New Tribological Ways Part 15 docx

New Tribological Ways 474 the mass-conserving lubrication problem have been proved, while an original approach to the thermal problem has been explained. The numerical examples show how the quasi-3D approach has enhanced the reliability of the mass- and energy-conserving lubrication analysis proposed by Kumar and Booker. Indeed, TEHD models are very sensitive to boundary conditions, which choice is particularly difficult in all of the multi-physics simulations. Future work will adapt the devised method to detailed transient analyses and it will further extend the model flexibility by including advanced turbulent lubrication theory. 8. Appendix Let f and F be scalar and vector-valued functions respectively. A variant of the divergence theorem states ( ) ff d f d ΩΓ ∇ +∇ Ω= Γ ∫∫ FF Fnii i (A1) where Γ is the boundary of Ω oriented by the outward-pointing unit normal n. If V Γ is the Eulerian velocity at the boundary Γ, the Reynolds transport theorem generalizes the Leibniz’s rule to multidimensional integrals as follows ( ) f f ddfd tt ΩΩΓ ∂ ∂ Ω =Ω+ Γ ∂∂ ∫∫∫ Γ Vni (A2) 9. References Banwait, SS. & Chandrawat, HN. (1998). Study of thermal boundary conditions for a plain journal bearing. Tribol. Int., Vol. 31, No. 6, pp. 289–296, ISSN: 0301-679X Bathe, K J. (1996). Finite Element Procedures, Prentice-Hall, ISBN: 0-13-301458-4 1, Upper Saddle River, New Jersey Booker, J. F. & Huebner, K. H. (1972). Application of Finite Element Methods to Lubrication: An Engineering Approach. ASME J. Lubr. Technol., Vol. 94, pp. 313–323 , ISSN: 0022-2305 Bonneau, D. & Hajjam, M. (2001). Modélisation de la rupture et de la formation des films lubrifiants dans les contacts élastohydrodynamiques. Revue Européenne des Eléments Finis, Vol. 10, No. 6-7, pp. 679-704, ISSN : 1250-6559 Bouyer, J. & Fillon, M. (2004). On the Significance of Thermal and Deformation Effects on a Plain Journal Bearing Subjected to Severe Operating Conditions. ASME J. Tribol., Vol. 126, No. 4, pp. 819-822, ISSN: 0742-4787 Brugier, D. & Pasal, M.T. (1989). Influence of elastic deformations of turbo-generator tilting pad bearings on the static behavior and on the dynamic coefficients in different designs. ASME J. Tribol., Vol. 111, No. 2 , pp. 364–371, ISSN: 0742-4787 Chang, Q.; Yang, P.; Meng, Y. & Wen, S. (2002). Thermoelastohydrodynamic analysis of the static performance of tilting-pad journal bearings with the Newton–Raphson method. Tribol. Int., Vol. 35, No. 4, pp. 225-234, ISSN: 0301-679X Dowson, D. (1967). A Generalized Reynolds Equation for Fluid-Film Lubrication. Int. J. Mech. Sci. , Pergamon Press Ltd., Vol. 4, pp. 159-170 FEM Applied to Hydrodynamic Bearing Design 475 Floberg, L. & Jakobsson, B. (1957). The finite journal bearing considering vaporization. Transactions of Chalmers University of Technology, Vol. 190, Gutenberg, Sweden Fatu, A.; Hajjam, M. & Bonneau, D., (2006). A new model of thermoelastohydrodynamic lubrication in dynamically loaded journal bearings. ASME J. Tribol., Vol. 128, pp. 85–95, ISSN: 0742-4787 Glavatskikh, S. (2001). Steady State Performance Characteristics of a Tilting Pad Thrust Bearing, ASME J. Tribol., Vol. 123, No. 3, pp. 608-616, ISSN: 0742-4787 Kelly, D.W.; Nakazawa, S. & Zienkiewicz, O.C. (1980). A