EVALUATION OF CONTACT FORCE AND ELASTIC FOUNDATION MODELS FOR WEAR ANALYSIS OF MULTIBODY SYSTEMS

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1 Poceedngs of the 34 th Annual Mechansm and Rootcs Confeence IDTC/CI 1 August 15-18, 1, Monteal, Canada DTC1-875 VALUATION OF CONTACT FORC AND LASTIC FOUNDATION MODLS FOR WAR ANALYSIS OF MULTIBODY SYSTMS Saad M. Mukas Mechancal ngneeng Qassm Unvest Buadah, Qassm, Saud Aaa mukas@qec.edu.sa Nathan A. Mauntle Mechancal & Aeospace ng. Unvest of Floda Ganesvlle, Fl, 3611 mauntle@ufl.edu Nam Ho Km Mechancal & Aeospace ng. Unvest of Floda Ganesvlle, Fl, 3611 nkm@ufl.edu Ton L. Schmtz Mechancal & Aeospace ng. Unvest of Floda Ganesvlle, Fl, 3611 tschmtz@ufl.edu W. Gego Sawe Mechancal & Aeospace ng. Unvest of Floda Ganesvlle, Fl, 3611 wgsawe@ufl.edu ABSTRACT In ths pape, two pocedues to analze plana multod sstems expeencng wea at a evolute ont ae compaed. In oth pocedues, the evolute ont of nteest ncludes a cleaance whose shape and sze ae dctated wea. The pocedues consst of coupled teatve analses etween a dnamc analss of the sstem wth non-deal onts and wea pedcton to detemne the evoluton of the ont cleaance. In the fst pocedue, ont foces and contact pessue ae estmated usng the elastc foundaton model (FM) wth hsteess dampng va the dnamc analss. In the second pocedue, a contact foce model wth hsteess dampng s used to estmate the ont foces. In the latte case, howeve, the contact pessue s estmated usng a fnte element method (FM). Compason n pefomance of the two models s facltated though the use of an expemental slde-cank mechansm n whch wea s pemtted to occu at one of the onts. It s oseved that the two pocedues povde smla estmates fo the dnamcs esponse and wea volumes ut sustantall dffeent pedctons on the wea pofles. Addtonall, expemental esults show that whle pedctons on the wea volume fom oth models ae easonal accuate, the FM-ased model poduced moe accuate pedctons on the wea pofle. 1. INTRODUCTION Mechancal/multod sstems consst of a nume of components that ae nteconnected onts. These onts tpcall consst of at least two components that ae n contact and expeence elatve moton. It s theefoe nevtale that ont wea wll occu. Dependng on the sstem confguaton and the amount of wea at the onts, the pefomance of the sstem ma e advesel affected. It s theefoe no supse that a consdeale amount of effot has een expended to develop was to account fo wea n the desgn pocess. One appoach s to develop pocedues that can e used to estmate wea efoehand. A tend that has developed, n ths egad, s the use of Achad s [1] wea model n an teatve wea pedcton scheme. In ths scheme, ncemental wea s estmated ased on the elatve sldng, the contact pessue and tologcal data (tpcall n the fom of a wea coeffcent) that desces the opeatng condtons. The geomet s then updated to eflect wea and the ncemental wea s ecomputed and accumulated n susequent teatons. Ths pocedue has een used n a nume of applcatons ncludng wea pedcton n cam and followe components [-5], evolute ont components [6], helcal geas [7] and spu geas [8, 9]. It has also een used n medcal applcatons to pedct wea n hp athoplast [1, 11]. Despte the nceasng use of the teatve wea pedcton pocedue, fo cetan cases the pocedue ma eld less than accuate pedctons. Ths s ecause the pocedue s a component level pedcton n whch the component s solated fom the ente sstem and the wea estmate s ased on ntal sstem dnamcs. Two possle eos ma e assocated wth the component level assumpton. Fst, as the wea at the 1 Copght 1 ASM

2 contact nteface of the component evolves, the sstem dnamcs ma also evolve. Thus, the ont foces esponsle fo the wea ae alteed as the wea pogesses. Dependng on the degee of evoluton, t ma not e suffcent to use the ntal dnamcs n the wea pedcton. Ths ssue was dentfed and documented Blanchet [1] and Sawe [13] fo a scotch oke mechansm and Dckell et al. [14] fo cam wea. Secondl, wthout pefomng a dnamc analss t ma not e possle to coectl detemne the contact locatons n the ont and the pedctons ma lead to ncoect wea pofles. Ths aspect of wea pedcton was also dentfed and dscussed Mukas et al. [15] and Floes [16]. In Mukas et al. [15], a pocedue to analze multod sstems wth non-deal onts ased on a contact foce model was coupled wth the pevousl dscussed wea pedcton pocedue. Ths couplng accounts fo changes n the dnamcs due to ont wea. In addton, the contact locatons and, susequentl, the wea egons can e pedcted n a multod dnamc famewok. The pocedue was used to estmate the wea at a evolute ont n a sldecank mechansm. Results fom the model showed ageement wth expement. A smla pocedue was pesented Floes [16]. In ths pape, a pocedue to analze plana multod sstems wth evolute ont wea, smla to that used n Mukas et al. [15] and Floes [16], s pesented. Howeve, the pocedue s ased on the elastc foundaton model (FM), athe than the contact foce model. A smla concept was emploed Be et al. [17] n multod analss of knee contact and Fegl et al. [18] n wea pedcton of a knee eplacement. The pocedue pesented hee emplos the FM to smultaneousl estmate the ont eacton foce, equed n the dnamc analss, and contact pessue dstuton at the ont, whch s equed fo wea analss. The FM-ased pocedue s then compaed wth the contact foce model appoach. The advantages