Bundang-gu, Seongnami-si, , South Korea. YongIn-si, Gyeonggi-do, , South Korea

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1 Appled Mechancs and Materals Onlne: ISSN: , Vols , pp do: / Trans Tech Publcatons, Swtzerland Numercal Model of Journal Bearng Lubrcaton Consderng a Bendng Stffness Effect Juhwan Cho 1,a, Seong Su Km 1,b, Sungsoo Rhm 2,c and Jn Hwan Cho 2,d 1 FunctonBay, Inc., 5F, Pangyo Seven Venture Valley 1 danj 2 dong, 625, Sampyeong-dong, Bundang-gu, Seongnam-s, , South Korea 2 Department of Mechancal Engneerng, KyungHee Unversty, 1 Seochun-dong, Khung-gu, YongIn-s, Gyeongg-do, , South Korea a juhwan@functonbay.co.kr, b seongsu@functonbay.co.kr, c ssrhm@khu.ac.kr, d jhcho@khu.ac.kr Keywords: Journal Bearng, Lubrcaton, Elastohydrodynamc, MFBD, Bendng Stffness Effect. Abstract. An analyss for operatng characterstcs of journal bearng lubrcaton system s performed based on the numercal model. Dynamc bearng lubrcaton characterstcs such as ol flm pressure and thckness dstrbuton can be analyzed through a numercal model wth an ntegraton of elastohydrodynamcs and mult-flexble-body dynamcs (MFBD). In partcular, the ol flm thckness varaton by elastc deformaton s consdered n the elastohydrodynamc analyss by applyng the bendng stffness effect of journal. And the ol flm thckness varaton by the bendng stffness effect s appled to the flud governng equatons to calculate the ol flm pressure n the elastohydrodynamc lubrcaton regon. A seres of process proposed n ths study s avalable for the analyss of realstc elastohydrodynamc lubrcaton phenomenon. Also, a numercal example for the journal bearng lubrcaton system s demonstrated and compared wth the expermental results. The numercal results consderng the bendng stffness effect show a good agreement wth the expermental results. Introducton The journal bearngs, whch s the one of the wdely used machne elements, transmt the power whle reducng the frcton and resstng the external loads. In partcular, n the nternal combuston engne whch s frequently used for power generaton, the varous journal bearngs are used between the pston, pston pn, connectng rod, crankshaft, and engne block. These journal bearngs, whch are under the alternatng loads caused by the gas forces of the nternal engnes, guarantee the smooth operaton of the engne and are tghtly related to the durablty of the engne system. In order to acheve the hgh-performance output and to reduce the engne weght, the mportance of the bearng lubrcaton analyss has been ncreased [1-3]. The study of bearng lubrcaton s based on the Reynolds equaton [4] whch descrbes the thckness and pressure of flud flm generated by the relatve moton of objects. Accordngly, to estmate the lubrcaton flm characterstcs such as the ol flm thckness and pressure, a multflexble-body dynamcs analyss (MFBD) [5-7] s needed to obtan the nformaton for the relatve moton between journal and bearng and the elastohydrodynamc lubrcaton analyss s also needed to get lubrcaton characterstcs. Generally, elastohydrodynamc lubrcaton can be classfed by two types based on the relatonshp between surface roughness and ol flm thckness. One type s the full-flm lubrcaton. It has been wdely used when the lubrcant flm s suffcently thck n whch there s no sgnfcant asperty contact. In ths case, the pressure s only governed by Reynolds equaton whch s frst establshed by Reynolds [4]. The other type s the mxed lubrcaton. When the lubrcant flm s not enough to thck, the asperty contacts between two bodes can occur [8, 9]. Therefore, n mxed lubrcaton regon, the total pressure should be treated as the sum of the pressure nduced by the flud flow and the asperty contact. Ths study uses the Reynolds equaton to obtan the hydrodynamc pressure and Greenwood and Trpp s asperty contact model [9] to obtan asperty contact pressure. All rghts reserved. No part of contents of ths paper may be reproduced or transmtted n any form or by any means wthout the wrtten permsson of Trans Tech Publcatons, (ID: , Pennsylvana State Unversty, Unversty Park, USA-05/03/16,07:15:12)

