Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches

402 Pages·2011·3.799 MB·English

by  Vasile Marinca, Nicolae Herisanu (auth.)

Nonlinear Dynamical Systems in Engineering
.
Vasile Marinca (cid:129) Nicolae Herisanu
Nonlinear Dynamical
Systems in Engineering
Some Approximate Approaches
VasileMarinca NicolaeHerisanu
PolitehnicaUniversityofTimisoara PolitehnicaUniversityofTimisoara
DepartmentofMechanicsandVibrations DepartmentofMechanicsandVibrations
Bd.MihaiViteazu1 Bd.MihaiViteazu1
300222Timisoara 300222Timisoara
Romania Romania
RomanianAcademy,TimisoaraBranch, RomanianAcademy,TimisoaraBranch,
CenterforAdvancedandFundamental CenterforAdvancedandFundamental
TechnicalResearch, TechnicalResearch,
Bd.M.Viteazu, 24 Bd.M.Viteazu, 24
300223Timis¸oara 300223Timis¸oara
Romania Romania
[email protected] [email protected]
ISBN978-3-642-22734-9 e-ISBN978-3-642-22735-6
DOI10.1007/978-3-642-22735-6
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011941777
# Springer-VerlagBerlinHeidelberg2011
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Preface
Thepractice ofsciencecouldbedescribedastheprocessofobservation followed
bytheconstructionofverbalormathematicalmodelstoexplaintheobservations,or
vice-versa. The use of the mathematical models implies that the observations are
quantitative, that they involve numbers used to specify the observations. The
successfulapplicationofanysciencedependsontheuseofanappropriatemixture
of principles, techniques and approaches. Sometimes principles are easily under-
stood,sometimessodifficultastobeopaque.Oftensomerejectiontakesrootinthe
drymathematicalforminwhichthesubjectispresented,unrelatedtoobservation,
quiteirrelevanttolife,andlackinganyformofinterest.Settingproblemsbasedona
wide variety of experience should engage interest, challenge and intelligence and
evenstimulatecuriosity.Thewealthofdetailofferedshouldnotlullthereaderinto
thinking that the material can be meaning by leafing through the book. As much
timeaspossibleitshouldbedevotedtogoingthroughthecalculationsandsolving
problems.
Analytical solutions to nonlinear differential equations or linear differential
equationswithvariablecoefficientsplayanimportantroleinthestudyofnonlinear
dynamical systems, but sometimes it is difficult to find these solutions, especially
for nonlinear problems with strong nonlinearity. In general, the known analytical
methodsarerestrictedtolimitedcasesdependingontheparameterswhichappear
in the governing equations and are valid only for nonlinear problems with weak
nonlinearity.
Dynamicalsystemsareavastsubject.Itisoftenfoundthatthegoinggetseasier
as one goes deeper, learning the mathematical connections tying together the
variousphenomena.Thematerialofthisbookcanbeincludedincoursescovering
thetheoryofnonlinearoscillations,thetheoryofelectricalmachines,classicaland
fluidmechanics,thermodynamicsorevenbiology.
Theprerequisitesforstudyingdynamicalsystemsusingthisbookareundergrad-
uate courses inlinear algebra, real and complexanalysis,calculus, dynamics,and
ordinary differential equations, classical physics of oscillations, knowledge of a
computer language would be essential. Also, it is assumed that the reader knows
basic notions about nonlinear systems of differential equations as well as the
v
vi Preface
plottingofphaseportraits,analysisofnonlinearsystems,andgraphicalrepresenta-
tionoferrorsandsoon.
Thisbookisinformedbytheresearchinterestoftheauthors,whicharecurrently
nonlinear differential equations. Some references include recently published
research articles. Our work has very hands-on approaches and takes the reader
fromthebasicmethodsrightthroughtorecentlypublishedresearchmaterial.
Most of the material in every chapter is at postgraduate level and has been
influenced by the authors’ own research interest. These chapters are especially
usefulasreferencematerialforseniorundergraduateprojectwork.
The text is aimed to undergraduate students in accelerated programs, working
scientists in various branches of engineering, natural scientists or applied
mathematicians.
