A Smooth and Discontinuous Oscillator: Theory, Methodology and Applications PDF
Preview A Smooth and Discontinuous Oscillator: Theory, Methodology and Applications
Springer Tracts in Mechanical Engineering Qingjie Cao Alain Léger A Smooth and Discontinuous Oscillator Theory, Methodology and Applications Springer Tracts in Mechanical Engineering Board of editors Seung-Bok Choi, Inha University, Incheon, South Korea Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, P.R. China Yili Fu, Harbin Institute of Technology, Harbin, P.R. China Carlos Guardiola, Universitat Politècnica de València, València, Spain Jian-Qiao Sun, University of California, Merced, USA About this Series Springer Tracts in Mechanical Engineering (STME) publishes the latest develop- ments in Mechanical Engineering - quickly, informally and with high quality. The intentistocoverallthemainbranchesofmechanicalengineering,boththeoretical and applied, including: (cid:129) Engineering Design (cid:129) Machinery and Machine Elements (cid:129) Mechanical structures and stress analysis (cid:129) Automotive Engineering (cid:129) Engine Technology (cid:129) Aerospace Technology and Astronautics (cid:129) Nanotechnology and Microengineering (cid:129) Control, Robotics, Mechatronics (cid:129) MEMS (cid:129) Theoretical and Applied Mechanics (cid:129) Dynamical Systems, Control (cid:129) Fluids mechanics (cid:129) Engineering Thermodynamics, Heat and Mass Transfer (cid:129) Manufacturing (cid:129) Precision engineering, Instrumentation, Measurement (cid:129) Materials Engineering (cid:129) Tribology and surface technology Within the scopes of the series are monographs, professional books or graduate textbooks, edited volumes as well as outstanding PhD theses and books purposely devoted to support education in mechanical engineering at graduate and post- graduate levels. More information about this series at http://www.springer.com/series/11693 Qingjie Cao é Alain L ger A Smooth and Discontinuous Oscillator Theory, Methodology and Applications 123 Qingjie Cao AlainLéger Schoolof Astronautics Laboratoire deMécaniqueet d’Acoustique, Harbin Institute of Technology CNRS Harbin Marseille Cedex13 China France ISSN 2195-9862 ISSN 2195-9870 (electronic) SpringerTracts inMechanical Engineering ISBN978-3-662-53092-4 ISBN978-3-662-53094-8 (eBook) DOI 10.1007/978-3-662-53094-8 LibraryofCongressControlNumber:2016947202 ©Springer-VerlagBerlinHeidelberg2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagGmbHBerlinHeidelberg The original version of the book was revised: For detailed information please see erratum. The erratum to this book is available at DOI 10.1007/978-3-662-53094-8_17 Foreword I am proud to write the Foreword for this volume by Prof. Qingjie Cao and his co-author.IcametoknowProf.QingjieCaofromthelastdecadebeginningofthis centuryatAberdeen,UK,Kyoto,Japan,St.Petersburg,Russian,andHarbinChina. Iobservedhissincereattitudefornonlineardynamics,especiallythecreationofthe smooth and discontinuous (SD) oscillator. This work opened my eyes to the phe- nomena in both smooth and discontinuous dynamics. As for my knowledge of nonlinear dynamics, I felt something fresh in the creation of a discontinuous oscillator and a generalized Hooke’s law due to hyperbolicity transition. So, per- hapsapersonalwordisinorderhereonthecreationoftheSDoscillator,thetheory and applications of which will give a better knowledge of the natural world. Manyworkshavebeenpublished,manybooksandmanypapersaswell,which deal with the qualitative dynamics of systems arising from any field of natural science, either in the case of simple models, or in the case of large-scale systems, even in continuous media such as those dealt with in fluid mechanics. This book comes back to a very simple mass spring oscillator, followed to wisdom “To know something new to visit the old.” A case at smaller strains is so simple that it might be included in any undergraduate class on vibrations, oscilla- tions, pre-buckling, etc. A case at larger strains is that nonlinear terms are usually introducedwhichcoulddeservetorepresentanymodelofnonlinearity.Thisallows to extend the analysis of nonlinear vibrations or post-buckling theories. In the present case, the nonlinearity arises from the changes of the geometry. What was new and really impressive in the so-called SD oscillator was the simple idea that taking the exact nonlinearity without any approximation might lead not only to a betterapproximationbutalsotonewphenomena.Atfirst,norestrictionisrequired fortheanalysistogivetheexacttrajectory.Thentheauthorsobservethatchanging a parameter from a given nonzero value to zero, which amounts to modifying the geometry and is associated with the name “Smooth to Discontinuous” changes the qualitative properties. This oscillator seems to have really new properties and gives a new step in the understanding of nonlinear phenomena. During the last decade, the authors vii viii Foreword accumulateahugequantityoffactsandstatisticscollectedtogetherforreferenceor analysis due to a large number of theoretical and experimental studies. The book they wrote is aimed to share some of these findings. Itstartswiththebasicdescriptionofthemechanicalsystemwithasurveyofits new properties. Then it investigates in detail each of the announced properties, backsuptheresultsbyseveraldifferentwaysfornumericalcalculations,andcloses by experiments and engineering applications. Thebookuseseithersimpletoolsaslongastheyaresufficienttogiveperfectly rigorous results, or difficult and recent mathematical concepts when necessary. It