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ProceedingsofACOUSTICS2016 9-11November2016,Brisbane,Australia

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A numerical approach to modelling the effect of a static pressure field on the frequency response of a clamped thin circular plate

Kyle Saltmarsh1, Ali Karrech2, David Matthews1,3 and Jie Pan1

1SchoolofMechanical&ChemicalEngineering,TheUniversityofWesternAustralia,Perth,Australia2SchoolofCivil,EnvironmentalandMiningEngineering,TheUniversityofWesternAustralia,Perth,Australia

3UnderwaterSensorsGroup,DefenceScienceandTechnologyGroup,HMASStirling,Australia

ABSTRACT Adetailedunderstandingofthevibrationalpropertiesofsubmergedobjectssuchassubmarinesisessential inordertominimisesoundemissionandoptimisesonarperformance.Complicationsarisewhenhollowstructuresaresubmergedinwater producing loading on one side. Not only does thewater dampen the resonances of the structure but also theincrease in depth producesmechanical strainwithin the structure that changes the resonant frequencies. In previouswork, the effect of hydrostatic pressure loading on one side of a rigidly clamped plate was investigated. The modalcouplingbetweenthewatertankandthevibratingplatehinderedtheabilitytoobservetheexperimentaldependenceoftheplate'sfrequencyresponseontheappliedhydrostaticpressure.Inthispapersomeresultsarepresentedontheeffectofstaticpressureloadingproducedbyairontheplate,removingthecomplicationsintroducedbyfluidloadingandmodalcoupling. The aim of thiswork is to accuratelymeasure these effects experimentally and identify similarities and anyinterestingdiscrepancieswiththenumericalmodelsusedforprediction.ThefiniteelementpackageABAQUSwasusedtomodelthefrequencyresponseoftheplatesubjectedtoastaticpressurefieldproducedbyafiniteaircavity.

1. INTRODUCTION Adetailedunderstandingofthevibrationalpropertiesofsubmergedobjectssuchassubmarinesisessentialin order tominimize soundemission andoptimise sonarperformance. Theeffect of hydrostatic pressureon themodalfrequenciesofsubmergedhollowstructuresisacomplexproblem.Modellingsucheffectsusingconventionalfiniteelementmethodsandboundaryelementmethodsisdifficult.Theaccuracyofthesemodelsisheavilyrelianton user intuition and non-systematic methods. Significant discrepancies between results and predictions werefoundduringthefrequencyandmodeshapeanalysisofatorpedomodelsubmergedinwater(PanandMatthews,2011).Effortswerethenmadetoexperimentallycharacterisetheeffectsofhydrostaticpressureonthefrequencyresponseofa thin rigidly clampedcircularplate (Saltmarshetal, 2015). This investigationyielded interestingyetambiguousresultsduetothepresenceofacoustic-structuralcoupling.Thepresenceoftankandcoupledplate-tankmodesmade itdifficult to identify theplatemodedependenceon thehydrostaticpressure.Observationsdrawnfrom the frequency response of the plate-tank rig yielded both increasing and decreasing modes, with varyingdependencies. Experimental or coupled theoretical-experimental work in this area is scarce but there aremanyapplicationswheresucheffectsareveryimportantandneedtobeunderstood.

Inthiswork,someresultsarepresentedontheeffectsofastaticpressurefieldproducedbyanaircavityonthe frequency response of the thin rigidly clamped circular plate. Some preliminary experimental results arepresentedandcomparedwiththefiniteelementmodel.Significantfocushasbeenappliedtotheconstructionandverificationofthefiniteelementmodel,withadetaileddescriptionoftheanalysismethodsused.

2. EXPERIMENTAL SETUP Figure1showsaschematicdiagramoftheplatetestrigusedforexperimentalanalysis.Athinsteelplateisclampedrigidlybetweentwosteelflangeswithamuchgreatermassdisparitytosuppresstheeffectsofstructuralcoupling between the plate and the rig. By using such a configuration, it was possible to avoid some of theenvironmentalcomplicationsencounteredinthehydrostaticmeasurementsmentionedabove.Asteelbackingplateisboltedtooneoftheflangestoproduceanenclosedacousticaircavityononesurfaceoftheplate.Thebackingplate possesses a valve to allow the adjustable pressurization of the air cavity. Thin 1mm rubber gasketswere

