中文6680字TheTwo-DimensionalDynamicBehaviorofConveyorBeltsIr.G.Lodewijks,DelftUniversityofTechnology,TheNetherlands1.SUMMARY1--------Inthispaperanewfiniteelementmodelofabelt-conveyorsystemwillbeintroduced.Thismodelhasbeendevelopedinordertobeabletosimulateboththelongitudinalandtransversedynamicresponseofthebeltduringstartingandstopping.Applicationofthemodelinthedesignstageoflongoverlandbelt-conveyorsystemsenablestheengineer,forexample,todesignproperbelt-conveyorcurvesbydetectingprematureliftingofthebeltofftheidlers.Italsoenablesthedesignofoptimalidlerspacingandtroughingconfigurationinordertoensureresonancefreebeltmotionbydetermining(standing)longitudinalandtransversebeltvibrations.Applicationoffeed-backcontroltechniquesenablesthedesignofoptimalstartingandstoppingprocedureswhereasanoptimalbeltcanbeselectedbytakingthedynamicpropertiesofthebeltintoaccount.2.INTRODUCTION2--------TheNetherlandshaslongbeenrecognisedasacountryinwhichtransportandtranshipmentplayamajorroleintheeconomy.TheportofRotterdam,inparticularisknownasthegatewaytoEuropeandclaimstohavethelargestharboursystemintheworld.Besidesthelargenumbersofcontainers,alargevolumeofbulkgoodsalsopassesthroughthisport.NotallthesegoodsareintendedfortheDutchmarket,manyhaveotherdestinationsandaretranshippedinRotterdam.Goodexamplesoftypicalbulkgoodsthataretranshippedarecoalandironore,asignificantpartofwhichisintendedfortheGermanmarket.Inordertohandlethebulkmaterialsawiderangeofdifferentmechanicalconveyorsincludingbelt-conveyorsisused.3--------Thelengthofmostbelt-conveyorsystemserectedintheNetherlandsisrelativelysmall,sincetheyaremainlyusedforin-plantmovementofbulkmaterials.Thelongestbelt-conveyorsystem,whichisabout2kmlong,issituatedontheMaasvlakte,partoftheportofRotterdam,whereitisusedtotransportcoalfromabulkterminaltoanelectricitypowerstation.Inadditiontodomesticprojects,anincreasingnumberofDutchengineeringconsultanciesparticipatesininternationalprojectsforthedevelopmentoflargeoverlandbelt-conveyorsystems.Thisdemandstheunderstandingoftypicaldifficultiesencounteredduringthedevelopmentofthesesystems,whicharestudiedintheDepartmentofTransportTechnologyoftheFacultyofMechanicalEngineering,DelftUniversityofTechnology,oneofthethreeDutchUniversitiesofTechnology.4--------Theinteractionbetweentheconveyorbeltproperties,thebulksolidsproperties,thebeltconveyorconfigurationandtheenvironmentallinfluencetheleveltowhichtheconveyor-systemmeetsitspredefinedrequirements.Someinteractionscausetroublesomephenomenasoresearchisinitiatedintothosephenomenawhichcausepracticalproblems,[1].Onewaytoclassifytheseproblemsistodividethemintothecategorywhichindicatetheirunderlyingcausesinrelationtothedescriptionofbeltconveyors.5---------Thetwomostimportantdynamicconsiderationsinthedescriptionofbeltconveyorsarethereductionoftransientstressesinnon-stationarymovingbeltsandthedesignofbelt-conveyorlay-outsforresonance-freeoperation,[2].Inthispaperanewfiniteelementmodelofabelt-conveyorsystemwillbepresentedwhichenablesthesimulationofthebelt'slongitudinalandtransverseresponsetostartingandstoppingproceduresandit'smotionduringsteadystateoperation.It'sbeyondthescopeofthispapertodiscusstheresultsofthesimulationofastart-upprocedureofabelt-conveyorsystem,thereforeanexamplewillbegivenwhichshowsomepossibilitiesofthemodel。3.FINITEELEMENTMODELSOFBELT-CONVEYORSYSTEMS6--------Ifthetotalpowersupply,neededtodriveabelt-conveyorsystem,iscalculatedwithdesignstandardslikeDIN22101thenthebeltisassumedtobeaninextensiblebody.ThisimpliesthattheforcesexertedonthebeltduringstartingandstoppingcanbederivedfromNewtonianrigidbodydynamicswhichyieldsthebeltstress.Withthisbeltstressthemaximumextensionofthebeltcanbecalculated.Thiswayofdeterminingtheelasticresponseofthebeltiscalledthequasi-static(design)approach.Forsmallbelt-conveyorsystemsthisleadstoanacceptabledesignandacceptableoperationalbehaviorofthebelt.Forlongbelt-conveyorsystems,however,thismayleadtoapoordesign,highmaintenancecosts,shortconveyor-componentlifeandwellknownoperationalproblemslike:excessivelargedisplacementoftheweightofthegravitytake-updeviceprematurecollapseofthebelt,mostlyduetothefailureofthesplicesdestructionofthepulleysandmajordamageoftheidlersliftingofthebeltofftheidlerswhichcanresultinspillageofbulkmaterialdamageandmalfunctioningof(hydrokinetic)drivesystemsManyresearchersdevelopedmodelsinwhichtheelasticresponseofthebeltistakenintoaccountinordertodeterminethephenomenaresponsiblefortheseproblems.Inmostmodelsthebelt-conveyormodelconsistsoffiniteelementsinordertoaccountforthevariationsoftheresistance'sandforcesexertedonthebelt.Theglobalelasticresponseofthebeltismadeupbytheelasticresponseofallitselements.Thesefiniteelementmodelshavebeenappliedincomputersoftwarewhichcanbeusedinthedesignstageoflongbelt-conveyorsystems.Thisiscalledthedynamic(design)approach.Verificationoftheresultsofsimulationhasshownthatsoftwareprogramsbasedonthesekindofbelt-modelsarequitesuccessfulinpredictingtheelasticresponseofthebeltduringstartingandstopping,seeforexample[3]and[4].Thefiniteelementmodelsasmentionedabovedetermineonlythelongitudinalelasticresponseofthebelt.Thereforetheyfailintheaccuratedeterminationof:themotionofthebeltovertheidlersandthepulleysthedynamicdrivephenomenathebendingresistanceofthebeltthedevelopmentof(shock)stresswavestheinteractionbetweenthebeltsagandthepropagationoflongitudinalstresswavestheinteractionbetweentheidlerandthebelttheinfluenceofthebeltspeedonthestabilityofmotionofthebeltthedynamicstressesinthebeltduring.passageofthebeltovera(driven)pulleytheinfluenceofparametricresonanceofthebeltduetotheinteractionbetweenvibrationsofthetakeupmassoreccentricitiesoftheidlersandthetransversedisplacementsofthebeltthedevelopmentofstandingtransversewavestheinfluenceofthedampingcausedbybulkmaterialandbythedeformationofthecross-sectionalareaofthebeltandbulkmaterialduring,passageofanidlertheliftingofthebeltofftheidlersinconvexandconcavecurvesThetransverseelasticresponseofthebeltisoftenthecauseofbreakdownsinlongbelt-conveyorsystemsandshouldthereforebetakenintoaccount.Thetransverseresponseofabeltcanbedeterminedwithspecialmodelsasproposedin[5]and[6],butitismoreconvenienttoextendthepresentfiniteelementmodelswithspecialelementswhichtakethisresponseintoaccount.3.1THEBELTAtypicalbelt-conveyorgeometryconsistingofadrivepulley,atailpulley,averticalgravitytake-up,anumberofidlersandaplatesupportisshowninFigure1.Thisgeometryistakenasanexampletoillustratehowafiniteelementmodelofabeltconveyorcanbedevelopedwhenonlythelongitudinalelasticresponseofthebeltisofinterest.Sincethelengthofthebeltpartbetweenthedrivepulleyandthetake-uppulley,Is,isnegligiblecomparedtothelengthofthetotalbelt,L,thesepulleyscanmathematicallybecombinedtoonepulleyaslongasthemassinertia'softhepulleysofthetake-upsystemareaccountedfor.Sincetheresistanceforcesencounteredbythebeltduringmotionvaryfromplacetoplacedependingontheexactlocal(maintenance)conditionsandgeometryofthebeltconveyor,theseforcesaredistributedalongthelengthofthebelt.Inordertobeabletodeterminetheinfluenceofthesedistributedforcesonthemotionofthebelt,thebeltisdividedintoanumberoffiniteelementsandtheforceswhichactonthatspecificpartofthebeltareallocatedtothecorresponding,element.Iftheinterestisinthelongitudinalelasticresponseofthebeltonlythenthebeltisnotdiscreditedonthoseplaceswhereitissupportedbyapulleywhichdoesnotforceitsmotion(slippossible).Thelaststepinbuilding,themodelistoreplacethebelt'sdriveandtensioningsystembytwoforceswhichrepresentthedrivecharacteristicandthetensionforces.Theexactinterpretationofthefiniteelementsdependsonwhichresistance'sandinfluencesoftheinteractionbetweenthebeltanditssupportingstructurearetakenintoaccountandthemathematicaldescriptionoftheconstitutivebehaviorofthebeltmaterial.Dependingonthisinterpretation,theelementscanberepresentedbyasystemofmasses,springsanddashpotsasisshowninFigure1,[9],wheresuchasystemisgivenforonefiniteelementwithnodalpointscandc+1.ThespringsKanddashpotHrepresentthevisco-elasticbehaviorofthebelt'stensilemember,Grepresentsthebelt'svariablelongitudinalgeometricstiffnessproducedbytheverticalactingforcesonthebelt'scrosssectionbetweentwoidlers,Vrepresentthebeltsvelocitydependentresistance's.Figure1:Fiveelementcompositemodel[9].3.1.1NONLINEARTRUSSELEMENTIfonlythelongitudinaldeformationofthebeltisofinterestthenatrusselementcanbeusedtomodeltheelasticresponseofthebelt.AtrusselementasshowninFigure2hastwonodalpoints,pandq,andfourdisplacementparameterswhichdeterminethecomponentvectorx:xT=[upvpuqvq]

