# Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws
## 1 Introduction  
With the improvement of machining efficiency and machining accuracy of the NC machine tools, high-speed ball screws (BSs) are widely used in the machining field [1, 2]. However, there is a calculation error problem of the Hertzian elastohydrodynamic contact stress due to the asymmetry of the contact region of the helix raceway. Thus, it is essential to investigate the non-Hertzian elastohydrodynamic contact stress of the BSs in these properties, such as kinematics, force balance, and elastohydrodynamic contact mechanics. In terms of the force and deformation characteristics of the double-nut ball screws (DNBSs), many researchers studied the force balance equation and the torque balance equation of the working ball. [PERSON] et al. [3] proposed a static load distribution model of the preloaded BSs considering geometric errors, which takes into account the influence of non-loaded balls and the interaction between the elastic deformation of a screw/nut and the Hertz contact forces of screw-ball/nut-ball contact areas. [PERSON] et al. [4] established a low order static load distribution model for BSs that incorporates lateral deformation and geometric error effects to optimize the design parameters of BSs. [PERSON] et al. [5] presented a novel method to analyze the static load distribution of BSs, which takes into account the nut position variation. The effects of axial load and original contact angle on load distribution are obtained. [PERSON] et al. [6] used the machine learning method to diagnose the preload state of the BSs by monitoring the mechanical signals, such as the driving motioncurrent, ball screw speed, linear scale positioning, and vibration. [PERSON] et al. [7] studied the contact stress and fatigue life of the BSs with the outer cycling device, considering the dimension errors of balls. [PERSON] et al. [8] designed a ball screw with smaller balls, which can decrease both the torque fluctuation and position deviation. [PERSON] et al. [9] established a multibody dynamic model of BSs to study the influence factors of the friction torque, such as operating speed, temperature, and geometry, and an experimental test bench was constructed to validate the model results. However, the non-Hertzian normal contact stress of the BSs can be calculated more accurately by using the minimum excess principle, which is not explored in the above research. In the case of the elastohydrodynamic contact mechanics of the BSs, many scholars studied the elastohydrodynamic lubrication (EHL) characteristics [10; 11] of the lubricant film. [PERSON] et al. [12] analyzed the elastohydrodynamic lubrication characteristics of gears, considering the involute profile error of the surface, and the analyzed results quantified the effects of measured profile deviations on the EHL lubricant film developed between helical gears. [PERSON] et al. [13] proposed a wear prediction model for worm gears based on the Archard wear law and determined the wear rate, considering the variation of stress and film thickness over the contact region. In addition, the detailed wear pattern of worm gear teeth [14] was obtained by the proposed model, and the wear scars on the gear teeth were produced by the development way of the wear pattern. [PERSON] et al. [15] analyzed the kinematics and the transmission efficiency considering the oil lubrication of a single-nut double-cycle ball screw, and the simulated results are consistent with the experimental results. [PERSON] et al. [16] studied the shear properties of the polyisobutylene lubricant under the different sliding speeds and the contact stresses. [PERSON] et al. [17] proposed an approach to increase the lifetime of the BSs by using the optimal amounts of lubricant. The results were lubricant savings and significant increases in component-lifetimes of approximately 70%. [PERSON] et al.