Ordinary Differential Equations and Mechanical Systems

621 Pages·2014·10.204 MB·English

by  Jan Awrejcewicz (auth.)

Jan Awrejcewicz
Ordinary Diff erential
Equations and
Mechanical Systems
Ordinary Differential Equations and Mechanical
Systems

Jan Awrejcewicz
Ordinary Differential
Equations and Mechanical
Systems
123
JanAwrejcewicz
DepartmentofAutomation,
BiomechanicsandMechatronics
Łódz´UniversityofTechnology
Łódz´,Poland
ISBN978-3-319-07658-4 ISBN978-3-319-07659-1(eBook)
DOI10.1007/978-3-319-07659-1
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014947361
MathematicsSubjectClassification(2010):34-xx,70-xx,37-xx
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Preface
Chapter1describesbasicconceptsofthetheoryofordinarydifferentialequations.
Namely,solutionsto ODEs,a notionofextendedphasespace,orbit(phasetrajec-
tory),motion,cascadesandflows,trajectory,wanderingpoint,˛.!/-limitingpoint,
attractorsandrepellersaredefined,amongothers.Picard’sandPeano’stheoremsare
introduced.Furthermore,globalexistenceand uniquenessof solutionsare defined
andproved,andexamplesillustratingtheintroduceddefinitionsareadded.
Chapter 2 deals with ODEs of the first order. First, a general introduction to
nonlinearODEsisgiven,andafewillustrativeexamplesareprovided.Definitions
andexamplesregardingseparablehomogenousandlinear,aswellastheexactand
implicitdifferentialequationsaregiven.
Chapter 3 is concerned with second-orderODEs. Firstly, we begin with linear
homogenousODEswithtime-dependentcoefficients,andfourillustrativeexamples
are added. Then the hyper-geometric (Gauss) equation, Legendre equation and
Besselequationsupplementedbyexamplesarestudied.Particularattentionispaid
toODEswithperiodiccoefficientsincludingtheHillequation,Meissnerequation,
Ince–Struttand Kotowski diagrams. In addition, modelling of a generalized para-
metric oscillator using the nonlinear Milne–Pinney equation is carried out. Then
ODEswithconstantcoefficientsarebrieflyrevisited.
Variational Hamiltonian principle, exhibiting physical aspects of a studied
dynamicalsystem, is applied to derive the Lagrangeequations. Next, in Sect.3.4,
a reductionof the second-orderODEsto thatof the firstorderis presented.Then,
the canonical (Jacobi) form after application of a dual (Legendre) transformation
is derived and discussed. Both Lagrangian and Hamiltonian functions, and their
mutual relationships are presented including the illustrative geometric interpreta-
tion. Canonical transformations,Poisson brackets, as well as generatingfunctions
are described in Sect.3.5, and two illustrative examples are provided (Sect.3.6).
Section 3.7 deals with normal forms of Hamiltonian systems. Furthermore, it is
shown how to solve the Hamiltonian equations with damping terms. Problems
relatedtoRiemannianformulationofdynamicsandchaosexhibitedbyHamiltonian
v
vi Preface
systemsusingtheexampleofaswingingpendulumaredescribedinSect.3.8.The
Jacobi–Levi–Civita(JLC) equationis derivedgoverningthe evolutionof geodesic
separation. The original 2-DOF system oscillations have been reduced to consid-
eration of one second-order homogenous ODE with time-dependent coefficient.
Numericalresultsincludingthe quasi-periodicandchaotic dynamicsare reported.
Next,thesofarappliedgeometricanalysishasbeenextendedtostudydoublependu-
lumdynamics.Inparticular,aconnectionbetweentheJLCequationandthetangent
dynamicsequationisillustratedallowingustodefinetheLyapunovexponentsvia
the Riemannian geometry approach. This qualitatively different explanation and
interpretationofchaoticdynamicsasaparametricinstabilityofgeodesicshasbeen
illustrativelysupportedby numericalresults. Section 3.10is devotedto a studyof
thelinearsecond-orderODEswithconstantcoefficientspresentedinamatrixform.
Bothconservativeandnon-conservativenon-autonomoussystemsareanalysedand
importantquestionregardingtheirdecouplingisrigorouslyconsidered.Then,afew
problemsofmodalanalysisandidentificationareaddressed.
