Dynamic Model of the Axially Moving Viscoelastic Belt System with Tensioner Pulley Yanqi Liu1, a, Hongyu Wang2, b, Dongxing Cao3, c, Xiaoling Gai1, d

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1 Inernaonal Indsral Informacs and Comper Engneerng Conference (IIICEC 5) Dynamc Model of he Aally Movng Vscoelasc Bel Sysem wh Tensoner Plley Yanq L, a, Hongy Wang, b, Dongng Cao, c, Xaolng Ga, d Bejng Key Laboraory of Envronmen Nose and Vbraon, Bejng Mncpal Inse of Labor Proecon, Bejng, 5, Chna Sae Key Laboraory of Trbology, Tsngha Unversy, Bejng, 8, Chna College of Mechancal Engneerng and Appled elecroncs, Bejng Unversy of Technology, Bejng,, Chna a emal: jdyql@6.com, bemal: aal888@6.com, cemal: caosar@bj.ed.cn, demal: gldynmg@6.com Keywords: Movng Bel; Nonlnear vbraon; Dynamcal model; Vscoelasc Absrac. Dynamc model of he movng bel sysem wh ensoner plley s sded n hs paper. Bel s consdered elasc connos elemen, and plley and ensoner arm are dscree elemen. Consderng he geomercal nonlneary and Kelvn vscoelasc consve model, he knec energy, poenal energy and work done by eernal forces of movng bel sysem wh ensoner plley are gven respecvely. Then, he nonlnear eqaons of moon for he movng vscoelasc bel sysem wh ensoner plley are obaned by sng he Hamlon s prncple. Noe ha nonlnear oscllaons of he bel and plley oscllaons are copled. Inrodcon The movng bel sysem wh ensoner plley has wde applcaons n he areas of engneerng. As an aally movng maeral, he ransverse vbraon of movng bel has been nvesgaed eensvely [-]. For movng bel sysem wh ensoner plley, oscllaons of he bel and plley oscllaons are copled and ehb comple dynamcal eqaon and dynamcal characerscs. However, he mos sdes neglec he vscoelascy of he bel and regard bel as elasc maeral. In hs paper, consderng he vscoelascy of bel maeral, nonlnear eqaons of moon of movng bel sysem wh ensoner plley are derved. Oscllaons of he bel and plley oscllaons are copled and he dynamcal eqaons are comple. Fg. Three-plley drve sysem wh enson plley 5. The ahors - Pblshed by Alans Press 5

2 Dynamc Model of he Movng Bel Sysem wh Tensoner Plley In hs paper, s assmed ha he bel enson domnaes he ransverse sffness of he bel. Then, an aally movng vscoelasc bel can be modeled as a srng, and he plleys are consdered dscree elemen, as shown n Fg.. For srng model, he effecs of he momen of nera of he cross-seconal area and shear deformaon can be gnored. The sran of srng cased by aal dsplacemen s gven by, ε =, w ( =,,) () here, and w are he longdnal and ransverse deflecons n span from eqlbrm. The bel s consdered homogeneos vscoelasc maeral and obeys he lnear vscoelasc dfferenal consve relaon. Takng no accon he Kelvn vscoelasc consve model, he dynamc ensons n he bel spans become,, w pd = EA, w, ( =,,) () where, E s he sffness consan of bel, s he dynamcal vscos coeffcen of bel. In he followng analyss, he eended Hamlon s prncple s sed o derve he nonlnear governng eqaons of moon of he bel and plley copled sysem. The knec energy, T, of he movng vscoelasc bel and plley copled sysem s c L T = J θ, m ( w,, ) (,, ) c d r c L J θ, m ( w,, ) (,, ) c d r () L c m ( w,, ) (,, ) c d J θ, J θ, r where J, r andθ are he mass momen of nera, rads, and roaon angle from eqlbrm of he h dscree elemen, respecvely. m s mass of bel, c s he bel aal velocy, and L s he lengh of bel span. Sbscrps, or, denoe he paral dervave wh respec o ha varable, represens a spanwse coordnae, and s me. The poenal energy of he bel and plley copled sysem s EA φ P ( l ) σ P ( ) P ( l ) U = Kr( θ θr ) rdσ EA φ EA EA EA L P EA φp ( l ) σ P ( ) P ( l ), w, d r dσ EA EA φ EA EA () EA L P EA P ( l φ ) σ P ( ) P ( l ), w, d rd EA σ EA φ EA EA EA L P, w, d EA where σ and φ are he coordnae and wrap angles on plley respecvely, and θ s he r ensoner sprng deflecon n he reference poson. Here he oal enson of bel span s 55

