Thermal elastohydrodynamic lubrication analysis of the oil film temperature in tapered roller bearings

Tapered roller bearings often experience thermal deformation and stress concentration due to high temperature during prolonged operation. However, the traditional model for calculating bearing heat does not consider the thermal elastohydrodynamic behavior of the lubricant. To address this issue, this paper proposes a finite line contact model based on thermal elastohydrodynamic lubrication analysis for calculating the temperature distribution of the lubricant within tapered roller bearings. The temperature of the lubricant is solved using the column-by-column technique. The results show that the temperature distributions along the thickness of the oil film are symmetrical, with the highest temperature occurring at the middle layer. Furthermore, the temperature at the middle layer increases with both the bearing speed and the concentrated load. These findings provide a theoretical framework for calculating the temperature distribution of lubricated bearings, offering valuable insights for practical applications.


Introduction
Tapered Roller Bearings (TRBs) play a critical role in aircraft engines, automobile transmissions, and industrial machinery due to their ability to handle rotational motion in demanding environments characterized by high speed, temperature, and load.However, these working conditions expose the bearings to thermal deformation and stress concentration over prolonged operation, which significantly affects their performance and lifespan.
Based on the Palmgren frictional heating theory, Gao et al. [1] conducted finite element thermal simulations on angular contact ball bearings under different speeds and loads, but this study neglected the influence of lubricant on heat generation.Jia et al. [2] studied the thermal characteristics of doublerow TRBs used in wind turbine main shafts.They observed that the bearing temperature distribution is influenced by the load distribution.However, the model considering the viscosity-temperature effect of the lubricating grease leads to a slight decrease in overall heat generation and temperature rise in the bearings.Wang et al. [3] systematically studied the thermal characteristics of the contact region in double-row TRBs based on the roller modification.They found that when the modification factor is greater than 0.8, it can eliminate the influence of stress concentration at the ends.Kyrkou et al. [4] conducted fluid-structure interaction simulations to examine the lubricant interface with the bearing inner ring in journal bearings.They explored the impact of two lubricant types on the pressure and temperature distributions at the interface under adiabatic and isothermal conditions.Du et al. [5] simulated the oil film pressure and temperature inside a sliding bearing, considering cavitation, using Fluent software.They observed a weak relationship between the inlet pressure, maximum pressure, and inlet temperature of the oil film with the bearing's rotational speed.
However, the aforementioned literature, in addressing the thermal issues of bearings, does not sufficiently consider the effect of thermal elastohydrodynamic lubricant (EHL) in their models.The purpose of this paper is to calculate the oil film temperature in TRBs based on thermal EHL analysis.It also considers the influence of bearing speed and concentrated load on the oil film temperature.The findings can provide valuable theoretical references for the calculation of temperature fields and other characteristics in tapered roller bearings.

Theoretical model
The tapered roller contact model can be regarded as a finite line contact model consisting of a tapered roller and an infinite plane [6], as shown in Figure 1, where, is the concentrated load, is the length of the roller generating line, is the roller big end radius, is the roller small end radius, so the equivalent cylindrical contact radius , is the roller big end repair radius, is the roller small end repair radius, is the half cone angle, is the rotational speed of the roller, and the roller rotates around the y-axis in the direction of rotation also shown in the Figure 1.The corresponding Reynolds equation is (1) where, is the density of the lubricant, is the pressure of the lubricant, is the film thickness of the lubricant, is the dynamic viscosity of the lubricant, is the entrainment velocity.
The Film thickness equation is (2) where, is the distance given by the approximation of the two bodies in contact, is the geometric film thickness of the lubricant, is the change in film thickness of the lubricant during elastic deformation of the rigid body.The specific expressions are as follows: (3) where, (4) The elastic deformation term of the oil film is (5) where, is the composite elastic modulus.The pressure integral equation of the oil film must satisfy the load balance equation: (6) The viscosity-pressure-temperature equation given by Roelands [7] is (7) where, is the ambient viscosity of the lubricant, the viscous pressure coefficient of the lubricant is taken as 0.6, is the operating temperature, is the ambient temperature of the lubricant.
The density-pressure-temperature equation [8] is (8) where, is the ambient density of the lubricant, the temperature density coefficient D is taken as 0.00065K -1 .
The expression of the energy equation is (9) where, is the specific heat capacity of the lubricant, is the thermal conductivity of the lubricant, is the viscous heat dissipation of the lubricant, its expression is (10) where, where, and are the velocities of the raceway and the roller body, respectively.Using the velocities and to solve for the velocity along the film thickness direction through the continuity equation: (13)

Boundary conditions
The pressure boundary conditions are (14) where, , , and are the boundary coordinates of the computational domain, which are the inlet and outlet of the lubricant along the x-axis, and the boundary coordinates in the y-axis direction, respectively.
The thermal boundary conditions are where, and are the thermal conductivity of the raceway and the roller body, and are the density of the raceway and the roller body, and and are the specific heat capacity of the raceway and the roller body, respectively.

