A CYLINDRICAL CONTACT MODEL FOR TWO DIMENSIONAL MULTIASPERITY PROFILES

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1 Poceedings of 003 STLE/ASME Intentionl Joint Tibology Confeence Ponte Ved Bech, loid USA, Octobe 6 9, TIB-69 A CYLINDICAL CONTACT MODEL O TWO DIMENSIONAL MULTIASPEITY POILES John J. Jgodnik nd Sinn Müftü Nothesten Univesity Detment of Mechnicl Engineeing Boston, MA 05 ABSTACT In ctice, multi-seity contct oblems e often solved s two dimensionl (D) lne oblems the thn tue thee dimensionl (3D) oblems. This is ccomlished by ssuming tht ech ek on D scnned ofile is the inncle of hlf shee. Hetz contct equtions e then used to solve fo the dius of contct nd essue ofile. In elity, the locl mximum in the lne my not be the mximum in the unmesued deth diection, ceting inheent eos in the contct model. This eo is shown to be significnt in contct oblems when estimting the e of contct nd totl contct foce ove single seity. The essue coected Stenbeg- Tuteltub model is intoduced, in which cylinde is used to model shee in lne. This model is shown to imove the contct e nd totl foce estimtes fo nge contct metes. INTODUCTION Tyiclly, multi-seity contct oblems e solved by ssuming tht ech seity is efect shee. Hetzs equtions fo contcting shees gives good estimtes of the contct essue ofile, if the dius of the shee is known []. Howeve, the tooghy of the sufce is often mesued by using stylus tye ofile, which scns single lne. In elity, the locl mximum in the scnned lne my not be the mximum in the unmesued deth diection. Using this dius in the sheicl Hetz contct model cetes inheent eos. In this e the significnce nd esoning behind these eos e intoduced nd coections, including modified fom of the cylindicl contct model, e oosed to educe the eo in lne oximtions. Plne Aoximtions Multi-seity contct oblems cn be solved in two o thee dimensions. A sufce ofile cn be used to detemine the sufce height in both the x nd y diections (ig ). Then, the ovell equilibium of the two contcting sufces could be found by diect modeling of ech seity s Hetz contcts [], o by sttisticl techniques []. o these oblems, ech seity is teted s shee nd Hetz contct equtions e lied. Tyiclly, 3D solutions equie significnt comuttionl time. It is lso difficult to nlyze comlex oblems involving sliding, comosite lyes, etc. in 3D [4]. Two dimensionl (D) lne oximtions ovide n ltentive, whee sufce ofile could be used to obtin the sufce heights long the x diection, on single x-z lne, s shown in ig. Simil ofiles cn be obtined fo genel sufce using ndom ofile geneto [3]. Tyiclly ech ek on the ofile is teted s the inncle of shee nd Hetz contct equtions e lied. In this e, we will investigte the eo fom ignoing the y diection effects when D ofile is nlyzed, the thn full 3D sufce. THEOY Single Aseity Anlysis Hee single seity the thn n entie sufce o ofile is nlyzed. The seity is teted s efect hlf shee, whose cente lies in lne- s shown in ig. The shee is subected to intefeence fom flt igid unch. It will be ssumed tht the sufce ofile scnned the seity long lne-, which is llel to lne-, but offset by distnce y. ocusing on font view of the seity in ig b, it is igue 3D lne ofile, nd the D ofile in the x-z scn lne Coyight 003 by ASME

2 used in the ST equtions. This modified cylindicl contct model is clled the ST-model, hee. noted tht if the scn lne is identicl to lne-, the tue dius of the shee () nd the tue mximum intefeence (δ ) will be found. If y is not zeo, diffeent vlue fo the dius ( ) nd mximum intefeence ( δ ) will be obseved. Clely ssuming tht the obseved hlf cicle on the scnned ofile is the inncle of shee will led to some mount of eo deending on the vlue of y. ocusing on ig c, the eltionshis between the tue nd obseved dii nd mximum intefeences e found s follows, ( ) δ =δ (), = y (b) () Anlysis Ove The Entie Aseity In ode to gin vluble infomtion bout the eo ssocited with using the metes obtined fom line scn, the entie seity will be nlyzed, the thn simly nlyzing single y vlue. In ode to do this, n (=00) eqully sced y vlues e selected. These y vlues cove the entie nge, whee the unch intefees with the undefomed shee. o ech y vlue the contct oblem is solved on lne- using