Tolerance Analysis of Cylindrical Roller Bearing under Combined Radial and Axial Loads

. In order to analyze the infuence of tolerance values of key parameters for cylindrical roller bearings under combined axial and radial loads, a coupled model, incorporating a dynamic model of cylindrical roller bearings, contact model, and fatigue life model, is developed to investigate the efect of fange angle, roller-end radius, interval of roller length gauge, and roller profle on contact performance and fatigue life. Te results show that the grouping design of fange angle and roller-end radius in the tolerance range was helpful for reducing contact ellipse truncation. Te diference of the roller length would change the axial load distribution of the bearings. For the longest roller located in the bottom position, the bigger the diference, the bigger the roller tilt angle and carried-axial load. Te (0, +2.5 μ m) tolerance range of the crown drop can limit the diference of the fatigue life within 20% in the current analysis.


Introduction
Cylindrical roller bearings can support axial loads if they are used as semi-locating or locating bearings. Tese axial loads are transferred via fanges and roller-end faces. Terefore, the design of the fange and roller-end geometry is very important which directly afects the contact condition and determines the axial load carrying capacity of the bearing. Many bearing manufacturers [1][2][3] and scholars [4][5][6] have tried to improve the design. Some geometrical optimization of fange-roller end contact truly improves the axial load carrying capacity. However, the manufaturing tolerance is inevitable. Te results from reference [4] have shown that the location of fange-roller end contact points was sensitive to the variation of fange angle and roller-end spherical radius, and the location of contact points would afect sliding velocity and roller tilt, further afecting heat generation and fatigue life. In addition, the interval of roller length gauge and roller profle tolerance also afects the axial load distribution and anti-tilt capacity. Terefore, tolerance analysis around the optimal design would be signifcant in predicting the life of the bearing. Studies [7][8][9][10][11][12][13][14][15] investigating the efect of manufaturing error on bearing performance have been widely conducted. Chen et al. [7,8] analyzed the efect of of-sized roller and its arrayed order on load distribution in a cylindrical roller bearing (CRB). It was shown that one of-sized roller has a signifcant infuence on load distribution. Tong and Hong [9] examined the efect of distributed roller diameter error on the fatigue life of tapered roller bearings. Te results showed that the mean value of the fatigue life exhibits a signifcant decrease with increasing roller diameter deviation. Yu et al. [10,11] found that the amplitude and order of the roundness error in the inner and outer raceways have a signifcant impact on the motion of the CRB. Ono and Takahasi [12,13] analyzed the efect of outer ring waviness on ball bearing vibrations. Fujiwara and Yamauchi [14] defned the tolerance of logarithmic crowning applicable to cylindrical bearing rollers. Liu et al. [15] proposed an analytical method for calculating the friction torque of needle roller bearings with roundness error.
In summary, some research eforts were made to relate the manufacture error with bearing contact and dynamic performance. However, little work has been done on the CRBs under combined axial and radial loads, and most of the work focused on a single tolerance factor. In practice, different kinds of rolling bearing have their specifc concerns. For example, for CRBs under combined axial and radial loads, the match between fange angle tolerance and rollerend radius tolerance requires additional attention. Recently, the authors have developed a dynamic simulation model of CRB under combined axial and radial loads [16], which incorporates a contact solver with skew angle consideration. Tis model will be further developed in the current article to further include the evaluation of fatigue life. Te authors hope that the efect of tolerance on the fatigue life of CRB under axial and radial loads would be thoroughly analyzed by the extended model. It is also hoped that this theoretical analysis can provide some insights into the CRB design for engineers.

Brief Description of the Model
Te model used in the current analysis is extended from the model developed by Wang et al. [16]. Some key features of the previous model are as follows. (a) A dynamic simulation model of CRBs under combined axial and radial loads includes the friction factor and can be applied to evaluate roller skew angle. (b) A contact solver with the consideration of skew angle was developed to analyze contact pressure distribution. Here, in order to investigate the efect of manufacture tolerance on the fatigue life of CRB, the fatigue life model of CRB is added. Te numerical solution scheme is shown in Figure 1, and for more numerical details, the readers are referred to [16].