Numerical solution methods
In order to make the calculation process easy, the above equations need to be dimensionless and discretized.Afterwards, we use the multigrid method [9] to solve for the oil film pressure , use the multilevel multi-integration method [10] to solve for the elastic deformation term in equation (5).The balance of load and pressure is satisfied by adjusting in equation (2).The number of mesh layers divided by the methods is 3, and the convergence accuracy of pressure and load is 1e-3.The columnby-column technique [11] is used to solve for the temperature of the lubricant.The temperature distribution along the X-axis of the lubricant flow direction and along the Z-axis of the film thickness direction is solved, and then the column-by-column scanning along the tapered roller's generating line direction is used, to obtain the temperature distribution of the whole oil film with different layers along the film thickness direction.The number of nodes divided along the film thickness direction in this method is 5, and the convergence accuracy of temperature is 1e-3.

Results and discussion
The properties of the lubricant and solids used in this paper are: , , , , and the computational domain boundaries are: , , , .

Characteristic analysis of the oil film temperature distributions
The temperature of the oil film is solved for different layers along the film thickness direction, and the results of the solution are shown in Figures 2 to 6.It can be seen from the figures that the temperature of the lubricant is not high in the X-inlet region and the X-outlet region, and the temperature rises faster in the X-middle region, which is because the lubricant produces a large amount of dissipative heat under the violent extrusion and shearing effects in the X-middle region.From the figures, it can be found that the temperature at the two ends of the upper and lower walls along the Y direction, i.e., the big and small ends of the roller, has a larger peak compared with that in the Y-middle region, which is caused by the end leakage of the roller.The lubricant, due to the high pressure in the Y-middle region, flows out from the region to the big and small ends along the Y direction, taking away part of the heat, resulting in a decrease in the temperature of its Y-middle region.
The oil film temperature distribution of each layer at the nondimensional Y = 0 of the tapered roller is shown in Figure 7.It can be seen that the temperature at the middle layer is the highest, while the temperature at the upper and lower walls shows symmetrical distribution.The longitudinal section is made at the nondimensional X = 0, and the temperature distribution of each layer of the longitudinal section is obtained, as shown in Figure 8, it is observed that the oil film temperature of the roller big end appears higher, which is due to the bigger Hertzian contact area at the big end than that at the small end, where the lubricant accumulates and its thermal effect acts more strongly, and this effect in this working condition exceeds that effect by the pressure at the small end is bigger than that at the big end.

Effects of concentrated load.
Changing the magnitude of different concentrated loads, the oil film temperature distribution of each layer at the nondimensional Y = 0 and the nondimensional X = 0, respectively, is calculated, as shown in Figures 9 and 10, from which it can be seen that the temperature of each layer of the lubricant increases as the concentrated load increases, which is due to the increase in the oil film pressure, which intensifies the effects of shearing and squeezing of the lubricant, causing the overall oil film temperature to rise.

Effects of bearing speed.
Changing the size of different bearing speed parameters, the oil film temperature distribution of each layer at the nondimensional Y = 0 and the nondimensional X = 0 is calculated, respectively, as shown in Figure 11 and Figure 12, from which it can be seen that the oil film temperature of each layer increases with the increase of bearing speed, which is due to the flow speed of the lubricant is accelerated, which intensifies the collision and extrusion effects between the lubricant and the solids, and increase the overall temperature of each layer of the oil film.

Conclusions
In this paper, a finite line contact EHL model considering thermal effect is built.The control equations including the energy equation are dimensionless and discretized, and the oil film temperatures of five layers are calculated using the column-by-column technique: the lower wall temperature, the near lower wall temperature, the middle layer temperature, the near upper wall temperature and the upper wall temperature.It is found that the temperatures are symmetrically distributed along the film thickness direction as well as the temperature at the middle layer is the highest, and the temperature at the middle layer increases with the increase of speed and load.

Figure 1 .
Figure 1.Tapered roller and infinite plane contact schematic.

Figure 2 .
Figure 2. Temperature distribution on the lower wall.

Figure 8 .
Figure 8. Temperature distribution of each layer in the longitudinal section.

Figure 9 .
Figure 9. Temperature distribution of transverse section for different loads.

Figure 10 .
Figure 10.Temperature distribution of longitudinal section for different loads.

Figure 11 .
Figure 11.Temperature distribution of transverse section for different speeds.

Figure 12 .
Figure 12.Temperature distribution of longitudinal section for different speeds.