Chapter 4 is devoted to linear ODEs. Firstly, we show how a single nth order
ODEisreducedtofirst-orderODEs.Then,normalandsymmetricformsofODEs
are defined, and five illustrative examples are added. In Sect.4.3 local solutions
behaviourregardingtheexistence,extensionandstraightnessaredescribed.Classi-
caltheoremsare formulatedincludinga fewwell-knownproofs.First-orderlinear
ODEswithvariablecoefficientsare studiedinSect.4.4.Itcontainssomeclassical
theoremswiththeirproofs,fundamentalmatrixofsolutions,homogenousandnon-
homogenousdifferentialequationsandnumerousexamples.Particular attentionis
paid to homogenous and non-homogenous ODEs with periodic coefficients, the
Floquettheory,characteristicmultipliersandexponentsstructureofsolutions,etc.
Chapter 5 focuses on higher order ODEs of a polynomial form. First, the
Peano and Cauchy–Picard theorems are revisited. Second, a linear homogenous
nth order equation is studied. Third, it is shown how an nth order differential
equation is reduced to the nth order algebraic equation. Distinct and multiple
rootsofcharacteristicequationarediscussed.Thischapterincludesalsonumerous
examples.
Chapter 6 describes systems. It includes the system definition, as well as the
asymptotic relations between Newton’s and Einstein’s theory,classical mechanics
andquantummechanics,andothers.
Theory and criteria for similarity are introduced and discussed in Chap.7.
Geometrical, kinematic and dynamic similarities are illustrated. Three different
approaches of obtaining similarity criteria are outlined. Several examples serving
asaguideforintuitionandphysicalinterpretationsaregiven.
Chapter 8 is devoted to a model definition and modelling. After an introduc-
tory Sect.8.1, the mathematical modelling characterized by interdisciplinary and
universal features is described (Sect.8.2). Section 8.3 focuses on the modelling
in mechanics, whereas the next section deals with general characteristics of
mathematicalmodelling.Modellingapproachesapplied in a controlare described
in Sect.8.5. Ordinary,adaptive and distributed control systems are discussed, and
Preface vii
associated block diagrams are given. The mechanicalengineeringoriented results
ofcontrolsystemsandtheirblockdiagramsarereportedanddiscussed.
Chapter 9 deals with a phase plane and phase space based on the first-order
ODEs.First,ageneralintroductionofthephaseplaneconceptisgiveninSect.9.1.
SingularpointsarestudiedinSect.9.2,wherealsoaclassificationofphaseportraits
is given. Section 9.3 yields the classification of singular points with a use of
“Mathematica”. An analysis of singular points governed by the three first-order
ODEs is carried out in Sect.9.4, and analytical and numerical solutions of phase
spacetrajectoriesintheneighbourhoodofsingularpointsarederived.
ProblemsofstabilityarestudiedinChap.10.IntroductorySect.10.1givesafew
classical stability definitions of Lyapunov,Lagrange,exponential,conditionaland
technicalstability.
In addition, the limiting sets, attractors and repellers are also defined. Finally,
a concept of Zhukovskiy’s stability is illustrated and its impact on the stability
of quasi-periodic and chaotic orbits is discussed. Lyapunov functions and second
Lyapunov methods are discussed in Sect.10.2, where also the so-called first and
secondLyapunovtheoremsofstability/instabilityareformulated.Thenafewother
theoremsareintroducedincludingtheseofChetayevandBarbashin–Krasovski,and
two illustrative examples are added as well. Section 10.3 deals with the classical
theoriesofstabilityandtheirimpactonchaoticdynamics.
Chapter 11 is focused on modelling via perturbation methods. First section
describes advantages and disadvantages of asymptotic methods. Section 11.2
presents some perturbation techniques including the Krylov method and Krylov–
Bogolubov–Mitropolskiy method applied to autonomous and non-autonomous
oscillators.Inthelattercaseresonanceandnon-resonanceoscillationsarestudied.
ContinualizationanddiscretizationapproachesaredescribedinChap.12.Intro-
duction is followed by a study of the 1D chain of coupled oscillators, which is
thenconvertedtoaPDEgoverningpropagationofwaves(Sect.12.2).Theapproach
is extendedto a modelof planar hexagonalnet of coupledoscillators(Sect.12.3).
The discretization approach is briefly commented on the basis of vibrations of
a nonlinear shell with imperfections (Sect.12.4). Finally, the modelling of 2D-
structuresgovernedbytheamplitudeequationispresented,andthelatterequation
isstudiedusingaperturbationmethod.
