VibrationofContinuous Systems
SecondEdition
SingiresuS.Rao
UniversityofMiami
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Names:Rao,SingiresuS.,author.
Title:Vibrationofcontinuoussystems/SingiresuSRao,UniversityofMiami.
Description:Secondedition.|Hoboken,NJ,USA:JohnWiley&SonsLtd, [2019]|Includesbibliographicalreferencesandindex.| Identifiers:LCCN2018041496(print)|LCCN2018041859(ebook)| ISBN9781119424253(AdobePDF)|ISBN9781119424277(ePub)| ISBN9781119424147(hardcover)
Subjects:LCSH:Vibration—Textbooks.|Structuraldynamics—Textbooks. Classification:LCCTA355(ebook)|LCCTA355.R3782019(print)| DDC624.1/71—dc23
LCrecordavailableathttps://lccn.loc.gov/2018041496
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Prefacexv Acknowledgmentsxix AbouttheAuthorxxi
1Introduction:BasicConcepts andTerminology1
1.1ConceptofVibration1
1.2ImportanceofVibration4
1.3OriginsandDevelopmentsinMechanics andVibration5
1.4HistoryofVibrationofContinuous Systems7
1.5DiscreteandContinuousSystems12
1.6VibrationProblems15
1.7VibrationAnalysis16
1.8Excitations17
1.9HarmonicFunctions17
1.9.1RepresentationofHarmonic Motion19
1.9.2Definitionsand Terminology21
1.10PeriodicFunctionsandFourier Series24
1.11NonperiodicFunctionsandFourier Integrals25
1.12LiteratureonVibrationofContinuous Systems28 References29 Problems31
2VibrationofDiscreteSystems: BriefReview33
2.1VibrationofaSingle-Degree-of-Freedom System33
2.1.1FreeVibration33
2.1.2ForcedVibrationunderHarmonic Force36
2.1.3ForcedVibrationunderGeneral Force41
2.2VibrationofMultidegree-of-Freedom Systems43
2.2.1EigenvalueProblem45
2.2.2OrthogonalityofModal Vectors46
2.2.3FreeVibrationAnalysisofan UndampedSystemUsingModal Analysis47
2.2.4ForcedVibrationAnalysisofan UndampedSystemUsingModal Analysis52
2.2.5ForcedVibrationAnalysis ofaSystemwithProportional Damping53
2.2.6ForcedVibrationAnalysis ofaSystemwithGeneralViscous Damping54
2.3RecentContributions60 References61 Problems62
3DerivationofEquations:Equilibrium Approach69
3.1Introduction69
3.2Newton’sSecondLawofMotion69
3.3D’Alembert’sPrinciple70
3.4EquationofMotionofaBarinAxial Vibration70
3.5EquationofMotionofaBeam inTransverseVibration72
3.6EquationofMotionofaPlateinTransverse Vibration74
3.6.1StateofStress76
3.6.2DynamicEquilibrium Equations76
3.6.3Strain–Displacement Relations77
3.6.4Moment–Displacement Relations79
3.6.5EquationofMotioninTerms ofDisplacement79
3.6.6InitialandBoundary Conditions80
3.7AdditionalContributions81 References81 Problems82
4DerivationofEquations:Variational Approach87
4.1Introduction87
4.2CalculusofaSingleVariable87
4.3CalculusofVariations88
4.4VariationOperator91
4.5FunctionalwithHigher-Order Derivatives93
4.6FunctionalwithSeveralDependent Variables95
4.7FunctionalwithSeveralIndependent Variables96
4.8ExtremizationofaFunctionalwith Constraints98
4.9BoundaryConditions102
4.10VariationalMethodsinSolid Mechanics106
4.10.1PrincipleofMinimumPotential Energy106
4.10.2PrincipleofMinimum ComplementaryEnergy107
4.10.3PrincipleofStationaryReissner Energy108
4.10.4Hamilton’sPrinciple109
4.11ApplicationsofHamilton’s Principle116
4.11.1EquationofMotionforTorsional VibrationofaShaft(Free Vibration)116
4.11.2TransverseVibrationofaThin Beam118
4.12RecentContributions121
Notes121 References122 Problems122
5DerivationofEquations:IntegralEquation Approach125
5.1Introduction125
5.2ClassificationofIntegralEquations125
5.2.1ClassificationBasedonthe NonlinearAppearance of ��(��) 125
5.2.2ClassificationBasedonthe LocationofUnknownFunction ��(��) 126
5.2.3ClassificationBasedontheLimits ofIntegration126
5.2.4ClassificationBasedontheProper NatureofanIntegral127
5.3DerivationofIntegralEquations127
5.3.1DirectMethod127
5.3.2DerivationfromtheDifferential EquationofMotion129
5.4GeneralFormulationoftheEigenvalue Problem132
5.4.1One-DimensionalSystems132
5.4.2GeneralContinuous Systems134
5.4.3Orthogonalityof Eigenfunctions135
5.5SolutionofIntegralEquations135
5.5.1MethodofUndetermined Coefficients136
5.5.2IterativeMethod136
5.5.3Rayleigh–RitzMethod141
5.5.4GalerkinMethod145
5.5.5CollocationMethod146
5.5.6NumericalIntegration Method148
5.6RecentContributions149 References150 Problems151
6SolutionProcedure:EigenvalueandModal AnalysisApproach153
6.1Introduction153
6.2GeneralProblem153
6.3SolutionofHomogeneousEquations: Separation-of-VariablesTechnique155
6.4Sturm–LiouvilleProblem156
6.4.1ClassificationofSturm–Liouville Problems157
6.4.2PropertiesofEigenvaluesand Eigenfunctions162
6.5GeneralEigenvalueProblem165
6.5.1Self-AdjointEigenvalue Problem165
6.5.2Orthogonalityof Eigenfunctions167
6.5.3ExpansionTheorem168
6.6SolutionofNonhomogeneous Equations169
6.7ForcedResponseofViscouslyDamped Systems171
6.8RecentContributions173
References174
Problems175
7SolutionProcedure:IntegralTransform Methods177
7.1Introduction177
7.2FourierTransforms178
7.2.1FourierSeries178
7.2.2FourierTransforms179
7.2.3FourierTransformofDerivatives ofFunctions181
7.2.4FiniteSineandCosineFourier Transforms181
7.3FreeVibrationofaFiniteString184
7.4ForcedVibrationofaFiniteString186
7.5FreeVibrationofaBeam188
7.6LaplaceTransforms191
7.6.1PropertiesofLaplace Transforms192
7.6.2PartialFractionMethod194
7.6.3InverseTransformation196
7.7FreeVibrationofaStringofFinite Length197
7.8FreeVibrationofaBeamofFinite Length200
7.9ForcedVibrationofaBeamofFinite Length201
7.10RecentContributions204
References205
Problems206
8TransverseVibrationofStrings209
8.1Introduction209
8.2EquationofMotion209
8.2.1EquilibriumApproach209
8.2.2VariationalApproach211
8.3InitialandBoundaryConditions213
8.4FreeVibrationofanInfiniteString215
8.4.1Traveling-WaveSolution215
8.4.2FourierTransform-Based Solution217
8.4.3LaplaceTransform-Based Solution219
8.5FreeVibrationofaStringofFinite Length221
8.5.1FreeVibrationofaString withBothEndsFixed222
8.6ForcedVibration231
8.7RecentContributions235
Note236
References236 Problems237
9LongitudinalVibrationofBars239
9.1Introduction239
9.2EquationofMotionUsingSimple Theory239
9.2.1UsingNewton’sSecondLaw ofMotion239
9.2.2UsingHamilton’s Principle240
9.3FreeVibrationSolutionandNatural Frequencies241
9.3.1SolutionUsingSeparation ofVariables242
9.3.2Orthogonalityof Eigenfunctions251
9.3.3FreeVibrationResponsedue toInitialExcitation254
9.4ForcedVibration259
9.5ResponseofaBarSubjectedto LongitudinalSupportMotion262
9.6RayleighTheory263
9.6.1EquationofMotion263
9.6.2NaturalFrequenciesandMode Shapes264
9.7Bishop’sTheory265
9.7.1EquationofMotion265
9.7.2NaturalFrequenciesandMode Shapes267
9.7.3ForcedVibrationUsingModal Analysis269
9.8RecentContributions272
References273 Problems273
10TorsionalVibrationofShafts277
10.1Introduction277
10.2ElementaryTheory:Equationof Motion277
10.2.1EquilibriumApproach277
10.2.2VariationalApproach278
10.3FreeVibrationofUniformShafts282
10.3.1NaturalFrequenciesofaShaft withBothEndsFixed283
10.3.2NaturalFrequenciesofaShaft withBothEndsFree284
10.3.3NaturalFrequenciesofaShaft FixedatOneEndandAttachedtoa TorsionalSpringatthe Other285
10.4FreeVibrationResponseduetoInitial Conditions:ModalAnalysis295
10.5ForcedVibrationofaUniformShaft:Modal Analysis298
10.6TorsionalVibrationofNoncircularShafts: Saint-Venant’sTheory301
10.7TorsionalVibrationofNoncircularShafts, IncludingAxialInertia305
10.8TorsionalVibrationofNoncircularShafts: TheTimoshenko–GereTheory306
10.9TorsionalRigidityofNoncircular Shafts309
10.10Prandtl’sMembraneAnalogy314
10.11RecentContributions319 References320 Problems321
11TransverseVibrationofBeams323
11.1Introduction323
11.2EquationofMotion:TheEuler–Bernoulli Theory323
11.3FreeVibrationEquations331
11.4FreeVibrationSolution331
11.5FrequenciesandModeShapesofUniform Beams332
11.5.1BeamSimplySupportedatBoth Ends333
11.5.2BeamFixedatBothEnds335
11.5.3BeamFreeatBothEnds336
11.5.4BeamwithOneEndFixed andtheOtherSimply Supported338
11.5.5BeamFixedatOneEndandFree attheOther340
11.6OrthogonalityofNormalModes345
11.7FreeVibrationResponseduetoInitial Conditions347
11.8ForcedVibration350
11.9ResponseofBeamsunderMoving Loads356
11.10TransverseVibrationofBeamsSubjected toAxialForce358
11.10.1DerivationofEquations358 11.10.2FreeVibrationofaUniform Beam361
11.11VibrationofaRotatingBeam363
11.12NaturalFrequenciesofContinuousBeams onManySupports365
11.13BeamonanElasticFoundation370