Thewholebookconsistsofconcreteexamplesfromvariousdomainsofnonlin-
eardynamicalsystems.Theauthorsbelievethattheproblemofmotionofdifferent
dynamical systems can be assimilated only by working with the differential
equations applied to concrete examples. Nearly all the sections of this book are
followedbycomparisonswithnumericalresultsorwithotherknownresultsinthe
literature.
The aim of this book is to present and extend different known methods in the
literature,especiallyLindstedt-Poincare´ method,themethodofharmonicbalance,
the method of Krylov-Bogolyubov and the method of multiple scales, to solve
differenttypesofstrongnonlinearities.Abetterknowledgeofthesemethodsleadto
abetterchoiceoftheso-called“basefunctions”whichareabsolutelynecessaryto
obtaintheauxiliaryfunctionspresentinthelastChapters,devotedtosomeoptimal
analytical approaches. These auxiliary functions are cornerstone of the optimal
methodsandalso,ensuretheconditionsofconvergenceofthesolutionsobtainedby
different approaches. Unlike all previous analytic approaches, these few optimal
methodsprovideuswithasimplewaytocontrolandadjusttheconvergenceregion
ofsolutionsofnonlineardynamicalsystems.Thesenewoptimalmethodsshowone
stepintheattempttodevelopanewnonlinearanalyticaltechniqueworkinginthe
absenceofsmallorlargeparameters.Actually,thecapitalstrengthofouroptimal
procedures is the fast convergence, since after only two or three iterations, or
sometimesafteronlyoneiteration,itconvergestotheexactsolution,whichproves
thattheseoptimalmethodsareveryefficientinpractice.
The text begins with some known procedures, presented in Chaps. 1–4: the
Lindstedt-Poincare´ method, the method of harmonic balance, the method of
Krylov-Bogolyubov and the method of multiple scales. All these techniques sup-
pose the presence of a small parameter into the governing nonlinear equations.
There are presented some alternatives and examples to each of these approaches,
suchastheuseofperturbationmethodforstrongparameter,therationalharmonic
balancemethod,acombinationofthemethodofKrylov-Bogolyubovanditeration
method.Thelastfourchapters,from5to8aredevotedtooptimalapproachessuch
as: the Optimal Homotopy Asymptotic Method, the Optimal Homotopy Perturba-
tionMethod,theOptimalVariationalIterationMethodandtheOptimalParametric
Iteration Method. The validity of the proposed procedures has been demonstrated
Preface vii
onsomerepresentativeexamplesandverygoodagreementwasfoundbetweenthe
approximate analytic results and numerical simulations. The convergence of the
approximatesolutionsobtainedbyeachofthesenewmethodsisgreatlyinfluenced
bytheconvergence-controlconstantswhichareoptimallydetermined.
Theexamplespresentedinthisbookleadtotheconclusionthattheaccuracyof
theobtainedresultsisgrowingalongwithincreasingthenumberofconstantsinthe
auxiliary functions. These methods are very rapid and effective and show their
validity and potential for the solution of nonlinear problems arising in dynamical
systems.
Finally,ourmainaimistoinspirethereadertoappreciatethebeautyaswellas
the usefulness of the optimal analytical techniques in the study of nonlinear
dynamicalsystems.
Timis¸oara VasileMarinca
NicolaeHeris¸anu
.
