Wear depth calculation and influence factor analysis for groove ball bearing

. Based on Hertz contact theory and load distribution, the formulas for contact stress cycle times, slip distance and wear depth measurements are derived, and the influences of load, curvature coefficient, roll body diameter and friction coefficient on the contact region wear depth and distributions are thoroughly analyzed. The results show that the wear depth is zero at the pure rolling point and the long half-axle terminals of contact ellipse, and reaches maximum value near by the long half-axle terminals of the contact ellipse, and further shows that the wear depth increases with increase of the load and friction coefficient, however decreases with increase of the curvature coefficient and roll body diameter.


Introduction
Higher speed roll bearing in the aviation engine and its transmission system has merits of low friction coefficient, small starting moment and strong anti-off oil performance, etc. The groove ball bearing could not only suffer radial load, but also suffer the radial load and a certain amount of the axial load. Whereas, the roll body can occur direct metal contact between inner and outer race which could lead to severe friction and wear under heavy load condition at high speed, especially under starved lubrication. Consequently, the geometry and drive precision could be severely affected, and even slight geometric change would also affect the stress distribution and service life of the bearing. Moreover, the bearing clearance would also increase with increase of the wearing depth which may further result in the decrease of operation and sustaining precisions and the increase of vibration and noise.
In order to well master operation performance and structure design of the groove ball bearing, scholars have focused on the friction and wearing studies in recent years. Fitzsimmons [1] investigated the influence of impurity density, particle size and hardness in lubricants on wear of taped roller bearing. Olofsson，et al. [2][3][4][5][6][7] derived the wear value of spherical thrust roller bearing under boundary lubrication according to Archard wear theory, and conducted the wear experiment. The test results show better consistence with that of the theoretic calculation. Miki, et al. [8] studied the influence of rotation speed and lubricant viscosity on the wear value. Value. Zhang Yicheng, et al. [10] proposed a formula for slip wear of the high pair based upon the energy intensity theory, which well improved the wear where, Σρ is the sum of main curvatures, E' and Q are the comprehensive elastic module and maximum load of roller element respectively. Herein, the coefficients of a * , b * , δ * could be obtained by employing the calculation method or parameter table [12] .

Calculation method of contact stress cycle times
For the total wear, the roller element number passing on a point of the inner or outer race in unit time should be obtained in advance. Assuming the roller element number is Z in the groove ball bearing, passing on the inner and outer race is The roller element numbers passing on one point are not surely equal to the stress cycle times at this point. If the included angle of load distribution region is 2ΨL and the inner race rotation is n, the stress cycle times passing on one point of the inner and outer race per minute respectively are The stress in the race contact region may occur a certain of variations in the cyclic process because the roll bearing suffers load distribution in Cosine function. When the race rotates to the maximum loaded area, the contact stress is the maximum one. Whereas, when the race rotates to other loaded area, the contact stress behaves in Cosine distribution. Assuming a roller bearing has 5 loaded roller elements, the inner race stress, the near two roller element stress and the other two roller element stress are p0 , p1 and p2. Consequently, one point stress cycle times on the inner race is Furthermore, when the loaded roller element is in condition of odd pressure, and the loaded roller element number is z0, as well as kn=(z0-1)/2, then, we obtain If the loaded roller element is in condition of even pressure, as well as kn=z0/2, such that where, bx is the half length of point (x, y) in perpendicular direction with respect to the x axis, and V0 is the sliding speed of contact region, i. e., the relative speed of roller element with respect to race along the short half-axis. According to distribution law of the elliptical contact and speed, Vx is formulated as

Calculation of wear depth
According to [10], the wear rate of ball bearing can be obtained by where, HB is the hardness of material, ε0 is the extension rate when the material is broken, p is the contact stress, and r, Rmax are the classic curvature radius and maximum height of micro-bulge part on the contact surface, ψ, χ are the curvature coefficients of support surfaces, k, m, c are the experiential values, respectively. Considering of the influence of lubricant on the contact region, ξ(f) f is taken values of f, 1+f or (1+f 2 ) 1/2 . Setting where, G0 is the wear constant.
As a result, the contact ellipse wear depth, at a distance x with respect to the half axle, can be calculated as

Analysis of influence factor on wear depth
It is found that the wear depth is zero at the pure rolling point x=0.348a and at the terminals of ellipse long half-axis from the Figs.2-5, in which the wear depth initially increases to maximum value and then decreases to zero at the terminals, as well as the wear depth shows nearly parabolic distribution between two pure rolling points. Fig.2-4 further show the wear depth affected by the load, curvature coefficient and roller element diameter when the friction coefficient is 0.007, respectively. Amongst, the wear depth is strongly affected by the load and rapidly increases with the increase of load, as shown in Fig.2, because the load affects the wear depth with m power and also affects the slip distance. Fig.3 illustrates that the roller element diameter yields slight influence on the wear depth, which decreases with the increase of roller element diameter, and Fig.4 shows that the wear depth is also slightly affected by the curvature coefficient and decreases with the increase of curvature coefficient, however, the curvature coefficient affects the wear depth more heavy than that of the roller element diameter. Fig.5 further shows the influence of friction coefficient on the wear depth, from which it is found that the wear depth is strongly affected by the friction coefficient with m power of friction coefficient.  (1) The formulas are derived for calculating the stress cycle times for one point of race passing through different loaded regions, and for calculating the slip distance and total wear depth under a single stress action.
(2) It is concluded that wear depth holds a smallest value of zero at the pure rolling point and at the ellipse long half-axis terminals, and the maximum wear depth does not locate in the center of elliptical contact but nearby the long half-axis terminals.