1-s2.0-S0022460X15002151-main

潜艇推进系统的减振研究

JournalofSoundandVibration347(2015)150–168

ContentslistsavailableatScienceDirect

JournalofSoundandVibration

journalhomepage:http://wendang.chazidian.com/locate/jsvi

Vibro-acousticcouplingdynamicsofafinitecylindricalshell

underarotor-bearing-foundationsystem'snonlinear

vibrationexcitation

QizhengZhoun,DeshiWang

DepartmentofWeaponryEngineering,NavalUniversityofEngineering,Wuhan,HubeiProvince430033,People'sRepublicofChina

articleinfo

Articlehistory:

Received7October2013

Receivedinrevisedform

29January2015

Accepted26February2015

HandlingEditor:K.Worden

http://wendang.chazidian.comingrotordynamics,structuredynamicsandacousticaltheory,thevibro-acousticcouplingequationofacylindricalshellunderrotor-bearing-foundationsystem'snonlinearvibrationexcitationsisderivedbasedonvariationalprinciple.Inordertosolvethecouplingequation,theinfluencesoftheshell'svibrationtotherotor-bearingsystemareneglected,andthenthedynamicalequationisreduced.Thenonlinearforces

transmittedtothecylinderarefittedintheFourierseriesbythefastFouriertranslation

andtheharmonicbalancemethod,andthentheanalyticalsolutionofthevibro-acoustic

couplingequationofthecylinderisderived.Basedoninherentassumption,theanalytical

expressionsoftheacousticradiationpowerandthesurfacevelocityaregiven.Then

bifurcationdiagrams,phasediagrams,timehistorydiagramsandspectrumgraphsare

employedtostudythenonlinearvibrationcharacteristicsandtheacousticradiation

characteristics.Itisinferredthatthepresentworkproposesasemi-analyticalandsemi-

numericalmethodforthenonlinearvibro-acousticcouplingsystem.Themotionsofthe

forcestransmittedtothecylinderareperiodicmotions,quasi-periodicmotions,andsoon.

Thevibro-acousticcharacteristicsofshellaredominatedbytherotationfrequencyofthe

rotor,whiletherearesomeharmoniccomponentsdominatingthevibro-acoustic

characteristicsatresonantfrequencies.

&2015ElsevierLtd.Allrightsreserved.

1.Introduction

Acousticradiationfromtheunderwaterelasticstructurewithinternalsubstructuresisofgreatimportancefortheacousticstealthdesignofvehicles.Duetotheinteractionofthestructuralvibrationandtheacousticpressureinmedium,theacousticpressureintegralexiststhesingularityonthesurfacebetweenthestructureandthefluidmedium,hencethemodelandsolutionofunderwaterstructurewithinternalsubstructureshasbeenofgreatinterestformanyyears.Theradiationacousticfieldcanbetreatedanalyticallyforsimpleregularstructureswithsubstructuresattachedalongcoordinatelines.Whenconsidermoreengineeringfactors,suchastheinnerrotor-bearingsystem'snonlinearexcitations,itishardtoestablishthevibro-acousticcouplingmodelandgivetheanalyticallysolutionfortheunderwaterstructure.nCorrespondingauthor.

E-mailaddresses:zqizheng@http://wendang.chazidian.com(Q.Zhou),wdeshi@http://wendang.chazidian.com(D.Wang).

http://wendang.chazidian.com/10.1016/j.jsv.2015.02.040

0022-460X/&2015ElsevierLtd.Allrights

reserved.

潜艇推进系统的减振研究

Q.Zhou,D.Wang/JournalofSoundandVibration347(2015)150–168151

Althoughtheboundaryorfiniteelementtechniquescouldbeemployedtoderivethenumericalsolution,thenumericalmethodscannotbeusedtoanalyzethenatureoflaw,becauseachangeofthestructuralparametersorthephysicalparameterswouldinflecttheessencepropertiesoftheradiatedstructure.Therefore,theinvestigationonthesemi-analyticalsolutionoftheacousticradiationandvibrationofthefinitecylindricalshellundertherotor-bearing-foundationsystem'snonlinearvibrationexcitationsissignificantinboththeoryandpractice.

