Vibro-Acoustics, Volume 2

453 Pages·2016·18.4 MB·English

by  Anders Nilsson

Anders Nilsson · Bilong Liu
Vibro-
Acoustics,
Volume 2
Second Edition
Vibro-Acoustics, Volume 2
Anders Nilsson Bilong Liu
(cid:129)
Vibro-Acoustics, Volume 2
Second Edition
123
Anders Nilsson Bilong Liu
MWL-Marcus WallenbergLaboratory Key Laboratoryof Noise andVibration
forSound andVibration Research Research
KTH,The RoyalInstitute of Technology Institute of Acoustics, ChineseAcademy
Stockholm ofSciences
Sweden Beijing
China
SupportedbyNational 973Program of China(2012CB720204)
ISBN978-3-662-47933-9 ISBN978-3-662-47934-6 (eBook)
DOI 10.1007/978-3-662-47934-6
JointlypublishedwithSciencePress,Beijing
ISBN:978-7-03-039150-6SciencePress,Beijing
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2ndedition:©SciencePress,BeijingandSpringer-VerlagBerlinHeidelberg2016
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Preface
ThevibrationofsimplestructureslikebeamsandplateswasdiscussedinVolumeI
of Vibro-Acoustics. Various wave types and their propagation in simple structures
were investigated. The coupling effects between the basic wave types propagating
in real constructions were demonstrated in the first volume.
In Volume II the response and dynamic characteristics of more complicated
structureslikeshells,compositepanelsandframereinforcedplatestypicalofmany
types of vehicles are derived. The acoustic coupling between these structures and
the energy flow between them are important parts of Volume II. The interaction
between structures and fluids is also highlighted.
VolumeIIofthetextincludesachapteronvariationalmethods.Thetechniqueis
usedforthederivationof equations governingthevibrationofsandwich andother
composite elements and some simple shell elements. In the following chapter the
couplingbetweenmechanicalsystemsisexplored.Thisincludesanintroductionto
the vibration of rubber mounts, resilient mountings and the design of engine
foundations. Then follow chapters on waves in fluids including outdoor sound
propagation androom acoustics. Couplingeffects between vibrating structuresand
asurroundingfluidorquitesimplysoundradiationandsoundtransmissionlossand
fluid loading of structures are the subjects of the next chapter. Discussions on
randomexcitationofstructuresfollow.Finally,somemethods,includingStatistical
EnergyAnalysis,forthepredictionofexcitationandpropagationofstructure-borne
sound in large built-up structures are investigated.
Anumberofproblemsareformulatedattheendofeachchapter.Solutionstothe
problemsaregiveninVolumeIII.Asummaryofsomebasicequationspresentedin
the first two volumes are summarized in Volume III. Many results discussed in
Volume II are verified by model and full-scale measurements.
Genova, Italy Anders Nilsson
August 2015
v
Contents
9 Hamilton’s Principle and Some Other Variational Methods . . . . . 1
9.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
9.2 Flexural Vibrations of Slender Beams . . . . . . . . . . . . . . . . . 4
9.3 Equation of Motion for Honeycomb Beams in Flexure . . . . . 7
9.4 Plates with Constrained Viscoelastic Layer. . . . . . . . . . . . . . 14
9.5 Timoshenko Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
9.6 Mindlin Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.7 Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9.8 Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.9 Garlekin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.10 An Example Using Garlekin’s Method. . . . . . . . . . . . . . . . . 36
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
10 Structural Coupling Between Simple Systems. . . . . . . . . . . . . . . . 41
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
10.2 Coupled Mass–Spring Systems. . . . . . . . . . . . . . . . . . . . . . 43
10.3 Coupled Systems with Losses. . . . . . . . . . . . . . . . . . . . . . . 47
10.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
10.5 Rubber Mounts, Some Material Parameters . . . . . . . . . . . . . 53
10.6 Wave Propagation in Rubber Mounts, Approximate
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10.7 Equivalent Stiffness of Simple Mounts—Approximate
Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.8 Static Deflection of Cylindrical Rubber Mounts . . . . . . . . . . 65
10.9 Wave Propagation in Circular Rods, Exact Solutions. . . . . . . 66
10.10 Measurements of Effective Stiffness of Mounts. . . . . . . . . . . 75
10.11 Structural Coupling Via Resilient Mounts . . . . . . . . . . . . . . 80
10.12 Simple Transmission Model. . . . . . . . . . . . . . . . . . . . . . . . 85
10.13 Multi-point Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
vii
viii Contents
10.14 Multi-point Coupling, Low and High Frequency Limits. . . . . 96
