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Vibro-Acoustics

Noise pollution is a general problem.Structures excited by dynamic forces radiate noise.The art of noise reduction requires an understanding of vibro-acoustics.This topic describes how structures are excited,energy flows from an excitation point to a sound radiating surface,and finally how a structure radiates noise to a surrounding fluid.The aim of this text is to give a fundamental analysis and a mathematical presentation of these phenomena.The text is intended for graduate students,researchers and engineers working in the field of sound and vibration.

Preface

Notations

Chapter 1 MECHANICAL SYSTEMS WITH ONE DEGREE OF FREEDOM

1.1 A simple mass-spring system

1.2 Free vibrations

1.3 Transient vibrations

1.4 Forced harmonic vibrations

1.5 Fourier series

1.6 Complex notation

Problems

Chapter 2 FREQUENCY DOMAIN

2.1 Introduction

2.2 Frequency response

2.3 Correlation functions

2.4 Spectral density

2.5 Examples of spectral density

2.6 Coherence

2.7 Time averages of power and energy

2.8 Frequency response and point mobility functions

2.9 Loss factor

2.10 Response of a 1-DOF system,a summary

Problems

Chapter 3 WAVES IN SOLIDS

3.1 Stresses and strains

3.2 Losses in solids

3.3 Transverse waves

3.4 Longitudinal waves

3.5 Torsional waves

3.6 Waves on a string

3.7 Bending or flexural waves-beams

3.8 Waves on strings and beams-a comparison

3.9 Flexural waves-plates

3.10 Orthotropic plates

3.11 Energy flow

Problems

Chapter 4 INTERACTION BETWEEN LONGITUDINAL AND TRANSVERSE WAVES

4.1 Generalised wave equation

4.2 Intensity

4.3 Coupling between longitudinal and transverse waves

4.4 Bending of thick beams/plates

4.5 Quasi longitudinal waves in thick plates

4.6 Rayleigh waves

4.7 Sandwich plates-general

4.8 Bending of sandwich plates

4.9 Equations governing bending of sandwich plates

4.10 Wavenumbers of sandwich plates

4.11 Bending stiffness of sandwich plates

4.12 Bending of I-beams

Problems

Chapter 5 WAVE ATTENUATION DUE TO LOSSES AND TRANSMISSION ACROSS JUNCTIONS

5.1 Excitation and propagation of L-waves

5.2 Excitation and propagation of F-waves

5.3 Point excited infinite plate

5.4 Spatial Fourier transforms

5.5 Added damping

5.6 Losses in sandwich plates

5.7 Coupling between flexural and inplane waves

5.8 Transmission of F-waves across junctions,diffuse incidence

5.9 Transmission of F-waves across junctions,normal incidence

5.10 Atenuation due to change of cross section

5.11 Some other methods to increase attenuation

5.12 Velocity level differences and transmission losses

5.13 Measurements on junctions between beams

Problems

Chapter 6 LONGITUDINAL VIBRATIONS OF FINITE BEAMS

6.1 Free longitudinal vibrations in finite beams

6.2 Forced longitudinal vibrations in finite beams

6.3 The mode summation technique

6.4 Kinetic energy of vibrating beam

6.5 Mobilities

6.6 Mass mounted on a rod

6.7 Transfer matrices

Problems

Chapter 7 FLEXURAL VIBRATIONS OF FINITE BEAMS

7.1 Free flexural vibrations of beams

7.2 Orthogonality and norm of eigenfunctions

7.3 Forced excitation of F-waves

7.4 Mode summation and modal parameters

7.5 Point mobility and power

7.6 Transfer matrices for bending of beams

7.7 Infinite periodic structures

7.8 Forced vibration of periodic structures

7.9 Finite composite beam

Problems

Chapter 8 FLEXURAL VIBRATIONS OF FINITE PLATES

8.1 Free vibrations of simply supported plates

8.2 Forced response of a simply supported plate

8.3 Forced excitation of a rectangular plate with two opposite sides simply supported

8.4 Power and energy

8.5 Mobility of plates

8.6 The Rayleigh-Ritz method

8.7 Application of the Rayleigh-Ritz method

8.8 Non flat plates

8.9 The effect of an added mass or mass-spring system on plate vibrations

8.10 Small disturbances

8.11 Plates mounted on resilient layers

8.12 Vibration of orthotropic plates

