Available online at ScienceDirect. Procedia Engineering 150 (2016 )

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1 Available online at ScienceDirect Procedia Engineering 150 (016 ) International Conference on Indstrial Engineering, ICIE 016 Research of Dynamics of the Rotor with Three-film bearings A. Fischer, A. Doiin*, A. Rlevsiy Soth Ural State University, 76, Lenin Avene, Chelyabins, , The Rssian Federation Abstract Rotating floating rings in the rotor bearings are widely sed for decrease of rotor vibrations in small rotor machines. Sch bearings are named two-film, becase have two lbricant films. Dynamics of the rotor with floating ring bearings is investigated in details in [1-3]. The charging system also se non-rotating floating rings in trbochargers of internal combstion engines [4]. In some combinations of design and condition parameters of the rotor bearing system, the dimension of rotor and rings motion trajectory can essentially increase, thogh it is still limited [1-3]. To decrease jornal and ring oscillation amplitdes and loadings affecting the bearing, the design with three lbricant films can be sed. It is achieved throgh installation of the floating ring pacage into a rotor machine case. Floating ring bearing pacage consists of one non-rotating floating ring (NF ring) and two rotating floating rings (RF ring). 016 The Athors. Pblished by by Elsevier Elsevier Ltd. Ltd. This is an open access article nder the CC BY-NC-ND license ( Peer-review nder responsibility ofthe organizing committee of ICIE 016. Peer-review nder responsibility of the organizing committee of ICIE 016 Keywords: dynamics of rotor; floating rings; lbricant film; trbocharger 1. Theoretic analysis The scheme of rotor with floating rings pacage bearing is shown in Fig. 1. Each of two parts (either left part or right part on fig. 1) of rotor bearing contains three lbricant films. The first film is restricted by rotor jornal srface and RF rings inner srfaces, the second one is restricted by RF ring oter srfaces and NF ring inner srface, the third one is restricted by rotor-machine case srface and NF ring oter srface. NF ring is fixed from rotation by pin. * Corresponding athor. Tel.: ; fax: address: doiinaa@ss.r The Athors. Pblished by Elsevier Ltd. This is an open access article nder the CC BY-NC-ND license ( Peer-review nder responsibility of the organizing committee of ICIE 016 doi: /j.proeng

2 636 A. Fischer et al. / Procedia Engineering 150 ( 016 ) Nomenclatre i B width of bearing in i -th film i J a inertia moment of rotating floating ring i c clearances g gravity acceleration j shoc acceleration stiplated by shoc loading on the rotor case in a direction of coordinate Ox i m mass i P hydrodynamic pressre in i -th film i r oter radises of bearing elements t time Rx, R y oil film reactions (hydrodynamic forces) M moments de to lbricant film shear stresses acting to jornal and ring dynamic viscosity of oil i nmber of a lbricant film 1 index reference to a jornal and to a bearing The ndertaen bearings have circmferential grooves (oil spply sorces) on inner and oter srfaces of RF rings and NF ring. The oil spply is as follows: the groove of the third film filled with oil from external system pmp, and then the second and first films filled from third film throgh radial hole in grooves. In the designed software the possibility of researching of both other manners of oil spply and distribtion in films is provided. There is accepted that rotor is rigid and symmetric, loads acted to the left and to the right parts of bearings are eqal. It is allow considering only simple scheme of one film spport shown in Fig. dring prediction of oscillation of jornal and rings in every lbricant film of bearing. The ring fixed comptational coordinate system XO1Y rotates with an anglar speed 1 relative to inertial coordinate system xoy. Position the jornal center O is determined by eccentricity e OO1 and polar angle. The jornal rotates in the comptational coordinate system with a relative anglar speed 1. Projections of jornal center relative velocity to centers line and perpendiclar direction are assigned throgh de dt and ed dt. Fig. 1. Scheme of three-films rotor bearings with pacage of floating rings.

