AN ANALYTICAL SOLUTION OF THE REYNOLDS EQUATION FOR THE FINITE JOURNAL BEARING AND EVALUATION OF THE LUBRICANT PRESSURE

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1 AN ANALYTICAL SOLUTION OF THE REYNOLDS EQUATION FOR THE FINITE JOURNAL BEARING AND EVALUATION OF THE LUBRICANT PRESSURE Dimitis Sfyis Uivesity of Aegea 84 Aegea, Geece Athaasios Chasalevis Damstadt Uivesity of Techology Dept. of Mechaical Egieeig, Ist. fo Stuctual Dyamics 6487 Damstadt, Gemay ABSTRACT The Reyolds equatio fo the pessue distibutio of the lubicat i a joual beaig with fiite legth is solved aalytically. Usig the method of the sepaatio of vaiables i a additive ad i a multiplicative fom a set of paticula solutios of the Reyolds equatio is added i the geeal solutio of the homogeous Reyolds equatio ad a closed fom expessio fo the defiitio of the lubicat pessue is peseted. The Reyolds equatio is split i fou liea odiay diffeetial equatios of secod ode with o costat coefficiets ad togethe with the bouday coditios they fom fou Stum-Liouville poblems with the thee of them to have diect foms of solutio ad oe of them to be cofoted usig the method of powe seies. I this pat of the wok, the mathematical pocedue is peseted up to the poit that the applicatio of the boudaies fo the pessue distibutio yield the fial defiitio of the solutio with the calculatio of the costats. The distibutios of the pessue give fom the paticula solutio ad the solutio of the homogeeous Reyolds equatio ae peseted togethe with the esultig pessue. Also, the esults of a appoximate aalytical solutio usig Bessels fuctios ad lieaizatio of the fluid film thickess fuctio ae also peseted togethe with the esults of the umeical solutio usig the fiite diffeeces method. Diagams fo the pessue pofiles ude the cuet study ae compaed with those fom the appoximate aalytical ad the umeical solutio. The locatios i which the maximum, the zeo, ad the miimum pessue ae peseted ae give as a fuctio of ecceticity ate with closed fom expessios. KEYWORDS Joual beaig, Reyolds equatio, pessue distibutio, aalytical solutio, sepaatio of vaiables, Stum-Liouville, powe seies. INTRODUCTION The exact aalytical solutio of Reyolds equatio fo lubicatio of joual beaigs [] is up to owadays a poblem ude ivestigatio. Most ivestigatos achieved to defie fuctios of pessue distibutio P( θ, x) ude appoximate solutios of Reyolds Eq. (.). h t P x h t P x h h t x x R t 3 3 ( θ, ) (, θ) ( θ, ) (, θ) ( θ) ( θ, ) + =Ω + 6µ θ 6µ θ θ (.) The appoximate aalytical solutios of the Reyolds equatio ae based o the assumptios that oe of the two tems i the left side of Eq. (.) ca be eglected. The fist tem ca be eglected whe the joual beaig is cosideed as a beaig with high legth to diamete atio (log beaig, L/D>> []) ad the secod tem whe the joual beaig is cosideed as a beaig with low legth to diamete atio (shot beaig L/D< [3-8]). Such solutios fo pessue distibutio give cosideably simplified mathematical expessios. They ae exact solutios i the sese that they satisfy Eq. (.) fo the case of ifiite axial legth o of ifiite shot beaig. Howeve they ae egaded as appoximate solutios whe they ae used to detemie the pessue distibutio i beaigs of fiite legth. The eseach cotibutios i the aalytical calculatio of the pessue distibutio i a joual beaig with fiite legth ae ot may ad they have ot bee made ecetly. Some of them ae efeeced i bief i the cuet wok. Kigsbuy [9] detemied the pessue distibutio by a expeimetal electical aalogy. Chistopheso [] detemied the pessue distibutio by utilizig the mathematical model of elaxatio. Vogelpohl [-3] achieved to give closed foms of the fuctios fo the pessue distibutio alog both diectios of the joual beaig, axial ad cicumfeetial. Vogelpohl [-3] assumed a patial solutio of the Reyolds equatio that coespods to a log beaig appoximatio ad the he used the techique of the sepaatio of vaiables fo the solutio of the homogeous Reyolds equatio. By addig the two solutios, the pessue distibutio was defied i closed fom. Cameo ad Wood [4] had exteded the wok of Chistopheso [] to show the effect of legth to diamete atio o ecceticity atio, attitude agle ad fictio coefficiet. I all cases, these solutios ae give to expess atual pheomea i the oil film o the

