## 4 Discussion  
Test results show that the friction under the action of particles has pulsating characteristics, it can be speculated that the spool in the movement process there must be particles of rotation and spool posture changes; of course, the microscopic roughness of the clearance wall is also a factor that cannot be ignored. Let us analyze the whole process of particles from the invasion of the clearances to producing pulsating friction: the spool is in the original position, the clearance is large, the particles invade, the spool moves in the axial direction, the spool axial friction is small, the spool posture changes, microscopically a side of the clearances size decreases, the reduced clearances will have particles stuck in the rough wall (the \"three-body contact\"), as shown in Figure 3. The mechanism of clamping of a single square particle. ), the spool continues to move, the spool and the wall of the valve body produce relative motion, driving the particles to rotate, and the spool axial friction increases abruptly. This process occurs repeatedly, resulting in a non-periodic pulsation characteristic of the friction force. In practical work, the material of particles relates to the material of spool valves and the working conditions. The diameter and concentration of particles mainly depend upon the degree of the contamination of the filter and the hydraulic system, while the spool posture upon its processing, assembly process, and service. There are some methods to be chosen to avoid spool clamping triggered by solid particles, such as thorough cleaning in processing and dust-free operation in assembling, and the effective isolation between the hydraulic system and polluted environment in the working process. Furthermore, it is necessary to choose wear-resistant materials to make hydraulic valves, and further research on new valve structures with the function of adsorbing, dredging, and filtering particles will be very valuable. There are some shortcomings in this study, such as the following: in the experimental study, the addition of oil-containing particles can only be carried out by dropping them into the clearance inlet through the test tubes, which cannot be performed automatically, and, therefore, a certain degree of manual error will occur, and in the future, we can increase the automatic particle mixing and adding device to reduce the manual error. In mathematical modeling, due to the limited space of the test sample, the accuracy of the empirical formulas of the base amount of spool valve friction and jumping amount obtained is limited, and it is necessary to improve its accuracy through more test data. Conceptualization, H.J. ; methodology, Q.C. ; validation, M.L. ; formal analysis, W.C.; investigation, Q.C. ; resources, C.L. ; data curation, M.L. ; writing--original draft preparation, Q.C. ; writing--review and editing, M.L. ; visualization, W.C.; supervision, H.J. ; project administration, H.J. ; funding acquisition, W.C. and C.L. All authors have read and agreed to the published version of the manuscript. This study was funded by \"The National Natural Science Foundation of China\", with grant number 51575254. Moreover, it also funded by the \"Major Training Program of University Research and Innovation Platform of Gansu Provincial Department of Education\", with grant number No. 2024 CXPT-09. This study also received support from the project of Guizhou Honglin Aero Engine Control Technology Co., Ltd.--Research on Anti pollution Design Technology for *** System, with grant number \"Design [2022]-8\". Not applicable. Data Availability Statement: The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. Conflicts of Interest: Authors [PERSON] and [PERSON] were employed by the company Guizhou Honglin Aero Engine Control Technology Co., Ltd.