Intherotor-bearing-foundation-cylindersystem,thenonlinearforceduetotherotatingoftherotor-bearing-foundationsubsystemexcitesthevibrationofthecylinder,thesurfacewaveofthecylinderdisturbsthesurroundingacousticmedium,andthentheacousticpressureiscomeintobeing.Hence,theresearchofthenonlinearvibro-acousticcouplingsystemisaninterdisciplinaryissuebetweenthebranchofthemechanicsandtheacoustics.Thevibro-acousticcouplingmodelandsolutionoftherotor-bearing-foundation-cylindersystemneedthestructuralacousticradiationtheory,structuraldynamicsandnonlinearvibration.Instructuralvibro-acousticcouplingfield,theanalyticalmethod,numericalmethodandexperimentationareusedtoinvestigatethevibro-acousticcharacteristicsofdifferentstructureforms.Forunderwaterregularstructures,sincetheresearchonvibrationsofelasticshellsinafluidmediumandtheassociatedradiationofsoundbyJunger[1],thereareanumberofresearchontheanalyticalsolutionofvibrationandsoundradiationwhichwerederivedbymodalexpansionmethodandFouriertransformmethodbasedonfoundationalvibrationequationofDonnell,Flügge,LoveandSanders[2–4].Stepanishen[5]utilizedacombinedGreen'sfunctionandFourierintegraltechniquetoevaluatetheradiationloadingandradiatedpowerfromanon-uniformharmonicallyvibratingsurfaceonaninfinitecylinder,andpresentedthegeneralintegralexpressionsfortheacousticimpedanceswhichincludedself-radiationandinteractionimpedance.Basedonthisresult,LaulagneandGuyader[6]studiedtheacousticradiationinlightandheavyfluidsfromashell.Forvibro-acousticcouplingtheory,theprophaseworkmainlystudiedthebasicstructures,whilethepostmainlyconsideredthestiffenedstructures.Fortheinfinitesingleshellwithregularstiffeners,theanalyticalsolutionorsemi-analyticalsolutionalsocouldbegiven.Bernblit[7]studiedthesoundradiationfromaribbedcylindricalshell.Burroughs[8]studiedtheacousticradiationfromfluid-loaded,ribbedcylindricalshellsexcitedbydifferenttypesofconcentratedmechanicaldrives.Crighton[9]investigatedthesoundradiationfromtheinfinitereinforcedplate,andCrightonmadeaconclusionabouttheseworkandresults.Itisnotedthatthevibro-acousticcouplingofthestructureunderwaterisequaltothevibrationanalyzeiftheacousticradiationimpedanceofthestructureandthemechanicimpedanceofthestiffenerisderived.Hence,theacousticradiationimpedanceortheequivalentacousticimpedanceisthekeypointofthevibro-acousticcouplinginvestigation.

Theunderwatervehicleneedsreinforcementmeasurestosatisfythestrengthandstability,somanyresearchershaveinvestigatedtheeffectsofforms,propertiesandinternalconnectionsofstiffenerstothevibro-acousticcharacteristicsofshellwithinternalstructures.Ei-RahebandWagner[10]studiedthefrequencyresponseandacousticradiationfromashellwithconical,sphericalsegmentsstiffenedbydiscreteringsandbulkheads,whileamajorissuefacingthemethod'streatmentofthefluidwouldbelackofexistenceorencounteredintheuncoupled,externalacousticproblem.HarariandSandman[11],LaulagnetandGuyader[12]studiedthevibrationandacousticradiationofsubmergedcylindricalstiffenedshells.Theyderivedthesolutionintheformofmatrixequationbyconsideringtheforcesandmomentsin4Â4impedancematrixandtheboundaryconditionaboutstiffeners,whiletheresultsneededanumberofnumericalcalculation,anditisnoteasytoanalysisthecontrolmechanism.Liao[13]developedanapproximateanalyticmethodtostudysoundradiationcharacteristicsofafinitesubmergedaxialperiodicallystiffenedcylindricalshellexcitedbyradialharmonicforcesbasedonmodalanalysis.Guo[14]studiedsoundradiationfromcylindricalshellsduetointernalsources,andpresentedastandardmethodforsolvingsuchcoupledproblemsintermsofreceptancesoftheshellandtheplate,whichrelatedtheresponsesofthetwotocouplingforcesandbendingmomentsattheconnections.Lee[15]studiedsoundtransmissionthroughadouble-walledcylindricalshellbycombiningthesolutionsoftwodifferentmodels,andthetransmissionlossescalculatedfromthetwomodelswerecombinedtorepresentthesystemresponseintheentirefrequencyrange.Itisnotedthatthemodalexpansionmethodisthefoudationoftheanalyticalsolution,whichcouldbeeasilyusedtoderivethestructuralvibro-acousticlaw.Andthevariationalmethodcouldbeemployedtoestablishthevibro-acousticcouplingmodeloftherotor-bearing-foundation-cylindersystem,andthemodalexpansionmethodcouldbeusedtoderivethesemi-analyticalsolution.