11.13.1FreeVibration370 11.13.2ForcedVibration372
11.13.3BeamonanElasticFoundation SubjectedtoaMoving Load373
11.14Rayleigh’sTheory375 11.15Timoshenko’sTheory377
11.15.1EquationsofMotion377 11.15.2EquationsforaUniform Beam382
11.15.3NaturalFrequenciesof Vibration383
11.16CoupledBending–TorsionalVibration ofBeams386
11.16.1EquationsofMotion387 11.16.2NaturalFrequencies ofVibration389
11.17TransformMethods:FreeVibration ofanInfiniteBeam391
11.18RecentContributions393 References395 Problems396
12VibrationofCircularRingsandCurved Beams399
12.1Introduction399
12.2EquationsofMotionofaCircular Ring399
12.2.1Three-DimensionalVibrations ofaCircularThinRing399
12.2.2AxialForceandMomentsinTerms ofDisplacements401
12.2.3SummaryofEquationsand Classificationof Vibrations403
12.3In-PlaneFlexuralVibrationsof Rings404
12.3.1ClassicalEquationsof Motion404
12.3.2EquationsofMotionthat IncludeEffectsofRotary InertiaandShear Deformation405
12.4FlexuralVibrationsatRightAnglestothe PlaneofaRing408
12.4.1ClassicalEquationsof Motion408
12.4.2EquationsofMotionthat IncludeEffectsofRotary InertiaandShear Deformation409
12.5TorsionalVibrations413
12.6ExtensionalVibrations413
12.7VibrationofaCurvedBeamwithVariable Curvature414
12.7.1ThinCurvedBeam414
12.7.2CurvedBeamAnalysis,Including theEffectofShear Deformation420
12.8RecentContributions423 References424 Problems425 13VibrationofMembranes427
13.1Introduction427
13.2EquationofMotion427
13.2.1EquilibriumApproach427
13.2.2VariationalApproach430
13.3WaveSolution432
13.4FreeVibrationofRectangular Membranes433
13.4.1MembranewithClamped Boundaries434
13.4.2ModeShapes438
13.5ForcedVibrationofRectangular Membranes444
13.5.1ModalAnalysisApproach444
13.5.2FourierTransform Approach448
13.6FreeVibrationofCircular Membranes450
13.6.1EquationofMotion450
13.6.2MembranewithaClamped Boundary452
13.6.3ModeShapes454
13.7ForcedVibrationofCircular Membranes454
13.8MembraneswithIrregularShapes459
13.9PartialCircularMembranes459
13.10RecentContributions460 Notes461 References462
Problems463
14TransverseVibrationofPlates465
14.1Introduction465
14.2EquationofMotion:ClassicalPlate Theory465
14.2.1EquilibriumApproach465
14.2.2VariationalApproach466
14.3BoundaryConditions473
14.4FreeVibrationofRectangular Plates479
14.4.1SolutionforaSimplySupported Plate481
14.4.2SolutionforPlateswithOther BoundaryConditions482
14.5ForcedVibrationofRectangular Plates489
14.6CircularPlates493
14.6.1EquationofMotion493
14.6.2Transformationof Relations494
14.6.3MomentandForce Resultants496
14.6.4BoundaryConditions497
14.7FreeVibrationofCircularPlates498
14.7.1SolutionforaClamped Plate500
14.7.2SolutionforaPlatewithaFree Edge501
14.8ForcedVibrationofCircularPlates503
14.8.1HarmonicForcing Function504
14.8.2GeneralForcingFunction505
14.9EffectsofRotaryInertiaandShear Deformation507
14.9.1EquilibriumApproach507
14.9.2VariationalApproach513
14.9.3FreeVibrationSolution519
14.9.4PlateSimplySupportedonAll FourEdges521
14.9.5CircularPlates523
14.9.6NaturalFrequenciesofaClamped CircularPlate528
14.10PlateonanElasticFoundation529
14.11TransverseVibrationofPlatesSubjectedto In-PlaneLoads531
14.11.1EquationofMotion531
14.11.2FreeVibration536
14.11.3SolutionforaSimplySupported Plate536
14.12VibrationofPlateswithVariable Thickness537
14.12.1RectangularPlates537
14.12.2CircularPlates539
14.12.3FreeVibrationSolution541
14.13RecentContributions543
References545
Problems547
15VibrationofShells549
15.1IntroductionandShellCoordinates549
15.1.1TheoryofSurfaces549
15.1.2DistancebetweenPointsinthe MiddleSurfacebefore Deformation550
15.1.3DistancebetweenPointsAnywhere intheThicknessofaShellbefore Deformation555
15.1.4DistancebetweenPointsAnywhere intheThicknessofaShellafter Deformation557
15.2Strain–DisplacementRelations560
15.3Love’sApproximations564
15.4Stress–StrainRelations570
15.5ForceandMomentResultants571
15.6StrainEnergy,KineticEnergy,andWork DonebyExternalForces579
15.6.1StrainEnergy579
15.6.2KineticEnergy581
15.6.3WorkDonebyExternal Forces581
15.7EquationsofMotionfromHamilton’s Principle582
15.7.1VariationofKinetic Energy583
15.7.2VariationofStrainEnergy584
15.7.3VariationofWorkDoneby ExternalForces585
15.7.4EquationsofMotion585
15.7.5BoundaryConditions587
15.8CircularCylindricalShells590
15.8.1EquationsofMotion591
15.8.2Donnell–Mushtari–Vlasov Theory592
15.8.3NaturalFrequenciesofVibration AccordingtoDMV Theory592
15.8.4NaturalFrequenciesofTransverse VibrationAccordingtoDMV Theory594
15.8.5NaturalFrequenciesofVibration AccordingtoLove’s Theory595
15.9EquationsofMotionofConicaland SphericalShells599
15.9.1CircularConicalShells599
15.9.2SphericalShells599
15.10EffectofRotaryInertiaandShear Deformation600
15.10.1DisplacementComponents600
15.10.2Strain–Displacement Relations601
15.10.3Stress–StrainRelations602
15.10.4ForceandMoment Resultants602
15.10.5EquationsofMotion603
15.10.6BoundaryConditions604
15.10.7VibrationofCylindrical Shells605
15.10.8NaturalFrequenciesofVibration ofCylindricalShells606
15.10.9AxisymmetricModes609
15.11RecentContributions611
Notes612
References612
Problems614
16VibrationofCompositeStructures617
16.1Introduction617
16.2CharacterizationofaUnidirectionalLamina withLoadingParalleltotheFibers617
16.3DifferentTypesofMaterial Behavior619
16.4ConstitutiveEquationsorStress–Strain Relations620
16.4.1AnisotropicMaterials620
16.4.2OrthotropicMaterials623
16.4.3IsotropicMaterials624
16.5CoordinateTransformationsforStresses andStrains626
16.5.1CoordinateTransformation RelationsforStressesinPlane StressState626
16.5.2CoordinateTransformation RelationsforStrainsinPlaneStrain State628
16.6LaminawithFibersOrientedatan Angle632
16.6.1TransformationofStiffnessand ComplianceMatricesforaPlane StressProblem632
16.6.2TransformationofStiffnessand ComplianceMatricesforan OrthotropicMaterialinThree Dimensions633
16.7CompositeLaminainPlaneStress634
16.8LaminatedCompositeStructures641
16.8.1ComputationoftheStiffness MatricesoftheIndividual Laminas642
16.8.2Strain–DisplacementRelations oftheLaminate643
16.8.3ComputationofMid-PlaneStrains forEachLamina649
16.8.4ComputationofMid-PlaneStresses InducedinEachLamina650
16.8.5ComputationofMid-PlaneStrains andCurvatures656
16.8.6ComputationofStressesinthe Laminate657
16.9VibrationAnalysisofLaminatedComposite Plates659
16.10VibrationAnalysisofLaminatedComposte Beams663
16.11RecentContributions666 References667 Problems668
17ApproximateAnalyticalMethods671
17.1Introduction671
17.2Rayleigh’sQuotient672
17.3Rayleigh’sMethod674
17.4Rayleigh–RitzMethod685
17.5AssumedModesMethod695
17.6WeightedResidualMethods697
17.7Galerkin’sMethod698
17.8CollocationMethod704
17.9SubdomainMethod709
17.10LeastSquaresMethod711
17.11RecentContributions718 References719 Problems721
18NumericalMethods:FiniteElement Method725
18.1Introduction725
18.2FiniteElementProcedure725
18.2.1DescriptionoftheMethod727
18.2.2ShapeFunctions728
18.3ElementMatricesofDifferentStructural Problems739
18.3.1BarElementsforStructures SubjectedtoAxialForce739
18.3.2BeamElementsforBeams SubjectedtoBending Moment743
18.3.3ConstantStrainTriangle(CST) ElementforPlatesUndergoing In-planeDeformation747
18.4DynamicResponseUsingtheFinite ElementMethod753
18.4.1UncouplingtheEquations ofMotionofanUndamped System754
18.5AdditionalandRecent Contributions760 Note763
References763 Problems765
ABasicEquationsofElasticity769
A.1Stress769
A.2Strain–DisplacementRelations769
A.3Rotations771
A.4Stress–StrainRelations772
A.5EquationsofMotioninTerms ofStresses774
A.6EquationsofMotioninTerms ofDisplacements774
BLaplaceandFourierTransforms777
Index783
Preface
Thisbookpresentstheanalyticalandnumericalmethodsofvibrationanalysisofcontinuousstructuralsystems,includingstrings,bars,shafts,beams,circularringsandcurved beams,membranes,plates,shells,andcompositestructures.Theobjectivesofthebook are(1)tomakeamethodicalandcomprehensivepresentationofthevibrationofvarious typesofstructuralelements,(2)topresenttheexactanalytical,approximateanalytical, andapproximatenumericalmethodsofanalysis,and(3)topresentthebasicconcepts inasimplemannerwithillustrativeexamples.Favorablereactionsandencouragement fromprofessors,studentsandotherusersofthebookhaveprovidedmewiththeimpetus topreparethissecondeditionofthebook.
