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manual

MeasurementProcedureandProgramCalculationsforDeterminationofDynamicPropertiesforSandwichbeams

EvaNilsson

CONTENTS

Abstract3

Dynamicpropertiesofsandwichbeam3

Bendingstiffness3

Wavenumbers3

Lossfactors6

Measurementprocedure7

Determiningtheeigenfrequencies7Boundaryconditions8

Typeofexcitation9

Vibrationmeasurement9

Measurementandexcitationpoints10

Repeatability10

Effectoflowdamping10

Laminatesmovinginphase10

Addedmassduetosurroundingfluid12

Programcalculations13

Leastsquaremethodlimitations13

ABSTRACT

Thefrequencydependentbendingstiffness,E-modulusandshearmoduluscanbedeterminedforsandwichstructuresusingsimplemeasurements.Themeasurementsarenon-destructiveandthecharacteristicsofeachlayercanbedeterminedmakingmeasurementsontheentirestructure.Themodelcanalsoeasilybeusedforparameterstudies.Themethodisshowntoworkforfoamaswellashoneycombcores.

ThisisabriefdiscussiononthemeasurementprocedureusedtodeterminesomedynamicpropertiesofasandwichstructureandtheMATLABprogramusedforthecalculations.Thetheoryispresentedinref.[1].

DYNAMICPROPERTIESOFBEAM

Bendingstiffness

Forabeamwithboundaryconditionswelldefined,thebendingstiffnesscanbedeterminedbymeansofsimplemeasurements.Theapparentbendingstiffness

formodenhavingtheeigenfrequency

isforabeam,length

andmassperunitarea

givenby

forn=1,2,3….

（1）

wherenisgiveninTable1forthreeboundaryconditions.

TABLE1

n

1

2

3

4

5

n>5

free-free

clamped-clamped

n

4.73

7.85

11.00

14.14

17.28

n

+

/2

free-clamped

n

1.88

4.69

7.85

11.0

14.14

n

-

Forasimplysupportedbeamn=n.Measurementsrevealthefirstfeweigenfrequenciesofthebeam.Thebendingstiffnessofacompositepanelisstronglyfrequencydependent.Dxcanbedeterminedfrom

（2）

where

;

;

（3）

Howeverwithinthefrequencyrangeofinterest,hereupto4kHz,theparametersD1,D2and

inequation（3）areassumedtobeconstantforthestructuresinvestigated.UsingthemeasureddatatheparametersA,BandCcanbedeterminedbymeansoftheleastsquaremethod.ThequantityQ1isdefinedby

（4）

whereDxiisthemeasuredbendingstiffnessatthefrequency

formodei.TheparametersA,BandCarechosentogivetheminimumofQ1.Theshearmodulus

andthebendingstiffnesses

andD2canbedetermined,oncetheparametersA,BandCarecalculatedfromequation

（2）.

Thedynamicperformanceofabeamisdeterminedbytheexpression

（2）.TheparametersA,BandCinequation

（2）arefunctionsoftheshearmodulus

andthebendingstiffnessesD1andD2.Foreacheigenfrequencyfnthecorrespondingapparentbendingstiffness

isdeterminedfromequation

（1）.Atleastthreesetsofdata,fnandDxn,determinetheparametersA,BandCandthereby

D1andD2asdiscussedin[1].

Figure1.Predictedbendingstiffness.

______,BeamCy;-.-.,BeamCa,increasedshear;

---BeamCb,increasedbendingstiffnessoflaminates;

***,measuredresults,BeamCy.

Byusingequation

（2）todeterminetheparametersA,BandCthelowfrequencymeasurementswilldominate.TheparameterA,whichdependsonthestaticbendingstiffnessD1isthereforedeterminedfromequation

（2）.Ifinsteadthequantitytobeminimisedisdefinedas

（5）

thehighfrequencyresponseofthebeamwilldominate.WhenusingAderivedfromequation

（2）theparameterCisdeterminedfromequation（5）bysettingQ2/B=0andQ2/C=0.InthemidfrequencyregionQiswrittenas

（6）

whereAandCarenowknownparameters.TheparameterBisthendeterminedbysettingQ3/B=0.WithknownconstantsA,BandCthedynamicparameterscanbedeterminedaccordingtoequation3.

Wavenumbers

（27）

Thewavenumber

forthefirstpropagatingwavecanbewrittenas

whereDxistheapparentbendingstiffnessofthestructure.ConsequentlyDxcan,asafirstapproximation,bedefinedasthebendingstiffnessofasimplehomogeneousbeamwhichatacertainfrequencyhasthesamedynamicpropertiesasthehoneycombstructure.Byinsertingthedefinitionof

inthewaveequationanequationin

isobtained.Theresultingexpressioncangenerallybesimplifiedwhenever

and

.Forthestructuresdiscussedheretheseassumptionsholdforfrequenciesbelow4kHzasshownin[1].Theapparentbendingstiffness

isconsideringtheseapproximationsobtainedasthesolutiontotheequation

（7）

Inthelowfrequencyrange,oras

thefirstpartoftheequationdominateswhy

.Thebendingstiffnessisconsequentlydeterminedbypurebendingofthebeam.Inthehighfrequencyrange,when

.Thelaminatesareassumedtomoveinphase.Inthisfrequencyrangethebendingstiffnessfortheentirebeamisequaltothesumofthebendingstiffnessofthelaminates.