2 basis of Reyolds assumptios egadig lubicatio, the most impotat assumptio beig that cetai tems i the geealized Navie-Stokes equatios fo flow i a viscous fluid may be eglected. I the cuet wok the Reyolds equatio of the fom of Eq. (.) is teated similaly to the path that Vogelpohl [-3] followed, but thee ae cucial diffeeces that have to do with the way that the paticula solutio is obtaied ad with the solutios of the odiay diffeetial equatios that ae yielded duig the pocedue. I the begiig of the cuet wok the Reyolds equatio is classified. As expected it tus out that it is a elliptic equatio, which is easoable sice it defies a static poblem (the time is ot cosideed as a idepedet vaiable but as a paamete that gives the ecceticity ad its ate of chage) The stategy fo obtaiig the aalytical solutio is based i the applicatio of the poweful method of sepaatio of vaiables. The cucial step is a splittig of the solutio i to two pats. The oe satisfies the homogeeous Reyolds equatio; amely, Reyolds equatio without the secod pat that the pessue does ot itefee. Fo solvig this pat we assume a multiplicative sepaatio of the idepedet vaiables so we obtai two Stum-Liouville poblems. I this poit, the cotibutio of the cuet wok is that the use of the powe seies method is used i ode to obtai the eigefuctios of the oe Stum-Liuville poblem while i the liteatue the coespodig cofotmet of this poblem is made with the liea appoximatio of the fluid film thickess fuctio ad the expessio of the eigefuctios usig Bessel fuctios o omal siusoidal fuctios. I the cuet wok, the easy stum-liuville poblem is solved ad the eigevalues of it ae icopoated i the fuctios defied with powe seies. The secod pat is a paticula solutio of the Reyolds equatio itself. Fo fidig a paticula solutio we assume a additive splittig of the idepedet vaiables ad two Stum-Liuville poblems ae obtaied. The fist has to do with the pessue distibutio alog the cicumfeetial coodiate ad it has a diect solutio with a closed fom expessio take fom the liteatue fo ODE teatmet. The othe poblem has to do with the pessue distibutio alog the axial coodiate ad the bouday coditios which ae chose yield a tivial solutio, without this to be poblematic fo the futhe pogess of the solutio. The cuet paticula solutio is also a cotibutio of the cuet wok sice it is actually a set of paticula solutios that ca be diffeet fom the solutio of the ifiitely log beaig as used i []. The cuet paticula solutio yields the log beaig pessue distibutio as a sub case. The pape is ogaised as follows. The sectio cotais the basic igediets of Reyolds equatio togethe with the classificatio of it ad the cucial step whee the ukow pessue P is split ito two fuctios g ad u. The Sectio 3 is elated with the evaluatio of the paticula solutio of the Reyolds equatio, u. I Sectio 4 we peset a aalysis fo evaluatig the fuctio g usig agai the method of sepaatio of vaiables, but ow i a multiplicative fom. The teatmet with the Bessel fuctios ad the appoximatig aalytical solutio is also give. The last sectio, sectio 5 deals with the bouday value poblems. The esultig pessue is evaluated fo a specific set of values of the physical ad geometical joual beaig chaacteistics ad is compaed with the aalytical appoximatig solutio ad the umeical solutio usig fiite diffeeces method. The aticle cocludes i Sectio 6 whee also the fothcomig esults ae descibed. REYNOLDS EQUATION: CLASSIFICATION AND SPLITTING OF THE SOLUTION The poblem of the lubicatio of joual beaigs with fiite legth is defied i this wok as the calculatio of the pessue distibutio of the Newtoia lubicat that is assumed to flow ude lamia, isoviscous, ad isothemal coditios i betwee the otatig joual ad the static beaig. The joual of adius R ad legth Lb is assumed to be otatig with a costat otatioal speed Ω (coute clockwise) ad to be costatly located i a poit of ecceticity e with espect to the geometic cete of the beaig of adius R+ c ad legth Lb afte a applicatio of a costat vetical loadw as show i Fig.. The joual ecceticity is assumed to have ate of chage descibed by the velocity e. The joual ad the beaig ae supposed to be i paallel (aliged beaig) ad the fluid film thickess h becomes a fuctio of the uique paameteθ fo a time momet of costat e ad e which meas that the fuctio fo the fluid film thickess is h = c + ecos( θ ) ad its time deivative as h/ t = e cos( θ ). The dyamic viscosity of the lubicat is assumed to be costat ad equal to µ though the etie cotol volume (otified with shadow i Fig. ) that is defied fom the beaig ad the joual sufaces. The attitude agle of the joual is defied as ϕ with espect to the vetical coodiate axis (see Fig. ). The bouday coditios fo the pessue distibutio ae left to be defied i the last sectio whee the umeical applicatio is made.

3 Fig.. Defiitio of the coodiate system ad of the paametes of opeatio ad desig i a plai cylidical joual beaig. The statig poit is the equatio of Reyolds which is expessed as i Eq. (.). 3 3 h P ( x, θ) h P ( x, θ) h + h =Ω + (.) x 6µ x R θ 6µ θ θ t The costat magitudes Ω, R, µ, c, e, e, W epeset the otatioal speed of the joual, the adius of the joual, the lubicat dyamic viscosity, the beaig adial cleaace, the joual ecceticity, the joual ecceticity ate of chage ad the exteal loadig foce coespodigly, see Fig.. Afte substitutig the fluid film thickess fuctio of Eq. (.) ito Eq. (.) ad pefomig the deivatios oe will aive at Eq. (.3). h = c + ecos( θ ) (.) ( c + ecos( θ) ) 3( c + ecos( θ) ) esi ( θ) ( c + ecos( θ) ) 3 3 P( θ, x) P( θ, x) P( θ, x) + = eωsi 6µ x 6µ R θ 6µ R θ ( θ ) e cos( θ ) + (.3) Eq. (.3) is the oe that we ae goig to wok with. This is a o homogeous liea patial diffeetial equatio of the secod ode fo the ukow fuctio Pxθ (, ) with tigoometic coefficiets. Befoe embakig o the coe of ou aalysis we classify Eq. (.3). Fo doig so we eed to evaluate the discimiat as i Eq. (.4). h h h (.4) 6µ R 6µ 36R µ = B AΓ= = < Whateve the expessios of hrµ,, ae, we speak about a elliptic patial diffeetial equatio. The cucial step fo obtaiig the solutio is based o the followig splittig of the solutio. We assume that the ukow fuctio ca be witte i the fom of Eq. (.5). (, θ ) = (, θ) + (, ) P x u x g x θ (.5) paticula hom ogeic 3

