vibrations. . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.1 Feedback Control of Linear Systems . . . . . . . . . . . . . . . 26
1.4.2 Feedback Control of Nonlinear Systems. . . . . . . . . . . . . 30
1.5 Continuous Structural Elements. . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Smart Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 Chapter Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Nonlinear Vibration Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 State Space Analysis of Dynamical Systems . . . . . . . . . . . . . . . 37
2.1.1 Harmonically Forced Linear Oscillator. . . . . . . . . . . . . . 38
2.1.2 Equilibrium Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.3 Local Linear Approximation Near Equilibrium Points . . . 45
2.2 Systems with Two States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Equilibrium Points for Linear Harmonic Oscillator . . . . . 47
2.3 The Link Between State Space and Mechanical Energy . . . . . . . 51
2.3.1 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Multiple Solutions, Stability and Initial Conditions. . . . . . . . . . . 58
2.4.1 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
viii Contents
2.5 Periodic and Non-periodic Oscillations. . . . . . . . . . . . . . . . . . . 61
2.6 Parameter Variation and Bifurcations . . . . . . . . . . . . . . . . . . . . 65
2.6.1 The Onset of Oscillations via a Hopf Bifurcation . . . . . . 71
2.6.2 Bifurcations in Forced Nonlinear Oscillations . . . . . . . . . 74
2.7 Systems with Harsh Nonlinearities. . . . . . . . . . . . . . . . . . . . . . 81
2.7.1 Friction Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7.2 Impact Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.8 Nonlinear Phenomena in Higher Dimensions. . . . . . . . . . . . . . . 87
2.8.1 The Fermi-Pasta-Ulam Paradox. . . . . . . . . . . . . . . . . . . 88
2.8.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.8.3 Modelling Approaches. . . . . . . . . . . . . . . . . . . . . . . . . 89
2.9 Chapter Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3 Control of Nonlinear Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1 Control Design for Nonlinear Vibrations. . . . . . . . . . . . . . . . . . 97
3.1.1 Passive Vibration Control. . . . . . . . . . . . . . . . . . . . . . . 98
3.1.2 Nonlinear Passive Vibration Isolators. . . . . . . . . . . . . . . 105
3.2 Semi-active Vibration Control. . . . . . . . . . . . . . . . . . . . . . . . . 106
3.3 Active Vibration Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.3.1 Observability and Controllability. . . . . . . . . . . . . . . . . . 111
3.3.2 Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.4 Stability Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.4.1 Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.4.2 Bounded Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.5 Linearisation Using Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.5.1 Input-Output Linearisation . . . . . . . . . . . . . . . . . . . . . . 126
3.6 Control of Multi-Degree-of-Freedom Systems . . . . . . . . . . . . . . 130
3.6.1 Modal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.7 Adaptive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.7.1 Adaptive Feedback Linearisation. . . . . . . . . . . . . . . . . . 136
3.8 Chapter Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4 Approximate Methods for Analysing Nonlinear Vibrations. . . . . . . 145
4.1 Backbone Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.2 Harmonic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.1 Forced Vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.3 Averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.3.1 Free Vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.3.2 Forced Vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.4 Perturbation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.4.1 Regular Perturbation Theory. . . . . . . . . . . . . . . . . . . . . 163
4.4.2 Multiple Scales Method. . . . . . . . . . . . . . . . . . . . . . . . 167
Contents ix
4.5 Normal Form Transformations. . . . . . . . . . . . . . . . . . . . . . . . . 171
4.5.1 Free Vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.5.2 Higher Order Accuracy . . . . . . . . . . . . . . . . . . . . . . . . 184
4.5.3 Forced Vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.5.4 Steady-State Stability. . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.6 Chapter Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5 Modal Analysis for Nonlinear Vibration . . . . . . . . . . . . . . . . . . . . 211
5.1 Modal Behaviour in Vibrating Systems. . . . . . . . . . . . . . . . . . . 212
5.2 Modal Decomposition Using Linear Techniques . . . . . . . . . . . . 213
5.2.1 Discrete Linear Systems. . . . . . . . . . . . . . . . . . . . . . . . 213
5.2.2 State Space Form for Discrete Linear Systems . . . . . . . . 216
5.2.3 Continuous Linear Systems . . . . . . . . . . . . . . . . . . . . . 219
5.3 Modal Decomposition for Nonlinear Systems . . . . . . . . . . . . . . 226
5.3.1 Nonlinear Normal Modes. . . . . . . . . . . . . . . . . . . . . . . 227
5.3.2 Internal Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.3.3 The Geometry of Nonlinear Modal Response . . . . . . . . . 235
5.4 Backbone Curves from Normal Form Transformations. . . . . . . . 236
5.4.1 Single Mode Backbone Curves. . . . . . . . . . . . . . . . . . . 237
5.4.2 Multi-mode Backbone Curves and Bifurcations. . . . . . . . 243
5.4.3 Nonlinear Mode Shape Analysis. . . . . . . . . . . . . . . . . . 247
5.4.4 Backbone Curves in the Symmetry Breaking Case. . . . . . 250
5.5 Application to Larger Scale Systems . . . . . . . . . . . . . . . . . . . . 257
5.6 Chapter Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
6.1 Small-Deflection Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . 261
6.1.1 The Euler-Bernoulli Equation . . . . . . . . . . . . . . . . . . . . 262
6.1.2 The Galerkin Method. . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.1.3 Initial Conditions and Forcing. . . . . . . . . . . . . . . . . . . . 268
6.1.4 Collocation Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.2 Nonlinear Beam Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.2.1 Large Deflections of Thin Beams . . . . . . . . . . . . . . . . . 277
6.2.2 Nonlinear Beam Equations with Axial Loading. . . . . . . . 278
6.2.3 Stretching of a Constrained Beam. . . . . . . . . . . . . . . . . 292
6.3 Case Study of Modal Control Applied to a Cantilever Beam. . . . 299
6.3.1 Modal Control of a Beam. . . . . . . . . . . . . . . . . . . . . . . 299
6.3.2 Vibration Suppression Using Piezoelectric Actuation. . . . 303
6.3.3 Positive Position Feedback. . . . . . . . . . . . . . . . . . . . . . 304
6.3.4 PPF for Nonlinear Vibration. . . . . . . . . . . . . . . . . . . . . 308
6.4 Chapter Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311