Withknownmaterialparametersthewavenumberforthestructurecanbedetermined.Theeigenfrequenciescanbedeterminedusingthepropagatingwavenumberforknownboundaryconditions.Theseeigenfrequenciescaninturnbeusedtodeterminethebendingstiffnessofthebeam.Forthecalculationofthewavenumberthemomentofinertianeedstobedetermined.Todothisthecoreandlaminatedensitiesshouldbeknown.Ifthecoreorlaminatedensitiescannotbedeterminedtheycanbesettozero,whichgivesI=0.WhenusingI=0theresultfor2isimaginaryintheentirefrequencyrangeinsteadofshiftingfromimaginarytorealat

.Forthebendingstiffnesscalculationsthemomentofinertiacanbeomitted.Thoughfortheinformationforthewavenumberandtherebythewavepropagationinthestructurethemomentofinertiashouldbeincluded.

Lossfactors

Thelossfactorisdeterminedaccordingtothehalfpowerbandwidthmethoddirectlyfromthetransferfunction.Alineisdrawn3dBdownfromthemaximumamplitudeofthepeak.Thenthelossfactorisdeterminedas

.ThisisshowninFigure2.

ddf

frequency（Hz）

H（dB）

Figure2Halfpowerbandwidthmethod

Therepeatabilityforthelossfactordeterminationisverypoor.Thelossfactorcanthereforeonlybeestimated,basedonanaverageofseveralmeasurements.Forsandwichstructuresthelossfactorisfrequencydependent.

MEASUREMENTPROCEDURE

Measurementsshouldbemadeonbeamstodeterminetheeigenfrequencies.Usingthetheorypresentedinthepreviouschaptersthefrequencydependentcharacteristicscanbedetermined.Limitationsandproblemsduringthemeasurementsarediscussedhere.

Fororthotropicstructuresbeamscanbecutinthetwomaindirectionsofthestructure.Withknowncharacteristicsforthetwomaindirectionstheplatecharacteristicscanbedeterminedasshowninref.[1].

Determiningtheeigenfrequencies

TheeigenfrequenciesaredeterminedfromtheFRFmeasuredforthebeam.TheFRFisdeterminedfortherelativelevelofthevelocitydividedbytheforceusedfortheexcitation.Foreachoftheeigenfrequencypeakstheapparentbendingstiffnessisdeterminedaccordingtoequation

（1）forknownboundaryconditions.Thebendingstiffnessofthesandwichbeamshouldbesteadilydecreasing.Thepeaksgivingdiscontinuouslydecreaseshouldthereforebeexcluded.Thesepeaksmightrepresenteigenfrequenciesfortheentiremeasurementsystemincludingstringsandmeasurementequipmentandtwistingmodes.Thesepeaksshouldbeexcludedwhendeterminingthebendingstiffnessofthebeam.AsseenfromFigure3wheretheeigenfrequenciesforbendingwavesforthebeamaremarked.

Figure3Frequencyresponsefunctionforbeam.

Forlowfrequenciesthepeaksareusuallyeasilyresolved.Dependingonthelengthofthebeamandthetypeofexcitationanumberofeigenfrequenciescanbedetermined.Ifthebeamistooshortthefirsteigenfrequencyishighupinfrequencyandthebendingstiffnesscannotbedeterminedinthelowfrequencyrangeexceptfromanapproximation.

Boundaryconditions

Usuallythefree–freeboundaryconditionsaretheeasiesttomakerealistic.ThiscouldbedoneusingtwostringsasshowninFigure4.Thestringsshouldbelongascomparedtothewidthofthebeamtoallowthebeamtovibratefreely.

Figure4Mountingofbeamduringmeasurement.

Theboundaryconditionshavecertaineffectonthebendingstiffnessofthestructure.Thisisduetothefactthatshearinthebeamisinducedbyclampedboundariestoalargeextentascomparedtothecasewithfreeboundaries.

Thefigure4showsthemeasuredapparentbendingstiffnessforthesamebeamwithvariousboundaryconditions.

Figure5Measuredbendingstiffnessforahoneycombbeam

o-free-freeand*-clamped-free

Typeofexcitation

Forlightweightstructuressuchashoneycombstructuressolidlymountedshakershouldnotbeused.Itwouldchangethecharacteristicsofthestructureconsiderably.Animpacthammeristhereforepreferablyusedtoexcitethestructure.Nomassisthenaddedtothestructure.Thestructureisgivenoneimpactstartingthemeasurementperiod.Averagesaremadefor5-12differentexcitationpoints.Thedrawbackwiththeimpacthammeristhoughthedifficultytoexcitethehighermodesofthebeam.AsseenfromtheFRFthismakesitdifficulttoseparatetheeigenfrequenciesforincreasingfrequencies.

Foragoodmeasurementresultitisimportanttohitthestructureperpendicularwiththehammer.Incaseofdoublehits,thehammerisbouncingonthebeam,themeasurementshouldnotbeincludedinthemeasurementseries.

Vibrationmeasurement

Thevibrationsofthebeamcanbedeterminedusingasmallaccelerometer.Forlightweightstructuresitisthoughpreferabletouseanon-contactmeasurementsystemsuchasalase