Correction of Radial Load Distribution Integral for Radial Bearings

 Abstract: The radial load distribution integral is corrected for radial bearings. The error of Harris integral method for calculating the radial load distribution is analysed. The error is found absolutely caused by the inaccurate values of the radial load distribution integral given by Harris. Based on the extent of the load zone, the radial load distribution integral is corrected by three stages. The corrected radial load distribution integral is capable of calculating the load distribution of a bearing with a small load zone caused by a light external load or a great radial clearance. The corrected radial load distribution integral is found affected by the number of rolling elements. According to the variation of the number of rolling elements participating in the radial load transfer, the corrected radial load distribution integral can be divided into different phases. Some specific numerical examples are shown to illustrate the performance of the corrected radial load distribution integral. The comparison between the results obtained from the corrected radial load distribution integral and Harris integral shows the higher accuracy and superiority of the corrected radial load distribution


Introduction
When a radial ball or roller bearing is subjected to a radial load, the load is transferred from one raceway to another through the rolling elements. In this transfer process, the rolling elements are not equally loaded. This uneven load distribution is affected by the amplitude of the external load and the geometry of the rolling bearing [1] and plays a significant role on the operating characteristics of a bearing [2], such as the static carrying capacity [3,4], dynamic vibration [5][6][7][8], radial stiffness [5,[9][10][11][12] and fatigue life [1,3,[13][14][15]. To determine the radial load distribution, Stribeck [16] first derived an equation to calculate the maximum load on a rolling element of a ball bearing with zero radial clearance. A radial load distribution integral was proposed by Sjövall [17] to calculate the load distribution for nonzero clearances. The integral value is affected by the type of contact in a bearing and the ratio of the radial clearance to the ring radial shift so that the variables of the elliptic integral were introduced into the calculation. This integral is referred to as the Sjövall integral. Based on Sjövall's research, Harris [1] proposed a load distribution factor and adjusted the values of the radial load distribution integral by a numerical method which is called Harris integral in this paper. To obtain the integral value more easily, Oswald et al. [14] and Houpert [18] used different polynomials to fit the relation of integral vs. the load distribution factor. A coefficient for the boundary external radial load was defined by Tomović [2,19] to ensure the number of the active rolling elements participating in the external load transfer. Without a calculation of the contact stiffness, Ren et al. [20] proposed a mathematical model in which only the deformations and geometric parameters of the bearing were used to determine the radial load distribution. Moreover, the bearing was considered a mass-spring-damper system, and the load distribution was calculated by multi-body nonlinear dynamic models by Petersen et al. [5,11] and Sawalhi et al. [21,22]. In addition, a comprehensive explicit dynamic finite element model [23] was used to analyse the dynamic contact forces in rotational bearings. Although all the methods are capable of calculating the radial load distribution, most of these methods have specific application conditions. For example, the load zone is assumed to be π and fixed in Stribeck's equation [16], which means Stribeck's method is not suitable for the calculation 1 of the bearings with a nonzero clearance. The discrete models proposed by Tomović [2,19] and Ren et al. [20] cannot be used to obtain the angle scope of the load zone while the dynamic models [11,21] are mostly used for vibration analysis and defect identification. Due to its simplicity, the method based on the load distribution integral has been widely used in many aspects of bearings [14][24] [18] [25].
However, several open questions remain for calculating the load distribution using the existing integral values. First, in the load distribution result obtained by the values of Sjövall integral or Harris integral, the sum of the vertical components of the calculated rolling element loads is not equal to the external radial load [1,19,20]. This calculation error cannot be neglected if the calculated load is used to predict some critical parameters such as fatigue life [25]. Second, when the load distribution factor is greater than 0 and less than 0.1, the value of the Harris integral or the way to calculate the integral are not given in previous studies. It indicates that the current integral method is not applicable for the small load zone situation caused by the light external load or great clearance [1]. The skidding of the rolling elements often occurs and results in smearing type of surface damage under light load situation [26], while the bearing tends to early failure operating with a great clearance [14].
In this paper, the correction for radial load distribution integral is proposed. The radial load distribution for point and line contacts are accurately obtained especially for the light load or great clearance conditions. Several numerical examples of commercial bearings are given to illustrate the impact of corrected load distribution integral. Figure 1 shows the geometries of the radial ball and roller bearings to calculate the radial load distribution. To achieve a practically applicable mathematical model and reduce ambiguous issues, some necessary assumptions should be presented:

