Table Of ContentNonlinear Dynamic Phenomena in Mechanics
SOLIDMECHANICSANDITSAPPLICATIONS
Volume181
SeriesEditor: G.M.L.GLADWELL
DepartmentofCivilEngineering
UniversityofWaterloo
Waterloo,Ontario,CanadaN2L3GI
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Jerzy Warminski, Stefano Lenci,
Matthew P. Cartmell, Giuseppe Rega,
and Marian Wiercigroch (Eds.)
Nonlinear Dynamic
Phenomena in Mechanics
ABC
Editors
JerzyWarminski GiuseppeRega
LublinUniversityofTechnology SapienzaUniversityofRome
Nadbystrzycka36 DepartmentofStructuraland
20-618Lublin GeotechnicalEngineering
Poland ViaAntonioGramsci53
E-mail:[email protected] 00197Roma
Italy
StefanoLenci E-mail:[email protected]
PolytechnicUniversityofMarche
DepartmentofArchitecture, MarianWiercigroch
BuildingsandStructures UniversityofAberdeen
ViaBrecceBianche SchoolofEngineering
60131Ancona CentreforAppliedDynamics
Italy Research(CADR)
E-mail:[email protected] King’sCollege
AB243UEAberdeen
MatthewP.Cartmell UnitedKingdom
UniversityofGlasgow E-mail:[email protected]
SchoolofEngineering
JamesWattSouthBuilding
G128QQGlasgow
UnitedKingdom
E-mail:[email protected]
ISBN978-94-007-2472-3 e-ISBN978-94-007-2473-0
DOI10.1007/978-94-007-2473-0
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(cid:2)c 2012Springer-VerlagBerlinHeidelberg
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Preface
Nonlinearphenomenashouldplayacrucialroleinthedesignandcontrolofengi-
neeringsystemsandstructuresastheycandrasticallychangetheprevailingdynam-
icalresponses.Forexample,bifurcationsortransitionstoirregularity(chaos)may
completely alter intuitively expected behaviour. Dealing with nonlinear dynamic
phenomenarequires special analytical treatment and dedicated control techniques
to harness the effects of unpredictable behaviour. In many cases, formulating an
appropriate nonlinearmathematicalmodel of a real structure or system would be
essentialtoobtainabroadknowledgeoftherelevantresponseandtheinfluenceof
itsparameters.
Nonlinearmechanicsisclassicalin origin,howeveritsapplicationsaremodern
andtheyarevastinscienceandengineering.Inspiteofthefactthatlinearmodels
areincommonuse,inmanypracticalproblemscertainkeyeffectscanbeexplained
onlybyexplorationofnonlinearmodels.
The origin of this book is a series of lectures given in the frame of the Trans-
fer of Knowledge Project led by Prof. Tomasz Sadowski of Lublin University of
Technology on ’Modern Composite Materials Applied in Aerospace, Civil and
SanitaryEngineering:TheoreticalModellingandExperimentalVerification’(con-
tract MTKD-CT-2004-014058) and the FP7 Project ’Centre of Excellence for
Modern Composites Applied in Aerospace and Surface Transport Infrastructure’
(CEMCAST,FP7-REGPOT-20091,grantagreementNo:245479).Thementioned
projectsaccommodatedtwogroupsofresearchers:one,workinginthefieldofmod-
ernmaterialsmechanics,andtheother,workinginnonlineardynamics,bifurcation,
chaos theory and control. Some of the results obtained by the latter are included
in this book, whose scope covers theoretical and applications-based problems of
nonlineardynamics.Inthepresentedchaptersthenewestmethodsofnonlinearme-
chanicsareappliedtoelucidatearichvarietyoffeaturesofsystemresponseandthe
latest controltechniquesare used to enhancethe dynamicsor to reduceundesired
responses. Besides composite structures and systems with controllable and adap-
tiveproperties,flexiblestructuresandnon-smoothproblemsareaddressed,paying
properattentiontorealapplications.
VI Preface
So called small nonlinearities very often result in large qualitative and quan-
titative changes in structural dynamics. As a classical example we may mention
pendulum-likesystems addressed in Chap. 1. In practice, a pendulumcan be used
asadynamicabsorbermountedinhighbuildings,bridgesorchimneys.Swingsof
the pendulumcan suppress oscillations of the primary structure which then oscil-
lateswithaverysmallamplitudeornotatall.However,geometricalnonlinearities
introducedby pendulummotionmay changethe system dynamics,and instead of
theexpectedresponsewecanobservearapidincreaseoftheoscillationsofboththe
pendulumandthestructure,leadingtofullpendulumrotationorchaoticdynamics.