Thefollowingchangeshavebeenmadefromthefirstedition:
•Somesectionswererewrittenforbetterclarity.
•Somenewproblemsareadded.
•Theerrorsnotedinthefirsteditionhavebeencorrected.
•Somesectionshavebeenexpanded.Thechapteron“ElasticWavePropagation” hasbeendeleted.
•Anewchapteron“VibrationofCompositeStructures”hasbeenadded.
•Anewchapterentitled,“ApproximateNumericalMethods:FiniteElement Method,”isaddedtocomplementtheexistingchapteron“ApproximateAnalyticalMethods.”
Continuousstructuralelementsandsystemsareencounteredinmanybranchesof engineering,suchasaerospace,architectural,chemical,civil,ocean,andmechanical engineering.Thedesignofmanystructuralandmechanicaldevicesandsystemsrequires anaccuratepredictionoftheirvibrationanddynamicperformancecharacteristics.The methodspresentedinthebookcanbeusedintheseapplications.Thebookisintended toserveasatextbookforadual-levelorfirstgraduatedegree-levelcourseonvibrationsorstructuraldynamics.Morethanenoughmaterialisincludedforaone-semester course.Thechaptersaremadeasindependentandself-containedaspossiblesothata coursecanbetaughtbyselectingappropriatechaptersorthroughequivalentself-study. Asuccessfulvibrationanalysisofcontinuousstructuralelementsandsystemsrequiresa knowledgeofmechanicsofmaterials,structuralmechanics,ordinaryandpartialdifferentialequations,matrixmethods,variationalcalculus,andintegralequations.Applications ofthesetechniquesarepresentedthroughout.Theselection,arrangement,andpresentationofthematerialhavebeenmadebasedonthelecturenotesforacoursetaughtbythe author.Thecontentsofthebookpermitinstructorstoemphasizeavarietyoftopics,such asbasicmathematicalapproacheswithsimpleapplications,barsandbeams,beamsand plates,orplatesandshells.Thebookwillalsobeusefulasareferencebookforpracticing engineers,designers,andvibrationanalystsinvolvedinthedynamicanalysisanddesign ofcontinuoussystems.
OrganizationoftheBook
Thebookisorganizedinto18chaptersandtwoappendices.ThebasicconceptsandterminologyusedinvibrationanalysisareintroducedinChapter1.Theimportance,origin,and abriefhistoryofvibrationofcontinuoussystemsarepresented.Thedifferencebetween discreteandcontinuoussystems,typesofexcitations,descriptionofharmonicfunctions, andbasicdefinitionsusedinthetheoryofvibrationsandrepresentationofperiodicfunctionsintermsofFourierseriesandtheFourierintegralarediscussed.Chapter2provides abriefreviewofthetheoryandtechniquesusedinthevibrationanalysisofdiscrete systems.Freeandforcedvibrationofsingle-andmultidegree-of-freedomsystemsare outlined.Theeigenvalueproblemanditsroleinthemodalanalysisusedinthefreeand forcedvibrationanalysisofdiscretesystemsarediscussed.
Variousmethodsofformulatingvibrationproblemsassociatedwithcontinuous systemsarepresentedinChapters3,4,and5.Theequilibriumapproachispresentedin Chapter3.UseofNewton’ssecondlawofmotionandD’Alembert’sprincipleisoutlined, withapplicationtodifferenttypesofcontinuouselements.Useofthevariationalapproach inderivingequationsofmotionandassociatedboundaryconditionsisdescribedin Chapter4.Thebasicconceptsofcalculusofvariationsandtheirapplicationtoextreme valueproblemsareoutlined.Thevariationalmethodsofsolidmechanics,includingthe principlesofminimumpotentialenergy,minimumcomplementaryenergy,stationary Reissnerenergy,andHamilton’sprinciple,arepresented.TheuseofHamilton’sprinciple intheformulationofcontinuoussystemsisillustratedwithtorsionalvibrationofashaft andtransversevibrationofathinbeam.TheintegralequationapproachfortheformulationofvibrationproblemsispresentedinChapter5.Abriefoutlineofintegralequations andtheirclassification,andthederivationofintegralequations,aregiventogetherwith examples.Thesolutionofintegralequationsusingiterative,Rayleigh–Ritz,Galerkin, collocation,andnumericalintegrationmethodsisalsodiscussedinthischapter.
Thecommonsolutionprocedurebasedoneigenvalueandmodalanalysesforthe vibrationanalysisofcontinuoussystemsisoutlinedinChapter6.Theorthogonalityof eigenfunctionsandtheroleoftheexpansiontheoreminmodalanalysisarediscussed. Theforcedvibrationresponseofviscouslydampedsystemsarealsoconsideredinthis chapter.Chapter7coversthesolutionofproblemsofvibrationofcontinuoussystems usingintegraltransformmethods.BothLaplaceandFouriertransformtechniquesare outlinedtogetherwithillustrativeapplications.
ThetransversevibrationofstringsispresentedinChapter8.Thisproblemfinds applicationinguywires,electrictransmissionlines,ropesandbeltsusedinmachinery, andthemanufactureofthread.Thegoverningequationisderivedusingequilibriumand variationalapproaches.Thetraveling-wavesolutionandseparationofvariablessolution areoutlined.Thefreeandforcedvibrationofstringsareconsideredinthischapter.The longitudinalvibrationofbarsisthetopicofChapter9.EquationsofmotionbasedonsimpletheoryarederivedusingtheequilibriumapproachaswellasHamilton’sprinciple.The naturalfrequenciesofvibrationaredeterminedforbarswithdifferentendconditions. Freevibrationresponseduetoinitialexcitationandforcedvibrationofbarsareboth presented,asisresponseusingmodalanalysis.Freeandforcedvibrationofbarsusing RayleighandBishoptheoriesarealsooutlinedinChapter9.Thetorsionalvibrationof shaftsplaysanimportantroleinmechanicaltransmissionofpowerinprimemoversand otherhigh-speedmachinery.Thetorsionalvibrationofuniformandnonuniformrodswith
bothcircularandnoncircularcross-sectionsisdescribedinChapter10.Theequationsof motionandfreeandforcedvibrationofshaftswithcircularcross-sectionarediscussed usingtheelementarytheory.TheSaint-VenantandTimoshenko–Geretheoriesareconsideredinderivingtheequationsofmotionofshaftswithnoncircularcross-sections. Methodsofdeterminingthetorsionalrigidityofnoncircularshaftsarepresentedusing thePrandtlstressfunctionandPrandtlmembraneanalogy.
Chapter11dealswiththetransversevibrationofbeams.Startingwiththeequation ofmotionbasedonEuler–Bernoulliorthinbeamtheory,naturalfrequenciesandmode shapesofbeamswithdifferentboundaryconditionsaredetermined.Thefreevibrationresponseduetoinitialconditions,forcedvibrationunderfixedandmovingloads, responseunderaxialloading,rotatingbeams,continuousbeams,andbeamsonanelastic foundationarepresentedusingtheEuler–Bernoullitheory.Theeffectsofrotaryinertia (Rayleightheory)androtaryinertiaandsheardeformation(Timoshenkotheory)onthe transversevibrationofbeamsarealsoconsidered.Thecoupledbending-torsionalvibrationofbeamsisdiscussed.Finally,theuseoftransformmethodsforfindingthefreeand forcedvibrationproblemsisillustratedtowardtheendofChapter11.In-planeflexural andcoupledtwist-bendingvibrationofcircularringsandcurvedbeamsisconsideredin Chapter12.Theequationsofmotionandfreevibrationsolutionsarepresentedfirstusing asimpletheory.Thentheeffectsofrotaryinertiaandsheardeformationareconsidered. Thevibrationofringsfindsapplicationinthestudyofthevibrationofring-stiffened shellsusedinaerospaceapplications,gears,andstatorsofelectricalmachines.