2 Appled Mechancs and Materals Vols To obtan more reasonable results for lubrcaton characterstcs, t s also mportant to consder the varaton for ol flm thckness and ol flm pressure resultng from the elastc deformaton of flexble bodes. To consder ths effect on the elastohydrodynamc analyss, ths study proposes a numercal model whch can express the bendng effect of journal n the journal bearng system. The proposed bendng effect s based on the beam theory. Elastohydrodynamc (EHD) Governng Equaton of Hydrodynamcs. Fg. 1 shows a schematc dagram for relatve moton and dmensons between a bearng and a journal. In the fgure, R s the journal radus and C r s the clearance n the journal bearng lubrcaton problems. The governng equaton for the flud flow becomes the Couette-Poseulle flow equaton [10-12]. Then, f the mass or flow rate conservaton law s appled, the Reynolds equaton for the hydrodynamc problems can be expressed as Eq. (1). 3 p p Γ + Γ = 12V + 6U H + 6 W H, Γ = H x x z z x z µ (1) Here, U, V, and W are x, y, and z components of the relatve velocty of the journal surface (at y = H ), respectvely, wth respect to the bearng. H s the ol flm thckness consderng the bendng effect and µ s the dynamc vscosty. The Eq. (1) s solved teratvely wth successve over-relaxaton method [13]. The ol flm thckness wthout bendng effect s defned as Eq. (2). ( ) H0 θ = Cr excosθ eysnθ (2) Addtonally, to support the general-purpose EHD soluton, groove and ol hole effects are also mplemented as the pressure boundary condtons. Journal e ω R θ H x,u y,v θ =0 Cr Bearng Fgure 1. The schematc dagram of a journal bearng. Asperty Contact. When the ol flm thckness s not enough to thck compared to the surface roughness, the contact pressure resultng from the aspertes between bodes should be consdered. In ths study, the asperty contact model by [9] s used to model the mxed lubrcaton regon. In Greenwood and Trpp s model, the asperty contact pressure p a can be calculated as follows:

3 856 Innovaton for Appled Scence and Technology a ( ) = KE' F ( H / σ ) p H F H 5/2 ( / ) 5/2 s H H , f < 4 σs = σs σs 0, otherwse (3) where K s the elastc factor and σ s s the root mean square of the asperty summt heghts and E ' s the composte elastc modulus [14]. Bendng Stffness Effect Bendng Effect of Journal. When a hgh external load s appled to the journal bearng, the journal can be bent and the varaton of gap δ between journal and bearng occurs n axal drecton as shown n Fg. 2. Because ths gap s related to the ol flm thckness, we need to consder ths gap for the ol flm thckness evaluaton to analyze more accurate lubrcaton characterstcs. The ol flm thckness consderng the bendng of journal can be defned as follows: ( ) H ( ) H θ = 0 θ δxcosθ δycosθ (4) where H 0 s defned n Eq. (2) and H s used to solve Reynolds equaton of Eq. (1). In ths study, the bendng stffness effect model based on beam theory s proposed to obtan the gap δ. Bearng δ Journal Y External Force Fgure 2. Bendng phenomenon of journal. Bendng Stffness Effect Model. To express the bendng phenomenon of journal, two dmensonal beam model s ntroduced as shown n Fg. 3. In ths numercal model, translatonal and rotatonal sprngs are used to adjust the degree of restrcton for both ends of beam. If the sprng coeffcents for translatonal and rotatonal sprng are nfnte, both ends of beam can be regarded as fxed. Addtonally, pressure appled to journal s equvalent to lne dstrbuted load for beam. Z 1 N K Trans L K Trans K Rot N-1 w L Fgure 3. Two dmensonal beam model equvalent to journal. K Rot Z Y