Fatigue Life
Model. According to [17,18], the L 10 fatigue life of a roller-raceway line contact subjected to normal load Q may be estimated by where L 10 is in millions of revolutions and Q c is the basic dynamic load rating of inner or outer ring.
where D is the roller diameter, c � D/dm, and dm is the pitch diameter of the bearing. Te upper signs refer to the inner ring, and the lower signs refer to the outer ring. Te basic of load rating of a slice is expressed as follows: Te equivalent load of a slice is calculated as where p ij denote the maximum contact pressures at the i th slice and j th roller of inner or outer race, which are available from the pressure analysis. Tus, the basic reference rating life, L 10r , is It should be pointed out that the fatigue life introduced in reference [17] is applied in the conditions of time-invariant load distribution. Here, the internal load distribution of cylindrical roller bearings varies periodically due to tolerance. Te Palmgren-Miner linear cumulative damage theory can be used to evaluate fatigue life of cylindrical roller bearings [9].
where N denotes the number of load cases.

Results and Discussion
In this section, the authors attempt to design fange angle tolerance, roller-end radius tolerance, interval of roller length gauge, and roller profle tolerance for a NJ213ECP cylindrical roller bearing. Te basic geometry and operation parameters are shown in Table 1, which are identical to those in reference [16]. Here the fange angle υ and roller-end radius R s are assumed to be 30′ and 650 mm, respectively, which are considered to be in the reasonable range. Tese values would be used as the basic values in this article to design tolerance range. Some principles are set for optimizing the tolerance range. First, no contact ellipse truncation and edge loading exist for any dimension values in the tolerance range. Second, the scatter of evaluated fatigue life is not more than 20% for any dimension values in the tolerance range in order to improve the reliability of the bearings.

Tolerance Analysis of Flange Angle and Roller-End Radius.
From reference [16], it can be found that the values of fange angle and roller-end radius directly afect the location of contact points and roller tilt angle, which further afect the contact ellipse and contact pressure. Tus, the efect of tolerance of fange angle and roller-end radius on contact ellipsis and fatigue life is discussed in this section. Here, the fange angle varies from 25 minutes to 35 minutes, the rollerend radius varies from 550 mm to 750 mm, and the fange height is 3.0 mm. In addition, the half width of the contact ellipse is nearly 1.0 mm in current working conditions. Tus, the location of the contact point should be in the range of 1 mm to 2 mm in order to avoid contact ellipse truncation. Figure 2 shows the efect of fange angle and roller-end radius on the location of contact point. From Figure 2, it can be found that when the fange angle is fxed, the location of contact point decreases with increasing roller-end radius and vice versa. Moreover, the efective values of fange angle and roller-end radius are in the dotted box. Generally, manufacturing error of fange angle and roller-end radius can be controlled in the 5 minutes and 100 millimeters range, respectively. However, the set of these tolerance values may lead to contact ellipse truncation. Two methods can be applied to eliminate this risk. First, the manufacturing precision should be further improved. For example, the error of fange angle and roller-end radius can be controlled in the 2.5 minutes and 50 millimeters range. In this situation, the tolerance range of fange angle and roller-end radius can be set as (30′, 32.5′) and (650 mm, 700 mm), respectively. Apparently, this method put more challenges on machining and process control. Te other method is grouping the roller-end radius and fange angle. For example, the tolerance range of fange angle can be sorted into two groups, (30′, 32.5′) and (32.5′, 35′), and the roller-end radius can be sorted into two groups, (600 mm, 650 mm) and (650 mm, 700 mm), as well. When assembling a CRB, fange in the group of (30′, 32.5′) should pair with rollers in the (650 mm, 700 mm) group, and fange in the group of (30′, 32.5′) should be pair with rollers in the (650 mm, 700 mm) group. Figure 3 gives the relative fatigue life of CRBs in the groups above. Here, the fatigue life at the fange angle 30 minutes and roller-end radius 650 mm is set as the benchmark. From Figure 3, it can be found that the diference between the maximum and the minimum fatigue life in each group is approximately 35% in current working conditions, and the average fatigue life is about 82.5% of the benchmark. Te minimum fatigue life is achieved at larger fange angle and larger roller-end radius. Tis is because the location of the contact point is relatively low which can cause roller tilt. Terefore, the results in Figures 2 and 3 hint that for the CRBs, the values of fange angle and roller-end radius have significant infuence on the location of contact point. In order to avoid the contact ellipse truncation, the fange angle and roller-end radius should be sorted and assembled according to the detailed dimension gap. Furthermore, the fner the grouping, the less the discreteness of the performance.

