Abstract: The calculation method of fatigue life of super-large double-row four-point contact ball slewing bearing under the combined action of radial, axial load and overturning moment is proposed. Contact load under different clearances and loads, then according to the Lundberg-Palmgren theory, calculate the equivalent rolling element load of each ferrule, and calculate the rated life of each channel from the rated rolling element load and equivalent rolling element load of the ferrule, Then, the fatigue life of the inner and outer rings of the pitch bearing and the fatigue life of the entire bearing are calculated, and the relationship between the bearing rated life, bearing clearance and groove curvature radius is analyzed. The results show that with the increase of the absolute value of the negative clearance of the bearing, the rated life of the bearing first increases and then decreases; the larger the bearing groove curvature radius coefficient, the smaller the fatigue life.
Wind turbine pitch bearings generally use extra-large double-row or single-row four-point contact ball slewing bearings, which can bear axial load, radial load and overturning moment at the same time. Pitch bearings are generally installed at an altitude of more than 20 meters, and the cost of replacement and maintenance is relatively high . Therefore, in the design and selection process of the pitch bearing of the wind turbine, it is necessary to use the reliability method to calculate its life under a certain reliability.
At present, there are many researches on the fatigue life of single-row four-point contact ball bearings at home and abroad [2-5], but less research on the life of multi-row four-point contact ball bearings [6-7]. Reference  compares and analyzes various methods for calculating the fatigue life of bearings, and concludes that the stress-life method is to some extent an effective method for calculating the fatigue life of rolling bearings. Reference  presents a method for calculating the fatigue life of a three-row cylindrical roller slewing bearing under the combined action of radial, axial loads and overturning moments. Reference  gave a theoretical formula for estimating the life of a multi-row roller slewing bearing, but did not consider the effect of clearance and the load on the bearing life.
In this paper, an extra-large double-row four-point contact ball slewing bearing is taken as an example to calculate the fatigue life of the bearing under the combined load. The key to the life calculation of extra-large double-row four-point contact ball slewing bearing under combined load lies in the calculation of the internal load distribution of the bearing; the speed of the pitch bearing is generally low, and the static analysis of the bearing can meet the requirements. In this paper, the static model of the pitch bearing is established, and the Newton-Raphson iteration method is used to solve it, and the internal load distribution of the bearing is obtained according to the Hertz contact theory. On this basis, the equivalent rolling element load of the ring is calculated. The life of the whole set of bearings is calculated based on the rated rolling element load. Finally, the influence of the bearing clearance and groove curvature radius on its life is analyzed.
1 Pitch bearing load distribution
In the static analysis, it is assumed that the outer ring of the bearing is fixed, and external forces (axial force Fa, radial force Fr and overturning moment M) act on the inner ring. The bearing is in four-point contact. Here, the contact pairs that bear the main axial force are called contact pair 1 (upper row) and contact pair 3 (lower row), and the other two corresponding contact pairs are called contact pair 2 (upper row) respectively. ), contact pair 4 (lower row), as shown in Figure 1. In the figure, Dpw is the pitch circle diameter of the bearing ball group, and dc is the center distance between the two rows of steel balls.
In the formula: fi is the curvature radius coefficient of the inner groove; Dw is the diameter of the steel ball; fe is the outer groove curvature radius coefficient; Ga is the axial clearance of the bearing; α0 is the contact angle before the bearing is not loaded.
