In the rollerraceway contacts of the radial cylindrical roller bearing used in the axle boxes of a railway bogie, pressure accumulation may occur, reducing the fatigue life. This accumulation can be eliminated by applying logarithmic correction of generators and in particular varieties of the modified logarithmic correction. The correction parameters should be adapted to the operating conditions of the bearing. This article presents a comparison of the predicted fatigue life of an axle box bearing on correctly selected correction parameters with bearing life, in which correction of roller generators was used, typical for cylindrical roller bearings of general application. The finite element method was used to determine the subsurface stress distributions necessary to calculate the fatigue life.
Keywords
 rolling bearing
 correction of rolling elements generators
 fatigue life
 finite element method
At the ends of the contact lines of bodies with different curvatures and rectilinear generators, there is pressure accumulation and, as a consequence, the accumulation of subsurface stresses determines the fatigue life. Such a phenomenon may occur in the contact of rollers with raceways in rolling bearings. The only way to eliminate or reduce stress accumulation is by giving a proper shape to generators of the mating elements. In modern cylindrical roller bearings, the logarithmic correction of roller generators is used. In the second half of the past century, its simplified equivalent, the chordarch (ZB) correction, was widely used [1].
The problem of determining the shape of a roller generator, ensuring the pressure distribution closest to the distribution for the Hertzian linear contact, was first undertaken by Lundberg [2]. Aiming to achieve an even distribution of pressures along with the generator, Lundberg proposed the logarithmic shape of the generator of a rolling element using the following formula:
In formula (1),
From formula (1), it follows that at the ends of the generator the amount of correction
Logarithmic correction ensures advantageous pressure distribution along the contact line for only one pressure value
For generalpurpose radial cylindrical roller bearings, KrzemińskiFreda [3, 4] proposed the following parameter values in the modified logarithmic correction formula:
correction selection pressure:
relative maximum correction coordinate:
exponent in the correction formula:
In many devices, rolling bearings work at significantly lower loads. This occurs in the case of the axle box roller bearing of the railway carriage. Figure 1 shows a crosssection of a typical rolling bearing assembly of a twoaxle Intercity passenger carriage bogie. The bearing assembly is made of two FAG radial cylindrical roller bearings, one of the type WJ 130 × 240TVP and the other of type WJP 130 × 240PTVP [5, 6]. The bearings are positioned in such a way that an integral flange of the WJ bearing inner ring and a loose flange ring of the WJP bearing are located on the outer sides of the bearing system.
According to the studies of the FAG company [6], the load of a single bearing of the assembly occurring during operation is equal to
Warda [7] presents the results of analyses of the influence of modified logarithmic correction parameters on the fatigue life of the WJ 130 × 240TVP axle box bearing. The computer simulations carried out showed that the largest forecasted bearing fatigue life is ensured by the following correction parameters:
correction selection pressure:
relative maximum correction coordinate:
exponent:
For calculations, a computer program was used to determine the load distribution on radial cylindrical roller bearings (ROLL1), a program for calculating predicted bearing fatigue life (ROLL2), and a program that allows finding pressure and subsurface stress distributions in roller contacts with raceways by solving the Boussinesq problem for elastic halfspace (ROLL4). The programs were formulated based on the methodology described by Warda [8] and Warda and Chudzik [9].
This study presents the results of numerical analysis of pressure distribution and the corresponding stress distribution. Numerical calculations were obtained using modern computational techniques. The paper presents a comparison of the results of the predicted fatigue life of a bearing with correction typical for generalpurpose bearings, whose parameters were determined for
Determination of fatigue life of a radial cylindrical roller bearing in which correction of rolling elements generators has been applied is possible only with the use of computer technology. In this case, the method of predicting durability requires finding local values of subsurface stress on which the bearing life depends.
The most accurate information about the nature of the pressure distribution and the corresponding subsurface stress distribution is obtained by the finite element method (FEM). For this purpose, a solid FEM model of the bearing should be built that accurately reflects the internal geometry of the bearing, taking into account the bearing clearances and the relative position of the bearing rings and rollers resulting from the load, as well as resulting from assembly errors or shaft deflection.
