## 2 Materials and Methods  
### Establishment of Mechanical Model  
#### 2.1.1 Fundamental Theories  
The spool valves, valve clearance, and sensitive particles are three major subjects of this study. The sensitive particles possess such primary characteristics as material, size, and concentration in oils, the clearance clamping risk zone is defined by the particle size, any single factor will affect the performance of spool clamping. By measuring and analyzing the clamping force under particle contamination, a theoretical model was established for the influential factors of clamping force under the action of particles. The theoretical framework of this study is shown in Figure 2.  
The action of particles onto the spool clamping was taken as a stacking effect of each tiny particle. The tension characteristics of a single particle were analyzed, and the actions of single particles were superimposed. Then, the basic model of multiple particles acting on the clamping force can be established.  
The essential cause for spool clamping by solid particles is the rough morphology of the spool surface at the micro level; a three-dimensional surface topography instrument was used to measure the rough morphology of the surface of a spool (the diameter is 12 mm), as Figure 3a shows, the local contour of the valve core surface was extracted along the red line, and then the rough contour line along the axial direction of the valve core was formed. The profile of the spool clearance is drawn as irregular according to the measurement result, and the solid particle is abstracted as square according to the observation result, as Figure 3b shows. A single particle would produce a turning moment and rotate under the action of spool motion when its two edges are attached to the spool and the valve body concurrently. Moreover, under the yield stress, the particle would deform when being extruded by the spool and the valve body. Let us assume the strain of a particle after being extruded to be \(\varepsilon\) (unit: mm/mm) and the contact area between the particle and the spool to be \(\delta\) (unit: mm\({}^{2}\)). If the elastic modulus of the particle is E (unit: N/mm\({}^{2}\)), then the clamping force \(\sigma_{i}\) that a single particle produces on the spool can be expressed as Equation (1).  
\[\sigma_{i}=\sigma_{z}=\mathrm{E}\varepsilon\delta\cos\beta \tag{1}\]  
After the particle material was confirmed to be an elastic modulus \(E_{k}\), the strain \(\varepsilon\) of a single particle and the contact area \(\delta\) between the particle and the spool held a direct correlation with the particle size \(a\) (unit: mm). Given the clearance size \(u_{0}\) (unit: mm) in a perfect case, a dimensionless variable \(e=a/u_{0}\) (unit: mm/mm) and the function \(g_{1}(e)\) of clamping force changes caused by this variable were introduced and can be expressed as Equation (2):  
\[\sigma_{i}=E_{k}g_{i}(e)\quad(e\leq 1) \tag{2}\]  
The dimensionless variable \(c\%\) (m\({}^{3}\)/m\({}^{3}\)) that denotes the volume fraction of particles in oil and the spool clamping risk zone \(A_{i}\) were introduced as parameters for the calculation of the number of sensitive particles. Let us suppose that there are \(n\) sensitive particles  
Figure 2: Establishment of spool clamping model under the action of sensitive particles.  
acting on the spool, then the clamping force \(F_{clamping}\) on the spool is the superimposing effect of these sensitive particles. It can be expressed as Equation (3).  
\[F_{clamping}=\sum_{i=1}^{n}\sigma_{i}=E_{k}\sum_{i=1}^{n}g_{i}(e)=F(k,c,e) \tag{3}\]  
In Equation (3), \(k\) denotes the material kind, which corresponds to the elastic modulus \(E_{k}\) of the material. Besides, both \(e\) and \(c\%\) are inherent attributes of particles in the oil and can be considered known. In the following section, the calculation equation of \(A_{i}\) was deduced to obtain the expressions of \(n\), \(E_{k}g_{i}(e)\), and \(F_{clamping}\).