Table Of ContentConference Proceedings of the Society for Experimental Mechanics Series
Gaetan Kerschen · M. R. W. Brake
Ludovic Renson Editors
Nonlinear
Structures and
Systems, Volume 1
Proceedings of the 37th IMAC, A Conference and
Exposition on Structural Dynamics 2019
Conference Proceedings of the Society for Experimental Mechanics
Series
SeriesEditor
KristinB.Zimmerman,Ph.D.
SocietyforExperimentalMechanics,Inc.,
Bethel,CT,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/8922
Gaetan Kerschen • M. R. W. Brake (cid:129) Ludovic Renson
Editors
Nonlinear Structures and Systems,
Volume 1
Proceedings of the 37th IMAC, A Conference and Exposition
on Structural Dynamics 2019
123
Editors
GaetanKerschen M.R.W.Brake
SpaceStructures&SystemsLab.,BldgB52/3 RiceUniversity
UniversityofLiége,Space Houston,TX,USA
Liége,Belgium
LudovicRenson
UniversityofBristol
Bristol,UK
ISSN2191-5644 ISSN2191-5652 (electronic)
ConferenceProceedingsoftheSocietyforExperimentalMechanicsSeries
ISBN978-3-030-12390-1 ISBN978-3-030-12391-8 (eBook)
https://doi.org/10.1007/978-3-030-12391-8
©SocietyforExperimentalMechanics,Inc.2020
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Preface
Nonlinear Structures and Systems represents one of eight volumes of technical papers presented at the 37th IMAC, A
Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in
Orlando, Florida, on January 28–31, 2019. The full proceedings also include volumes on Dynamics of Civil Structures;
ModelValidationandUncertaintyQuantification;DynamicsofCoupledStructures;SpecialTopicsinStructuralDynamics
&ExperimentalTechniques;RotatingMachinery,OpticalMethods&ScanningLDVMethods;SensorsandInstrumentation,
Aircraft/Aerospace,EnergyHarvesting&DynamicEnvironmentsTesting;andTopicsinModalAnalysis&Testing.
Eachcollectionpresentsearlyfindingsfromexperimentalandcomputationalinvestigationsonanimportantareawithin
structuraldynamics.Nonlinearityisoneoftheseareas.
Thevastmajorityofrealengineeringstructuresbehavenonlinearly.Therefore,itisnecessarytoincludenonlineareffects
inallthestepsoftheengineeringdesign:intheexperimentalanalysistools(sothatthenonlinearparameterscanbecorrectly
identified)andinthemathematicalandnumericalmodelsofthestructure(inordertorunaccuratesimulations).Insodoing,
itwillbepossibletocreateamodelrepresentativeoftherealitywhich,oncevalidated,canbeusedforbetterpredictions.
Severalnonlinearpapersaddresstheoreticalandnumericalaspectsofnonlineardynamics(coveringrigoroustheoretical
formulationsandrobustcomputationalalgorithms)aswellasexperimentaltechniquesandanalysismethods.Therearealso
papersdedicatedtononlinearityinpracticewherereal-lifeexamplesofnonlinearstructuresarediscussed.
Theorganizerswouldliketothanktheauthors,presenters,sessionorganizers,andsessionchairsfortheirparticipationin
thistrack.
Liége,Belgium G.Kerschen
Houston,TX M.R.W.Brake
Bristol,UK LudovicRenson
v
Contents
1 NonsmoothModalAnalysisofaNon-internallyResonantFiniteBarSubjecttoaUnilateralContact
Constraint ................................................................................................................ 1
CarlosYoongandMathiasLegrand
2 ANewIwan/PalmovImplementationforFastSimulationandSystemIdentification........................... 11
DrithiShettyandMatthewS.Allen
3 AnalysisofTransientVibrationsforEstimatingBoltedJointTightness.......................................... 21
M.Brøns,J.J.Thomsen,S.M.Sah,D.Tcherniak,andA.Fidlin
4 SpiderConfigurationsforModelswithDiscreteIwanElements................................................... 25
AabhasSingh,MitchellWall,MatthewS.Allen,andRobertJ.Kuether
5 PredictingS4BeamJointNonlinearityUsingQuasi-StaticModalAnalysis...................................... 39
MitchellWall,MatthewS.Allen,andImanZare
6 TheBestLinearApproximationofMIMOSystems:FirstResultsonSimplifiedNonlinearityAssessment.. 53
PéterZoltánCsurcsia,BartPeeters,andJohanSchoukens
7 ForcedResponseofNonlinearSystemsUnderCombinedHarmonicandRandomExcitation ................ 65
AlwinFörster,LarsPanning-vonScheidt,andJörgWallaschek
8 GerrymanderingforInterfaces:ModelingtheMechanicsofJointedStructures................................ 81
T.Dreher,NidishNarayanaaBalaji,J.Groß,MatthewR.W.Brake,andM.Krack
9 AnAnalysisoftheGimballedHorizontalPendulumSystemforUseasaRotaryVibrationalEnergy
Harvester................................................................................................................. 87
D.Sequeira,J.Little,andB.P.Mann
10 OntheDynamicResponseofFlow-InducedVibrationofNonlinearStructures................................. 91
BanafshehSeyed-Aghazadeh,HamedSamandari,andRezaAbrishamBaf
11 PotentialandLimitationofaNonlinearModalTestingMethodforFriction-DampedSystems ............... 95
MarenScheel,TobiasSchulz,andMalteKrack
12 DynamicsofaMagneticallyExcitedRotationalSystem ............................................................ 99
Xue-SheWangandBrianP.Mann
13 ExperimentalNonlinearDynamicsofaPost-buckledCompositeLaminatePlate............................... 103
John I. Ferguson, Stephen M. Spottswood, David A. Ehrhardt, Ricardo A. Perez, Matthew P. Snyder,
andMatthewB.Obenchain
14 SimulationofaSelf-ResonantBeam-Slider-SystemConsideringGeometricNonlinearities ................... 115
FlorianMüllerandMalteKrack
15 Reinforcement Learning for Active Damping of Harmonically Excited Pendulum with Highly
NonlinearActuator...................................................................................................... 119
JamesD.Turner,LeviH.Manring,andBrianP.Mann
vii
viii Contents
16 InvestigationofNonlinearDynamicPhenomenaApplyingReal-TimeHybridSimulation..................... 125
MarkusJ.HochrainerandAntonM.Puhwein
17 ExperimentalandNumericalAeroelasticAnalysisofAirfoil-AileronSystemwithNonlinear
EnergySink .............................................................................................................. 133
Claudia Fernandez-Escudero, Miguel Gagnon, Eric Laurendeau, Sebastien Prothin, Annie Ross,
andGuilhemMichon
18 OntheModalSurrogacyofJointParameterEstimatesinBoltedJoints ......................................... 137
NidishNarayanaaBalajiandMatthewR.W.Brake
19 VehicleEscapeDynamicsonanArbitrarilyCurvedSurface....................................................... 141
LeviH.ManringandBrianP.Mann
20 NonlinearDynamicalAnalysisforCoupledFluid-StructureSystems............................................. 151
Q.Akkaoui,E.Capiez-Lernout,C.Soize,andR.Ohayon
21 ExperimentalNonlinearVibrationAnalysisofaShroudedBladedDiskModelonaRotatingTestRig...... 155
FerhatKaptan,LarsPanning-vonScheidt,andJörgWallaschek
22 TheMeasurementofTangentialContactStiffnessforNonlinearDynamicAnalysis............................ 165
C.W.SchwingshacklandD.Nowell
23 InvestigatingNonlinearityinaBoltedStructureUsingForceAppropriationTechniques...................... 169
BenjaminR.Pacini,DanielR.Roettgen,andDanielP.Rohe
24 TechniquesforNonlinearIdentificationandMaximizingModalResponse ...................................... 173
