Mixed Lubrication of Coupled Journal-Thrust-Bearing Systems Including Mass Conserving Cavitation

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1 Yansong Wang Q. Jane Wang Department of Mechanical Engineering, Northwestern University, Evanston, IL Chih Lin Baker Hughes, Inc., Houston, TX Mixed Lubrication of Coupled Journal-Thrust-Bearing Systems Including Mass Conserving Cavitation A mixed lubrication model is developed to investigate the lubrication of coupled journalthrust-bearing systems. The governing equations are mapped into a computational domain with a conformal mapping technique developed in a previous study by Wang et al. (2002). In order to study the influence of the boundary conditions on the performance of the bearing system, either the Reynolds boundary conditions or the JFO s mass conserving conditions are applied to the governing Reynolds equations. The performance of a typical coupled journal-thrust bearing system is numerically investigated and the effects of viscosity, misalignment angle and feeding scheme are discussed. DOI: / Introduction The end face of the shaft in a journal bearing is often utilized to form a hydrodynamic thrust bearing. The combination of a journal bearing with the end-face thrust bearing creates a coupled journalthrust-bearing system. The angular misalignment between the centerlines of the shaft and the journal bearing produces a geometric condition conducive for the thrust bearing hydrodynamics, and the side-leakage flow from the journal bearing provides a natural lubricant supply to the thrust bearing. Most of the research effort in this area has been concentrated on aerostatic journal-thrust bearings, or Yates bearings 1,2, for hard-disk drive applications. Usually, the bearing system for harddisk drives consists of small journal bearings and small thrust bearings for light loads 3. The characteristics of this kind of journal-thrust bearing pair have been studied for full-film hydrodynamic lubrication under perfectly aligned geometric conditions 4 7. Recently, the authors 8 published a model for the coupled journal-thrust-bearing under mixed lubrication for heavy-duty machinery applications. The analyses based on this model reveal that, when the journal bearing and the thrust bearing are hydrodynamically coupled, an intensification of the hydrodynamic pressure may exist in both bearings. So far Reynolds boundary conditions were applied to the model of the coupled bearing system to treat the downstream pressure condition, which do not accurately describe the downstream flow condition, i.e., the formation of cavitation and mass conservation in the cavitation region. In order to fully understand the performance of the coupled bearing system, mass conserving cavitation needs to be properly taken into account. This paper presents a mixed lubrication model for coupled journal-thrust-bearing systems with an angular misalignment including mass conserving cavitation. The conformal mapping technique that facilitates a universal flow description for the coupled bearing system is extended to the current cavitation model. The performance of a typical coupled journal-thrust-bearing system is numerically investigated and the effects of viscosity, misalignment angle and feeding scheme are discussed. Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division February 20, 2002; revised manuscript received January 21, Associate Editor: C. H. Venner. 2 Description of the Coupled Journal-Thrust-Bearing System Figure 1a illustrates the geometry of a journal-thrust-bearing system that consists of a shaft and a bearing cylinder with an open end. The bearing system is subject to both radial and thrust loads. The angular misalignment, caused by improper assembly practices, shaft centerline deflection, and structural deformation, defines the configuration of the thrust bearing. Two wedges that are responsible for the hydrodynamic pressure generation are formed in the thrust bearing: one in the radial direction and the other in the circumferential direction. A positive misalignment angle,, is shown in Fig. 1. The wedge in the radial direction of the thrust bearing, or the face width direction, as shown in Fig. 1b, is called the face wedge. The wedge