Numericalalgorithmisanapproximatesolutiontothemathematicalproblem.Duetothepuretheoreticalsolutionsnotsatisfyingthephysicalrealityinengineeringapplications,therearesomedifficultiesinfindingtheexactsolutionofthefinite,multipleexcitations,andstiffenedshell.Therefore,consideringmoreengineeringfactors,thenumericalsolutionmethodsuchasdiscreteelementmethodandenergymethodisusedtoderivethedynamiccharacteristicofcoupledsystem.Thediscreteelementmethodincludingfiniteelementmethod,infiniteelementmethodandboundaryelementmethodisusedtostudythevibro-acousticcouplingproblemsinmiddleandlowfrequencyranges[16–19].Themethodgenerallyutilizesthefiniteelementmethodtodiscretethestructureandthencalculatetheacousticfield,whiletheboundaryelementmethodalwaysischosentodealwiththestructuralexterioracousticfields.Theenergymethodincludingstatisticalenergyanalysismethod,energyfiniteelementmethodandpowerflowmethodisusedtostudythevibro-acousticcouplingproblemsinmiddleandhighfrequencyranges[20–23].Itisnotedthattheindirectboundaryelementmethod(orwavesuperpositionmethod)whichproposedbyKoopmann[24]ismoreefficiencytocalculatetheacousticfield.ThemethodutilizesthelinearizedEulerequationsanddipolematrixcomposedbypointsoundsourcestosolvethesingularityofintegralequationsintheboundaryelementmethod.AndthenFahnlineandConstans[25–28]usedthismethodtodealwithstructuralnoisecontrolsuchasthepiezoelectriccontrolofstructuralnoise,structuraltailorandsoundpoweroptimization.Itcoulddrawaconclusionfromtheseworksthatthewavesuperpositionmethodnotonlycouldbeusedto

潜艇推进系统的减振研究

152Q.Zhou,D.Wang/JournalofSoundandVibration347(2015)150–168

reconstructtheacousticfield,butalsocouldbeusedtothestructuralacousticdesignandoptimization.Whileduetoachangeofthestructuralparametersorthephysicalparameterswouldinflecttheessencepropertiesoftheradiatedstructure,thesenumericalmethodscannotbeusedtoanalyzethenatureoflaw.

Experimentationisanotherscientificmethodtoinvestigatetheobjectivelawofthing.Theassumptionsinnumericalcalculationinducetosomeerrorsbetweenthephysicalmodelandengineeringapplication,sotheexperimentalmeasurementmethodincludingthesoundpressuremethod,thesoundintensitymethodandthenear-fieldacousticalholographyisneededtostudythestructuralvibrationandacousticradiation.Thereisfewworkspublicizedduetotheresultsrelatedtothemilitarysecrecy.HarariandSandman[11]utilizedanalyticalandexperimentaldeterminationofthevibrationandpressureradiationfromasubmerged,stiffenedcylindricalshellwithtwoendplates.Gilroy[29,30]adoptedexperimentalmeasurementmethodtostudythering-stiffenedcylindricalshellfortheDREAstructuralacousticdesignprogram.Plona[31]utilizedinaxialequalintervalreceiverslocaledatshellwallandhydrophonestomeasuretheshell'svibrationwaveformintimedomain,andgavethedispersioncurve.Ekimov[32]didexperimentalmeasuretoinvestigatedlocalmassinfluenceonsoundradiationfromathinlimitedcylindricalshell.ThestructuralvibrationandtheacousticradiationofunderwatercomplicatedshellstructurewithmultipleexcitationswereinvestigatedbyCheng[33],andtheexperimentalresultsshowedthatsoundpressurecausedbymultipleexcitationscanbeapproximatedastheincoherentsuperpositionofthosecausedbyeachexcitationseparately.Theexperimentalresultscouldgivesomeinstructiontotheresearchonthestructuralvibrationandacusticradiation,whileitalsocouldnotbeusedtoanalyzethestructuralacousticradiationnatureoflaw.