and dsadvantages of the two pocedues ae then dscussed. The am of ths pape s to pesent a fa compason etween the two pocedues to enale uses the oppotunt to make an nfomed choce.. WAR PRDICTION PROCDUR In ths wok, an teatve wea pedcton pocedue, smla to the one used n the woks of Poda et al. [19, ], s emploed. The pocedue s ased on Achad s wea model whch can e expessed n dffeental fom as follows: dh kp s ds. (1) In q. (1), h s the wea depth, s and p ae the elatve sldng dstance and contact pessue etween the contactng odes and k s the wea coeffcent. If the two odes ae made of dffeent mateals, each od wll have a sepaate wea coeffcent. The wea depth ma e estmated usng a fnte dffeence appoach n whch a tempoal dscetzaton of the elatve moton of the odes n contact elds the followng updatng fomula: h h kp s. () 1 In q. (), p, h and s ae the contact pessue, wea depth th and ncemental sldng dstance at the ccle and h 1 s the wea depth at the pevous ccle. The poduct on the ght-hand sde of q. () s efeed to as the ncemental wea depth. In ths tem, the ncemental sldng dstance s usuall specfed whle the wea coeffcent s detemned though expements [1 3] and the contact pessue s detemned usng numecal technques such as FM o the fnte element method (FM). Accuate wea pedctons eque that the geomet e updated to eflect the evolvng contact condtons. One wa to acheve ths s dsplacng the contact ounda n the decton of the suface nomal an amount equal to the wea estmated q. (). In addton, these pocedues must ncopoate some stateges, such as extapolaton, to mnmze computatonal costs. Moe detaled studes on wea pedcton on geometes, such as the evolute onts ncludng geomet update and the use of extapolatons, can e found n Mukas et al. [6] and Km et al. [1]. 3. ANALYSIS OF MULTIBODY SYSTMS In ode to pedct ont wea wthn a sstem, t s necessa to fomulate t wthn the multod dnamc analss of the sstem. The analss of multod sstems wth ethe deal o non-deal onts nvolves the asseml and soluton of a set of dffeental algeac equatons of moton (DA). A ef dscusson on how to pefom a dnamc analss s pesented n ths secton. A multod sstem conssts of seveal components (odes) that ae nteconnected onts whch mpose estctons on the elatve moton of the odes. The descpton of nteconnectons s expessed though algeac equatons known as knematc constants. The knematc constant can e expessed as [4, 5]: Φq,t, (3) whee q s the set of genealzed coodnates that unquel defne the poston and oentaton of all odes n the sstem and t s tme. quaton (3) s usuall efeed to as the poston equaton. It s dffeentated once to otan the veloct equaton and twce fo the acceleaton equaton. These equatons can, espectvel, e expessed as: Φ q Φ, (4) q t t tt Φ q Φ q q Φ q Φ γ. (5) q q q q Fo a popel constaned sstem, qs. (3), (4) and (5) can e solved to detemne the poston, veloct and acceleaton of the sstem components. Ths analss, efeed to as a knematc analss, povdes onl nfomaton on the moton of the sstem. A dnamc analss s equed n ode to detemne the Copght 1 ASM

3 dnamc esponse of a multod sstem. Ths nvolves fomulatng and solvng the DA of moton. The DA can e otaned comnng the equatons of moton fo the sstem and the acceleaton equaton (q. (5)). Fo a constaned gd multod sstem, the equaton on moton fo a multod sstem can e expessed as [4, 5]: T A Mq Φqλ Q, (6) whee M s the genealzed mass matx consstng of masses and moments of neta fo the sstem components, q and Φ ae the genealzed acceleaton vecto and Jacoan of the constants, espectvel, λ s a vecto of Lagange multples, A and Q s a vecto of extenall appled loads. The second tem on the left hand sde of q. (6) s the vecto of eacton foces. Fo an unconstaned sstem, the vecto of Lagange multples s zeo and ths tem dsappeas. quatons (5) and (6) can then e comned to esult n a mxed sstem of dffeental algeac equatons (DA) of moton. The equatons ae expessed as: T A M Φ q q Q. (7) Φq λ γ quaton (7) can then e solved to eveal the dnamcs of the sstem. The soluton pocedue fo ths equaton s well documented n the lteatue [4 6]. Pn Bod- Bod- Cleaance - c Bushng Fgue 1: A evolute ont wth cleaance. Pn T q 4. ANALYSIS OF MULTIBODY SYSTMS WITH NON- IDAL JOINTS In ths secton the pocedue to model non-deal evolute onts usng the contact foce model and the FM s pesented. The analss of a sstem wth such onts can then e pefomed wth the pocedue outlned n secton 3. It s assumed n ths wok that the evolute ont conssts of two components, a pn and a ushng. The pn and the ushng ae gdl attached to the two odes that shae the ont. Ths s depcted n Fg. 1 whee the pn and the ushng ae, espectvel, attached to od and od. The deal evolute ont, whch allows elatve otaton etween two odes n the sstem, assumes that no cleaance s pesent at the ont and the pn and ushng cente concde at all tme. In the case of a nondeal evolute ont, cleaance s pesent at the ont so that the pn and ushng centes do not necessal concde. The ont components (pn and ushng) of a non-deal ont can assume one of thee confguatons. The thee confguatons ae: feefall, when the components ae not n contact; mpact, when contact s estalshed; and the followng moton, whch togethe desce the duaton when the ont components ae n contact and n elatve moton. It s possle to model the effect of a non-deal ont ensung that the moton of the