Note on Upwinding and Anisotropic Balancing Dissipation in Finite Element Approximations to Convective Diffusion Problems. Int. J. Numer. Meth. Eng., Vol. 15, pp. 1705-1711, ISSN: 0029- 5981 Kim, B.J. & Kim, K.W. (2001). Thermo-elastohydrodynamic analysis of connecting rod bearing in internal combustion engine, ASME J. Tribol., Vol. 123, pp. 444–454, ISSN: 0742-4787 Khonsari, M.M. & Booser, E.R. (2008). Applied tribology: bearing design and lubrication, Second Edition, Wiley & Sons, ISBN: 9780470057117, Chichester, UK Kumar, A. & Booker, J.F. (1991). A finite element cavitation algorithm: Application/validation. ASME J. Tribol., Vol. 107, pp. 253-260, ISSN: 0742-4787 Kumar, A. & Booker, J.F. (1994). A Mass and Energy Conserving Finite Element Lubrication Algorithm. ASME J. Tribol., Vol. 116 , No. 4, pp. 667-671, ISSN: 0742-4787 LaBouff, G.A. & Booker, J.F. (1985). Dynamically Loaded Journal Bearings: A Finite Element Treatment for Rigid and Elastic Surfaces. ASME J. Tribol., Vol. 107, pp. 505-515, ISSN: 0742-4787 Lund, J.W. & Tonnesen J. (1984). An approximate analysis of the temperature conditions in a journal bearing. Part II: Application. ASME J. Tribol., Vol. 106, pp. 237–245, ISSN: 0742-4787 Kucinski, B.R.; Fillon, M.; Frêne, J. & Pascovici, M. D., (2000). A transient Thermoelastohydrodynamic study of steadily loaded plain journal bearings using finite element method analysis, ASME J. Tribol., Vol. 122, pp. 219-226, ISSN: 0742- 4787 Murty, K.G. (1974). Note on a Bard-type Scheme for Solving the Complementarity Problem. Opsearch, Vol. 11, pp. 123-130 Olsson, K. O. (1965). Cavitation in dynamically loaded bearing. Transactions of Chalmers University of Technology , Vol. 308, Guthenberg, Sweden Piffeteau, S.; Souchet, D. & Bonneau, D. (2000). Influence of Thermal and Elastic Deformations on Connecting-Rod End Bearing Lubrication Under Dynamic Loading. ASME J. Tribol., Vol. 122, No. 1, pp. 181-191, ISSN: 0742-4787 Robinson, C.L. & Cameron, A. (1975). Studies in hydrodynamic thrust bearings. Philos. Trans ., Vol. 278, No. 1283, pp. 351–395, ISSN: 1364-503X Stefani, F. & Rebora, A. (2009). Steadily loaded journal bearings: Quasi-3D mass–energy- conserving analysis. Tribol. Int., Vol. 42, No. 3, pp. 448-460, ISSN: 0301-679X Stieber, W. (1933). Das Schwimmlager. VDI, Berlin Swift, H. W. (1932). The stability of lubricating films in journal bearings. Proc. Inst. Civil Eng., Vol. 233, pp. 267–288 New Tribological Ways 476 Tezduyar, T. & Sunil, S. (2003). Stabilization Parameters in SUPG and PSPG formulations. Journal of Computational and Applied Mechanics, Vol. 4, No. 1, 7 pp. 1-88, ISSN: 15862070 Wang, Y.; Wang, Q.J. & Lin, C. (2003). Mixed Lubrication of Coupled Journal-Thrust-Bearing Systems Including Mass Conserving Cavitation. ASME J. Tribol., Vol. 125, pp. 747- 756, ISSN 0742-4787 Wendt, F. (1933). Turbulente stromungen zwischen zwei rotierenden konaxialen zylindern. Ingenieur-Archiv, Vol. 4, No. 3, pp. 577–595 23 Comparison between Different Supply Port Configurations in Gas Journal Bearings Federico Colombo, Terenziano Raparelli and Vladimir Viktorov Politecnico di Torino, Department of Mechanics Italy 1. Introduction Because of their precision, gas bearings are widely used for very high speed spindle applications. Compared