ThetransversevibrationofmembranesisthetopicofChapter13.Membranesfind applicationindrumsandmicrophonecondensers.Theequationofmotionofmembranes isderivedusingboththeequilibriumandvariationalapproaches.Thefreeandforced vibrationofrectangularandcircularmembranesarebothdiscussedinthischapter. Chapter14coversthetransversevibrationofplates.Theequationofmotionandthe freeandforcedvibrationofbothrectangularandcircularplatesarepresented.The vibrationofplatessubjectedtoin-planeforces,platesonelasticfoundation,andplates withvariablethicknessisalsodiscussed.Finally,theeffectofrotaryinertiaandshear deformationonthevibrationofplatesaccordingtoMindlin’stheoryisoutlined.The vibrationofshellsisthetopicofChapter15.First,thetheoryofsurfacesispresented usingshellcoordinates.Thenthestrain-displacementrelationsaccordingtoLove’s approximations,stress–strain,andforceandmomentresultantsaregiven.Thenthe equationsofmotionarederivedfromHamilton’sprinciple.Theequationsofmotionof circularcylindricalshellsandtheirnaturalfrequenciesareconsideredusingDonnel–Mushtari–VlasovandLove’stheories.Finally,theeffectofrotaryinertiaandshear deformationonthevibrationofshellsisconsidered.
Chapter16presentsvibrationoffiber-reinforcedcompositestructuresandstructuralmembers.Thecompositematerialmechanicsoflaminatesincludingconstitutive relations,stressanalysisunderin-planeandtransverseloadsaswellasfreevibration analysisofrectangularplatesandbeamsarepresentedinthischapter.Chapter17is devotedtotheapproximateanalyticalmethodsusefulforvibrationanalysis.ThecomputationaldetailsoftheRayleigh,Rayleigh-Ritz,assumedmodes,weightedresidual, Galerkin,collocation,subdomaincollocation,andleastsquaresmethodsarepresented alongwithnumericalexamples.Finally,thenumericalmethods,basedonthefiniteelementmethod,forthevibrationanalysisofcontinuousstructuralelementsandsystems
areoutlinedinChapter18.Thedisplacementapproachisusedinderivingtheelement stiffnessandmassmatricesofbar,beam,andlineartriangle(constantstraintriangleor CST)elements.Numericalexamplesarepresentedtoillustratetheapplicationofthefinite elementmethodforthesolutionofsimplevibrationproblems.
AppendixApresentsthebasicequationsofelasticity.LaplaceandFouriertransformpairsassociatedwithsomesimpleandcommonlyusedfunctionsaresummarized inAppendixB.
Introduction:BasicConcepts andTerminology
1.1CONCEPTOFVIBRATION
Anyrepetitivemotioniscalled vibration or oscillation.Themotionofaguitarstring, motionfeltbypassengersinanautomobiletravelingoverabumpyroad,swayingoftall buildingsduetowindorearthquake,andmotionofanairplaneinturbulencearetypical examplesofvibration.Thetheoryofvibrationdealswiththestudyofoscillatorymotion ofbodiesandtheassociatedforces.TheoscillatorymotionshowninFig.1.1(a)iscalled harmonicmotion andisdenotedas
where X iscalledthe amplitudeofmotion, �� isthe frequencyofmotion,and t isthetime. ThemotionshowninFig.1.1(b)iscalled periodicmotion,andthatshowninFig.1.1(c) iscalled nonperiodic or transientmotion.ThemotionindicatedinFig.1.1(d)is random or long-durationnonperiodicvibration.
Thephenomenonofvibrationinvolvesanalternatinginterchangeofpotential energytokineticenergyandkineticenergytopotentialenergy.Hence,anyvibrating systemmusthaveacomponentthatstorespotentialenergyandacomponentthatstores kineticenergy.Thecomponentsstoringpotentialandkineticenergiesarecalleda spring or elasticelement anda mass or inertiaelement,respectively.Theelasticelementstores potentialenergyandgivesituptotheinertiaelementaskineticenergy,andviceversa,in eachcycleofmotion.Therepetitivemotionassociatedwithvibrationcanbeexplained throughthemotionofamassonasmoothsurface,asshowninFig.1.2.Themassis connectedtoalinearspringandisassumedtobeinequilibriumorrestatposition1. Letthemass m begivenaninitialdisplacementtoposition2andreleasedwithzero velocity.Atposition2,thespringisinamaximumelongatedcondition,andhencethe potentialorstrainenergyofthespringisamaximumandthekineticenergyofthemass willbezerosincetheinitialvelocityisassumedtobezero.Becauseofthetendency ofthespringtoreturntoitsunstretchedcondition,therewillbeaforcethatcausesthe mass m tomovetotheleft.Thevelocityofthemasswillgraduallyincreaseasitmoves fromposition2toposition1.Atposition1,thepotentialenergyofthespringiszero becausethedeformationofthespringiszero.However,thekineticenergyandhence thevelocityofthemasswillbemaximumatposition1becauseoftheconservationof energy(assumingnodissipationofenergyduetodampingorfriction).Sincethevelocity
ismaximumatposition1,themasswillcontinuetomovetotheleft,butagainstthe resistingforceduetocompressionofthespring.Asthemassmovesfromposition1to theleft,itsvelocitywillgraduallydecreaseuntilitreachesavalueofzeroatposition3. Atposition3,thevelocityandhencethekineticenergyofthemasswillbezeroandthe
Figure1.1 Typesofdisplacements(orforces):(a)periodic,simpleharmonic;(b)periodic, nonharmonic;(c)nonperiodic,transient;(d)nonperiodic,random.
Di
Di
Displacement (or force), x(t) (d)
(continued ) (a) m k
Position 1 (equilibrium) x(t) m k
Position 2 (extreme right)
Position 3 (extreme left)
(b) m k (c)
Figure1.2 Vibratorymotionofaspring–masssystem:(a)systeminequilibrium(spring undeformed);(b)systeminextremerightposition(springstretched);(c)systeminextremeleft position(springcompressed).
deflection(compression)andhencethepotentialenergyofthespringwillbemaximum. Again,becauseofthetendencyofthespringtoreturntoitsuncompressedcondition, therewillbeaforcethatcausesthemass m tomovetotherightfromposition3.The velocityofthemasswillincreasegraduallyasitmovesfromposition3toposition1.
Figure1.1
Atposition1,allthepotentialenergyofthespringhasbeenconvertedtothekinetic energyofthemass,andhencethevelocityofthemasswillbemaximum.Thus,the masscontinuestomovetotherightagainstincreasingspringresistanceuntilitreaches position2withzerovelocity.Thiscompletesonecycleofmotionofthemass,andthe processrepeats;thus,themasswillhaveoscillatorymotion.
Theinitialexcitationtoavibratingsystemcanbeintheformofinitialdisplacementand/orinitialvelocityofthemasselement(s).Thisamountstoimpartingpotential and/orkineticenergytothesystem.Theinitialexcitationsetsthesystemintooscillatorymotion,whichcanbecalled freevibration.Duringfreevibration,therewillbean exchangebetweenthepotentialandthekineticenergies.Ifthesystemisconservative,the sumofthepotentialenergyandthekineticenergywillbeaconstantatanyinstant.Thus, thesystemcontinuestovibrateforever,atleastintheory.Inpractice,therewillbesome dampingorfrictionduetothesurroundingmedium(e.g.air),whichwillcausealossof someenergyduringmotion.Thiscausesthetotalenergyofthesystemtodiminishcontinuouslyuntilitreachesavalueofzero,atwhichpointthemotionstops.Ifthesystemis givenonlyaninitialexcitation,theresultingoscillatorymotioneventuallywillcometo restforallpracticalsystems,andhencetheinitialexcitationiscalled transientexcitation andtheresultingmotioniscalled transientmotion.Ifthevibrationofthesystemisto bemaintainedinasteadystate,anexternalsourcemustcontinuouslyreplacetheenergy dissipatedduetodamping.
1.2IMPORTANCEOFVIBRATION
Anybodythathasmassandelasticityiscapableofoscillatorymotion.Infact,most humanactivities,includinghearing,seeing,talking,walking,andbreathing,alsoinvolve oscillatorymotion.Hearinginvolvesvibrationoftheeardrum,seeingisassociatedwith thevibratorymotionoflightwaves,talkingrequiresoscillationsofthelarynx(tongue), walkinginvolvesoscillatorymotionoflegsandhands,andbreathingisbasedonthe periodicmotionofthelungs.Inengineering,anunderstandingofthevibratorybehavior ofmechanicalandstructuralsystemsisimportantforthesafedesign,construction,and operationofavarietyofmachinesandstructures.
Thefailureofmostmechanicalandstructuralelementsandsystemscanbeassociatedwithvibration.Forexample,thebladeanddiskfailuresinsteamandgasturbines andstructuralfailuresinaircraftareusuallyassociatedwithvibrationandtheresulting fatigue.Vibrationinmachinesleadstorapidwearofparts,suchasgearsandbearings,to looseningoffasteners,suchasnutsandbolts,topoorsurfacefinishduringmetalcutting, andexcessivenoise.Excessivevibrationinmachinescausesnotonlythefailureofcomponentsandsystemsbutalsoannoyancetohumans.Forexample,imbalanceindiesel enginescancausegroundwavespowerfulenoughtocreateanuisanceinurbanareas. Supersonicaircraftcreatesonicboomsthatshatterdoorsandwindows.Severalspectacularfailuresofbridges,buildings,anddamsareassociatedwithwind-inducedvibration, aswellasoscillatorygroundmotionduringearthquakes.
