Wave Propagation in Structures (Mechanical Engineering Series)

432 Pages·2020·18.056 MB·English

by  James F. Doyle

Mechanical Engineering Series
James F. Doyle
Wave Propagation
in Structures
Third Edition
Mechanical Engineering Series
SeriesEditor
FrancisA.Kulacki
DepartmentofMechanicalEngineering
UniversityofMinnesota
Minneapolis,MN,USA
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James F. Doyle
Wave Propagation
in Structures
Third Edition
JamesF.Doyle
SchoolofAeronautics&Astronautics
PurdueUniversity
WestLafayette,IN,USA
ISSN0941-5122 ISSN2192-063X (electronic)
MechanicalEngineeringSeries
ISBN978-3-030-59678-1 ISBN978-3-030-59679-8 (eBook)
https://doi.org/10.1007/978-3-030-59679-8
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Thisbookisdedicatedto
myfatherandmother,
PatrickandTeresaDoyle.
Thanksforthedreams.
HadItheheavens’embroideredcloths,
Enwroughtwithgoldenandsilverlight,
Theblueandthedimandthedarkcloths
Ofnightandlightandthehalflight,
Wouldspreadtheclothsunderyourfeet:
ButI,beingpoor,haveonlymydreams;
Ihavespreadmydreamsunderyourfeet;
Treadsoftlybecauseyoutreaduponmydreams.
—W.B.Yeats[1]
Preface
Thestudyofwavepropagationseemsveryremotetomanyengineers,eventothose
who are involved in structural dynamics. One of the reasons for this is that the
examples usually taught in school are either so simple as to be not applicable
to real world problems or so mathematically abstruse as to be intractable. This
book contains an approach, spectral analysis or frequency domain synthesis, that
I have found to be very effective in analyzing waves. What has struck me most
about this approach is how I can use the same analytical framework to examine
the experimental results as well as to manipulate the experimental data itself.
As an experimentalist, I had found it very frustrating having my analytical tools
incompatiblewithmyexperiments.Forexample,itisexperimentallyimpossibleto
generate a step function wave and yet that is the type of analytical solution often
available.
Spectral analysis is very encompassing—it touches on analysis, numerical
methods,andexperimentalmethods.Iwantthisbooktodojusticetoitsversatility,
somanysubjectsareintroduced.Asaresult,someareasmayseemalittlethin,butI
dohope,nonetheless,thatthebiggerpicture,theunity,comesacross.Furthermore,
spectralanalysisisnotsomuchasolutiontechniqueasitisadifferentinsightinto
the wave mechanics; consequently, in most of the examples an attempt is made to
maketheconnectionbetweenthefrequencydomainandtimedomains.
In writing the second edition, I strived to keep what was good about the
first edition—that is, the combination of experimental and analytical results—but
incorporated more recent developments and extensions at that time. The question
not fully articulated in the first edition is: What should be different about a book
on waves in structures? This is the question that guided my reorganization of the
material as well as the selection of new topics. It had become clearer to me that
theessenceofastructureisthecouplingofsystemsand,thisshouldbethecentral
theme of a book on waves in structures. There are two readily recognized forms
of coupling: mechanical coupling “at the ends” such as when two bars are joined
at an angle, and differential coupling as when two bars are connected uniformly
alongtheirlengthsbysprings.Bothcouplingsareintimatelyrelatedtoeachotheras
seenfromtheexampleofacurvedbeam:itcanbemodeledasacollectionofsmall
vii
viii Preface
straightsegmentsconnectedendtoend,ordirectlyintermsofcoupleddifferential
equations. The two approaches, ultimately, give the same results but, at the same
time, give quite different insights into the system behavior. The former leads to
richer system response functions and its ultimate form is in the spectral element
method. The latter leads to richer differential equations, which is manifested in
very interesting spectrum relations. The variety of examples were chosen so as to
illustrateandelaborateonthisdualaspectofcoupling.
Inwritingthisthirdedition,againIstrivedtokeepintactwhatisgoodaboutthe
secondeditionbutaddrecentdevelopments.Thetwomostimportantdevelopments
overthelasttwodecadesistheintroductionofengineeredmaterialsintheformof
metamaterialsandnanostructures.Theyaddanewleveltothemeaningofstructure.
The other development is the almost universal adoption of finite element (FE)
methodsasthe“goto”toolforsolvingstructuraldynamicsproblems.Iseespectral
analysisasatooltogiveinsightintodatabeitobtainedfromphysicalexperiment
or computer experiment. In this edition, some tools are developed to extend the
postprocessingofFEdatausingspectralanalysisideas.
Theorganizationofthechaptersismostlysimilartothatofthesecondedition,
buteachchapterisrevisedandupdatedwithsomenewexamplesandreferencesto
the literature. One completely new chapter has been added. Chapter 9 deals with
discreteanddiscretizedstructures.Spectralanalysismethodsareaverynaturalway
toanalyzethebehaviorofthesesystems.Examplesaretakenfromnanotechnology
andmoleculardynamics.
In recent years, desktop computers have become incredibly powerful and very
affordable. Thus, problems that would not be tackled in the past can be accom-
plishedveryquickly,andsolutionschemesrejectedinthepastarenowfeasible.For
example,inthefirstedition,thedoublesummationwasbarelyintroduced,butnow
it plays a central role in the dynamics of plate and shell structures. Similarly, the
eigenvibrationanalysisoflargestructureswaslimitedtothelowestmodesbutnow
canbeappliedtotheveryhigh-frequencymodesofrelevancetowavepropagation
in discrete structures. Therefore, not only is the use of a computer implicit in all
theexamples,thesolutionstrategiesandtechniquesarealsocomputeroriented.In
asimilarvein,Ihavetriedtosupplementeachchapterwithacollectionofpertinent