along the circumferential direction, as shown in Fig. 1c and defined by the extended perimeters of a pair of circles of the end face of the shaft and the outer edge of the thrust bearing, is called the circumferential wedge. The wedge is said to be convergent if the flow is from the larger clearance side to the smaller clearance side. In the mathematical development of the model for the coupled bearing system described above, the lubricant is assumed to be an incompressible fluid with a constant density in the pressurized region. Lubricant flows are treated as steady-state laminar flows. Moreover, the hydrodynamic pressure is considered to be constant across the film thickness. Usually, the gap of the interface of the journal bearing and the thrust bearing is much smaller than the lateral dimensions. Therefore, the Reynolds equation is applicable here. A special case arises when the gap is sufficiently large as if there were several lubricant exit holes. Under this circumstance, the journal bearing and the thrust bearing may perform independently without any hydrodynamic coupling effect even though they are still geometrically coupled. For this case, the governing equations for the journal and thrust bearings should be solved separately. 2.1 Governing Equations of the Bearing Systems. The average Reynolds equation derived by Patir and Cheng 9 is employed to describe the pressure-film thickness relation for the thrust bearing in mixed lubrication. This average Reynolds equation in cylindrical coordinates has the following form: Journal of Tribology Copyright 2003 by ASME OCTOBER 2003, Vol. 125 Õ 747

2 L D m 2c1 2 sin 2 1/2 cos and its value can vary from 0.0 to 1.0. max is defined by (6) max 21 2 sin 2 1/2 cos (7) The influences of the three misalignment parameters, D m,, and should be analyzed separately. Usually, is fixed at a prescribed value 13,14. Therefore, the analysis is carried out for 0. Combining Eqs. 4 through 7, the film thickness can be expressed in terms of the misalignment angle,, as hc1 coszl/2cos (8) Asperity interaction is one of the main features of mixed lubrication. Because the film thickness varies circumferentially, a relationship between the asperity contact pressure and the film thickness in terms of the average gap is needed in order to compute the asperity contact load. The rough surface contact model by Lee and Ren 15 is employed in this study. Fig. 1 Schematic of the journal bearing and the end thrust bearing formed by shaft misalignment: a External boundary conditions; b Face wedge; and c Circumferential wedge. r 1 r h 6r c s (1) h r 3 r p r h 3 p where r and are the pressure flow factors and s is the shear flow factor defined by Patir and Cheng 9, c is the asperity contact factor 10. The film thickness is described by the following formula hr,h 0 r tan cos (2) In Eq. 2, h 0 is the average film thickness of thrust bearing measured at the center of the thrust bearing and is a misalignment angle. The governing equation for the journal bearing is expressed in the Cartesian coordinate system where xr i. h x 3 p x 12 x h z 3 p z 12 z U c h U 2 x 2 s x (3) The film thickness of the journal bearing with misalignment can be defined by three basic parameters, viz.,,, and 11,12 and expressed by hc1 coscz/l1/2cos (4) where c is the radial bearing clearance, the eccentricity ratio, the loading angle, the magnitude of the projection of the entire journal axis onto the mid plane, L the effective length of the journal bearing, and the angle between the projection of the journal rear center line onto the mid plane of the bearing and the eccentricity vector defined by and the bearing angle at the mid plane. can be written as D m max (5) where D m is known as the degree of misalignment and max is the maximum possible value of at which contact between journal and bearing takes place. D m is defined by 2.2 Coupling With Different Hydrodynamic Boundary Conditions. Four pressure boundary conditions involved in the coupled bearing system, as shown in Fig. 1a, are needed to solve the boundary value problem: BC1, the lubricant supply pressure of bearings, p s Here lubricant feeding holes are used for lubricant replenishment; BC2, the film rupture pressure, p rup ; BC3, the