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Abstract: A dynamic calculating model for angular contact ball bearings is established based on analysis of interaction among parts of bearings． Combing with calculation method for fatigue life based on contact load on micro area，taking the angular contact ball bearings for rotor system of gas － turbine engine as examples，the influence of rotation mode for rings on performance of bearings is analyzed，which provide reference for optimal design of bearings.
Key words: angular contact ball bearing; ring; rotation mode; dynamics; fatigue life
Under high-speed operating conditions, angular contact ball bearings often experience steel ball slippage, unstable retainers, and other issues, resulting in excessive temperature rise and frequent impacts, leading to early bearing failure. To study the potential performance degradation of rolling bearings under high-speed conditions, many scholars have established bearing dynamics analysis methods through theoretical analysis and modeling. Famous international bearing companies have also developed professional bearing calculation and analysis software. At present, these analysis methods and calculation software are widely used for bearing failure analysis and structural design, but in studying the impact of operating conditions on bearing performance, they are mostly focused on external loads and speeds.
The actual bearing speed refers to the relative speed of the inner and outer rings of the bearing, and their respective changes in motion will also affect the bearing's load-bearing performance and dynamic performance. Therefore, it is necessary to further study the influence of the rotation form of the ring on the bearing's fatigue life and dynamic performance. In the article, the dynamic model of angular contact ball bearings and the fatigue life calculation method based on bearing micro contact are established to analyze the effects of four different rotation methods on bearing performance: fixed inner ring rotation, fixed outer ring rotation, co rotating inner and outer rings, and reverse rotating inner and outer rings.
1. Dynamic model and fatigue life calculation of angular contact ball bearings
1.1 Dynamic Model of Angular Contact Ball Bearing
The internal forces of angular contact ball bearings include the interaction forces between parts, as well as the blocking force and torque of lubricating oil on the steel ball. Considering the force situation of all loads on the ball, as shown in Figure 1, the coordinate system Ob xb yb zb is the body coordinate system of the steel ball, and the origin Ob is located at the center of mass of the steel ball, The xb axis is parallel to the bearing axis, The zb axis points from the center of the bearing towards the center of the steel ball, The direction of the yb axis is determined according to the right-hand rule of the coordinate system.
Figure 1 Force and torque acting on the ball
According to the force analysis of the ball, taking into account the combined force and moment of the j-th ball, its motion equation is:
In the formula, F μ and Fbp represent the drag force between the channel/ball and the force between the cage pocket/ball, respectively; M μ and Mbp are the drag torque between the channel/ball and the action torque between the cage/ball, respectively; FD and Me respectively represent the blocking force and torque of lubricating oil on the ball, which can be calculated based on Schlichting's fluid theory; Mb and Ib are the mass and moment of inertia of the ball, respectively; ω · is the angular acceleration; Rb is the radius of revolution of the ball; The subscript j represents the jth ball; Subscript x, y. Z represents the components of the relevant parameters in three coordinate directions.
The drag force F μ between the channel and the ball is
In the formula, s is the slip to roll ratio, calculated from kinematic analysis.
The contact load Q between the channel and the ball is calculated using Hertz contact theory, and the contact load between the j-th ball and the channel is
In the formula, the stiffness K is calculated according to reference [10]; The contact deformation δ is determined based on the positional relationship between the ball and the channel in Figure 2, where the coordinate system Oxyz, O2 x2 y2 z2 and Obj xbj ybjzbj are the fixed coordinate system of the bearing, the inner ring coordinate system, and the body coordinate system of the j-th ball, respectively.
The transformation matrix from the fixed coordinate system to the inner circle coordinate system can be obtained based on the deflection angles θy and θz of the inner circle relative to the outer circle
The contact deformation δ j between the jth ball and the channel is
In the formula, lrb is the distance from the center of the channel to the center of the ball; F is the groove curvature coefficient of the ring; Dw is the diameter of the steel ball.