thee methods. ist, the full 3D model is solved using sheicl Hetz contct equtions with the tue dius nd tue mximum intefeence fom lne-. The elevnt infomtion fo lne- is then extcted fom the 3D solution. This och is clled the Hetz (H) model. The second method fo solving the lne oblem is to simly ly the sheicl Hetz contct equtions to the nd intefeence eceived dius δ vlue, obtined fom the ofile. This och, which is bsed on incomlete infomtion on sufce tooghy is clled the incomlete Hetz () model. The thid method involves using modified fom of the cylindicl contct model bsed on Stenbeg nd Tuteltubs (ST) wok [5]. This model ovides eltion between the unch enettion δ nd the contct dius, nd cylinde dius. The modified ST model is intoduced lte in the e. The vlues of nd ) b) igue : ) Thee dimensionl model of n seity; The font b) nd side c) views; nd d) Detils of the el contct e. δ obtined fom the ofile e c) d) Equtions fo the Thee Models The focus of the nlysis is to detemine the cumultive effect of ndomly chosen lne- (y ) loctions on the snned contct e (A ) nd totl foce ( ), using the thee methods. A diffeent subscit used on the vibles, to indicte the method. The subscit H is dded fo the el 3D Hetz solution. The subscits nd ST e used fo the incomlete Hetz model nd fo the cylindicl contct with Stenbeg- Tutlebums modifictions, esectively. The dius nd unch enettion vlues in the lst two models e obtined in lne-. Equtions fo the el 3D Hetz Solution The eltion between the tue contct dius nd the eceived contct dius cn be found fom ig. d s, = y. () Substituting the well-known Hetz [] eltionshi between contct dius nd mximum intefeence ( = δ ), the el contct dius on ny lne cn be found to be, = δ y. (3) Using the essue distibution fom Hetz contct, the eqution fo the essue in the contct length (x) of the scn lne is, Eδ x + y x = ( ) H π( ν ) The foce e unit width on the scn lne ( H ) H =. (4) is found s, Eδ y, (5) ν by integting H ove the el contct length fo the scn lne-. The snned contct e (A H ) is found by numeiclly integting the contct length though the y diection with, n AH = y δ yi. (6) whee n is the numbe of line scn loctions. Similly, the totl foce lied to the seity is found by numeiclly integting though the y diection with, H n Eδ y y i = ( ν ). (7) In these summtions, y is the scing between scn lnes in the y diection. It should lso be noted tht the Hetz equtions cn be lied diectly to find the tue contct e nd foce suoted by the seity without numeicl integtion. These equtions cn be used to veify tht smll enough y vlue hs been selected. Coyight 003 by ASME

3 Equtions fo the Incomlete Hetz Model o the Hetz shee solution, the dius of contct on ny scn lne is given s, = δ. (8) whee nd δ e given in Eqns () nd (b). The contct e is found by integting in the y diection in the sme mnne tht ws followed fo the 3D solution, giving, n / A = y ( δ ). (9) The essue t given x vlue in the contct length on scn lne is, ( ) i = Eδ x ( x) = π ν. (0) The foce e unit width on the scn lne cn be found by integting ove the contct length fo the scn lne. Pefoming this integtion leds the foce e unit width, given s, Eδ =. () ν Numeicl integtion though the entie seity in the y diection leds to the totl foce on the seity, given s, n E y = δi ν. () Equtions fo the Cylindicl Contct with Stenbeg- Tuteltub Modifictions Stenbeg nd Tuteltub nlyzed n infinitely long cylinde with finite dius comessed distnce δ between two flt igid unches [5]. om this nlysis the following equtions wee obtined unde the ssumtion tht dius of the cylinde is much gete thn the hlf-contct length, 4 δ γ 4 9γ = ln + γ 64 ϕγ =γ 4 3 s 3γ υ o =γ+ 6 ( ) (3,b,c) s whee o is the mximum contct essue, γ = ( υp π) /, υ= ν E, nd ϕ= ( ν) ( ν ). Using symmety bout the mid-lne between the two unches llows the oblem to be teted s hlf cylinde. In this wok Eqns (3) nd (3b) e solved numeiclly by using ν = 0.3, nd cuve fit eltionshi between,, nd δ is obtined, ˆ m m = C s δ (4) with C s = 0.96 nd m = Note tht this eqution is vlid in the nge δ/ Substituting eqns () nd (b) yield the hlf-contct length, = 0.96 δ. (5) ST The contct e is obtined, by numeiclly integting though the y diection, n AST =.94 y