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

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. Copyright ©2021 Advanced Engineering Sciences. All rights reserved.