pn s confned wthn the nne pemete of the ushng. Ths can e acheved usng a foce constant n place of a knematc constant to desce the ont. The foce constant nvolves applng a foce on the ont components eve tme contact s estalshed (ths foce s equvalent to the ont eacton). Ths pocedue has pevousl een used Ravn [7] and Floes et al. [8]. O A x Fgue : Geometc descpton of a non-deal evolute ont wth eccentc vecto e. Befoe the constant/contact foce can e appled, t s necessa to detemne the decton n whch the contact foce s actng. In Fg., a dagam of two odes constaned a non-deal evolute ont s shown. Bod coodnates x - and x - ae fxed to the cente of masses of the odes and, espectvel. The coodnates ae oented at angles and elatve to the gloal x-axs. The pont of contact C s defned as the cente of contact egon etween the pn and the ushng. Ths pont can e located usng the eccentc vecto e whch s a vecto connectng the ushng cente D and the pn cente B. At the tme of contact the eccentc vecto ponts n the decton of the contact. It s assumed that ths s also the decton n whch the ont eacton wll act. Ths vecto s expessed as: s e A s A s, (8) whee and C B D e Bod- ae vectos lnkng the gloal ogn and the cente o masses of the odes, s and s ae vectos n the local coodnate sstem that lnk the cente of masses to the pn and ushng centes espectvel, and A and A ae matces that tansfom a vecto fom the local coodnate sstem to the gloal sstem. In ths case, the tansfom vectos s and s s x x x Bod- x 3 Copght 1 ASM

4 nto the gloal equvalent. A unt vecto n the decton of the eccentc vecto can e defned : e n ; e e. (9) e The constant/contact foce can then e appled n the decton of the aove unt vecto. The foces can e estmated usng a contact fom model wth hsteess dampng [9] as dscussed Ravn [7] and Floes et al. [8]. The FM ma also e used to detemne the constant foce. The two technques wll e dscussed n the followng susectons Contact Foce Model fo Modelng Non-Ideal Revolute Jonts It s assumed that, whle the ont components ae complant, the odes to whch the components ae attached ae gd. It s then possle to estmate the constant foce usng a contact foce model. A wdel used model n multod analss, whch was deved fo colldng sphees, s the Hetz model wth hsteess dampng [9]. It assumes a nonlnea elaton etween the contact foce and the penetaton esultng fom the contact. Usng ths model, the foce nomal to the plane of contact can e expessed as: n FN K D, (1) whee n = 1.5 and K s a constant that epesents complance. The constant K depends on the elastc constants and as well as the ad ( R and R ) of the sphees: K 4 3 whee R 1, (11) RR 1k R ; k ; k,. R R k (1) In the case of confomal contact as shown n Fg. Fgue, the adus of the second od (R ) should take a negatve value. Dampng s ncluded n the contact model n the fom of a dampng coeffcent D and the penetaton veloct. The dampng coeffcent s expessed as: 3K 1 e n D, (13) 4 whee e and ae the coeffcent of esttuton and the ntal penetaton veloct upon mpact. The coeffcent of esttuton plas an mpotant ole when components mpact at hgh elatve veloctes esultng n ouncng of the components and elasto-plastc defomaton [44, 45]. Howeve, n the evolute ont, the nteacton etween the ont components s manl smooth contact wth contnuous sldng. The contact suface wll expeence onl elastc defomaton. In such a case, the effect of the coeffcent of esttuton s mnmal. The model desced q. (1) s vald fo colldng sphees whose contact aea s small and ccula. Whle seveal expessons have een poposed to model the colldng clndes [3, 31] (as s the case fo evolute onts), ths model (q. (1)) has een used n multod dnamc analss a nume of eseaches [7, 8, 3] to estmate the contact foce etween colldng clndes. In addton, the Hetz contact model ma lead to eoneous esults when the confomt s lage [43]. The ustfcatons fo usng the expesson n q. (1) ae that (1) the depth of evolute ont s elatvel small compaed to the damete of the ont and () the eo n q. (1) s acceptale as long as the contact foce s calculated accuatel fom the dnamc analss ecause the penetaton wll e e-calculated fom ethe the FM o FM model. A moe detaled dscusson of the ustfcaton s pesented n the lteatue [6 8]. In ths wok, q. (1) s thus used to estmate the contact foce at the evolute ont. It wll e shown that ths appoach elds smla esults to expement when used n multod analss. In ode to evaluate the contact foce n q. (1), the penetaton has to e detemned. In the case of the non-deal evolute ont, the penetaton etween the pn and ushng dung contact s computed as the dffeence etween the eccentct and cleaance: e c, (14) whee the cleaance s defned as the dffeence etween the ushng and pn ad, c R R. When the pn and ushng ae not n contact, the eccentct s smalle than the cleaance and the penetaton has a negatve value. Howeve, when the value of the penetaton s equal to o geate than zeo, contact s estalshed. Thus, fo a value of geate than zeo, a contact foce s appled etween the odes. The contact foce vanshes when s equal to o less than zeo. stmates of the ont foces ma e enhanced ncludng fcton. In ths wok, Coulom fcton s appled. The fcton foce s expessed as: F F, (15) f k N whee k s the coeffcent of fcton whch can e detemned though expements as dscussed Schmtz [33] and F N s the nomal foce as defned n q. (1). 4.. FM fo Modelng Non-Ideal Revolute Jonts In the FM the contact suface s modeled as a set of spngs (spead ove the contact suface). The spngs epesent the elastc lae and the thckness of the lae s composed of the thckness of one o oth odes (dependng on whethe one of the odes s defned as gd). 4 Copght 1 ASM

