Nonlinear Vibration with Control

Nonlinear Vibration with Control

SOLID MECHANICS AND ITS APPLICATIONS Volume 170 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?,andHow much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of 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, kinematics and dynamics of rigid and elastic bodies: 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. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For other titles published in this series, go to www.springer.com/series/6557

David Wagg Simon Neild Nonlinear Vibration with Control For Flexible and Adaptive Structures

Professor David Wagg Department of Structural Dynamics Department of Mechanical Engineering University of Bristol Queen s Building University Walk Bristol, BS8 1TR, UK Dr Simon Neild Senior Lecturer in Dynamics and Control Department of Mechanical Engineering University of Bristol Queen s Building University Walk Bristol, BS8 1TR, UK Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands In association with Canopus Academic Publishing Limited, 15 Nelson Parade, Bedminster, Bristol, BS3 4HY, UK www.springer.com and www.canopusbooks.com ISBN 978-90-481-2836-5 e-isbn 978-90-481-2837-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009937609 Canopus Academic Publishing Limited 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface Identifying, modelling and controlling nonlinear vibrations is becoming increasingly important in a range of engineering applications. This is particularly true in the design of flexible structures such as aircraft, satellites, bridges, sports stadia and 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 flexibility and other higher levels of performance requirements. It is increasingly common for structures to have integrated actuator and sensor networks to carry out tasks such as limiting unwanted vibrations, detecting damage and in some cases changing the shape of the structure. These types of structures have become known 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 common material and geometric nonlinearities. By their nature, these structures are typically made up of highly flexible continuous elements such as beams, cables and plates. They are also required to operate in a dynamic environment and, as a result, understanding the vibration behaviour of the structures is critically important. The focus of this book is first to give a comprehensive treatment of nonlin- ear multi-modal structural vibration problems, and secondly to show how (a limited set 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 extensive numerical simulation. It should be emphasized that there is no attempt to 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 examv

vi Preface ple problems are included at the ends of Chaps. 2 to 4. The second half of the book, Chaps. 5, 6, 7 and 8, has more of a research emphasis, with case studies and research examples shown where appropriate. Chapters 1 to 3 contain introductory material on nonlinear vibration phenomena and control methods for nonlinear vibration. Chapter 4 introduces the approximate techniques such as harmonic balance, and perturbation methods which can be used 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 to 8 is dedicated to a particular type of structural element. Chap. 6 is focused on beams, Chap. 7 on cables and Chap. 8 on plates and shells. In these chapters a selection of nonlinear vibration case studies is presented. Discussions of control methods are also included where appropriate. This book has only been possible with the generous help and support of many colleagues and collaborators. In particular we would like to acknowledge the work of Andres Arrieta Diaz, Alicia Gonzalez-Buelga, Nihal Malik, Claire Massow and Jack Potter, who carried out some of the original work which is presented in this book. For informed discussion on the scope of the book and feedback on the draft manuscript, we would like to thank Nick Alexander, Alex Carrella, Mike Davies, David Ewins, Peter Gawthrop, Dan Inman, Bernd Krauskopf, Lawrie Virgin and Paul Weaver. We would also like to thank Keith Worden and Series Editor, Graham Gladwell, for their detailed technical comments on the draft manuscript. In addition, we are very grateful to Paul Neild, who meticulously proof read the manuscript. Finally we would like to thank Robin Rees and Tom Spicer at Canopus books for their help and support. Bristol, June 2009 David Wagg Simon Neild

Contents 1 Introduction to Nonlinear Vibration and Control.................. 1 1.1 Vibration of Flexible Structures............................... 1 1.2 Causes of Nonlinear Vibration................................ 4 1.2.1 Material Properties................................... 4 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.............................. 24 1.4.1 Feedback Control of Linear Systems.................... 25 1.4.2 Feedback Control of Nonlinear Systems................. 29 1.5 Continuous Structural Elements.............................. 31 1.6 Smart Structures........................................... 31 1.7 Chapter Notes............................................. 32 References..................................................... 33 2 Nonlinear Vibration Phenomena................................. 35 2.1 State Space Analysis of Dynamical Systems.................... 35 2.1.1 Equilibrium Points................................... 38 2.1.2 Local Linear Approximation Near Equilibrium Points..... 42 2.2 The Link Between State Space and Mechanical Energy........... 49 2.2.1 Potential Functions................................... 50 2.3 Multiple Solutions, Stability and Initial Conditions.............. 55 2.4 Periodic and Non-Periodic Oscillations........................ 58 2.5 Parameter Variation and Bifurcations.......................... 62 2.5.1 The Onset of Oscillations via a Hopf Bifurcation.......... 68 2.5.2 Bifurcations in Forced Nonlinear Oscillations............ 71 vii