Insomeengineeringapplications,vibrationsserveausefulpurpose.Forexample,in vibratoryconveyors,sieves,hoppers,compactors,dentistdrills,electrictoothbrushes, washingmachines,clocks,electricmassagingunits,piledrivers,vibratorytestingof materials,vibratoryfinishingprocesses,andmaterialsprocessingoperations,suchas castingandforging,vibrationisusedtoimprovetheefficiencyandqualityoftheprocess.
1.3ORIGINSANDDEVELOPMENTSINMECHANICS ANDVIBRATION
Theearliesthumaninterestinthestudyofvibrationcanbetracedtothetimewhenthefirst musicalinstruments,probablywhistlesordrums,wereinvented.Sincethattime,people haveappliedingenuityandcriticalinvestigationtostudythephenomenonofvibration anditsrelationtosound.Althoughcertainverydefiniteruleswereobservedintheartof music,eveninancienttimes,theycanhardlybecalledscience.TheancientEgyptians usedadvancedengineeringconcepts,suchastheuseofdovetailedcrampsanddowels, inthestonejointsofthepyramidsduringthethirdandsecondmillenniabc.
Asfarbackas4000bc,musicwashighlydevelopedandwellappreciatedinChina, India,Japan,andperhapsEgypt[1,6].Drawingsofstringedinstrumentssuchasharps appearedonthewallsofEgyptiantombsasearlyas3000bc.TheBritishMuseumalso hasananga,aprimitivestringedinstrumentdatingfrom155bc.Thepresentsystemof musicisconsideredtohaveariseninancientGreece.
Thescientificmethodofdealingwithnatureandtheuseoflogicalproofsforabstract propositionsbeganinthetimeofThalesofMiletus(624–546bc),whointroducedthe term electricity afterdiscoveringtheelectricalpropertiesofyellowamber.Thefirst persontoinvestigatethescientificbasisofmusicalsoundsisconsideredtobetheGreek mathematicianandphilosopher Pythagoras(c.570–c.490bc). Pythagorasestablished the Pythagoreanschool,thefirstinstituteofhighereducationandscientificresearch. Pythagorasconductedexperimentsonvibratingstringsusinganapparatuscalledthe monochord. Pythagorasfoundthatiftwostringsofidenticalpropertiesbutdifferent lengthsaresubjecttothesametension,theshorterstringproducesahighernote,and inparticular,ifthelengthoftheshorterstringisone-halfthatofthelongerstring,the shorterstringproducesanoteanoctaveabovetheother.Theconceptofpitchwasknown bythetimeof Pythagoras;however,therelationbetweenthepitchandthefrequencyof asoundingstringwasnotknownatthattime.Onlyinthesixteenthcentury,aroundthe timeofGalileo,didtherelationbetweenpitchandfrequencybecomeunderstood[2].
Daedalusisconsideredtohaveinventedthependuluminthemiddleofthesecond millenniumbc.Oneinitialapplicationofthependulumasatimingdevicewasmadeby Aristophanes(450–388bc).Aristotlewroteabookonsoundandmusicaround350bc anddocumentshisobservationsinstatementssuchas“thevoiceissweeterthanthe soundofinstruments”and“thesoundofthefluteissweeterthanthatofthelyre.”Aristotlerecognizedthevectorialcharacterofforcesandintroducedtheconceptofvectorial additionofforces.Inaddition,hestudiedthelawsofmotion,similartothoseofNewton. Aristoxenus,whowasamusicianandastudentofAristotle,wroteathree-volumebook called ElementsofHarmony.Thesebooksareconsideredtheoldestbooksavailableon thesubjectofmusic.AlexanderofAfrodisiasintroducedtheideasofpotentialandkinetic energiesandtheconceptofconservationofenergy.Inabout300bc,inadditiontohis contributionstogeometry,Euclidgaveabriefdescriptionofmusicinatreatisecalled IntroductiontoHarmonics.However,hedidnotdiscussthephysicalnatureofsound inthebook.Euclidwasdistinguishedforhisteachingability,andhisgreatestwork,the Elements,hasseennumerouseditionsandremainsoneofthemostinfluentialbooks ofmathematicsofalltime.Archimedes(287–212bc)iscalledbysomescholarsthe fatherofmathematicalphysics.Hedevelopedtherulesofstatics.Inhis OnFloating Bodies,Archimedesdevelopedmajorrulesoffluidpressureonavarietyofshapesand onbuoyancy.
Chinaexperiencedmanydeadlyearthquakesinancienttimes.ZhangHeng,ahistorianandastronomerofthesecondcenturyad,inventedtheworld’sfirstseismographto measureearthquakesinad132[3].Thisseismographwasabronzevesselintheformof awinejar,withanarrangementconsistingofpendulumssurroundedbyagroupofeight levermechanismspointingineightdirections.Eightdragonfigures,withabronzeballin themouthofeach,werearrangedoutsidethejar.Anearthquakeinanydirectionwould tiltthependuluminthatdirection,whichwouldcausethereleaseofthebronzeballin thatdirection.Thisinstrumentenabledmonitoringpersonneltoknowthedirection,time ofoccurrence,andperhaps,themagnitudeoftheearthquake.
Thefoundationsofmodernphilosophyandsciencewerelaidduringthesixteenth century;infact,theseventeenthcenturyiscalledthe centuryofgenius bymany.Galileo (1564–1642)laidthefoundationsformodernexperimentalsciencethroughhismeasurementsonasimplependulumandvibratingstrings.Duringoneofhistripstothechurch in Pisa,theswingingmovementsofalampcaughtGalileo’sattention.Hemeasured theperiodofthependulummovementsofthelampwithhispulseandwasamazedto findthatthetimeperiodwasnotinfluencedbytheamplitudeofswings.Subsequently, Galileoconductedmoreexperimentsonthesimplependulumandpublishedhisfindings in DiscoursesConcerningTwoNewSciences in1638.Inthiswork,hediscussedthe relationshipbetweenthelengthandthefrequencyofvibrationofasimplependulum,as wellastheideaofsympatheticvibrationsorresonance[4].
AlthoughthewritingsofGalileoindicatethatheunderstoodtheinterdependence oftheparameters—length,tension,density,andfrequencyoftransversevibration—of astring,hedidnotofferananalyticaltreatmentoftheproblem.MarinusMersenne (1588–1648),amathematicianandtheologianfromFrance,describedthecorrectbehaviorofthevibrationofstringsin1636inhisbook HarmonicorumLiber.Forthefirsttime, byknowing(measuring)thefrequencyofvibrationofalongstring,Mersennewasableto predictthefrequencyofvibrationofashorterstringhavingthesamedensityandtension. Heisconsideredtobethefirstpersontodiscoverthelawsofvibratingstrings.Thetruth wasthatGalileowasthefirstpersontoconductexperimentalstudiesonvibratingstrings; however,publicationofhisworkwasprohibiteduntil1638,byorderoftheInquisitorof Rome.AlthoughGalileostudiedthependulumextensivelyanddiscussedtheisochronismofthependulum,ChristianHuygens(1629–1695)wastheonewhodevelopedthe pendulumclock,thefirstaccuratedevicedevelopedformeasuringtime.Heobserved deviationfromisochronismduetothenonlinearityofthependulum,andinvestigated variousdesignstoimprovetheaccuracyofthependulumclock.
TheworksofGalileocontributedtoasubstantiallyincreasedlevelofexperimental workamongmanyscientistsandpavedthewaytotheestablishmentofseveralprofessionalorganizations,suchastheAcademiaNaturaeinNaplesin1560,theAcademiadei LinceiinRomein1606,theRoyalSocietyinLondonin1662,theFrenchAcademyof Sciencesin1766,andtheBerlinAcademyofSciencein1770.
TherelationbetweenthepitchandfrequencyofvibrationofatautstringwasinvestigatedfurtherbyRobertHooke(1635–1703)andJosephSauveur(1653–1716).Thephenomenonofmodeshapesduringthevibrationofstretchedstrings,involvingnomotionat certainpointsandviolentmotionatintermediatepoints,wasobservedindependentlyby SauveurinFrance(1653–1716)andJohnWallis(1616–1703)inEngland.Sauveurcalled pointswithnomotion nodes andpointswithviolentmotion, loops.Also,heobserved thatvibrationsinvolvingnodesandloopshadhigherfrequenciesthanthoseinvolvingno
nodes.Afterobservingthatthevaluesofthehigherfrequencieswereintegralmultiples ofthefrequencyofsimplevibrationwithnonodes,Sauveurtermedthefrequencyof simplevibrationthe fundamentalfrequency andthehigherfrequencies,the harmonics. Inaddition,hefoundthatthevibrationofastretchedstringcancontainseveralharmonicssimultaneously.ThephenomenonofbeatswasalsoobservedbySauveurwhen twoorganpipes,havingslightlydifferentpitches,weresoundedtogether.Healsotried tocomputethefrequencyofvibrationofatautstringfromthemeasuredsagofits middlepoint.Sauveurintroducedtheword acoustics forthefirsttimeforthescience ofsound[7].