(1)Forthein-planemotionofthetrusselementtherearethreeindependentrigidbodymotionsthereforeonedeformationparameterremainswhichdescribesFigure2:Definitionofthedisplacementsofatrusselementthechangeoflengthoftheaxisofthetrusselement[7]:ε1=D1(x)=∫1ods2-ds2odξ

(2)2ds2owheredsoisthelengthoftheundeformedelement,dsthelengthofthedeformedelementandξadimensionlesslengthcoordinatealongtheaxisoftheelement.Figure3:StaticsagofatensionedbeltAlthoughbending,deformationsarenotincludedinthetrusselement,itispossibletotakethestaticinfluenceofsmallvaluesofthebeltsagintoaccount.Thestaticbeltsagratioisdefinedby(seeFigure3):K1=δ/1=q1/8T

(3)whereqisthedistributedverticalloadexertedonthebeltbytheweightofthebeltandthebulkmaterial,1theidlerspaceandTthebelttension.Theeffectofthebeltsagonthelongitudinaldeformationisdeterminedby[7]:εs=8/3K2s

(4)whichyieldsthetotallongitudinaldeformationofthenonlineartrusselement:3.1.2BEAMELEMENTFigure4:Definitionofthenodalpointdisplacementsandrotationsofabeamelement.Ifthetransversedisplacementofthebeltisbeingofinterestthenthebeltcanbemodelledbyabeamelement.Alsoforthein-planemotionofabeamelement,whichhassixdisplacementparameters,therearethreeindependentrigidbodymotions.Thereforethreedeformationparametersremain:thelongitudinaldeformationparameter,ε1,andtwobendingdeformationparameters,ε2andε3.Figure5:ThebendingdeformationsofabeamelementThebendingdeformationparametersofthebeamelementcanbedefinedwiththecomponentvectorofthebeamelement(seeFigure4):xT=[upvpμpuqvqμq]