to conventional oil bearings, gas bearings generate less heat and do not pollute the environment. Air viscosity is three orders of magnitude lower than oil, so the power dissipated in gas bearings is very low. The major disadvantage of these bearings is rotor whirl instability, which restricts the possible range of applications. Researchers have studied this problem using different methods since the '60s. Gross first applied a perturbation method to evaluate the stability of an infinitely long journal bearing (Gross & Zachmanaglou, 1961). Galerkin’s method was used by others to calculate rotor speed and mass at the stability threshold (Cheng & Pan, 1965). Lund investigated the stiffness and damping coefficients of hydrostatic gas bearing, and used these coefficients to investigate whirl instability (Lund, 1968). Wadhwa et al. adapted the perturbation method to calculate the dynamic coefficients and to study the stability of a rotor supported by orifice compensated gas bearings (Wadhwa et al., 1983). Results show that aerostatic bearings have a larger load capacity and higher stability than plain journal bearings. Han et al. proved that more circumferential supply ports result in increased stiffness coefficient but reduced damping (Han et al., 1994). Others found that orifice-compensated and shallow-pocket type hybrid gas journal bearings offer better stability than eight-orifice type bearings (Zhang & Chang, 1995). Also porous journal bearings were studied (Sun, 1975) and compared against hybrid gas bearings with multi-array entries (Su & Lie, 2006), (Heller et al., 1971). Despite the fact that damping is generally higher in porous bearings than in aerostatic bearings, the results of (Su & Lie, 2006) suggest that at high operating speeds, multi-array entry bearings are more stable than porous bearings. Other studies (Andres, 1990), (Sawcki et al., 1997), (Yoshikawa et al., 1999) considered various pressurized air compensated configurations, but very few papers analysed the influence of the number and location of entry ports. In (Su & Lie, 2003) hybrid air journal bearings with multi-array supply orifices were compared to porous bearings. One to five rows of orifices were considered. It was found that five rows of supply orifices perform as well as porous bearings, whilst supply orifice feeding has the advantage of consuming less power than porous feeding. Paper (Yang et al., 2009) compared bearing systems with double-array orifice restrictions to three and six entry New Tribological Ways 478 systems. Results show that the stability threshold is better with six-ports than with three ports. In (Colombo et al., 2009) the authors analysed two externally pressurized gas bearings, one with a central row of supply orifices, the other with a double row. The supply port downstream pressure was found to be proportional to the critical mass. At this pressure reading, the second bearing type was 30% stiffer and 50% more stable. The aim of this work is to compare three externally pressurized gas journal bearings at given air consumption rates. The idea was to investigate which offers the best spatial distribution of supply orifices under the same pneumatic power. The study compared radial stiffness and pressure distribution for the three bearing types, also evaluating the damping factor and the whirl ratio of the shaft. The stability threshold was calculated for different restriction parameters so that the proposed bearing types could be compared. 