Table Of ContentTrimSize:170mmx244mm Blevins ffirs.tex V3-09/29/2015 5:24P.M. Pagei
FORMULAS FOR
DYNAMICS, ACOUSTICS
AND VIBRATION
TrimSize:170mmx244mm Blevins ffirs.tex V3-09/29/2015 5:24P.M. Pageiii
FORMULAS FOR
DYNAMICS, ACOUSTICS
AND VIBRATION
RobertD. Blevins
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Thiseditionfirstpublished2016
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LibraryofCongressCataloging-in-PublicationData
Blevins,RobertD.
Formulasfordynamics,acousticsandvibration/RobertD.Blevins.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-119-03811-5(cloth)
1.Engineeringmathematics–Formulae.2.Dynamics–Mathematics.3.Vibration–Mathematical
models.I.Title.
TA332.B592015
620.001′51–dc23
2015015479
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
Setin10/12ptTimesLTStdbySPiGlobal,Chennai,India
1 2016
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Contents
Preface xi
1 Definitions,Units,andGeometricProperties 1
1.1 Definitions 1
1.2 Symbols 8
1.3 Units 11
1.4 MotionontheSurfaceoftheEarth 18
1.5 GeometricPropertiesofPlaneAreas 19
1.6 GeometricPropertiesofRigidBodies 30
1.7 GeometricPropertiesDefinedbyVectors 40
References 41
2 DynamicsofParticlesandBodies 43
2.1 KinematicsandCoordinateTransformations 43
2.2 Newton’sLawofParticleDynamics 50
2.2.1 ConstantMassSystems 50
2.2.2 VariableMassSystems 57
2.2.3 ParticleTrajectories 58
2.2.4 WorkandEnergy 63
2.2.5 Impulse 65
2.2.6 Armor 68
2.2.7 GravitationandOrbits 71
2.3 RigidBodyRotation 73
2.3.1 RigidBodyRotationTheory 73
2.3.2 Single-AxisRotation 73
2.3.3 Multiple-AxisRotation 84
2.3.4 GyroscopicEffects 85
References 87
3 NaturalFrequencyofSpring–MassSystems,Pendulums,Strings,and
Membranes 89
3.1 HarmonicMotion 89
3.2 SpringConstants 91
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3.3 NaturalFrequenciesofSpring–MassSystems 99
3.3.1 Single-Degree-of-Freedom 99
3.3.2 Two-Degree-of-FreedomSystem 113
3.4 ModelingDiscreteSystemswithSpringsandMasses 117
3.4.1 SpringswithMass 117
3.4.2 Bellows 118
3.5 PendulumNaturalFrequencies 119
3.5.1 MassPropertiesfromFrequencyMeasurement 120
3.6 TensionedStrings,Cables,andChainNaturalFrequencies 121
3.6.1 EquationofMotion 121
3.6.2 CableSag 123
3.7 MembraneNaturalFrequencies 126
3.7.1 FlatMembranes 126
3.7.2 CurvedMembranes 131
References 132
4 NaturalFrequencyofBeams 134
4.1 BeamBendingTheory 134
4.1.1 Stress,Strain,andDeformation 134
4.1.2 SandwichBeams 136
4.1.3 BeamEquationofMotion 137
4.1.4 BoundaryConditionsandModalSolution 137
4.1.5 BeamsonElasticFoundations 141
4.1.6 SimplificationforTubes 141
4.2 NaturalFrequenciesandModeShapesofSingle-SpanandMultiple-Span
Beams 142
4.2.1 Single-SpanBeams 142
4.2.2 Orthogonality,Normalization,andMaximumValues 150
4.2.3 BeamsStress 150
4.2.4 Two-SpanBeams 151
4.2.5 MultispanBeams 151
4.3 AxiallyLoadedBeamNaturalFrequency 158
4.3.1 UniformAxialLoad 158
4.3.2 LinearlyVaryingAxialLoad 159
4.4 BeamswithMasses,TaperedBeams,BeamswithSpringSupports,and
ShearBeams 162
4.4.1 BeamswithMasses 162
4.4.2 TaperedandSteppedBeams 162
4.4.3 Spring-SupportedBeams 167
4.4.4 ShearBeams 167
4.4.5 EffectofShearingForceontheDeflectionsofBeams 170
4.4.6 RotaryInertia 170
4.4.7 MultistoryBuildings 174
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Contents vii
4.5 TorsionalandLongitudinalBeamNaturalFrequencies 176
4.5.1 LongitudinalVibrationofBeamsandSprings 176
4.5.2 TorsionalVibrationofBeamsandShafts 179
4.5.3 CircularCrossSection 179