Bifurcation phenomena are the subject of Chap.13. Introductory Sect.13.1
presentsarelationbetweenbifurcationsandODEs,stableandunstablemanifolds,
global and local bifurcation diagrams, as well as the classification of isolated
solutions. Singular points of 1D and 2D vector fields are illustrated and analysed
in Sect.13.2. Examples taken from mechanics are given. Local bifurcations of
hyperbolicand non-hyperbolicfixed pointsare studied, and nextthe double Hopf
bifurcation is addressed. Section 13.3 deals with fixed points of maps and the
associated bifurcations. Continuation (path following) approach using either the
Galerkin approximation or shooting method is described in Sect.13.4. An illus-
trative example from biomechanicsis added. The next section aims at illustrating
basicfeaturesofglobalbifurcations.Section13.6copeswithbifurcationsexhibited
by piece-wise smoothdynamicalsystems. First, their importanceis describedand
viii Preface
thenstabilityofdiscontinuoussystemsusefulfornumericalapplicationsisgiven.In
thenextsectionorbitsexhibitingadegeneratedcontactwithdiscontinuitysurfaces,
bifurcations in Filippov’s systems, bifurcations of stationary points and periodic
orbitsareanalysed.
The optimization of systems is a subject of Chap.14. Firstly, historical roots
are briefly revisited and variational principles are mentioned. Secondly, simple
examplesofoptimizationarereported(Sect.14.2).Thirdly,conditionalextremesare
defined(Sect.14.3),andthenstaticoptimizationproblemsarerevisited(Sect.14.4),
including local function approximation and definition of stationary points and
quadratic forms. In the fourth place, Sect.14.5 addresses problems associated
with convexity of the sets of functions with a few definitions and theorems.
Next, optimization without constraints and with conditions of local optimality
is given in Sect.14.6. Subsequently, optimality conditions regarding quadratic
forms are presented in Sect.14.7, whereas Sect.14.8 deals with the equivalence
constraints.Afterward,bothLagrangefunctionandLagrangemultipliers,including
their geometrical interpretations, are discussed in Sect.14.9. Then, constraints of
inequivalenttypesaredescribedinSect.14.10.Finally,theshockworkoptimization
isdiscussedinSect.14.11.
Chapter 15 describes phenomena of chaos and synchronization. Introductory
Sect.15.1 deals with a historicalbackgroundand intuitiveunderstandingof chaos
and synchronization occurring in Nature as well as in pure and applied sciences.
ModellingandidentificationofchaosispresentedinSect.15.2,whereasSect.15.3
describes the role of Lyapunov exponents and their geometrical interpretation in
quantifyingchaoticorbits.Then,afrequencyspectrumandautocorrelationfunction
aredescribedinSects.15.4and15.5,respectively.Modellingofnonlineardiscrete
systems with an emphasis on the chaotic dynamics is addressed in Sect.15.6.
It consists of an introduction, Bernoulli map, logistic map, map of a circle into
circle, devil’s stairs, Farey tree, Fibonacci numbers, Henon map and Ikeda map.
Section 15.7 focuses on modelling chaotic ODEs and it includes a study of non-
autonomous oscillator with different potentials, Melnikov’s function approach,
externally driven van der Pol’s oscillator, Lorenz ODEs and their derivation. The
synchronization phenomena of a mechanical system consisting of coupled triple
pendulumswithtime-periodicmassdistributionsareanalysedin Sect.15.8,where
many different kinds of synchrony as well as rich nonlinear dynamical effects
includingsynchronizationbetweenchaoticaswellaschaoticandregulardynamics
havebeennumericallyreported.
Finally, in Sect.15.9 chaotic vibrations of flexible spherical rectangular shells
loaded harmonicallyvia the boundaryconditionsare investigated.In the first part
one-layershell made froman isotropic and homogeneousmaterialis studied.The
secondpartaddressesnonlineardynamicsofmulti-layershells,takingintoaccount
gapsbetweenthelayers(designnonlinearity).Phaseportraits,Fourierpowerspectra
andwaveletspectraareconstructedandinvestigated.Analysisoftheshellcurvature
andshelldesignnonlinearityonthesynchronizationphenomenaiscarriedoutusing
also the phase difference as a new characteristic for monitoring and quantifying
nonlinearvibrations.
Preface ix
The author wishes to express his thanks to T. Andrysiak, M. Kaz´mierczak, G.
KudraandJ.Mrozowskifortheirhelpinthebookpreparation.
Finally, I acknowledge the financial support of the National Science Centre of
PolandunderthegrantMAESTRO2,No.2012/04/A/ST8/00738,foryears2013–
2016.
Łódz´,Poland JanAwrejcewicz