2. Description of the problem The object of the study was a rigid rotor supported by two identical gas journal bearings situated symmetrically with respect to the journal centre. The rotor, with diameter D=50 mm, was considered to be perfectly balanced. The radial air clearance was h 0 =20 µm and the bearings had L/D ratio equal to unity. Three bearing types were considered, as illustrated in figure 1. Bearing type 1 featured four supply ports situated in the centre plane of the bearing; bearing type 2 featured two sets of supply ports, situated at z=L/4 and z=3L/4; bearing type 3 also featured a central vented circumferential chamber. The three bearing types were comparable in terms of stiffness and damping coefficients, air consumption and stability. In (Colombo et al., 2009) the authors compared bearing types 1 and 2 (see figure 1) considering the same supply port diameter d s . The bearing with double array entries (bearing type 2) was found to be 30% stiffer than the one with a single central array (bearing type 1) but the air consumption was two times as much. Moreover, bearing 2 was more stable: the rotor mass at incipient whirl instability was about 50% greater. Another point of interest was which bearing type was to be preferred for the same level of air consumption. In this paper the bearings illustrated in figure 1 were compared considering different supply port diameters in order to have the same air consumption. 3. Lubrication analysis 3.1 Mathematical model The two-degree-of-freedom rotor equations of motion are shown in (1). The rotor mass is m. As the shaft was assumed to have cylindrical motion, gyroscopic effects and tilting inertia moments are non-existent. The second member of the equations is zero because the rotor was assumed to be perfectly balanced and there were no external forces applied to it. This was the most unstable condition, as shown in (Belforte et al., 1999). () () 2 00 2 00 2,cos 0 2,sin 0 L L mx p z rd dz my p z rd dz π π θθθ θθθ ⎧ + = ⎪ ⎪ ⎨ ⎪ + = ⎪ ⎩ ∫∫ ∫∫   (1) Comparison between Different Supply Port Configurations in Gas Journal Bearings 479 Fig. 1. Bearing types under study The pressure distribution in clearance h was calculated solving the distributed parameters problem described by the Reynolds equation for a compressible-fluid-film journal bearing (2), assuming isothermal gas expansion. ( ) ( ) 3300 12 6 12 p hph pp G ph ph R T z z r r rdrd t μμωμ θθ θ θ ∂∂ ∂∂ ∂∂ ⎛⎞⎛ ⎞ ++ =+ ⎜⎟⎜ ⎟ ∂∂∂ ∂ ∂ ∂ ⎝⎠⎝ ⎠ (2) Mass flow rate G at supply orifice was calculated in accordance with the isentropic expansion formula (3), corrected by experimentally identified discharge coefficient c d , expressed by eq. (4). Reynolds number at the supply hole was calculated as per equation (5). Formula (4) is the result of an extensive set of experimental tests carried out on air pads with different inherence parameters (Belforte et al., 2008). 