journal-bearing environment pressure, p j, at the open end of the journal bearing; and BC4, the thrust-bearing environment pressure, p t, at the inner edge of the annular end face of the bearing that forms the thrust bearing. There is also an internal boundary along the interface between the journal bearing and the thrust bearing where the continuity of flow rates and pressures must be satisfied. Each bearing has two flows: the circumferential flow and the side flow in the journal bearing and the circumferential flow and the radial flow in the thrust bearing. The side flow in the journal bearing is connected to the radial flow of the thrust bearing. Because the flow continuity in the circumferential direction is automatically satisfied, only the continuity in the radial direction of the thrust bearing and the width direction of the journal bearing needs to be treated. This condition may be expressed as 3 h j p 0 p t h t 3 p j p 0 (9) 12 r 12 z The Reynolds boundary conditions imply that the fluid pressure becomes zero at a downstream location where the pressure derivative is zero. Jakobsson, Floberg 16 and Olsson 17 JFO proposed a set of self-consistent conditions for cavitation to be applied to the Reynolds equation. These conditions properly account for mass conservation in the cavitation region. Elrod and Adams 18 and Elrod 19 proposed a cavitation algorithm to implement the JFO theory, which automatically predicts any cavitation regions. Brewe 20 applied Elrod s algorithm to model vaporous cavitation in dynamically loaded bearings. Vijayaraghavan and Keith 21 proposed a modification to Elrod s algorithm, specifically to the shear-flow term, by introducing a type-difference procedure. Kumar and Booker 22 proposed another cavitation algorithm that facilitates the implementation of the finite-element method. Payvar and Salant 23 applied Elrod s cavitation algorithm to the analysis of a wavy mechanical seal by controlling the cavitation index to ensure convergence. In the flow region where the pressure is below a given cavitation pressure, the fluid may form striations. When this occurs, both liquid and vapor phases co-exist under a uniform pressure which is set to zero in this study. In such cavitation regions, the pressure gradient terms in Eq. 3 disappear, and the equation reduces to 748 Õ Vol. 125, OCTOBER 2003 Transactions of the ASME

3 Table 1 Exponential conformal mapping pairs h c s 0 (10) Payvar and Salant s approach to analyze the cavitation pattern is adopted in the current model. The cavitation index, G, isde- fined as G1, p/6 0 R 1 / 2 0 G0, in the noncavitation region / c 1 0 in the cavitation region 2.3 A Conformal Mapping Method and a Universal Equation for the Coupled Bearing System. A conformal mapping is utilized to transform the end face of the thrust bearing from its annular physical domain to a rectangular computational domain formed by the extended geometry of the journal bearing so that a universal equation for the journal-thrust-bearing system can be obtained. A conformal mapping pair formed by an exponential function and a logarithmic function given in Table 1 facilitates such domain conversion. By using the transformation function given by Nehari 24, ln(r/r 1 ), and, a physical point, r,, on the thrust bearing is mapped onto a mapping point,,, in the computational domain. The end face of the thrust bearing can then be mapped into a rectangular shape whose edges conform well with the geometry of the extended journal surface. The correspondence between the mapped domain and the physical domain is shown in Fig. 2. Thus, the problem for the coupled journal-thrust-bearing is simplified from a three-dimensional space domain into a two-dimensional plane domain. In the mapping, r has two expressions, rr 1 for 0, and r R 2 for ln(r 2 /R 1 ). By changing the variables, applying the chain rule of differentiation, and using the inverse Jacobian matrix, we arrive at a universal governing equation for the journalthrust-bearing system. Fig. 2 Conformal mapping that transfers the thrust bearing surface into a rectangular computational domain which is connected to the journal bearing domain Using the following nondimensionalized terms,, z/l for journal Bearing, / 0, h h/, / 