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placedbetweentheplate,flangeandbackingplatetoensurethatthepressurizedaircavityiseffectivelysealed.Apositive pressure pump was used to increase the static gauge pressure from 0 Pascal to 10 Kilopascal in 0.5Kilopascalincrements.Animpacthammerwasusedtoexcitetheplate.Thismethodofexcitationwasselectedoveramechanicalshakertoeliminatetheeffectsofmassloading.Asingleaccelerometer(PCB352C67)ofmass5gwasusedtomeasuretheaccelerationoftheplateduetothemechanicalexcitation.Boththestrikeandaccelerometerlocationswereselectedasirrationalratiosoftheplate’sradiustoensurenospecificmodeswerebeingsuppressedorexplicitlyexcited.Figure1showstheexperimentalsetupforthetestrig.

Figure1:Schematicoftestrigdetailingcomponents

ThesoftwarepackagePULSE(Brüel&Kjær,2015)wasusedtomeasuretheaccelerometeroutputandimpactforce. The frequency response functions of the plate were then generated through computing the ratio of theaccelerationtoimpactforce.Thisquantityisdefinedasthetransferaccelerance.Thissystemwaslimitedtoa1Hzresolution.

3. FINITE ELEMENT MODEL The finiteelementmodelwasconstructedusing thesoftwarepackageABAQUS (DassaultSystèmes,2014).

Themodelincludestheplate,rubbergaskets,flanges,bolts,backingplateandtheaircavityasdepictedinfigure1.

3.1 Geometric And Material Properties Thegeometricandmaterialpropertiesfortheentiretyofthetestrigaredetailedintable1.Thesewerethe

valuesthatwereusedinthefiniteelementmodelfortheoreticalcomparisontoexperimentalresults.Thedensity,Young’smodulus,Bulkmodulus,Poisson’s ratio,outsidediameter, innerdiameterand thicknessaregivenby thesymbolsρ,E,K,v,OD,IDandt,respectively.



Table1:Experimentaltestrigproperties

3.2 Element Type Due to the disparity between the plate’s thickness and planar dimensions, the shell planar elements S4Rwereused formodelling.Theelement typeS4R isa reduced integrated,general-purpose, finite-membrane-strainshellelement.Theelement'smembraneresponseistreatedwithanassumedstrainformulationthatgivesaccuratesolutionstoin-planebendingproblems,isnotsensitivetoelementdistortion,andavoidsparasiticlocking.Reducedintegration is suitable due to minimal deflection amplitude with hourglass control activated to avoid spuriousdeformationwithinthemesh. Theflanges,bolts,rubbergasketsandbackingplatewereallmodelledusingthesolidelementC3D8R.Theelement typeC3D8R is a reduced integrated,hourglass controlled general purpose3-D stress solid element. Thecoupledmass-stiffnessmatrixfortheflangesandbackingplateeffectivelyresultinminimalstressesandstrainsinthesegeometries. The air cavity was modelled utilising the acoustic element type AC3D8. This element type allows fluid-structural coupling between the air cavity and the test rig. The acoustic element type can only sustain pressuredifferentials due to acoustic loading and hence does not deform due to the plate’s vibration. Energy balancecontributionsarepresentastheacousticenergyisdistributedbetweentheplategeometryandtheaircavity.

3.3 Procedural Steps Theanalysiscomprisedthreeseparateprocedures.Stepsmayfallintoeitherthegeneralanalysisorthelinearperturbation procedural types. The classification of the analysis is determined by the linearity of the response.Simulationsutilising linearandnon-linearanalysiswereapplied todetermine theappropriateclassificationwhenconsideringtheinducedstressanddeflectionoftheplate.Thedeflectionsatthecentreoftheplateasafunctionofpressureareshowninfigure2forthelinearsolution,non-linearsolutionandthetheoreticalsolutionforaloadedclampedplate.Thediscrepancybetweenthetheoreticalandlinearfiniteelementsolutionisduetothenatureofthe clamped boundary condition assumed by eachmodel. The theoretical solution assumes a perfectly clampededgewhilst the finiteelementmodelhasassumeda finitevalueof stiffness for theboundary condition thatwillresultinlargerdisplacementswhichisdepictedbytheplot.Duetothecurvatureofthenon-linearsolutionitwasdeterminedthattheapplicationof thepressurefieldtothesurfaceof theplateovertherangefrom0to10kPayieldsanon-linearresponse.