(5)andthedeformedconfigurationasshowninFigure5:ε2=D2(x)=e2p1pq

(6)1oε3=D3(x)=-eq21pq1o3.2THEMOVEMENTOFTHEBELTOVERIDLERSANDPULLEYSThemovementofabeltisconstrainedwhenitmovesoveranidlerorapulley.Inordertoaccountfortheseconstraints,constraint(boundary)conditionshavetobeaddedtothefiniteelementdescriptionofthebelt.Thiscanbedonebyusingmulti-bodydynamics.Theclassicdescriptionofthedynamicsofmulti-bodymechanismsisdevelopedforrigidbodiesorrigidlinkswhichareconnectedbyseveralconstraintconditions.Inafiniteelementdescriptionofa(deformable)conveyorbelt,wherethebeltisdiscretisedinanumberoffiniteelements,thelinksbetweentheelementsaredeformable.Thefiniteelementsareconnectedbynodalpointsandthereforesharedisplacementparameters.Todeterminethemovementofthebelt,therigidbodymodesareeliminatedfromthedeformationmodes.Ifabeltmovesoveranidlerthenthelengthcoordinateξ,whichdeterminesthepositionofthebeltontheidler,seeFigure6,isaddedtothecomponentvector,e.g.(6),thusresultinginavectorofsevendisplacementparameters.Figure6:Beltsupportedbyanidler.Therearetwoindependentrigidbodymotionsforanin-planesupportedbeamelementthereforefivedeformationparametersremain.Threeofthem,ε1,ε2andε3,determinethedeformationofthebeltandarealreadygivenin3.1.Theremainingtwo,ε4andε5,determinetheinteractionbetweenthebeltandtheidler,seeFigure7.Figure7:FEMbeamelementwithtwoconstraintconditions.Thesedeformationparameterscanbeimaginedasspringsofinfinitestiffness.Thisimpliesthat:ε4=D4(x)=(rξ+uξ)e2-rid.e2=0ε5=D5(x)=(rξ+uξ)e1-rid.e1=0

(7)Ifduringsimulationε4>0thenthebeltisliftedofftheidlerandtheconstraintconditionsareremovedfromthefiniteelementdescriptionofthebelt.3.3THEROLLINGRESISTANCEInordertoenableapplicationofamodelfortherollingresistanceinthefiniteelementmodelofthebeltconveyoranapproximateformulationforthisresistancehasbeendeveloped,[8].Componentsofthetotalrollingresistancewhichisexertedonabeltduringmotionthreepartsthataccountforthemajorpartofthedissipatedenergy,canbedistinguishedincluding:theindentationrollingresistance,theinertiaoftheidlers(accelerationrollingresistance)andtheresistanceofthebearingstorotation(bearingresistance).Parameterswhichdeterminetherollingresistancefactorincludethediameterandmaterialoftheidlers,beltparameterssuchasspeed,width,material,tension,theambienttemperature,lateralbeltload,theidlerspacingandtroughangle.Thetotalrollingresistancefactorthatexpressestheratiobetweenthetotalrollingresistanceandtheverticalbeltloadcanbedefinedby:ft=fi+fa+fb

(8)wherefiistheindentationrollingresistancefactor,fatheaccelerationresistancefactorandfbthebearingsresistancefactor.Thesecomponentsaredefinedby:Fi=CFznzhnhD-nDVbnvK-nkNTnT

