## 2 Materials and Methods
### Establishment of Mechanical Model
#### 2.1.2 Mathematical Model of Spool Clamping Risk Zone \(A_{i}\)
The curve \(d\Phi(a)/da\) was fitted by a normal distribution function, and the obtained curve \(\varphi(a)\) was denoted by the red dotted line. It can be found that the curves \(d\Phi(a)/da\) and \(\varphi(a)\) are highly coincided. This indicates that the differential form of the function \(\Phi(a)\) can be approximately expressed as the probability density curve \(\varphi(a)\) which fulfills the law of normal distribution. The expression of \(\varphi(a)\) is  
Figure 6: Morphological statistics of some particles in samples. shown in Equation (13). In Figure 7, the \(\Phi(a)\) can be expressed as Equation (14) and the \(\Phi(u_{0})-\Phi(u_{\text{min}})\) as Equation (15). In Equations (13)-(15), the \(\mu\) and \(\sigma\) denote the mean and variance of normal distribution, respectively. In Equation (15), the \([u_{i},u_{j}]\) denotes the size range of contaminant particles, which is \([0.2,0.7]\) in this statistical sample. The \(u_{0}\) and \(u_{min}\), respectively, refer to the maximum and the minimum clearance size in the clamping risk zone; the \(\Phi(u_{0})-\Phi(u_{\text{min}})\) denotes the proportion of sensitive particles in the total particles. If \(\circ\%\) denotes the volume concentration of contaminant particles within \([u_{i},u_{j}]\) in the oil, then the concentration of sensitive particles can be expressed as \(c_{0}\%\), as shown in Equation (16). The number of sensitive particles \(n\) in the clamping risk zone \(A_{i}\) can be obtained by Equation (17). Given the pulsation clamping force generated by particle rotation, the spool clamping force was separated into a basic term \(f(E_{k},c,e)\) and a pulsation term \(f^{\prime}(E_{k},c,e)\). To sum up, there is the following:  
\[\varphi(a)=\frac{1}{\sqrt{2\pi\sigma}}\exp\biggl{[}-\frac{(a-\mu)^{2}}{2\sigma ^{2}}\biggr{]}\quad\left(\sigma>0,\int_{-\infty}^{+\infty}\varphi(a)da=1\right) \tag{13}\]  
\[\Phi(a)=\int_{-\infty}^{a}\frac{1}{\sqrt{2\pi\sigma}}\exp\biggl{[}-\frac{(a- \mu)^{2}}{2\sigma^{2}}\biggr{]}da\quad(\sigma>0) \tag{14}\]  
\[\Phi(u_{0})-\Phi(u_{\text{min}})=\int_{u_{\text{min}}}^{u_{0}}\frac{1}{\sqrt{2 \pi\sigma}}\exp\biggl{[}-\frac{(a-\mu)^{2}}{2\sigma^{2}}\biggr{]}da\ (\sigma>0,u_{\text{min}}\in[u_{i},u_{j}]) \tag{15}\]  
Figure 8: Probability density function of particle distribution. Figure 7: Statistical histogram and probability distribution curve of particle sizes. \[c_{0}=c\times[\Phi(u_{0})-\Phi(u_{\text{min}})]=c\times\int_{u_{\text{min}}}^{u_{ 0}}\frac{1}{\sqrt{2\pi\sigma}}\exp\Biggl{[}-\frac{(a-\mu)^{2}}{2\sigma^{2}} \Biggr{]}da\quad(\sigma>0,u_{\text{min}}\in[u_{i},u_{j}]) \tag{16}\]  
\[n=c_{0}\cdot A_{i}\cdot(u_{0}+u_{\text{min}})/2a^{3} \tag{17}\]  
\[F_{clamping}=E_{k}\underset{i=1}{\overset{n}{\sum}}g_{i}(e)=F(k,c,e)=f(k,c,e)+ f^{\prime}(k,c,e) \tag{18}\]  
\[F_{clamping}= \begin{cases}&F_{Ek}=f_{E}(k,\overline{c},\overline{e})+f_{E}^{ \prime}(k,\overline{c},\overline{e})\quad(c=const,e\in[e_{1},e_{2}])\\ &F_{c}(c)=f_{c}(\overline{k},c,\overline{e})+f_{c}^{\prime}(\overline{k},c, \overline{e})\quad(k=const,e\in[e_{1},e_{2}])\\ &F_{e}(e)=f_{e}(\overline{k},\overline{c},e)+f_{e}^{\prime}(\overline{k}, \overline{c},e)\quad(k=const,c=const)\end{cases}  \tag{19}\]  
Since the elastic modulus \(E_{k}\) of the particles of a specific material \(k\) is a constant rather than a variable, \(F_{Ek}\) is a value instead of a function. In particular, after the Expression (19) of function \(F_{clamping}\) was obtained, \(F_{Ek}\) was combined with Equations (9), (17)-(19). Then, the tension that a single particle of a certain material and a certain size receives in the clamping risk zone can be obtained and expressed as Equation (20). Next, the clamping force \(F_{clamping}\) of the spool in particulate oil was measured by the single variable method, with the material \(k\), volume concentration \(c\%\), and particle size \(a\) as the variables. Then, empirical equations for \(F_{c}(c)\) and \(F_{e}(e)\) were obtained.