problemsplusspecificreferencesthatcanformthebasisforfurtherstudy.
A book like this is impossible to complete without the help of many people,
but it is equally impossible to properly acknowledge all of them individually.
However, I would like to single out helpers on the first and second editions:
Brian Bilodeau, Albert Danial, Sudhir Kamle, Lance Kannal, Matt Ledington,
Mike Martin, Steve Rizzi, Gopal Srinivasan, and Hong Zhang. Thank you guys.
Theerrorsandinaccuraciesinalleditionshavebeenpurelymyowndoing.
WestLafayette,IN,USA JamesF.Doyle
August,2020
Contents
Introduction ...................................................................... 1
References..................................................................... 6
1 SpectralAnalysisofWaveMotion ......................................... 7
1.1 FourierTransforms..................................................... 7
1.1.1 ContinuousFourierTransforms............................... 8
1.1.2 DiscreteFourierTransform ................................... 12
1.1.3 FastFourierTransformAlgorithm............................ 18
1.1.4 SpaceDistributionsUsingFourierSeries .................... 20
1.2 ApplicationsUsingtheFFTAlgorithm ............................... 22
1.2.1 ExplorationsoftheProperties ................................ 23
1.2.2 ExperimentalAspectsofWaveSignals....................... 26
1.3 SpectralAnalysisofWaveMotion .................................... 31
1.3.1 GeneralFunctionsofSpace-TimeandSpectrumRelations. 31
1.3.2 SomeWaveExamples......................................... 34
1.3.3 GroupSpeed................................................... 37
1.3.4 SummaryofWaveRelations.................................. 39
1.4 PropagatingandReconstructingWaves............................... 40
1.4.1 BasicAlgorithm ............................................... 40
1.4.2 IntegrationofSignals.......................................... 44
FurtherResearch .............................................................. 46
References..................................................................... 47
2 LongitudinalWavesinRods................................................ 49
2.1 ElementaryRodModeling............................................. 49
2.1.1 EquationofMotionandSpectralAnalysis................... 50
2.1.2 BasicSolutionforWavesinRods ............................ 53
2.2 DissipationinRods..................................................... 56
2.2.1 DistributedConstraint......................................... 56
2.2.2 ViscoelasticRod............................................... 59
2.3 CoupledThermoelasticWaves......................................... 61
2.3.1 GoverningEquations.......................................... 62
ix
x Contents
2.3.2 TheSpectrumRelation........................................ 64
2.3.3 BlastLoadingofaRod........................................ 65
2.4 ReflectionandTransmissionofWaves................................ 68
2.4.1 ReflectionfromanElasticBoundary......................... 69
2.4.2 ReflectionfromanOscillator ................................. 71
2.4.3 ConcentratedMassConnectingTwoRods................... 72
2.4.4 InteractionsataDistributedElasticJoint .................... 74
2.5 DistributedLoading.................................................... 76
2.5.1 PeriodicallyExtendedLoadModel........................... 76
2.5.2 ConnectedWaveguideSolution............................... 79
FurtherResearch .............................................................. 80
References..................................................................... 82
3 FlexuralWavesinBeams.................................................... 85
3.1 Bernoulli–EulerBeamModeling...................................... 85
3.1.1 EquationsofMotionandSpectralAnalysis.................. 86
3.1.2 BasicSolutionforWavesinBeams .......................... 89
3.1.3 BeamwithAxialLoad ........................................ 93
3.2 Bernoulli–EulerBeamwithConstraints............................... 94
3.2.1 BeamonanElasticFoundation............................... 95
3.2.2 CoupledBeamStructure ...................................... 98
3.3 ReflectionandTransmissionofFlexuralWaves...................... 104
3.3.1 ReflectionsfromSimpleEnds ................................ 105
3.3.2 ReflectionsandTransmissionsataGeneralJoint............ 107
3.4 CurvedBeams.......................................................... 112
3.4.1 DeformationofCurvedBeams ............................... 112
3.4.2 SpectrumRelation............................................. 115
3.4.3 ImpactofaCurvedBeam..................................... 119
FurtherResearch .............................................................. 121
References..................................................................... 122
4 HigherOrderWaveguideModels.......................................... 123
4.1 WavesinInfiniteandSemi-InfiniteMedia............................ 123
4.1.1 Navier’sEquationsandHelmholtzPotentials................ 124
4.1.2 ConstructingPotentialsAppropriateforBoundaries ........ 127
4.1.3 ForcedResponseofaSemi-InfinitePlane.................... 132
4.1.4 Free-EdgeWaves:RayleighSurfaceWaves.................. 135
4.2 WavesinDoublyBoundedMedia..................................... 137
4.2.1 ForcedResponses.............................................. 137
4.2.2 SpectrumRelationsforFree-WaveResponses............... 141
4.2.3 DiscussionofLambWavesinStructuralWaveguides....... 145
4.3 VariationalFormulationofDynamicEquilibrium.................... 146
4.3.1 WorkandStrainEnergyinaGeneralBody.................. 146
4.3.2 VirtualWorkandHamilton’sPrinciple....................... 148
4.3.3 IllustrativeApplicationoftheRitzSemi-directMethod..... 152