Introduction to finite element vibration analysis

518 Pages·2010·3.829 MB·English

by  M Petyt

This page intentionally left blank
INTRODUCTION TO FINITE ELEMENT VIBRATION
ANALYSIS, SECOND EDITION
There are many books on finite element methods but few give more
thanabriefdescriptionoftheirapplicationtostructuralvibrationanal-
ysis. This book presents an introduction to the mathematical basis
of finite element analysis as applied to vibrating systems. Finite ele-
ment analysis is a technique that is very important in modelling the
responseofstructurestodynamicloads.Althoughthisbookassumes
nopreviousknowledgeoffiniteelementmethods,thosewhodohave
knowledge will still find the book to be useful. It can be utilised
by aeronautical, civil, mechanical and structural engineers as well as
navalarchitects.Thissecondeditionincludesinformationonthemany
developments that have taken place over the last 20 years. Existing
chapters have been expanded,where necessary, and three new chap-
tershavebeenincludedthatdiscussthevibrationofshellsandmulti-
layeredelementsandprovideanintroductiontothehierarchicalfinite
elementmethod.
MauricePetytisanEmeritusProfessorofStructuralDynamics,Insti-
tute of Sound and Vibration Research, University of Southampton.
He has also held appointments as a Research Professor at George
WashingtonUniversityandVisitingProfessorattheUniversityd’Aix-
MarseilleII,France,andtheNationalUniversityofSingapore.Heisa
ChartedMathematician,aFellowoftheInstituteofMathematicsand
Its Applications, a Fellow of the International Institute of Acoustics
and Vibration, and a Fellow of the Institute of Acoustics. Formerly
European Editor and Editor-in-Chief, he is now Editor Emeritus of
theJournalofSoundandVibration.

