工程科学与技术   2021, Vol. 53 Issue (2): 178-186

1. 北京工业大学 材料与制造学部 先进制造技术重点实验室，北京 100124;
2. 电火花加工技术北京市重点实验室，北京 100191;
3. 中国铁道科学研究院 标准计量研究所，北京 100081

Non-Hertzian Contact Stress Calculation of High-speed Ball Screw Mechanism
WANG Min1,2, SUN Tiewei1, DONG Zhaoyang1, KONG Deshun3, GAO Xiangsheng1
1. Beijing Key Lab. of Advanced Manufacturing Technol., Faculty of Materials and Manufacturing, Beijing Univ. of Technol., Beijing 100124, China;
2. Beijing Key Lab. of Electro-Machining Technol., Beijing 100191, China;
3. Inst. of Standard Metrology, Chinese Academy of Railway Sciences, Beijing 100081, China
Abstract: During the movement of the ball screw mechanism (BSM), the friction and wear between the balls and raceway cause the accuracy to deteriorate and reduce its accuracy retention. For the asymmetry of the contact area of the helical raceway of the high-speed BSM, there is a big error in the calculation of the contact geometry and relationship between the screw and the nut raceway by using the traditional Hertzian contact theory. Based on the minimum excess principle, a method was proposed to analyze and calculate the normal contact stress between the ball and the raceway of the high-speed BSM. Firstly, a BSM coordinate system was established based on the Frenet-Serret coordinate conversion formula to geometrically describe the contact between the ball and the raceway at the contact point, and then re-gridded by a two-dimensional interpolation algorithm to facilitate subsequent contact mechanics analysis; the contact area and control equation between the ball and raceway were discretized to improve the accuracy of numerical calculations. To solve the complex computational solution of the contact problem and the slow convergence, the contact problem was transformed by applying the variational fraction principle. The polar value problem was solved by using the conjugate gradient method for cyclic iteration to achieve fast convergence and improve the computational efficiency of the contact problem numerical solution, solving the contact stress distribution between the ball and the raceway. Since the calculation of the contact surface deformation was equivalent to the convolution between the influence coefficient matrix and the normal compressive stress, the two-dimensional convolution calculation, and the corresponding two-dimensional fast Fourier transform were used to solve the numerical solution of the contact problem. In numerical solutions, the main computational effort was focused on the computation of elastic deformations using the two-dimensional fast Fourier transform (FFT) technique Easy and fast calculation of elastic deformation in the contact area. Applying the minimum excess principle and Hertzian contact theory to solve the elastic contact stress distribution between a smooth ball and a plane, respectively, The results of the two calculation methods are compared, and the calculated values overlap well, which verifies the correctness of solving the contact stress distribution based on the principle of minimum excess; Based on the principle of minimum excess to analyze the influence of helix angle on the contact area and elastic deformation at the contact point of ball and raceway, and the results obtained by this method and Hertzian contact theoretical calculations for comparative analysis, as the BSM raceway helix angle increases, the error at the screw and ball contact point A is always greater than the error at the contact point of the nut and ball contact point B. The non-Hertzian solution contact stress peak at the contact point of the screw raceway and ball contact point A gradually becomes smaller, the non-Hertzian solution contact stress peak at the contact point of the nut raceway and ball contact point B gradually increases, and the stress peak at the contact point A is always greater than the peak at the contact point B. The non-Hertzian solution contact stress peak at the contact point of the nut raceway and ball contact point B gradually increases. The minimum excess principle non-Hertzian contact calculation method accurately calculates the stress distribution in the contact area of the ball and raceway and allows comprehensive and accurately calculate the width and depth of the wear zone of the raceway. Therefore, using the minimum excess principle non-Hertzian contact solution can improve the calculation accuracy of the contact stress distribution of the ball screw mechanism, and ensure the precision prediction retention of its accuracy.
Key words: ball screw mechanism    contact stress    minimum excess principle    conjugate gradient method    Hertzian contact

1 接触几何描述

 图1 滚珠丝杠副系统的坐标系 Fig. 1 Coordinate system of ball screw mechanism

 图2 滚珠丝杠副接触坐标系 Fig. 2 Contact coordinate system of ball screw mechanism

 ${\textit{z}}_{k}={r}_{{\rm{b}}}-\sqrt{{r}_{{\rm{b}}}{}^{2}-{x}_{k}{}^{2}-{y}_{k}{}^{2}}, k=A,B$ (1)

 \begin{aligned}[b] &{{P}}_{{A{\rm{S}}}}^{\rm{H}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {{r_{\rm{S}} }\cos\; {\beta _{A\rm{S}}} - ({r_{\rm{S}} } - {r_{\rm{b}}})\cos \;{\beta _A }} \\ {{r_{\rm{S}}}\sin \;{\beta _{A\rm{S}}} - ({r_{\rm{S}} } - {r_{\rm{b}} })\sin \;{\beta _A }} \\ 1 \end{array}} \right],\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;{\beta _A} - {\beta _{A0}} \le {\beta _{A{\rm{S}} }} \le {\beta _A } + {\beta _{A0}} \end{aligned} (2)

 \begin{aligned}[b] &{{P}}_{A{\rm{S}}}^A = {\left( {{{T}}_{\rm{S}}^{\rm{H}}({\theta _{\rm{C}}}){{T}}_{\rm{H}}^A} \right)^{ - 1}}\left( {{{T}}_{\rm{S}}^{\rm{H}}{{P}}_{A{\rm{S}}}^{\rm{H}}} \right),\\ &{\beta _A} - {\beta _{A0}} \le {\beta _{A{\rm{S}}}} \le {\beta _A} + {\beta _{A0}} \end{aligned} (3)

 \begin{aligned}[b] &{{P}}_{B{\rm{S}}}^B = {\left( {{{T}}_{\rm{S}}^{\rm{H}}({\theta _{\rm{C}}}){{T}}_{\rm{H}}^B} \right)^{ - 1}}\left( {{{T}}_{\rm{S}}^{\rm{H}}{{P}}_{{B\rm{S}}}^{\rm{H}}} \right),\\ &{\beta _B} - {\beta _{B0}} \le {\beta _{B{\rm{S}}}} \le {\beta _B} + {\beta _{B0}} \end{aligned} (4)