Thereasonforsuchbehaviourisrelatedtoautoparametriccouplingresultinginthe
occurrenceofinstabilityzones.Toavoidsuchdangeroussituationstheproperselec-
tionofparametersortheintroductionofsemi-activemagnetorheologicaldampingis
proposed.OnthebasisoftheanalyticalsolutionsofanonlineartwoDOFmodelthe
resonanceandinstabilityregionsaredetected,andthenchaoticoscillations,bifurca-
tionpointsandtransitionpathsfromregulartochaoticvibrationsaredeterminedby
numericaltechniques.Theoreticalresultsarevalidatedbyrealexperimentaltests.
Nonlinearmechanicsalso hasto beused toexplainundesiredresponseinslen-
der footbridges. Motivation for research in this topic was the famous example of
the London Millenium Bridge event. Strong horizontalvibrations, caused by syn-
chronisationofpedestrianmotions,wereinducedonitsopeningday.Theproblem
of pedestrian inducedlateral vibrationsmay occurin bridgesof variousstructural
typesandmaterials.TheparametricstudypresentedinChap.2allowsabetterun-
derstandingofthestructuralmechanicsandalsothedetectionofregionsofincreas-
ingvibration.Theobservedphenomenacanbeexplainedbyananalyticalnonlinear
discrete-time model based on the stroboscopic Poincare´ map which then enables
the location of instability regionsand the predictionof the number of pedestrians
required to trigger synchronisation of the structure. The analytical formula gives
reliable values which are in goodqualitative and quantitative agreementwith real
examplesandobservations.
Smart active or semi-active elements, like for example: magnetorheological
dampers,piezoelectricpatchesorshapememoryalloysactuatorsembeddedinside
thestructure,togetherwithrobustcontrolalgorithms,mayeliminateregionsofdan-
gerousbehaviour.Alsowemaytakeadvantageofthenonlinearphenomenatode-
sign an active structure to work more effectively. Shape memory alloys (SMAs)
exhibitsveryinterestingnonlinearthermo-mechanicalpropertiessuchastheshape
memoryeffectandsuperelasticity.Methodologiesforintegratingshapememoryal-
loy elements are based on active property tuning (APT) and active strain energy
tuning(ASET).Chapter3presentsdetailsofthemodellingoftheSMAeffectand
applicationsforSMAwiresembeddedinmechanicalstructurestocontroltheirdy-
namics.SMAelementsintegratedwithincompositebeamsorplatescanbeusedfor
activemodificationofstructurepropertiese.g.byaffectingtheirnaturalfrequencies.
Itisshownthattheresonantcharacteristicsofsuchhostscanbesignificantlyaltered
byactivationoftheembeddedelements.Thisconceptisextendedtoshell-likestruc-
tures, specifically tubular bearing housingsused to locate flexible rotors, and also
Preface VII
tomorecomplexplategeometriesinwhichtheSMAisarrangedinaperiodicand
repeatingstructureinordertocontrolmultiplemodesofvibration.
Recent advancementsin the theoretical and experimentalresearch on the finite
amplitude,resonant,forceddynamicsofsagged,horizontalorinclined,elasticca-
blesarepresentedinChap.4,byconsideringmodelling,analysis,response,and,in
particular,nonlinear/nonregularphenomena.Asymptoticsolutionsanda richvari-
etyoffeaturesofnonlinearmultimodalinteractionoccurringinvariousresonance
conditions are comparatively discussed. Dynamical and mechanical characteris-
ticsof someofthe principal,experimentallyobserved,responsesaresummarized,
along with the relevant robustness, spatio-temporal features, and dimensionality.
Challenging issues arising in the characterization of involved bifurcation scenar-
iosresultingintransitionto complexdynamicsare addressed,andhintsforproper
reduced-ordermodelling in cable nonlinear dynamics are obtained based on both
asymptoticsolutionsandexperimentalinvestigations,in the perspectiveof a prof-
itablecross-validationoftheobservednonlinearphenomena.
The importance of non-smooth dynamical systems, which are very common in
engineeringpracticeisdiscussedinChap.5.Mathematically,suchsystemscanbe
consideredaslocallysmoothandthereforetheglobalsolutionisobtainedbystitch-
inglocalsolutions,whichcanbedeterminedbystandardmethods.Ifthedynamical
systemispiecewiselinearthenanimplicitglobalanalyticalsolutioncanbegiven,
howevertheoccurrencesofnon-smoothnesshavefirsttobedetermined.Thisleads
tothenecessityofsolvingasetofnonlinearalgebraicequations.Toillustratenon-
smoothproblemsandthemethodologyforsolvingthem,threemechanicalengineer-
ingexamplesarestudied:(i)avibro-impactsystemintheformofamolingdevice,
(ii) the influenceof the openingandclosing of afatigue crack on the hostsystem
dynamics,and(iii)nonlinearinteractionsbetweenarotorandsnubberringsystem.