4 The fuctio u( x, θ ) is a paticula solutio of Eq. (.3) while the fuctio g ( x, θ ) descibes the set of solutios fo the homogeeous Reyolds equatio, amely, Eq. (.3) without the ight had side tem. I ode to see moe clealy that istead of seekig P( x, θ ) we ca seek fo the fuctios g ( x, θ ) ad u( x, θ ) oe may wite the diffeetial opeato of Reyolds equatio as i Eq. (.6). 3 ( c + ecos( θ) ) 3( c + ecos( θ) ) esi ( θ) ( c + ecos( θ) ) 3 + 6µ x 6µ R θ 6µ R θ (.6) If oe applies this opeato to both sides of Eq. (.5) he will immediately veify that equivaletly oe may seek fo the, u x, θ. fuctios g ( x θ ) ad 3 EVALUATION OF THE PARTICULAR SOLUTION I ode to evaluate the fuctio u( x, ) i the followig additive fom of Eq. (3.). (, θ ) ϕ(, θ) ψ ( θ) θ a paticula solutio of Eq. (.3) is eeded. Fo doig so we assume that (, ) u x θ ca be split u x = x + x, (3.) We ae lookig fo a solutio whee the idepedet vaiables ca be split i the above fom. If we substitute Eq. (3.) i Eq. (.3) afte some calculatios Eq. (3.) is obtaied. d ϕθ ( ) Ω ( ) 3 ( θ) ( ) d ψ x 3esi θ dϕθ 6µ e si θ µ ecos = + dx R dθ R c + ecos θ dθ c + ecos θ c + ecos θ 3 (3.) By ispectig the above equatio oe obseves that the ight had side is a fuctio of θ while the left had side is a fuctio of x oly. So, the equality will be feasible oly whe both sides ae equal to the same costat C. Fom the latte equatio we obtai ψ x. These ae odiay diffeetial equatios witte as i Eqs. (3.3) ad (3.4). two equatios fo the fuctios ϕ ( θ ) ad ( x) d ψ = C (3.3) dx ( ) Ω ( ) 3 ( ) d ϕθ 3esi θ dϕθ 6µ e si θ µ ecos θ + R dθ R c + ecos θ dθ c + ecos θ c + ecos θ Eq. (3.3) ca be solved diectly to give Eq = C (3.4) 3 Cx ψ ( x) = + cx + c (3.5) The costats c, cae abitay costats of itegatio. Fo solvig Eq. (3.4) oe obseves that we talk about a liea odiay diffeetial equatio with o-costat coefficiets whee the ukow fuctio z θ as ϕ ( θ ) is ot peset explicitly. So, by settig i Eq. (3.6), istead of Eq. (3.4) we may solve the followig liea odiay diffeetial equatio defied i Eq. (3.7) of the fist ode fo the fuctio z ( θ ). dϕθ = z ( θ ) (3.6) dθ 4

5 ( ) Ω ( ) 3 ( ) dz( θ ) 3esi θ 6µ e si θ µ ecos θ z( θ ) + R dθ R c + ecos θ c + ecos θ c + ecos θ = (3.7) The geeic set of solutios of the last equatio is give i Eq. (3.8) [8]. 3esi( θ) 3esi( θ) dθ dθ c cos cos 6 si cos e θ c e θ µ Re θ µ Re θ + z( θ ) e + Ω Ω = c3 + e + + C dθ (3.8) 3 3 ( c + ecos( θ) ) ( c + ecos( θ) ) All the itegals i Eq. (3.8) ca be evaluated usig diect closed fom expessios. By evaluatig the itegals that appea i the ϕ θ, Eq. (3.9). 3 C expessio fo z ( θ ) oe ca itegate the outcome oe time ad obtai the fuctio ϕθ = z θdθ+ c (3.9) 4 The sum of ϕ θ ad ( x) ψ edes classes of paticula solutios by choosig values fo the costats Cc,, c, c, c. By choosig C = we obtai Eqs. (3.) ad (3.). 3 4 ψ ( x) = cx + c (3.) ( ) ( cos( θ )) ( θ) 3 ( ce + e) ( c + e ( θ )) B 8ce R µ Ω+ c + e c3 3e cr µ Ω cc3 si θ c + e er µ e 6cR µ Ω c3 si ϕθ = c / c cos + e c e c + e (3.) Fo the pessue distibutio yielded fom the paticula solutio, the bouday coditios ae settig the pessue equal to zeo at the L b both eds of the beaig i axial diectio x =± ad at the cicumfeetial locatio of θ =. With the followig bouday coditios i Eqs. (3.-3.5) the costats c, c, c3, c4 ca be defied as i Eqs. ( ). The fomula i Eq. (3.5) does ot coespod to a bouday coditio but it expesses the symmety of pessue distibutio to the hoizotal axis of Figue ad it also coicides with the bouday coditio at θ = π. The most kow assumptios fo pessue distibutio though the cicumfeetial ϕ = ad ϕ( π ) =, b) the diectio (alog θ ) that ae give i the liteatue ae: a) the Sommefeld coditio i which Reyolds coditio i which ( ) ϕθ =, ad c) the Gumbel coditio i which ϕ ( ) =, ϕ( π ) = ad ϕ = ad ϕ θ = fo π < θ < π ( π film beaig). Whe e = the the Eq. (3.5) yields the Reyolds coditio withθ = π. Whe e it is cosequetly θ ad fo this easo the fomula of Eq. (3.5) is used, ad istead of pedefiig the agleθ ad use it as a bouday of zeo pessue it is let to be defied by the global pessue distibutio though θ givig to the cuet aalysis a futhe geeality. The values of θ as a fuctio of e ae peseted i the sectio 5 because θ is lightly effected also fom the values of the, ϕ θ. pessue yielded by the homogeeous solutio g ( x θ ) that is added to L b ψ = L b ψ = (3.) (3.3) ϕ ( ) = (3.4) 5