0, p p/6 0 (R 1 /) 2, the universal equation for the coupled bearing system becomes h 3 G h 3 G 1 G 11G h s G 2 c G 2 (11) The cavitation index, G, and the cavitation parameter,, are redefined as G1, G0, p 0 in the noncavitation region / c 1 0 in the cavitation region For G 1 (R 2 /L) 2 and G 2 (R 2 /R 1 ) 2, the universal equation, Eq. 11, reduces to the following equation for the journal bearing h 3 G R 2 R 2 2 R 1 L 2 h 3 c 11G h R 2 G R 1 2 s (12) For G 1 1, G 2 e 2ln(r/R 1 ), the universal equation, Eq. 11, reduces to the following equation for the thrust bearing h 3 G h 3 G 11G h e 2 c e s 2 (13) If G is set to be unity, Eq. 13 becomes the governing equation with the Reynolds boundary conditions. A finite difference method is used to discretize the governing equation. A set of results from this cavitation model for the journal bearing is compared with the results published by Vijayaraghavan and Keith 21, and good agreement in pressure distributions has been obtained. The parameters used in this case study are listed as the followings: both the bearing length and diameter are 62.8 mm, eccentricity ratio is 0.6, velocity is 19.7 m/s, and the viscosity is Pa.s. Equation 13 includes the interface of the journal and thrust bearings as a part of the computational domain. Thus, the conditions for pressure and flow rate continuity are satisfied automatically. The mapping greatly simplifies the solution process, and the entire bearing system can be solved simultaneously. The solution accuracy is controlled as maxp (n1) 0 p (n) 0 /maxp (n) Both force equilibrium and moment balance are satisfied in the simulation. The current model formulation has been verified by solving separately a journal bearing and a thrust bearing problem. Journal of Tribology OCTOBER 2003, Vol. 125 Õ 749

4 3 Results and Discussion A set of journal and thrust bearings, coupled and uncoupled, with the Reynolds boundary conditions and the cavitation consideration are numerically investigated. In the following case studies, the outer radius, R o, inner radius, R i, and the width of the journal bearing, L, are chosen to be mm, mm, and mm, respectively. The outer radius of the thrust bearing is R 2 (R i ), and the recess radius, R 1, is 0.25R 2. The journal bearing radial clearance is 0.025R i. Under the steady-state conditions, the journal bearing eccentricity ratio,, the nondimensionalized misalignment angle, L/2c, and the nondimensionalized thrust bearing average film thickness, h 0h 0 /c, are three independent parameters that are specified for each case analyzed. In most of the analyses, is set at The root-mean square roughness of Fig. 3 The thrust bearing performance with different feeding conditions and different boundary conditions: a Contour of the pressure distribution with inner feeding and the Reynolds boundary condition; b Contour of the pressure distribution with outer feeding and the Reynolds boundary condition; c Contour of the pressure distribution with inner feeding and the cavitation consideration; d Contour of the pressure distribution with outer feeding the cavitation consideration; e Cavitation index for the outer feeding; and f Cavitation index for the outer feeding. 750 Õ Vol. 125, OCTOBER 2003 Transactions of the ASME

5 Fig. 4 Performance of the journal-thrust bearing with the Reynolds boundary conditions in hydrodynamic lubrication Ä0.8, h 0Ä0.395, misalignment angle Ä0.089 : a With coupling; and b Without coupling. the surfaces studied in this paper is 0.5 m with a transverse orientation at 1/3, where is the asperity aspect ratio. The hardness of the shaft surface is 3.15 GPa. 3.1 End-Face Thrust Bearing and the Effect of the Feeding Direction. Consider first the pressure distribution only in the end-face thrust bearing formed by an angular misalignment. The lubricant in the thrust bearing may be fed in two ways: 1 inner feeding where the lubricant is fed along the thrust-bearing recess circumference, creating an outward radial lubricant flow and; 2 outer feeding where the lubricant is fed along the thrust-bearing outer circumference, creating an inward radial lubricant flow. In the simulation, the journal was kept stationary, while the bearing was rotating at 1000 rpm, creating a