This paper developed a coupled model, incorporating the quasistatic model, fatigue life model, and mixed lubrication model, to investigate the effect of misalignment angle on high-speed cylindrical roller bearings. The model is verified by comparing with the published literature results. Then, a parametric analysis is carried out. The results show that as the misalignment increases, the load distribution is basically unchanged, but the fatigue life of the roller bearing decreases due to the variation of contact pressure, and the skewing moment of single roller contact pair increases. Furthermore, the optimal design of roller profile needs to consider the effect of lubrication in order to improve the fatigue life of roller bearings. In general, the optimal crown drop is too small according to the design from the slicing technique.
Cylindrical roller bearings are often used in high-speed engine applications. In these applications, cylindrical roller bearings are very sensitive to the misalignment of the ring which can lead to the reduction of fatigue life and the cage instability. However, in some demanding applications, the misalignment can be easily caused by improper assembly or shaft bending under load. Thus, it is necessary to study the performance of high-speed cylindrical roller bearings under misalignment and put forward some corresponding optimal designs.
An early study about the effect of misalignment on fatigue life of cylindrical roller bearings was investigated by Harris [
The object of this work is to investigate the effect of misalignment on the lubrication and life of high-speed cylindrical roller bearings. The coupled quasistatic and mixed lubrication model will be applied to evaluate the load distribution, lubrication condition, fatigue life, and skewing moments of the single roller contact pair under a misaligned operating condition. Some optimum geometric conditions for reliable operation of high-speed cylindrical roller bearings under misalignment will be established.
In this part, the coupled model, the quasistatic model, and mixed lubrication model will be described. First, the quasistatic model is used to calculate the load distribution among the rolling elements and the magnitude of misalignment of the individual roller. Then, these results are applied to the mixed lubrication model which considers the effect of roughness and elastoplastic deformation of asperities. By the mixed lubrication model, the film thickness distribution along the roller axial directions is calculated. Last, the fatigue life considering lubrication will be evaluated by the fatigue life model. The solution process of the couple model is illustrated in Figure
Flowchart of numerical solution.
In this section, the quasistatic model developed by Liu [
Suppose that the contact is divided into
The roller profile.
Figure
Misalignment of bearing inner ring.
Due to radial displacement
Designate the roller tilt angle as
Thus, the load intensity on the roller at
The static equilibrium of the roller is
The equilibrium equation of the bearing is
These nonlinear equations can be solved by the Newton–Raphson iterative technique. The detailed modeling process and numerical solution can be found in reference [
According to [
The basic of load rating of a slice is expressed as follows:
The equivalent load of a slice is calculated as
Thus, the basic reference rating life,
Zaretsky et al. [
The effect of film parameter on bearing fatigue life.
It should be noted that when the film parameter is less than 0.6, the predicted error may increase. In this situation, the life of bearings mainly depends on the properties of the boundary films. Thus, the fatigue life may be more reliable when the film parameter is larger than 0.6 in the current model.
The modified Reynolds equation derived by Patir and Cheng [
For an isotropic surface with Gaussian distribution of the surfaces heights, the pressure and the shear flow factors are given by Patir and Cheng [
For the Gaussian distribution of the surfaces heights,
In equation (
The lubricant film thickness
The elastic-plastic microcontact model developed by Zhao et al. [
The total load
In order to evaluate skewing moment more accurately, the friction coefficient in each contact grid needs to be calculated. Based on Johnson et al.’s load-sharing concept [
The film shear stress
The skewing moment is caused by nonuniform shear stress, as shown in Figure
Skewing moment.
Once these three models are connected, the coupled model outputs the following useful variables to evaluate the tribological performance of high-speed cylindrical roller bearings under misalignment: The inner/outer ring contact load in different misalignment angles The film pressure/thickness profile in different misalignment angles The fatigue life of high-speed cylindrical roller bearing under misalignment The skewing moment of single roller contact pairs under misalignment
The coupled model mainly includes two parts: quasistatic model and mixed lubrication model under misaligned operating conditions. The results of the quasistatic model are compared with the results from reference [
Verification of the quasistatic model.
A high-speed cylindrical roller bearing used in a helicopter turbine engine is analyzed. Its geometrical and material parameters are listed in Table
The related data of cylindrical roller bearing.
Inner raceway diameter, mm | 51.92 |
Outer raceway diameter, mm | 79.95 |
Roller diameter, mm | 13.99 |
Roller effective length, mm | 14.68 |
Number of rollers | 12 |
Roughness of raceway, | 0.05 |
Roughness of roller, | 0.06 |
Elastic moduli of roller and rings, MPa | 206000 |
Poisson’s ratios of roller and rings | 0.3 |
Density of rollers, kg/m3 | 7850 |
Figures
Outer raceway contact load versus roller location.
Contact load variation along roller axial direction.
The effect of misalignment on fatigue life (without considering the effect of lubrication).
In this section, the results from the heaviest loaded roller are analyzed. The basic operating parameters are listed in Table
The basic operating parameters in EHL analysis.
Misalignment angle | Load (N) | Roller relative misalignment angle | The equivalent radius (mm) | Velocity (m/s) | |
---|---|---|---|---|---|
0′ | Inner contact | 3073 | 0 | 5.512 | 16.48 |
Outer contact | 3158 | 0 | 8.482 | 16.48 | |
2′ | Inner contact | 3064 | 1.01 | 5.512 | 16.48 |
Outer contact | 3149 | 0.99 | 8.482 | 16.48 | |
4′ | Inner contact | 3034 | 2.01 | 5.512 | 16.48 |
Outer contact | 3133 | 1.99 | 8.482 | 16.48 |
The effect of misalignment on the contact pressure and film thickness.
Figure
The effect of lubrication on film parameter and asperity-carrying load.
Figure
The effect of lubrication on fatigue life.
Figure
The effect of misalignment on skewing moment.
From the above analysis, it can be found that in large misaligned operating condition, the fatigue life would be greatly reduced due to lubrication failure. In order to improve this situation, roller profile is an effective method. Lots of theoretical and experimental works [
The effect of crown drop on fatigue life.
In this study, a coupled model for evaluating the lubrication condition and fatigue life of high-speed cylindrical roller bearings under misaligned operating conditions was established. This model incorporates the quasistatic model, fatigue life model, and mixed lubrication model. A parametric analysis was carried out to investigate the effect of misalignment on load distribution, fatigue life, and skewing moment. Some conclusions are drawn as The misalignment of rings has little influence on load distribution. However, it can greatly reduce fatigue life. Lubrication plays an important role in operating the roller bearing reliably. For the misaligned operating conditions, the minimum film thickness occurs in the load-carrying end of the roller. As the misalignment angle increases, the minimum film thickness decreases. In current analysis, when the misalignment angle is larger than 6 minutes, the minimum film thickness is equal to zero. The optimal design of roller profile needs to consider the influence of lubrication. If the effect of lubrication is not considered, the designed crown drop may be too small. As the misalignment angle increases, the skewing moment of single roller contact pair increases.
All data generated or analyzed during this study are included in this article. The code used during the current study is available from the corresponding author on reasonable request.
The authors declare that they have no potential conflicts of interest with respect to the research, authorship, and publication of this article.
This study was financially supported by the Scientific Research Foundation of Changzhou University.