4 Appled Mechancs and Materals Vols Based on above mentoned features, we can make a general beam model whch has N nodes and N 1 elements. In accordance wth beam theory, stffness matrx for each element except for elements at both ends s defned as follows: L L L L L L L L K = EI, for 2 N 2 (5) L L L L L L L L where K and L denotes the stffness matrx and the length of -th element except for elements at both ends, respectvely. E s the Young s modulus and I s the area moment of nerta of journal body. For the both end elements, stffness matrx s defned as follows: KTrans 0 KTrans 0 0 KRot 0 K Rot K End = (6) KTrans 0 KTrans 0 0 KRot 0 KRot where K Trans and K Rot are translatonal and rotatonal sprng coeffcent, respectvely. Lne dstrbuted load w can be calculated as follows: F w = L (7) where F s the external force appled to journal resultng from the ol flm pressure and L s the length of part n whch ol flm pressure occurs. Accordngly, force vector for an -th element can be defned as follows: T 2 2 wl wl wl wl f = (8) Once we defne the stffness matrx and force vector for the all elements, we can create a global stffness matrx and force vector by superposng the element stffness matrx and force vector as follows: f = Kδ (9) where K s a global stffness matrx and f s a global force vector. δ denotes the normal and rotatonal dsplacement vector for all nodes. If we get the dsplacement vector δ for all nodes by solvng Eq. (9), the gap δ mentoned n Secton 4.1 can be evaluated.

5 858 Innovaton for Appled Scence and Technology Numercal Analyss Procedure In ths study, EHD and MFBD solvers are used together to analyze the flud lubrcaton and flexble multbody dynamc characterstcs of the journal bearng [14]. In EHD solver, pressure dstrbutons are calculated by consderng the bendng stffness effect of the journal. Frst, the relatve velocty of the journal surface wth respect to the bearng and ol flm thckness s calculated over the EHD grd ponts usng the nformaton for the postons and veloctes of journal and bearng. Addtonally, ol flm thckness that s calculated prevously s updated by consderng the varaton of ol flm thckness resultng from the bendng of the journal. Fnally, Reynolds equaton s solved wth the asperty contact force for the gven boundary condtons. And then, the calculated pressure feld and the resultng force and torque are transmtted to the MFBD solver. In the MFBD solver, the transmtted force and torque data are used as the external forces or torques actng on the journal and bearng. Then, from the MFBD analyss, postons and veloctes of journal and bearng are calculated. These data are transmtted to the EHD solver agan. Fg. 4 shows the procedure of the flud-structure nteracton solvng method between EHD and MFBD solvers. Numercal Example Fgure 4. Flud-structure nteractons between EHD and MFBD solver. To mplement the EHD module consderng the bendng stffness effect wth MFBD solver together, ths study used the RecurDyn TM [15] MFBD envronment. To valdate the numercal results of ths study, the expermental results of Okamoto et al. [16] are used. The detaled explanaton about the numercal model s descrbed well n [16]. Fg. 5 shows the numercal model and measured ponts of the ol flm pressure. The rotatonal speed of shaft s 3250 rpm. Table 1. The parameters of numercal model Parameters Values Mesh sze (crcum. depth) Journal Dameter 53 [mm] Bearng Wdth 17 [mm] Clearance [mm] Dynamc Vscosty 3.5e-2 [Pa s] Roughness 5.e-4 [mm] Composte Elastc Modulus [MPa] Elastc Factor 3.e-3 Area Moment of Inerta of Journal [mm 4 ] Translatonal Sprng Coeffcent 10e+7 [N/mm] Rotatonal Sprng Coeffcent 3.62e+5 N mm/rad Table 1 shows the smulaton parameters used n the numercal model. Numercal results are compared wth the expermental results of [16] at measured ponts to valdate the model. Fg. 6 shows a pressure dstrbuton whch has a steep slope around the edge and a flat slope around the

