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VIBRO-IMPACT
DYNAMICS
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VIBRO-IMPACT
DYNAMICS
Albert C. J. Luo
Yu Guo
SouthernIllinoisUniversity,USA
A John Wiley & Sons, Ltd., Publication
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Thiseditionfirstpublished2013
(cid:2)C 2013JohnWiley&Sons,Ltd
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ISBN:9781118359457
Typesetin10/12ptTimesbyAptaraInc.,NewDelhi,India
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Contents
Preface ix
1 Introduction 1
1.1 DiscreteandDiscontinuousSystems 1
1.1.1 DiscreteDynamicalSystems 1
1.1.2 DiscontinuousDynamicalSystems 4
1.2 FermiOscillatorsandImpactProblems 7
1.3 BookLayout 9
2 NonlinearDiscreteSystems 11
2.1 Definitions 11
2.2 FixedPointsandStability 13
2.3 StabilitySwitchingTheory 22
2.4 BifurcationTheory 38
3 CompleteDynamicsandFractality 47
3.1 CompleteDynamicsofDiscreteSystems 47
3.2 RoutestoChaos 54
3.2.1 One-DimensionalMaps 54
3.2.2 Two-DimensionalSystems 58
3.3 CompleteDynamicsoftheHenonMap 59
3.4 SimilarityandMultifractals 64
3.4.1 SimilarStructuresinPeriodDoubling 65
3.4.2 FractalityofChaosviaPDBifurcation 68
3.4.3 AnExample 70
3.5 CompleteDynamicsofLogisticMap 72
4 DiscontinuousDynamicalSystems 85
4.1 BasicConcepts 85
4.2 G-Functions 88
4.3 PassableFlows 91
4.4 Non-PassableFlows 95
4.5 GrazingFlows 107
4.6 FlowSwitchingBifurcations 119
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viii Contents
5 NonlinearDynamicsofBouncingBalls 131
5.1 AnalyticDynamicsofBouncingBalls 131
5.1.1 PeriodicMotions 133
5.1.2 StabilityandBifurcation 136
5.1.3 NumericalIllustrations 140
5.2 Period-mMotions 143
5.3 ComplexDynamics 152
5.4 ComplexPeriodicMotions 156
6 ComplexDynamicsofImpactPairs 165
6.1 ImpactPairs 165
6.2 Analytical,SimplestPeriodicMotions 168
6.2.1 AsymmetricPeriod-1Motion 170
6.2.2 StabilityandBifurcation 172
6.2.3 NumericalIllustrations 175
6.3 PossibleImpactMotionSequences 179
6.4 GrazingDynamicsandStickMotions 182
6.5 MappingStructuresandPeriodicMotions 189
6.6 StabilityandBifurcation 192
7 NonlinearDynamicsofFermiOscillators 203
7.1 MappingDynamics 203
7.2 AFermiOscillator 209
7.2.1 AbsoluteDescription 210
7.2.2 RelativeDescription 212
7.3 AnalyticalConditions 214
7.4 MappingStructuresandMotions 218
7.4.1 SwitchingSetsandGenericMappings 218
7.4.2 MotionswithMappingStructures 221
7.4.3 PeriodicMotionsandLocalStability 223
7.5 PredictionsandSimulations 226
7.5.1 BifurcationScenario 226
7.5.2 AnalyticalPrediction 229
7.5.3 NumericalSimulations 236
Appendix7.A 248
References 253
Index 259
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Preface
Thisbookisaboutthedynamicsofvibro-impactoscillators.Vibro-impactsystemsextensively
existinengineeringandphysics.Suchvibro-impactsystemspossessthecontinuouscharac-
teristicsascontinuousdynamicalsystemsanddiscretecharacteristicsbyimpactdiscontinuity.
Such properties require an appropriate development of discrete maps for such vibro-impact
systemstoinvestigatethecorrespondingcomplexmotions.Therichdynamicalbehaviorsin
vibro-impact systems drew the authors’ attention on nonlinear dynamical systems. In addi-
tion,abetterunderstandingofsuchvibro-impactsystemshelpsonestudynonlineardynamical
systemswithdiscontinuityinengineeringandphysics.
In1964,ProfessorWeiwuDengexperimentallystudiedthelathevibrationreductionthrough
impact dampers, which originated from the flutter reduction of airplane wines in Russia in
the1930s.ProfessorDengfoundtheoptimalvibrationreductionofthelathesisbetween0.6
and 0.8 of the impact restitution coefficient with potential maximum energy dissipation. To
furtherunderstandthedynamicalmechanismofsuchimpactdampersandextendapplications
inengineering,in1987ProfessorDenginvitedthefirstauthortoworkonthisproblemwith
him.Afterliteraturesurveyandexperimentalsetup,itwascrucialtodevelopanappropriate
mathematicalmodeltodescribetheimpactdampersandtocatchallpossiblecomplexmotions.