Searching for load distribution on bearing rolling elements using FEM, while taking into account contact phenomena occurring in the contacts, is timeconsuming. Usually, the authors of papers regarding the determination of load and stress distributions in roller bearings used twodimensional FEM models [10,11,12], and thus did not allow for taking into account the shape of the mating elements. Often, to simplify the threedimensional bearing model, rollers were replaced by socalled superelements. Superelements were used by Golbach [13] and Kania [14] in their research, which allowed them to consider the effect of corrections of rolling element generators on load and stress distribution in bearings. In a few cases, the authors used the threedimensional FEM model of the entire bearing but did not include the complex profile of the generators. This is what Blanusa et al. [15, 16] and Ke et al. [17] did in their work on thermal phenomena occurring in bearings of the axle box. On the other hand, whole bearing models are often used for ball bearing analysis [18].
Researchers generally used numerical methods other than the FEM to determine load distribution on rolling elements [19,20,21], while solid FEM models of a bearing fragment were used to analyze phenomena occurring in a single contact [22,23,24,25,26].
The authors of this article used a combination of two methods in their earlier work on the prediction of fatigue life of radial cylindrical roller bearings – the FEM for a bearing section to find stresses and the numerical solution of equilibrium equations of rings and rollers to determine load distribution on rolling parts [8, 27,28,29]. This approach significantly reduced the computational time and allowed them to be carried out on standard computer equipment.
The fatigue life of the axle box bearing was determined using the method described by Warda [8] and Warda and Chudzik [9] and also used in papers [27,28,29]. This method uses the basic assumptions of the fatigue life prediction model developed by Lundberg and Palmgren [30, 31].
The durability of the inner ring rotating relative to the load is calculated from the formula determining the inverse logarithm of the probability of durability (_{i}as:
The fatigue life
In the above formulas,
To determine the values of subsurface stress in the tested bearing, a numerical 3D solid model mapping of the bearing's internal geometry was built introducing modified logarithmic correction of roller generators. Calculations were carried out for comparative purposes for rollers under load during bearing operation when
To analyze the subsurface stress in the tested model, it is extremely important to properly model the contact zone of the roller with the raceway. Nonlinear phenomena appear during bearing operation. For this reason, the creation of contact pairs and the appropriate setting of contact parameters is a key element to obtain the correct calculation results.
In the numerical analysis, the results of calculations mainly depend on the number of elements resulting from the division of the tested model into finite elements. The contact model was used for calculations in the contact zone: the raceway surface was adopted as the target surface (TARGE type), the roller surface was adopted as the contact surface (CONTA type). The contact surface was modeled using contact elements CONTA175. The target surface was modeled using elements TARGE 170. For selected roller volumes in the contact zone, the roller contact edge with the raceway is divided into equal parts with a length of 0.05 mm. The edge of selected volumes of contact between the raceway and the roller is divided into equal parts with a length of 0.1 mm.
In the numerical model of the roller bearing part not involved in contact with rolling elements, 10node solid elements of the TET187 type were used to divide into finite elements. Due to the large dimensions of the rolling elements of the tested bearing, the adopted model differs from the models used by the authors in previous works, in which 8node SOLID 185 solid elements were used. As a result, the examined numerical model was divided into 803,145 solid finite elements. Possible shape errors have been eliminated by adjusting the distance angle between the raceway nodes and the roller. To reflect the actual operating conditions of the bearing, the possibility of displacement in all directions of the outer surfaces of the outer and inner race was taken. The lateral surfaces of the outer and inner races are deprived of the possibility of displacement in the direction of the
The main parameters of the analyzed axle box bearing are shown in Table 1.
Parameters of the WJ 130 × 240TVP (WJP 130 × 240PTVP) axle box bearing [5].
Dynamic load rating  

Bearing bore diameter  
Bearing outside diameter  
Bearing width  
Diameter of the outer ring raceway  
Diameter of the inner ring raceway  
Roller diameter  
Roller length  
Roller chamfer dimension  
Number of rollers in the bearing 
The profile of the roller generator with modified logarithmic correction can also be described by the formula:
Table 2 summarizes the values of the parameters
Modified logarithmic correction parameters for the assumed correction selection pressure
Correction selection pressure 
2100 MPa  900 MPa 

Lundberg correction coordinate 
0.0202 mm  0.0043 mm 
Relative maximum correction coordinate 
3  2 
Maximum correction coordinate  
0.0606 mm  0.0086 mm  
3  3  
0.011695 mm  0.001432 mm  
23 mm  23 mm  
0.193  0.166  
6  6 
The first part of the calculations determining the predicted fatigue life of the axle box bearing was to determine the load distributions for individual bearing rollers. The calculations were carried out assuming the bearing load radial force
The results of the load distribution calculations obtained using the ROLL1 program [9] are shown in Table 3. As the table shows, due to the symmetry of the distribution, radial load carries 7 out of 17 bearing rollers simultaneously.