Inordertosatisfythemotionandworkrequirements,therearemanyrotatingmachineryintheshell,suchastherotor-bearing-foundationsystem.Therearemanyfactors(includingthenonlinearoil-filmforce,thenonlinearsupported)couldinducethenonlinearvibrationofrotor.Andthesefactorscouldinducenonlinearconstrainedforcesbetweenrotorandadjacentstructures,whichwouldactivizethestructuralvibrationandthenradiatenoise.Numerousarticlesfocusonthenonlinearrotortouncoverthecomplicateddynamicalcharacteristicsofrotorsystem.Theresultsincludethevibrationandstabilityofrotorunderoilfilmoscillation,rubandcollision,misalignmentandevencracks[34–39].Therearesomanycomplicatedvibrationmotionsincludingbifurcationandchaosinthenonlinearvibratingsystemthattheforcestransmitedtotheshellcannotbeexpressedbyasimpleharmonicfunction.Andalthoughthesevibrationcouldsupplyplentyacousticsourcecharacteristicstothestructure,thestructuralnoisecontrolmechanismofstructureunderrotor-bearingsystem'snonlinearvibrationexcitationisnotyetclear,andtheworkonthisproblemisscarcelyreported.

Actually,therearemanyrotatingmachineryintheunderwatervehicle,sothevehiclealwaysworksundermultipleexcitations.However,untilnow,thereislittleresearchonthevibrationandacousticradiationofvehicleundermultipleexcitations.Inthispaper,avibro-acousticanalysistodescribetheacousticradiationandvibrationoffinitecylindricalshellunderrotor-bearing-foundationsystem'snonlinearvibrationexcitationsisdeveloped.Thenextsectiondescribestheproposedshell/rotor/bearing/foundationmodelbasedonHamiltonprincipleisderivedindetails.Anexampleforillustratingthenonlinearvibro-acousticcouplingmodelsolutioncanbefoundinSection3.Thelastsectionconcludesthepaperwiththesummary.

2.Vibro-acousticcouplingdynamicsanalysis

2.1.Physicalmodel

Thevibro-acousticcouplingmodelofcylindricalshellunderrotor-bearing-foundationsystem'svibrationexcitationsandcoordinatessystemofrotor-bearing-foundationsystemisshowninFig.1.a,handLare,respectively,theradius,thicknessandlengthofcylindricalshell.E,ρ,ηandυare,respectively,theYoung'smodulus,density,lossfactorandPoissonratioofmaterial.Andρ0,c0are,respectively,thedensityandacousticvelocityofmedium.

InFig.1,m1isthemassofrotor,o1isthegeometriccenterofrotor,oeisthecentricofrotor,andeistheeccentricdistanceoftherotor.c1isthedampingofcirculardisk,2kpisthestiffnessofshaft,andx1,y1arethedisplacementsofrotor.m2,o2aretheportionofrotormassandthecenterofliftjournalbearing,Fx2andFy2arethenonlinearoil-filmforcecomponents,andx2,y2arethedisplacementcomponents.m3,o3aretheportionofrotormassandthecenterofrightjournalbearing,Fx3and

Fig.1.(a)Vibro-acousticcouplingmodelofcylindricalshellunderrotor-bearingsystemvibrationexcitations;(b)coordinatesofrotor-bearingsystem.