4.5.4 NoncircularCrossSections 182
4.6 WavePropagationinBeams 183
4.7 CurvedBeams,Rings,andFrames 184
4.7.1 CompleteRings 184
4.7.2 StressandStrainofArcs 189
4.7.3 SupportedRingsandHelices 190
4.7.4 CircularArcs,Arches,andBends 190
4.7.5 LowestFrequencyIn-PlaneNaturalFrequencyofanArc 196
4.7.6 ShallowArc 197
4.7.7 PortalFrames 198
References 199
5 NaturalFrequencyofPlatesandShells 203
5.1 PlateFlexureTheory 203
5.1.1 StressandStrain 203
5.1.2 BoundaryConditions 203
5.1.3 PlateEquationofMotion 205
5.1.4 SimplySupportedRectangularPlate 206
5.1.5 PlatesonElasticFoundations 207
5.1.6 SandwichPlates 207
5.1.7 ThickPlatesandShearDeformation 207
5.1.8 MembraneAnalogyandIn-PlaneLoads 208
5.1.9 Orthogonality 208
5.2 PlateNaturalFrequenciesandModeShapes 209
5.2.1 PlateNaturalFrequencies 209
5.2.2 CircularandAnnularPlates 209
5.2.3 SectorialandCircularOrthotropicPlates 214
5.2.4 RectangularPlates 214
5.2.5 Parallelogram,TriangularandPoint-SupportedPlates 215
5.2.6 RectangularOrthotropicPlatesandGrillages 215
5.2.7 StiffenedPlates 231
5.2.8 PerforatedPlates 232
5.3 CylindricalShells 234
5.3.1 DonnellThinShellTheory 235
5.3.2 NaturalFrequenciesofCylindricalShells 237
5.3.3 InfinitelyLongCylindricalShellModes(j=0) 241
5.3.4 SimplySupportedCylindricalShellswithoutAxialConstraint 243
5.3.5 CylindricalShellswithOtherBoundaryConditions 246
5.3.6 Free–FreeCylindricalShell 248
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5.3.7 CylindricallyCurvedPanels 249
5.3.8 EffectofMeanLoadonNaturalFrequencies 250
5.4 SphericalandConicalShells 250
5.4.1 SphericalShells 250
5.4.2 OpenShellsandChurchBells 252
5.4.3 ShallowSphericalShells 252
5.4.4 ConicalShells 254
References 254
6 AcousticsandFluids 260
6.1 SoundWavesandDecibels 260
6.1.1 SpeedofSound 260
6.1.2 AcousticWaveEquation 264
6.1.3 DecibelsandSoundPowerLevel 276
6.1.4 StandardsforMeasurement 277
6.1.5 AttenuationandTransmissionLoss(TL) 278
6.2 SoundPropagationinLargeSpaces 285
6.2.1 AcousticWavePropagation 285
6.2.2 SoundPressureonRigidWalls 288
6.2.3 MassLawforSoundTransmission 289
6.3 AcousticWavesinDuctsandRooms 289
6.3.1 AcousticWavesinDucts 289
6.3.2 MufflersandResonators 298
6.3.3 RoomAcoustics 302
6.4 AcousticNaturalFrequenciesandModeShapes 305
6.4.1 Structure-AcousticAnalogy 306
6.5 FreeSurfaceWavesandLiquidSloshing 310
6.6 ShipsandFloatingSystems 319
6.6.1 ShipNaturalFrequencies(1/Period) 319
6.7 AddedMassofStructureinFluids 321
6.7.1 AddedMassPotentialFlowTheory 328
6.7.2 AddedMass 329
6.7.3 AddedMassofPlatesandShells 330
References 331
FurtherReading 335
7 ForcedVibration 336
7.1 Steady-StateForcedVibration 336
7.1.1 Single-Degree-of-FreedomSpring–MassResponse 336
7.1.2 Multiple-Degree-of-FreedomSpring–MassSystemResponse 344
7.1.3 ForcedHarmonicVibrationofContinuousSystems 347
7.1.4 GeneralSystemResponse 357
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Contents ix
7.2 TransientVibration 359
7.2.1 TransientVibrationTheory 359
7.2.2 ContinuousSystemsandInitialConditions 365
7.2.3 MaximumTransientResponseandResponseSpectra 371
7.2.4 ShockStandardsandShockTestMachines 374
7.3 VibrationIsolation 374
7.3.1 Single-Degree-of-FreedomVibrationIsolation 374
7.3.2 Two-Degree-of-FreedomVibrationIsolation 377
7.4 RandomVibrationResponsetoSpectralLoads 379
7.4.1 PowerSpectralDensityandFourierSeries 380
7.4.2 ComplexFourierTransformandRandomResponse 381
7.5 ApproximateResponseSolution 385
7.5.1 EquivalentStaticLoads 389
7.5.2 ScalingModeShapestoLoad 389
References 391