Figure 2 Schematic diagram of the position of the channel and steel ball
The force Fbp between the ball and the pocket hole can be determined based on the relationship between the root mean square value Δ t of the surface roughness of the ball and the pocket hole of the cage, as well as the minimum gap Δ bh between the two objects. When Δ bh>Δ t, the action between the ball and the pocket hole is the fluid dynamic pressure, Fbp can be calculated by solving the Reynolds equation; When δ bh<Δ t, it is considered that the interaction between the ball and the pocket is Hertz contact, Fbp is calculated using Hertz line contact theory.
The relative position relationship between the ball and the cage pocket hole is shown in Figure 3, and the minimum gap δ bh between the ball and pocket hole is
In the formula: Opj Obj is the sum of vectors from the center of the cage pocket to the center of the ball; Δ bh is determined by the initial gap between the cage pocket and the ball.
Figure 3 Relationship between the relative position of the cage pocket hole and the ball
Similarly, based on the force on the cage, the differential equation of motion is derived as
In the formula, Fl and Ml respectively represent the force and torque between the cage and the guide ring; MC and Ic are the mass and moment of inertia of the cage; Dc is the diameter of the cage; Z is the number of steel balls; θ j is the azimuth angle of the j-th ball;
The action of the retainer and guide ring is shown in Figure 4, and the force between the two can be approximated as the fluid dynamic pressure borne by the short sliding bearing, i.e
In the formula, ε and Cg respectively represent the eccentricity of the cage and the bearing guide clearance; η is the viscosity of the lubricating oil; R and B are the radius and width of the guiding surface, respectively; ω c is the bearing speed; Ω 1 is the outer ring speed; Omega 2 is the inner ring speed. When the cage adopts the inner ring guidance method, the symbol before the calculation formula is taken as positive; When using the outer circle guidance method, the sign before the calculation formula is taken as negative. In addition, the result calculated by equation (8) is the component of the cage in the dynamic coordinate system, which can be converted into the component Fly in the inertial coordinate system by coordinate transformation based on the angle of action (d), Flz and Ml.
a) Internal guidance (b) External guidance
Figure 4 Schematic diagram of the interaction between the cage and guide surface. The dynamic model of the angular contact ball bearing established was solved using the fourth-order Runge Kutta method with adaptive step size through the motion differential equations of the joint ball and cage.
1.2 Fatigue life calculation based on micro contact of bearings
According to the L-P fatigue life theory, the overall life of a bearing can be equivalently represented by the contact fatigue life of each point in the contact area as
In the formula, ij is the ratio of the inner ring speed to the rotational speed of the j-th ball; Lc1j and Lc2j are the fatigue lives of the points in contact with the j-th ball on the outer and inner raceways, respectively; Lb1j and Lb2j are the fatigue lives of the jth ball at the contact points with the outer and inner raceways, respectively. The required parameters for calculation, such as internal contact load, contact angle, and steel ball speed of the bearing, are obtained from the dynamic model of the angular contact ball bearing. E is generally taken as 3.
2. The influence of ring rotation method on bearing performance
Taking the front pivot rolling bearing of a certain gas turbine engine rotor system as an example, the material adopts a hybrid ceramic ball design, and the lubrication uses 4050 aviation lubricating oil. The bearing bears an axial load of 400 N and a radial load of 350 N. Four different rotation methods are used: outer ring fixed inner ring rotation, inner ring fixed outer ring rotation, inner and outer ring rotating in the same direction, and inner and outer ring rotating in the opposite direction.
The maximum contact load and contact stress inside the bearing calculated under different ring rotation modes are shown in Figure 5. As shown in the figure, the influence of the rotation mode of the ring on the maximum contact load and contact stress inside the bearing is mainly reflected between the outer ring and the steel ball. The reason is that different rotation modes of the ring cause changes in the rotation speed of the steel ball, resulting in significant changes in the contact load and contact stress between the steel ball and the outer ring groove under the action of centrifugal force. In addition, the influence of the rotation method of the ring on the contact stress of the inner and outer ring channels is also inconsistent. In working condition 4, there was a phenomenon where the contact stress of the inner ring was greater than that of the outer ring. The reason is that the steel ball and the inner ring are in reverse curved contact, while the outer ring is in same curved contact. The main curvature of the contact pair and the difference in the contact result in inconsistent stress changes.