Radial load distributions on WJ 130 × 240TVP bearing rollers.
Roller ( 


1  13,490 
2  11,831 
3  7,215 
4  877 
5  0 
The second stage of calculations, which is necessary to predict the bearing's fatigue life, is to determine the maximum von Mises stress distribution
The basic package of the ANSYS program allows to create graphs showing the maximum stress distributions along the roller contact line with the raceway; however, it is not possible to obtain the depth distributions of these stresses. But it is possible to determine the distribution of subsurface stress in a plane perpendicular to the axis of the roller set by the user at any point along the contact line. By analyzing changes in subsurface stresses in selected sections, the distribution of the depth of occurrence of maximum subsurface stresses can be determined. The method of determining distributions has been described by Chudzik and Warda [28, 29].
Combining the characteristics of stress distribution
Values of von Misses stress
Roller No. 1  0.13981  557.89  0.14200  0.148810  604.39  0.14901 
11.88423  557.94  0.14200  10.92539  604.45  0.15408  
17.89625  552.69  0.14196  14.11846  588.22  0.15219  
19.64394  543.08  0.14192  16.81260  532.79  0.14150  
20.76245  524.77  0.13700  18.20957  458.38  0.11150  
21.88097  484.53  0.10700  19.20740  353.56  0.06150  
22.51013  375.73  0.05700  
Roller No. 2  0.20972  524.12  0.13300  
11.88423  526.61  0.13302  0.24853  559.63  0.14300  
17.68653  517.63  0.13296  10.42656  559.58  0.14596  
19.71384  504.88  0.13291  13.61966  550.46  0.14300  
20.69254  493.54  0.12800  16.51342  514.68  0.13300  
21.32171  479.37  0.12300  17.91040  464.49  0.11300  
21.53143  465.21  0.11800  19.20760  368.44  0.05300  
22.09069  416.79  0.08800  
22.44023  318.53  0.03800  
Roller No. 3  0.20972  419.27  0.10457  
12.16386  426.40  0.10457  
14.68052  419.11  0.10457  0.14874  460.36  0.11550  
16.42820  416.43  0.10457  12.52203  455.45  0.11650  
18.10598  406.09  0.10457  15.41579  418.42  0.11250  
19.29440  389.62  0.10457  16.21406  393.11  0.10750  
20.27310  365.17  0.09850  17.51126  323.63  0.08750  
20.97217  335.37  0.08850  
21.60134  289.88  0.06850  
Roller No. 4  0.13982  81.96  0.02500  
7.06080  82.65  0.02500  
9.78724  82.83  0.02632  0.14868  121.62  0.03650  
11.88451  90.96  0.02695  9.82791  118.94  0.03696  
12.44378  91.80  0.02500  10.82578  114.80  0.03616  
13.28269  88.42  0.02550  12.12299  101.82  0.03526  
14.05169  88.77  0.02499  13.22064  81.232  0.03250  
15.44986  84.39  0.02499  14.41807  39.520  0.01850  
17.26750  73.24  0.02250  
18.38604  60.15  0.01250 
As can be seen from the graphs, the value of the correction selection pressure
Regardless of the generator shape, each of the rollers is loaded with the same radial force
Graphs similar to those presented in Section 6 were also prepared for roller contacts with the outer ring raceway, and the data contained in the files based on which the charts were made were used to calculate the predicted fatigue life of the axle box bearing. The ROLL2 computer program [9] was used to calculate the predicted bearing fatigue life. Figure 9 shows a comparison of the bearing fatigue life for both types of corrections considered.
Calculations of the predicted fatigue life of the axle box bearing show how much impact the selection of appropriate correction parameters of roller generators has on the bearing durability. The generator profile, which was determined for the correction selection pressure
A modified logarithmic correction can provide a favorable pressure distribution along the contact line of the mating elements for a much wider load range than the logarithmic correction proposed by Lundberg; however, it requires a suitable selection of the parameters. Correction parameters should be different for standard cylindrical roller bearings operating under a load representing 30–40% of the dynamic bearing capacity of the bearing, and different for specialpurpose bearings, such as railway carriage axle box bearings.