Initial Assumptions
1) Deformation only occurs at the individual contacts between the rolling elements and raceways. It complies with the Hertz elastic contact theory, other parts are absolutely rigid [1].
2) The bearing is subjected to a static load. The dynamic load distribution is considered the same as the static load distribution because, in most applications, the speeds of rotation are almost not as great as to cause ball or roller inertial force of sufficient magnitude to significantly affect the load distribution [13].
3) An external radial load r F is downward applied to the inner ring of the bearing, and the outer ring is radially fixed. 4) For purposes of identifying and locating the position of the rolling elements, the bottom rolling element is set to 0, and the others are numbered clockwise as 1, 2,  , j ,  , Z-1, where Z is the total number of rolling elements.
The rolling element 0 is located in the line of the loading direction of r F .

5)
The unit azimuth angle  is constant for a bearing with fixed number of rolling elements, which is separated by a cage and calculated by the following formula: According to the above assumptions, the load distribution is symmetrical with respect to the line of the loading direction, and the most loaded rolling element is the rolling element 0.
 is defined as the load distribution factor. From Eq. (4), the angular extent of the half load zone l  is determined by the diametral clearance as follows: According to the load-deflection relationship of a rolling element-raceway contact expressed by   n j j F K (6) and max max where j F is the radial load on the rolling element j and max F is the maximum distributing radial load and is equal to 0 F here. Number 3 2 n  for ball bearings, and 10 9 n  for roller bearings. K is a series of stiffnesses of the inner and outer raceway contact deformation. The calculation of K is illustrated in the Appendix.
From Eqs. (3), (6) and (7), the rolling element load j F at any rolling element position can be given in terms of the maximum rolling element load max For static equilibrium, the external radial load must equal the sum of the vertical components of the rolling element loads: where     (12) In Harris' procedure, the radial integral ( ) r J  has been given numerically for various values of the load distribution factor  and shown in Table 7.1 and Figure  7.2 in Ref. [1]. From Eqs. (2) and (6), max F can be expressed as follows: Therefore, Eq. (13) is introduced into Eq. (11), For a given bearing with a given clearance under a given load, r  is the only unknown value. Harris et al. [1] presented that Eq. (14) may be solved by trial and error. A value of r  is first assumed, and  is calculated from Eq.

Error of Harris Model and Integral
In Harris model mentioned above, three formulas, Eqs. (4) (12) and (14), significantly determine the radial load distribution in a radial bearing. During the calculation using these formulas, as input parameters, the external radial load r F , total number of rolling elements Z, radial clearance d P and contact stiffness K do not affect the accuracy of the calculation. However, some calculation errors exist and vary with these input parameters during the calculation of radial load distribution by Harris integral model [1,19,20].   Figure 2. It is found that the errors of Harris method for both roller and ball bearings become smaller with the increase of the total number of the rolling elements overall, while the values and fluctuation of the error for roller bearing are greater than for ball bearing. The maximum error of these numerical examples is up to nearly 12% when the bearing is a roller bearing with 6 rollers. Moreover, according to the different number of loaded rolling elements, the errors can be divided into different phases. For example, 3 rollers participate in the radial load transfer and the error shows a negative-positive-negative trend when Z=6-12 for the roller bearing. In all of the phases, the variation trends of the errors vs. the rolling element number Z are absolutely the same. [20], however, some different trends are indicated in Figure  3. According to different load zone extents, the load distribution in a radial bearing with 10 rollers is divided into six phases as shown in Figure 4. Phase I to VI, respectively corresponding to only one rolling element participating in load transfer to all rolling elements participating in load transfer, introduce the evolution of the load distribution with the load distribution factor  . These phases are marked in   With the load distribution factor  as the independent variable and the error as the dependent variable, Figure 5 shows the error varying with  for a roller bearing with