5 The FM assumes that the spngs ae ndependent fom each othe and thus the shea foce etween them s neglected. The consequence of ths assumpton s that the model does not account fo how pessue appled at one locaton affects the defomaton at othe locatons. Ths s conta to what s expeenced n elastc contact whee the dsplacement at one locaton s a functon of the pessue appled at othe locatons. Although ths smplfng assumpton volates the ve natue of contact polems, some enefts can e deved fom ts use. In patcula, the smplfed model povdes a cheape altenatve fo estmatng contact pessue (compaed to othe computatonal technques such as the FM) and facltates the analss of confomal geomet, laeed contact and nonlnea mateals [17]. The contact pessue fo an spng (spng ) n the elastc foundaton can e calculated fom [19]: p, (16) W L whee p s the contact pessue, W s the elastc modulus fo the elastc lae, L s the thckness of the elastc lae and s the defomaton of the spng. When oth odes ae defomale W s a composte of the elastc modulus and Posson s ato fo the two odes. The pocedue to detemne the composte modulus s dscussed Poda [19] and n moe detal Johnson [34]. Fo the pupose of llustaton, t s assumed that onl one of the odes n contact s defomale. Fo ths case, a common expesson fo s gven [17, 18, 35-38]: W 1 11 W, (17) whee and ae the elastc modulus and Posson s ato of the defomale od, espectvel. The contact pessue fo the spng can then e detemned fom: p 1 11, (18) L The total load suppoted the elastc lae (o the ont eacton foce) can then e computed summng the contact foces on all elements n the decton of the md-suface nomal. The esultant foce has a magntude equal to the ont eacton wth decton nomal to the contact plane (as defned n q. (9)). The magntude of the esultant can e expessed as the sum of the poduct of the element pessue p and the element aea A poected n the decton of load applcaton n (o decton of eccentc vecto; see q. (9)): F p A. (19) N In the case of the evolute ont, the element aea n q. (19) can e estmated usng the followng expesson: 1 A R d pn ushng. () whee R pn s the pn adus, dushng s the ushng depth and s the angle etween two dscetzed ponts on the ushng measued fom the pn cente. To enale a fa compason etween the contact foce model and FM, a dampng tem, smla to the one used n the contact foce model, should e added to q. (19). The FM model wth dampng can e deved notng that ndvdual spng foce wthout dampng can e expessed as: F p A, (1) N The contact pessue n q. (18) can e susttuted nto q.(1) to esult n the followng expesson fo the ndvdual spng foce: F N 1 11 KA whee the spng constant K s K 1 11 A L, (). (3) L The FM model wth dampng can then e expessed as: F K D N 31e K K 4 1 A 1 31 A e 1 1 L 1 1 L e A 1 1 L 1 1 L 4.(4) A Notng that the contact pessue s gven q.(18), q.(4) can e wtten as: 31e FN p A p A. (5) 4 The total contact foce can then e otaned as the sum of the ndvdual foces. The contact foce can then e expessed as: 5 Copght 1 ASM

6 31e FN p A p A. (6) 4 Once the contact foces ae detemned usng ethe of the two pocedues (contact foce model o FM), the ma e assemled nto the DA (q. (7)) as appled loads. The DA ma then e solved to eveal the sstem dnamcs xample: Slde-Cank Mechansm wth Jont Cleaance The slde-cank mechansm shown n Fg. Fgue 3 s used to llustate the pocedue to model plana sstems wth nondeal evolute onts. The mechansm conssts of two deal evolute onts, one etween the gound and the cank and the othe etween the followe and the slde, a non-deal ont etween the cank and the followe and an deal tanslatonal ont etween the slde and the gound. The dmenson and mass popetes of the slde-cank ae lsted n Tale Tale 1. In Tale If t s assumed that the cank s dven at a constant angula veloct of the knematc constants can e fomulated usng methods dscussed n Nkavesh [4] and Haug [5]. Fo ths sstem, the knematc constants can e wtten as: x1 l1 cos1 1 l1 sn 1 x x3 l cos Φ 3 l sn, (7) t whee l 1 and l ae the lengths of the cank and the followe and x, and ae the genealzed coodnates. In q. o! the fst fou equatons desce the two deal evolute onts, the 5th and 6th equatons epesent the tanslatonal ont and the last equaton desces the dvng constant that specfes the veloct of the cank. It should e emphaszed that, snce the ont etween the cank and the followe s non-deal, t does not appea n the set of knematc constants. The non-deal ont s nstead desced usng a foce constant that s pesent n the DA as appled loads. Tale, the mateal popetes and dmensons fo the ont components ae povded. The pn and ushng ae assumed to e made of steel and poltetafluooethlene (PTF), espectvel. Jont cleaance Gound Gound Fgue 3: Slde-cank mechansms wth ont cleaance etween the cank and followe. Tale 1: Dmenson and mass paamete fo slde-cank mechansm Length (m) Mass Moment of (g) neta (kg-m ) Cank Connectng Rod Slde If t s assumed that the cank s dven at a constant angula veloct of the knematc constants can e fomulated usng methods