This paper presents the analysis method of stress distribution at the contact point of extra-large negative clearance four-point contact ball bearing, and calculates the contact stress and contact between the ball and the inner and outer rings of a certain type of slewing bearing with different negative clearances under no-load conditions District area. The results show that when there is no load, the negative clearance bearing has four-point contact, the contact areas at the four points are approximately equal, the contact stress distribution is approximately the same, and the contact stress on the inner ring is slightly larger than that on the outer ring. The larger the absolute value of the negative clearance, the larger the contact area and the larger the contact stress. The bearing friction torque can be increased by increasing the absolute value of the negative clearance.

**0 Introduction**

Slewing bearing is a kind of large ball bearing, which is widely used in heavy machinery such as cranes, wind power generators and excavators. The slewing bearing we study in this paper is used in wind energy generators, which requires the bearing to have a large starting friction torque. At this time, a negative clearance bearing must be used, that is, a certain amount of axial preload is given to the bearing. There are many theoretical analysis and experimental studies on small and medium-sized two-point or three-point contact ball bearings at home and abroad [1-5], and less research on large two-point, three-point or four-point contact ball bearings [6-8] ]. At present, the theoretical analysis of large-scale four-point contact ball bearings with negative clearance is still a blank at home and abroad. The calculation of contact stress is the basis of rolling bearing analysis. When considering the load distribution, stiffness, lubrication, friction, vibration and bearing life in the bearing, the contact stress must be calculated first. The contact stress has an important influence on the contact fatigue and wear of the bearing, and determines the life of the bearing to a large extent. Based on the quasi-static analysis of the extra-large four-point contact ball bearing, the analysis method of the contact stress at the four contact points of the bearing with negative clearance under no-load condition is given, and the contact stress of a certain type of slewing bearing is calculated. The shape and area of stress and contact zone provide theoretical basis and important data for bearing design, and lay a foundation for further theoretical research of negative clearance bearing.

**1 Geometric Analysis**

In the bearing analysis, the coordinate system is shown in Figure 1, and the X axis is along the axis of the bearing. The position of each ball in the bearing, Figure 2 shows the schematic diagram of the geometric relationship of the radial contact between the ball and the inner ring (or outer ring) before the bearing is deformed. The ball contacts the inner ring and the outer ring at two points. and the radial contact angle between the outer ring and the ball. The subscript i or o represents the raceway of the inner ring and the outer ring, respectively, go is the axial distance between the left half outer ring curvature center Col and the right half outer ring curvature center Cor, gi is the axial distance between the left half inner ring curvature center Cil and the right half inner ring curvature center Cir, ri and ro are the groove curvature radii of the inner and outer rings, respectively, and Db is the roller diameter. According to Figure 2 and the Pythagorean theorem,

In this paper, what we study is the contact problem of the bearing with negative axial clearance under no-load, that is, the outer ring of the bearing is fixed, only bears the gravity of the inner ring (along the axial direction), and is not subject to other external loads, so the center of curvature of the bearing raceway is Only axial displacement occurs. Figure 3 shows the relative positions of the center of the steel ball and the center of curvature of the ferrule at the angular position j before and after the bearing is deformed. Col′and Cor′are the curvature centers of the left half outer ring and the right half outer ring after the bearing is deformed, respectively. O and O’ are the centers of the balls before and after loading, respectively. Cil and Cir are the curvature centers of the left half inner ring and the right half inner ring respectively before the bearing is deformed, Cil′ and Cir′ are the curvature centers of the left half inner ring and the right half inner ring respectively after the bearing is deformed, αil, αir, αol and αor are the contact angles between the four ferrules and the ball after the bearing is deformed (subscripts l and r represent left and right, respectively, and subscripts i and o represent inner and outer, respectively).

Finally, when the bearing is only axially deformed and only subjected to axial external load, the balance of the bearing can be described by equations (8), (10) and (12), a total of 8Z+1 equations. The unknowns in equations (8), (10) and (12) are αorj, αolj, αirj, αilj, δorj, δolj, δirj, δilj and δx, a total of 8Z+1 unknowns. Using the Newton-Raphson method to solve the above equations, δorj, δolj, δirj and δilj can be obtained. Substituting into equation (9) can obtain the value of the contact force Q at the four contact points.

**2 Calculation of contact stress**

According to the Hertz theory, the contact area between the rolling element and the inner and outer rings is an ellipse, and the surface pressure conforms to the semi-ellipsoid distribution. In formula (15), ∑ρ is the principal curvature sum at the contact point between the steel ball and the ferrule; is the elastic shear modulus of the ball and raceway; υ1 and υ2 are the Poisson’s ratios of the ball and raceway, respectively; ma and mb are related to the ellipticity

**3 Results and analysis **

Taking a certain type of slewing bearing as an example to calculate, the bearing parameters are shown in Table 1. Figures 5 and 6 show the area of the contact area between the ball and the inner and outer rings when the bearing axial clearance is 0 and -0.06mm.

It can be seen from the figure that when the clearance is 0, the ball contacts and deforms with the right half inner ring and outer ring, and contacts with the left half inner ring, but does not deform. The contact area of the ball with the right half inner ring and the left half outer ring is basically equal, 0.24mm2 and 0.26mm2 respectively, and the contact area with the right half outer ring is very small, 0.01mm2, which is the result of the inner ring gravity. When the clearance is -0.06mm, the contact ellipse areas of the ball and the right half inner ring, left half outer ring, right half outer ring and left half inner ring are 6.22mm2, 6.19mm2, 6.30mm2 and 6.33mm2, respectively. The clearance changes from 0 to -0.06mm, and the contact area increases by more than 20 times. It can be seen that the negative clearance can greatly improve the contact deformation of the bearing, thereby increasing the friction torque of the bearing. In addition, it can be seen from the figure that when the clearance is 0, the areas at the four contact points are quite different, and when the clearance is negative, the contact areas at the four contact points are approximately equal.

Figure 7 shows the contact stress distribution between the ball and the left half inner ring of the bearing when the clearance is -0.06mm. The shape of the contact stress distribution between the ball and the right half inner ring, right half outer ring and left half outer ring is similar to the above figure, but the stress values are different. The peak stress at the contact center of the ball and the right half inner ring and the left half inner ring are 982.7MPa and 975.5MPa respectively, the contact stress between the ball and the left half outer ring and the right half outer ring is slightly smaller, and the peak stress at the contact center is 974.9MPa and 968.1MPa. For other negative clearances, the distribution of the contact stress is similar to the above-mentioned stress distribution, but the value of the stress is different. The greater the absolute value of the negative clearance, the greater the contact stress. The small change of the negative clearance has a great influence on the contact stress. For example, when the negative clearance changes from 0 to -0.06mm, the peak value of the contact stress increases from 274.1MPa to 982.7MPa, an increase of nearly 4 times. If it is necessary to ensure that the bearing has a certain starting friction torque or greater stiffness, it can be achieved by increasing the absolute value of the negative clearance.

**4 Conclusions **

(1) When there is no load, the negative clearance bearing has four-point contact, the contact area is elliptical, the contact stress is ellipsoidal distribution, the contact area at the four points is approximately equal, and the contact stress distribution shape is approximately the same. The contact stress between the ball and the outer ring is slightly greater than the contact stress with the inner ring. (2) The larger the absolute value of the negative clearance, the larger the contact area and the larger the contact stress. (3) The bearing can be guaranteed to have a certain starting friction torque or greater stiffness by increasing the absolute value of the negative clearance.