Table Of ContentMECHANICAL
VIBRATIONS
MECHANICAL
VIBRATIONS
THEORY AND APPLICATION TO
STRUCTURAL DYNAMICS
ThirdEdition
MichelGéradin
UniversityofLiège,Belgium
DanielJ.Rixen
TechnischeUniversitätMünchen,Germany
Thiseditionfirstpublished2015
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SecondEditionpublishedin1997
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LibraryofCongressCataloging-in-PublicationData
Géradin,Michel,1945–
Mechanicalvibrations:theoryandapplicationtostructuraldynamics/MichelGéradin,
DanielJ.Rixen.–Thirdedition.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-90020-8(hardback)
1. Structuraldynamics. I.Rixen,Daniel.II.Title.
TA654.G452014
624.1’76–dc23
2014014588
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-1-118-90020-8
Typesetin10/12ptTimesbyLaserwordsPrivateLimited,Chennai,India
1 2015
Contents
Foreword xiii
Preface xv
Introduction 1
SuggestedBibliography 7
Listofmainsymbolsanddefinitions 9
1 AnalyticalDynamicsofDiscreteSystems 13
Definitions 14
1.1 Principleofvirtualworkforaparticle 14
1.1.1 Nonconstrainedparticle 14
1.1.2 Constrainedparticle 15
1.2 Extensiontoasystemofparticles 17
1.2.1 VirtualworkprincipleforN particles 17
1.2.2 Thekinematicconstraints 18
1.2.3 Conceptofgeneralizeddisplacements 20
1.3 Hamilton’sprincipleforconservativesystemsandLagrangeequations 23
1.3.1 Structureofkineticenergyandclassificationofinertiaforces 27
1.3.2 Energyconservationinasystemwithscleronomicconstraints 29
1.3.3 Classificationofgeneralizedforces 32
1.4 Lagrangeequationsinthegeneralcase 36
1.5 Lagrangeequationsforimpulsiveloading 39
1.5.1 Impulsiveloadingofamassparticle 39
1.5.2 Impulsiveloadingforasystemofparticles 42
1.6 Dynamicsofconstrainedsystems 44
1.7 Exercises 46
1.7.1 Solvedexercises 46
1.7.2 Selectedexercises 53
References 54
2 UndampedVibrationsofn-Degree-of-FreedomSystems 57
Definitions 58
2.1 Linearvibrationsaboutanequilibriumconfiguration 59
vi Contents
2.1.1 Vibrationsaboutastableequilibriumposition 59
2.1.2 Freevibrationsaboutanequilibriumconfigurationcorresponding
tosteadymotion 63
2.1.3 Vibrationsaboutaneutrallystableequilibriumposition 66
2.2 Normalmodesofvibration 67
2.2.1 Systemswithastableequilibriumconfiguration 68
2.2.2 Systemswithaneutrallystableequilibriumposition 69
2.3 Orthogonalityofvibrationeigenmodes 70
2.3.1 Orthogonalityofelasticmodeswithdistinctfrequencies 70
2.3.2 Degeneracytheoremandgeneralizedorthogonalityrelationships 72
2.3.3 Orthogonalityrelationshipsincludingrigid-bodymodes 75
2.4 Vectorandmatrixspectralexpansionsusingeigenmodes 76
2.5 Freevibrationsinducedbynonzeroinitialconditions 77
2.5.1 Systemswithastableequilibriumposition 77
2.5.2 Systemswithneutrallystableequilibriumposition 82
2.6 Responsetoappliedforces:forcedharmonicresponse 83
2.6.1 Harmonicresponse,impedanceandadmittancematrices 84
2.6.2 Modesuperpositionandspectralexpansionoftheadmittancematrix 84
2.6.3 Staticallyexactexpansionoftheadmittancematrix 88
2.6.4 Pseudo-resonanceandresonance 89
2.6.5 Normalexcitationmodes 90
2.7 Responsetoappliedforces:responseinthetimedomain 91
2.7.1 Modesuperpositionandnormalequations 91
2.7.2 Impulseresponseandtimeintegrationofthenormalequations 92
2.7.3 Stepresponseandtimeintegrationofthenormalequations 94
2.7.4 Directintegrationofthetransientresponse 95
2.8 Modalapproximationsofdynamicresponses 95
2.8.1 Responsetruncationandmodedisplacementmethod 96
2.8.2 Modeaccelerationmethod 97
2.8.3 Modeaccelerationandmodelreductiononselectedcoordinates 98
2.9 Responsetosupportmotion 101
2.9.1 Motionimposedtoasubsetofdegreesoffreedom 101
2.9.2 Transformationtonormalcoordinates 103
2.9.3 Mechanicalimpedanceonsupportsanditsstatically
exactexpansion 105
2.9.4 Systemsubmittedtoglobalsupportacceleration 108
2.9.5 Effectivemodalmasses 109
2.9.6 Methodofadditionalmasses 110
2.10 Variationalmethodsforeigenvaluecharacterization 111
2.10.1 Rayleighquotient 111