dscussed n Nkavesh [4] and Haug [5]. Fo ths sstem, the knematc constants can e wtten as: x1 l1 cos1 1 l1 sn 1 x x3 l cos Φ 3 l sn, (7) t whee l 1 and l ae the lengths of the cank and the followe and x, and Cank Followe 1 x 1 x Slde ae the genealzed coodnates. In q. o! Refeence souce not found. the fst fou equatons desce the two deal evolute onts, the 5th and 6th equatons epesent the tanslatonal ont and the last equaton desces the dvng constant that specfes the veloct of the cank. It should e emphaszed that, snce the ont etween the cank and the followe s non-deal, t does not appea n the set of knematc constants. The non-deal ont s nstead desced usng a foce constant that s pesent n the DA as appled loads. Tale : Mateal popetes and dmensons fo the ont components Pn (steel) Bushng (PTF) Young s modulus 6.8 GPa.5 GPa Posson ato.9.38 Radus mm.3, 3 mm x 3 x 3 6 Copght 1 ASM

7 -coodnate -coodnate (mm) -coodnate -coodnate (mm) The DA fo ths sstem can easl e assemled and solved. Fo a cank speed of 3 pm, epesentatve esults of the dnamc analss ae shown n Fgues 4 7. In Fg. 4, compasons of the ont eacton etween the deal and nondeal onts ae povded fo two cleaance szes when the contact foce model s used. Fo a cleaance of.3 mm the two cases pedct almost dentcal values fo the ont eacton Fg. 4(a)). Ths smlat n ont eacton pedctons s expected snce the cleaance s small enough so that the nondeal ont ehaves essentall lke an deal ont. In Fg. 4(), a plot of the ont eacton of the non-deal ont fo a cleaance of 3 mm s shown. Fom the two plots t s clea that as the cleaance s nceased, the cuve of the ont eacton evolves fom a smooth one to one chaactezed peak foces. The locaton of these peaks can e explaned notng that the occu ght afte the ont eacton attans a mnmum value o when contact etween the ont components s tempoal lost. When contact s eestalshed, thee s an mpact whch causes the peaks foces. Smla esults ae otaned when the FM s used to model the non-deal ont except that hghe peak foces ae oseved (see Fg. 5). Locus of the Cente of Regon Contact (Pn) x-coodnate Locus of the Cente of Regon Contact (Bushng) x-coodnate (mm) a Fgue 6: Locus of contact pont C fo a complete cank ccle ased on the contact foce model. a) Locus fo the pn. ) Locus fo the ushng. In Fg. 6(a), the locus of the cente ponts of the contact egon (contact pont C as defned n Fg. Fgue ) fo a complete cank ccle, measued fom the pn cente, s shown. The coespondng locus of ponts fo the ushng, measued fom the ushng cente, s shown n Fg. 6(). Fo the pn (Fg. 6(a)), the locus contact ponts C ae concentated on the left half, wheeas fo the ushng (Fg. 6()) the contact pont C s concentated on oth the left and the ght sde of the ushng. Ths coesponds to ushng angula coodnates of appoxmatel and π adans as defned n Fg. 6(). Smla esults ae otaned when the non-deal ont s modeled usng the FM pocedue (Fg. 7). θ x Locus of the Cente of Regon Contact (Pn) 3 Locus of the Cente of Regon Contact (Bushng) 3 a Fgue 4: Compason of eacton etween the deal and the non-deal onts fo vaous ont cleaances fo the contact foce model θ x x-coodnate x-coodnate (mm) a Fgue 7: Locus of contact pont C fo a complete cank ccle ased on FM. a) Locus fo the pn. ) Locus fo the ushng. a Fgue 5: Compason of eacton etween the deal and the non-deal onts fo vaous ont cleaances fo FM. The mpotance of the pont C cannot e oveemphaszed. Ths pont s a efeence pont n the dnamc analss when usng the pevousl outlned pocedues to model sstems nondeal evolute onts. It dentfes the locatons whee the constant foce wll e appled and thus the egons that wll expeence wea. It would e dffcult to pedetemne the locatons that would wea wthout knowledge of ths pont; ths hghlghts the need fo a sstem level wea pedcton. 7 Copght 1 ASM

8 5. WAR ANALYSIS OF MULTIBODY SYSTMS WITH NON-IDAL RVOLUT JOINT In secton, a pocedue to pedct wea at the nteface of two odes that ae n contact and elatve moton was pesented. Late, two dnamc analss pocedues to analze multod sstems wth non-deal evolute onts wee dscussed. Next, the dnamc analss pocedues and the wea pedcton pocedue must e ntegated to enale the analss of multod sstems wth ont wea. Ths tpe of analss allows an changes n the sstem dnamcs caused ont wea to e captued. In addton, the analss allows the coect contact locaton (whee wea occus) to e dentfed n accodance wth the wea pedcton pocedue. In the ntegated model, pevousl dscussed n Mukas et al. [15], a dnamc analss s pefomed on the sstem fo a complete ccle n ode to detemne the ont eactons as well as the ncemental sdng dstance at the weang onts. Ths nfomaton s otaned at each tme ncement of the dscetzed ccle. Thus, fo a non-deal ont, the contact and fcton foce at tme ncement t can e expessed as: F F N, t N, t, t F F n n, t k N, t, t, (8) whee F N s the contact foce detemned va the contact foce model o FM and n s a unt nomal vecto pontng n the decton of contact. The coespondng ncemental sldng dstance can also e desced as: s t R t t, (9) 1 whee R s the ushng adus, t s the angula dffeence (n adans) etween the local x- axes of the two odes and that shae the evolute ont at a cuent tme, and t 1 coesponds to the dffeence at a pevous tme. The ont eacton and the ncemental sldng dstance can then e used to estmate the ont wea usng the