circumferential relative motion between these two faces. A feeding pressure of 0.1 MPa was applied, the nondimensionalized average thrust-bearing film thickness, h 0, was 0.53, and the nondimensionalized misalignment angle,, was set at Figures 3a and 3c show the results for the cases under the inner feeding, while Figs. 3b and 3d are for the results of the cases under the outer feeding. With the inner feeding, the pressure distributions corresponding to the Reynolds boundary conditions and the cavitation conditions are very similar, referring to Figs. 3a and 3c. However, when the lubricant is supplied from the outer circumference, referring to Figs. 3b and 3d, the thrust bearing under the cavitation condition has a different hydrodynamic pressure distribution than that under the Reynolds condition. With the cavitation condition, the rupture location can be identified precisely. The outer feeding causes a 7% smaller cavitation area, as shown in Fig. 3f, compared to that due to the inner feeding which is shown in Fig. 3e. Note that these figures are plotted in the mapped computational domain. 3.2 Coupling Effect With the Reynolds Boundary Conditions. If the lubricant supply to the coupled bearing comes from the open end of the journal bearing, the thrust bearing is fed along its outer circumference through the interface. Figures 4 and 5 show the pressure distributions for the coupled journal-thrustbearing system obtained under the Reynolds boundary conditions with and without the hydrodynamic coupling effect. The results shown in Fig. 4 are for a case of fully hydrodynamic lubrication without asperity contact, while those shown in Fig. 5 are examples of a system under mixed lubrication with asperity contact. Pressure distributions are plotted in the computational domain: the radial hydrodynamic pressure distribution in the journal-bearing region and the axial or thrust hydrodynamic pressure distribution in the thrust-bearing region. The integration of the pressure in the journal-bearing region yields the radial load and that of the pressure in the thrust-bearing region gives the thrust load. For the case shown in Fig. 4, the journal-bearing eccentricity ratio is 0.8, and the non-dimensionalized average thrust-bearing film thickness is The hydrodynamic pressure in the journal bearing rises towards the open end of the journal bearing, where the film thickness is the minimum. The hydrodynamic pressure in the thrust bearing is generated by the convergent circumferential wedge in the region corresponding to the pressurized side of the journal bearing and is further intensified by the pressurized flow from the journal bearing due to hydrodynamic coupling. When the two bearings are decoupled, they work as two separated bearings although they are geometrically connected. Figure 4a indicates that, when the two bearings are hydrodynamically coupled, the load supported by the journal bearing is increased by 75% in the example. Meanwhile, the load supported by the thrust bearing increases by about 150%. The increased load capacity is a result of the retained internal flow between the journal bearing and thrust bearing due to the hydrodynamic coupling along their common border. Increasing the journal-bearing eccentricity ratio to 0.91 and reducing the non-dimensionalized thrust bearing average film thickness to further boosts the loads supported by each of the bearings. However, asperity contact occurs in the journal bearing, as shown in Fig. 5, even in the case of hydrodynamic coupling. The hydrodynamic coupling produces a 26% increase in the journal-bearing load and a 25% increase in the thrust-bearing load. 3.3 Coupling Effect With the Mass Conserving Cavitation Conditions. Consider now the influence of the mass conserving cavitation conditions on the performance of the journal-thrustbearing system with and without the hydrodynamic coupling effect. The results are plotted in Figs. 6 and 7. In Fig. 6, the eccentricity ratio is 0.8 and the nondimensionalized thrust-bearing average film thickness is while in Fig. 7, the eccentricity ratio is increased to 0.91 and the average thrust-bearing film thickness is reduced to The bearing performance in terms of pressure distributions is similar to the cases