3 P P Pd P EA, w, = =, w, (5) The work done by eernal forces ncldes ha done by he racve enson P and vscoelasc dampng. The epresson for eernal work s ε W = P ε (6) = = Accordng o he geomercal nonlneary and Kelvn vscoelasc consve model, he knec energy, poenal energy and work of he movng vscoelasc bel sysem wh ensoner plley are gven respecvely. Eqaons of Moon of Movng Vscoelasc Bel Sysem wh Tensoner Plley In hs secon, he eqaons of moon of he movng bel and plley wll be obaned by Hamlon s prncple. Sbsng eqaons (), () and (6) no Hamlon s prncple, ( d T d U d W ) d = (7) Here T, U andδ W are he knec energy, he poenal energy, and he vral work done by he eernal forces, respecvely. Afer negrang by pars where needed, he governmen eqaons of moon for he bel spans are m mc mc = P P =,, (8),,, d, ( ) ( ) mw, m, mc w, = Pd P w, =,, (9), Noe ha each feld eqaon has hree acceleraon erms: he local acceleraon w or,, he, Corols acceleraon or, c, and he cenrpeal acceleraon or, c., Owng o longdnal vbraon compared o ransverse vbraon s very small, as a conseqence, one may neglec he longdnal vbraon. Under he qas-sac srechng assmpon, he dynamc enson n he bel span P d become L (, ) (, ) L w, d EA L pd = ( L, ) (, ) w, d L () and are nform hrogho he span. Wh hs assmpon, he eqaons for he ransverse vbraon of movng bel redce o mw m mc w = P P w =,, (),,, d, The eqaons of moon for he dscree elemens are obaned as follows: For plley P P r P P r = Jθ (a) d d o o, For plley ( Pd Pd) r ( P o Po ) r = J θ, (b) For plley ( Pd Pd ) r ( P o Po ) r = J θ, (c) For he ensoner arm 56

4 [ m, ( l ) mc ( Pd Po ) w, ( l )] r cos( θ α) ( mc ( P P )) r sn( θ α ) [ m ( ) ( P P mc ) w ( ) ] r cos( θ α ), d o d o, (d) ( Pd Po mc ) r sn( θ α ) M kr ( θ θ r ) = J θ For ease, he moon eqaons of he dscree elemens n erms of dsplacemens along he arc-lenghs are mw m mc w = P P w =,, (a),,, d, m c ( k k ) c k c k c k cosϕc c c c cosϕc, = Pd NL Pd NL L,,,, L L L L m c k c ( k k ) c k c ( k cosϕ k cosϕ ) c c c,,, L L L c cosϕ cosϕ c = P P L L L,, d NL dnl m c k c k c ( k k ) c k c cosϕ c c c cosϕc, = Pd NL Pd NL L,,,, L L L L cos ( ) cos( ) m c, m, l P w, l θ α m, P w, θ α k,,, w, l L k ( c cosϕ c c ) ( c cosϕ c c ) ( ) cos( θ α ) ( c c c cosϕ ) ( c c c cosϕ ) w ( ) cos( θ α ),,,, L, (b) (c) (d) (e) θr k r c = Pd NLw, ( l ) cos( θ α ) Pd NLw, ( ) cos( θ α ) r where m = J r, χ = rθ. Eqaon (a)-(e) are he nonlnear eqaons of moon for he movng vscoelasc bel sysem wh ensoner plley. They represen a se of paral and ordnary dfferenal eqaons descrbng he copled ransverse vbraon of he movng bels and he roaonal moons of he plleys. Conclson The sysem of movng vscoelasc bel wh ensoner plley s dvded elasc connos elemen, bel, and dscree elemen, plley and ensoner arm. Consderng Kelvn vscoelasc consve model, he knec energy, poenal energy and work done by eernal forces of movng bel sysem wh ensoner plley are gven respecvely. Then, he nonlnear eqaons of moon of he vscoelasc movng bel sysem wh ensoner plley are derved by sng he Hamlon s prncple. Noe ha nonlnear oscllaons of bel and plley oscllaons are copled. Acknowledgemen The research s sppored by Naral Scence Fondaon of Chna (9,76), 57

5 Naral Scence Fondaon of Bejng (Z5) and Yoh Backbone Fondaon of Bejng Academy of Scence and Technology. References [] L.Zhang, J.W.Z. One-o-one ao-paramerc resonance n serpenne bel drve sysems [J]. Jornal of Sond and Vbraon, () [] R.G. Parker. Effcen egensolon,dynamc response,and egensensvy of serpenne bel drves [J]. Jornal of Sond and Vbraon, [] L. Kong, R. G. Parker. Eqlbrm and bel-plley vbraon coplng n serpenne bel drves [J]. Jornal of Appled Mechancs, 7(5)