Before the bearing is loaded, when the clearance between the steel ball and the groove is 0, the distance between the inner and outer ring groove curvature centers of any steel ball position contact pair
Under the action of axial load Fa, radial load Fr and overturning moment M, the inner ring will produce axial, radial and angular displacements relative to the outer ring, which are δa, δr, θ respectively, as shown in Figure 2 . After the bearing is loaded, due to the displacement of the inner ring, the center distance of the groove curvature of all contact pairs has changed, as shown in Figure 3. In the figure, C1el, C1er, C2el, and C2er are the curvature centers of the left and right outer channels in the upper and lower rows, respectively, and C1il, C1ir, C2il, and C2ir are the curvature centers of the left and right inner channels in the upper and lower rows before loading, respectively. C’1il, C’1ir, C’2il, C’2ir are the curvature centers of the left and right inner channels of the upper and lower rows after loading, respectively, O1, O2 are the ball centers of the upper and lower rows of steel balls before loading, O’1, O’2 are the ball centers of the upper and lower rows of steel balls after being loaded. α1, α2, α3, α4 are the contact angles of contact pairs 1, 2, 3, and 4 after loading, respectively. Then the groove center distance Ajφ of the contact pair j (j=1, 2, 3, 4) at the position angle φ
At the position angle φ of the contact pair j, the total elastic contact deformation δjφ between the steel ball and the channel is δjφ=Ajφ-A0. (8) After the inner ring is displaced, the contact angle αjφ of the contact pair j at the position angle φ is divided into
In the formula: Kn is the total load deformation constant of the steel ball and the inner and outer rings, which is determined by the material and geometric parameters of the bearing . At the angular position φk, the inner ring is subjected to axial load, radial load, overturning moment and the contact load of the steel ball to the inner channel, as shown in Figure 4.
Equations (14) to (16) constitute a three-variable nonlinear equation system in which δa, δr and θ are unknown quantities. Solving this system of equations can obtain the contact load distribution Qjφ of the steel ball.
2 Calculation of bearing life based on load distribution
According to the Lundberg-Palmgren theory, when calculating the rated life of the pitch bearing, first calculate the life of each channel, then calculate the rated life of a single ring, and finally fit the rated life of the entire bearing. In ordinary bearings, when the ball contacts the inner and outer channels, it is generally 2-point contact. Since the bearing channel discussed here is a peach-shaped channel, as shown in Figure 5, when a steel ball contacts the inner and outer rings, it is 4 points. point contact. Then the rated life of a single ferrule is the fitting value of the two channels in contact with the steel ball, and the channels on the contact pairs 1, 2, 3, and 4 are named as channels 1, 2, 3, and 4, respectively.
3 Calculation example and result analysis
Taking a certain type of double-row four-point contact ball slewing bearing as an example to calculate the fatigue life, its structural parameters are: ball group pitch circle diameter Dpw=2215mm, steel ball diameter Dw=44.45mm, initial contact angle α0=45°, two The center distance between the rows of steel balls is dc=69mm, the curvature radius coefficient of the inner and outer ring grooves is fi=fe=0.525, and the total number of steel balls is Z=256. The steel ball and ferrule are made of 42CrMo steel, Poisson’s ratio ν=0.3, and elastic modulus E=207GPa. When the bearing speed ni=0.1r/min, the axial load Fa=250kN, the radial load Fr=140kN, and the overturning moment M=1300kN·m, calculate the fatigue life, take the bearing clearance as 0, -0.01 , -0.02, -0.03, -0.04, -0.05, -0.06 and -0.1mm, substitute the above parameters into equations (14) to (16), and use the Newton-Raphson iteration method Calculate δa, δr and θ, and calculate the normal contact load of the bearing at different position angles according to formula (13). Substitute the required results and parameters into equations (17) to (25) to calculate the rated life of the bearing ring and the complete set of bearings. The results are shown in Table 1. It can be seen from the table that with the increase of the absolute value of the negative clearance of the bearing, the rated life of the pitch bearing first increases and then decreases. The pitch bearing generally requires a service life of 20 years , which is equivalent to 175200h, and can meet the requirements when the bearing clearance is -0.04 ~ 0mm. Therefore, in order to meet the requirements of bearing life when designing pitch bearings, a reasonable bearing clearance should be selected.
When the bearing clearance is 0 and different bearing groove curvature radii are used, the calculation results of the rated life of the ring and the entire bearing are shown in Table 2. It can be seen from Table 2 that the larger the curvature radius coefficient of the pitch bearing groove, the shorter the bearing life. This is mainly because with the increase of the groove curvature radius coefficient, the tightness of the steel ball and the groove decreases, and the contact ellipse area is relatively small under the same load, and the contact stress is large, thereby reducing the bearing life.
With the increase of the absolute value of the negative clearance of the pitch bearing, the rated life first increases and then decreases. When the pitch bearing bears the combined load, the calculation method in this paper can be used to calculate the fatigue life and select a reasonable bearing clearance. When other structural parameters are determined, the larger the bearing groove curvature radius coefficient is, the smaller the rated life of the bearing will be.