viii Contents 2.6 Nonlinear Phenomena in Higher Dimensions................... 76 2.7 Chapter Notes............................................. 77 References..................................................... 77 3 Control of Nonlinear Vibrations................................. 81 3.1 Control Design for Nonlinear Vibrations....................... 81 3.1.1 Semi-Active Vibration Control......................... 82 3.1.2 Active Vibration Control.............................. 85 3.2 Stability Theory............................................ 90 3.2.1 Lyapunov Functions.................................. 91 3.2.2 Bounded Stability.................................... 94 3.3 Linearization Using Feedback................................ 98 3.3.1 Input-Output Linearization............................ 101 3.4 Control of Multi-Degree-of-Freedom Systems.................. 105 3.4.1 Modal Control....................................... 105 3.5 Adaptive Control........................................... 110 3.5.1 Adaptive Feedback Linearization....................... 111 3.6 Chapter Notes............................................. 115 References..................................................... 115 4 Approximate Methods for Analysing Nonlinear Vibrations......... 119 4.1 Backbone Curves........................................... 119 4.2 Harmonic Balance.......................................... 122 4.2.1 Forced Vibration..................................... 125 4.3 Averaging................................................. 127 4.3.1 Free Vibration....................................... 128 4.3.2 Forced Vibration..................................... 131 4.4 Perturbation Methods....................................... 135 4.4.1 Regular Perturbation Theory........................... 135 4.4.2 Multiple Scales Method............................... 139 4.5 Normal Form Transformations............................... 143 4.5.1 Free Vibration....................................... 144 4.5.2 Forced Vibration..................................... 155 4.6 Chapter Notes............................................. 169 References..................................................... 169 5 Modal Analysis for Nonlinear Vibration.......................... 173 5.1 Modal Behaviour in Vibrating Systems........................ 173 5.2 Modal Decomposition Using Linear Techniques................. 175 5.3 Modal Decomposition for Nonlinear Systems................... 186 5.3.1 Nonlinear Normal Modes............................. 188 5.3.2 Internal Resonance................................... 190 5.4 Normal Form Transformations............................... 195 5.4.1 Dealing with Internal Resonance....................... 204

Contents ix 5.4.2 Comparison Between Similar Nonlinear Normal Modes and Normal Forms................................... 207 5.5 Chapter Notes............................................. 211 References..................................................... 212 6 Beams........................................................ 215 6.1 Small-Deflection Beam Theory............................... 215 6.1.1 The Euler-Bernoulli Equation.......................... 217 6.1.2 The Galerkin Method................................. 219 6.1.3 Initial Conditions and Forcing......................... 222 6.1.4 Collocation Method.................................. 225 6.2 Nonlinear Beam Vibration................................... 229 6.2.1 Large Deflections for Thin Beams...................... 231 6.2.2 Nonlinear Beam Equations with Axial Loading........... 232 6.2.3 Stretching of a Constrained Beam...................... 239 6.3 Case Study of Modal Control Applied to a Cantilever Beam....... 243 6.3.1 Modal Control of a Beam............................. 243 6.3.2 Vibration Suppression Using Piezoelectric Actuation...... 246 6.3.3 Positive Position Feedback (PPF)....................... 248 6.3.4 PPF for Nonlinear Vibration........................... 252 6.4 Chapter Notes............................................. 254 References..................................................... 254 7 Cables........................................................ 257 7.1 Horizontal Cable Vibration.................................. 257 7.1.1 Cable Sag.......................................... 258 7.1.2 Static Deflection Due to Sag........................... 259 7.1.3 Dynamic Deflection.................................. 262 7.2 Inclined Cable Vibration..................................... 264 7.2.1 Force Balance....................................... 265 7.2.2 Excitation.......................................... 267 7.2.3 Quasi-Static Motion.................................. 268 7.2.4 Modal Motion....................................... 270 7.3 Nonlinear Cable Dynamics.................................. 276 7.3.1 Compatibility....................................... 277 7.3.2 Out-of-Plane Motion................................. 278 7.3.3 In-Plane Motion..................................... 280 7.3.4 Modal Interaction.................................... 283 7.4 Case Study of Analysis of Cable Response..................... 285 7.4.1 Harmonic Balance................................... 287 7.4.2 Averaging.......................................... 289 7.4.3 Multiple Scales...................................... 292 7.4.4 Normal Forms....................................... 294 7.5 Chapter Notes............................................. 300 References..................................................... 300

x Contents 8 Plates and Shells............................................... 303 8.1 Vibration of Plates.......................................... 303 8.1.1 Force Moment Relations.............................. 304 8.1.2 Strain-Displacement Relations......................... 308 8.1.3 Stress-Strain Relations................................ 311 8.1.4 Force Balance and Compatibility....................... 313 8.2 Small Amplitude Vibration.................................. 315 8.3 Vibration with Axial Loading................................ 320 8.4 Vibration of Shells.......................................... 323 8.5 Case Study of Nonlinear Shell Vibration....................... 327 8.5.1 Description of Case Study............................. 328 8.5.2 Governing Equations for Composite Shells............... 331 8.5.3 Galerkin Decomposition.............................. 333 8.5.4 Three-Mode Model.................................. 336 8.5.5 Subharmonic Resonance.............................. 339 8.6 Adaptive Structure Applications.............................. 344 8.6.1 Multi-Form Shell Structures........................... 344 8.7 Chapter Notes............................................. 346 References..................................................... 346 Index............................................................. 349