## 2 Materials and Methods
### Experiment Design and Result Analysis
#### 2.2.4 Influences of Particle Size on the Clamping Force  
Taking the clearance size of \(u_{0}\) = 0.6 mm as a reference, aluminum particles in feature size \(a\) of 0.12 mm, 0.18 mm, 0.24 mm, 0.30 mm, 0.36 mm, 0.42 mm, 0.48 mm, and 0.54 mm were selected, and their corresponding \(e=a/u_{0}\) is 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90%, respectively, with a particle size error less than 5%. In order to minimize the influence of concentration and to highlight the influence of size change on the clamping force, the particles of different sizes were separately proportioned into hydraulic oil with a volume concentration of \(c=1\%\) for the experiment. Figure 15a shows the micrographs of the particles of four different sizes. Figure 15b shows the spool clamping force curve of hydraulic oil containing aluminum particles of different sizes, which was measured by a clamping force experiment device. It can be found that the clamping force curves at different sizes all present aperiodic pulsation. As the particle size enlarges, the clamping force \(F_{e}(e)\) increases and reaches its peak of 21.5 N when \(a=0.54\) (\(e=0.9\)). To further investigate the relationship between particle size and clamping force, the mean and maximum clamping forces at different particle sizes were compared. The comparative analysis result is shown in Figure 16.  
According to Figure 16, as the particle size \(a\) increases from 0.12 mm (\(e=0.2\)) to 0.54 mm (\(e=0.9\)), both the mean clamping force and the maximum clamping force grow, and so does the pulsation increment of the maximum clamping force. The fitting curve diagram shows that the mean clamping force is fitted into curve \(f_{e}(e)\) and the pulsation increment force \(F_{e}(e)-f_{e}(e)\) into curve \(f_{e}^{\prime}(e)\). Both \(f_{e}(e)\) and \(f_{e}^{\prime}(e)\) keep a relatively low value within an extensive size range \(e\in[0.2,\,0.7]\). When \(e\geq 0.7\), however, \(f_{e}(e)\) and \(f_{e}^{\prime}(e)\) increase rapidly in the manner of a quadratic curve, with \(f_{e}^{\prime}(e)\) increasing obviously faster than \(f_{e}(e)\). The fitting equations of \(f_{e}(e)\) and \(f_{e}^{\prime}(e)\) are shown as Equations (28) and (29). \[f_{e}(e)=\left\{ \begin{array}{rl}1.155e&(e<0.7)\\ 172.143e^{2}-250e+91.17&(e\geq 0.7)\end{array} \right. \tag{28}\]  
\[f_{e}^{\prime}(e)=\left\{ \begin{array}{rl}2.51e+0.93&(e<0.7)\\ 318e^{2}-445e+158.37&(e\geq 0.7)\end{array} \right. \tag{29}\]  
The appearance of \(f_{e}^{\prime}(e)\) is a random event with a certain probability. With the datum force \(f_{e}(e)\) as a reference, then the pulsation increment force \(f_{e}^{\prime}(e)\) can be expressed as:  
\[f_{e}^{\prime}(e)=\left\{ \begin{array}{rl}&2.173f_{e}(e)+0.93&(e<0.7)\\ &\left(1.3567\sqrt{f_{e}(e)}+0.425\right)^{2}+2.12&(e\geq 0.7)\end{array} \right. \tag{30}\]  
Figure 14: Average value, maximum value, and pulsation comparison of clamping force in oils at different particle concentrations. The measurement results show that when the ratio of particle size to clearance size is less than 0.7, the risk of spool clamping is relatively low; when the ratio of particle size to clearance size is greater than 0.7, the spool clamping force rises rapidly as the particle size approaches the clearance size. The results further confirm that the particles in the size of 0.7-1 clearance (\(e\in[0.7,1]\)) are sensitive. That is because when the ratio reaches 0.7, the size of the particle is close to the clearance size, which will cause contact among the valve core surface, particle, and valve hole surface, named \"three-body contact\". This situation will increase friction force sharply. Equations (28)-(30) can be taken as the empirical equations of spool clamping force with varying particle sizes. The clamping force of a single particle can be obtained by introducing the values of dependent variables \(f_{e}(e)\) and \(f_{e}^{\prime}(e)\) into Equation (20).