Thermohydrodynamic analysis of a worn plain journal bearing

Tribology International 37 (2004) 129 136 www.elsevier.com/locate/triboint Thermohydrodynamic analysis of a worn plain journal bearing M. Fillon, J. Bouyer Université de Poitiers, Laboratoire de Mécanique des Solides, U.M.R., C.N.R.S 6610, SP2MI, Bd Pierre et Marie Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France Abstract Hydrodynamic journal bearings are widely used in industry because of their simplicity, efficiency and low cost. They support rotating shafts over a number of years and are often subjected to many stops and starts. During these transient periods, friction is high and the bushes become progressively worn, thus inducing certain disabilities. This paper seeks to present the thermohydrodynamic performance of a worn plain journal bearing. The study deals with a 100 mm diameter bearing, submitted to a static load varying from 5000 to 30,000 N with a rotational speed varying from 1000 to 10,000 rpm. The defects caused by wear are centered on the load line and range from 10% to 50% of the bearing radial clearance. Our main focus was on hydrodynamic pressure, temperature distributions at the film/bush interface, oil flow rate, power losses and film thickness. Defects caused by wear of up to 20% have little influence on bearing performance whereas above this value (30 to 50%) they can display an interesting advantage: a significant fall in temperatures, due to the tendency of the bearing to go into the footprint created by the wear. Thus, the worn bearing presents not only some disadvantages but also advantages, such as lower temperature, since in certain cases of significant defects due to wear the geometry approaches that of a lobe bearing. 2003 Elsevier Ltd. All rights reserved. Keywords: Hydrodynamic bearing; Wear; Thermal effects 1. Introduction Wear in journal bearings is a phenomenon that often occurs in bearings that have been working over long periods (10 years), on huge machinery. Investigation into this phenomenon, which is more prevalent than might have been thought, was first undertaken within the industry because it was a real problem for bearing users. In 1957, Duckworth and Forrester [1] analyzed wear in lubricated bearings. Forrester [2] and Katzenmeier [3] analyzed several examples of damage due to wear quantitatively as well as qualitatively. Another study concerning the problem of wear was led by Dufrane et al. [4] in 1983 who analyzed a worn bearing in a steam turbine. They were the first to propose a geometrical model taking into account the worn region of a bearing, in order to include it in calculations. They paid particular attention to the mechanisms that lead to wear, for a bearing operating at low speed. They showed that wear most Corresponding author. Tel.: +33-5-49-49-65-43; fax: +33-5-49-49-65-04. E-mail address: fillon@lms.univ-poitiers.fr (M. Fillon). often occurs symmetrically on the bottom of the bearing, and that even bearings equipped with an hydrostatic pocket were subjected to wear if the pocket was badly dimensioned. The first people to be interested in the consequences of a wear defect were Hashimoto et al. [5]. They analyzed the influence of wear defect on the pressure field and on the eccentricity ratio and showed that wear defect damages bearing stability and that weak L/D ratio bearings were less sensitive to a defect. Vaidyanathan and Keith [6] studied the performance of four geometrically different bearings, one of which was a worn bearing. They were interested in the influence of wear defect on parameters such as friction, pressure or the Sommerfeld number compared to the eccentricity ratio. In 1991, Scharrer et al. [7] worked on the dynamic coefficients of a hydrostatic bearing and showed that small wear defects have only a slight influence on bearing performance. The stability of a worn bearing was then analyzed by Suzuki and Tanaka [8] in 1995. The last work done on the topic is that of Kumar and Mishra [9] who made a study approaching that of Suzuki et al.; they showed that the defect decreases the stability of the hydrostatic bearing when it is submitted to a light load. In the same year, they continued with their work [10] 0301-679X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/s0301-679x(03)00051-3

