带式输送机 外文翻译 外文文献 英文文献 输送带的二维动态特性.docx

带式输送机 外文翻译 外文文献 英文文献 输送带的二维动态特性.docx

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带式输送机外文翻译外文文献英文文献输送带的二维动态特性

TheTwo-DimensionalDynamicBehaviorofConveyorBelts

3.1.1NONLINEARTRUSSELEMENT

Ifonlythelongitudinaldeformationofthebeltisofinterestthenatrusselementcanbeusedtomodeltheelasticresponseofthebelt.AtrusselementasshowninFigure2hastwonodalpoints,pandq,andfourdisplacementparameterswhichdeterminethecomponentvectorx:

xT=[upvpuqvq]            

（1）

Forthein-planemotionofthetrusselementtherearethreeindependentrigidbodymotionsthereforeonedeformationparameterremainswhichdescribes

Figure2:

Definitionofthedisplacementsofatrusselement

thechangeoflengthoftheaxisofthetrusselement[7]:

ε1=D1（x）=∫¹o

ds²-ds²o

dξ            

（2）

2ds²o

wheredsoisthelengthoftheundeformedelement,dsthelengthofthedeformedelementandξadimensionlesslengthcoordinatealongtheaxisoftheelement.

Figure3:

Staticsagofatensionedbelt

Althoughbending,deformationsarenotincludedinthetrusselement,itispossibletotakethestaticinfluenceofsmallvaluesofthebeltsagintoaccount.Thestaticbeltsagratioisdefinedby（seeFigure3）:

K1=δ/1=q1/8T          （3）

whereqisthedistributedverticalloadexertedonthebeltbytheweightofthebeltandthebulkmaterial,1theidlerspaceandTthebelttension.Theeffectofthebeltsagonthelongitudinaldeformationisdeterminedby[7]:

εs=8/3K²s             （4）

whichyieldsthetotallongitudinaldeformationofthenonlineartrusselement:

3.1.2BEAMELEMENT

Figure4:

Definitionofthenodalpointdisplacementsandrotationsofabeamelement.

Ifthetransversedisplacementofthebeltisbeingofinterestthenthebeltcanbemodelledbyabeamelement.Alsoforthein-planemotionofabeamelement,whichhassixdisplacementparameters,therearethreeindependentrigidbodymotions.Thereforethreedeformationparametersremain:

thelongitudinaldeformationparameter,ε1,andtwobendingdeformationparameters,ε2andε3.

Figure5:

Thebendingdeformationsofabeamelement

Thebendingdeformationparametersofthebeamelementcanbedefinedwiththecomponentvectorofthebeamelement（seeFigure4）:

xT=[upvpµpuqvqµq]        （5）

andthedeformedconfigurationasshowninFigure5:

ε2=D2（x）=

e2p1pq

        （6）

1o

ε3=D3（x）=

-eq21pq

1o

3.2THEMOVEMENTOFTHEBELTOVERIDLERSANDPULLEYS

Themovementofabeltisconstrainedwhenitmovesoveranidlerorapulley.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=0

ε5=D5（x）=（rξ+uξ）e1-rid.e1=0              （7）

Ifduringsimulationε4>0thenthebeltisliftedofftheidlerandtheconstraintconditionsareremovedfromthefiniteelementdescriptionofthebelt.

3.3THEROLLINGRESISTANCE

Inordertoenableapplicationofamodelfortherollingresistanceinthefiniteelementmodelofthebeltconveyoranapproximateformulationforthisresistancehasbeendeveloped,[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  ∂t²

fb=

    Mf    

  Fzbri

whereFzisdistributedverticalbeltandbulkmaterialload,hthethicknessofthebeltcover,Dtheidlerdiameter,Vbthebeltspeed,KNthenominalpercentbeltload,Ttheambienttemperature,mredthereducedmassofanidler,bthebeltwidth,uthelongitudinaldisplacementofthebelt,Mfthetotalbearingresistancemomentandritheinternalbearingradius.Thedynamicandmechanicpropertiesofthebeltandbeltcovermaterialplayanimportantroleinthecalculationoftherollingresistance.Thisenablestheselectionofbeltandbeltcovermaterialwhichminimisetheenergydissipatedbytherollingresistance.

3.4THEBELT'SDRIVESYSTEM

Toenablethedeterminationoftheinfluenceoftherotationofthecomponentsofthedrivesystemofabeltconveyor,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

∂²øj

+Cik

∂øk

  Kilø                （13）

∂t²

∂t

whereTisthetorquevector,Itheinertiamatrix,Cthedampingmatrix,Kthestiffnessmatrixandøtheangleofrotationofthedrivecomponentaxis's.

Tosimulateacontrolledstartorstopprocedureafeedbackroutinecanbeaddedtothemodelofthebelt'sdrivesysteminordertocontrolthedrivetorque.

3.5THEEQUATIONSOFMOTION

Theequationsofmotionofthetotalbeltconveyormodelcanbederivedwiththeprincipleofvirtualpowerwhichleadsto[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.

3.6EXAMPLE

ApplicationoftheFEMinthedesianstageoflongbeltconveyorsystemsenablesitsproperdesign.Theselectedbeltstrength,forexample,canbeminimisedbyminimising,themaximumbelttensionusingthesimulationresultsofthemodel.Asanexampleofthefeaturesofthefiniteelementmodel,thetransversevibrationofaspanofastationarymovingbeltbetweentwoidlerstationswillbeconsidered.Thisshouldbedeterminedinthedesignstageoftheconveyorinordertoensureresonancefreebeltsupport.

Theeffectoftheinteractionbetweenidlersandamovingbeltisimportantinbelt-conveyordesign.Geometricimperfectionsofidlersandpulleyscausethebeltontopofthesesupportstobedisplaced,yieldingatransversevibrationofthebeltbetweenthesupports.Thisimposesanalternatingaxialstresscomponen