10.15 Source Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11 Waves in Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.1 Wave Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.2 Energy and Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.3 Losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
11.4 Basic Solutions to Wave Equation. . . . . . . . . . . . . . . . . . . . 113
11.5 Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
11.6 Dipole and Other Multipole Sources . . . . . . . . . . . . . . . . . . 121
11.7 Additional Sources and Solutions . . . . . . . . . . . . . . . . . . . . 123
11.8 Moving Monopole Sources. . . . . . . . . . . . . . . . . . . . . . . . . 129
11.9 Reflection from a Plane Surface . . . . . . . . . . . . . . . . . . . . . 133
11.10 Reflection from a Water Surface. . . . . . . . . . . . . . . . . . . . . 141
11.11 Influence of Temperature and Velocity Gradients . . . . . . . . . 143
11.12 Acoustic Fields in Closed Rooms . . . . . . . . . . . . . . . . . . . . 146
11.13 Geometrical Acoustics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
11.14 Near and Reverberant Acoustic Fields in a Room. . . . . . . . . 155
11.15 Measurement of the Sound Transmission Loss of a Wall. . . . 157
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
12 Fluid Structure Interaction and Radiation of Sound. . . . . . . . . . . 163
12.1 Radiation and Fluid Loading of Infinite Plates . . . . . . . . . . . 163
12.2 Radiation—General Formulation. . . . . . . . . . . . . . . . . . . . . 169
12.3 Green’s Function—Rigid Plane Boundary . . . . . . . . . . . . . . 171
12.4 Spatial Fourier Transforms—Several Variables . . . . . . . . . . . 174
12.5 Radiation from Infinite Point-Excited Plates. . . . . . . . . . . . . 177
12.6 Mobilities of Fluid-Loaded Infinite Plates. . . . . . . . . . . . . . . 181
12.7 Discussion of Results—Infinite Fluid-Loaded Plates . . . . . . . 184
12.8 Radiation from Finite Baffled Plates . . . . . . . . . . . . . . . . . . 186
12.9 Radiation Ratios—Finite Baffled Plates . . . . . . . . . . . . . . . . 192
12.10 Radiation from Point-Excited Plates. . . . . . . . . . . . . . . . . . . 197
12.11 Sound Radiation Ratios—Cylinders. . . . . . . . . . . . . . . . . . . 200
12.12 Losses Due to Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 202
12.13 Radiation from Fluid-Loaded Finite Plates . . . . . . . . . . . . . . 204
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
13 Sound Transmission Loss of Panels. . . . . . . . . . . . . . . . . . . . . . . 215
13.1 Sound Transmission Through Infinite Flat Panels . . . . . . . . . 216
13.2 Plate Velocity Induced by an Acoustic Field. . . . . . . . . . . . . 223
13.3 Sound Transmission Between Rooms Separated
by a Single Leaf Panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
13.4 Sound Transmission Between Equal Rooms. . . . . . . . . . . . . 234
Contents ix
13.5 Sound Transmission Between Irregular Rooms. . . . . . . . . . . 236
13.6 Effect of Boundary Conditions of Plate on Sound
Transmission Loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
13.7 Effect of a Baffle on Sound Transmission Loss. . . . . . . . . . . 243
13.8 Measurement Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
13.9 Loss Factors and Summary. . . . . . . . . . . . . . . . . . . . . . . . . 253
13.10 Sound Transmission Through Complex Structures. . . . . . . . . 256
13.11 Flanking Transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
13.12 Sound Transmission Through Fluid Loaded Plates . . . . . . . . 261
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
14 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
14.2 Structural Waveguides. . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14.3 Coupled Structural Waveguides . . . . . . . . . . . . . . . . . . . . . 270
14.4 Measurements and Predictions . . . . . . . . . . . . . . . . . . . . . . 276
14.5 Composite, Sandwich, and Honeycomb Plates . . . . . . . . . . . 289
14.6 Flexural Vibrations of Honeycomb/Sandwich Beams. . . . . . . 294
14.7 Wavenumbers, Sandwich/Honeycomb Beams. . . . . . . . . . . . 296
14.8 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
14.9 Dynamic Properties of Sandwich Beams . . . . . . . . . . . . . . . 302
14.10 Bending Stiffness of Sandwich Plates . . . . . . . . . . . . . . . . . 305
14.11 Response of Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . 307
14.12 Energy Flow in Sandwich Beams . . . . . . . . . . . . . . . . . . . . 312
14.13 Energy Flow Across Pinned Junctions. . . . . . . . . . . . . . . . . 314
14.14 Wave Propagation on Infinite Cylinders. . . . . . . . . . . . . . . . 316
14.15 Vibration of Open Circular Cylindrical Shells. . . . . . . . . . . . 322
14.16 Sound Transmission Loss of Shallow Shell Segments . . . . . . 324
14.17 Comparison Between Measured and Predicted TL. . . . . . . . . 329
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