130 M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 Nomenclature C radial clearance (m) C p fluid specific heat (J/kg K) D bearing diameter (m) D b bush outer diameter (m) d d defect depth (m) e eccentricity of the shaft (m) g switch function h film thickness (m) h 0 unworn bearing film thickness (m) K thermal conductivity of the fluid (W/m K) L bearing length (m) R shaft radius (m) R b bush radius (m) R b,q,z b cylindrical coordinates in the bush (m, rad) T fluid temperature ( C) T b bush temperature ( C) u,v,w velocity components in the fluid film (m/s) x,y,z cartesian coordinates (m) d d dimensionless defect depth f attitude angle (rad) g bulk modulus in Elrod s model (Pa) m fluid dynamic viscosity (Pa s) r fluid density (kg/m 3 ) q,y,z shaft coordinates system (rad, m) q b beginning angle of the footprint (rad) q e end angle of the footprint (rad) q g Angular extent of groove (rad) density ratio in the full-film region and mass fraction in the rupture zone w angular velocity of the shaft (rad/s) by analyzing noncircular bearings in turbulent flow submitted to a wear defect in steady-state conditions. They concluded that wear increases the friction as well as the flow rate and reduces the load capacity of the bearing. The few studies carried out on worn bearings are essentially theoretical and do not take into account the thermal effects. The aim of this work is to analyze the influence of a wear defect ranging from 10% to 50% of the bearing radial clearance on the characteristics of the bearing such as the temperature, the pressure, the eccentricity ratio, the attitude angle or the minimal thickness of the lubricating film. 2. Thermohydrodynamic analysis 2.1. Basic equations and boundary conditions The thermohydrodynamic study presented here was carried out on a plain journal bearing including one axial feeding groove. The corresponding thermal analysis has already been presented elsewhere by Pierre et al. [11,12]. The pressure field was determined by resolving the generalized Reynolds Eq. (1), by using the Elrod Algorithm [13] in laminar regime. This algorithm is based on mass conservation and takes into account the rupture and reformation of the fluid film, thus defining both an active and an inactive zone. 1 R q ggg q z ggg w z q I 2 2 J 2 (1) with y G h m y I h h 2 y I J 2 2 dy m dy J dy 2 m. 0 0 In this equation, is the variable introduced by Elrod; it represents the local proportion of fluid in the inactive zone and the density ratio in the active zone. The compressibility variation in the active zone is characterized by g. The temperature field in the lubricant film is calculated from the energy Eq. (2). 0

rc T P u v T w T x y z K 2 T y m u 2 y w y 2. (2) 2 In the inactive zone, it is necessary to determine equivalent characteristics of the air lubricant mixture such as the dynamic viscosity, the volumic mass, the specific heat or the thermal conductivity coefficient. The analysis also takes into account the heat transfer by conduction within the bush. The temperature field in the solid is determined by solving the Laplace Eq. (3) written in cylindrical coordinates: M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 131 2 T b r 2 b 1 T b 1 2 T b r b r b r 2 b q 2 T b 0. (3) 2 z 2 b To solve the thermal problem, we had to impose realistic boundary conditions. The inlet temperature was calculated by means of a balance of heat fluxes in the inlet zone, which takes into account the mixing of the feeding and recirculating oil. The continuity of heat flux was applied at the film bush interface and the fluid temperature was considered to be the same as that of the bush. Due to the fact that the shaft was rotating, its temperature was considered to be constant along q but varying axially. It was determined by assuming that the balance of the thermal flux at the film shaft interface is null. Finally, the external surface of the bush was considered as being submitted to a free convection thermal exchange with the external environment. All these equations were solved numerically by a finite difference method and an iterative process. 