Calculation of Load Distribution Integral by Discrete Method
A discrete method is used to correct the radial load distribution integral here. The equilibrium between the external radial load and the sum of the vertical components of the rolling element loads can be expressed in two forms: discrete form as Eq. (10) and integral form as Eq. (11). Comparing these two formulas, the radial load distribution integral can be expressed by discrete form: As long as the value of k is ensured, the value of ( ) r J  can be calculated by Eq. (17). From Eq. (4) and Eq.
Furthermore, because the angle between two adjacent rollers is the unit azimuth angle  , the value of k depends on how many  are contained in l  : , there is only one rolling element participating in the load transfer and 0 k  . Introducing 0 k  into Eq. (17), the values of ( ) r J  when only one rolling element participates in the load transfer can be calculated as follows: Eq. (23) Figure 7 shows  Figure 8b shows the difference between the corrected radial load distribution integral and Harris integral which represents the degree of correction to Harris integral. It is found that the variation trend of the difference ( ) Figure 8b is absolutely the same with that of the  increases with the increase of Z . Moreover, although the same load phase tends to be narrower with the increase of Z , for a fixed number of rolling element, different phases correspond to a same angle range of 2 except for the last load phase. As shown in Figure 10, according to the assumption 4, there is a rolling element between two adjacent unit azimuth angles located at the top of a radial bearing with even numbers of rolling element, while there is a unit azimuth angle between two adjacent rolling elements located at the top of a radial bearing with odd numbers of rolling element. Therefore, the angle range of the last load phases in Figure 9(a)-(h) is equal to 2 for even numbers of rolling element and equal to  for odd numbers of rolling element. According to the statements above, there are some characteristics of the corrected integral:

Examples for the Corrected Radial Load Distribution Integral
1) The corrected radial load distribution integral makes it possible to calculate the load distribution of a bearing with a small load zone (less than 73.74 degrees) caused by a light external load or a great radial clearance.
2) The corrected radial load distribution integral is affected by the total number of rolling element in a radial bearing.
3) For a bearing with a given number of rolling element, the corrected radial load distribution integral shown in Figure 9 can be used to intuitively ensure the number of rolling element participating in the load transfer and the extent of the load zone.

Numerical Examples and Discussion
Some specific numerical examples for the calculation of the radial load distribution are given to illustrate the performance of the proposed correction of the radial load distribution integral. The conditions of the radial load distribution in these examples cover both radial ball and roller bearings with positive and negative clearances under different loads. Moreover, the results of the calculation by the corrected radial load distribution integral are compared with those calculated by Harris integral to show the improvement of the calculation accuracy.
A single row deep groove ball bearing 6209 with odd number of rolling elements and a single row cylindrical roller bearing NU209 with even number of rolling elements are considered in the numerical examples. The parameters of the ball and roller bearings for the numerical calculation are shown in Table 1. The radial clearances shown in Table  1 are the initial clearance chosen from the Rolling Bearing Catalogue of SKF [27]. An interference fitting clearance of -0.010 mm is chosen as the negative clearance for both ball and roller bearings in the numerical examples. Three values of the external radial load r F are introduced to the calculation. According to these parameters in Table 1 Figure 7.2 in Ref. [1]. Moreover, as shown in Table 4 Under the situation of a small load zone caused by a light external load or a great radial clearance, the load distribution plays a significant role in the studies of skidding behaviour [26,28] and the fatigue life [14] of bearings. The sliding velocities of the rollers are considered to exist under light external loads and vary with the radial load distribution in the bearing by Tu et al. [26] and Han et al. [28]. Moreover, the skidding of the roller at the entry of the loaded zone is likely to cause wear on the raceway surface [26]. Thus, accurate calculation of the load distribution under light load condition can be used to ensure the skidding states of the rollers and the locations of the possible wear. Furthermore, a greater radial clearance will cause a kind of load distribution with a smaller load zone and a greater maximum rolling element load. Then, the fatigue life of the bearing will decrease under this kind of load distribution [14]. Therefore, accurate calculation of the load distribution under great clearance condition contributes to avoiding early failure caused by improper assembly. Besides, according to the calculation method of fatigue life proposed by Lundberg and Palmgren [25], the basic dynamic capacity of a bearing is calculated by the radial load distribution and the radial load distribution integral. It reveals that the errors of radial load distribution and radial load distribution integral will be transferred and accumulated in the calculation of the fatigue life of bearings. Thus, the correction of the radial load distribution integral and its contribution on the accurate calculation of the load distribution are significant for predicting the fatigue life accurately especially for the light load or great clearance conditions.

Conclusions
Based on the analysis of the error from Harris method to calculate the radial load distribution in radial bearings, the radial load distribution integral for radial bearings was corrected in this paper. The calculation of the corrected ( ) r J  was divided into three stages according to the values of the load distribution factor  . Some numerical examples were used to illustrate the characteristics and application of the corrected radial load distribution integral. The following conclusions were obtained:  The errors of these equations were analysed. For 1 10    , the errors in the calculation of  are less than 3%. The errors on E are essentially nil except at 1   and in the vicinity where they are less than 2%. The errors on F are essentially nil except at 1   and in the vicinity where they are less than 2.6%.
x R and y R are the directional equivalent radii of the major axis direction and minor axis direction of the contact, respectively. For the given ball bearing shown in Figure 1 .