IsaacNewton(1642–1727)studiedatTrinityCollege,Cambridge,andlaterbecame professorofmathematicsatCambridgeandpresidentoftheRoyalSocietyofLondon. In1687,hepublishedthemostadmiredscientifictreatiseofalltime, Philosophia NaturalisPrincipiaMathematica.Althoughthelawsofmotionwerealreadyknown inoneformorother,thedevelopmentofdifferentialcalculusbyNewtonandLeibnitz madethelawsapplicabletoavarietyofproblemsinmechanicsandphysics.Leonhard Euler(1707–1783)laidthegroundworkforthecalculusofvariations.Hepopularized theuseoffree-bodydiagramsinmechanicsandintroducedseveralnotations,including �� = 2.71828 ...,�� (��), Σ, and �� = √ 1.Infact,manypeoplebelievethatthecurrent techniquesofformulatingandsolvingmechanicsproblemsareduemoretoEuler thantoanyotherpersoninthehistoryofmechanics.Usingtheconceptofinertia force,JeanD’Alembert(1717–1783)reducedtheproblemofdynamicstoaproblemin statics.JosephLagrange(1736–1813)developedthevariationalprinciplesforderiving theequationsofmotionandintroducedtheconceptofgeneralizedcoordinates.He introduced Lagrangeequations asapowerfultoolforformulatingtheequationsof motionforlumped-parametersystems.CharlesCoulomb(1736–1806)studiedthe torsionaloscillationsboththeoreticallyandexperimentally.Inaddition,hederivedthe relationbetweenelectricforceandcharge.
ClaudeLouisMarieHenriNavier(1785–1836)presentedarigoroustheoryforthe bendingofplates.Inaddition,heconsideredthevibrationofsolidsandpresentedthe continuumtheoryofelasticity.In1882,AugustinLouisCauchy(1789–1857)presented aformulationforthemathematicaltheoryofcontinuummechanics.WilliamHamilton (1805–1865)extendedtheformulationofLagrangefordynamicsproblemsandpresented apowerfulmethod(Hamilton’sprinciple)forthederivationofequationsofmotionof continuoussystems.HeinrichHertz(1857–1894)introducedtheterms holonomic and nonholonomic intodynamicsaround1894.JulesHenri Poincaré(1854–1912)mademany contributionstopureandappliedmathematics,particularlytocelestialmechanicsand electrodynamics.Hisworkonnonlinearvibrationsintermsoftheclassificationofsingularpointsofnonlinearautonomoussystemsisnotable.
1.4HISTORYOFVIBRATIONOFCONTINUOUSSYSTEMS
Theprecisetreatmentofthevibrationofcontinuoussystemscanbeassociatedwiththe discoveryofthebasiclawofelasticitybyHooke,thesecondlawofmotionbyNewton, andtheprinciplesofdifferentialcalculusbyLeibnitz.Newton’ssecondlawofmotion isusedroutinelyinmodernbooksonvibrationstoderivetheequationsofmotionofa vibratingbody.
Strings Atheoretical(dynamical)solutionoftheproblemofthevibratingstringwas foundin1713bytheEnglishmathematicianBrookTaylor(1685–1731),whoalsopresentedthefamousTaylortheoremoninfiniteseries.Heappliedthefluxionapproach, similartothedifferentialcalculusapproachdevelopedbyNewtonandNewton’ssecondlawofmotion,toanelementofacontinuousstringandfoundthetruevalueof thefirstnaturalfrequencyofthestring.ThisvaluewasfoundtoagreewiththeexperimentalvaluesobservedbyGalileoandMersenne.TheprocedureadoptedbyTaylorwas perfectedthroughtheintroductionofpartialderivativesintheequationsofmotionby DanielBernoulli,JeanD’Alembert,andLeonhardEuler.Thefluxionmethodprovedtoo clumsyforusewithmorecomplexvibrationanalysisproblems.Withthecontroversy betweenNewtonandLeibnitzastotheoriginofdifferentialcalculus,patrioticEnglishmenstucktothecumbersomefluxionswhileotherinvestigatorsinEuropefollowedthe simplernotationaffordedbytheapproachofLeibnitz.
In1747,D’Alembertderivedthepartialdifferentialequation,laterreferredtoasthe waveequation,andfoundthewavetravelsolution.AlthoughD’Alembertwasassisted byDanielBernoulliandLeonhardEulerinthiswork,hedidnotgivethemcredit.With allthreeclaimingcreditforthework,thespecificcontributionofeachhasremained controversial.
Thepossibilityofastringvibratingwithseveralofitsharmonicspresentatthesame time(withdisplacementofanypointatanyinstantbeingequaltothealgebraicsumof displacementsforeachharmonic)wasobservedbyBernoulliin1747andprovedbyEuler in1753.ThiswasestablishedthroughthedynamicequationsofDanielBernoulliinhis memoir,publishedbytheBerlinAcademyin1755.Thischaracteristicwasreferredtoas the principleofthecoexistenceofsmalloscillations,whichisthesameasthe principle ofsuperposition intoday’sterminology.Thisprincipleprovedtobeveryvaluablein thedevelopmentofthetheoryofvibrationsandledtothepossibilityofexpressingany arbitraryfunction(i.e.,anyinitialshapeofthestring)usinganinfiniteseriesofsineand cosineterms.Becauseofthisimplication,D’AlembertandEulerdoubtedthevalidity ofthisprinciple.However,thevalidityofthistypeofexpansionwasprovedbyFourier (1768–1830)inhis AnalyticalTheoryofHeat in1822.
ItisclearthatBernoulliandEuleraretobecreditedastheoriginatorsofthemodal analysisprocedure.TheyshouldalsobeconsideredtheoriginatorsoftheFourierexpansionmethod.However,aswithmanydiscoveriesinthehistoryofscience,thepersons creditedwiththeachievementmaynotdeserveitcompletely.Itisoftenthepersonwho publishesattherighttimewhogetsthecredit.
TheanalyticalsolutionofthevibratingstringwaspresentedbyJosephLagrangein hismemoirpublishedbytheTurinAcademyin1759.Inhisstudy,Lagrangeassumedthat thestringwasmadeupofafinitenumberofequallyspacedidenticalmassparticles,and heestablishedtheexistenceofanumberofindependentfrequenciesequaltothenumber ofmassparticles.Whenthenumberofparticleswasallowedtobeinfinite,theresultingfrequencieswerefoundtobethesameastheharmonicfrequenciesofthestretched string.Themethodofsettingupthedifferentialequationofmotionofastring(calledthe waveequation),presentedinmostmodernbooksonvibrationtheory,wasdevelopedby D’AlembertanddescribedinhismemoirpublishedbytheBerlinAcademyin1750.
Bars Chladniin1787,andBiotin1816,conductedexperimentsonthelongitudinal vibrationofrods.In1824,Navierpresentedananalyticalequationanditssolutionfor thelongitudinalvibrationofrods.
Shafts
CharlesCoulombdidboththeoreticalandexperimentalstudiesin1784onthe torsionaloscillationsofametalcylindersuspendedbyawire[5].Byassumingthat theresultingtorqueofthetwistedwireisproportionaltotheangleoftwist,hederivedan equationofmotionforthetorsionalvibrationofasuspendedcylinder.Byintegratingthe equationofmotion,hefoundthattheperiodofoscillationisindependentoftheangleof twist.Thederivationoftheequationofmotionforthetorsionalvibrationofacontinuous shaftwasattemptedbyCaughyinanapproximatemannerin1827andgivencorrectlyby Poissonin1829.Infact,Saint-Venantdeservesthecreditforderivingthetorsionalwave equationandfindingitssolutionin1849.
Beams
Theequationofmotionforthetransversevibrationofthinbeamswasderived byDanielBernoulliin1735,andthefirstsolutionsoftheequationforvarioussupportconditionsweregivenbyEulerin1744.Theirapproachhasbecomeknownasthe Euler–Bernoulli or thinbeamtheory.Rayleighpresentedabeamtheorybyincludingthe effectofrotaryinertia.In1921,StephenTimoshenkopresentedanimprovedtheoryof beamvibration,whichhasbecomeknownasthe Timoshenko or thick beamtheory,by consideringtheeffectsofrotaryinertiaandsheardeformation.
Membranes In1766,Eulerderivedequationsforthevibrationofrectangularmembraneswhichwerecorrectonlyfortheuniformtensioncase.Heconsideredtherectangularmembraneinsteadofthemoreobviouscircularmembraneinadrumhead,because hepicturedarectangularmembraneasasuperpositionoftwosetsofstringslaidin perpendiculardirections.Thecorrectequationsforthevibrationofrectangularandcircularmembraneswerederivedby Poissonin1828.Althoughasolutioncorrespondingto axisymmetricvibrationofacircularmembranewasgivenby Poisson,anonaxisymmetric solutionwaspresentedby Paganiin1829.
Plates Thevibrationofplateswasalsobeingstudiedbyseveralinvestigatorsatthis time.BasedonthesuccessachievedbyEulerinstudyingthevibrationofarectangular membraneasasuperpositionofstrings,Euler’sstudentJamesBernoulli,thegrandnephewofthefamousmathematicianDanielBernoulli,attemptedin1788toderivean equationforthevibrationofarectangularplateasagridworkofbeams.However,the resultingequationwasnotcorrect.Asthetorsionalresistanceoftheplatewasnotconsideredinhisequationofmotion,onlyaresemblance,nottherealagreement,wasnoted betweenthetheoreticalandexperimentalresults.