pocedue outlned s secton. It s, howeve, necessa to fst detemne the contact pessue dstuton at the ont. Fo the dnamcs analss pocedue ased on the FM, the contact pessue s eadl avalale fom the dnamc analss, wheeas fo the contact foce ased pocedue, FM s used to detemne the contact pessue dstuton. The wea s estmated usng q. () and the geomet s updated to eflect the wea. The evolvng geomet must also e eflected n the dnamc analss. Ths can e done notng that the value of the cleaance c wll no longe e a constant ut wll depend on the contact pont C as defned n Fg. Fgue. The new value fo the cleaance c can e detemned the followng expesson: whee B and D ae the poston vectos of the pn and ushng centes, espectvel. In the case of the contact foce model, once the value of the cleaance c at the efeence contact pont C has een otaned (usng q. (3)), the new value of R at contact pont C can e detemned usng the expesson c R R (see q. (14)). In ths case t s assumed that the value of R emans constant. The contact foce model can then e updated fo wth new value of R. In the case of the FM, the new value of the cleaance c at the efeence contact pont C s also otaned usng q. (3). The penetaton at that pont can then e otaned usng q. (14). Once the pn and ushng geometes have een updated and the cleaance sze s adusted to eflect the wea, a dnamc analss fo the next ccle s pefomed. The wea s then computed ased on the esults of the new dnamc analss followed an update on the geomet and cleaance sze. Ths pocess s epeated untl the desed nume of ccles has een completed. Flowchats showng the ntegated models ased on the contact foce model and the FM model ae shown n Fg. 8 and Fg. Fgue 9, espectvel. 6. COMPARISON BTWN TH FM AND A CONTACT FORC MODL IN WAR ANALYSIS OF MULTIBODY SYSTMS Two pocedues to analze multod sstems wth ont wea wee pesented n secton 5. In ths secton the two pocedues ae used to smulate an actual slde-cank mechansm n whch wea s allowed to occu at one of the onts. The wea esults fom the two pocedues ae compaed. In addton, the pefomance of the two pocedues s assessed compang smulaton esults to esults fom the actual mechansm. Fgue 8: Integaton of wea analss nto sstem dnamcs analss ased on the contact foce model c, (3) B D 8 Copght 1 ASM

9 Fgue 9: Integaton of wea analss nto sstem dnamcs analss ased on the FM method. A dagam of the slde-cank mechansm s shown n Fg. Fgue 1. The mechansm was desgned and constucted to allow onl the ont etween the cank and the followe to wea. Povsons wee made to ensue that all othe onts ehaved as deal onts. The components of the weang ont wee made of a hadened steel pn whch s assumed to e had enough so that no appecale wea occus on ts suface and a PTF ushng whch s soft and expeences consdeale wea. Futhemoe, a spng was attached to the slde whch seved as a means to acceleate the wea on the ushng nceasng the ont eacton foce. The ont eacton foce at the ont of nteest (ont etween the cank and the followe) s measued usng a load cell ult onto a necked poton of the ont s steel pn. The load cell, povded Deee & Compan, uses two full-dge stan gage ccuts to measue the tansvese shea on the pn whch can then e tanslated nto ont eacton foce. Detals of the coespondng nstumentaton and foce extacton can e found n the wok of Mauntle [39-41]. adus of.9mm wee machned nto the ushng. A dagam of the ushng s shown n Fg. Fgue 11. The dmensons and mass popetes fo the slde-cank ae povded n Tale Tale 3. The dmensons, the mateal popetes, the fcton coeffcent and the wea coeffcent of the ont components (ushng and pn) ae lsted n Tale The slde-cank mechansm was opeated fo 1,4 ccles at a constant veloct of 3 pm. A spng wth a spng constant of 55 N/m was used. Smulatons ased on these condtons wee conducted usng the two ntegated models outlned n the pevous secton and shown n Fg. 8 and Fg. Fgue 9. Fo the FM-ased model the fnte element model shown n Fgue 1 was used. In ths model, a gd element was used to model the pn. Ths follows the assumpton that the pn s much hade than the ushng so that t defomaton would e neglgle compaed to that of the ushng. Futhemoe, t s assumed that the pn has neglgle wea and as a esult etans ts ognal shape. The ushng geomet s, howeve, modeled usng 8-node quadlateal element and the coespondng potental contact suface (nne ushng suface) s modeled usng 3-node contact elements. The nodes on the oute suface of the ushng ae constaned fom tanslaton. Contact etween the two components s geneated applng the ont eacton foce on the plot node that s attached to the gd pn element. Tale 4. Futhe detals of the slde-cank mechansm ncludng ts constucton and nstumentaton can e found n the woks of Mauntle et al. [39 41]. Wea Des goove Fgue 11: Bushng wth des gooves. Tale 3: Dmenson and mass popetes of the slde-cank mechansm Length (m) Mass (kg) Ineta x1-6 (kg.m ) Cank Connectng od Slde Fgue 1: Slde-cank mechansm used n ths stud. To allow wea des to ext the contact aea and pevent t fom affectng the pogesson of wea, clndcal gooves wth The slde-cank mechansm was opeated fo 1,4 ccles at a constant veloct of 3 pm. A spng wth a spng constant of 55 N/m was used. Smulatons ased on these condtons wee conducted usng the two ntegated models outlned n the pevous secton and shown n Fg. 8 and Fg. Fgue 9. Fo the FM-ased model the fnte element model shown n Fgue 1 was used. In ths model, a gd element was 9 Copght 1 ASM