3 A. Fischer et al. / Procedia Engineering 150 ( 016 ) Fig.. Scheme of one-filmed bearing. Film thicness and its derivative h t with sal approximation degree can be written as [6-1]: h de d hcecos ; cos e sin. t dt dt (1) The hydrodynamic pressre determined by an integration of the Reynolds eqation: 1 r 1 z1 z t 3 3 h p h p h h () at bondary conditions: p(, z) p(, z); p(, z B ) p ; p(, z) 0, (3) a where p a atmospheric pressre; z coordinate of a point in a lbricant film conted in a direction of an axis Oz, which is laying on a ring srface. In oil spply sorces area, 1,... the constant oil spply pressre p s is set. Lbricant sorces in jornal are relocated to the ring. Anglar speed of non-rotated element of bearing system is set eqal to zero. Let's remar that: h( ) c( x x )cos ( y y )sin. (4) Here x, y 1 1 is coordinates vector of points O in xoy inertial coordinate system. Obviosly

4 638 A. Fischer et al. / Procedia Engineering 150 ( 016 ) e x x y y ; de 1 d d dt e dt dt d 1 d d e x x x x y y y y x x y y y y x x dt dt dt ;. (5) The oil film reactions (hydrodynamic forces) acted to a jornal in projections on an axis of coordinate system XOY are determined from eqations: 1 RX r pcos ddz; R r psin ddz. (6) Y p0 p0 Here integration area, in which p 0, inclde sbregions of oil spply sorces. Rotor jornal motion eqation i 1 : i i d i i i, m R m g j F dt (7) where x y F m cos t; F m sin t. (8) RF ring motion eqations i : d ( i1) m R m g jr dt d ( i1) J M M dt (9) NF single-ring motion eqation i 3: d ( i1) m R m g j R dt (10). Soltion The eqation () was written in term of finite differences and integrated by a Gass-Seidel method sing of three levels of grids. The finest grid contains 96 intervals on an axis and 5 ones on an axis z. Each time, when condition p 0 dring integrating is not satisfied in any point of a difference grid, pressre in this point is setting eqal to zero. All integrals were evalated by Simpson method. The motion eqations given as a system of first-order eqations, which was integrated sing Rnge-Ktta method with Merson correction. The lbricant films temperatres were calclated from thermal balances eqations. Thermal balances means that an average amont of heat prodced in some time interval in i -th lbricant film is eqal to amont of heat carried ot by axial oil flow.

5 A. Fischer et al. / Procedia Engineering 150 ( 016 ) The heat exchange between films was not taen into consideration. The oil viscosity is determined by Vogel s formlas. Following three design of rotor bearing system sed in small trbocharger TKR-8.5S for internal combstion engines was analyzed. The first two-filmed bearing systems is bearings with FR rings applied in serial trbocharger models (sch as TKR-8.5S) and the second one is bearing with FN single ring as alternative variant for the same trbocharger model. The third three-filmed bearing system (pacage bearing) is also applicable as alternative variant for this trbocharger (Fig. 1). Oil spply from center circmference grooves on ring srfaces is considered in all investigated variants. 3. Trajectories of bearing elements Fig. 3 present obtained by compter images of motion trajectories of the jornal centre on time interval t ** at two vales of rotor anglar speed. It is clear from figre the magnitde of jornal motion in three-films pacage bearing is essentially lower than one in two- films bearings with only RF or NF rings. Dependencies of motion trajectories dimensions and jornal relative eccentricities vale verss rotor anglar speed are shown in Fig. 4. It is seen that the jornal center in pacage bearing even at high rotor anglar speeds has a position close to central (jornal relative eccentricity is close to zero). Fig. 3. Jornal centre motion trajectories at anglar speeds 7000 s (1) 1 (a) and s (1) 1 (b). As a rle, the high-speed rotor, for example trbocharger rotor, is exposed to carefl balancing. The admissible displacement of the centre of masses 1 6 is 3, 0 10 m. Predictions shown that three types of bearings design remain efficient even at two-mltiple magnification of misbalance. Maximm dimensions of rotor and ring trajectories, their relative eccentricities in the pacage bearing are significantly less, than ones at two-films bearings. In some cases these characteristic in pacage bearing are p to ten time less. 4. Conclsion The presented reslts testify that the floating ring pacage bearings can find application as rotor spport in small rotor-machines. Its main advantage consists in essentially smaller magnitdes of rotor and ring vibrations. In threefilmed bearing the jornal center has a position close to central even at two-mltiple magnification of a rotor misbalance. The antivibration characteristics of bearings with RF and NF rings are close one to another. The energy losses in the pacage bearing exceed losses in two-filmed bearings only, bt losses in three-filmed bearing are twice smaller, than in one-filmed bearing with non-floating ring.

