## 2 Materials and Methods
### Experiment Design and Result Analysis
#### 2.2.3 Influences of Particle Concentration on Clamping Force  
This section focuses on the influences of particle concentration on the clamping force. Since a good suspension is characteristic in hydraulic oils, aluminum particles neither sink nor float in those oils in a short time, which is favorable for the experiment. According to the previous research (as Figure 7 shows), aluminum particles in size within \(a\in[0.2,\,0.7]\) (unit: mm) were chosen. Based on previous low-resolution predictive experiments, it was found that the sudden increase in friction occurs when the volume concentration is between 1% and 10%. So, the particle volume concentration of 1%, 2%, 3%, 4%, 5%, 6%, 7%, 8%, 9%, and 10% were selected to prepare experimental oils, respectively, as shown in Figure 13a. Figure 13b shows the spool clamping force curves of aluminum particulate oils at different volume concentrations, measured by a clamping force experimental device. It can be found that the clamping force curves at different concentrations present aperiodic pulsation. The pulsation force is relatively large at the initial moment and then gradually decreases. Such change is particularly evident when the concentration is 9% and 10%, suggesting a relatively large spool clamping force at the starting moment. Moreover, the clamping force \(F_{c}(c)\) is found to grow gradually as the particle concentration increases, and reaches its peak of 15.98 N when \(c=9\%\). To further investigate the relationship between concentration and clamping force, the mean and maximal clamping forces at different concentrations were compared. The comparative analysis results are shown in Figure 14. It can be seen from Figure 14 that as the particle volume concentration \(c\%\) increases from 1% to 10%, both the mean and maximum clamping forces increase, and the pulsation increment of the maximum clamping force rises at an increasing amplitude. Since the clamping pulsation has a certain randomness, the maximum clamping force shows a decrease at the volume concentration of 10%. According to the fitting curve diagram, the mean clamping force is fitted into the curve \(f_{c}(c)\), and the clamping pulsation increment \(F_{c}(c)-f_{c}(c)\) into the curve \(f_{c}^{\prime}(c)\). It can be found that when the particle volume concentration \(c\%<5\%\), both the values of the curves \(f_{c}(c)\) and \(f_{c}^{\prime}(c)\) are relatively small and increase slowly and linearly with the concentration. When the particle volume concentration \(c\%\geq 5\%\), both curves \(f_{c}(c)\) and \(f_{c}^{\prime}(c)\) increase rapidly in the manner of a quadratic curve and their increase rates are rather equal. That is because when \(c\%\geq 5\%\), the number of particles invading into clearance significantly increases, which  
Figure 12: Measurement results of spool clamping force in hydraulic oil containing aluminum, gravel, and iron particles. (**a**) Spool clamping force curve in hydraulic oil containing aluminum, gravel, and iron particles. (**b**) Average value, maximum value, and pulsation frequency of spool clamping force in hydraulic oil containing aluminum, gravel, and iron particles. will accumulate in clearance. When the valve core slides in the valve hole, the gathered particles extremely easily cause greater friction force. The fitting equations for \(f_{c}(c)\) and \(f_{c}^{\prime}(c)\) are shown as (25) and (26). \[f_{c}(c)=\left\{ \begin{array}{cc}6.043c&(c<5\%)\\ 2472.7c^{2}-202.46c+4.4&(c\geq 5\%)\end{array} \right. \tag{25}\]  
\[f_{c}^{\prime}(c)=\left\{ \begin{array}{cc}20.6903c+0.8537&(c<5\%)\\ 2747.6c^{2}-211c+5.67&(c\geq 5\%)\end{array} \right. \tag{26}\]  
The appearance of \(f_{c}^{\prime}(c)\) is a random event with a certain probability. With basic force \(f_{c}(c)\) as a reference, then the pulsation increment force \(f_{c}^{\prime}(c)\) can be expressed as follows:  
\[f_{c}^{\prime}(c)\,\sim\left\{ \begin{array}{cc}3.42f_{c}(c)+0.8537&(c<5\%) \\ f_{c}(c)+1.27&(c\geq 5\%)\end{array} \right.