## 2 Materials and Methods
### Experiment Design and Result Analysis
Of the spool, the upper chamber was at an atmospheric pressure \(P_{0}\), and the lower chamber was sealed. A vertically upward force \(F_{x}\) was imposed on the spool, making the spool move upward at a constant speed. In this process, the volume of the lower chamber enlarged, and a relatively negative pressure \(P_{f}\) was produced. This relatively negative pressure \(P_{f}\) served to drive the oil in the valve clearance towards the lower chamber under the pressure difference, thereby bringing the particles into the clearance. The resistance that the negative pressure \(P_{f}\) produced is expressed as \(F_{y}\). Besides, the spool was subject to gravity \(G\), inherent friction \(F_{o}\) (including the viscous friction in the test spool clearance and the friction on the direction spool wall), as well as the clamping force \(F_{clumping}\) that the particles engendered in the test spool clearance. The forces on the spool are expressed as Equation (21). With no particles added, there is \(F_{clamping}=0\) and the time-varying inherent friction \(F_{o}(t)\) is expressed as Equation (22). In Equation (22), \(F_{y}(t)=P_{f}(t)\times S\), \(S=\pi d^{2}/4\), and \(G=mg\), where the spool mass \(m=0.2983\) kg and g = 10 m/s\({}^{2}\), and '\(S\)' denotes the area of negative pressure on the lower spool chamber. Let \(\overline{F}\) denote the mean of inherent friction within a period. By measurement, the curves of \(F_{x}(t)\) and \(P_{f}(t)\) were obtained, and then, the valve of \(\overline{F}_{o}\) was calculated, as shown in Equation (23). Introducing \(\overline{F}_{o}\) into Equation (22), the clamping force curve \(F_{clamping}(t)\) of the particles invading the clearance was obtained and expressed as Equation (24). \[F_{x}=F_{clamping}+F_{y}+F_{o}+G \tag{21}\]  
\[F_{o}(t)=F_{x}(t)-F_{y}(t)-G \tag{22}\]  
\[\overline{F}_{o}=\frac{\int_{t_{1}}^{t_{2}}F_{o}(t)dt}{t}=\frac{\int_{t_{1}}^{ t_{2}}\left(F_{x}(t)-F_{y}(t)-G\right)dt}{t}=\frac{\int_{t_{1}}^{t_{2}} \left(F_{x}(t)-P_{f}(t)\times\pi d^{2}/4-mg\right)dt}{t} \tag{23}\]  
\[F_{clamping}(t)=F_{x}(t)-F_{y}(t)-\overline{F}_{o}-G \tag{24}\]  
Figure 9b shows the basic sizes of the spool and valve body. In the experiment, in order to enable the particles to enter the clearance, the valve clearance \(u_{0}\) was expanded properly to 0.6 mm. The coarseness of the spool and valve body was set at 0.8 in order to highlight the influence of the particulate characteristics on the clamping force and reduce the influence of surface coarseness on the clamping force. In light of the measurement  
Figure 9: Structure, size, and force analysis of experimental spool. (**a**) Force analysis of experimental spool; (**b**) Parameter about experimental spool. principles and the dimension structure, a clamping force experimental device was devised, as shown in Figure 10.  
As Figure 10 illustrates, the rotational motion is converted to a linear motion of the spool which is driven by a stepper motor. The spool tension \(F_{x}\) was measured by a tension sensor, and the negative pressure \(F_{y}\) measured by a negative pressure sensor. Moreover, the spool motion speed was controlled by the controller of the stepper motor, and the data were collected and processed through the tension detection system, the negative pressure monitoring system, and the data acquisition system. The range of the tension sensor is 50 N, the accuracy of the measurement is 0.1 N, and the measurement time is uniformly 30 s. The \(F_{x}(t)\) and \(P_{y}(t)\) without the particles were measured by the established experimental device to calculate \(F_{y}(t)\). Then, according to Equations (2) and (3), the clamping force curve \(F_{o}(t)\) and its mean \(\overline{F}_{o}\) were obtained, as shown in Figure 11. HM46 hydraulic oil was used as the experimental oil. At last, the inherent friction \(\overline{F}_{o}\) of the spool valve in purified oil was calculated to be 0.43404 N.  
Figure 11: Inherent friction calculated curve of spool.