Themethodofplacingsandonavibratingplatetofinditsmodeshapesandtoobserve thevariousintricatemodalpatternswasdevelopedbytheGermanscientistChladniin 1802.Inhisexperiments,Chladnidistributedsandevenlyonhorizontalplates.During vibration,heobservedregularpatternsofmodesbecauseoftheaccumulationofsand alongthenodallinesthathadnoverticaldisplacement.NapoléonBonaparte,whowasa trainedmilitaryengineer,waspresentwhenChladnigaveademonstrationofhisexperimentsonplatesattheFrenchAcademyin1809.NapoléonwassoimpressedbyChladni’s demonstrationthathegaveasumof3000francstotheFrenchAcademytobepresented tothefirstpersontogiveasatisfactorymathematicaltheoryofthevibrationofplates. Whenthecompetitionwasannounced,onlyoneperson,SophieGermain,hadentered thecontestbytheclosingdateofOctober1811[8].However,anerrorinthederivation ofGermain’sdifferentialequationwasnotedbyoneofthejudges,Lagrange.Infact, Lagrangederivedthecorrectformofthedifferentialequationofplatesin1811.Whenthe Academyopenedthecompetitionagain,withanewclosingdateofOctober1813,
Germainenteredthecompetitionagainwithacorrectformofthedifferentialequation ofplates.Sincethejudgeswerenotsatisfied,duetothelackofphysicaljustificationof theassumptionsshemadeinderivingtheequation,shewasnotawardedtheprize.The AcademyopenedthecompetitionagainwithanewclosingdateofOctober1815.Again, Germainenteredthecontest.Thistimeshewasawardedtheprize,althoughthejudges werenotcompletelysatisfiedwithhertheory.Itwasfoundlaterthatherdifferential equationforthevibrationofplateswascorrectbuttheboundaryconditionsshepresented werewrong.Infact,Kirchhoff,in1850,presentedthecorrectboundaryconditionsfor thevibrationofplatesaswellasthecorrectsolutionforavibratingcircularplate.
ThegreatengineerandbridgedesignerNavier(1785–1836)canbeconsideredthe originatorofthemoderntheoryofelasticity.Hederivedthecorrectdifferentialequation forrectangularplateswithflexuralresistance.Hepresentedanexactmethodthattransformsthedifferentialequationintoanalgebraicequationforthesolutionofplateand otherboundaryvalueproblemsusingtrigonometricseries.In1829, Poissonextended Navier’smethodforthelateralvibrationofcircularplates.
Kirchhoff(1824–1887),whoincludedtheeffectsofbothbendingandstretching inhistheoryofplatespublishedinhisbook LecturesonMathematicalPhysics,is consideredthefounderoftheextendedplatetheory.Kirchhoff’sbookwastranslated intoFrenchbyClebschwithnumerousvaluablecommentsbySaint-Venant.Love extendedKirchhoff’sapproachtothickplates.In1915,Timoshenkopresentedasolution forcircularplateswithlargedeflections.Fopplconsideredthenonlineartheoryofplates in1907;however,thefinalformofthedifferentialequationforthelargedeflection ofplateswasdevelopedbyvonKármánin1910.Amorerigorousplatetheorythat considerstheeffectsoftransverseshearforceswaspresentedbyReissner.Aplatetheory thatincludestheeffectsofbothrotatoryinertiaandtransversesheardeformation,similar totheTimoshenkobeamtheory,waspresentedbyMindlinin1951.
Shells ThederivationofanequationforthevibrationofshellswasattemptedbySophie Germain,whoin1821publishedasimplifiedequation,witherrors,forthevibrationof acylindricalshell.Sheassumedthatthein-planedisplacementoftheneutralsurface ofacylindricalshellwasnegligible.Herequationcanbereducedtothecorrectformfor arectangularplatebutnotforaring.Thecorrectequationforthevibrationofaringhad beengivenbyEulerin1766.
Aron,in1874,derivedthegeneralshellequationsincurvilinearcoordinates,which wereshowntoreducetotheplateequationwhencurvaturesweresettozero.The equationswerecomplicatedbecausenosimplifyingassumptionsweremade.Lord Rayleighproposeddifferentsimplificationsforthevibrationofshellsin1882and consideredtheneutralsurfaceoftheshelleitherextensionalorinextensional.Love,in 1888,derivedtheequationsforthevibrationofshellsbyusingsimplifyingassumptions similartothoseofbeamsandplatesforbothin-planeandtransversemotions.Love’s equationscanbeconsideredtobemostgeneralinunifyingthetheoryofvibration ofcontinuousstructureswhosethicknessissmallcomparedtootherdimensions. Thevibrationofshells,withaconsiderationofrotatoryinertiaandsheardeformation, waspresentedbySoedelin1982.
ApproximateMethods LordRayleighpublishedhisbookonthetheoryofsoundin 1877;itisstillconsideredaclassiconthesubjectofsoundandvibration.Notableamong themanycontributionsofRayleighisthemethodoffindingthefundamentalfrequency
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the following passage: “Before the universe was created, there was only the Almighty and His name.” Observe how clearly the author states that all these appellatives employed as names of God came into existence after the Creation. This is true; for they all refer to actions manifested in the Universe. If, however, you consider His essence as separate and as abstracted from all [91]actions, you will not describe it by an appellative, but by a proper noun, which exclusively indicates that essence. Every other name of God is a derivative, only the Tetragrammaton is a real nomenproprium, and must not be considered from any other point of view. You must beware of sharing the error of those who write amulets (kameot). Whatever you hear from them, or read in their works, especially in reference to the names which they form by combination, is utterly senseless; they call these combinations shemot(names) and believe that their pronunciation demands sanctification and purification, and that by using them they are enabled to work miracles. Rational persons ought not to listen to such men, nor in any way believe their assertions. No other name is called shemha-meforashexcept this Tetragrammaton, which is written, but is not pronounced according to its letters. The words, “Thus shall ye bless the children of Israel” (Num. vi. 23) are interpreted in Siphri as follows: “ ‘Thus,’ in the holy language; again ‘thus,’ with the Shemha-meforash.” The following remark is also found there: “In the sanctuary [the name of God is pronounced] as it is spelt, but elsewhere by its substitutes.” In the Talmud, the following passage occurs: “ ‘Thus,’ i.e., with the shem ha-meforash.—You say [that the priests, when blessing the people, had to pronounce] the shemha-meforash; this was perhaps not the case, and they may have used other names instead.—We infer it from the words: ‘And they shall put My name’ (Num. vi. 27), i.e., My name, which is peculiar to Me.” It has thus been shown that the shemha-meforash(the proper name of God) is the Tetragrammaton, and that this is the only name which indicates nothing but His
essence, and therefore our Sages in referring to this sacred term said “ ‘Myname’ means the one which is peculiar to Me alone.”
In the next chapter I will explain the circumstances which brought men to a belief in the power of Shemot(names of God); I will point out the main subject of discussion, and lay open to you its mystery, and then not any doubt will be left in your mind, unless you prefer to be misguided.
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CHAPTER LXII
We were commanded that, in the sacerdotal blessing, the name of the Lord should be pronounced as it is written in the form of the Tetragrammaton, the shemha-meforash. It was not known to every one how the name was to be pronounced, what vowels were to be given to each consonant, and whether some of the letters capable of reduplication should receive a dagesh. Wise men successively transmitted the pronunciation of the name; it occurred only once in seven years that the pronunciation was communicated to a distinguished disciple. I must, however, add that the statement, “The wise men communicated the Tetragrammaton to their children and their disciples once in seven years,” does not only refer to the pronunciation but also to its meaning, because of which the Tetragrammaton was made a nomenpropriumof God, and which includes certain metaphysical principles.
Our Sages knew in addition a name of God which consisted of twelve letters, inferior in sanctity to the Tetragrammaton. I believe that this was not a single noun, but consisted of two or three words, the sum of their letters being twelve, and that these words were used by our Sages as a substitute [92]for the Tetragrammaton, whenever they met with it in the course of their reading the Scriptures, in the same manner as we at present substitute for it aleph, daleth, etc. [i.e., Adonay, “the Lord”]. There is no doubt that this name also, consisting of twelve letters, was in this sense more distinctive than the name Adonay: it was never withheld from any of the students; whoever wished to learn it, had the opportunity given to him without any reserve: not so the Tetragrammaton; those who knew it did not communicate it except to a son or a disciple, once in seven years. When, however, unprincipled men had become
acquainted with that name which consists of twelve letters and in consequence had become corrupt in faith—as is sometimes the case when persons with imperfect knowledge become aware that a thing is not such as they had imagined—the Sages concealed also that name, and only communicated it to the worthiest among the priests, that they should pronounce it when they blessed the people in the Temple; for the Tetragrammaton was then no longer uttered in the sanctuary on account of the corruption of the people. There is a tradition, that with the death of Simeon the Just, his brother priests discontinued the pronunciation of the Tetragrammaton in the blessing; they used, instead, this name of twelve letters. It is further stated, that at first the name of twelve letters was communicated to every man; but when the number of impious men increased it was only entrusted to the worthiest among the priests, whose voice, in pronouncing it, was drowned amid the singing of their brother priests. Rabbi Tarphon said, “Once I followed my grandfather to the daïs [where the blessing was pronounced]; I inclined my ear to listen to a priest [who pronounced the name], and noticed that his voice was drowned amid the singing of his brother priests.”