8.13 Circular and homogeneous plates

8.14 Bending of plates in tension

Problems

References

Index

Anders C.Nilsson holds MSc in Engineering Physics from University of Lund and Dr.Tech.in Sound and Vibration from Chalmers University in Sweden.Anders C.Nilsson worked with problems on the propagation of sound and sonic booms at Boeing Co.,Seattle,USA.Later he moved to Norway and the Research Division of Det Norske Veritas.At Veritas Anders C.Nilsson worked on the propagation of structureborne sound in large built up structures and on the excitation of plates from flow and cavitation.Anders C.Nilsson then transferred to Denmark and was head of the Danish Acoustical Institute for four years.His main activities in Denmark concerned building acoustics.In 1987,Anders C.Nilsson was appointed professor of Applied Acoustics at KTH in Stockholm,Sweden.He was also the head of the Department of Vehicle Engineering and the founder and head,until 2002,of the Marcus Wallenberg Laboratory of Sound and Vibration Research (MWL).Anders C.Nilsson has been a guest professor at James Cook University,Australia,INSA-Lyon in France and at the Institute of Acoustics,Chinese Academy of Sciences in Beijing and is professor emeritus at MWL,KTH since 2008.His main interests are problems relating to composite structures as well as vehicle acoustics.Bilong Liu received his PhD in acoustics at the Institute of Acoustics,Chinese Academy of Sciences in 2002.Then he worked on noise transmission through aircraft structures at MWL,KTH,Sweden till 2006.

Bilong Liu also holds PhD in applied acoustics from MWL,KTH.During Aug.2004 to Jan.2005,he worked on pipe and pump noise at the University of Western Australia in Perth.From 2007 he has been working as a research professor at the Institute of Acoustics,Chinese Academy of Sciences,and from 2011 he has been acting as an associate editor for an Elsevier journal-Applied Acoustics.His main interests include vibro-acoustics,acoustics materials,fluid-structure interaction,duct acoustics,active noise control,smart acoustic materials and structures.

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Chapter 1

MECHANICAL SYSTEMS WITH ONE DEGREE OF FREEDOM

Innoisereducingengineeringtheconsequencesofchangesmadetoasystemmustbeunderstood.Questionsposedcouldbeonthee.ectsofchangestothemass,sti.nessorlossesofthesystemandhowthesechangescanin.uencethevibrationofornoiseradiationfromsomestructures.Realconstructionscertainlyhavemanyorinfactin.nitemodesofvibration.However,toacertainextent,eachmodecanoftenbemodelledasasimplevibratorysystem.Themostsimplevibratorysystemcanbedescribedbymeansofarigidmass,mountedonaverticalmasslessspring,whichinturnisfastenedtoanin.nitelysti.foundation.Ifthemasscanonlymoveintheverticaldirectionalongtheaxisofthespring,thesystemhasonedegreeoffreedom(1-DOF).Thisisavibratorysystemneveractuallyencounteredinpractice.However,certaincharacteristicsofsystemswithmanydegreesoffreedom,orrather,continuoussystemswithanin.nitedegreeoffreedom,canbedemonstratedbymeansoftheverysimple1-DOFmodel.Forthisreason,thebasicmassspringsystemisusedinthischaptertoillustratesomeofthebasicconceptsconcerningfreevibrations,transient,harmonicandothertypesofforcedexcitation.Kineticandpotentialenergiesarediscussedaswelltheirdependenceontheinputpowertothesystemanditslosses.

1.1 A simple mass-spring system

A simple mass-spring system is shown in Fig. 1-1. The mass is m and thespringconstantk0.Thefoundationtowhichthespringiscouplediscompletelysti.andunyielding.ItisassumedthatthespringismasslessandthatthespringforcefollowsthesimpleHooke’slaw.Thus,whenthespringiscompressedthedistancex,thereactingforcefromthespringonthemassisequaltok0x.ThedampingforceduetolossesinthesystemisdenotedFd.Whenthesystemisatrest,thedampingforceFdisequaltozero.Thestaticloadonthespringismgwheregistheaccelerationduetogravity.Duetothestaticload,thespringiscompressedthedistanceΔx.Thereactingforceonthemassisk0Δx.Thus

?

Δxk0=mg(1-1)

?

