## 2 Materials and Methods
### Establishment of Mechanical Model
#### 2.1.2 Mathematical Model of Spool Clamping Risk Zone \(A_{i}\)
\[\omega_{\max}=\arcsin\frac{d_{0}-l}{\sqrt{h^{2}+d^{2}}}-\arctan\tfrac{d}{h} \quad(0\leq l<\tfrac{d_{0}-d}{2}) \tag{5}\]  
\[\tan\theta=\frac{2l}{d}\cdot\frac{1}{\sin\omega}-\tan\frac{\omega}{2}\ (0\leq \omega<\omega_{\max},0\leq l<\tfrac{d_{0}-d}{2}) \tag{6}\]  
\[A_{\omega}=\frac{\pi d^{2}}{8}\tan\theta=\frac{\pi d^{2}}{8}\tan\frac{\omega} {2}\ (0<\omega\leq\omega_{\max}l=0) \tag{7}\]  
\[A_{\omega l}=\left|\tfrac{\pi d^{2}}{8}\tan\theta\right|=\tfrac{\pi d^{2}}{8} \Big{|}\Big{(}\tfrac{2l}{d}\cdot\tfrac{1}{\sin\omega}-\tan\tfrac{\omega}{2} \Big{)}\Big{|}\quad(0<\omega\leq\omega_{\max},0<l<\tfrac{d_{0}-d}{2}) \tag{8}\]  
Figure 5: Computation of a spatial sector in clamping risk zones resulting from spool tilting and off-center postures. (**a**) Parameters used in the calculation of ‘\(S\)’; (**b**) Parameters used in the calculation of ‘\(A_{3}\)’; (**c**) Parameters used in the calculation of ‘\(A_{4}\)’. \[A_{i}=\left\{ \begin{array}{rl}&A_{1}=0\quad(\omega=0,l=0)\\ &A_{2}=\frac{\pi}{8}hd^{2}\quad(\omega=0,\ 0<l\leq\frac{4q_{0}-d}{2})\\ &A_{3}=\frac{1}{4}A-A_{\omega}=\frac{\pi}{16}hd^{2}-\frac{\pi d^{2}}{8}\tan \frac{\omega}{2}\quad(0<\omega\leq\omega_{\max},l=0)\\ &A_{4}=\frac{1}{4}A+A_{\omega l}=\frac{\pi}{16}hd^{2}+\frac{\pi d^{2}}{8} \Big{(}\frac{2l}{4}\cdot\frac{1}{\sin\omega_{\max}}-\tan\frac{\omega}{2}\Big{)} \quad(0<\omega<\omega_{\max},0<l<\frac{4q_{0}-d}{2})\\ &u_{1}=\frac{d_{0}\cos\omega-d}{2\cos\omega}-l\quad(0\leq\omega<\omega_{\max 2 },0<l\leq\frac{4q_{0}-d}{2})\\ &u_{2}=\frac{d_{0}-d\cos\omega}{2}-l\quad(0\leq\omega<\omega_{\max 2 },0<l\leq\frac{4q_{0}-d}{2})\\ &u_{\min}=\frac{1}{2}(d_{0}-d\cos\omega-l-h\sin\omega)\quad(0\leq\omega<\omega _{\max 2},0<l\leq\frac{4q_{0}-d}{2})\end{array} \right. \tag{9}\]  
Calculation of the Number of Sensitive Particles n, and Deduction of the Expressions of Joint Force \(F_{clamping}\) and Single Force \(\sigma_{i}\)  
Since the irregular shapes of tiny particles, the connection line between the closest two points on the particle surface was taken as the short axis and expressed by \(a\). When \(a\to u_{0}\) and \(a<u_{0}\), some particles may not only invade in the clearance but also bring about spool clamping, and those particles are known as \"sensitive particles\". Since the clearance size varies as the spool posture changes, the size of sensitive particles lies in a certain range. Setting the clearance size at \(u_{0}\) = 0.6 mm, a cluster of aluminum particles in the size of \(a\in[0.2,\,0.7]\) (unit: mm) were selected for the size measurement and morphological analysis. Figure 6 shows the micromorphology of some particles after magnification. It can be seen that the shapes of most particles can be abstracted into squares or rectangles, of which the square particles account for 2/3 or so. Therefore, the simplification of the particles into square particles in Figure 3 tallies with the actual situation. The statistical result histogram is shown below. The probability distribution curves of the particles of different sizes were fitted, and the probability density function \(\varphi(a)\) of these particles was obtained. Finally, the expression \(\Phi(a)\) of probability distribution curves of the particles of different sizes was derived inversely. The particle size distribution histogram was obtained by statistics. Figure 7 shows that the particles in the size within \(a\in[0.4,\,0.5]\) are the most in quantity, accounting for 41.3% of the total samples. The particle sizes are mainly in the range of [0.3, 0.6], accounting for 87% of the total samples. By the statistical results, the probability scatter diagram of the particles smaller than a certain size was obtained, and the probabilities were fitted into a probability density curve \(\Phi(a)\) of particle sizes. Then, \(\Phi(u_{0})-\Phi(u_{\min})\) can be used to denote the particle probability of \(a\in[u_{min},\,u_{0}]\). The expression of function \(\Phi(a)\) is derived as follows. In Figure 8, the green solid line denotes the curve \(d\Phi(a)/da\) obtained by the differentiation of the distribution curves.