shown in Figs. 4 and 5, where the Reynolds boundary conditions are used. Under the cavitation conditions, the increases in load capacity due to hydrodynamic coupling at 0.8 and h are about 76% and 165% for the journal bearing and the thrust bearing, respectively. Whereas the increase in load capacity due to hydrodynamic coupling at 0.91 and h is 26% for the journal bearing and 33% for the thrust bearing. Asperity contact occurs at 0.91 and h These results in comparison with those in the previous section demonstrate that most of the steady-state results obtained under the Reynolds boundary conditions 1 are still valid as the trends in both case studies are similar. However, the application of the cavitation conditions produces a higher load carrying capacity in the thrust bearing. 3.4 Misalignment Effect on Bearing Load Capacities. The load carrying capacity of the coupled journal-thrust-bearing is of the greatest interest to bearing designers. Because the end thrust bearing is formed by the angular misalignment, it is crucial to study the effect of the misalignment angle. In Fig. 8, the performance of the coupled bearing in terms of the nondimensionalized load carrying capacity, or the Sommerfeld number, is plotted Journal of Tribology OCTOBER 2003, Vol. 125 Õ 751

6 Fig. 5 Performance of the journal-thrust bearing with the Reynolds boundary conditions in mixed lubrication Ä0.91, h 0Ä0.263, misalignment angle Ä0.089 : a Hydrodynamic pressure with coupling; b Hydrodynamic pressure without coupling; c Asperity contact pressure with coupling; and d Asperity contact pressure without coupling. as a function of the bearing misalignment angle at an eccentricity ratio of 0.8. The non-dimensionalized average thrust bearing film thickness is chosen to be Under the controlled eccentricity ratio and average thrust bearing film thickness, a larger misalignment angle produces smaller film thickness, intensifies the system hydrodynamic pressure and improves the load carrying capacity of both journal and thrust bearings in the range of the misalignment angles studied, as shown in Fig. 8a. Load ratios as defined by the load capacity with coupling over the load capacity without coupling are plotted in Fig. 8b. The increase in the load capacity of the journalbearing in the cases with coupling is nearly proportional to the corresponding cases without coupling. Therefore, the load ratio is nearly constant at a value of 1.6 with respect to the angular misalignment. The misalignment angle has a much stronger effect on the load carrying capacity of the thrust bearing in the coupled bearing system. As the misalignment angle increases, the benefit of hydrodynamic coupling diminishes, as evident by the decrease in load ratio in Fig. 8b. 3.5 The Effect of Viscosity on the Performance of the Coupled Bearing System. With the improved model, it is now possible to study the flow state in the coupled bearing system with certainty and explore how the cavitation region varies due to the change in lubrication conditions. As viscosity of a lubricant is one of the most crucial parameters that determine the lubrication conditions, the effect of viscosity on the performance of a coupled bearing system is studied. The hydrodynamic pressure and density distributions at a viscosity of 0.03 Pa.s and 0.09 Pa.s are calculated respectively. With the three-fold increase in lubricant viscosity, the load capacity of both the journal and thrust bearings is shown to increase drastically. The load capacity in both the radial direction and the thrust direction increases by more than three times. The density of the lubricant in the bearing system is an indicator of the flow state. When the non-dimensionalized density becomes less than unity, cavitation occurs. In the cavitation region, the density of the fluid/gas mixture appears to be lower for the case with the higher viscosity. Therefore, the higher the lubricant viscosity, the stronger the cavitation becomes. 3.6 The Effect of Feeding Scheme on the Performance of the Coupled Bearing System. There are many different schemes to feed lubricant to the coupled journal-thrust-bearing system. Feeding from the recess circumference of the thrust bearing inner feeding and feeding from the open end of the journal bearing open end feeding are the two most common choices and are studied here. The simulation results show that the hydrodynamic pressure distribution in the thrust bearing has a shape similar to that in Fig. 752 Õ Vol. 125, OCTOBER 2003 Transactions of the ASME