Therefore, in principle if the static

de.ection is known, then the spring

constant can be determined. How

ever, for real systems, the static and

dynamic sti.ness are not necessarily

equal. In particular, this is quite

evidentforvarioustypesofrubberFig. 1-1 Simple mass-spring system springs. In addition, a real mount has a certain mass.

Fig.1-1showsasimplemass-springsystemexcitedbyanexternalforceF(t)andadampingforceFd.Theequationofmotionforthissimplesystemis

mx¨+k0x+Fd=F(t)(1-2)

Thedeviationofthemassfromitsequilibriumpositionisx=x(t).Whenthemassisatrest,thenx=0.ThedampingforceFdisdeterminedbythelossesinthesystem.Variousprocessescancausetheselosses.Someexamplesofoften-usedsimpletheoreticalmodelsare:

i) Viscous damping;

ii) Structural or hysteretic damping;

iii) Frictional losses or Coulomb damping;

iv) Velocity squared damping.

Thedampingforcescanbeillustratedbyassumingasimpleharmonicdisplacementofthemass.Themotionisgivenbyx=x0sin(ωt).Heretime

?

istandω=2πfwhereωistheangularfrequencyandfthecorrespondingfrequency.Thedampingforcesforthefourcasesare:

i) Viscous damping.

Fd=cx?=cx0ωcos(ωt)(1-3)

?

Thedampingforceisproportionaltothevelocity?xofthemass,cisaconstant.Theenergydissipatedpercycle,i.e.inatimeintervalt0.t.t0+T,whereωT=2π,dependslinearlyonω,theangularfrequencyofoscillation.Thistypeofdampingoccursforsmallvelocitiesforasurfaceslidingona.uid.lmandfordashpotsandhydraulicdampers.

ii) Structural damping.

αα

Fd=x?=x0cos(ωt)(1-4)πω ? π ??

Theamplitudeofthedampingforceisproportionaltotheamplitudeofthedisplacementbutindependentoffrequencyforharmonicoscillations.Theenergydissipatedpercycleofmotionisfrequencyindependentoverawidefrequencyrangeandproportionaltothesquareoftheamplitudeofvibration.Thelossesinsolidscanoftenbedescribedinthisway.Ineq.(1-4)αisaconstant.

iii) Frictional damping.

Fd = ±F (1-5)

Thefrictionalforcehasaconstantmagnitude.Theplusorminussignshouldbedeterminedsothatthefrictionalforceiscounteractingthemotionofthemass.Thefrictionalforcecanbeduetoslidingbetweendrysurfaces.

iv) Velocity squared damping.

Fd = ±qx?2 = ±q(x0ωcos(ωt))2 (1-6)

? ??

Thedampingforceisproportionaltothevelocitysquaredandthesignshouldbechosensothattheforceagainiscounteractingthemotionofthemass.Ineq.(1-6)qisaconstant.Abodymovingfairlyrapidlyina.uidcouldcausethisdampingforce.

Thestructuraldampingofeq.(1-4)onlyappliesforaharmonicdis-placementofthemass.AmoregeneraldescriptionofstructuraldampingispresentedinChapter3.Examplesofenergydissipationduetoviscous,structuralandfrictionallossesaregiveninproblems1.1to1.3.

Theforcerequiredmovingthemassofasimple1-DOFsystemdependofthetypedampinginthespring.Formaintainingamotionx=x0sin(ωt)

?

ofthemass,theforcewhichmustbeappliedtothemassisobtainedfromeq.(1-2)asF=mx¨+k0x+Fd.TwocasesareillustratedinFigs.1-2and1-3.Inthe.rstexample,Fig.1-2,thedampingforceisviscous.Anellipserepresentstheforce-displacementrelationshipforthiscase.Themi-noraxisoftheellipseisproportionaltotheangularfrequencyωandtheparametercofeq.(1-3).Structuraldampinggivesthesametypeofforce-displacementcurve.However,forthiscasetheminoraxisoftheellipseisjustproportionaltothecoe.cientα,eq.(1-4),andnottotheangularfre-quency.Inthesecondexample,Fig.1-3,themassisexposedtofrictionaldamping.Thearrowinthediagramindicateshowforceanddisplacementvaryastimeincreases.Theareaenclosedbyoneloopisequivalenttotheenergyrequiredtoperformonecycleofmotionofthemass.Forasystemwithstructuraldamping,theenergydissipatedpercycleisindependentoftheangularfrequency.ThisisnotthecasewhenthelossesareviscousasdiscussedinProblems1.1and1.2.