(9)fa=Mred?2u

Fzb

?t2fb=

Mf

FzbriwhereFzisdistributedverticalbeltandbulkmaterialload,hthethicknessofthebeltcover,Dtheidlerdiameter,Vbthebeltspeed,KNthenominalpercentbeltload,Ttheambienttemperature,mredthereducedmassofanidler,bthebeltwidth,uthelongitudinaldisplacementofthebelt,Mfthetotalbearingresistancemomentandritheinternalbearingradius.Thedynamicandmechanicpropertiesofthebeltandbeltcovermaterialplayanimportantroleinthecalculationoftherollingresistance.Thisenablestheselectionofbeltandbeltcovermaterialwhichminimisetheenergydissipatedbytherollingresistance.3.4THEBELT'SDRIVESYSTEMToenablethedeterminationoftheinfluenceoftherotationofthecomponentsofthedrivesystemofabeltconveyor,onthestabilityofmotionofthebelt,amodelofthedrivesystemisincludedinthetotalmodelofthebeltconveyor.Thetransitionelementsofthedrivesystem,asforexamplethereductionbox,aremodelledwithconstraintconditionsasdescribedinsection3.2.Areductionboxwithreductionratioicanbemodelledbyareductionboxelementwithtwodisplacementparameters,μpandμq,onerigidbodymotion(rotation)andthereforeonedeformationparameter:εred=Dred(x)=iμp+μq=0

(10)Todeterminetheelectricaltorqueofaninductionmachine,theso-calledtwoaxisrepresentationofanelectricalmachineisadapted.Thevectorofphasevoltagesvcanbeobtainedfrom:v=Ri+ωsGi+L?i/?t

(11)Ineq.(11)iisthevectorofphasecurrents,Rthematrixofphaseresistance's,Cthematrixofinductivephaseresistance's,Lthematrixofphaseinductance'sandωstheelectricalangularvelocityoftherotor.Theelectromagnetictorqueisequalto: Tc=iTGi

(12)Theconnectionofthemotormodelandthemechanicalcomponentsofthedrivesystemisgivenbytheequationsofmotionofthedrivesystem:Ti=Iij?2?j+Cik??k

Kil?

(13)?t2?twhereTisthetorquevector,Itheinertiamatrix,Cthedampingmatrix,Kthestiffnessmatrixand?theangleofrotationofthedrivecomponentaxis's.Tosimulateacontrolledstartorstopprocedureafeedbackroutinecanbeaddedtothemodelofthebelt'sdrivesysteminordertocontrolthedrivetorque.3.5THEEQUATIONSOFMOTIONTheequationsofmotionofthetotalbeltconveyormodelcanbederivedwiththeprincipleofvirtualpowerwhichleadsto[7]:fk-Mkl?2x1/?t2=σ1Dik