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 PerturbationMethod:Lindstedt-Poincare´. . . . . . . . . . . . . . . . . . . . 9
2.1 TheOscillatorwithCubicElasticRestoringForce. . . . . . . . . . . 14
2.1.1 TheExactSolutionofDuffingEquation. . . . . . . . . . . . . 14
2.1.2 UseofthePerturbationMethodforDuffingOscillator
withSmallParameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 UseofthePerturbationMethodforDuffingOscillators
withStrongParameter. . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 TheMethodofHarmonicBalance. . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 FreeVibrationsofCantileverBeam. . . . . . . . . . . . . . . . . . . . . . 35
3.2 RationalHarmonicBalanceMethod. . . . . . . . . . . . . . . . . . . . . 40
4 TheMethodofKrylovandBogolyubov. . . . . . . . . . . . . . . . . . . . . . 47
4.1 OscillatorwithLinearandCubicElasticRestoringForce
andWeakAsymmetricQuadraticDamping. . . . . . . . . . . . . . . . 53
4.2 UseoftheMethodofKrylov-BogolyubovandIteration
MethodtoWeaklyNonlinearOscillators. . . . . . . . . . . . . . . . . . 60
4.2.1 “Nonresonance”Case(o6¼pO). . . . . . . . . . . . . . . . . . . 61
q
4.2.2 “Resonance”Caseo(cid:2)pO. . . . . . . . . . . . . . . . . . . . . . 66
q
4.2.3 NumericalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 TheMethodofMultipleScales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 DuffingOscillatorwithSofteningNonlinearity. . . . . . .. . . . . .. 89
5.2 AParametricSystemwithCubicNonlinearityCoupled
withaLanchesterDamper. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 TheOptimalHomotopyAsymptoticMethod. . . . . . . . . . . . . . . . . . 103
6.1 BasicIdeaofOHAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 DuffingOscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.1 NumericalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
ix
x Contents
6.3 ThinFilmFlowofaFourthGradeFluidDown
aVerticalCylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4 DampedOscillatorwithFractional-OrderRestoringForce. . . . . 120
6.4.1 NumericalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5 NonlinearEquationsArisinginHeatTransfer. . . . . . . . . . . . . . 127
6.5.1 CoolingofaLumpedSystemwith
VariableSpecificHeat. . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5.2 TheTemperatureDistributionEquationina
ThickRectangularFinRadiationtoFreeSpace. . . . . . . . 131
6.6 Blasius’Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.7 OscillationsofaUniformCantileverBeamCarrying
anIntermediateLumpedMassandRotaryInertia. . . . . . . . . . . . 143
6.8 OscillationsofanElectricalMachine. . . . . . . . . . . . . . . . .. . . . 151
6.9 OscillationsofaMassAttachedtoaStretchedElasticWire. . . . 156
6.10 NonlinearOscillatorwithDiscontinuities. . . . . . . . . . . . . . . . . . 162
6.11 NonlinearJerkEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.12 TheMotionofaParticleonaRotatingParabola. . . . . . . . . . . . 173
6.13 NonlinearOscillatorwithDiscontinuities
andFractional-PowerRestoringForce. . . . . . . . . . . . . . . . . . . . 184
6.14 OscillationsofaFlexibleCantileverBeam
withSupportMotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 191
6.15 TheJeffery-HamelFlowProblem. . . . . . . . . . . . . . . . . . . . . . . 197
6.15.1 NumericalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . 206
7 TheOptimalHomotopyPerturbationMethod. . . . . . . . . . . . . . . . . 211
7.1 HomotopyPerturbationMethod. . . . . . . . . . . . . . . . . . . . . . . . 211
7.2 ModifiedHomotopyPerturbationMethod. . . . . . . . . . . . . . . . . 214
7.3 BasicIdeaofOptimalHomotopyPerturbationMethod
andSomeApplications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.4 AHeatTransferProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.5 ThinFilmFlowofaFourthGradeFluidDown
aVerticalCylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.6 NonlinearDynamicsofanElectricalMachine
Rotor-BearingSystem. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . 240
7.7 ANon-conservativeOscillatorySystemofaRotating
ElectricalMachine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8 TheOptimalVariationalIterationMethod. . . . . . . . . . . . . . . . . . . . 259
8.1 TheVariationalIterationMethodandApplications. . . . . . . . .. . 259
8.1.1 NonlinearOscillatorwithQuadratic
andCubicNonlinearities. . . . . . . . . . . . . . . . . . . . . . . . 264
8.1.2 AFamilyofNonlinearDifferentialEquations. . . . . . . . . 270
8.1.3 TheDuffingEquation. . . . . . . . . . . . . . . . . . . . . . . . . . 271
8.2 MathematicalDescriptionoftheOptimalVariational
IterationMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 275