The Method of Normal Forms

342 Pages·2011·1.558 MB·English

by  Ali Hasan Nayfeh

AliHasanNayfeh
TheMethodofNormalForms
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Ali Hasan Nayfeh
The Method of Normal Forms
Second,UpdatedandEnlargedEdition
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TomyyoungestsonNader

VII
Contents
Preface XI
Introduction 1
1 SDOFAutonomousSystems 7
1.1 Introduction 7
1.2 DuffingEquation 9
1.3 RayleighEquation 13
1.4 Duffing–Rayleigh–vanderPolEquation 15
1.5 AnOscillatorwithQuadraticandCubicNonlinearities 17
1.5.1 SuccessiveTransformations 17
1.5.2 TheMethodofMultipleScales 19
1.5.3 ASingleTransformation 21
1.6 AGeneralSystemwithQuadraticandCubicNonlinearities 22
1.7 ThevanderPolOscillator 24
1.7.1 TheMethodofNormalForms 25
1.7.2 TheMethodofMultipleScales 26
1.8 Exercises 27
2 SystemsofFirst-OrderEquations 31
2.1 Introduction 31
2.2 ATwo-DimensionalSystemwithDiagonalLinearPart 34
2.3 ATwo-DimensionalSystemwithaNonsemisimpleLinearForm 39
2.4 Ann-DimensionalSystemwithDiagonalLinearPart 40
2.5 ATwo-DimensionalSystemwithPurelyImaginaryEigenvalues 42
2.5.1 TheMethodofNormalForms 43
2.5.2 TheMethodofMultipleScales 47
2.6 ATwo-DimensionalSystemwithZeroEigenvalues 48
2.7 AThree-DimensionalSystemwithZero
andTwoPurelyImaginaryEigenvalues 52
2.8 TheMathieuEquation 54
2.9 Exercises 57
VIII Contents
3 Maps 61
3.1 LinearMaps 61
3.1.1 CaseofDistinctEigenvalues 62
3.1.2 CaseofRepeatedEigenvalues 64
3.2 NonlinearMaps 66
3.3 Center-ManifoldReduction 72
3.4 LocalBifurcations 76
3.4.1 FoldorTangentorSaddle-NodeBifurcation 76
3.4.2 TranscriticalBifurcation 79
3.4.3 PitchforkBifurcation 80
3.4.4 FliporPeriod-DoublingBifurcation 81
3.4.5 HopforNeimark–SackerBifurcation 85
3.5 Exercises 91
4 BifurcationsofContinuousSystems 97
4.1 LinearSystems 97
4.1.1 CaseofDistinctEigenvalues 98
4.1.2 CaseofRepeatedEigenvalues 99
4.2 FixedPointsofNonlinearSystems 100
4.2.1 StabilityofFixedPoints 100
4.2.2 ClassificationofFixedPoints 101
4.2.3 Hartman–GrobmanandShoshitaishviliTheorems 102
4.3 Center-ManifoldReduction 103
4.4 LocalBifurcationsofFixedPoints 107
4.4.1 Saddle-NodeBifurcation 108
4.4.2 NonbifurcationPoint 110
4.4.3 TranscriticalBifurcation 111
4.4.4 PitchforkBifurcation 113
4.4.5 HopfBifurcations 114
4.5 NormalFormsofStaticBifurcations 117
4.5.1 TheMethodofMultipleScales 117
4.5.2 Center-ManifoldReduction 126
4.5.3 AProjectionMethod 132
4.6 NormalFormofHopfBifurcation 137
4.6.1 TheMethodofMultipleScales 138
4.6.2 Center-ManifoldReduction 141
4.6.3 ProjectionMethod 144
4.7 Exercises 146
5 ForcedOscillationsoftheDuffingOscillator 161
5.1 PrimaryResonance 161
5.2 SubharmonicResonanceofOrderOne-Third 164
5.3 SuperharmonicResonanceofOrderThree 167
5.4 AnAlternateApproach 169
5.4.1 SubharmonicCase 171
5.4.2 SuperharmonicCase 172
Contents IX
5.5 Exercises 172
6 ForcedOscillationsofSDOFSystems 175
6.1 Introduction 175
6.2 PrimaryResonance 176
6.3 SubharmonicResonanceofOrderOne-Half 178
6.4 SuperharmonicResonanceofOrderTwo 180
6.5 SubharmonicResonanceofOrderOne-Third 182
7 ParametricallyExcitedSystems 187
7.1 TheMathieuEquation 187
7.1.1 FundamentalParametricResonance 188
7.1.2 PrincipalParametricResonance 190
7.2 Multiple-Degree-of-FreedomSystems 191
7.2.1 TheCaseofΩ Nearω Cω 194
2 1
7.2.2 TheCaseofΩ Nearω (cid:1)ω 194
2 1
7.2.3 TheCaseofΩ Nearω Cω andω (cid:1)ω 194
2 1 3 2
7.2.4 TheCaseofΩ Near2ω andω Cω 195
3 2 1
7.3 LinearSystemsHavingRepeatedFrequencies 195
7.3.1 TheCaseofΩ Near2ω 198
1
7.3.2 TheCaseofΩ Nearω Cω 199
3 1
7.3.3 TheCaseofΩ Nearω (cid:1)ω 200
3 1
7.3.4 TheCaseofΩ Nearω 200
1
7.4 GyroscopicSystems 205
7.4.1 TheCaseofΩ Near2ω 208
1
7.4.2 TheCaseofΩ Nearω (cid:1)ω 208
2 1
7.5 ANonlinearSingle-Degree-of-FreedomSystem 208
7.5.1 TheCaseofΩ Awayfrom2ω 209
7.5.2 TheCaseofΩ Near2ω 211
7.6 Exercises 212
8 MDOFSystemswithQuadraticNonlinearities 217
8.1 NongyroscopicSystems 217
8.1.1 Two-to-OneAutoparametricResonance 220
8.1.2 CombinationAutoparametricResonance 222
8.1.3 SimultaneousTwo-to-OneAutoparametricResonances 223
8.1.4 PrimaryResonances 223
8.2 GyroscopicSystems 225
8.2.1 PrimaryResonances 226
8.2.2 SecondaryResonances 227
8.3 TwoLinearlyCoupledOscillators 229
8.4 Exercises 232
9 TDOFSystemswithCubicNonlinearities 235
9.1 NongyroscopicSystems 235
9.1.1 TheCaseofNoInternalResonances 236
9.1.2 Three-to-OneAutoparametricResonance 238