Fig.1-2Force-displacementcurveforsinusoidalmotionofa1-DOFsystemwithviscousdamping

Fig.1-3Force-displacementcurveforsinusoidalmotionofa1-DOFsystemwithfrictionaldamping

Realstructuressubjectedtovibrationstendtoshowaforce-displacementbehaviourshowninFig.1-4.Theforce-displacementcurvefollowsadis-tortedhysteresisloop,whichisnotreadilydescribedmathematicallyorphysically.However,ingeneral,theenergydissipatedpercycleratherthantheexactforce-displacementrelationshipisofprimaryimportanceforrealvibratorysystems.Therefore,formostpracticalpurposesviscousormate-rialorforthatmatterafrequencydependentdampingcanbeassumedforthesimpleharmonicmotionofastructure.

Frictionallossesandvelocitysquareddampingresultinnonlinearequa-tionswhenintroducedineq.(1-2).Examplesofnon-linearequationsandtheirsolutionsarepresentedinforexamplerefs.[1]?[3].TheRunge-Kuttamethod,usedfornumericalsolutionsinthetimedomain,isdiscussedinrefs.[1],[4]and[5].

Fig.1-4Force-displacementcurveforsinusoidalmotionofa1-DOFsystemwithdistortedstructuralorhystereticdamping

Forlinearproblems,dampingisoftendescribedasviscousorstructural.Inpractice,thisisnotnecessarilythecase.However,if,forexamplethelossesaresmallandalmoststructural,theparameterαineq.(1-4)canbeallowedtodependonfrequencyforharmonicorapproximatelyharmonicmotion.Thistypeofdampingmodelisdiscussedinsubsequentchapters.Furtherandmostimportantly,ifviscousorstructuraldampingisintroducedineq.(1-2),theresultingequationofmotionislinear.

Forsomeapplications,likeexperimentalmodalanalysisand.niteele-mentcalculations,acertainformofdampingisoftenassumed,seeChapter

10. Aspeci.cdampingmodelmightbenecessaryformathematicalornu-mericalreasons.However,themodelcouldviolatethephysicalcharacteris-ticsofthedampingmechanism.

1.2 Free vibrations

Freevibrationsofasystemoccurifforexamplethesystematacertaininstantisgivenadisplacementfromitsequilibriumpositionoraninitialvelocityatthatparticulartime.Afterthisinitialexcitation,noexternalforcesareappliedtothesystem.Theresultingmotionofthesystemisduetofreevibrations.Forallnaturalsystems,thereisalwayssometypeofdampingpresent.Forsuchsystems,thefreevibrationsdieoutafteracertainlengthoftime.Theinitialenergyofthesystemisabsorbedbylosses.

Forasimplemass-springsystem(1-DOF)withviscouslossestheequa-tionofmotionforfreevibrations,F(t)=0,givenbyeqs.(1-2)and(1-3)as

mx¨+cx?+k0x=0(1-7)

Thegeneralboundaryconditionsorinitialvaluesare

x(0)=x0,x?(0)=v0(1-8)

The traditional approach to solving this equation is to assume a solution of the form

x(t)=Aeλt (1-9)

?

Theeigenvalueλisobtainedbyinsertingthisexpressionineq.(1-7).Con-sequently

λ2 m + λc + k0 = 0 (1-10)

Itisconvenienttode.nethefollowingparameters:

ω02 =k0/m,β=c/(2m)(1-11)

Using these parameters, the solution to eq. (1-10) is

λ1,2=.β ± .β2 . ω02 orλ1,2=.β ± i.ω02 . β2 (1-12)

with i = √.1. In general, there are two solutions to eq. (1-7). The expression for the displacement x is therefore of the form

x(t)=A1eλ1t + A2 eλ2t (1-13)

??

The parameters A1 and A2 are determined from the initial condition (1-8).

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前言

书籍介绍

《振声学(第1卷)》振声学主要研究典型结构在各种激励下的振动响应、振动传递以及声辐射的一般性规律。《振声学(第1卷)》第一卷首先回顾了单自由度系统和频域分析，接着重点讨论了固体中的波传播、纵波和横波的耦合关系、结构阻尼和结构连接引起的波衰减、以及梁和板的弯曲振动等，在此基础上讨论了变分分析、弹性支撑、流体介质中的波、声振耦合以及能量分析方法等内容。