 图3 滚道曲面插值结果 Fig. 3 Interpolation results of raceway surface

2 接触区域的法向应力分析 2.1 接触区域的离散化

 \begin{aligned}[b] {{\rm{c}}_{i,j}} = {d_0} + {{\rm{c}}_{gi,j}} + {{\rm{c}}_{ri,j}} + {V_{i,j}}, 0\le i\le M, \;0\le j\le N \end{aligned} (5)

 ${V_{i,j}} = \sum\limits_{k = 0}^M {} \sum\limits_{l = 0}^N {{K_{i - k,j - l}}} {p_{i,j}}$ (6)

 ${c_{i,j}} \geq 0,\;{p_{i,j}} \ge 0$ (7)

 ${c_{i,j}}{p_{i,j}} = 0$ (8)
2.2 接触变形的计算方法

2.3 共轭梯度法求解接触压力分布

 ${{c}} = \tilde c + {{KP}}$ (9)
 ${c_{i,j}} \ge 0,{p_{i,j}} \ge 0,{c_{i,j}}{p_{i,j}} = 0$ (10)

 $W({{P}}) = {{{c}}^{\rm{T}}}{{P}} + \frac{1}{2}{{{P}}^{\rm{T}}}{{KP}}$ (11)

3 计算结果验证与分析 3.1 计算结果验证

 $p(r) = \frac{{3{F_\textit{z}}}}{{2\text{π}{a^2}}}{\left[ {1 - {{\left( {\frac{r}{a}} \right)}^2}} \right]^{1/2}}$ (12)

 ${p_{\max }} = p(0) = \frac{{3{F_{\textit{z}}}}}{{2\text{π}{a^2}}}$ (13)

 ${F_{\textit{z}} } = \frac{4}{3}{E^*}{R^*}^{1/2}{\delta _{\textit{z}} }^{3/2}$ (14)

 $\frac{1}{{{E^*}}} = \frac{{1 - v_1^2}}{{{E_1}}} + \frac{{1 - v_2^2}}{{{E_2}}},\;\frac{1}{{{R^*}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}$

 ${a^2} = {R^*}{\delta _{\textit{z}}}$ (15)

 图4 球与平面弹性接触压力分布的非赫兹解 Fig. 4 Comparison of the non-Hertzian and Hertzian solutions for the pressure distribution of the elastic contact between a ball and a plane

3.2 接触应力计算

 图5 螺旋升角为32.48°时接触区域应力分布赫兹解 Fig. 5 Hertzian solution of stress distribution in contact area with helix rise angle of 32.48°

 图6 螺旋升角为32.48°时接触区域应力分布非赫兹解 Fig. 6 Non-Hertzian solution of stress distribution in contact area with helix rise angle of 32.48°

 图7 螺旋升角为32.48°时接触点处应力的非赫兹解与赫兹解之差的等高线图 Fig. 7 Contour map of the difference between the non-Hertzian solution and the Hertzian solution of the stress at the contact point with helix rise angle of 32.48°

3.3 接触区域分析

 图8 赫兹解的接触区域 Fig. 8 Contact area calculated by Hertzian method

 图9 非赫兹解的接触区域 Fig. 9 Contact area calculated by non-Hertzian method

3.4 螺旋升角对接触应力误差的影响分析

 图10 丝杠–滚珠接触点A处应力的非赫兹解与赫兹解之差 Fig. 10 Difference between non-Hertzian solution and Hertzian solution of stress at contact point A

 图11 赫兹解相对于非赫兹解的接触应力误差 Fig. 11 Contact stress error of Hertzian solution relative to non-Hertzian solution

 图12 接触点处非赫兹解和赫兹解的接触应力峰值 Fig. 12 Peak value of contact stress of non-Hertzian solution and Hertzian solution at the contact point

4 结　论

1）赫兹接触解与最小余能非赫兹接触精确解相比，当螺旋升角增加时，丝杠滚道和螺母滚道与滚珠接触点处接触应力误差逐渐增大。而且，随着滚珠丝杠副螺旋升角的增大，丝杠滚道与滚珠接触点处的误差始终大于螺母滚道与滚珠接触点处的误差。

2）赫兹解和非赫兹解两种方法计算所得滚珠与滚道接触区域的面积和长短轴有所不同。准确计算滚珠和滚道接触区域应力分布可以更加全面和准确的计算滚道磨损带的宽度和深度。

3）随着数控机床高速高精化发展，具有大螺旋升角的高速滚珠丝杠副应用越来越广泛。由于高速滚珠丝杠副接触区域非对称性大，有必要采用最小余能原理进行接触应力分析，以便更准确的分析高速滚珠丝杠副的磨损机理和预测其精度保持性。