7 6a where the system is fed from the recess circumference of the thrust bearing in addition to the feeding from the feeding holes specified as BC1. The density of the fluid/gas mixture decreases in the non-pressurized region near the outer circumference of the thrust bearing and across the entire journal-bearing width. The trend is reversed when the additional lubricant is supplied from the open end of the journal bearing. In this case, the journalbearing hydrodynamic pressure increases with the reduced cavitation area. Fig. 6 Performance of the journal-thrust bearing including mass conserving cavitation Ä0.8, h 0Ä0.395, misalignment angle Ä0.089 : a With coupling; and b Without coupling. 4 Conclusions A mixed lubrication model capable of handling different hydrodynamic boundary conditions is developed to investigate the lubrication of coupled journal-thrust-bearing systems. A conformal mapping technique is implemented in the model formulation to facilitate a universal flow description. The JFO cavitation algorithm is incorporated into the model to maintain mass conservation in the downstream region. Cases with and without the cavitation algorithm are numerically analyzed and compared. Based on the numerical results, the following conclusions can be drawn: 1. A coupled journal-thrust bearing system can be created by utilizing the end face of the shaft in a journal bearing. The hydrodynamic coupling and the journal angular misalignment are two essential components of the coupled bearing Fig. 7 Performance of the journal-thrust bearing including mass conserving cavitation Ä0.91, h 0Ä0.263, misalignment angle Ä0.089 : a Hydrodynamic pressure with coupling; b Hydrodynamic pressure without coupling; c Asperity contact pressure with coupling; and d Asperity contact pressure without coupling. Journal of Tribology OCTOBER 2003, Vol. 125 Õ 753

8 applied, the thrust bearing yields a notable increase in the load carrying capacity compared to that which is calculated with the Reynolds boundary conditions for the same eccentricity ratio. It is therefore important to consider cavitation conditions in a coupled journal-thrust-bearing system. 4. Increasing the viscosity of the lubricant reduces the fluid/gas mixture density in the cavitation area but it increases the load carrying capacity of both journal and thrust bearings significantly. 5. Feeding through the open end of the journal bearing yields a smaller cavitation area, a higher radial load and a lower thrust load as compared to the case that feeds through the thrust-bearing recess. Acknowledgment The authors would like to express their sincere gratitude to Baker Hughes, Inc. for the financial support and permission to publish the paper. Y. Wang and Q. Wang would also like to thank the US National Science Foundation for support through Grant CMS Fig. 8 Misalignment effect on the performance of coupled journal-thrust including mass conserving cavitation: a Load carrying capacity as a function of the misalignment angle; and b Load ratio as a function of the misalignment angle. system. The hydrodynamic coupling encourages the communication of lubricant flows in the journal and thrust bearings. The increase of the misalignment angle improves the load carrying capacity of both journal and thrust bearings for the same journal bearing eccentricity ratio and average film thickness. The effect of the hydrodynamic coupling is prominent even at zero misalignment. 