D.Roettgen,B.R.Pacini,andR.Mayes
25 InfluencesofModalCouplingonExperimentallyExtractedNonlinearModalModels......................... 189
Benjamin J. Moldenhauer, Aabhas Singh, Phil Thoenen, Daniel R. Roettgen, Benjamin R. Pacini,
RobertJ.Kuether,andMatthewS.Allen
26 DynamicResponseofaCurvedPlateSubjectedtoaMovingLocalHeatGradient............................. 205
DavidA.Ehrhardt,B.T.Gockel,andT.J.Beberniss
27 ATest-CaseonContinuationMethodsforBladed-DiskVibrationwithContactandFriction................. 209
Z.Saeed,G.Jenovencio,S.Arul,J.Blahoš,A.Sudhakar,L.Pesaresi,J.Yuan,F.ElHaddad,H.Hetzler,
andL.Salles
28 DynamicsofGeometrically-NonlinearBeamStructures,Part1:NumericalModeling......................... 213
D.Anastasio,J.Dietrich,J.P.Noël,G.Kerschen,S.Marchesiello,J.Häfele,C.G.Gebhardt,andR.Rolfes
29 DynamicsofGeometrically-NonlinearBeamStructures,Part2:ExperimentalAnalysis ...................... 217
D.Anastasio,J.Dietrich,J.P.Noël,G.Kerschen,S.Marchesiello,J.Häfele,C.G.Gebhardt,andR.Rolfes
30 Constructing Backbone Curves from Free-Decay Vibrations Data in Multi-Degrees of Freedom
OscillatorySystems...................................................................................................... 221
MattiaCenedeseandGeorgeHaller
31 Nonlinear3DModelingandVibrationAnalysisofHorizontalDrumTypeWashingMachines ............... 225
CemBaykal,EnderCigeroglu,andYigitYazicioglu
32 ComparisonofLinearandNonlinearModalReductionApproaches............................................. 229
ErhanFerhatoglu,TobiasDreher,EnderCigeroglu,MalteKrack,andH.NevzatÖzgüven
33 ReducedOrderModelingofBoltedJointsinFrequencyDomain ................................................. 235
GokhanKarapistikandEnderCigeroglu
34 ComparisonofANMandPredictor-CorrectorMethodtoContinueSolutionsofHarmonicBalance
Equations................................................................................................................. 239
LukasWoiwode,NidishNarayanaaBalaji,JonasKappauf,FabiaTubita,LouisGuillot,ChristopheVergez,
BrunoCochelin,AurélienGrolet,andMalteKrack
Contents ix
35 APrioriMethodstoAssesstheStrengthofNonlinearitiesforDesignApplications............................. 243
E.Rojas,S.Punla-Green,C.Broadman,MatthewR.W.Brake,B.R.Pacini,R.C.Flicek,D.D.Quinn,
C.W.Schwingshackl,andE.Dodgen
36 PredictiveModelingofBoltedAssemblieswithSurfaceIrregularities............................................ 247
Matthew Fronk, Gabriela Guerra, Matthew Southwick, Robert J. Kuether, Adam Brink, Paolo Tiso,
andDaneQuinn
37 ANovelComputationalMethodtoCalculateNonlinearNormalModesofComplexStructures.............. 259
HamedSamandariandEnderCigeroglu
38 Experimental-NumericalComparisonofContactNonlinearDynamicsThroughMulti-levelLinear
ModeShapes ............................................................................................................. 263
ElvioBonisoli,DomenicoLisitano,andChristianConigliaro
39 DynamicBehaviorandOutputChargeAnalysisofaBistableClamped-EndsEnergyHarvester............. 273
MasoudDerakhshaniandThomasA.Berfield
Chapter 1
Nonsmooth Modal Analysis of a Non-internally Resonant Finite
Bar Subject to a Unilateral Contact Constraint
CarlosYoongandMathiasLegrand
Abstract The present contribution describes a numerical technique devoted to the nonsmooth modal analysis (natural
frequencies and mode shapes) of a non-internally resonant elastic bar of length L subject to a Robin condition at x = 0
and a frictionless unilateral contact condition at x = L. When contact is ignored, the system of interest exhibits non-
commensurate linear natural frequencies, which is a critical feature in this study. The nonsmooth modes of vibration are
defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the
boundary conditions. In order to find a few of the above families, the unknown displacement is first expressed using the
well-known d’Alembert’s solution incorporating the Robin boundary condition at x = 0. The unilateral contact constraint
atx = LisreducedtoaconditionalswitchbetweenNeumann(opengap)andDirichlet(closedgap)boundaryconditions.