10 Jont Foce (N) used to model the pn. Ths follows the assumpton that the pn s much hade than the ushng so that t defomaton would e neglgle compaed to that of the ushng. Futhemoe, t s assumed that the pn has neglgle wea and as a esult etans ts ognal shape. The ushng geomet s, howeve, modeled usng 8-node quadlateal element and the coespondng potental contact suface (nne ushng suface) s modeled usng 3-node contact elements. The nodes on the oute suface of the ushng ae constaned fom tanslaton. Contact etween the two components s geneated applng the ont eacton foce on the plot node that s attached to the gd pn element. Tale 4: Popetes of the pn and ushng Pn Bushng Bushng Inne adus mm Oute adus mm Depth mm Posson ato.9.38 Denst 7.8 g/cm 3.5g/cm 3 Young s Modulus 6.8 GPa.5 GPa Fc. coef. (steel-ptf) [33].13 Wea coef. (steel-ptf) [] 5.5x1-4 mm 3 /Nm The slde-cank mechansm was opeated fo 1,4 ccles at a constant veloct of 3 pm. A spng wth a spng constant of 55 N/m was used. Smulatons ased on these condtons wee conducted usng the two ntegated models outlned n the pevous secton and shown n Fg. 8 and Fg. Fgue 9. Fo the FM-ased model the fnte element model shown n Fgue 1 was used. In ths model, a gd element was used to model the pn. Ths follows the assumpton that the pn s much hade than the ushng so that t defomaton would e neglgle compaed to that of the ushng. Futhemoe, t s assumed that the pn has neglgle wea and as a esult etans ts ognal shape. The ushng geomet s, howeve, modeled usng 8-node quadlateal element and the coespondng potental contact suface (nne ushng suface) s modeled usng 3-node contact elements. The nodes on the oute suface of the ushng ae constaned fom tanslaton. Contact etween the two components s geneated applng the ont eacton foce on the plot node that s attached to the gd pn element. Fgue 1: Fnte element model fo the pn and ushng. Fgues Fgue 13 and Fgue 14 show epesentatve esults fom the ntal sstem dnamcs. Thee plots of the ont eacton foce fom: 1) the expement, ) the model ased on the FM, and 3) the model ased on the contact foce model ae shown n Fg. Fgue 13. The two models, whch ae dentcal n ths case, pedct the ont eacton foce easonal well ove the ente cank ccle except at aout π adans. At ths locaton, the measued foce exhts hgh fequenc oscllaton fo a shot duaton. The locaton of these oscllatons coesponds to one half of the cank otaton when the slde changes decton. It s eleved that these hghe ode dnamcs ae a esult of the change n the decton of the slde whch most lkel nvolves a slght otaton of the slde and thus a moment of ef mpact wth the sldng al. It should e mentoned that, although the magntude of these oscllatons s lage, the effects on the wea pedcton s qute small. Ths s ecause the coespondng ncemental sldng dstance s also qute small Compason Jont Foce Between Models and xpt FM Contact Model xpement Cank Poston (ad) Fgue 13: Compason of the ntal ont eacton foce etween the two models and the expement. 1 Copght 1 ASM

11 Wea on Bushng (mm) Wea on Bushng (mm) Wea on Bushng (mm) -coodnate (mm) -coodnate (mm) Cente of Regon Contact (Contact Foce Model) 1 5 Pn Bushng Locus of the Cente of Regon Contact (FM) 1 5 Pn Bushng and the contact pessue dung the dnamc analss. Howeve, n the case of the FM-ased model the contact foce s detemned n the dnamc analss whch, n tun, s used to detemne the contact pessue θ x Compason of Wea on Bushng Usng FM and FM.5 FM FM x-coodnate (mm) x-coodnate (mm) a Fgue 14: Locus of the cente of the contact egon. a) Pedcton ased the elastc foundaton model. ) Pedcton ased on the contact foce model. In Fg. Fgue 14 the locus of the cente of the egon of contact fo the pn and ushng (measued fo the pn and ushng centes espectvel) s plotted. Fgue Fgue 14(a) shows the locus of ponts when the model ased n the FM s used. It s seen that the ente pn suface expeences contact, whle the cente of the egon of contact on the ushng s concentated on the left sde of the ushng. Ths means that onl one sde of the ushng wll expeence wea and that the maxmum wea wll occu at the cente of the egon of contact. The concentaton of the contact pont n ths locaton s easonale ecause the spng estcts the moton of the pn elatve to the ushng. FgueFgue 14() dsplas the locus of the cente of the egon of contact when the contact foce model s used. It s clea that the two models eld neal dentcal esults. The wea pedcted the contact foce model (n FM FM Dff. Wea Volume mm mm 3.