## 1 Introduction  
Rolling bearings are widely used in machine tools, mineral mining, automotive and wind power generation, and other important industries. They are the most commonly used parts of rotating machinery and are also some of the most vulnerable parts. Their reliability and safety play a vital role in the whole life service process of rotating machinery [1, 2, 3]. Due to the complex working conditions of rolling bearings, working environment, bearing a high load, and uninterrupted work, rolling bearings are particularly prone to spalling failure and will directly affect the stability of the equipment and even cause damage to the entire equipment [4, 5, 6]. Previous solutions were generally based on empirical judgment or periodic replacement, but this method's judgment error is large and will cause economic waste. Therefore, the failure dynamics model established based on bearing running parameters provides important engineering significance in monitoring the running status of bearings [7, 8, 9]. Deep groove ball bearings are usually not designed to bear axial forces, but in practical applications, they may be subjected to slight axial impacts. When a deep groove ball bearing is subjected to a certain axial force, the raceways of the inner and outer ringsand the rolling elements will undergo a relative displacement to adjust in the centripetal direction, forming a new equilibrium state [10; 11]. In this case, a contact angle will be generated, similar to the working mode of angular contact ball bearings. But the contact angle of the deep groove ball bearing is usually small and returns to the original working mode after the impact ends. Research has found that material defects in the manufacturing process are also one of the main causes of spalling failures, so peeling failures may occur in non-center positions of the raceway. At this point, the bearing is no longer symmetrical in the radial view, and the rolling elements will make uneven contact with the raceway when moving between them, resulting in derived axial impact and reducing the axial stability and service life of the bearing system [12; 13; 14]. Nowadays, to establish a dynamic model that is more in line with actual needs, it is mainly divided into two aspects. Firstly, it is the simulation of the actual internal state of the bearing. Scholars have carried out research on the contact characteristics of the internal components of the bearing and their mutual influences: Regarding the conduction mode of the internal load of the bearing, the influence of the curvature factor between the geometric shapes of the force-bearing components is considered [15; 16; 17; 18]; regarding the calculation of the equivalent damping and equivalent stiffness of the bearing, the material properties and the factors of elastohydrodynamic lubrication are considered; regarding the load distribution mode of the bearing, the radial clearance of the bearing and the coordinated deformation after deformation are considered, but the influence of spalling defects on these factors is ignored [19]. On the other hand, a model under specific conditions for the actual working conditions is established. Regarding the mass and stiffness of the bearing under the actual working conditions, considering that the bearing is tightly installed on the equipment and can be regarded as one entity, the material properties of the fixing device are also included in the dynamic model [20; 21; 22; 23]; studies on the effects of different loads and rotational speeds have been conducted. Considering the load variation patterns and the weights of influencing factors at different rotational speeds, models that meet the requirements have been established. However, these models can only describe radial impacts and do not take into account the possible effects of axial impacts under actual complex working conditions. In summary, the purpose of this paper is to analyze the impact response characteristics of the bearing system caused by spalling faults at non-central positions of the raceway.