There was also a name of forty-two letters known among them. Every intelligent person knows that one word of forty-two letters is impossible. But it was a phrase of several words which had together forty-two letters. There is no doubt that the words had such a meaning as to convey a correct notion of the essence of God, in the way we have stated. This phrase of so many letters is called a name because, like other proper nouns, they represent one single object, and several words have been employed in order to explain more clearly the idea which the name represents; for an idea can more easily be comprehended if expressed in many words. Mark this and observe now that the instruction in regard to the names of God extended to the signification of each of those names, and did not confine itself to the pronunciation of the single letters which, in
themselves, are destitute of an idea. Shemha-meforashapplied neither to the name of forty-two letters nor to that of twelve, but only to the Tetragrammaton, the proper name of God, as we have explained. Those two names must have included some metaphysical ideas. It can be proved that one of them conveyed profound knowledge, from the following rule laid down by our Sages: “The name of forty-two letters is exceedingly holy; it can only be entrusted to him who is modest, in the midway of life, not easily provoked to anger, temperate, gentle, and who speaks kindly to his fellow men. He who understands it, is cautious with it, and keeps it in purity, is loved above and is liked here below; he is respected by his fellow men; his learning remaineth with him, and he enjoys both this world and the world to come.” So far in the Talmud. [93]How grievously has this passage been misunderstood! Many believe that the forty-two letters are merely to be pronounced mechanically; that by knowledge of these, without any further interpretation, they can attain to these exalted ends, although it is stated that he who desires to obtain a knowledge of that name must be trained in the virtues named before, and go through all the great preparations which are mentioned in that passage. On the contrary, it is evident that all this preparation aims at a knowledge of Metaphysics, and includes ideas which constitute the “secrets of the Law,” as we have explained (chap. xxxv.). In works on Metaphysics it has been shown that such knowledge, i.e., the perception of the active intellect, can never be forgotten; and this is meant by the phrase “his learning remaineth with him.”
When bad and foolish men were reading such passages, they considered them to be a support of their false pretensions and of their assertion that they could, by means of an arbitrary combination of letters, form a shem(“a name”) which would act and operate miraculously when written or spoken in a certain particular way. Such fictions, originally invented by foolish men, were in the course
of time committed to writing, and came into the hands of good but weak-minded and ignorant persons who were unable to discriminate between truth and falsehood, and made a secret of these shemot (names). When after the death of such persons those writings were discovered among their papers, it was believed that they contained truths; for, “the simple believeth every word” (Prov. xiv. 15).
We have already gone too far away from our interesting subject and recondite inquiry, endeavouring to refute a perverse notion, the absurdity of which every one must perceive who gives a thought to the subject. We have, however, been compelled to mention it, in treating of the divine names, their meanings, and the opinions commonly held concerning them. We shall now return to our theme. Having shown that all names of God, with the exception of the Tetragrammaton (Shemha-meforash), are appellatives, we must now, in a separate chapter, speak on the phrase EhyehasherEhyeh, (Exod. iii. 14), because it is connected with the difficult subject under discussion, namely, the inadmissibility of divine attributes.
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CHAPTER LXIII
Before approaching the subject of this chapter, we will first consider the words of Moses, “And they shall say unto me, What is His name? what shall I say unto them?” (Exod. iii. 13). How far was this question, anticipated by Moses, appropriate, and how far was he justified in seeking to be prepared with the answer? Moses was correct in declaring, “But, behold, they will not believe me, for they will say, The Lord hath not appeared unto thee” (ib. iv. 1); for any man claiming the authority of a prophet must expect to meet with such an objection so long as he has not given a proof of his mission. Again, if the question, as appears at first sight, referred only to the name, as a mere utterance of the lips, the following dilemma would present itself: either the Israelites knew the name, or they had never heard it; if the name was known to them, they would perceive in it no argument in favour of the mission of Moses, his knowledge and their knowledge of the divine name [94]being the same. If, on the other hand, they had never heard it mentioned, and if the knowledge of it was to prove the mission of Moses, what evidence would they have that this was really the name of God? Moreover, after God had made known that name to Moses, and had told him, “Go and gather the elders of Israel, … and they shall hearken to thy voice” (ib. xvi. 18), he replied, “Behold, they will not believe me nor hearken unto my voice,” although God had told him, “And they will hearken to thy voice”; whereupon God answered, “What is that in thine hand?” and he said, “A rod” (ib. iv. 2). In order to obviate this dilemma, you must understand what I am about to tell you. You know how widespread were in those days the opinions of the Sabeans; all men, except a few individuals, were idolaters, that is to say, they believed in spirits, in man’s power to direct the influences of the heavenly bodies, and in the effect of talismans. Any one who
in those days laid claim to authority, based it either, like Abraham, on the fact that, by reasoning and by proof he had been convinced of the existence of a Being who rules the whole Universe, or that some spiritual power was conferred upon him by a star, by an angel, or by a similar agency; but no one could establish his claim on prophecy, that is to say, on the fact that God had spoken to him, or had entrusted a mission to him; before the days of Moses no such assertion had ever been made. You must not be misled by the statements that God spoke to the Patriarchs, or that He had appeared to them. For you do not find any mention of a prophecy which appealed to others, or which directed them. Abraham, Isaac, or Jacob, or any other person before them did not tell the people, “God said unto me, you shall do this thing, or you shall not do that thing,” or “God has sent me to you.” Far from it! for God spoke to them on nothing but of what especially concerned them, i.e., He communicated to them things relating to their perfection, directed them in what they should do, and foretold them what the condition of their descendants would be; nothing beyond this. They guided their fellow-men by means of argument and instruction, as is implied, according to the interpretation generally received amongst us, in the words “and the souls that they had gotten in Haran” (Gen. xii. 5). When God appeared to our Teacher Moses, and commanded him to address the people and to bring them the message, Moses replied that he might first be asked to prove the existence of God in the Universe, and that only after doing so he would be able to announce to them that God had sent him. For all men, with few exceptions, were ignorant of the existence of God; their highest thoughts did not extend beyond the heavenly sphere, its forms or its influences. They could not yet emancipate themselves from sensation, and had not yet attained to any intellectual perfection. Then God taught Moses how to teach them, and how to establish amongst them the belief in the existence of Himself, namely, by saying EhyehasherEhyeh, a name derived from the verb hayahin
the sense of “existing,” for the verb hayahdenotes “to be,” and in Hebrew no difference is made between the verbs “to be” and “to exist.” The principal point in this phrase is that the same word which denotes “existence,” is repeated as an attribute. The word asher, “that,” corresponds to the Arabic illadiand illati, and is an incomplete noun that must be completed by another noun; it may be considered as the subject of the predicate which follows. The first noun which is to be described [95]is ehyeh; the second, by which the first is described, is likewise ehyeh, the identical word, as if to show that the object which is to be described and the attribute by which it is described are in this case necessarily identical. This is, therefore, the expression of the idea that God exists, but not in the ordinary sense of the term; or, in other words, He is “the existing Being which is the existing Being,” that is to say, the Being whose existence is absolute. The proof which he was to give consisted in demonstrating that there is a Being of absolute existence, that has never been and never will be without existence. This I will clearly prove (II. Introd. Prop. 20 and chap. i.).
God thus showed Moses the proofs by which His existence would be firmly established among the wise men of His people. Therefore the explanation of the name is followed by the words, “Go, gather the elders of Israel,” and by the assurance that the elders would understand what God had shown to him, and would accept it, as is stated in the words, “And they will hearken to thy voice.” Then Moses replied as follows: They will accept the doctrine that God exists convinced by these intelligible proofs. But, said Moses, by what means shall I be able to show that this existing God has sent me? Thereupon God gave him the sign. We have thus shown that the question, “What is His name?” means “Who is that Being, which according to thy belief has sent thee?” The sentence, “What is his name” (instead of, Who is He), has here been used as a tribute of praise and homage, as though it had been said, Nobody can be
ignorant of Thy essence and of Thy real existence; if, nevertheless, I ask what is Thy name, I mean, What idea is to be expressed by the name? (Moses considered it inappropriate to say to God that any person was ignorant of God’s existence, and therefore described the Israelites as ignorant of God’s name, not as ignorant of Him who was called by that name.)—The name Jahlikewise implies eternal existence. Shadday, however, is derived from day, “enough”; comp. “for the stuff they had was sufficient” (dayyam, Exod. xxxvi. 7); the shinis equal to asher, “which,” as in she-kebar, “which already” (Eccles. ii. 16). The name Shadday, therefore, signifies “he who is sufficient”; that is to say, He does not require any other being for effecting the existence of what He created, or its conservation: His existence is sufficient for that. In a similar manner the name ḥasin implies “strength”; comp. “he was strong (ḥason) as the oaks” (Amos ii. 9). The same is the case with “rock,” which is a homonym, as we have explained (chap. xvi.). It is, therefore, clear that all these names of God are appellatives, or are applied to God by way of homonymy, like ẓurand others, the only exception being the tetragrammaton, the Shemha-meforash(the nomenpropriumof God), which is not an appellative; it does not denote any attribute of God, nor does it imply anything except His existence. Absolute existence includes the idea of eternity, i.e., the necessity of existence. Note well the result at which we have arrived in this chapter.
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