21 2 2   41 k kk cc c s ds ss s pp p d k Gc p if b kp p RTp π + ⎡⎤ ⎛⎞ ⎛⎞ ⎢⎥ = −≥ ⎜⎟ ⎜⎟ ⎢⎥ − ⎝⎠ ⎝⎠ ⎢⎥ ⎣⎦ 2 2 1 00 22 if 411 k c s ds s p d k Gc p b kk p RT π − ⎛⎞ = < ⎜⎟ ++ ⎝⎠ (3) New Tribological Ways 480 () 8.2 0.001 0.85 1 1  s h d Re d cee − − ⎛⎞ ⎜⎟ =− − ⎜⎟ ⎝⎠ (4) 4 s G Re d π μ = (5) Assuming a cylindrical shaft motion, the clearance may be expressed by the following: ( ) 0 () 1 cos sin xy hz h ε θε θ =− − (6) 3.2 Solution method The Reynolds equation was discretized using a finite difference method along directions z and θ for integration over the fluid film. A rectangular grid with equi-spaced nodes in both directions was considered. The number of nodes in the axial (index i) and circumferential (index j) directions were n and m respectively. Equation (2) may be written for each node as follows: ( ) ( ) ( ) ( ) ()( ) 2222 1, , , 1, , , , 1 , , , 1 , , 1 ,, 2 ,, , ,1 ,1, , , 1 00 ,, ,, 224 24 24 i j ij ij i j ij ij ij ij ij ij ij ij tt ij ij ij ij ij ij ij ij ij ij tt ij ij t ij ij pabpabpcdpcd hh pa c p p e p g t pp RT Gh rz t μ μ μ θ +− +− − +− + + +−+++−+ ⎛⎞ − ⎜⎟ −++− −+ + ⎜⎟ Δ ⎝⎠ − += Δ ΔΔ (7) where, 32 ,, ,, 2 , 32 ,, ,, 22 2 , , ,, , 3 2 3 2 612 ij ij ij ij i j ij ij ij ij i j ij ij ij i j hh h ab zz z hh h cd rr h h eg θ θθ μω μω θθ ∂ ⎛⎞ == ⎜⎟ Δ∂ Δ ⎝⎠ ∂ ⎛⎞ == ⎜⎟ ∂ ΔΔ ⎝⎠ ∂ ⎛⎞ == ⎜⎟ Δ∂ ⎝⎠ At the supply port G i,j was calculated using equation (3), whereas elsewhere it was zero. The boundary conditions imposed were: • p=p a at z=0 and z=L; for bearing type 3 p=p a also at z=L/2 • periodic condition at θ=0 and θ=2π. The Euler explicit method was used, so equation (7) becomes: 11 ,, ,,1,11,1,,, ,, , ,,,,,, ,  tt tt ttttttt ij ij ij ij ij i j i j ij ij i j i j hh pp tfppppphh z θ +− +−+− ⎡ ⎤ ∂∂ ⎛⎞⎛⎞ =+Δ⋅ ⎢ ⎥ ⎜⎟⎜⎟ ∂∂ ⎝⎠⎝⎠ ⎢ ⎥ ⎣ ⎦ (8) The system of nxm equations (8) was solved together with equations (3) to (6) and rotor equations of motion (1). Comparison between Different Supply Port Configurations in Gas Journal Bearings 481 The solution procedure started with a set of input data (shaft diameter, radial clearance, bearing axial length, position and diameter of supply orifices, shaft speed). To calculate the static pressure distribution, h was maintained constant in time and the system was solved with initial condition p i,j =p a for each node. Pressure distribution was evaluated at each time step and the bearing forces acting on the shaft were updated in equation (1). Thus, the rotor trajectory was determined starting with the initial static pressure distribution and using the following set of initial conditions: ( ) ( ) 0 00 x xh ε = ; ( ) ( ) 0 00 y yh ε = ( ) 0 0(0) x xh ε =  ; ( ) 0 0(0) y yh ε =  3.3 Mesh size and time step definition Calculations were made with different mesh sizes and the results were compared for optimum trade-off between computational time and accuracy of the solution. The grids are detailed in table 1. nxm Δz (mm) rΔθ (mm) 13x24 4.17 6.54 17x32 3.12 4.91 25x48 2.08 3.27 49x96 1.04 1.64 Table 1. Mesh sizes used in calculations; r=25 mm, L/D=1 Figure 2 shows the axial and circumferential pressure distributions obtained for bearing type 1 with different numbers of grid points. If the number of grid points is increased, the pressure distribution becomes more clearly defined, but the difference is almost negligible. Only at the supply ports, where pressure gradients are high, the difference is more marked. The grid selected for calculation was n=49, m=96. 