6 π ϕθ dθ= e =, π lim ϕθ dθ=, e e (3.5) c = (3.6) c = (3.7) 3 c e c e 3 c c e c e 6R µ π ( c e) c + e e + 3ce Ω Log Log + 3ce accos Log Log Ω e c + e c + e c + e c + e c 3 = c + e c e c c e c e e( c + e ) + + π + Log Log + accos Log + Log c + e c e e c + e + e e c + e (3.8) c 6eR µ = ec ( + e) 4 (3.9) The cuet paticula solutio coespods to the pessue distibutio developed i a ifiitely log beaig ad fo a umeical applicatio with the paamete values of Table the pessue distibutio alog the cicumfeetial diectio is show i Figue fo thee cases of the ecceticity velocity e. The cuet paticula solutio gives a futhe cotibutio i Reyolds equatio teatmet because the additive sepaatio of vaiables ca yield the solutios of the log o of the shot beaig without the eed of easig the oe of two left had tem i Eq. (.). Fig.. The pessue distibutio alog the agula coodiateθ give fom the paticula solutio ϕ ( θ ). R =.5m e.7c = Ω= ad/s c = 5µm L =.m µ =.5Pa s b Table. Defiitio of the values of the geometical ad the physical paametes of the cuet joual beaig. Fig. 3. The pessue distibutio alog the agula coodiateθ give fom the paticula solutio ϕ ( θ ). As show i Fig. the pessue distibutio of the paticula solutio becomes zeo at the agle θ that is a fuctio of the ecceticity velocity. The pessue distibutio give by the paticula solutio is symmetic to the hoizotal axis at y = fo the case of e =. The pessue distibutio is itesively affected fom e ad obtais much highe maximum values eve fo small 6

7 values of e such as e=.ωr while the domai with egative pessue has a icemet i its miimum pessue. 4 EVALUATION OF THE HOMOGENEOUS SOLUTION g x, θ of Eq. (.5) is pefomed i this sectio. As it was claimed this should be the geeic set of solutios fo The evaluatio of the homogeeous Reyolds equatio, amely fo Eq. (4.). ( c + ecos( θ) ) θ 3( c + ecos( θ) ) esi ( θ) θ ( c + ecos( θ) ) 3 3 gx (, ) gx (, ) gx (, θ) + = (4.) 6µ x 6µ R θ 6µ R θ We assume that the idepedet vaiables of the fuctio g ( x, θ ) ca be sepaated i the multiplicative fom of Eq. (4.). ( θ) g( x, θ) = f m x (4.) By usig this assumptio i Eq. (4.) oe will aive at the followig outcome of Eq. (4.3). ( θ ) 3 si ( + cos( θ )) ( θ ) m x f θ e θ f θ = m x R f R c e f (4.3) By ispectig the latte equatio oe ealizes that the left had is a fuctio of x while the ight a fuctio of θ oly. So, i ode this equality to be feasible both sides should be equal to the same costat, set λ. We thus obtai Eqs. (4.4) ad (4.5). ( x) m = λ (4.4) m x ( θ) R R c e 3 si ( + cos( θ )) ( θ) f θ e θ f θ f f = λ (4.5) The pimes deote odiay deivative with espect to the fuctios agumets. Fo teatig Eq. (4.4) we distiguish the followig cases: a) If λ = Eq. (4.4) becomes as i Eq. (4.6) which ca be solved to give Eq. (4.7). ( x) m = (4.6) m( x) = c5x+ c 6 (4.7) b) If λ > the λ = k ad the solutio is as i Eq. (4.8). cos si m x = c k x + c k x (4.8) 5 6 c) If λ < the λ = k ad oe will fially aive to the followig solutio as i Eq. (4.9). kx 5 6 m x c e c e kx = + (4.9) 7

8 Eq. (4.5) ca vey easily be solved if λ = so as to make Eq. (4.5) a fist ode diffeetial equatio (usig the coespodig tasfomatio as i Eq. (3.6)) ad the to be teated as Eq. (3.7). Such a assumptio would yield liea distibutio of the pessue though the axial coodiate, sice it would be m( x) = c5x+ c 6, thus the case of λ = is ot accepted. If λ the the diffeetial equatio of Eq. (4.5) has ot diect closed fom of solutio. Vogelpohl i [] icopoates i detail the teatmet of the homogeous Reyolds poblem with the assumptios made by Michell [5] ad Duffig [5] fo the lieaizatio of the fluid film thickess fuctio so as Eq. (4.5) to be solvable. The subsectios 4. ad 4. show how the cuet poblem was teated i the past ad how it is teated i the cuet pape. Two ways ae peseted i this pape i ode to cotiue with the solutio of Eq. (4.9) ewitte as i Eq. (4.). The fist way is to appoximate the tigoometic coefficiets, itoduced by the tem 3 h / hi Eq. (4.), with a liea fuctio of fist ode as Michell [5] ad Duffig [5] did so as to obtai a solutio usig Bessel s fuctios. This appoximatio is peseted i subsectio 4. with the use of a liea fuctio fo h. The secod way is to use the powe seies method ad to give the exact solutio of Eq. (4.) as a sum of ifiite seies. This way is peseted i detail i the subsectio 4.. 3h f ( θ) + f ( θ) λr f ( θ) = (4.) h 4. Liea appoximatio of the fluid film thickess ad use of the Bessel s fuctios fo the defiitio of f ( θ ) Michell i [5] uses the fomula fuctios. h = cost. θ i ode Eq. (4.) to become as i Eq. (4.) ad its solutio to be give usig Bessel 3 f ( θ) + f ( θ) λr f ( θ) = (4.) θ Duffig i [5] used also liea vayig fluid film thickess; he used the tasfomatio f ( θ) w( θ) / usig these tasfomatios the Eq. (4.) was witte as i Eq. (4.). Sice H eigefuctios of Eq. (4.) ca be of the fom w si ( λ θ ) i =. i = H, with H = h 3 / µ ; ad = cost. θ = a θ, this yield H = ad the the H w θ + λw θ = w( θ) (4.) H I the cuet aalysis the tigoometic fuctio h is appoximated with the liea fuctio h that is defied i Eq. (4.3) ad show i Fig. 4 fo the set of the values of the Table. e h = ( c + e) θ (4.3) π With the use of h the solutio of Eq. 4. is feasible usig Bessel fuctios ad the geeal solutio is peseted i Eq. (4.4) fo both cases of positive ad egative value of λ. 8

