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Abstract: After analysis on geometric parameters and contact stress inside a deep groove ball bearing,the current relation of contact stress Coe and raceway groove curvature radius coefficient is plotted according to calculation method for contact stress limit without regard to the bearing internal clearance and the selective method is explained for the raceway groove curvature radius coefficient.
Key words: deep groove ball bearing; raceway groove curvature radius coefficient ;geometry parameters; contact stress
1. Contact stress and deformation of ball bearings
Regardless of the bearing clearance, when the steel ball contacts the inner and outer grooves of the bearing, the geometric parameters near the contact point are specified as follows: dm is the diameter of the ball joint circle, Ri and Re are the curvature radii of the grooves of the inner and outer rings, respectively.
Dw - Steel ball diameter,
fi, fe - coefficient of groove curvature radius of the inner and outer rings.
The curvature and curvature difference functions at the contact between the steel ball and the channel are:
(3) The upper and lower operators in equation (4) are applicable to the inner and outer channels respectively. Under load, for steel bearings, the calculation formula for Hertz contact deformation and contact stress is
In the formula, Q - maximum radial load of the steel ball,
a、b - the major and minor axes of the contact ellipse,
σmax - maximum contact stress,
a *, b * - f(ρ) mapping of, where 
, 1＜k.
Because both a * and b * are f（ ρ) By combining equations (5), (6), and (7), we can obtain:
2. Coe curves corresponding to different groove curvature radius coefficients
Equation (8) can be transformed into:
Among them, 
. The range of f generally selected in China is 0.51~0.54; The positive correlation between the channel curvature function A* and (both a* and b* are positively correlated with the channel curvature function). For a certain bearing, Q can be regarded as a constant, then it can be known that σmax is only related to the changes in Coe, with 
as the horizontal axis and Coe as the vertical axis. Draw Coe for different fi, fe, and time to obtain Figure 1. From Figure 1, it can be observed that:
(1) Whenγcertain, the Coe of both the inner and outer channels increases with the increase of f; This can be explained as follows: when the coefficient of curvature radius of the groove is large, the groove has a larger groove radius, and the tightness between the groove and the steel ball is low (i.e. the contact area between the groove and the steel ball is small). Under the same pressure, the pressure is greater. At the same time, when the γ is equal to f, the outer channel Coe is smaller than the inner channel Coe, which can also be understood as the stress in the outer channel being smaller than the stress in the inner channel. This is because the outer channel is inscribed with the steel ball and the inner channel is inscribed.
(2) The Co e values of the inner and outer channels do not vary linearly, but decrease with the increase of f value; The Coe value of the internal channel varies decreases with γ increase. Both of these are due to reasons (3) and (4).
3. Determination of the curvature radius coefficient of the inner and outer grooves
For a preliminarily designed bearing, given the bearing load and the diameter of the steel ball used, a groove curvature coefficient can be determined based on Figure 1, and then another coefficient can be selected based on the principle of equal or close stress. When selecting, different interpolation choices should be made according to different interval ranges to select the appropriate groove curvature coefficient. Considering the influence of centrifugal force during bearing operation, it is generally necessary to ensure that the groove curvature coefficient obtained is such that the Coe value of the outer ring is smaller than the Coe value of the inner ring.
Figure 1: Different Coe under Different Curvature Radius Coefficients
When the speed is 2 000r/min, dm=65mm, the maximum rolling element load is 1 000N, Dw=12.7, and the bearing preload and bearing clearance are not considered, fi=0.515 and f e=0.53 are selected to ensure that the maximum stress does not exceed 1800 MPa. The maximum stresses in the inner and outer grooves are 1 766 MPa and 1 705 MPa, respectively. When the speed increases to 40000 r/min, if the above parameters are still used, the inner and outer ring stresses will be 1748 MPa and 2065 MPa respectively, which is not allowed under long-term working conditions. At this point, it is necessary to change the channel coefficient of the outer circle, and in this case, the formula
A centrifugal force of approximately 779N can be considered for calculation, using
Find fi=0.515, fe=0.514, and Coe is equal. The calculated stresses are 1766MPa and 1764MPa, respectively.
4. Conclusion
The influence of fi and fe on the performance of bearings is crucial, and the selection of fi and fe mainly considers that the ultimate stress and stress of the inner and outer grooves of the bearing are equal or similar. Generally, selecting a suitable set of fi and fe requires a lot of calculation. Figure 1 intuitively solves this problem. After determining the diameter of the rolling element and knowing the bearing speed, suitable fi and fe can be selected to maximize the design of the bearing to meet the usage requirements.
More about KYOCM Deep Groove Ball Bearing:
Deep groove ball bearings are the most widely used bearing type and are particularly versatile. They have low friction and are optimized for low noise and low vibration which enables high rotational speeds. They accommodate radial and axial loads in both directions, are easy to mount, and require less maintenance than other bearing types.
Features and benefits:
Low friction and running temperatures, low noise and vibration
High running speeds
High quality and performance capabilities for your application
Accommodate radial loads and axial loads in both directions
Available with a variety of greases for most conditions, including food quality grease, high temperature grease and Solid Oil
Increased reliability and provide long bearing and lubricant service life
http://www.kyocm.com/products/Deep-Groove-Ball-Bearing/732.html
2023-10-19