INTERACTION OF MECHANICS AND MATHEMATICS khanh chau le lu trong khiem nguyen Energy Methods in Dynamics Second Edition 123 Interaction of Mechanics and Mathematics Serieseditor LevTruskinovsky,LaboratoiredeMechaniquedesSolides,Palaiseau,France e-mail:[email protected] Forfurthervolumes: http://www.springer.com/series/5395 AboutthisSeries The Interaction of Mechanics and Mathematics (IMM) series publishes advanced textbooksandintroductoryscientificmonographsdevotedtomodernresearchinthe wideareaofmechanics.Theauthorsaredistinguishedspecialistswithinternational reputation in their field of expertise. The books are intended to serve as modern guidesintheirfieldsandanticipatedtobeaccessibletoadvancedgraduatestudents. IMMbooksareplannedtobecomprehensivereviewsdevelopedtothecuttingedge oftheirrespectivefieldandtolistthemajorreferences. AdvisoryBoard D.Colton,USA R.Knops,UK G.DelPiero,Italy Z.Mroz,Poland M.Slemrod,USA S.Seelecke,USA L.Truskinovsky,France IMMispromotedundertheauspicesofISIMM(InternationalSocietyforthe InteractionofMechanicsandMathematics). · Khanh Chau Le Lu Trong Khiem Nguyen Energy Methods in Dynamics Second Edition ABC KhanhChauLe RuhrUniversitätBochum LehrstuhlfürMechanikundMaterial Theorie Bochum Germany LuTrongKhiemNguyen RuhrUniversitätBochum LehrstuhlfürMechanikundMaterial Theorie Bochum Germany ISSN1860-6245 ISSN1860-6253 (electronic) ISBN978-3-319-05418-6 ISBN978-3-319-05419-3 (eBook) DOI10.1007/978-3-319-05419-3 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014933546 (cid:2)c SpringerInternationalPublishingSwitzerland2011,2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Hoa, Ly, and Jan To Khiem's parents Preface to the Second Edition Sincetheappearanceofthefirsteditionwehavereceivedvariousfeedbacksandsug- gestionsfromstudentsandteachers.Tomakethetextbookmoresuitableforlearn- ingandteachingpurposes,especiallytoshowtheenergyandvariational-asymptotic method“inaction”,wedecidedtoincludethesolutionstoallexercisesinthissec- ond edition (some of the exercises have been changed). Besides, we have found and corrected various small errors and misprints in the first edition. Finally, Sec- tion8.4isrewrittencompletelytakingintoaccountthenewdevelopmentsinampli- tude and slope modulationof nonlineardispersive waves. We would like to thank especially the editor of the series ”Interaction of Mechanics and Mathematics”, L.Truskinovsky,forhisvariousconstructivesuggestions.Theinterestandsupport ofthepublishingeditorofSpringerVerlag,Dr.T.Ditzinger,inthissecondedition isalsogratefullyacknowledged. Bochum,November2013 KhanhChauLe LuTrongKhiemNguyen Preface to the First Edition We live in the world surrounded by vibrations and waves. Some, such as sound, light,andelectricity,arefamiliarcompanionsin everydaylife,whilemanyothers, suchaslattice vibrations,vibrationsoffluidparticlesin a turbulentflow,or vibra- tionsoftemperatureaboutitsaveragevalue,cannotbeobservedbynakedeyesbut canbemeasuredbysensitivemoderninstruments.Vibrationsandwavesalsodom- inate in the micro-world: vibrations of atoms, waves of elementary particles, and onewouldberemisstofailtomentionthatthegoverningequationofquantumme- chanicsistheSchro¨dingerwaveequation.Vibrationsandwavesfindvariousappli- cationsinengineeringandinoureverydaylife,fromclocks,bells,lamps,musical instruments, to radio, television, laser, plasma, computer, internet etc. Sometimes vibrationsandwavesarenotasharmlessasonemightthink.Whoeverlivesneara highwayoranairportknows,howannoyingpermanentnoisecanbe.Medicaltech- niquesandhighprecisionequipmentsoftenrequirevibrationfreeframeswhichare notalwayseasilyrealizable.Thewingsofairplaneortherotorbladesofhelicopter may encounter strong vibrations during exploitation causing their quick damage. Variouscatastrophesinthenaturelikeearthquakeortsunamiarealsoduetovibra- tionsandwaves. Theaboveexamplesshouldmakeclearthenecessityofunderstandingthemech- anism of vibrations and waves in order to control them in an optimal way. How- ever, vibrations and waves are governed by differential equations which require, as a