willcertainlybeveryuseful.Lookingback,fiftyyearsafterthebeginningofchaos in a nonlinear circuit determined by a special Duffing system, I am extremely pleasedtoseethatprogressescontinuetobeingmade,eventhroughthebehaviorof very simple mechanical systems. As a last comment, the classical chapter on challenges and future works, which is often conventional and formal, includes here very interesting suggestions, some of them of a great theoretical importance. Kyoto Yoshisuke Ueda November 2016 Preface The most interesting phenomenon in nonlinear dynamics and certainly one of the most important changes in the understanding of the world in the last five or six decades, is the discovery of chaos. In 1961, Yoshisuke Ueda discovered the so-called Ueda Attractor as a young researcher using an analog computer when he was studying the responses of a nonlinear circuit described by a special type of Duffing oscillator. This discovery undoubtedly changed the way of looking at the world, which until then was perceived either deterministic or random. It is now in the common knowledge that it has had fundamental effects in many theories including oscillations, wave propagation and control, in any field of physics, in fluid mechanics, in meteorology, in astronomy, in biology, in economics, in pop- ulation dynamics and others. In essence, the world is really concerned with the discovery of chaos as a milestone in current knowledge and research perspectives. In 1963, Edward Lorenz proposed a new model of the weather system, which becameknownastheLorenzattractorbytryingtoextractthemainpropertiesofthe atmospheric turbulence through a truncation of Navier–Stokes equations. This modelofturbulencefolloweditspreviousobservationsoftheextremesensitivityto initial conditions. It is now associated with the so-called butterfly effect, which again brought a new outlook on what was previously taken as randomness. A few years later, a simplified model having the same objective of understanding turbu- lence has been proposed using iterates of maps and gave the Henon attractor. Lots of works then followed in the fields of mathematics and theoretical physics to improveourunderstandingofthemechanismswhichleadtochaos,perioddoubling bifurcations, transitions to torus of higher dimensions, intermittency, and others nonlinear phenomena. Anarchetypalmechanicaloscillatorwasfirstlyproposedin2006,whichhasthe fundamental property of being either continuous and smooth or discontinuous depending on the value of a certain parameter. This system is now defined as the SD oscillator to refer to this property with the generalized Hooke’s law where the stiffnesscanbepositive,negativeorzero(quasi-zero)dependingonthegeometrical configuration of the system. It is a simple mass-spring system in which the only ix x Preface difference with what has been done before, for example in vibration or buckling models, is that the change of the geometry during the motion was taken into account without any approximation whatever in the displacement or on the fre- quency,exactlyinthesamewayasiftheequationofasimplependulumkeepssinx in the right-hand side instead of any polynomial expansion. TheSDoscillatorcanbeseenasanarchetypalsystem,bywhichwemeanthatit isaprototypeofadynamicalsystemwherethenonlinearityisirrational.Thisfirstly follows from the fact that there are no approximations in the large displacements, and secondly that the nonlinearity is either smooth, as long as a geometrical parameter related to the relative positions of the spring and the mass is different from zero, or discontinuous when the geometrical parameter reaches zero. In the smooth regime, it bears significant similarities with classical types of nonlinear oscillators, but at the nonsmooth limit it involves substantial departures from the standard ones. The ability of the SD oscillator to provide a new outlook on non- linearphenomenareliesuponanyfeatureofqualitativeanalyses:setsandproperties of equilibria, periodic solutions, co-dimension bifurcations, and chaotic attractors, etc., all these features involving the transition when the nonlinearity changes from smooth to discontinuous. Another feature of this oscillator is the transition from a single stability to a bistability system as the geometrical parameter crosses the critical value, where the stiffness at the origin can be positive, or negative depending on the value of the geometrical parameter. This is always known as the generalized Hooke’s law. The main purpose of this book is to provide an unconventional way to under- standqualitativephenomenaofthenaturalworld,throughthedynamicsofasimple mechanicalsystemwhichmayhaveeitherasmoothoranonsmoothnonlinearity.In otherwords,followingthestepsofthediscoveryofchaos,itaimsatshowingthata deepanalysisofanappropriatesimplemodelmayhaveimportantconsequencesin the understanding of the world. This book is also strongly motivated by an invi- tation to professionals in science and engineering to pay special attention to the modeling, the analysis, and the applications of nonlinear phenomena. The authors would like to give a real encouragement to students and researchers who are interestedindevelopingacross-disciplineofnonlinearscienceattheintersectionof mathematics, theoretical physics, and engineering sciences. In addition to the collaborations indicated at the beginning of each chapter, the authors would like to express their utmost thanks to the following Doctors and Professors, whatever the specificity of their contribution: Enli Chen Yushu Chen Ming Feng Shengliang Fu Celso Grebogi Kai Guo Ning Han Zhifeng Hao
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