2.2. Worn zone model To model the worn zone, we used the geometry defined by Dufrane et al. [4], which supposes that the footprint created by the shaft in the bush is centered to the vertical direction (load direction). The value of the maximal defect is such that it represents 50% of the radial clearance of the bearing. The change in bush geometry is given by Eq. (4): dh d d C Ccos(q). (4) The dimensionless variable is used for calculations: d d ( = d d /C) which corresponds to the percentage of the wear defect in relation to the radial clearance. The defect value is added to the calculated film thickness (h = h+dh) with: dh C [d d 1 cos(q)] for q b q q e (5) dh 0 for q q b or q q e. The angles (q b ) and (q e ), corresponding to the beginning and the end of the footprint, respectively, are calculated from the equation: cos(q) = d d 1. Fig. 1. Worn bearing geometry. Fig. 1 gives a representation of the bearing geometry: O b is the bush center, O the shaft center, e the eccentricity and W the static load. 3. Results The numeric model described in Section 2 has been validated by comparison with results found in the literature. A good correlation has been observed with the published results of Hashimoto et al [5]. Further comparison has been made with the results of Vaidyanathan and Keith [6], which confirmed the accuracy of our numeric calculations. The aim of this section is to analyze the influence of a wear defect varying from 10% to 50% of the radial clearance of the bearing on several operating characteristics. These, as well as bearing characteristics, are given in Table 1. In this study, we have used the extreme configurations studied by Pierre et al. [11,12]: Table 1 Bearing and lubricant characteristics Bearing diameter D [mm] 100 Bearing length L [mm] 100 Radial clearance at 20 C C [µm] 75 Outer bush diameter D b [mm] 200 Angular extent of groove q g [ ] 18 Rotational speed N [rpm] 1000 & 10,000 Load W [N] 5000 & 30,000 Oil supply temperature T s [ C] 40 Oil supply pressure P s [MPa] 0.04 Lubricant viscosity at 40 C m 1 [Pa s] 0.03 Lubricant viscosity at 100 C m 2 [Pa s] 0.00405

132 M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 low speed (5.2 m/s) or high speed (52 m/s) and low specific pressure (0.5 MPa) or high specific pressure (3 MPa). Our main focus will be on the influence of wear on pressure, temperature and film thickness as well as on the axial flow rate, eccentricity ratio, attitude angle and dissipated power. 3.1. Influence on pressure The pressure field is modified when the bearing has a wear defect, as can be seen in Fig. 2, which represents the pressure fields without defect and with defects of 30% and 50% of the bearing radial clearance, for the four operating conditions. The wear defect leads to an extension of the active zone of the film; it remains very weak in the case of a high eccentricity of the shaft center (low speed and high load). For the 1000 rpm and 5000 N case, there is a fall in pressure in the divergent zone created by the wear defect, which reaches 0.15 MPa for a 50% wear defect. The same effect can be observed, although much more pronounced, when the rotational speed is high (10,000 rpm) with a fall in pressure to a null pressure, located in the divergent zone created by the defect. However, the defect also creates a second convergent zone: the bearing behaves in the same way as a two-lobe bearing, with two peaks of pressure. For the highly loaded cases, the influence of the divergent zone on the pressure field is small but the wear leads to a substantial fall in maximum pressure for the low speed (1.68 MPa for the 50% defect) and to a significant increase in it for the high speed. These results are given in Fig. 3, which gives the maximum pressure variation as a function of the wear degree of the bearing, ranging from 0% to 50% of the radial clearance. The maximum pressure increases with the wear defect, except for the low speed and heavy load case, where it decreases by 18%. In the other cases, an increase of 49% for the case 1000 rpm and 5000 N, 112% for the case 10,000 rpm and 5000 N and 26% for the case 10,000 rpm and 30,000 N can be seen. This can be explained by the fact that the 1000 rpm and 30,000 N case is the only one that leads to a slight increase in the minimum film thickness when the defect increases. The greater the wear defect, the greater the eccentricity of the bearing which tends to go into the defect footprint, so the film thickness increases in the minimal thickness zone and decreases in the convergent zone: this is Fig. 2. Pressure field in the mid-plane of the bearing for several cases of defect.