2. When only the thrust bearing and feeding from the outer circumference of the thrust bearing are considered, the mass conserving cavitation conditions lead to only a minor increase in the thrust load. 3. When the journal and thrust bearings are hydrodynamically coupled and the mass conserving cavitation conditions are Nomenclature L bearing length c journal bearing clearance, m Dm degree of misalignment G cavitation index h compliance, m h 0 thrust bearing average film thickness, m N rotation speed, rev/s p pressure, Pa R radius, m S Sommerfeld number, N/p(R/c) 2 U surface velocity, m/s r(x), z radial and width coordinates, m Greek letters misalignment angle circumferential coordinate, radians journal bearing eccentricity ratio at the mid-plane of bearing misalignment eccentricity ratio max maximum possible roughness, m flow factor bearing angle, radians density, kg/m 3 viscosity, PaS angular velocity, 1/s cavitation parameter, nondimensionalized coordinator asperity aspect ratio misalignment directional angle Subscripts 1,2 due to inner thrust, journal radius c contact, cavitation i due to journal bearing inner radius j journal bearing r,, z radial, circumferential, and width directions rup rupture s oil supply, shear flow t thrust bearing References 1 Tawfik, M., and Stout, K. J., 1981, Combined Radial and Thrust Aerostatic Bearings-A Summary, Paper 13, Proc. 8 th International Gas Bearing Symposium, April, BHRA Fluid Engineering, Cranfield, Bedford, England. 2 Pander, S. S., 1986, Analysis of Tapered Land Aerostatic Bearings for Com- 754 Õ Vol. 125, OCTOBER 2003 Transactions of the ASME

9 bined Radial and Thrust Loads Yates Configuration, Wear, 107, pp Zhang, Q. D., 1999, Design of A Hybrid Fluid Bearing System for HDD Spindles, IEEE Trans. Magn., 35, pp Pander, S. S., and Pandit, M. D., 1986, Analysis of Orifice Compensated Aerostatic Bearings for Combined Radial and Thrust Loads Yates Configuration, 12 th AIMTDR Conference, IIT, Delhi, pp Tieu, A. K., 1991, Hydrodynamic Thrust Bearings: Theory and Experiment, ASME J. Tribol., 113, pp Zang, Y., and Hatch, M. R., 1995, Analysis of Coupled Journal and Thrust Hydrodynamic Bearing Using a Finite-Volume Method, Advances in Information Storage and Processing Systems, conference proceedings, ISPS-Vol. 1, ASME, New York, pp Lie, Y., and Bhat, R. B., 1995, Coupled Dynamics of a Rotor-Journal Bearing System Equipped with Thrust Bearings, Shock and Vibration, 2, pp Wang, Y., Wang, Q., and Lin, C., 2002, Mixed Lubrication of Coupled Journal-thrust Bearing Systems, Computer Modeling in Engineering & Science, 3, pp Patir, N., and Cheng, H. S., 1978, An Average Flow Model for Determining Effects of Three-Dimensionalized Roughness on Partial Hydrodynamic Lubrication, ASME J. Lubr. Technol., 100, pp Wu, C. W., and Zheng, L. Q., 1989, An Average Reynolds Equation for Partial Film Lubrication with a Contact Factor, ASME J. Tribol., 111, pp Pinkus, O., and Bupara, S. S., 1979, Analysis of Misaligned Grooved Journal Bearing, ASME J. Lubr. Technol., 101, pp Vijayaraghavan, D., and Keith, T. G., 1989, Effect of Cavitation on the Performance of a Grooved Misaligned Journal Bearing, Wear, 134, pp Safar, Z. S., 1984, Energy Loss Due to Bearing Misalignment, Tribol. Int., 17, pp Safar, Z. S., and Riad, M. S. M., 1988, Prediction of the Coefficient of Friction of a Misaligned Turbulent Flow Journal Bearing, Tribol. Int., 21, pp Lee, S. C., and Ren, N., 1996, Behavior of Elastic-plastic Rough Surface Contacts as Affected by the Surface Topography, Load and Materials, STLE Tribol. Trans., 39, pp Jacobson, B. O., and Floberg, L., 1957, The Finite Journal Bearing Considering Vaporization, Report No. 190, Chalmers University of Technology, Goteborg, Sweden. 17 Olsson, K. O., 1965, Cavitation in Dynamically Loaded Journal Bearings, Report No. 308, Chalmers University of Technology, Goteborg, Sweden. 18 Elrod, H. G., and Adams, M. L., 1975, A Computer Program for Cavitation and Starvation Problems, Cavitation and Related Phenomena in Lubrication, Mechanical Engineering Publications, New York, pp Elrod, H. G., 1981, A Cavitation Algorithm, ASME J. Lubr. Technol., 103, pp Brewe, D. E., 1986, Theoretical Modeling of the Vapor Cavitation in Dynamically Loaded Journal Bearings, ASME J. Tribol., 108, pp Vijayraghavan, D., and Keith, T. G., Jr., 1989, Development and Evaluation of a Cavitation Algorithm, STLE Tribol. Trans., 32, pp Kumar, A., and Booker, J. F., 1991, A Finite Element Cavitation Algorithm, ASME J. Tribol., 113, pp Payvar, P., and Salant, R. F., 1992, A Computational Method for Cavitation in a Wavy Mechanical Seal, ASME J. Tribol., 114, pp Nehari, Z., 1975, Conformal Mapping, Dover Publications, New York. Journal of Tribology OCTOBER 2003, Vol. 125 Õ 755