6 640 A. Fischer et al. / Procedia Engineering 150 ( 016 ) The complex of software, designed dring exection of the present research can be sefl at the frther researches of bearings with floating rings and, in particlar, this software is applicable for the prpose of optimization of bearings design parameters, schemes of oil spply and distribtion in lbricant films. Fig. 4. Dependencies of jornal center trajectories dimensions (a) and jornal maximal eccentricities (b) verss rotor anglar speed. Acnowledgements This wor has been carried ot within financial spport of Rssian Fondation for Basic Research (project \16, project \16). References [1] M. Tanaa, Y. Hori, Stability Characteristics of Floating Bsh Bearings, Trans. ASME, J. Lbr. Tech. 94 (1971) [] C.-H. Li, Dynamic of Rotor Bearing System Spported by Floating Ring Bearings, in: Proceeding of ASLE/ASME Lbrication Conference, New Orleans. (1981) 81. [3] C.H. Li, S.M. Rohde, On the Steady State and Dynamic Performance Characteristics of Floating Ring Bearings, Trans. ASME, J. Lbr. Tech. 103 (1981) [4] V.N. Proopiev, V.V. Smirnov, A.K. Boyarshinova, Dynamic of High Speed Rotor with Floating Non-rotating Rings Bearings, Problems of mechanical engineering and machine steady. 5 (1995) 374. [5] A. Brandt, C.W. Cryer, Mltigrid Algorithms for the Soltion of Linear Complementarity, Problems Arising from Free Bondary Problems, SIAM/J.Sci. Stat.Compt. 4 (1983) [6] E.A. Zadorozhnaya, Docment Solving a thermohydrodynamic lbrication problem for complex-loaded sliding bearings with allowance for rheological behavior of lbricating flid, Jornal of Machinery Manfactre and Reliability. 44 (015) [7] P. Taraneno, O. Sliva, E. Zadorozhnaya, Dynamics analysis of flexible rotor spported by floating ring bearings, Mechanisms and Machine Science. 1 (015) [8] E.A. Zadorozhnaya, S.V. Cherneyo, M.I. Krochin, N.A. Lovich, A stdy the axial and radial rotor stability of the trbo machinery with allowance the geometry of the srface and properties of the lbricating flid, Tribology in Indstry. 37 (015) [9] V.N. Proop'ev, E.A. Zadorozhnaya, V.G. Karavaev, I.G. Levanov, Improvement of the comptation procedre for complex-loaded sleeve bearings lbricated with non-newtonian oils, Sorce of the Docment Jornal of Machinery Manfactre and Reliability. 39 (010) 555. [10] Y. Rozhdestvensy, E.A. Zadorozhnaya, A Simlation of the thermal state of heavily loaded tribo-nits and its evalation, Blletin of the Soth Ural State University, Series: Mathematical Modelling, Programming and Compter Software. 7 (014) [11] V.V. Rojdestvensy, K.V. Gavrilov, A.A. Doiin, Solving of optimization tas for tribological piston and rotor machines by sing the mass conservation algorithm, Friction and lbrication in machines and mechanisms. 8 (011) (in Rssian). [1] K.V. Gavrilov, A.A. Asalya, I.L. Kopyrin, Method based on the application of bacward differentiation formlas (bdf) and its modifications for calclating of dynamics for the tribosystem of piston engines, Jornal of Comptational and Engineering Mathematics. 1 (015) 310.