6 Appled Mechancs and Materals Vols center. Generally, f the flexblty of the journal and bearng s not consdered, the pressure dstrbuton shows a parabolc shape along the depth drecton. But, because of the flexblty such as bendng effect of journal or bearng, the pressure dstrbuton becomes a flat shape around the center. And a steep pressure dstrbuton occurs around the edge and ths can cause some wear phenomena whch can be modeled as an asperty contact model. On the other hand, n the case of expermental result, the pressure around the edge s hgher than that around the center. It may be that the bendng effect of both journal and bearng s consdered n the case of experment result but the bendng effect of journal s consdered only n the numercal result. But, the numercal result of current study shows a good agreement wth the tendency of the expermental results such as a flat phenomenon around the center and a steep slope around the edge. Fgure 5. Numercal model for journal bearng and measured ponts. Conclusons Fgure 6. Comparson between numercal and expermental results. In ths study, the elastohydrodynamc lubrcaton consderng the bendng effect of journal was coupled wth mult-flexble-body dynamcs (MFBD) to analyze dynamc bearng lubrcaton characterstcs such as the pressure dstrbuton and ol flm thckness. To solve coupled fludstructure nteracton system, ths study uses a MFBD solver and an EHD module teratvely. Especally, asperty contact model and bendng stffness effect model are ntroduced n EHD module respectvely. In the case of asperty contact model, asperty contact pressure s calculated from asperty contact model and then t s reflected on the ol flm pressure. In the case of bendng stffness effect model, the ol flm thckness s updated by usng the bendng stffness effect model. And then the updated ol flm thckness s used to solve Reynolds equaton. Addtonally, functons such as mesh grd control and ol hole and groove effects are also mplemented. Fnally, the numercal results are valdated and compared wth expermental data by usng the journal bearng example

7 860 Innovaton for Appled Scence and Technology References [1] C.M. Taylor: Engne Trbology, Elsever scence publshers B. V., pp (1993) [2] K.P. Oh and P.K. Goenka: The elastohydrodynamc soluton of journal bearngs under dynamc loadng, ASME, Journal of trbology, Vol. 107, No. 3, pp (1985) [3] G.A. Labouff and J.F. Booker: Dynamcally loaded journal bearngs: a fnte element treatment for rgd and elastc surfaces, ASME, Journal of trbology, Vol. 107, No. 4, pp (1985) [4] O. Reynolds: On the Theory of Lubrcaton and ts Applcaton to Mr. Beauchamp Tower s Experments, Includng an Expermental determnaton of the Vscosty of Olve Ol, Phl. Trans. Roy. Soc., Vol. 177, pp (1886) [5] D. Peskammer, H. Rener, M. Prandstotter and M. Stenbatz: Smulaton of motor components : ntegraton of EHD - MBS - FE - Fatgue, ADAMS User Conference. (2002) [6] H. Rener, M. Prandstotter and W. Wtteveen: Conrod Smulaton: Integraton on EHD - MBS - FE - Fatgue, ADAMS User Conference. (2001) [7] J. Cho: A Study on the Analyss of Rgd and Flexble Body Dynamcs wth Contact, PhD Dssertaton, Seoul Natonal Unversty, Seoul. (2009) [8] D. Zhu and H.S. Cheng: Effect of surface roughness on the pont contact EHL. Trans. ASME, J. Trbology, 110, (1998) [9] J.A. Greenwood and J.H. Trpp: The Contact of Two Nomnally Flat Rough Surfaces, Proc. Instn. Mech. Engrs., Vol. 185, Part 1, No. 48, pp (1971) [10] R.H. Sabersky, A.J. Acosta and E.G. Hauptmann: Flud Flow : A Frst Course n Flud Mechancs, Thrd Edton, Maxwell Macmllan Internatonal Edtons. (1989) [11] R. Gohar: Elastohydrodynamcs, Second Edton, Imperal College Press. (2001) [12] S. Jang and Y. Park: Study on the Effect of aerated lubrcant on the journal trace n the engne bearng clearance, Internatonal Journal of Automotve Technology, Vol. 6, No. 4, pp (2005) [13] S.V. Patankar: Numercal Heat Transfer and Flud Flow. Hemsphere. Washngton. (1980) [14] J. Cho, S.S. Km, S.S. Rhm and J.H. Cho: Numercal Modelng of Journal Bearng Consderng both Elastohydrodynamc Lubrcaton and Mult-Flexble-Body Dynamcs, Internatonal Journal of Automotve Technology, Vol. 13, No. 2, pp (2012) [15] RecurDyn TM Manual, FunctonBay, Inc. (2012) [16] Y. Okamoto, K. Ktahara, K. Ushjma, S. Aoyama, G.J. Jones and H. Xu: Effects of the Desgn Parameters on Wear and Fatgue of Engne Bearngs by EHL Analyss, Seoul 2000 FISITA World Automotve Congress, June (2000)

8 Innovaton for Appled Scence and Technology / Numercal Model of Journal Bearng Lubrcaton Consderng a Bendng Stffness Effect /