Geometry Material ρ(Kg/m3) E(GPA) K(GPA) v OD(mm) ID(mm) t(mm)CircularPlate MildSteel 7800 210 175 0.3 580 - 1.6Gasketseal Rubber 1000 1E-3 6.25E-3 0.47 580 412 2Flange MildSteel 7800 210 175 0.3 580 412 32

BackingPlate MildSteel 7800 210 175 0.3 580 - 32AcousticCavity Air 1.225 - 1.01E-4 - 412 - 32

Bolts MildSteel 7800 210 175 0.3 - - -



Figure2:Comparisonsforthecenterplatedeflectionsoverthepressurerange0to10kPa

The selectedanalysis stepwas then chosen tobeaStatic,General step. The resulting stressesand strainsproducedaretransferredtothebasestateforthefollowingstep.Theanalysisistimeindependentandhencetheloadisappliedincrementally,withthestiffnessandmassmatricesupdatedaccordingly.Thedetailsforthisstepareoutlinedintable2.

Table2:Static,Generalstepdetails

Parameter Value

ProcedureName Static

ProcedureType General

Non-LinearGeometry Yes

AutomaticStabilisation Yes

EnergyFraction 0.0002

Adaptivestabilisation Yes

MaxRatiotoStrainEnergy 0.05 The succeedingFrequency step is anEigenmodeextraction step to solve for the resonant frequencies andcorrespondingmodeshapesofthesystem.Thisstep isa linearperturbationstepthatyieldsthenaturalvibrationpropertiesofthesystemnotduetoexcitation.Theeigenvalueproblemforthenaturalfrequenciesofanundampedfiniteelementmodelisgivenbyequation1.

−𝜔2 𝑀𝑀𝑁 + 𝐾𝑀𝑁 𝜙𝑁 = 0 (1)



Where 𝑀!" is the mass matrix (symmetric and positive definite), 𝐾!" is the stiffness matrix, 𝜙! iseigenvector(themodeofvibration)andMandNaredegreesoffreedom.Thedetailsforthisstepareoutlinedintable3.

Table3:Frequencystepdetails

Parameter Value

ProcedureName Frequency

ProcedureType LinearPerturbation

Eigensolver Lanczos

Acoustic-StructuralCoupling Yes

EigenvectorNormalisation ByMass

MaxFrequency 1000Hz

ThefinalprocedureistheModalDynamicsstep.Thisisatransientlineardynamicsprocedurethatsolvesthe

systemthroughmodalsuperpositioning.TheEigenmodesused in thisanalysisarethoseofwhichweresolved intheFrequencystep.Animpulseloadisappliedtotheplateattheexperimentalstrikinglocationdepictedinfigure1.Theexcitationisgivenasafunctionoftimeandthatitsmagnitudevarieslinearlywitheachincrement.WhenthemodelisprojectedontotheEigenmodesthefollowingsetofequationsareobtainedattimet.

𝑞𝛽 + 𝐶𝛽𝛼𝑞𝛼 + 𝜔𝛽

2𝑞𝛽 = 𝑓𝑡 𝛽= 𝑓𝑡−Δ𝑡 +

ΔfΔ𝑡Δ𝑡 (2)

Where the𝛼 and𝛽 indices span the eigenspace,𝐶!" is the projected viscous damping matrix, 𝑞! is the

responseamplitudeofthemode𝛽,𝜔! = 𝑘!/𝑚! isthenaturalfrequencyoftheundampedmode𝛽, 𝑓! ! isthemagnitudeoftheloadingprojectedontothismodeandΔ𝑓isthechangeinfrequencyoverthetimeincrementΔ𝑡.Thesolution to theuncoupledequations isobtainedasaparticular integral for the loadingandasolution to thehomogenousequation.Thesesolutionscanbecombinedtogiveequation3.

𝑞!!!"𝑞!!!" =

𝑎!! 𝑎!"𝑎!" 𝑎!!

𝑞!𝑞! + 𝑏!! 𝑏!"

𝑏!" 𝑏!!𝑓!

𝑓!!!" (3)

Where𝑎!" and𝑏!" areconstantsastheloadingvarieslinearlyoverthetimeincrement.Thedynamicresponse

of the plate inclusive of stresses, strains and accelerations are then calculated for a specified time period. Thesampling frequencyandtimeperiodof theanalysiswereselectedtobe2Kilohertzand1secondrespectively, toproduceaNyquistfrequencyof1000Hzwitha1Hzresolutionakintotheexperiment.Thedetailsofthisstepareoutlinedintable4.



