## 7 Conclusions  
According to the movement characteristics of the ore-rock block in the vertical ore pass, the representation equation of the three-dimensional coordinates and the impact time of the ore-rock block moving in the ore pass are obtained. The angle between the movement direction of the ore-rock block and the shaft wall is proposed before and after the ore-rock block collides with the shaft wall.  
Based on the Hertz contact theory, a contact model of ore-rock blocks colliding with the ore pass wall is developed, and the process of the ore-rock blocks coming into physical contact with the shaft wall is categorized into three stages: elastic compression, elastic-plastic compression, and rebound of the ore pass wall material. The relationship between the normal impact force and the depth of the ore-rock block intruding into the shaft wall, as well as between the maximum contact radius and the depth of penetration, is determined. The NRC of the wall material is also acquired.  
Based on the tangential contact mechanics, the tangential cutting model along the shaft wall is developed. The tangential slipping distance of the ore-rock block along the shaft wall, the TRC, and the lost area and volume of the shaft wall material are derived based on the established model when the ore-rock block collides with the ore pass wall.  
The calculation model for the volume loss of the shaft wall material during the collision between the ore-rock block and the shaft wall is formulated, and the total lost volume during the collision is obtained through the analysis and resolution of the model.  
The reliability of the maximum intrusion depth of the ore-rock block into the ore pass wall, calculated using the formula proposed in this paper, is verified through simulation experiments using similar materials. The relative errors of the two depth results show that they are consistent, and the theoretical calculation can reflect the degree of impact damage exhibited by the ore pass wall.  
According to the specific characteristics and parameters of the ore pass in the case mine, the influence of the impact velocity and angle of collision of the ore-rock block on the restitution coefficient of the shaft wall material, the maximum normal intrusion depth of the ore-rock block into the ore pass wall, the maximum tangential displacement of ore-rock block along the shaft wall, and the volume loss of the shaft wall when the ore-rock block collides with the ore pass wall are analyzed by using the relevant formulas obtained in this paper, and the corresponding rules are obtained.  
Investigation, formal analysis, data curation, visualization, writing--original draft, Q.M.; investigation, resources, formal analysis, data curation, writing--original draft, C.M.; methodology, formal analysis, writing--review and editing, project administration, J.L.; conceptualization, methodology, writing--review and editing, supervision, project administration, fundingacquisition, Z.L.; data curation, formal analysis, writing--review and editing Z.X. All authors have read and agreed to the published version of the manuscript.  
The research was supported by the National Natural Science Foundation of China (Grant No. 51774176).  
The detailed data are available upon request. The data are not publicly available due to privacy.  
The authors declare no conflicts of interest.