% Max wea depth.4779 mm.463 mm 1.7% Computaton tme 11hs 7hs 4 hs conuncton wth FM) and the FM models ae compae n Fg. Fgue 15 and Tale 5. In Fg. Fgue 15, t can e seen that whle the FM-ased model pedcts a lage maxmum wea depth, the FM has a wde ase. The wde ase means that a wde egon n the ushng suface s won. An nteestng osevaton s that whle the wea depth fo the two models dffes, the wea pofle s such that the won volume s appoxmatel equal. Ths equalt s a manfestaton of the smlat n the foce pofle as shown n Fg. Fgue 13. In Tale 5, the computaton tme fo the wea pedcton ased on the two models s compaed. It can e oseved that the FM has a shote computatonal tme. Thee ae two easons fo ths osevaton: 1) evaluaton of the contact pessue usng FM s nexpensve ecause the elastc lae s composed of a ed of spngs that ae assumed to e ndependent, and ) n the case of the FM-ased model onl one set of analses fo each ccle s equed to smultaneousl detemne oth the contact foce Bushng Angula Coodnate (ad) () Fgue 15: Compason of the wea pedcton etween the models Compason of Wea (xpt and FM) xpt FM Bushng Angula Coodnate (ad) () Bushng Angula Coodnate (ad) () a Fgue 16: Compason of the wea pofle fo the models and the expement. a) Compason etween expement and FM. ) Compason etween expement and FM. Tale 5: Compason of wea esults fo FM and FM models (1,4 cank ccles) The wea esults fom expement and smulaton esults fo the two models ae compaed n Fg. Fgue 16, Tale 6 and Tale 7. Fom Fg. Fgue 16(a) and Tale 6, t can e seen that the maxmum wea depth, the wea pofle and the wea volume fom the expement ae accuatel pedcted the FM-ased model. Thee s, howeve, a dscepanc n the ange 4.5 < θ < 6.3 ad. Ths dscepanc could e attuted to eos encounteed dung measuement. It s possle, howeve, that the ae the esult of some wea mechansm that was not dentfed n the expement. Tale 6: Compason of wea esults etween test and FM model (1,4 cank ccles) Compason of Wea (xpt and FM) xpt FM 11 Copght 1 ASM

12 xpt. Smulaton (FM) Dff. Won mass.616 g.41 g 8.% Wea Volume mm mm 3 8.% Max wea depth mm 1.5% Tale 7: Compason of wea esults etween test and FM model (1,4 cank ccles) xpt. Smulaton (FM) Dff. Won mass.616 g.4 g 8.% Wea Volume mm mm 3 8.% Max wea depth mm 1.1% In the case of the FM model, the wea pofle and the maxmum wea depth ae ncoectl pedcted as shown n Fg. Fgue 16() and Tale 7. The locaton of maxmum wea and the wea volume ae, howeve, coectl pedcted as expected. 7. CONCLUDING RMARKS The oectve of ths wok was to pesent a compason etween two models that can e used to analze multod sstems wth ont wea. The compason was ased on wea esults of an expemental slde-cank mechansm that was ult to encouage wea at a sngle ont (a evolute ont etween the cank and the followe). Quanttes that wee consdeed n the compason ncluded the ont eacton, the contact/wea locaton and wea amount on the ushng. Fo the ntal dnamcs the two models povded easonal accuate pedcton of the contact foce. Ths ndcates that the sstem dnamcs ae nsenstve to the contact model used. The two models also poduced dentcal pedcton fo the locaton of maxmum wea, whch was vefed to e the coect locaton though the coespondng expement. Although ths locaton was coectl pedcted oth models, onl the FM-ased model gave an accuate pedcton of the wea pofle and maxmum wea depth. Pedcton fom the FM-ased model dffeed 6.7% fom expement, whle the pedcton fom the FM model dffeed 1.1% fom expement. The dffeence n the wea depth pedcton etween the FM- and FM-ased models s attuted to the assumpton that the spngs fomng the elastc foundaton ae ndependent (no shea foce etween the spngs). As was mentoned eale, the consequence of ths assumpton s that the model does not account fo how pessue appled at one locaton affects the defomaton at othe locatons [4]. Ths s conta to what s expeenced n elastc contact whee the dsplacement at one locaton s a functon of the pessue appled at othe locatons. Thus as s oseved, the FM-ased model povdes a ette pedcton of the wea depth. It should, howeve, e noted that, whle the pedctons on the wea pofle dffeed, the wea volume pedctons of the two models wee dentcal and easonal close to the expemental wea volume (8. %). Despte the poo pedcton on the wea pofle and maxmum wea depth, the FM-ased model had a shote computaton tme. The expement took aout 1 hous (excludng constucton and setup tme) and FM-ased model took 11 hous, whle the FM-ased model took onl 7 hous to complete. The speed of the FM-ased model s assocated wth the assumpton that the spngs n the elastc foundaton model ae ndependent and the fact the onl one analss s equed to detemne oth the contact pessue dstuton and the ont eacton. Fom the compason etween the esults of the two models and etween the esults of the two models and the expement, the followng conclusons can e made: 1) the two pocedues can accuatel pedct the contact foce, the contact locatons and the wea volume, ) the FM-ased pocedue s a ette pedcto of maxmum wea depth and the wea pofle than the FM-ased pocedue, and 3) the FM model s computatonall less expensve than the FM-ased model. It can e concluded that the FM-ased pocedue s pefeed fo the analss of multod sstems wth ont wea when the computatonal cost s not an ssue. If the cost s a concen, howeve, and onl qualtatve nfomaton aout the sstem s needed, then the FM-ased pocedue s a sutale choce. Othe scenaos wll eque a compomse on ethe accuac o computatonal costs. NOMNCLATUR A : Contact aea A : xtapolaton facto : Penetaton e : Coeffcent of esttuton : Young s modulus F : Nomal foce n the contact nteface N h k K λ M p q Q A s t Φ Φ q : Wea depth : Dmensoned wea coeffcent : lastc constant : Vecto of Lagange multples : Mass matx : Contact pessue : Poston vecto : Vecto of appled loads : Sldng dstance : Tme : Posons ato : Constant vecto : Jacoan matx : Cank veloct ACKNOWLDGMNTS The authos gatefull acknowledge suppot the Natonal Scence Foundaton (DMI-6375) and Deee & Compan. 1 Copght 1 ASM

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