15 Random Excitation of Structures. . . . . . . . . . . . . . . . . . . . . . . . . 339
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
15.2 Excitation of Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
15.3 Rain on the Roof Excitation of Plates . . . . . . . . . . . . . . . . . 346
15.4 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 350
15.5 TBL Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
15.6 Plate Response Due to TBL Excitation . . . . . . . . . . . . . . . . 356
15.7 Measurements of TBL-Induced Vibrations . . . . . . . . . . . . . . 362
15.8 Comparison Between Measured and Predicted
Velocity Levels Induced by TBL . . . . . . . . . . . . . . . . . . . . 366
15.9 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
15.10 Flow Noise Induced in Ships . . . . . . . . . . . . . . . . . . . . . . . 373
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
x Contents
16 Transmission of Sound in Built-Up Structures . . . . . . . . . . . . . . . 379
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
16.2 Statistical Energy Analysis, SEA. . . . . . . . . . . . . . . . . . . . . 382
16.3 Energy Flow Between Continuous Systems . . . . . . . . . . . . . 387
16.4 Coupling Between Acoustic Fields
and Vibrating Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . 390
16.5 Prediction of Sound Transmission Through
a Panel Using SEA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
16.6 Sound Transmission Through Double Walls. . . . . . . . . . . . . 395
16.7 Limitation of SEA-Derived Sound Transmission Loss . . . . . . 398
16.8 Coupling Between Vibrating Structures . . . . . . . . . . . . . . . . 400
16.9 Energy Flow in Large Structures, SEA . . . . . . . . . . . . . . . . 402
16.10 SEA Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
16.11 Ship Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
16.12 Waveguide Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
16.13 Noise Levels in Accommodation Spaces . . . . . . . . . . . . . . . 415
16.14 Source Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
16.15 Measured and Predicted Results . . . . . . . . . . . . . . . . . . . . . 418
16.16 Conclusions—Noise Prediction on Ships . . . . . . . . . . . . . . . 423
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Appendix A: Sound Transmission Loss of Single Leaf Panels . . . . . . . 425
Appendix B: Velocity Level of Single Leaf Panels Excited
by an Acoustic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Appendix C: Input Data for Noise Prediction on Ships . . . . . . . . . . . . 433
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
Chapter 9
Hamilton’s Principle and Some Other
Variational Methods
Manyproblemsinmathematicalphysicsandthusinvibro-acousticscannotbesolved
exactly.However,avariationaltechniquecanoftenbeusedtosufficientlywellfor-
mulatetheequationsgoverningtheresponseofastructureexcitedbyexternalforces.
Thetechniqueensuresthaterrorsareminimized.Variationaltechniquesareexcellent
toolsforsolvingdynamicproblemsforwhichexactsolutionscannotbeformulated.
ThewidelyusedFiniteElementMethodisbasedonHamilton’sprinciple,which
isaverypowerfulvariationalmethod.TheprinciplecanbeprovedbasedonNewton’s
law of motion. Inversely Newton’s law can be derived using Hamilton’s principle.
However,Hamilton’sprincipleismuchmoregeneralthanNewton’slawandforthis
reason,ithassurvivedtherevolutioninmechanicsbroughtbyEinstein.
The key problem for the successful application of any variational technique is
themathematicalformulationofthekineticandpotentialenergiesofasystem.This
formulation also requires a physical understanding of the mechanisms governing
the motion of a system. This can be illustrated by considering two different types
of three-layered beams. In one case, the structure consists of a beam with a con-
strainedviscoelasticlayer.Forthevibratingbeam,theshearforcesintheviscoelastic
layeralongtheaxisofthebeamareofimportance.Intheothercase,thecoreofa
three-layeredbeamconsistsofahoneycombstructure.Inthiscase,theshearforces
perpendiculartotheaxisofthebeamareofmajorimportanceforthedeflectionof
thebeam.Forthetwocases,theenergiesaremodeledindifferentwaysresultingin
two different equations as discussed in Sects.9.3 and 9.4. In each case, the results
areonlyvalidaslongasthebasicphysicalassumptionsaresatisfied.
Hamilton’sprincipleisinthischapterusedtoderivetheequations,whichupto
certain frequencies govern the flexural vibrations of thick beams or plates and of
cylindricalshells.TheLagrangeandGarlekinmethodsarealsodiscussed.Thelon-
gitudinalvibrationofthickbeamsorrodsisexaminedinChap.10inconnectionwith
discussions on various models describing the axial vibration of cylindrical rubber
mounts.
©SciencePress,BeijingandSpringer-VerlagBerlinHeidelberg2016 1
A.NilssonandB.Liu,Vibro-Acoustics,Volume2,
DOI10.1007/978-3-662-47934-6_9