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（1）

Forthein-planemotionofthetrusselementtherearethreeindependentrigidbodymotionsthereforeonedeformationparameterremainswhichdescribes

Figure2:

Definitionofthedisplacementsofatrusselement

thechangeoflengthoftheaxisofthetrusselement[7]:

ε1=D1（x）=∫¹

o

ds²

-ds²

o

dξ 

（2）

2ds²

wheredsoisthelengthoftheundeformedelement,dsthelengthofthedeformedelementandξadimensionlesslengthcoordinatealongtheaxisoftheelement.

Figure3:

Staticsagofatensionedbelt

Althoughbending,deformationsarenotincludedinthetrusselement,itispossibletotakethestaticinfluenceofsmallvaluesofthebeltsagintoaccount.Thestaticbeltsagratioisdefinedby（seeFigure3）:

K1=δ/1=q1/8T 

（3）

whereqisthedistributedverticalloadexertedonthebeltbytheweightofthebeltandthebulkmaterial,1theidlerspaceandTthebelttension.Theeffectofthebeltsagonthelongitudinaldeformationisdeterminedby[7]:

εs=8/3K²

s 

（4）

whichyieldsthetotallongitudinaldeformationofthenonlineartrusselement:

3.1.2BEAMELEMENT

Figure4:

Definitionofthenodalpointdisplacementsandrotationsofabeamelement.

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

thelongitudinaldeformationparameter,ε1,andtwobendingdeformationparameters,ε2andε3.

Figure5:

Thebendingdeformationsofabeamelement

Thebendingdeformationparametersofthebeamelementcanbedefinedwiththecomponentvectorofthebeamelement（seeFigure4）:

xT=[upvpµ

puqvqµ

q] 

（5）

andthedeformedconfigurationasshowninFigure5:

ε2=D2（x）=

e2p1pq

（6）

1o

ε3=D3（x）=

-eq21pq

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,belt