Comparative Analysis of Load Distribution and Dynamic Stiffness Characteristics of Back-to-Back Combined Ball Bearings
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摘要:
支承轴承的动态载荷分布及刚度特性是引起机床振动和切削稳定性的重要因素，为对比分析TBT （back-to-back）组合式球轴承左列和右列在不同工况下的动态载荷分布与动刚度特性，本文基于非线性弹性Hertz接触理论及Jones-Harris模型，结合滚动体-滚道接触状态判定准则，提出多列组合球轴承五自由度分析模型；采用改进的迭代算法求解，大大提高迭代计算在不同外界条件波动下的求解收敛性，得到恒定预紧力作用下TBT组合式轴承组内轴承动态载荷分布和动刚度特性. 研究结果表明：转速、径向载荷、轴向载荷均会改变组合轴承的载荷分布及动刚度特性，且对左列和右列轴承影响效果不同；转速增加会导致组内轴承内部产生非完全接触区域，使轴承载荷分布发生振荡；相较于单个轴承在等效外界载荷条件下，组合轴承的径向动刚度较轴向动刚度提升效果更明显.
Abstract:The dynamic load distribution and stiffness characteristics of bearings are important factors that cause vibration and cutting stability of machine tools. On the basis of nonlinear elastic Hertz contact theory and Jones-Harris model, combined with the judgment criterion of roller–raceway contact state, a five-degree-of-freedom analysis model of multi-row combined ball bearings was proposed to compare dynamic load distribution and dynamic stiffness characteristics of left and right rows in tandem back-to-back (TBT) combined ball bearings under different operating conditions. The improved iterative algorithm was adopted for solution, which greatly improved the convergence of the iterative calculation under the fluctuation of different external conditions and yielded the dynamic load distribution and dynamic stiffness characteristics within the TBT combined bearings under constant preload. The results show that the rotational speed, radial load, and axial load can change the load distribution and dynamic stiffness characteristics of combined bearings, and have different effects on the left-row and right-row bearings. The increase in rotational speed can lead to the incomplete contact area within the bearings, which makes the load distribution of the bearing oscillate. Compared with that of a single bearing under equivalent external load conditions, the radial dynamic stiffness of the combined bearing is more significantly improved than the axial dynamic stiffness.
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Key words:
- ball bearing /
- contact state analysis /
- dynamic load distribution /
- dynamic stiffness /
- contact angle
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图 3 滚动体-内滚道动态接触几何关系
Figure 3. Dynamic contact geometry of rolling body–internal raceway
图 4 接触状态时几何位移协调变形关系
Figure 4. Coordinated deformation relationship of geometric displacement in contact state
图 5 分离状态时几何位移协调变形关系
Figure 5. Coordinated deformation relationship of geometric displacement in separated state
图 9 TBT组合轴承动态载荷分布计算及动刚度求解流程
Figure 9. Dynamic load distribution calculation and dynamic stiffness solution process for TBT combined bearings
图 10 左列轴承接触角及载荷分布随转速变化曲线
Figure 10. Curves of contact angle and load distribution of left-row bearings with speed variation
图 11 右列轴承接触角及载荷分布随转速变化曲线
Figure 11. Curves of contact angle and load distribution of right-row bearing with speed variation
图 12 转速对TBT组合轴承径向动刚度影响
Figure 12. Influence of rotational speed on radial dynamic stiffness of TBT combined bearing
图 13 转速对TBT组合轴承轴向动刚度影响
Figure 13. Influence of rotational speed on axial dynamic stiffness of TBT combined bearing
图 14 转速对单个轴承径向动刚度影响
Figure 14. Influence of rotational speed on radial dynamic stiffness of single bearing
图 16 左列轴承接触角及载荷分布随径向载荷变化曲线
Figure 16. Curves of contact angle and load distribution of left-row bearing with radial load
图 17 右列轴承接触角及载荷分布随径向载荷变化曲线
Figure 17. Curves of contact angle and load distribution of right-row bearing with radial load
图 15 转速对TBT组合轴承轴向动刚度影响
Figure 15. Influence of rotational speed on axial dynamic stiffness of TBT combined bearing
图 18 径向载荷对TBT组合轴承径向动刚度影响
Figure 18. Influence of radial load on radial dynamic stiffness of TBT combined bearing
图 19 径向载荷对TBT组合轴承轴向动刚度影响
Figure 19. Influence of radial load on axial dynamic stiffness of TBT combined bearing
图 20 径向载荷对单个轴承径向动刚度影响
Figure 20. Influence of radial load on radial dynamic stiffness of single bearing
图 21 径向载荷对TBT组合轴承轴向动刚度影响图
Figure 21. Influence of radial load on axial dynamic stiffness of TBT combined bearing
图 22 左列轴承接触角及载荷分布随轴向载荷变化曲线
Figure 22. Curves of contact angle and load distribution of left-row bearing with axial load
图 23 右列轴承接触角及载荷分布随轴向载荷变化曲线
Figure 23. Curves of contact angle and load distribution of right-row bearing with axial load
图 24 轴向载荷对TBT组合轴承径向动刚度影响
Figure 24. Influence of axial load on radial dynamic stiffness of TBT combined bearing
图 25 轴向载荷对TBT组合轴承轴向动刚度影响
Figure 25. Influence of axial load on axial dynamic stiffness of TBT combined bearing
图 26 轴向载荷对单个轴承径向动刚度影响
Figure 26. Influence of axial load on radial dynamic stiffness of single bearing
图 27 轴向载荷对TBT组合轴承轴向动刚度影响
Figure 27. Influence of axial load on axial dynamic stiffness of TBT combined bearing
表 2 轴承刚度对比结果
Table 2. Comparison results of bearing stiffness
实验组别 ${K_{{\textit{zz}}}}$/（N·mm−1） ${K_{xx}}$/（N·mm−1） Kyy/（N·mm−1） ${K_{{\mathrm{M}}x}}$/（N·mm−1） KMy/（N·mm−1） 本文结果 226453.90 165949.90 165934.90 147401889.30 147401889.30 文献[16] 225376.50 166573.80 166573.80 144386234.10 144386234.10 文献[17] 226011.10 172131.60 155520.70 139304365.40 157160958.50 文献[16]误差/% −0.28 −3.23 7.11 3.65 −8.13 文献[17]误差/% −0.24 −3.59 6.70 5.81 −6.21 
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表 3 SKF 7012ACE/HCP4A轴承参数
Table 3. 7012ACE/HCP4A bearing parameters
轴承参数 符号 数值 轴承内滚道曲率半径/mm ri 4.08807 轴承外滚道曲率半径/mm ro 4.16745 初始接触角/（°） ${\alpha _0}$ 25 滚动体数量/个 Zr 25 滚动体直径/mm D 7.938 轴承节圆直径/mm Dm 77.591
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