9 Fig. 4. The tigoometic fuctio of fluid film thickess h ad the liea appoximatio of it h, as a fuctio of the agula coodiateθ. f ( θ ) π i c + e k R i c + e kπr BesselJ, ikrθ BesselY, ikrθ e e c 7 + c 8, λ = k cπ + eπ eθ cπ + eπ eθ = ( c + e) kπr ( c + e) kπr BesselJ, krθ BesselY, + krθ e e c 7 + c 8, λ = k cπ + eπ eθ cπ + eπ eθ (4.4) Sice o imagiay solutio ca be accepted, oly the case fo λ = k is accepted. The solutio of Eq. (4.) give by Eq. (4.5) will fom a bouday value poblem with esults plotted togethe with those fom the exact solutio give by the Powe Seies method, peseted i what follows. ( + ) π ( + ) c e k R c e kπr BesselJ, krθ BesselY, + krθ e e f ( θ ) = c + c c π + eπ eθ c π + eπ eθ 7 8 (4.5) 4. The use of the method of powe seies fo the defiitio of f ( θ ) The method of the Powe Seies [7] is used i this subsectio i ode to defie a solutio fo Eq. (4.). The fist step is to covet Eq. (4.) fom a liea ODE with tigoometic coefficiets i a liea ODE with polyomial coefficiets, thus the tasfomatio ξ f θ f ξ f ξ ae defied i Eqs. (4.6) ad (4.7). cos( θ ) = is used ad we ae lookig fo a fuctio =. The deivatives of df df = si ( θ ) = f ( ξ) ξ dθ dξ (4.6) 9

10 = si ( θ ) cos( θ) = f ( ξ)( ξ ) f ( ξ) ξ (4.7) dξ dξ d f d f df dθ By substitutig the last two expessios ito Eq. (4.) the followig diffeetial equatio i Eq. (4.8) is obtaied. ( αξ 3 βξ αξ β) f ( ξ) ( 4αξ βξ 3α) f ( ξ) ( 3αγξ 3βγ) ( ξ) f = (4.8) The costats α, βγ, ae defied as i Eq. (4.9). α β γ λ 3 3 = e, = c, = R (4.9) Study of f ( ξ ) i the odiay poits Fo all the odiay poits of the diffeetial equatio i Eq. (4.8) we assume that the solutio ca be witte i the fom of Eq. (4.). f ( ξ ) = δξ (4.) = The fist ad the secod deivative of the fuctio f ( ξ ) ae defied i Eq. (4.) ad (4.) fo the coespodigly. f ( ξ ) = δ ξ (4.) = ( ξ ) δ ( ) ξ f = (4.) = By substitutig the last thee expessios to the diffeetial equatio oe will fid out that the costats that appea i the powe seies expasio ae detemied by the followig fomulas as i Eq. (4.3). ( ) ( ) ( ) ( ) + + α α γ α δ β δ α α α δ β β β δ ξ α αγ δ = δ = = ( ) ( ) ( ) ( ) α + α + 3γ α δ + β δ α + 4α + 3α δ β + 3 β + β δ = + + (4.3) Study of f ( ξ ) i the sigula poits The diffeetial equatio of Eq. (4.8) is ow witte i the fom of Eq. (4.4). ( ξ)( ξ )( ξ )( αξ β) ( ξ) ( ( αξ β) 3α( ξ )( ξ ) ) ( ξ) ( 3 ( αξ β) ) f f x f d + = (4.4) The tem multiplied with the highe ode deivative has the followig oots as i Eq. (4.5).

11 β ξ =, ξ =, ξ = (4.5) α The fist coespods to the valueθ =, while the secod to the valueθ = π. The thid oe is ot accepted because it yields cos( θ ) <. So, we have two sigula poits, i the valuesξ = ±. Fo them, a sepaate aalysis has to be pefomed. Fo the sigula poitξ = we have that it is a omal sigula poit sice aalytic fuctios A ( ξ ), A ( ξ ) exist such that Eq. (4.6) to be satisfied. ( ξ ) ( x( aξ + β) + 3a( ξ )( ξ + ) ) = ( ξ )( ξ + )( aξ + β) A ( ξ) ( ξ ) ( 3γ ( aξ + β) ) = ( ξ )( ξ + )( aξ + β) A ( ξ) (4.6) The seies expasios of the fuctios A, A ξ ξ ae as i Eq. (4.7). A A ( ξ) = + 3 ( ξ + ) ( ξ) + + ξ + ξ + a a + β = = = ξ + = 3γ = + + (4.7) So, the idicative equatio of this omal sigula poit is as i Eq. (4.8) with the two oots as i Eq. (4.9). λ p λ = λ (4.8) λ =, λ = (4.9) Fo this sigula poit the solutios will be of the fom of Eq. (4.3). f f ( ξ) = δ ( ξ ) = ( ξ) = ( ξ ) ε ( ξ ) = (4.3) Afte a simila pocedue as the oe descibed fo the odiay poits, oe will be led to the followig fomulas of Eq. (4.3), fo the calculatio of the costats of the above powe seies i Eq. (4.3). δ = ( ( )) + + a γ + δ + + β a 3γ + 3 δ + + a+ b+ a δ = + (4.3) Fo the coefficiets of the othe idepedet solutio oe will be led to the followig fomulas of Eq. (4.3). ε = 3 3 a + 3γ 6+ ε + 3a β aγ βγ ( 3a+ 4β) + ( a+ 4β) ε ( a+ β) + ε (4.3)