Introduction to Finite Element Vibration
Analysis
Second Edition
Maurice Petyt
UniversityofSouthampton
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
Sa˜oPaulo,Delhi,Dubai,Tokyo,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521191609
(cid:2)C MauricePetyt1990,2010
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firsteditionpublished1990
Secondeditionpublished2010
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationdata
Petyt,M.
Introductiontofiniteelementvibrationanalysis/MauricePetyt.–2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-19160-9
1.Vibration. 2.Finiteelementmethod. I.Title.
TA356.P47 2010
624.1(cid:3)76–dc22 2010029494
ISBN978-0-521-19160-9Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pagexi
Notation xv
1 FormulationoftheEquationsofMotion . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 DynamicEquilibrium 1
1.2 PrincipleofVirtualDisplacements 3
1.3 Hamilton’sPrinciple 4
1.4 Lagrange’sEquations 8
1.5 EquationsofMotionforaSystemwithConstraints 12
Problems 15
2 ElementEnergyFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 AxialElement 20
2.2 TorqueElement 21
2.3 BeamBendingElement 23
2.4 DeepBeamBendingElement 25
2.5 MembraneElement 26
2.6 ThinPlateBendingElement 28
2.7 ThickPlateBendingElement 30
2.8 Three-DimensionalSolid 31
2.9 AxisymmetricSolid 33
2.10TheDissipationFunction 35
2.11EquationsofMotionandBoundaryConditions 36
Problems 40
3 IntroductiontotheFiniteElementDisplacementMethod . . . . . . . . . .45
3.1 Rayleigh–RitzMethod 45
3.2 FiniteElementDisplacementMethod 53
3.3 AxialVibrationofRods 56
3.4 TorsionalVibrationofShafts 70
3.5 BendingVibrationofBeams 72
3.6 VibrationofPlaneFrameworks 77
v
vi Contents
3.7 VibrationofThree-DimensionalFrameworks 84
3.8 TechniquesforIncreasingtheAccuracyofElements 92
3.9 ShearDeformationandRotaryInertiaEffects 95
3.10NumericalIntegration 101
3.11OtherConsiderationsforBeams 112
Problems 115
4 In-planeVibrationofPlates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.1 LinearTriangularElement 120
4.2 LinearRectangularElement 127
4.3 LinearQuadrilateralElement 132
4.4 AreaCoordinatesforTriangles 137
4.5 LinearTriangleinAreaCoordinates 138
4.6 IncreasingtheAccuracyofElements 139
Problems 144
5 VibrationofSolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.1 AxisymmetricSolids 148
5.2 AppliedLoading 149
5.3 Displacements 152
5.4 ReducedEnergyExpressions 152
5.5 LinearTriangularElement 153
5.6 CoreElements 160
5.7 ArbitraryShapedSolids 163
5.8 RectangularHexahedron 165
5.9 IsoparametricHexahedron 170
5.10RightPentahedron 174
5.11VolumeCoordinatesforTetrahedra 178
5.12TetrahedronElement 180
5.13IncreasingtheAccuracyofElements 182
Problems 190
6 FlexuralVibrationofPlates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.1 ThinRectangularElement(non-conforming) 193
6.2 ThinRectangularElement(conforming) 204
6.3 ThickRectangularElement 209
6.4 ThinTriangularElement(non-conforming) 215
6.5 ThinTriangularElement(conforming) 222
6.5.1 CartesianCoordinates 222
6.5.2 AreaCoordinates 228
6.6 ThickTriangularElement 234
6.7 OtherPlateBendingElements 237
Problems 245
7 VibrationofStiffenedPlatesandFoldedPlateStructures . . . . . . . . . 248
7.1 StiffenedPlatesI 248
Contents vii
7.2 StiffenedPlatesII 252
7.3 FoldedPlatesI 257
7.4 FoldedPlatesII 259
7.5 FoldedPlatesIII 262
Problems 265
8 VibrationofShells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.1 ThinShellElements 266
8.2 ThickShellElements 268
8.2.1 MiddleSurfaceShellElement 269
8.3 ThickAxisymmetricShellElements 277
8.3.1 MiddleSurfaceAxisymmetricShellElement 277
Problems 283
9 VibrationofLaminatedPlatesandShells . . . . . . . . . . . . . . . . . . . . .284
9.1 LaminatedPlateElements 285
9.1.1 ClassicalLaminatedPlateTheory 285
9.1.2 First-OrderShearDeformationPlateTheory 288
9.1.3 Third-OrderShearDeformationPlateTheory 290
9.2 LaminatedShellElements 295
9.3 SandwichPlateandShellElements 295
10 HierarchicalFiniteElementMethod . . . . . . . . . . . . . . . . . . . . . . . . 297
10.1PolynomialFunctions 297
10.1.1AxialVibrationofRods 298
10.1.2BendingVibrationofBeams 300
10.1.3FlexuralVibrationofPlates 303
10.1.4VibrationofShells 306
10.2TrigonometricFunctions 307
10.2.1AxialVibrationofRods 307
10.2.2BendingVibrationofBeams 309
10.2.3In-planeVibrationofPlates 310
10.2.4FlexuralVibrationofPlates 311
10.2.5VibrationofShells 314
11 AnalysisofFreeVibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
11.1SomePreliminaries 316
11.1.1OrthogonalityofEigenvectors 322
11.1.2TransformationtoStandardForm 322
11.2SturmSequences 326
11.3OrthogonalTransformationofaMatrix 334
11.4EigenproblemSolutionMethods 334
11.5ReducingtheNumberofDegreesofFreedom 336
11.5.1MakingUseofSymmetry 336
11.5.2RotationallyPeriodicStructures 338
viii Contents
11.5.3EliminationofUnwantedDegreesofFreedom 343
11.5.4ComponentModeSynthesis 349
11.5.4.1 FixedInterfaceMethod 349
11.5.4.2 FreeInterfaceMethod 352
Problems 355
12 ForcedResponseI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
12.1ModalAnalysis 357
12.2RepresentationofDamping 358
12.2.1StructuralDamping 358
12.2.2ViscousDamping 359
12.3HarmonicResponse 361
12.3.1ModalAnalysis 361
12.3.2DirectAnalysis 370
12.4ResponsetoPeriodicExcitation 377
12.5TransientResponse 381
12.5.1ModalAnalysis 381
12.5.1.1 CentralDifferenceMethod 384
12.5.1.2 TheHouboltMethod 389
12.5.1.3 TheNewmarkMethod 394
12.5.1.4 TheWilsonθ Method 400
12.5.2DirectAnalysis 402
12.5.2.1 CentralDifferenceMethod 403
12.5.2.2 TheHouboltMethod 408
12.5.2.3 TheNewmarkMethod 409
12.5.2.4 TheWilsonθ Method 409
12.5.3SelectingaTimeStep 410
12.5.4AdditionalMethods 411
Problems 411
13 ForcedResponseII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
13.1ResponsetoRandomExcitation 413
13.1.1RepresentationoftheExcitation 413
13.1.2ResponseofaSingleDegreeofFreedomSystem 423
13.1.3DirectResponseofaMulti-DegreeofFreedomSystem 426
13.1.4ModalResponseofaMulti-DegreeofFreedomSystem 430
13.2TruncationoftheModalSolution 431
13.2.1ModeAccelerationMethod 435
13.2.2ResidualFlexibility 437
13.3RitzVectorAnalysis 438
13.4ResponsetoImposedDisplacements 440
13.4.1DirectResponse 440
13.4.2ModalResponse 442
13.5ReducingtheNumberofDegreesofFreedom 443
13.5.1MakingUseofSymmetry 443
13.5.2RotationallyPeriodicStructures 444