M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 133 3.2. Influence on temperatures Fig. 3. Maximum pressure versus relative worn defect. characterized by a better distribution of the pressure field (decrease of the peak). The pattern of the temperature fields is only slightly modified by the presence of a wear defect. Fig. 4 gives the temperature fields without defect and with defects of 30% and 50% of the bearing radial clearance for the four cases studied. The wear defect of 50% leads to a systematic decrease in temperatures, up to 5 K for the case at high speed and heavy load, and up to 13 K for the light load case. When the eccentricity of the shaft is small, the film thickness increases, particularly the upstream minimum thickness, as a consequence of the wear defect: this leads to a significant decrease in shear rate and therefore less heating. The cooling of the 30% worn bearing is less, the decrease in temperature is not significant, and in several cases a slight increase in temperature can be observed. At low speed, there is a slight increase in the maximum temperature as can be seen in Fig. 5 which represents the variation in maximum temperature as a function of the wear defect magnitude. For high speed cases, the wear defect leads to a decrease in temperature as soon as the defect exceeds 10%. The wear defect of 50% leads to a fall in maximum temperature for all cases, especially for the 10,000 rpm and 5000 N case for which it reaches 7 K. For this case, the shaft Fig. 4. Temperature field in the mid-plane of the bearing for several cases of defect.

134 M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 Fig. 5. Maximum temperature versus relative worn defect. temperature, which is representative of the average operating temperature, decreases by 13 K. 3.3. Influence on film thickness Fig. 6 shows the evolution of the film thickness in the median cross-section of the bearing as a function of the value of the wear defect, for several operating conditions. As could have been predicted, the film geometry is significantly modified by the presence of a wear defect, with abrupt changes near the footprint. When the bearing is lightly loaded, this effect is more pronounced, and even more so if the speed is high. A decrease in the minimum film thickness and a break of slope can be observed, which are so pronounced in the case of a high speed and light load that the geometry becomes comparable to that of a two-lobe bearing: this configuration creates two distinct zones in the pressure field (Fig. 2). The minimum film thickness, presented in Fig. 7, decreases with the increase in defect except for the case 1000 rpm and 30,000 N, where it increases from 16.8 µm to 18.1 µm (8%). For the extreme case (10,000 rpm Fig. 6. Film thickness in the mid-plane of the bearing for several cases of defect.

M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 135 Fig. 7. Minimum film thickness versus relative worn defect. Fig. 8. Axial flow rate versus relative worn defect. and 30,000 N), it decreases by 5.3 µm (15%). This decrease goes up to 7.5 µm and 8.9 µm (12.5% and 18%) for the speeds of 1000 rpm and 10,000 rpm, respectively. The higher the speed, the greater the decrease in minimal film thickness. In the case of a high eccentricity of the shaft center (1000 rpm and 30,000 N), the increase in minimum film thickness and especially the decrease in the convergent slope lead to a better pressure distribution and therefore to a fall in the pressure peak (Fig. 2). 3.4. Influence on flow rate When the load is light, the increase in the axial flow rate is significant, as shown in Fig. 8. It increases by 51% and 80% for speeds of 1000 rpm and 10,000 rpm, respectively, for a static load of 5000 N. For a load of 30,000 N, the increase is less pronounced, 9% and 29% for the speeds of 1000 rpm and 10,000 rpm, respectively. 3.5. Influence on eccentricity Fig. 9. Eccentricity ratio versus relative worn defect. The behavior of the eccentricity ratio is the same as for the flow rate as can be seen in Fig. 9. For a light load, it increases by 51% for a speed of 1000 rpm and by 86% for a speed of 10,000 rpm, for the larger wear defect. When the load is high, the increase is only 41% and 33% for the speed of 1000 rpm and 10,000 rpm, respectively. It should be noted that for the highly loaded and low speed case, where the increase is 41%, the eccentricity ratio goes from 0.78 to 1.1 for the 50% defect: the shaft goes into the footprint when the wear defect is significant. 3.6. Influence on dissipated power As can be seen in Fig. 10, the dissipated power increases with the defect for high speed. There is an increase of 21% for the 5000 N case and 5% for a load of 30,000 N, with the 50% wear defect. For the low speed cases and for a 50% defect, it decreases by 4% and 8% for loads of 5000 N and 30,000 N, respectively. The 30% wear defect, in the 1000 rpm and 30,000 N case, leads to a greater decrease in dissipated power, which decreases by 11%. In high speed cases, the