2.10.2 Principleofbestapproximationtoagiveneigenvalue 112
2.10.3 Recurrentvariationalprocedureforeigenvalueanalysis 113
2.10.4 Eigensolutionsofconstrainedsystems:generalcomparison
principleormonotonicityprinciple 114
2.10.5 Courant’sminimaxprincipletoevaluateeigenvaluesindependently
ofeachother 116
Contents vii
2.10.6 Rayleigh’stheoremonconstraints(eigenvaluebracketing) 117
2.11 Conservativerotatingsystems 119
2.11.1 Energyconservationintheabsenceofexternalforce 119
2.11.2 Propertiesoftheeigensolutionsoftheconservativerotatingsystem 119
2.11.3 State-spaceformofequationsofmotion 121
2.11.4 Eigenvalueprobleminsymmetricalform 124
2.11.5 Orthogonalityrelationships 126
2.11.6 Responsetononzeroinitialconditions 128
2.11.7 Responsetoexternalexcitation 130
2.12 Exercises 130
2.12.1 Solvedexercises 130
2.12.2 Selectedexercises 143
References 148
3 DampedVibrationsofn-Degree-of-FreedomSystems 149
Definitions 150
3.1 Dampedoscillationsintermsofnormaleigensolutionsofthe
undampedsystem 151
3.1.1 Normalequationsforadampedsystem 152
3.1.2 Modaldampingassumptionforlightlydampedstructures 153
3.1.3 Constructingthedampingmatrixthroughmodalexpansion 158
3.2 Forcedharmonicresponse 160
3.2.1 Thecaseoflightviscousdamping 160
3.2.2 Hystereticdamping 162
3.2.3 Forceappropriationtesting 164
3.2.4 Thecharacteristicphaselagtheory 170
3.3 State-spaceformulationofdampedsystems 174
3.3.1 Eigenvalueproblemandsolutionofthehomogeneouscase 175
3.3.2 Generalsolutionforthenonhomogeneouscase 178
3.3.3 Harmonicresponse 179
3.4 Experimentalmethodsofmodalidentification 180
3.4.1 Theleast-squarescomplexexponentialmethod 182
3.4.2 DiscreteFouriertransform 187
3.4.3 Therationalfractionpolynomialmethod 190
3.4.4 Estimatingthemodesoftheassociatedundampedsystem 195
3.4.5 Example:experimentalmodalanalysisofabellmouth 196
3.5 Exercises 199
3.5.1 Solvedexercises 199
3.6 Proposedexercises 207
References 208
4 ContinuousSystems 211
Definitions 212
4.1 Kinematicdescriptionofthedynamicbehaviourofcontinuoussystems:
Hamilton’sprinciple 213
4.1.1 Definitions 213
viii Contents
4.1.2 Strainevaluation:Green’smeasure 214
4.1.3 Stress–strainrelationships 219
4.1.4 Displacementvariationalprinciple 221
4.1.5 Derivationofequationsofmotion 221
4.1.6 Thelinearcaseandnonlineareffects 223
4.2 Freevibrationsoflinearcontinuoussystemsandresponsetoexternal
excitation 231
4.2.1 Eigenvalueproblem 231
4.2.2 Orthogonalityofeigensolutions 233
4.2.3 Responsetoexternalexcitation:modesuperposition(homogeneous
spatialboundaryconditions) 234
4.2.4 Responsetoexternalexcitation:modesuperposition
(nonhomogeneousspatialboundaryconditions) 237
4.2.5 Reciprocityprincipleforharmonicmotion 241
4.3 One-dimensionalcontinuoussystems 243
4.3.1 Thebarinextension 244
4.3.2 Transversevibrationsofatautstring 258
4.3.3 Transversevibrationofbeamswithnosheardeflection 263
4.3.4 Transversevibrationofbeamsincludingsheardeflection 277
4.3.5 Travellingwavesinbeams 285
4.4 Bendingvibrationsofthinplates 290
4.4.1 Kinematicassumptions 290
4.4.2 Strainexpressions 291
4.4.3 Stress–strainrelationships 292
4.4.4 Definitionofcurvatures 293
4.4.5 Moment–curvaturerelationships 293
4.4.6 Frametransformationforbendingmoments 295
4.4.7 Computationofstrainenergy 295
4.4.8 ExpressionofHamilton’sprinciple 296
4.4.9 PlateequationsofmotionderivedfromHamilton’sprinciple 298
4.4.10 Influenceofin-planeinitialstressesonplatevibration 303
4.4.11 Freevibrationsoftherectangularplate 305
4.4.12 Vibrationsofcircularplates 308
4.4.13 Anapplicationofplatevibration:theultrasonicwavemotor 311
4.5 Wavepropagationinahomogeneouselasticmedium 316
4.5.1 TheNavierequationsinlineardynamicanalysis 316
4.5.2 Planeelasticwaves 318
4.5.3 Surfacewaves 320
4.6 Solvedexercises 327
4.7 Proposedexercises 328
References 333
5 ApproximationofContinuousSystemsbyDisplacementMethods 335
Definitions 337
5.1 TheRayleigh–Ritzmethod 339
5.1.1 Choiceofapproximationfunctions 339