(14)wherefisthevectorofresistanceforces,MthemassmatrixandσthevectorofmultipliersofLagrangewhichmaybeinterpretasthevectorofstressesdualtothevectorofstrainsε.Toarriveatthesolutionforxfromthissetofequations,integrationisnecessary.Howevertheresultsoftheintegrationhavetosatisfytheconstraintconditions.Ifthezeroprescribedstraincomponentsofforexamplee.g.(8)havearesidualvaluethentheresultsoftheintegrationhavetobecorrected,alsosee[7].Itispossibletousethefeedbackoptionofthemodelforexampletorestricttheverticalmovementofthetake-upmass.Thisinversedynamicproblemcanbeformulatedasfollows.Giventhemodelofthebeltanditsdrivesystem,themotionofthetake-upsystemknown,determinethemotionoftheremainingelementsintermsofthedegreesoffreedomofthesystemanditsrates.Itisbeyondthescopeofthispapertodiscussallthedetailsofthisoption.3.6EXAMPLEApplicationoftheFEMinthedesianstageoflongbeltconveyorsystemsenablesitsproperdesign.Theselectedbeltstrength,forexample,canbeminimisedbyminimising,themaximumbelttensionusingthesimulationresultsofthemodel.Asanexampleofthefeaturesofthefiniteelementmodel,thetransversevibrationofaspanofastationarymovingbeltbetweentwoidlerstationswillbeconsidered.Thisshouldbedeterminedinthedesignstageoftheconveyorinordertoensureresonancefreebeltsupport.Theeffectoftheinteractionbetweenidlersandamovingbeltisimportantinbelt-conveyordesign.Geometricimperfectionsofidlersandpulleyscausethebeltontopofthesesupportstobedisplaced,yieldingatransversevibrationofthebeltbetweenthesupports.Thisimposesanalternatingaxialstresscomponentinthebelt.Ifthiscomponentissmallcomparedtotheprestressofthebeltthenthebeltwillvibrateinit'snaturalfrequency,otherwisethebelt'svibrationwillfollowtheimposedexcitation.Thebeltcanforexamplebeexcitatedbyaneccentricityoftheidlers.Thiskindofvibrationsisparticularlynoticeableonbeltconveyorreturns.Sincethefrequencyoftheimposedexcitationdependsontheangularspeedofthepulleysandidlers,andthusonthebeltspeed,itisimportanttodeterminetheinfluenceofthebeltspeedonthenaturalfrequencyofthetransversevibrationofthebeltbetweentwosupports.Ifthefrequencyoftheimposedexcitationapproachesthenaturalfrequencyoftransversevibrationofthebelt,resonancephenomenaoccur.Theresultsofsimulationwiththefiniteelementmodelcanbeusedtodeterminethefrequencyoftransversevibrationofastationarymovingbeltspan.Thisfrequencyisobtainedaftertransformationoftheresultsofthetransversedisplacementofthebeltspanfromthetimedomaintothefrequencydomainusingthefastfouriertechnique.Besidesusingthefiniteelementmodelalsoananalyticalapproachcanbeused.Thebeltcanbemodelledasaprestressedbeam.Ifthebendingstiffnessofthebeltisneglected,thetransversedisplacementsaresmallcomparedtotheidlerspace,Ks<<1,andtheincreaseofthebeltlengthduetothetransversedisplacementisnegligiblecomparedtoitsinitiallength,thetransversevibrationofthebeltcanbeapproximatedbythefollowinglineardifferentialequation,alsoseeFigure5:?2v=(c22-C2b)?2v-2Vb?2v

(15)?t2?x2?x?twherevisthetransversedisplacementofthebeltandc2thewavespeedofthetransversewavesdefinedby,[1]:c2=√g1/8Ks

(16)ThefirstnaturaltransversefrequencyofthebeltspanofFigure5canbeobtainedfromeq.(16)ifitisassumedthatv(O,t)=v(l,t)=0:fb=

1

c2(1-?2)

(17)21where?isthedimensionlessspeedratiodefinedby:?=Vb/c2

(18)Thefrequencyfbisdifferentforeachindividualbeltspansincethebelttensionvariesoverthelengthoftheconveyor.Theexcitationfrequencyofanidlerwhichhasasingleeccentricityisequalto:fi=Vb/πD

(19)whereDisthediameteroftheidler.Inordertodesignaresonancefreebeltsupporttheidlerspaceissubjectedtothefollowingcondition:L≠πD(1-?2)