Analysis on the Interval of Roller Length Gauge.
For some specifc application requirements, the cylindrical rollers assembled in a bearing should be sorted in specifc length gauges. According to ISO 12997 [19], the interval of roller length gauge is 6 μm for the roller with diameter < 40 mm and length < 48 mm. Tat is to say, the length of each roller is diferent in a CRB, which would afect roller-end play and further afect roller tilt and axial load distribution. In this section, analysis is based on the severest situation for improving reliability. Te length of the most bottom roller is the longest, and others are identical in length. Figure 3 shows the efect of the length variation on radial and axial load distribution, where 0 μm represents that the length of all rollers is identical, 2 μm represents that the bottom roller is two microns longer than the others, and so on. From Figure 4, it can be found that the radial load distribution is nearly unchanged with the increase of the length of the bottom roller, while the axial load distribution changed. Although the number of loaded rollers remains the same, the axial load of the bottom roller increases from 372 N to 710 N. Tis is because the longest roller restricts the movement of the ring to some extent and avoids other rollers' early contact. For the bottom roller, it can be deduced that the contact ellipse would increase and bending force of the fange would increase which may cause crack formation of the fange or the fracture of the fange. Furthermore, the tilt angle of the bottom roller increases from       0.016 degrees to 0.036 degrees, as shown in Figure 5(a), due to the increase of the misalignment moment. Tis would lead to the occurrence of the edge loading and further reduce the fatigue life of CRBs, as shown in Figure 5(b). From Figure 5(

Tolerance Analysis of Roller Profle.
Although lots of studies have shown that the roller profle has signifcant infuence on bearing performance and given the optimization method of the roller profle, few works discussed the tolerance of roller profle required in engineering application. Fujiwara and Yamauchi [14] investigated the efect of crown drop on von Mises stress based on single rollerraceway contact and discussed the tolerance range of crown drop under the condition of a rate of stress <10%. Here, the relationship between the crown drops and fatigue life of CRBs was directly established. In order to defne the difference between logarithmic-profled rollers, the crown drop at the 0.47 l is set to be gauge points, where l is the efective length of the roller. Te crown drop varies from 1.30 μm to 5.03 μm, as shown in Figure 6. From Figure 7, it can be found that when the crown drop is 1.31 μm and 1.90 μm, the fatigue life is the longest and basically identical, and the fatigue life decreases with increasing crown drop. If the fatigue life is set as at least 80% of the longest fatigue life, the benchmark of crown drop can be set as 1.90 μm, and the tolerance range is (0, +2.5 μm). In this situation, the average fatigue life can be 90% of the longest fatigue life. It should be noted that although when the crown drop is 1.31 μm, the fatigue life is the longest in this analysis, the crown drop here is chosen from 1.90 μm since smaller crown drop may cause edge loading.

Conclusions
In this study, tolerance analyses of cylindrical roller under combined axial and radial loads were conducted to evaluate the infuence of tolerance range of key parameters, such as fange angle and roller-end radius, on contact and bearing performances. Some conclusions drawn from this study are as follows: (1) Te location of the contact point is very sensitive to the choice of fange angle and roller-end radius. When the fange angle and roller-end radius are controlled in the 5 minutes and 100 millimeters range, respectively, the contact ellipse of some sets would be truncated under the provided working conditions. Grouping the fange angle and roller-end radius and selective assembling would improve the contact performance.

Data Availability
All data generated or analyzed during this study are included within this article. Te code used during this study is available from the corresponding author on reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.  Mathematical Problems in Engineering