# Influence of raceway ovality on bearing fatigue life

**Abstract:** The gap distribution between the elliptical raceway and the rolling element enveloping circle is analyzed, and the contact load distribution of the rolling element is determined on this basis.

The theory of Lundberg and Palmgen calculates the rated fatigue life of the bearing, and finally gives an example of the calculation. Analysis and example calculations show that the ovality of the raceway has a significant impact on the fatigue life of the bearing and should be strictly controlled.

During the cold and hot working of the bearing ring and the installation and matching of the bearing, the deformation of the ring is inevitable. Deformation will cause the raceway diameter to deviate in the circumferential direction, affecting the load distribution of the rolling elements, thereby affecting the service life of the bearing. As a special case, the influence of raceway ovality on bearing life will be analyzed, and the requirements for raceway ovality control will be put forward.

**1 Radial gap between enveloping circle and ellipse**

Assume that the outer raceway of the bearing is an ellipse, and its long and short semi-axes are a and b respectively; the inner raceway is a circle, the rolling elements are attached to the inner raceway, and the outer enveloping circle radius of the rolling elements is b . Under the assumption of rigid ferrule and zero clearance, two typical cases of pure radial load acting on the major axis and minor axis are considered, as shown in Figure 1.

where: κ is the ellipse parameter, κ = b /a . Define ellipticity λ = 1 /κ = a /b , the closer λ is to 1, the closer the ellipse is to a circle; the larger λ is, the flatter the ellipse is.

**2 Contact displacement and contact load**

If the outer ring is fixed, the contact angle of the bearing is zero. Under the action of radial load Fr, the radial displacement of the inner ring is δr, then the radial contact displacement at the φ angle is [2] δφ = δrcos φ – Δφ.

**3 Bearing fatigue life**

According to the theory of Lundberg and Palmgen [3, 5], set the rated dynamic load of the bearing ring as Qci(e) (the subscript i and e represent the inner and outer rings respectively), for the radial ball bearing and the roller bearing, the formula In: f is the groove curvature radius coefficient; γ = Dwcos α/Dpw ; α is

Contact angle; Dpw is the diameter of the pitch circle of the rolling element group; Dw is the diameter of the rolling element; l is the length of the roller; the upper and lower operators apply to the inner and outer rings respectively. Let the equivalent load of the ferrule be Qeμ( ν) , and the subscripts μ and ν represent the rotating ferrule and the stationary ferrule, respectively.

bm is the life correction factor, and bm = 1 for ball bearings. 3, t = 10/9; for roller bearings bm = 1. 1, t = 9/8; L is in units of 106r. When the rolling element load Qφi is known, the bearing life can be calculated using the above formula.

**4 Calculation Examples**

Taking deep groove ball bearing 6010 as an example, it is known that Dw = 9 mm, Z = 13 , α = 0 , fi = 0. 515 , fe = 0. 525, Dpw = 65mm, Fr = 1 600 N; the inner channel is round, the outer channel is ellipse, the clearance is zero, and the inner ring rotates. Assuming that the minor semi-axis of the ellipse is unchanged, b = ( Dpw + Dw ) /2 = 37 mm , the major semi-axis a = λb , which changes with the ellipticity λ, the maximum diameter variation of the ellipse Δd = 2a – 2b = 2b( λ – 1) . According to the above method, first calculate the gap distribution between the inner and outer channels, then calculate the load distribution of the steel ball, and finally calculate the rated life of the bearing. lifespan. Figure 2 shows the corresponding change curve.

It can be seen from Figure 2 and Table 1 that when the load acts in the direction of the long axis of the ellipse (curve LA in Figure 2), the channel ellipticity λ =

1 to 1. 000 7, the bearing fatigue life is higher than the bearing life of the circular channel. If able to press the cart during manufacture and installation. It is speculated that the machining accuracy of the PI cage is not as high as that of the aluminum bronze cage; in addition, when the cage rotates at high speed, its center of mass is not in the center of the bearing, but moves radially [4]. Then reduce the outer diameter of the PI-1 cage by 0. 2 mm (PI – 2 cage), that is, the bearing design guide clearance increases 0. 6 mm, when the bearing speed is about 50 000r/min, the guide clearance of the bearing equipped with the PI-2 cage is 0. 58 mm, test for 15 minutes, the bearing runs smoothly, and the test parameters such as outer ring temperature, host current and host vibration are stable. Therefore, the design guide clearance of the PI cage should be 0. 9 mm, instead of aluminum bronze cage 0． 3 mm guide clearance.

**5 Conclusion**

Through the experimental analysis of the QJS207 bearing aluminum bronze cage and the PI cage with different guiding gaps, it is considered that the design guiding gap of the high temperature and high speed angular contact ball bearing is not only related to the bearing speed and working temperature, but also closely related to the cage material. The cage material determines the cage’s machining accuracy, centrifugal expansion, thermal expansion, and the movement of the cage’s center of mass. For the given working conditions of the QJS207 bearing, the design guide clearance of the aluminum bronze cage should be 0. 3 mm, the design guide clearance of the PI cage should be 0. 9 mm, but the pocket clearance of the cages of the two materials can take the same value, that is, both are 0. 3 mm.