0 20 40 60 1 1.2 1.4 1.6 1.8 x 10 5 z axis [mm] p [Pa] bearing 1 13x24 17x32 25x48 49x96 0 30 60 90 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x 10 5 circumferential axis [deg] p [Pa] bearing 1 13x24 17x32 25x48 49x96 Fig. 2. Axial and circumferential pressure distributions for bearing type 1 obtained with different mesh grids; h 0 =20 μm, p s =5·10 5 Pa rel., d s =0.1 mm, ω =60 krpm, ε =0 New Tribological Ways 482 Euler explicit method was used to solve the time progression of the system. The rotor trajectories obtained with different time steps Δt are compared in figure 3. The rotor initial conditions were: ( ) ( ) 00; 00 xy εε = = ( ) ( ) 00; 00 xy εε = =  The trajectories are increasingly adjacent with decreasing Δt. The time step used in the paper was Δt=10 -7 s. -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 ε x ε y n=25; m=48 dt=4e-7 dt=2e-7 dt=1e-7 dt=5e-8 Fig. 3. Rotor trajectories with bearing type 1 obtained with different time steps and grid 25x48; initial conditions specified by ε x (0)=0.05, ε y (0)=0, ( ) ( ) 00, 00 xy εε = =  , h 0 =20 μm, p s =5·10 5 Pa rel., d s =0.1 mm, ω =60 krpm 4. Discussion and results 4.1 Resistance analysis The air supply system may be described with an equivalent lumped parameters system, illustrated in figure 4. Orifice restriction resistance R s is related to the supply ports and decreases with increasing diameter d s . It may be calculated using linearizing expression (3) with respect to downstream pressure p c . Clearance resistance R h depends on clearance h 0 , on bearing dimensions size and on the arrangement of the supply ports. It is obtained by solving the distributed parameters problem and calculating pressure distribution in the clearance. Imposing mass continuity in the lumped parameters system of figure 4, supply port downstream pressure p c can be obtained by () s cs sa sh R pp pp RR =− − + (9) [...]... ds =0.05 mm type 3; ds =0.1 mm type 3; ds =0.2 mm 0 5 10 15 20 25 clearance [ μm] 30 35 Fig 5 Air consumption of the three bearings vs air clearance for different supply port diameters; calculations are for Λ=0 and with rotor in central position; ps=5·105 Pa rel 484 New Tribological Ways bearing type 1 2 3 1 2 3 1 2 3 2 3 2 3 diameter ds [mm] 0 .155 0.1 0.1 0.383 0.2 0.2 0.8 0.282 0.275 0.4 0.372 0.6... -5 β [deg] k* 2.5 2 -10 -15 1.5 -20 1 0.5 -1 10 0 1 10 10 -25 -1 10 2 10 Λ G=4.28e-4 kg/s 2 10 0 bearing 2 bearing 3 bearing 2 bearing 3 -10 k* β [deg] 3 2 -20 -30 1 0 -1 10 1 10 Λ G=4.28e-4 kg/s 5 4 0 10 0 1 10 10 Λ 2 10 -40 -1 10 0 1 10 10 2 10 Λ Fig 9 Non-dimensional bearing stiffness k* and attitude angle β vs bearing number Λ for the three bearings 490 New Tribological Ways The initial condition... bearings Precision Engineering, Vol 16, No 3, 164-173 Heller, S.; Shapiro, W.; Decker, O (1971) A porous hydrostatic gas bearing for use in miniature turbomachinery ASLE Transactions, 144 -155 498 New Tribological Ways Lund, J.W (1968) Calculation of stiffness and damping properties of gas bearings ASME Journal of Lubrication Technology, 793-803 Sawicki, J.T.; Capaldi, R.J.; Adams, M.L (1997) Experimental... x 10 20 30 z axis [mm] 50 bearing 1 bearing 2 bearing 3 3.5 3 40 G=2.14e-4 kg/s, ω=200 krpm 4 p [Pa] 4 p [Pa] 40 3 2.5 2 2 1.5 1 0 10 20 30 z axis [mm] 40 50 1 0 10 20 30 z axis [mm] 40 50 486 New Tribological Ways 5 4 G=2.94e-4 kg/s, ω=0 x 10 4 bearing 2 bearing 3 3.5 3 p [Pa] p [Pa] bearing 2 bearing 3 3.5 3 2.5 2.5 2 2 1.5 1.5 1 0 10 20 30 z axis [mm] 40 1 50 G=4.28e-4 kg/s, ω=0 5 4.5 x 10 0 10 5... Different Supply Port Configurations in Gas Journal Bearings -8 3 10 angular moment [kg*m2/s] attitude angle β [deg] 5 0 -5 m=9 kg m=9.5 kg m=11 kg -10 -15 -20 -25 0 0.005 0.01 t [s] 0. 