The calculated bearing fatigue life and steel ball speed under different ring rotation modes are shown in Figure 6. By comparing Figures 5 and 6, it can be inferred that the main reason for the variation in bearing fatigue life under different rotation modes of the ring is the difference in maximum contact stress of the outer ring groove. In addition, due to the different rotation modes of the ring and the different rotational speeds of the steel ball, the difference in contact frequency between the ring and the steel ball caused by this will also have an impact on the fatigue life of the bearing.
Figure 5: Effect of Ring Rotation Method on the Maximum Contact Load and Contact Stress in the Bearing Bearing Load Zone
Figure 6 Effect of Ring Rotation Method on Bearing Fatigue Life and Average Speed of Steel Balls
The calculated cage sliding rate under different ring rotation methods is shown in Figure 7. The difference in the sliding rate of the cage is caused by changes in the speed and centrifugal force of the steel ball under different rotation modes of the ring. The difference between condition 2 and condition 1 is mainly due to the fact that condition 2 uses the outer ring as the rotating sleeve, and the drag force of the outer ring groove on the steel ball increases due to the centrifugal force generated by the rotation of the steel ball, thereby improving the effective drag of the groove on the steel ball and ultimately reducing the sliding rate of the cage; Working condition 3, due to the simultaneous rotation of the inner and outer rings, causes the steel ball to be dragged by both the inner and outer rings, reducing the slip between the steel ball and the groove, ultimately maintaining a lower level of cage sliding rate; In working condition 4, although the dragging direction of the inner and outer rings on the steel ball is opposite, the dragging effect will be affected to a certain extent. However, the reverse rotation of the inner and outer rings also reduces the revolution speed of the steel ball, thereby controlling the sliding between the steel ball and the ring groove, and ultimately keeping the sliding rate of the retaining frame at a relatively low level.
Figure 7: The effect of ring rotation on the sliding rate of the cage
The vortex trajectory and velocity deviation of the cage center of mass under different ring rotation modes are shown in Figure 8. The deviation ratio of the vortex velocity at the center of mass of the cage is a quantitative criterion for determining the instability of the vortex at the center of mass of the cage. It is defined as the ratio of the standard deviation value of the velocity vector to its average value, i.e
In the formula: vi is the instantaneous velocity of the center of mass of the cage; VM is the average velocity of the center of mass of the cage.
Figure 8: The effect of ring rotation method on the vortex trajectory and vortex velocity deviation ratio of the cage center of mass
As shown in Figure 8, under the reverse rotation mode of the inner and outer rings, the lowest revolution speed of the steel ball is most conducive to the stability of the cage, which can reduce the premature failure caused by frequent impact between the cage pocket hole and the steel ball, as well as frequent collision and wear between the cage and the guide surface of the ring. Based on the above analysis, compared with the rotation method of inner ring rotation and outer ring stationary, the impact of three rotation methods of outer ring rotation and inner ring stationary, inner and outer ring co rotating, and inner and outer ring reverse rotating on bearing performance is studied. By comparison, the reverse rotation of the inner and outer rings is the most favorable rotation method for bearing performance.
3. Conclusion
By establishing a dynamic model of angular contact ball bearings and using a fatigue life calculation method based on bearing micro contact, the influence of ring rotation mode on the fatigue life and dynamic performance of the front fulcrum bearing in a certain gas turbine engine rotor system was analyzed as an example. The following conclusions were drawn:
1) The rotation mode of the ring has a significant impact on the fatigue life of the bearing and the sliding rate of the cage, mainly due to the difference in contact stress and frequency caused by the speed difference of the steel ball. The reverse rotation of the inner and outer rings of the bearing is most beneficial for improving the fatigue life of the bearing; The rotation of the inner and outer rings of the bearing in the same direction is most conducive to reducing the sliding rate of the bearing cage, thereby reducing the early failure caused by excessive temperature rise inside the bearing.
2) The reverse rotation of the inner and outer rings of the bearing is most beneficial for the stability of the bearing cage, thereby reducing premature failure caused by frequent impact between the cage pocket and the ball, as well as frequent collision and wear between the cage and the ring guide surface.
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2024-05-31