Energy Methods in Dynamics

419 Pages·2014·9.176 MB·English

by  Khanh Chau Le, Lu Trong Khiem Nguyen (auth.)

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. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
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