Table4:Modaldynamicsstepdetails

Parameter Value

ProcedureName ModalDynamics

ProcedureType LinearPerturbation

TimePeriod 1s

TimeIncrement 0.0005s

CriticalDampingFraction 0.002

LoadType Impulse

LoadTime 0.005s

3.4 Interactions Interaction pairs have been created between the plate, flanges, backing plate, bolts and air cavity. The

interactions between the bolts, flanges and backing plateweremodelled as surface-to-surface contact with thefriction coefficientof steelon steel as0.05. The rubber gaskets are connected toboth theplateand the flangesusing the tieconstraint thatensuresconstantcontact.Theaircavity isconnectedtoasinglesurfaceof theplateusingthetieconstraint.Theairmeshisselectedastheslavesurfacesothatthenodeswillconformtomatchthedeformationoftheplatenodes.

3.5 Boundary Conditions And Loads Fixedboundaryconditionswereappliedtotheinterfacebetweentherubbergasketsandsteelflanges,bolts

andthe interfacebetweentheplateandthebolts.Applicationof thefixedboundarytotherubbergasketratherthantheplate introducesdampingattheboundaryconditionsthat iswhat isobservedexperimentally.Fixingtheboltssupressesthevibrationofthevirtuallyrigidsteelflangesandbackingplate. The impulse loadapplied in theModalDynamics stepwasdesignedas a ramp loadover a timeperiodof0.005seconds.Theintegraloftheimpulseloadyieldsatotalappliedforceof1Ntosimplifyfindingtheresponse.

4. RESULTS Thepurposeof this finiteelementmodelling is to characterize theeffectsofa staticpressure fieldon the

frequency responseofplate structures, inparticular a circularplatewith clampededges.Comparisonsaremadebetweentheexperimentalresultsandfiniteelementmodelfirstlyforambientconditionswithnopressurefieldtoensuregoodagreementisfoundandtohencevalidatethemodel.Comparisonsarethenmadeforthephenomenonobservedwithinthefrequencyresponseoftheplatewiththeapplicationofthestaticpressurefield.

4.1 Plate Response at Ambient Pressure Figure2showsthetransferaccelerancefunctionsfortheplaterigforbothexperimentalandfiniteelementmethods at ambient pressure conditions. Good agreement is found between the two methods up to the 8thresonantfrequencyinbothpeaklocationandresponsefunctionshape.



Figure3:Transferaccelerancefromexperimentalandfiniteelementresultsatambientpressure

Table5detailsthecomparisonsoftheresonantfrequenciesforthefirst6peaks.Themodenumber(m,n)isalsogivenwheremandn represent thenumberofdiametralmodesandnumberofcircularmodes respectively.Generalised,mrepresentsthemodenumberwhilstnrepresentstheharmonicnumber.Thediscrepanciesbetweentheexperimentalresultsandfiniteelementmodelacrossthesepeaksrangedfrom0.58to3.82%,withanaverageof2.15%.Itcanbeseenthatthefiniteelementmodelcannotaccuratelymodeltheexperimentalresponsefunctionforhigherordermodesasitsaccuracydeclinesheavilybeyond800Hertz.Withtheexceptionofthemodelsplittingobservedintheexperimentalpeaksnumbered2and6,thecharacteristicsoftheresponsefunctionshapearealsomodeledaccurately.Thiscanbenoticedwhenobservingtheregionsbetweenpeaksaswellasthetrendinthepeakamplitudes themselves. This level of agreement validates the ability of the finite elementmodel to predict theresponseofthetestrigforthepressuresimulations.

Table5:Experimentalandfiniteelementresonantfrequencycomparison

Peak Mode Experimental(Hz) Numerical(Hz) Difference(%)

1 (0,1) 101 104 2.97

2 (1,1) 177,165 181 2.26

3 (2,1) 288 299 3.82

4 (0,2) 346 344 0.58

5 (3,1) 432 438 1.39

6 (1,2) 535,546 525 1.87



4.2 Plate Response With Applied Pressure The application of the static pressure field to one surface of the vibrating plate was found to producesignificantchangestothefrequencyresponsecharacteristicsforboththeexperimentalandfiniteelementmethods.Figures 3 and figure 4 depict the waterfall plots for the experimental and finite element transfer accelerancefunctionsrespectively,rangingfrom0Pascalto10Kilopascalin0.5Kilopascalincrements.