潜艇推进系统的减振研究

Q.Zhou,D.Wang/JournalofSoundandVibration347(2015)150–168153

Fy3arethenonlinearoil-filmforcecomponents,andx3,y3arethedisplacementcomponents.m4,o4arethemassandthecenterofliftjournalbearing,k4,c4arethestiffnessandthestructuraldampingoftheliftbearingsupportstructure,andx4,y4arethedisplacementcomponents.m5,o5arethemassandthecenterofliftbearingsleeve,k5,c5arethestiffnessandthestructuraldampingoftherightbearingsupportstructure,andx5,y5arethedisplacementcomponents.u,vandwaretheorthogonalcomponentsofdisplacementinz,φandrdirectionsrespectively.Inordertoacquirethevibro-acousticcouplingequationoffinitecylindricalshellunderthenonlinearvibrationexcitationsofrotor-bearing-foundationsystem,theassumptionsare

(1)Thethicknessoftheshellissmallcomparedwiththeotherdimensions,sotheclassicalthinshelltheoryhasenough

precisionbelowringfrequency.

(2)Thefluidsurroundingshellistheidealacousticmedium,thatis,themediumisisentropicandadiabatic,andneglectthe

viscidity,heatexchangeandrotation,sothewaveequationinidealacousticmediumcouldbegivenbyclassicalpotentialflowtheoryforthesmalldisplacement.

(3)Therearerigidinfinitecylindricalbaffle-platesatthetwoendoftheshell.

Theresponseofthesimplysupportedshellsubjectedtothepointloadwasdealtwithinthisstudy.Asinmoststudies,themodalanalysismethodwasusedtoobtaintheresponseoftheshell.Thecouplingvibrationofunderwatercylindercouldbesolvedbymodalexpansionmethodbasedonsuperpositionprinciple,theshelldisplacementcanbeexpandedasaseriesofitsnaturalmodalsasfollows[13]:

uðz;φ;tÞ¼

vðz;φ;tÞ¼??απ sinðωqtþφqÞUmnqcoskmzsinnφþα¼0q¼1m¼1n¼0Q111XXXX??απ VmnqsinkmzcosnφþsinðωqtþφqÞ2α¼0q¼1m¼1n¼0Q111XXXX

Q111XXXX(1)wðz;φ;tÞ¼Wmnqsinkmzsinnφþ??απ

2α¼0q¼1m¼1n¼0sinðωqtþφqÞ

whereα¼1(or0)representsthesymmetricalmode(ortheanti-symmetricalmode).Thesubscriptsrepresentthecircumferentialordern(thenumberofwavelengthinthecircumference),thelongitudinalorderm(thenumberofhalfwavelengthalongtheshellaxis),Qisthenumberofexcitationfrequencies.Forthesimplysupportedboundarycondition,km¼mπ=L.Theprocessfordrivingthevibro-acousticcouplingequationoffinitecylindricalshellunderthenonlinearvibrationexcitationsofrotor-bearingsystemisasfollowingforα¼1.

2.2.Energyanalysis

Forthevibro-acousticcouplingproblemofthefinitecylinderundernonlinearvibrationexcitationsofrotor-bearing-foundationsystem,theHamiltonvariationalprincipleisusedtoderivethegoverningequation.Thevariationofthestrainenergyofcylindricalshellis

δZt1

t0ZQ1X1nh?? XEhπLt1X2222Vcdt¼akþσnð1þβÞUmnqþankmσpVmnqmmt0q¼1m¼1n¼0?? iÂ23þβðσman2kmþa3kmÞÀυakmWmnqδUmnqþankmσpUmnq

????!?? 22223Àυ222WmnqδVmnqþnþσmakmð1þ3βÞVmnqþnþβakmn2???? ??22223Àυ233VmnqþβðσmankmþakmÞÀaυkmÞUmnqþnþβakmn2??hi io222222WmnqδWmnqsin2ðωqtþφqÞdtþ1þβðakmþnÞþ1À2n(2)

whereβ¼h=12a2,σm¼ð1ÀυÞ=2andσp¼ð1þυÞ=2.Neglectingtheeffectsofmomentofinertia,thekineticenergyoftheshellis

ZLZ2π"????2????2????2#1?u?v?wþþ(3)Tc¼ρhadφdz0022

Basedonintegrationbypartsandtimeboundaryconditions,Eq.(4)becomes

??Zt1??2Zt11?u?2v?2wδTcdt¼Àhδuþδvþδwadφdzdt?t?t?tt0t0(4)

DZͧ