## 2 Materials and Methods
### Establishment of Mechanical Model
#### 2.1.2 Mathematical Model of Spool Clamping Risk Zone \(A_{i}\)  
The spool posture in the valve hole directly affects the clearance height, shaping the clamping risk zones of different morphologies. The spool posture in the valve hole was categorized into three patterns: off-centered, tilting, and off-centered with tilt. When the spool moves upward, the clamping risk zone can be shown in Figure 4, where the red arrows are the movement direction of valve core. Let us suppose a cylindric spool with a diameter of \(d\) and a height of \(h\), and the superficial area of its side surface is \(A\), as shown in Figure 4a. This spool was assembled with a valve hole with a diameter of \(d_{0}\). Their central axes will coincide in the perfect case, shaping a unilateral clearance of \(u_{0}\) (\(u_{0}=(d_{0}-d)/2\)). If the feature size a of sensitive particles is close to \(u_{0}\) and \(a<u_{0}\), then these particles are likely to invade in the clearance. When the spool valve is off-centered or tilting in the valve hole, the clearance of some zones will be made \(u<u_{0}\), then shaping the clamping risk zones of different morphologies. And in this clearance, the sensitive particles will greatly increase the risk of spool clamping. Figure 4: Diagram of spool postures and clamping risk zones. (**a**) Ideal posture; (**b**) Off-centered posture; (**c**) Tilting posture; (**d**) Off-centered & Tilt posture. Figure 3: Mechanism of clamping of single square particle. (**a**) Micromorphology of spool mating surface. (**b**) Single square particle. In Figure 4a-d, the \(A_{1},A_{2}\), \(A_{3}\), and \(A_{4}\), respectively, represent the clamping risk zones of spools in perfect posture, off-centered posture \(l\), tilting posture \(\omega\), and off-centered with tilt posture \(l+\omega\). The calculation of the areas of \(A_{1}\) and \(A_{2}\) was relatively simple. \(A_{3}\) and \(A_{4}\) were both made of a \(1/4\) cylindric surface and part of a spatial fan-shaped surface. Their corresponding spatial fan-shaped surfaces were expressed as \(A_{\omega}\) and \(A_{\omega l}\). The geometric parameters required by the calculation are shown in Figure 5.  
It can be seen from Figure 5a that \(A_{\omega}\) or \(A_{\omega l}\) are actually planes which pass through the center point of the cylinder's bottom surface and shape a \(\theta\) angle with the bottom surface. The planes also shape spatial curved sector surfaces with the sides of the cylinder, and the areas of these curved surfaces are as follows:  
\[S=\int_{0}^{\pi}\!\left(\frac{d}{2}\right)^{2}\tan\theta\sin\alpha\mathrm{d} \alpha\quad\theta\in\left[0,\frac{\pi}{2}\right] \tag{4}\]  
It can be known from Equation (4) that the calculation of \(\theta\) is the key to calculate \(A_{\omega}\) and \(A_{\omega l}\). As shown in Figure 5b,c, since the zone of \(u<u_{0}\) is defined as a clamping risk zone, the value of \(\theta\) is determined by the location of \(u_{0}\), and the location of \(u_{0}\) is determined by \(\omega\) and \(l\). Let us suppose \(\omega_{max}\) to be the maximum \(\omega\), then the \(\omega_{\max}\) can be calculated by Equation (5) and \(\theta\) can be calculated by Equation (6). The areas of spatial sectors \(A_{\omega}\) and \(A_{\omega l}\) are expressed as Equations (7) and (8). According to Figure 5b,c, in fact, \(u_{1}<u_{0}<u_{2}\) means \(A_{3}=A/4-A_{\omega}\) and \(A_{4}=A/4-|A_{\omega l}|\), and \(u_{1}<u_{2}<u_{0}\), means \(A_{4}=A/4+|A_{\omega l}|\). To sum up, the area of \(A_{i}\) (\(i=1,2,3,4\)) can be expressed as Equation (9). In Figure 5b,c, \(u_{1}\) and \(u_{2}\) can be expressed as Equations (10) and (11), respectively, and \(u_{0}\) is the maximal clearance size of a clamping risk zone. The minimal clearance size \(u_{min}\) of a clamping risk zone can be calculated by \(u_{2}\), as shown in Equation (12). The \(u_{0}\) relates to the machining accuracy of spool valves, and \(u_{min}\) to the spool posture in the valve hole.