12 Fo the omal sigula poitξ = we have the followig idepedet solutios of Eqs. (4.33) ad (4.34). f ( ξ) = δ ( ) ξ + (4.33) = 3 = + ( + ) f ξ ξ ε ξ = (4.34) Fo the defiitio of the costats i Eqs. (4.33) ad (4.34), the fomulas of Eqs. (4.35) ad (4.36) ae used coespodigly. δ = ( ) ( ) + + a 3 3γ 3+ δ + 3aγ + β + a 3βγ + 3 δ a β + a+ β δ = + (4.35) ε = ε = 9 3aγ + + a + ε + a + 3γ 8 + β 3γ + 3+ ε 4 ( 5 + ) ( 4 + a+ a+ 4 β( + ) ) ε+ = 4 (4.36) 5 APPLICATION OF THE BOUNDARY CONDITIONS AND PRESSURE EVALUATION Two bouday value poblems ae teated i this sectio. The fist has to do with the distibutio of the pessue i the axial diectio of the beaig ad the secod with the distibutio i the cicumfeetial diectio f θ. The bouday value poblem fo m( x) m( x ) is defied is defied fom the ODE i Eq. (5.) ad the bouday coditios of Eqs. (5.) ad (5.3). m x k m x = (5.) L b m = L b m = (5.) (5.3) The explaatio fo the bouday coditios i Eqs. (5.) ad (5.3) is that the esultig pessue at both eds of the beaig P L /, θ u ± L /, θ = g ± L /, θ, that ( ± b ) has to be equal to zeo (the atmospheic pessue is ot icopoated) ad thus ( b ) ( b ) meas ϕ ( θ) = m( ± L /) f ( θ). b The geeal solutio of Eq. (5.) is give i Eq. (5.4). The geeal solutio i Eq. (5.) is substituted i the bouday coditios of Eqs. (5.) ad (5.3) ad oe ca defie the costats ad ad obtai the solutio as i Eq. (5.4). The eigefuctios m x ae c5 6 witte as i Eq. (5.5) with k to be the eigevalues of the poblem of Eqs. (5. 5.3). c Lk b Lk b kx e Lk b Lk b e m( x) = e + e + e + e kx (5.4)

13 Lk b Lk b e kx e = + Lk b Lk b m x e e + e + e π k =, =,,... L b kx, (5.5) The bouday value poblem fo f ( θ ) is teated i the two followig subsectios 5. ad 5. fo the cases that Bessel fuctios (Sectio 4.) ad Powe Seies (Sectio 4.) coespodigly. 5. Bouday value poblem fo f ( θ ) usig Bessel Fuctios The bouday value poblem is cosisted fom the ODE i Eq. (5.6) ad of the bouday coditios of Eqs. (5.7) ad (5.8). f θ is defied usig 3h f ( θ) + f ( θ) + k R f ( θ) = (5.6) h f ( ) = (5.7) f ( π ) = (5.8) The explaatio about the bouday coditios i Eqs. (5.7) ad (5.8) is that i the begiig of the oil film ( θ = ) the pessue P( x,) has to be zeo ad thus ϕ ( ) = m( x) f ( ). Sice ϕ ( ) is chose to be zeo (see sectio of the paticula solutio) the f ( ) =. Also, sice f ( θ ) is the homogeous solutio thee is o ifluece fom e (this measθ = π ) ad the pessue yielded fom the homogeeous poblem becomes also zeo at θ = θ = π. The geeal solutio of Eq. (5.6) is give i Eq. (5.9). The substitutio of the geeal solutio i the bouday coditios gives the system of equatios i Eq. (5.). ( + ) π ( + ) c e k R c e kπr BesselJ, krθ BesselY, + krθ e e f ( θ ) = c + c c π + eπ eθ c π + eπ eθ 7 8 (5.9) π c + e k R c + e kπr BesselJ, BesselY, e e c 7 + c 8 = cπ + eπ cπ + eπ ( c + e) kπr ( c + e) kπr BesselJ, krπ BesselY, + krπ e e c 7 + c 8 = cπ eπ cπ eπ (5.) The costats ad c ca be detemied as the solutio of the system of Eq. (5.) as i Eq. (5.). c7 8 3

14 8 c + e kπ R BesselY, e c7 = ( c + e) kπ R (5.) BesselJ, e c = The chaacteistic equatio of the system i Eq. (5.) is give i Eq. (5.) ad is plotted i Fig. 5 as a fuctio of k fo the set of values of Table. The eigevaluesκ ae detemied as the oots of it ad fo the cuet set of values of Table they ae vey well appoximated by the fomula κ =. The eigefuctios f ( θ ) ae give i Eq. (5.3) togethe with the eigevaluesκ fo the cuet set of values of Table. ( + ) π ( + ) c e k R c e kπr BesselJ, BesselY, e e c π + eπ c π + eπ π c + e k R c + e kπr BesselJ, krπ BesselY, + krπ e e c π eπ c π eπ = (5.) ( + ) κπ ( + ) c e R c e κπ R BesselJ, κrθ BesselY, + κrθ e e f ( θ ) = c + c c π + eπ eθ c π + eπ eθ κ =, =,, (5.3) Fig. 5. Values of the chaacteistic detemiat of the f θ as a fuctio of k, fo bouday value poblem fo the case of use of Bessel fuctios Fig. 6. The pessue distibutio alog the agula coodiateθ give fom the homogeous g θ, x fo vaiable values of axial solutio coodiate x ad fo the two diffeet solutios usig Bessel fuctios ad Powe Seies method. 4

15 Fig. 7. The pessue distibutio alog the axial coodiate x give fom the homogeous g θ, x fo vaiable values of agula solutio coodiateθ. Fig. 8. The pessue distibutio g ( θ, x) give fom the homogeous solutio usig the Powe Seies Method as a fuctio of agula ad axial coodiates. 5. Bouday value poblem fo f ( θ ) usig Powe Seies The cuet bouday value poblem is cosisted fom the ODE i Eq. (5.4) ad of the bouday coditios of Eqs. (5.5) ad (5.6). ( αξ 3 βξ αξ β) f ( ξ) ( 4αξ βξ 3α) f ( ξ) ( 3αγξ 3βγ) ( ξ) f = (5.4) f = (5.5) () f = (5.6) The explaatio of the cuet bouday coditios coespods to this of Eqs. (5.7) ad (5.8). Usig the solutio of Eq. (4.) ad π the eigevalues k =, =,,...the eigefuctios f ( cos( θ )) ae defied usig the iteative fomula Eq. (4.3) ad the two Lb bouday coditios fo evey =,,.... Fo the cuet evaluatio a total umbe of = was used. The aalysis aoud the f cos θ whe the highe eigevalues ae icopoated. sigula poits is used also i the evaluatio of the eigefuctios ( ) The omalized eigefuctios of both poblems (fo m( x) ad f ( θ ) ) ae give i Eqs. (5.7) ad (5.8) with the costats α oα ad β to be defied i Eqs. (5.9) ad (5.) coespodigly [7]. f ( θ ) = α f ( θ) o f ( θ ) = α f ( θ) (5.7) 5