8 PropertiesofSolids,Liquids,andGases 392
8.1 Solids 392
8.2 Liquids 402
8.3 Gases 405
8.3.1 IdealGasLaw 405
References 409
A ApproximateMethodsforNaturalFrequency 410
A.1 RelationshipbetweenFundamentalNaturalFrequencyand
StaticDeflection 410
A.2 RayleighTechnique 413
A.3 DunkerleyandSouthwellMethods 415
A.4 Rayleigh–RitzandSchmidtApproximations 415
A.5 GalerkinProcedureforContinuousStructures 416
References 417
B NumericalIntegrationofNewton’sSecondLaw 418
References 421
C StandardOctavesandSoundPressure 422
C.1 TimeHistoryandOverallSoundPressure 422
C.2 PeaksandCrest 423
C.3 SpectraandSpectralDensity 424
C.4 LogarithmicFrequencyScalesandMusicalTunings 424
C.5 HumanPerceptionofSound(PsychologicalAcoustics) 426
References 427
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D IntegralsContainingModeShapesofSingle-SpanBeams 429
Reference 429
E FiniteElementPrograms 435
E.1 Professional/CommercialPrograms 435
E.2 OpenSource/Low-CostPrograms 436
Index 439
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Preface
This book is an illustrated compilation of formulas for solving dynamics, acoustics, and
vibrationproblemsin engineering.Over1000formulas forsolutionofpracticalproblems
arepresentedin60tablesforquickreferencebyengineers,designers,andstudents.
Themajorityoftheformulasareexactsolutionsfordynamics,acoustics,andvibrations.
Theiroriginliesin thedevelopmentofcalculusandits applicationtophysicsbySirIsaac
NewtonandGottfriedWilhelmLeibnitzinthe17thcentury.SirIsaacNewton(1642–1727)
formulated the inverse square law of gravitation and proved all three of Kepler’s laws of
planetarymotion.LeonhardEuler(1707–1783)derivedtheequationsofrigidbodydynam-
icsandwavepropagation.In1732,DanielBernoullidescribedthenormalmodesofhanging
chainsintermsofBesselfunctions.GustavKirchhoffdevelopedthetheoryofplatevibra-
tionsin1850.Themodernageofexactclassicaldynamicsolutionsbeganwithpublication
ofLordRayleigh’sTheTheoryofSound(firstedition1877),followedbyA.E.H.Love’sA
TreatiseontheMathematicalTheoryofElasticity(firstedition1892–1893),HoraceLamb’s
TheDynamicalTheoryofSound(firstedition1910),andStephenTimoshenko’sVibration
Problems in Engineering (first edition 1928). These books are foundations of vibration
analysisandtheyarestillinprint.
FormulasinthisbookspanthetechnicalliteraturefromthesecondeditionofTheTheory
of Sound (1894) to technical journals of the 21st century. Most were generated by mod-
ern contributors including Arthur W. Leissa, H. Max Irvine, Phil McIver, Werner Soedel,
DanielJ. Gorman,and W. Kitpornchai. Systemstoo large andcomplexfor exactsolution
by formulas are analyzed approximately with numerical methods (Appendices A and B)
thatareformulated,interpreted,andcheckedwithexactclassicalformulasandreasoning.
Asstructuresbecomelighterandmoreflexible,theybecomemorepronetovibrationand
dynamicfailure.Vibrationfailuresincludewind-inducedvibrationoftheTacomaNarrows
suspensionbridgein1940andthesteamflow-inducedradiationleaksfromvibratingtubes
thatshutdowntheSanOnofreNuclearGeneratingStationin2012.Dynamicandvibration
analysisdiffersfromstaticanalysisbyincludingtimeandinertia.Dynamicanalysisstarts
with identification of a flexiblesystemin a dynamicenvironment. The nextstep is calcu-
lation of the system mass, stiffness, natural frequency, mode shape, and transmissibility.
Thethirdstepiscalculationofdynamicresponsetothetime-dependentloads.Thendesign
changescanbemadetoincreasereliabilityandreducenoiseandvibration.