δ. (6) The essue t ny oint in the contct length on scn lne is eesented by, whee, E 3 3 x ST ( x) = η+ η ν 4 ST η= ϕ ϕ ST / / (7). (8) The foce e unit width on the scn lne fo the ST cylinde is found in the sme mnne s the Hetz nlysis, E ST = 0.48π δ η+ η ν 4. (9) The totl foce on the seity is found s, n E y ST = 0.48π δ η+ i ηi ν 4. (0) Non-dimensionl Equtions All of the vibles involving length e scled with, y y o o δ δ δ δ o = () whee the subscit tkes the vlues H, nd ST s oite, nd the mximum essue is defined s, = Eδ ( ν ) o π. () This non-dimensionliztion esults in only thee indeendent metes tht contol the oblem, ν, nd y. ESULTS Contct Pessue The contct essue distibution fo the thee models e y vlues nd =0 in ig 3. This lotted fo thee diffeent figue shows tht the - nd the ST-models och the Hetz y 0 nd y, esectively. essue distibution s This indictes tht, s the line-scn moves futhe wy fom the inncle of the seity, the mtch between the ST nd Hetz models gdully imoves. 3 Coyight 003 by ASME

4 Contct dius Contct dius is lotted fo diffeent y loctions nd vlues in igue 4. The contct dius H fo the Hetz solution flls on unit cicle s exected. Howeve, the contct dius of the -model ( ) oveestimtes nd the contct dius of the ST model ( ST ) undeestimtes the Hetz solution. It is inteesting to note tht the effect of seity dius on the -model is so smll tht it cn not be detected in ig. 4. On the othe hnd the ST-model futhe undeestimtes the contct dius with incesing. Pessue, = / o 0.5 y = 0.5, t = 0 H ST Contct Ae Sn The obbility of scnning given seity t y loction is the sme fo ll y vlues. Theefoe, if we integte the e unde the cuve, given in ig 4, we get n estimte of the contct e sn A tht the thee models e cble of estimting. A is obtined numeiclly fom eqns (6), (9) nd (6). igue 4b dislys the A vlues fo ll thee methods. Note tht the snned e fo the Hetz model is the constnt π, nd is equl to the el contct e of the Hetz contct model with the non-dimensionl contct dius of. The snned e fo the -model is nely constnt with This vlue is 44% gete thn π. inlly, the snned e of the ST-model is vible s shown in ig 4b. The eo between snned e estimtes of the Hetz model (π) nd the ST-model vies between 3% nd 30%, whee the eo becomes smlle t smlle vlues of. When using lne oximtions, the - nd ST-models wongly ssume the shee o the cylinde is centeed on the scn lne nd thus the fist oint to touch the flt unch is the ek s seen in the scn lne. This esoning exlins the oveshoot e fo the - nd ST-models. The emining eo in the ST-model is simly ttibuted to modeling otion of shee with cylinde. While it my not be ent s to why the ST cylindicl model is being consideed fo modeling shees in lne, it is inteesting to note tht the this model ovides bette oximtion to the tue contct e in the nge of seity dii (5 50) consideed in ig. 4b. This is esult of the undeestimtion of the contct dius ne the cente of the shee, which to some extent coects the oveestimtion in the oveshoot e. This coection is lcking in the -model whee the contct dius is oveestimted t ll oints excet the cente whee the solution is ecise. oce e Width The foce e width exessions, fo the thee models, e given in equtions (5), () nd (9). These exessions e lotted in ig 5 s function of scn loction y, fo thee diffeent vlues of. This figue shows tht the foce e width is oveestimted by the -model s comed to the Hetz model; nd, it is undeestimted by the ST-model fo Pessue, = / o Pessue, = / o 0.5 Contct length, x = x/ Contct length, x = x/ Contct length, x = x/ lge section of the y nge. The -model is insensitive to chnges in, whees the estimte of the ST-model becomes wose s inceses. y = 0.7, t = 0 H ST y = 0.9, t = 0 H ST igue 3 Contct essue distibution fo nd =0. y =0.5, 0.7, Coyight 003 by ASME