Correctly selected parameters of the modified logarithmic correction should guarantee the greatest possible length of contact between the roller and the raceway while preventing pressure accumulation. Axle box bearing with such correction can achieve a life of 50% longer than a similar bearing with a correction typical for generalpurpose bearings.
The value that has the greatest impact on the effectiveness of the applied correction of the profile of the rolling element generators is the pressure for which the correction is selected. The value of the correction selection pressure should be close to the maximum pressure values occurring in the contact of the roller with the bearing raceways. What is also important are the other two parameters of the correction, the relative maximum correction coordinate and exponent of logarithmic function, determining the shape of the generator and pressure distribution at the edges of the roller.
To determine the predicted fatigue life of a bearing, it is necessary to know the subsurface stresses in the roller contacts with the bearing raceways. Stress distributions are most conveniently determined using the FEM. At the same time, load distribution on rolling elements should be determined. This can be done by performing FEM calculations for the complete bearing solid model, which is timeconsuming and requires the use of computers with very high computing power. Sufficiently accurate results can be obtained by combining the two methods: the FEM for a bearing section, and the numerical solution of equilibrium equations of rollers and bearing rings.
Parameters of the WJ 130 × 240TVP (WJP 130 × 240PTVP) axle box bearing [5].
Dynamic load rating  

Bearing bore diameter  
Bearing outside diameter  
Bearing width  
Diameter of the outer ring raceway  
Diameter of the inner ring raceway  
Roller diameter  
Roller length  
Roller chamfer dimension  
Number of rollers in the bearing 
Radial load distributions on WJ 130 × 240TVP bearing rollers.
Roller ( 


1  13,490 
2  11,831 
3  7,215 
4  877 
5  0 
Modified logarithmic correction parameters for the assumed correction selection pressure pK.
Correction selection pressure 
2100 MPa  900 MPa 

Lundberg correction coordinate 
0.0202 mm  0.0043 mm 
Relative maximum correction coordinate 
3  2 
Maximum correction coordinate  
0.0606 mm  0.0086 mm  
3  3  
0.011695 mm  0.001432 mm  
23 mm  23 mm  
0.193  0.166  
6  6 
Values of von Misses stress σ and the depth Z along the line of contact.
Roller No. 1  0.13981  557.89  0.14200  0.148810  604.39  0.14901 
11.88423  557.94  0.14200  10.92539  604.45  0.15408  
17.89625  552.69  0.14196  14.11846  588.22  0.15219  
19.64394  543.08  0.14192  16.81260  532.79  0.14150  
20.76245  524.77  0.13700  18.20957  458.38  0.11150  
21.88097  484.53  0.10700  19.20740  353.56  0.06150  
22.51013  375.73  0.05700  
Roller No. 2  0.20972  524.12  0.13300  
11.88423  526.61  0.13302  0.24853  559.63  0.14300  
17.68653  517.63  0.13296  10.42656  559.58  0.14596  
19.71384  504.88  0.13291  13.61966  550.46  0.14300  
20.69254  493.54  0.12800  16.51342  514.68  0.13300  
21.32171  479.37  0.12300  17.91040  464.49  0.11300  
21.53143  465.21  0.11800  19.20760  368.44  0.05300  
22.09069  416.79  0.08800  
22.44023  318.53  0.03800  
Roller No. 3  0.20972  419.27  0.10457  
12.16386  426.40  0.10457  
14.68052  419.11  0.10457  0.14874  460.36  0.11550  
16.42820  416.43  0.10457  12.52203  455.45  0.11650  
18.10598  406.09  0.10457  15.41579  418.42  0.11250  
19.29440  389.62  0.10457  16.21406  393.11  0.10750  
20.27310  365.17  0.09850  17.51126  323.63  0.08750  
20.97217  335.37  0.08850  
21.60134  289.88  0.06850  
Roller No. 4  0.13982  81.96  0.02500  
7.06080  82.65  0.02500  
9.78724  82.83  0.02632  0.14868  121.62  0.03650  
11.88451  90.96  0.02695  9.82791  118.94  0.03696  
12.44378  91.80  0.02500  10.82578  114.80  0.03616  
13.28269  88.42  0.02550  12.12299  101.82  0.03526  
14.05169  88.77  0.02499  13.22064  81.232  0.03250  
15.44986  84.39  0.02499  14.41807  39.520  0.01850  
17.26750  73.24  0.02250  
18.38604  60.15  0.01250 