Through inspectiontheresonant frequenciesof theresponse functionscorrespondingto themodesof thevibratingplate areobserved to increase. This is as expected as the applicationof thepressureproduces tensionwithintheplate(Leissa,1969).Unliketheresultsfromthehydrostaticpressure(Saltmarshetal,2015),theresultsobtainedhereallowthedirectcomparisonwiththeoreticalpredictionaseachmodecanbeconfidentlyidentified.Table6summarisestheobservedresonantfrequencymovementswithpressure.

Figure4:Waterfallplotdepictingthevariationinthetransferaccelerancefortheexperimentalmeasurements

rangingfrom0kPato10kPain0.5kPaincrements



Figure5:Waterfallplotdepictingthevariationinthetransferacceleranceforthefiniteelementsimulationranging

from0kPato10kPain0.5kPaincrements

Table6:Increasesinexperimentalandfiniteelementresonantfrequencycomparisonoveranairpressureincrease

of10kPa

Peak Mode ExperimentalShift(%) NumericalShift(%)

1 (0,1) 80.20 66.35

2 (1,1) 41.21,35.59 28.73

3 (2,1) 17.36 14.38

4 (0,2) 21.39 18.31

5 (3,1) 9.26 7.99

6 (1,2) 11.78,11.36 10.86



Thecomparisonbetweentheexperimentalandfiniteelementresultsshowsgoodagreementformodes3to6. The discrepancies found between the resonant frequency shifts for modes 1 and 2 are fairly considerable,howeverthetrendbetweenthedegreeofmovementofeachmodeis identical.Boththeexperimentalandfiniteelement results highlight that there is a clear dependence between the resonant frequencymovement and theassociatedmodenumber. It isobservedthatthere isadecreasingdependencywith increasingmodenumberm,whichisevidentwhencomparingtheshiftsharmonicmodes(0,2)and(1,2)withfundamentalmodes(2,1)and(3,1).

5. CONCLUSIONS It has been shown both experimentally and theoretically that the application of a static pressure field

producedbyanair cavity toonesurfaceofavibratingplate increases the resonant frequenciesof thestructure.Substantial increases were observed for relatively small pressures, however it is noted that the geometry andstiffness of the disc ultimately determine the subsequent stresses and strains produced in the plate. Thesepreliminaryresultshowevershowthecriticalityoftheseeffectsonthefrequencyresponseofplatestructures,andvalidatetheirtheoreticalcounterpartsproducedwithfiniteelementsoftwareABQUS.Furtherinvestigationintotheeffectsonthemodeshapes,theconsequentsoundradiatedpressureandthestrainproducedintheplatewillyieldanin-depthexperimentalandtheoreticalcharacterizationoftheeffectofpressurefieldsonthefrequencyresponseofvibratingstructures.

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DassaultSystèmes,2014,ABAQUS6.14-1,softwareprogram,DassaultSystèmes,Vélizy-Villacoublay,France.

Leissa, A.W. (1969), Vibration of Plates, Scientific and Technical Information Division, National Aeronautics andSpaceAdministration,Washington.

Liu,W,etal(2010),‘Measurementofsoundradiationfromtorpedo-shapedstructuresubjecttoanaxialexcitation’,Proceedingsof20thInternationalCongressonAcoustics.Availablefrom:ICAdigitallibrary.[August2010].

Pan,JandMatthews,D(2011)‘Analysisofunderwatervibrationofatorpedo-shapedstructuresubjectedtoaxialexcitation’,ProceedingsofAcoustics2011,the2011ConferenceoftheAustralianAcousticalSociety.Availablefrom:AASdigitallibrary.[November2011].

MatthewsD,etal(2014),‘Adetailedexperimentalmodalanalysisofaclampedcircularplate’,ProceedingsofInterNoise2014,the2014ConferenceoftheAustralianAcousticalSociety.Availablefrom:AASdigitallibrary.[November2014].

SaltmarshK,(2014),‘Theeffectsofsoundradiationenvironmentonstructuralvibration’,Honoursthesis,SchoolofMechanicalandCivilEngineering,TheUniversityofWesternAustralia.

SaltmarshK,etal(2015),‘Effectofwaterpressureonthefrequencyresponseofathinrigidlyclampedplate’,inYZhou,A.DLucey,YLiuandLHuang(eds),FSSIC2015:Fluid-Structure-SoundInteractionandControl;“Proceedingsofthe3rdSymposiumonFSSIC”,CurtinUniversity,Perth,WesternAustralia,pp.89-94.