高速滚珠丝杠副非赫兹接触应力计算  

Non-Hertzian Contact Stress Calculation of High-speed Ball Screw Mechanism

作　　者：王民[1,2] 孙铁伟 董朝阳 孔德顺 高相胜 WANG Min;SUN Tiewei;DONG Zhaoyang;KONG Deshun;GAO Xiangsheng(Beijing Key Lab.of Advanced Manufacturing Technol.,Faculty of Materials and Manufacturing,Beijing Univ.of Technol.,Beijing 100124,China;Beijing Key Lab.of Electro-Machining Technol.,Beijing 100191,China;Inst.of Standard Metrology,Chinese Academy of Railway Sciences,Beijing 100081,China)

机构地区：[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

出　　处：《工程科学与技术》2021年第2期178-186,共9页Advanced Engineering Sciences

基　　金：国家自然科学基金项目(51975020,51575014,51875008);北京市自然科学基金项目(3202005)。

摘　　要：在滚珠丝杠副运动过程中,滚珠与滚道之间的摩擦和磨损导致滚珠丝杠副精度衰退,降低其精度保持性。针对高速滚珠丝杠副螺旋线滚道接触区域的非对称性,采用传统的赫兹接触理论对丝杠和螺母滚道接触几何形状和关系的计算存在较大误差,提出了一种基于最小余能原理对高速滚珠丝杠副滚珠与滚道之间的法向接触应力进行分析计算的方法。首先,基于Frenet−Serret坐标转换公式建立滚珠丝杠副坐标系,对滚珠与滚道接触点处的接触情况进行几何描述,并通过2维插值算法重新进行网格划分以便于后续的接触力学分析;对滚珠与滚道之间的接触区域和控制方程进行离散化分析,提高数值计算的精确性;为了解决接触问题计算求解过程复杂、收敛速度缓慢,通过运用变分原理将接触问题转成极值问题,采用共扼梯度法进行循环迭代来实现快速收敛求解,提高接触数值解法的计算效率,求解出滚珠与滚道之间的接触应力分布;由于接触表面变形的计算相当于求影响系数矩阵与法向压应力之间的卷积,2维卷积运算和对应的2维快速傅里叶变换被用于求解接触问题的数值解。在数值求解中,主要的计算量集中在弹性变形的计算上,使用2维快速傅里叶变换(FFT)技术方便快速地计算接触区域弹性变形。将最小余能原理与赫兹接触理论分别用于求解光滑的球与平面之间的弹性接触应力分布,将两种计算方法得出的结果进行对比,计算数值重合度很好,验证了基于最小余能原理求解接触应力分布的正确性;基于最小余能原理分析螺旋升角对滚珠与滚道接触点处的接触面积和弹性变形的影响,并将该方法所得结果与赫兹接触理论计算结果进行对比分析,发现赫兹接触解与最小余能非赫兹接触精确解相比,随着滚珠丝杠副滚道螺旋升角的增大,丝杠与滚珠接触点A处的误差始终大于螺�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

关 键 词：滚珠丝杠副 接触应力 最小余能原理 共轭梯度法 赫兹接触 

分 类 号：TH132.1[机械工程—机械制造及自动化]