136 M. Fillon, J. Bouyer / Tribology International 37 (2004) 129 136 Fig. 10. Dissipated power versus relative worn defect. decrease in temperature in the fluid film leads to an increase in the dynamic viscosity and therefore in the dissipated power, although the film thickness is increased. At low speed, the thermal effects are weaker so the friction torque decreases as a consequence of the increase in film thickness. 4. Conclusion The wear defect would appear to be a drawback for plain journal bearing users. However, the conclusions of this study show that wear defect could lead to an increase in thermohydrodynamic performances. For highly loaded bearings operating at low speed, critical parameters are film thickness and maximum pressure: the wear defect allows the pressure peak to be minimized whilst maintaining or appreciably increasing the film thickness. The dissipated power is also decreased. For high speed operating bearings submitted to a light load, the shaft being almost centered, heating becomes very significant and maximum temperature becomes the critical parameter. The wear defect leads to a significant fall in maximum temperature and also to a decrease in average operating temperature. However, the dissipated power increases slightly. Moreover, it should be noted that several authors have shown that the dynamic stability of a worn bearing is, in certain cases (wear defect greater than 30% and high load), better than that of an unworn one. It would be interesting to continue by studying the influence of the wear defect on the behavior of the journal bearing during transient phases, such as, for example, start-up or stopping. References [1] Duckworth WE, Forrester PB. Wear of lubricated journal bearings. In: Proceedings of the Institution of Mechanichal Engineers Conference on Lubrication and Wear, London. 1957. p. 714 9. [2] Forrester PB. Bearing and journal wear. In: Symposium on Wear in the Gasoline Engine, Thornton Research Center. 1960. p. 75 91. [3] Katzenmeier G. The influence of materials and surface quality on wear behaviour and load capacity of journal bearings. In: Proceedings of the Institution of Mechanichal Engineers Tribology Convention. 1972. p. 80 4. [4] Dufrane KF, Kannel JW, McCloskey TH. Wear of steam turbine journal bearings at low operating speeds. Journal of Lubrication Technology 1983;105:313 7. [5] Hashimoto H, Wada S, Nojima K. Performance characteristics of worn journal bearings in both laminar and turbulent regime. Part 1: Steady-state characteristics. ASLE Transactions 1986;29:565 71. [6] Vaidyanathan K, Keith TG. Performance characteristics of cavitated noncircular journal bearings in the turbulent flow regime. Tribology Transactions 1991;34:35 44. [7] Scharrer JK, Hecht RF, Hibbs RI. The effects of wear on the rotordynamic coefficients of a hydrostatic journal bearing. ASME Journal of Tribology 1991;113:210 3. [8] Suzuki K, Tanaka M. Stability characteristics of worn journal bearing. In: Proceedings of the Asia Pacific Vibration Conference, Kuala Lumpur. 1995. p. 296 301. [9] Kumar A, Mishra SS. Stability of a rigid rotor in turbulent hydrodynamic worn journal bearings. Wear 1996;193:25 30. [10] Kumar A, Mishra SS. Steady-state analysis of non-circular worn journal bearings in non-laminar lubrication regimes. Tribology International 1996;29:493 8. [11] Pierre I, Bouyer J, Fillon M. Thermohydrodynamic study of misaligned plain journal bearings comparison between experimental data and theoretical results, Proceedings of the 2nd World Tribology Congress, 2002, International Journal of Applied Mechanical Engineering 7 (3) (in press). [12] Pierre I, Fillon M. Influence of geometric parameters and operating conditions on thermohydrodynamic behavior of plain journal bearings. Proc. Instn Mech. Engrs, Part J, Journal of Engineering Tribology 2000;214:445 57. [13] Elrod HG. A cavitation algorithm. ASME Journal of Lubrication Technology 1981;103:350 4.