16 m ( θ ) β m ( θ) = (5.8) π ( ) π ( ) α f θ dθ =, =,,... o α f θ dθ =, =,,... (5.9) π ( ) β m θ dθ =, =,,... (5.) 5.3 Evaluatio of the esultig pessue The solutio of g ( x, θ ) ca ow be witte as i Eq. (5.). The costats δ ae defied i Eq. (5.3) usig the bouday coditio fo zeo pessue at the eds of the beaig which yields ϕ ( θ) = m( ± L /) f ( θ) o sice m( ± L /) =, f ( θ ) = ϕθ see Eq. (5.). (, θ) ( δ fm) g x = g ( x θ) = o, = ( δ f m) = b (5.) Lb Lb Lb δf( θ) m δf( θ) m δ3f3( θ) m3... ϕ ( θ ) = fo f o f ad δ o δ. (5.) b δ = ϕ θ θ L π b f m d θ o π δ = ϕ ( θ ) f ( θ ) L m dθ (5.3) b The pessue distibutio defied i Eq. (5.) is plotted i Figs. 6-8 fo both cases of the eigefuctios f ad f ( θ ) θ (Bessel fuctios) (Powe Seies method). As it ca be see i Figs. 6 ad 7 the Bessel fuctios yield diffeeces i the pessue distibutio alog both the axial ad cicumfeetial coodiate. The two distibutios have simila maximum ad miimum pessue value but i diffeet locatios ofθ. The diffeeces i the pessue distibutio ae moe itese i the domai aoud the axial cete of the beaig ( x = ) but both distibutios tet to be equal ea the axial eds of the beaig ( locatios both distibutios ae foced to tet to ϕ ( θ ). x =± L b ) ad this is because i these The esultig fluid film pessue distibutio is the defied explicitly as i Eq. (5.4) ad is plotted i Figs. 9- though agula ad axial coodiate, togethe with umeical esults obtaied duig the cuet wok fom the solutio of Eq. (.) usig the Fiite Diffeeces Method. ( θ) = ϕ( θ) + ψ + δ ( θ) P x, x f m x (appoximate aalytical) = ( θ) = ϕ( θ) + ψ + δ ( θ) P x, x f m x (exact aalytical) = (5.4) I Figs. 9- it ca be see that the esultat pessue distibutio pesets diffeeces compaig the umeical solutio with the appoximate aalytical solutio (usig Bessel fuctios) ad the exact aalytical solutio (usig the Powe Seies method) especially i the values of maximum ad miimum pessue. The divegece i the value of the maximum pessue becomes moe itesive 6

17 whe the paamete of ecceticity velocity obtais highe values (see Fig. ) while the miimum pessue value is ot affected as much as this of the maximum. The esultig pessue distibutio though the axial coodiate is plotted i Fig. fo the thee cases of evaluatio. The diffeeces ae becomig moe itesive fo values ofθ whee the pessue tets to maximize (eg. θ =.8π ) while i the othe domais ofθ the pessue divege less. What those diffeeces mea fo the esultig impedace foces of the fluid film i the joual is a cocept that is left to be discussed i futue wok. A thee dimesioal plot fo the esultig pessue distibutio is give i Fig. 3. Havig also a look i Figs. 3 ad 8 oe ca ealize how thei distibutios cotibute i the esultig pessue of Fig. 3. Usig Eq. (5.4) the values fo the agles of the maximum ad the miimum pessue ca be calculated by fidig the oots of Eq. (5.5), while the oots of Eq. (5.6) ae the agles of zeo pessue. The oots of Eqs. (5.5) ad (5.6) ae evaluated umeically ad they ae plotted i Fig. 4 as fuctio of the paamete e fo the thee cases of study. The thee cases of evaluatio have diffeeces i whee the pessue obtais the maximum ad the zeo pessue especially as e takes highe values. d ϕθ ψ ( ) δ f( θ) m( ) dθ + + = (5.5) = ( ) f m( ) ϕθ + ψ + δ θ = (5.6) = 6 CONCLUDING REMARKS, CONTRIBUTION AND FUTURE CONCEPTS The cuet wok gives the path of obtaiig a exact aalytical solutio of the Reyolds equatio fo the lubicatio of joual beaigs with fiite legth. The cotibutio of the cuet wok ca be descibed with the followig commets which highlight the diffeeces to the solutios give i the past. a) A set of paticula solutios of the Reyolds equatio is give with a sub case of them to coespod to the pessue distibutio of the ifiitively log beaig without the demad of excludig ay patial deivative i Reyolds equatio. This meas that a paticula solutio that descibes eithe the ifiitely log o the ifiitely shot beaig ca be defied egadig the bouday coditios that ae applied i the fuctio of the paticula solutio fo the fiite beaig. b) The homogeeous Reyolds equatio is teated without the use of ay appoximatig fuctio fo the fluid film thickess ad the use of Powe Seies yield a closed fom solutio fo the homogeeous Reyolds Equatio fo the fiite beaig. c) The aalytical expessio of the esultig pessue as a fuctio of ecceticity ate of chage e gives the ability to defie a fomula that togethe with the iitial coditio fo the pessue distibutio i the cicumfeetial begiig of the film lets the agle of zeo pessue to be self defied. A additioal beefit is that thee exist closed fom fomulas fo the defiitio of the agles of maximum ad miimum pessue of the lubicat i the fiite beaig. A study of the pessue distibutio ude the appoximatio of the fluid film thickess fuctio with a liea fuctio was motivated fom the liteatue i ode to show the diffeeces betwee the appoximate aalytical solutio usig Bessel fuctios ad the exact aalytical solutio usig the method of powe seies. The cuet wok does ot itet to estimate the impotace of the diffeeces i the pessue distibutio betwee the methods of evaluatio (appoximatig aalytical o umeical) because thee ae cases of opeatioal coditios that these diffeeces ae miimized ad cases that ca yield highe divegece betwee the theoies. Two liea odiay diffeetial equatios of d ode ad with tigoometic coefficiets wee solved i this wok. They wee extacted ad fomed ude the pope assumptios of splittig the solutios ad the idepedet vaiables. The tigoometic coefficiets that wee itoduced by the fuctio fo the fluid film thickess itoduced the geat difficulty i solvig the Reyolds equatio. Sice the oe of them, which coespods to the paticula solutio, has diect fom of solutio whateve the thickess fuctio is, ad the othe, which coespods to the homogeous solutio, ca be cofoted usig the powe seies method, a ew cocept comes up ad has to do with the itoductio of a diffeet fuctio fo the fluid film thickess that coespods to a wo joual beaig. 7