Finally,T-periodicityisenforced.Itisalsoassumedthatonlyonecontactswitchoccurseveryperiod.Theabovesystemof
equationsisnumericallysolvedforthroughasimultaneousdiscretizationofthespaceandtimedomains,whichyieldsasetof
equationsandinequationsintermsofdiscretedisplacementsandvelocities.Theproposedapproachisnon-dispersive,non-
dissipativeandaccuratelycapturesthepropagationofwaveswithdiscontinuousfronts,whichisessentialforthecomputation
ofperiodicmotionsinthisstudy.Resultsindicatethatincontrasttoitslinearcounterpart(barwithoutcontactconstraints)
where modal motions are sinusoidal functions “uncoupled” in space and time, the system of interest features nonsmooth
periodic displacements that are intricate piecewise sinusoidal functions in space and time. Moreover, the corresponding
frequency-energy“nonlinear”spectrumshowsbackbonecurvesofthehardeningtype.Itisalsoshownthatnonsmoothmodal
analysis is capable of efficiently predicting vibratory resonances when the system is periodically forced. The pre-stressed
andinitiallygrazingbarconfigurationsarealsobrieflydiscussed.
Keywords Nonsmoothsystems · Modalanalysis · Internalresonance · Unilateralcontactconstraints · Waveequation
1.1 Introduction
The concept of linear modes (natural frequencies and mode shapes) is a widely studied subject in the field of structural
dynamics [7]. A possible extension of this notion to nonlinear conservative systems sees a mode of vibration as a one-
parameter continuous family of periodic orbits displaying similar qualitative features [5]. In the phase space, nonlinear
modesemergeasinvariantsurfacesofperiodictrajectories,referredtoasinvariantmanifolds[10],whereinvariantimplies
that the motion initiated on the manifold stays on it as time unfolds. To some extent, nonlinear modal analysis can be
employed for predicting vibratory resonances, computing the nonlinear spectra of vibration or performing model-order
reduction. Techniques traditionally employed for nonlinear modal analysis require a certain degree of smoothness in the
nonlinearities[11]andthusfailforsystemswithnonsmoothnonlinearitiessuchasunilateralcontactconstraints.Certainly,an
accuratecharacterizationofthevibratoryresponseofthesesystemsisessentialtoachievingenhancedandsaferengineering
applications [12]. Modal analysis of nonsmooth mechanical systems, also called nonsmooth modal analysis, has been
recentlyproposedforafiniteelasticbaroflengthLsubjecttoaDirichletboundaryconditionatx =0andaunilateralcontact
constraintatx =L[13].Thissystemsatisfiesacompleteinternalresonancecondition,i.e.alllinearnaturalfrequenciesare
commensuratewiththefirstone,whichhasdrasticconsequencesonthenonlinearmodalresponse.Despitethesimplicityof
thesystem,thecomputednonsmoothmodes(NSMs)indicatehighlyintricatevibratorybehaviour.Correspondingperiodic
displacements were observed to be unseparated piecewise linear functions of space and time, as opposed to their linear
counterpartswhicharesinusoidalfunctionsseparatedinspaceandtime.Moreover,forcertainNSMssuchinternalresonance
C.Yoong((cid:2))·M.Legrand
DepartmentofMechanicalEngineering,McGillUniversity,Montréal,QC,Canada
e-mail:[email protected]
©SocietyforExperimentalMechanics,Inc.2020 1
G.Kerschenetal.(eds.),NonlinearStructuresandSystems,Volume1,ConferenceProceedingsoftheSocietyforExperimental
MechanicsSeries,https://doi.org/10.1007/978-3-030-12391-8_1
