## 1 Introduction  
A ball screw is a mechanical part that converts rotational motion into translational motion. As shown in Figure 1, ball screws are composed of a shaft, a nut, a circulation part, balls, etc. It is designed in such a structure that the ball rolls between the grooves of threaded shaft and nut and returns to its initial position through the circulation part. As the ball screw adopts rolling friction and not sliding friction, it can have very low friction loss and high efficiency. These characteristics of ball screws are the reason they are used in precision mechanical devices such as artificial respiratory, precision positioning tables, machine tools, and guided missiles. However, when an excessive axial load is applied, the narrow contact surface between the ball and the ball groove is dented, leaving permanent deformation, which is fatal to the ball screw requiring precise driving [1, 2]. To determine whether such permanent deformation occurs, as it is necessary to know the change of the internal stress caused by the contacts of the components, many studies are being conducted to estimate this. The following are studies analyzing the rotating parts, and it was determined whether applied methodologies can observe stress changes occurring on the contact surface of the ball screw. Research to analyze rotating parts is divided into three main categories. The first is the theoretical method, the second is the FEA (Finite Element Analysis), and the third is the synthetic method that applies both theoretical methods and FEA. The study of [PERSON] and [PERSON] [3] and [PERSON] and [PERSON] [4] could theoretically represent the magnitude of the load applied to the contact surface of the rotating parts from an external force, but there was a limitation that the internal stress distribution could not be expressed. [PERSON] et al. [5] conducted a 2D FEA considering the adhesive force be-tween two contacting objects. However, as one of the two contacting objects was analyzed as a rigid flat surface, it is difficult to apply to the case of a ball screw whose both contacting objects are soft bodies and both contacting surfaces are curved. By applying the Laboratoire de Genie Mecanique de Toulouse model in [PERSON] et al. [6] and two nodes calculating physical proper-ties in [PERSON] et al. [7], load distribution and displacement were derived by FEA that replaced the ball between the inner and outer rings of the bearing. These methodologies were effective in expressing the rough behavior of the bearing, but still have difficulty in analyzing the stress distribution from the contact load. There are studies that have conducted FEA considering the shape and mesh of all ball bearing components, but this is a method that can only be applied to a specific shape when the degree of freedom of the ball is considerably limited [8; 9]. There are studies deriving the friction torque, rotational speed, and load applied to the contact surface through dynamic analysis of the ball screw, but there are also limits to deriving the stress distribution by the load applied to the contact surface [10; 11; 12]. There are various methodologies that can estimate the load applied to the contact surface of rotating parts and express the approximate movement. However, overall, all were limited in estimating how the stress distribution of the contact components changes inside the rotating parts that have complex shapes such as ball screws. This paper presents a new model that could observe the stress change of the internal contact components of the ball screw by applying both the theoretical method and the FEA. The components of the ball screw were separated one by one, and each FEA model was created and solved. To this end, the shape of the contact surface and the pressure applied to the contact surface were derived by a method based on Hertz contact theory. The shape of the contact surface appeared elliptical and was applied to the 3D modeling of ball screw.