Thetheoreticalresultshavebeenobtainedfromthedevelopedmathematicalmodels
andconfirmedbyexperimentaltests,withagooddegreeofcorrelation.
Thisbookisaimedatawide audienceof engineersandresearchersworkingin
the field of nonlinear structural vibrations and dynamics, and undergraduate and
postgraduatestudentsreadingmechanical,aerospaceandcivilengineering.
Lublin,Ancona,Glasgow,Roma,Aberdeen JerzyWarminski
December2010 StefanoLenci
MatthewCartmell
GiuseppeRega
MarianWiercigroch
Acknowledgements
”TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanUnion
SeventhFrameworkProgramme(FP7/2007-2013),FP7-REGPOT-2009-1,under
grantagreementNo:245479”.
Contents
AutoparametricVibrationsofaNonlinearSystemwithaPendulumand
MagnetorheologicalDamping ..................................... 1
JerzyWarminski,KrzysztofKecik
1 IntroductiontoAutoparametricVibrations .................... 2
2 ModeloftheNonlinearOscillatorwithanAttachedPendulum ... 3
2.1 DifferentialEquationsofMotionofaModelwitha
ViscousDamper .................................. 4
2.2 ApplicationofMagnetorheological(MR)Damper ...... 5
3 ApproximateAnalyticalSolutionsandTheirStability........... 7
3.1 HarmonicBalanceMethod.......................... 8
3.2 StabilityofAnalyticalSolutions ..................... 12
3.3 ModelwiththeInvertedPendulum ................... 14
4 Regularand Chaotic DynamicsNear the Main Parametric
Resonance ............................................... 16
4.1 InstabilityRegionandParametersInfluence ........... 16
4.2 Upside-DownPendulum............................ 21
5 ExperimentalSetupoftheSystemwithActiveMRDamper...... 24
5.1 LaboratoryRig.................................... 24
5.2 CharacteristicsoftheMagnetorheologicalDamperand
NonlinearSprings ................................. 25
6 DynamicsofanAutoparametricSystemwithMRDamper....... 27
6.1 RegularOscillations ............................... 27
6.2 InfluenceofMRDampingonPendulum’sRotation ..... 33
6.3 ChaoticMotionunderMRDampingInfluence ......... 38
7 InfluenceofaNonlinearSystem’sSuspensionontheInstability
Regions ................................................. 52
8 ConclusionsandRemarks .................................. 58
References..................................................... 59
XII Contents
On the Dynamics of Pedestrians-Induced Lateral Vibrations of
Footbridges .................................................... 63
StefanoLenci,LauraMarcheggiani
1 IntroductionandLiteratureReview .......................... 64
2 AContinuous-TimeModel:TheSAMEOModel............... 68
2.1 ParametricInvestigations:ModelImplementationand
ComputationalAspects............................. 71
2.2 NumericalSimulations............................. 77
3 ADiscrete-TimeModel.................................... 97
3.1 SingleDegreeofFreedomOscillator andDiscrete
DynamicModel................................... 98
3.2 InteractionOscillator-Pedestrians ....................102
3.3 FixedPoints......................................105
3.4 ACase-Study:TheLondonMillenniumFootbridge.....108
4 Conclusions..............................................109
References.....................................................112
ApplicationsforShape Memory AlloysinStructuralandMachine
Dynamics ...................................................... 115
MatthewP.Cartmell,ArkadiuszJ.Z˙ak,OlgaA.Ganilova
1 ReviewoftheLiteratureandIntroduction.....................115
2 ModellingoftheShapeMemoryEffect.......................118
3 DynamicsofCompositeBeamsandPlateswithIntegratedSMA
Elements ................................................128
4 ApplicationstoFlexibleRotors..............................139
5 AntagonisticActuationControlofVibrationinPlates ...........145
6 Conclusions..............................................152
References.....................................................154
TheoreticalandExperimentalNonlinearVibrationsofSaggedElastic
Cables......................................................... 159
GiuseppeRega
1 Introduction..............................................159
2 CableModellingandTheoreticalAnalysis ....................161
2.1 ContinuousModelling .............................161
2.2 StaticEquilibriumandPlanarLinearFreeDynamics....165
2.3 MultimodeDiscretizationforNonlinearDynamics......167
2.4 InternalResonancesandAsymptoticSolutions.........168
2.5 ModalInteractionCoefficientsasPredictiveToolsfor
ReliableNonlinearDynamicResponse................171
3 NonlinearPhenomenainForcedDynamicResponse............176
3.1 MultimodalInteractionandResonantVibrations .......177
3.2 Modulated,Non-regular,andMulti-harmonic
Responses........................................184
3.3 NonlinearDynamicDisplacementsandTensions .......187