Since then, the first author has been working on this topic. Herein he would like to share
what his group observed during the past 30 years with other scientists and engineers in
vibro-impactsystems.
Thisbookmainlyfocussedonanalyticalpredictionandphysicalmechanismsofcomplex
motionsinvibro-impactsystems.Afterliteraturesurvey,inthenexttwochapters,thetheory
for nonlinear discrete systems is presented from the recent development of the first author
primarily,includingtheYing-Yangtheoryofdiscretedynamicalsystemsbasedonthepositive
andnegativemapsindiscretedynamicalsystems.Thecompletedynamicsofnonlineardiscrete
dynamical systems is discussed and applied to one- and two-dimensional discrete systems,
and a geometric method is discussed for the fractality and complexity of chaos in discrete
dynamicalsystems.Fromtherecentdevelopmentofthefirstauthor,inChapter4,thetheory
of discontinuous dynamical systems is presented as a foundation of studying the dynamics
of vibro-impact systems. In Chapter 5, bouncing ball dynamics is discussed as one of the
simplest problems in vibro-impact systems to show the corresponding physical motions in
thissimplemodel.Thedynamicsforbouncinginitiationandimpactingchattervanishingwith
stick motion is presented for the first time, which is significant in engineering application.
Afterdiscussingthebouncingballwiththesinglemap,asimpleversionofanimpactdamper
ispresentedinChapter6toshowhowtodevelopthecomplexperiodicmotionsanalytically.
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x Preface
Themotionswitchingfromonemotiontoanotherisdiscussedthroughthegazingphenomena.
InChapter7,thenonlineardynamicsoftheFermioscillatorisdiscussedasanapplicationin
physics.Themethodologypresentedinthisbookcanbeappliedtoothervibro-impactsystems
ingeneral,anddiscontinuousdynamicalsystemsinscienceandengineering.
Finally,theauthorswouldliketothanktheirfamily’ssupportforthiswork,andthisbookis
alsodedicatedtoProfessorWeiwuDengasagoodteacher,colleagueandfriend.Theauthors
hopethematerialspresentedhereinwillprovedurableinthefieldofscienceandengineering.
AlbertC.J.Luo
YuGuo
Edwardsville,Illinois,USA
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1
Introduction
This bookisaboutthedynamicsofvibro-impactoscillators.Vibro-impactsystemsextensively
exist in engineering. Such vibro-impact systems possess the continuous characteristics such
as continuous dynamical systems and discrete characteristics by impact discontinuity. Such
propertiesrequireanappropriatedevelopmentofdiscretemapsforsuchvibro-impactsystems
toinvestigatethecorrespondingcomplexmotions.Inthisbook,asystematicwaywillbedevel-
opedthroughafewsimplevibro-oscillatorsinordertounderstandthephysicsofvibro-impact
systemsinengineering.Beforediscussingthenonlineardynamicalphenomenaandbehaviors
of vibro-impact oscillators, the theory for nonlinear discrete systems will be presented, and
thecompletedynamicsofnonlineardiscretedynamicalsystemswillbepresented,whichwill
beappliedtoone-andtwo-dimensionaldiscretesystems,andtheageometricmethodwillbe
presentedtodeterminethefractalityandcomplexityofchaosindiscretedynamicalsystems.
Thetheoryofdiscontinuousdynamicalsystemswillbepresentedasabasefromwhichtostudy
vibro-impactdynamicsinengineering.Bouncingballdynamicswillbeanalyticallydiscussed
firstandthephysicalmotionsshowninasimplemodel.Tounderstandthechaoticdynamics
ofabouncingball,complexmotionsinabouncingballwillbediscussed.Afterdiscussingthe
bouncingballwiththesinglemap,asimpleversionofanimpactdamperwillbepresentedto
showhowtodevelopthecomplexperiodicmotionsanalytically.TheFermi-acceleratorwillbe
discussedindetailforapplication.Inthischapter,abriefreviewaboutthediscreteanddiscon-
tinuousdynamicalsystemswillbegivenfirst,andabriefhistoryofFermi-oscillatorsandvibro-
impactoscillatorswillbepresented.Thebooklayoutandthechaptersummarywillbegiven.
1.1 DiscreteandDiscontinuousSystems
Abriefviewofrecentdevelopmentsindiscretedynamicalsystemsanddiscontinuousdynam-
icalsystemswillbepresentedherein.