015 0.02 x 10 m=9 kg m=9.5 kg m=11 kg 2.5 2 1.5 1 0.5 0 0.005 (a) 0.01 t [s] 0. 015 0.02 (b) Fig 11 Attitude angle vs time (a) and rotor angular moment vs time (b) with different rotor masses and initial condition x(0)=1 μm, dy/dt(0)=x(0)·(kxx/m)^(0.5);... bearing 2 bearing 3 -2 -3 -3 -2 -1 0 1 xg [μm] 2 3 4 Fig 12 Rotor trajectories with the three bearing types; m=1 kg, ω=50 krpm; initial conditions x(0)=1 μm and dy/dt(0)=x(0)·(kxx/m)^(0.5) 492 New Tribological Ways -8 x 10 80 bearing 1 bearing 2 bearing 3 40 2.5 angular moment [kg*m2/s] attitude angle [deg] 60 20 0 -20 -40 0 2 4 t [s] bearing 1 bearing 2 bearing 3 2 1.5 1 0.5 0 6 -3 x 10 (a) 1 2 3 t... 100 krpm 200 krpm 20 krpm 50 krpm 100 krpm 200 krpm 20 krpm 50 krpm 100 krpm 200 krpm 4 10 -1 0 10 10 1 10 m [kg] Fig 15 Whirl frequency ν vs m at different rotating speeds; G=0.5·10-4 kg/s 494 New Tribological Ways 20 krpm 50 krpm 100 krpm 200 krpm γ 0.55 0.5 0.45 0.4 0.5 1 1.5 2 m/mth Fig 16 Whirl ratio γ vs m/mth at different speeds; bearing type 1, G=0.5·10-4 kg/s indicated as mth Figure 16 shows... a) bearing 1 bearing 2 bearing 3 1.5 6 -4 x 10 1 0 2 4 air consumption [kg/s] 6 -4 x 10 b) Fig 18 Bearing stiffness k* vs air consumption for the three bearings; a) ω=0 rpm, b) ω=200 krpm 496 New Tribological Ways 0.24 0.22 0.2 0.18 mth 0.16 0.14 0.12 bearing 1 bearing 2 bearing 3 0.1 0.08 0.06 0.04 0.5 1 1.5 2 2.5 3 air consumption [kg/s] 3.5 4 4.5 -4 x 10 Fig 19 Rotor mass at stability threshold... unstable In the former case the rotor is attracted toward the centre of the bushing after initial disturbance; in the latter case the bearing forces move the rotor away from central position 488 New Tribological Ways k*(Λ>100)/k*(Λ=0) 2.62 2.71 3.75 2.54 1.82 2.4 5.8 2 2.2 2.5 2.26 5.33 3.08 bearing type 1 2 3 1 2 3 1 2 3 2 3 2 3 air flow G·104 [kg/s] 0.5 1.42 2.14 2.94 4.28 Table 3 Ratio k*(Λ>100)/k*(Λ=0)... -0.1 -0 .15 -1 0 10 1 10 10 2 10 m [kg] Fig 14 Damping factor vs rotor mass at different rotating speeds; G=0.5·10-4 kg/s bearing 1; bearing 1; bearing 1; bearing 1; bearing 2; bearing 2; bearing 2; bearing 2; bearing 3; bearing 3; bearing 3; bearing 3; 5 ν [rpm] 10 20 krpm 50 krpm 100 krpm 200 krpm 20 krpm 50 krpm 100 krpm 200 krpm 20 krpm 50 krpm 100 krpm 200 krpm 4 10 -1 0 10 10 1 10 m [kg] Fig 15 Whirl . rotor in central position; p s =5·10 5 Pa rel. New Tribological Ways 484 bearing type diameter d s [mm] air flow G·10 4 [kg/s] 1 0 .155 2 0.1 3 0.1 0.5 1 0.383 2 0.2 3 0.2 1.42. lubricating films in journal bearings. Proc. Inst. Civil Eng., Vol. 233, pp. 267–288 New Tribological Ways 476 Tezduyar, T. & Sunil, S. (2003). Stabilization Parameters in SUPG and. compared bearing systems with double-array orifice restrictions to three and six entry New Tribological Ways 478 systems. Results show that the stability threshold is better with six-ports