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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-updeviceprematurecollapseofthebelt,mostlyduetothefailureofthesplicesdestructionofthepulleysandmajordamageoftheidlersliftingofthebeltofftheidlerswhichcanresultinspillageofbulkmaterialdamageandmalfunctioningof(hydrokinetic)drivesystemsManyresearchersdevelopedmodelsinwhichtheelasticresponseofthebeltistakenintoaccountinordertodeterminethephenomenaresponsiblefortheseproblems.Inmostmodelsthebelt-conveyormodelconsistsoffiniteelementsinordertoaccountforthevariationsoftheresistance'sandforcesexertedonthebelt.Theglobalelasticresponseofthebeltismadeupbytheelasticresponseofallitselements.Thesefiniteelementmodelshavebeenappliedincomputersoftwarewhichcanbeusedinthedesignstageoflongbelt-conveyorsystems.Thisiscalledthedynamic(design)approach.Verificationoftheresultsofsimulationhasshownthatsoftwareprogramsbasedonthesekindofbelt-modelsarequitesuccessfulinpredictingtheelasticresponseofthebeltduringstartingandstopping,seeforexample[3]and[4].Thefiniteelementmodelsasmentionedabovedetermineonlythelongitudinalelasticresponseofthebelt.Thereforetheyfailintheaccuratedeterminationof:themotionofthebeltovertheidlersandthepulleysthedynamicdrivephenomenathebendingresistanceofthebeltthedevelopmentof(shock)stresswavestheinteractionbetweenthebeltsagandthepropagationoflongitudinalstresswavestheinteractionbetweentheidlerandthebelttheinfluenceofthebeltspeedonthestabilityofmotionofthebeltthedynamicstressesinthebeltduring.passageofthebeltovera(driven)pulleytheinfluenceofparametricresonanceofthebeltduetotheinteractionbetweenvibrationsofthetakeupmassoreccentricitiesoftheidlersandthetransversedisplacementsofthebeltthedevelopmentofstandingtransversewavestheinfluenceofthedampingcausedbybulkmaterialandbythedeformationofthecross-sectionalareaofthebeltandbulkmaterialduring,passageofanidlertheliftingofthebeltofftheidlersinconvexandconcavecurvesThetransverseelasticresponseofthebeltisoftenthecauseofbreakdownsinlongbelt-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:
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