1.1.1 DiscreteDynamicalSystems
Considerann-dimensionaldiscretedynamicalsystemdefinedbyanimplicitvectorfunction
f :D→D on an open set D⊂(cid:2)n, where the vector function is f = (f , f ,..., f )T ∈(cid:2)n
1 2 n
Vibro-impactDynamics,FirstEdition.AlbertC.J.LuoandYuGuo.
(cid:5)C 2013JohnWiley&Sons,Ltd.Published2013byJohnWiley&Sons,Ltd.
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2 Vibro-impactDynamics
andvariablevectorisxk = (xk1,xk2,...,xkn)T ∈D.Forxk,xk+1 ∈D,thereisadiscreterela-
tionas
f(xk,xk+1,p)=0 (1.1)
withaparametervectorp= (p ,p ,...,p )T ∈(cid:2)m.
1 2 m
From the aforementioned discrete dynamical system, nonlinear algebraic equations are
usedtodescriberelationsbetweentwostatesofdynamicalsystemsinphasespace.Usingsuch
discrete relations, if one of two states is given, the rest can be determined but is not unique
because the relations are given by nonlinear algebraic equations. In other words, if a final
stateisgiven,onecanfindmultipleinitialstatestosatisfysuchnonlinearalgebraicequations.
Ontheotherhand,ifaninitialstateisgiven,onecanfindmultiplefinalstatestosatisfythe
nonlinear algebraic equations. For a specific set of parameters, it is very difficult to find the
multipleinitialorfinalstatesglobally.Oneoftenusestheroughlyestimatedvaluesasguessed
values with linearization to resolve this puzzle. Such a computation can be done locally by
thecomputer.Forglobalbehaviors,thediscretestatesinnonlineardiscretesystemsbecome
morechaotic.
The nonlinear discrete systems are obtained from the nonlinear difference equations of
dynamicalsystems.Thecomplexdynamicalbehaviorsinsuchnonlineardiscretesystemsare
observedthroughthecascadeofstablesolutions.May(1976)usedaone-dimensionaldiscrete
map to describe the dynamical processes in biological, economic and social science. The
Henonmapinthediscrete-timedynamicsystemwasintroducedbyHenon(1976)tosimplify
the three-dimensional Lorenz equations as a Poincare map, and chaos in such a discrete
systemwasobservednumericallybyHenon.Suchnumericalresultsstimulatedmoreattention
on the Henon map. Feigenbaum (1978, 1980) discussed the universal behaviors of one-
dimensionalsystemsandqualitativelydeterminedtheuniversalconstantsforchaos.Marotto
(1979)mathematicallyprovedtheexistenceofchaoticbehaviorsoftheHenonmapforcertain
parameters. Curry (1979) used Lyapunov characteristic exponent and frequency spectrum
to measure the chaotic behaviors of the Henon map. Collet, Eckmann, and Koch (1981)
presentedageneralizedtheoryofperiod-doublingbifurcationsinhigh-dimensionaldynamical
systems. Cvitanovic, Gunaratne, and Procaccia (1988) investigated topologic properties and
multifractalityoftheHenonmap.LuoandHan(1992)presentedageometricapproachforthe
period doubling bifurcation and multifractality of a general one-dimensional iterative map.
Gallas(1993)numericallyinvestigatedtheparametermapsfortheHenonmap.Zhusubaliyev
et al. (2000) did the bifurcation analysis of the Henon map and presented a more detailed
parametermap.Theaforementionedinvestigationswerebasedonthenumericalcomputation.
Gonchenko, Meiss, and Ovsyannikov (2006) discussed the three-dimensional Henon map
generatedfromahomoclinicbifurcation.Hruska(2006)developedanumericalalgorithmto
model the dynamics of a polynomial diffeomorphism of C2 on its chain recurrent set, and
appliedthisalgorithmtotheHenonmap.Gonchenko,Gonchenko,andTatjer(2007)studied
thebifurcationbehaviorsofperiodicsolutionsofthegeneralizedHenonmap,andprovedthe
existenceofinfinitecascadesofperiodicsolutionsinageneralizedHenonmap.Lorenz(2008)
adoptedarandomsearchingproceduretodeterminetheparametermapsofperiodicwindows
embeddedinchaoticsolutionsofHenonmap.Luo(2005a)investigatedthemappingdynamics
ofperiodicmotionsinanon-smoothpiecewisesystem.Luo(2010)presentedtheYing-Yang
theory in nonlinear discontinuous dynamics. The solutions in nonlinear discrete dynamical