(20)2?Theresultsobtainedwiththelineardifferentialequation(16)howeverarevalidonlyforlowvaluesoftheratio?.Forhighervaluesof?,asisthecaseforhigh-speedconveyorsorlowbelttensions,thenon-lineartermsinthefullformofe.g.(16)becomesignificant.Thereforenumericalsimulationsusing,theFEMmodelhavebeenmadeinordertodeterminetheratiobetweenthelinearandthenon-linearfrequencyoftransversevibrationofabeltspan.Theserelationshavebeendeterminedfordifferentvaluesof?asafunctionofthesagratioKs.Theresultsforthetransversedisplacementsweretransformedtoafrequencyspectrumusingafast-fouriertechnique.Thefrequenciesobtainedfromthesespectrawerecomparedtothefrequenciesobtainedfrome.g.(18)whichyieldedthecurvesasshowninFigure8.Fromthisfigureitfollowsthatfor?smallerthat0.3thecalculationerrorsaresmall.Forhighervaluesof?thecalculationerrormadebyalinearapproximationismorethan10%.Applicationofafiniteelementmodelofthebeltwhichusesnon-linearbeamelementsthereforeenablesanaccuratedeterminationofthetransversevibrationsforhighvaluesof?.Forlowervaluesof?thefrequenciesoftransversevibrationcanalsobepredictedaccuratebye.g.(18).Howeveritisnotpossibletoanalyse,forexample,theinteractionbetweenthebeltsagandthepropagationoflongitudinalwavesortheliftingofthebeltofftheidlersascanbedonewiththefiniteelementmodel.Thedeterminedrelationbetweenthebeltstressandthefrequencyoftransversevibrationscanalsobeusedinbelttensionmonitoringsystems.Figure8:Ratiobetweenthelinearandthenon-linearfrequencyoftransversevibrationofabeltspansupportedbytwoidlers.4.EXPERIMENTALVERIFICATIONInordertobeabletoverificatetheresultsofthesimulations,experimentshavebeencarriedoutwiththedynamictestfacilityshowninFigure9.Figure9:DynamictestfacilityWiththistestfacilitythetransversevibrationofanunloadedflatbeltspanbetweentwoidlers,asforexampleareturnpart,canbedetermined.Anacousticdeviceisusedtomeasurethedisplacementofthebelt.Besidesthat,alsothetensioningforce,beltspeed,motortorque,idlerrotationsandidlerspacewereknownduringtheexperiments.5.EXAMPLESincethemostcost-effectiveoperationconditionsofbeltconveyorsoccurintherangeofbeltwidths0.6-1.2m[2],thebelt'scapacitycanbevariedbyvaryingthebeltspeed.Howeverbeforethebeltspeedisvariedtheinteractionbetweenthebeltandtheidlershouldbedeterminedinordertoensureresonancefreebeltsupport.Toillustratethisthetransversedisplacementofastationarymovingbeltspanbetweentwoidlershavebeenmeasured.ThetotalbeltlengthLwas52.7m,theidlerspaceIwas3.66m,thestaticsagratioKs2.1%,?was0.24andthebeltspeedVb3.57m/s.AftertransformationofthissignalbyafastfouriertechniquethefrequencyspectrumofFigure5wasobtained.InFigure5threefrequenciesappear.Thefirstfrequencyiscausedbythepassageofthebeltsplice:fs=Vb/L=0.067HzThesecondfrequency,whichappearsat1.94Hz,iscausedbythetransversevibrationofthebelt.Figure10:Frequenciesoftransversevibrationofastationarymovingbeltspansupportedbytwoidlers.Thethirdfrequencywhichappearsat10.5Hziscausedbytherotationoftheidlers.FromthenumericalsimulationsFigure11wasobtained.Figure11:Calculatedresonancezone'sfordifferentidlerdiametersD.Crossindicatesbeltspeedandidlerspaceduringexperiment.Figure11showsthezone'swhereresonancecausedbythebelt/idlerinteractionmaybeexpectedforthreeidlerdiameters.Theidlersofthebeltconveyorhadadiameterof0.108mthusresonancephenomenamaybeexpectednearbyabeltspeedof0.64m/s.Tocheckthis,themaximumtransversedisplacementofthebeltspanhasbeenmeasuredduringastart-upoftheconveyor.Figure12:Measuredratioofthestandarddeviationoftheamplitudeoftransversevibrationandthestaticbeltsag.AscanbeseeninFigure12themaximumamplitudeofthetransversevibrationoccuratabeltspeedof0.64m/saswaspredictedbytheresultsofsimulationwiththefiniteelementmodel.Thereforethebeltspeedshouldnotbechosennearby0.64m/s.Althoughaflatbeltisusedfortheexperimentsandthetheoreticalverification,theappliedtechniquescanalsobeusedfortroughedbelts.6.CONCLUSIONSApplicationofbeamelementsinfiniteelementmodelsofbeltconveyorsenablethesimulationofthetransversedisplacementofthebeltthusenablingthedesignofresonancefreebeltsupports.Theadvantageofapplyingbeamelementsforsmallvaluesof?insteadofusingalineardifferentialequationtopredictresonancephenomenaisthatalsotheinteractionbetweenthelongitudinalandtransversedisplacementofthebeltandtheliftingofthebeltofftheidlerscanbepredictedfromsimulation.7.REFERENCESLodewijks