18 This pat of the wok fo the aalytical solutio of the Reyolds equatio was plaed to give the fomulas of the pessue distibutio of the lubicat i a fiite joual beaig. These fomulas happe to have fuctios that ca be itegated i space domai with closed fom expessios so as to give the esultig impedace foces of the lubicat to the beaig. This fact gives the ability to defie the closed fom expessios fo the stiffess ad dampig coefficiets of the lubicat film ad this is a cocept to be made i the futue wok togethe with the defiitio of the fomulas fo the opeatioal paametes such as the ecceticity ad the attitude agle as a fuctio of Sommefeld umbe. Fig. 9. The pessue distibutio P( θ, x) alog the agula coodiate θ i the axial cete of the beaig ( x = ) fo e = Fig.. The pessue distibutio P( θ, x) alog the agula coodiateθ i the axial cete of the beaig ( x = ) fo e =.ΩR Figue. The pessue distibutio P( θ, x) alog the agula coodiateθ i the axial cete of the beaig ( x = ) fo e =.ΩR Fig.. The pessue distibutio P( θ, x) alog the axial coodiate x fo vaiable values of agula coodiateθ. 8

19 Fig. 3. The pessue distibutio P( θ, x) usig the powe seies method, as a fuctio of the agula ad the axial coodiate (i the case of e = ) Fig. 4. Values of the agles of the maximum pessueθ max, of the miimum pessueθ mi, ad of the zeo pessue θ as a fuctio of ecceticity velocity yielded fom the aalytical solutio usig Powe Seies method ad the umeical solutio with fiite diffeeces. ACKNOWLEDGMENTS D.-Ig. Athaasios Chasalevis would like to thak the Alexade vo Humboldt foudatio fo the fiacial suppot fo postdoctoal eseach. The cotibutio of Pof. D.-Ig Richad Maket, of Pof. D.-Ig Chis Papadopoulos ad of D. tech. Fadi Dohal with thei emaks i the pocedue of the solutio is gatefully appeciated. REFERENCES [] O. Reyolds, O the theoy of lubicatio ad its applicatio to M. Beauchamp Towe s expeimets, Icludig a expeimetal detemiatio of the viscosity of olive oil. Phil. Tas. Roy. Soc. (Lodo), se. A, vol. 77, (886), pp [] O. Pikus ad B. Stelicht, Theoy of hydodyamic lubicatio, McGaw-Hill Book Co. NY (96) [3] A. Sommefeld, The hydodyamic theoy of lubicatio fictio. Zs. Math. Ud Phys., vol. 5, os. ad, (94), pp [4] W. Haiso, The hydodyamical theoy of lubicatio with special efeece to ai as a lubicat. Poc. Cambidge Phil. Soc., vol., o. 3, (93), pp [5] A. Michell, Pogess i Fluid film lubicatio. Tas. ASME MSP-5-, vol. 5, (99), pp [6] F. Gadullo, Some pactical deductios fom theoy of lubicatio of shot cylidical beaigs. Tas. ASME, MSP-5-, vol. 5, (93), pp [7] G. Dubois ad F. Ocvik, Aalytical deivatio ad expeimetal evaluatio of shot-beaig appoximatio fo full joual beaigs. NACA TN 57, (95). [8] R. Kik ad E. Gute, Stability ad tasiet motio of a plai joual mouted i flexible damped suppots, J. Eg. fo Idusty, 976, p [9] A. Kigsbuy, Poblems i theoy of fluid film lubicatio with expeimetal method of solutio. Tas. ASME, APM-53-5, vol. 53, (93), pp [] D. Chistopheso, A ew mathematical method fo the solutio of film lubicatio poblems. Poc. Istitutio Mech. Eg. (Lodo), vol. 46, o. 3, (94), pp [] G. Vogelpohl, Zu Itegatio de Reyoldssche Gleichug fü das Zapfelage edliche Beite. Igeieu Achiv (I Gema) (943) [] G. Vogelpohl, Übe die Tagfähigkeit vo Gleitlage ud ihe Beechug. Westdeutsche Velag / Köl ud Oplade (I Gema) (956) [3] G. Vogelpohl, Beitäge zu Ketis de Gleitlageeibug. VDI-Fosch.-Heft 386 (I Gema) (937) S

20 [4] A. Cameo ad W. L. Wood, The full joual beaig. Poc. Istitutio Mech. Eg. (Lodo), vol. 6, W.E.P. Nos , (949), pp [5] G. Duffig, Die Schmiemitteleibug bei Gleitfläche vo edliche Beite. Habuchde phys. U. tech. Mechaik vo Auebach-Hot, Bd. 5, S Leipzig: Bath (93). [6] R. Couat ad D. Hilbet, Methods of mathematical physics. Vol. II. Wiley-VCH (989) [7] W. Boyce, R. DiPima, Elemetay diffeetial equatios ad bouday value poblems. J. Wiley, (997) [8] A. Polyai ad V. Zaitsev, Hadbook of exact solutios fo odiay diffeetial equatios. Chapma & Hall/CRC, (3)