rule, rather complicated mathematicalmethodsfor their analysis. The aim of this textbook is to help students acquire both a good grasp of the first principles from which the governingequationscan be derived,and the adequate mathemati- cal methods for their solving. Its distinctive features, as seen from the title, lie in the systematic and intensive use of Hamilton’s variationalprincipleand its gener- alizationsforderivingthegoverningequationsofconservativeanddissipativeme- chanicalsystems, and also in providingthe direct variational-asymptoticanalysis, wheneveravailable,oftheenergyanddissipationforthesolutionoftheseequations. It will be demonstratedthat many well-known methodsin dynamicslike those of Lindstedt-Poincare´,Bogoliubov-Mitropolsky,Kolmogorov-Arnold-Moser(KAM), andWhithamarederivablefromthisvariational-asymptoticanalysis. X PrefacetotheFirstEdition This book grew up from the lectures given by the author in the last decade at theRuhrUniversityBochum,Germany.Sincevibrationsandwavesareconstituents of various disciplines (physics, mechanics, electrical engineering etc.) and cannot be handled in a single textbook,I have restricted myself mainly to vibrationsand wavesofmechanicalnature.Unfortunately,duetothetimeconstraintIhadtoleave out the most exciting and quickly developingpart of dynamics, namely the deter- ministicchaos.Chapter7canserveasanintroductiontothisfascinatingtopic.The materialofthisbookcanberecommendedforaoneyearcourseinhigherdynamics for graduate students of mechanical and civil engineering.For this circle of read- ers, the emphasis is made on the constructive methodsof solution and not on the rigorousmathematicalproofsofconvergence.Ascompensation,variousnumerical simulationsoftheexactandapproximatesolutionsareprovidedwhichdemonstrate vividlythevalidityoftheusedmethods.Tohelpstudentsbecomemoreproficient, eachchapterendswithexercises,ofwhichsomecanbesolvedeffectivelybyusing Mathematica. I wouldliketo thankfirst ofallK.Hackl,whoinsisted thatthisone-yeargrad- uatecourseinhigherdynamicsatRuhrUniversityBochummustbetaughtbyme, andV.Berdichevsky,C.Gu¨nther,D.Hodges,M.Kaluza,R.Knops,G.Schmid,L. Truskinovsky,andmanyotherfriendsandcolleaguesfortheircommentsanduseful discussions. Bochum,May2011 KhanhChauLe Contents PartI: LinearTheory 1 SingleOscillator .............................................. 3 1.1 HarmonicOscillator ....................................... 3 1.2 DampedOscillator ........................................ 9 1.3 ForcedOscillator.......................................... 20 1.4 HarmonicExcitationsandResonance......................... 26 1.5 Exercises ................................................ 33 2 CoupledOscillators ........................................... 45 2.1 ConservativeOscillators.................................... 45 2.2 DissipativeOscillators ..................................... 52 2.3 ForcedOscillatorsandVibrationControl...................... 59 2.4 VariationalPrinciples ...................................... 65 2.5 OscillatorswithnDegreesofFreedom........................ 70 2.6 Exercises ................................................ 78 3 ContinuousOscillators ........................................ 99 3.1 ChainofOscillators ....................................... 99 3.2 String ................................................... 104 3.3 Beam.................................................... 110 3.4 Membrane ............................................... 114 3.5 Plate .................................................... 120 3.6 GeneralContinuousOscillators.............................. 126 3.7 Exercises ................................................ 135 4 LinearWaves................................................. 151 4.1 HyperbolicWaves ......................................... 151 4.2 DispersiveWaves ......................................... 160 4.3 ElasticWaveguide......................................... 169 4.4 EnergyMethod ........................................... 179 4.5 Exercises ................................................ 188