5 .5.00 Non-dimensionl contct dius, = / 0.5 t 0, 50 H ST Non-dimensionl foce ewidth.50 t 0, 50 H S T Scn loction, y = y /.50 Scn loction, y = y / Non-dimensionl e sn, A A H A A ST Totlfoce, H S T Totl oce As ws the cse fo the snned contct e, we cn view the effects of seity dius on the totl snned foce. Eqns (6), (3) nd (0), which give the exessions fo the thee models, e lotted in ig 5b. This figue shows tht H =.05 fo the Hetz model, nd tht it is indeendent of. The totl foce fo the -model is nely constnt with the vlue =.49. Howeve, the totl foce fo the ST-model Non-dimensionl seity dius, t = t / igue 4 ) Non-dimensionl contct dius fo diffeent vlues. b) Contct e sn vition with fo the thee models. vies with s shown in ig 5b. The eo between the STmodel nd the Hetz solution vies between 5 nd 7%. The eo between the totl foce estimtes cn be educed by intoducing coection fctos fo the essue estimtes given by eqns (0) nd (7) fo the - nd ST-models, esectively. ig 5b suggests tht constnt coection fcto of C = 0.7 should be used fo the -model. On the othe hnd the coection fcto C ST fo the ST-model deends on the vlue of. o exmle, we found tht fo =0 the coection fcto should be C ST =.8. The vlue of =0 eesents, fo Non-dimensionl foce ewidth Non-dimensionl seity dius, t = t / t = 0.50 Scn loction, y = y / exmle, = µm nd δ = 0 nm, which e metes tyicl of mgnetic te lictions [4]. ig 5c shows the effect of using the coection fctos indicted bove on the estimtes of foce e width of the seity. Both the - nd ST-models follow vey simil H C C ST S T igue 5 ) Dimensionless foce e width s function of scn loction y. b) Totl snned seity foce s function of seity dius. c) Estimtes of the nd ST models with coection fctos C nd C ST. 5 Coyight 003 by ASME

6 tends, nd the es unde the two modified cuves nely equls tht unde the H-model. It is lso imotnt to note tht the essue coection fctos do not effect the contct e nlysis stted bove. Theefoe, the eos in contct e estimtion emin s they e given in ig 4b. As the e estimte eos e smlle fo the ST-model, s comed to the -model use of the STmodel my be n ttctive choice. DISCUSSION Qulittive Anlysis of Multi-seity Contct In the evious sections contct ove single seity ws nlyzed. While this nlysis ovides insight into the esoning behind the use of the essue coected ST model, the el uose fo using eithe model is to nlyze single lne fom multi-seity sufce. In the evious section single seity ws boken into 00 thin stis. Ech sti ws nlyzed nd then combined to find the esulting e of contct nd totl foce oximtions. The behvio of lne fom multi-seity contct oblem closely llels the single seity nlysis. Let us conside scn lne, which intesects n contcting seities, nd ssume the y coodintes fo the centes of ech seity e ndom nd uncoelted. If this is the cse thee is n equl obbility fo ny one of the 00 thin stis of given seity flls within the scn lne. This mens tht if n is lge, we exect to see n/00 sti s, n/00 sti s, etc. This is close to nlyzing n/00 single seities, with the only diffeence being the vition of δ fom seity to seity due to the ndom ntue of the sufce. Neglecting the effects δ of the vition in, the conclusions eched fo single seity cn be diectly lied to the multi-seity contct cse. While this qulittive nlysis es esonble, moe detiled quntittive nlysis of multi-seity lne oximtions will be the subect of futue e. SUMMAY AND CONCLUSION While the ccucy of the Hetz contct equtions e uncontested in 3D nlysis, often the equtions e used with infomtion bsed on D lne scns. In these nlyses, the locl eks e ssumed to be the inncles of shee, when in fct they my not be the inncle hd the deth diection been consideed. The eos fo this ssumtion hve been nlyzed fo single seity nd shown to be substntil when estimting the contct e nd totl foce ove the seity. Coection fctos fo the contct essue e discussed. In ticul, modified (ST) cylindicl contct model is intoduced. This model is shown to be bette in edicting contct e. A moe obust model fo detemining the essue coection fcto nd moe vigoous multi-seity contct nlysis will be the subect of futue es. EEENCES. Johnson, K.L. (985) Contct Mechnics, Cmbidge Univesity Pess.. Geenwood, J.A. nd Willimson, J.B.P. (966) "Contct of Nominlly lt Sufces," Poceedings, oyl Society, A95, Pti, N. (978) Numeicl Pocedue fo ndom Genetion of ough Sufces, We, v 47, n, Bhushn, B. (996) Tibology nd Mechnics of Mgnetic Stoge Devices, Singe- Velg New Yok Inc., New Yok (Ch. nd 3). 5. Gldwell, G.M.L. (980) Contct Poblems in the Clssicl Theoy of Elsticity, Sithoff & Noodhoff Int. Publishes, The Nethelnds (Ch. 8). 6 Coyight 003 by ASME