## 3 Case Application
In the multi-process processing of complex parts, the fault phenomenon is that the tool holder system cannot realize the tool change for parts processing, but in-depth analysis to find the reason determines that the PLC system in the numerical control system is faulty, and the tool change instruction cannot be issued, so there is an arrow pointing from NC to M. The PLC control program in the CNC system can realize the auxiliary function control of the lathe. When it fails, the lubrication system in the auxiliary system may not work normally. When the CNC lathe cannot be lubricated, it will cause the spindle parts in the main driving system and the table in the base piece mechanical to wear, which reduces the life of the CNC lathe. Thus, the failure of the CNC system is transmitted to the main driving system and the base piece through the auxiliary system. On the basis of the operation fault information of CNC lathe, the calculation method and operation process in Section 2.2.1 are used to evaluate the dynamic fault rate of each component. The Johnson method is used to correct the order, and the hypothetical distribution is identified. We use the two-parameter Weibull distribution to build the reliability model, and the least squares method to estimate the parameters; the linear correlation test and the goodness of fit test are then carried out. Finally, the dynamics fault rate function of each component of the CNC lathe can be obtained in Table 3 below. Figure 3: Fault propagation structure diagram of each component of CNC lathe. (**a**) NC as a starting point, (**b**) F as a starting point, (**c**) M as a starting point, (**d**) E as a starting point, (**e**) S as a starting point, (**f**) T as a starting point, (**g**) H as a starting point, (**h**) A as a starting point, (**i**) B as a starting point. \begin{table}
\begin{tabular}{c c} \hline \hline
**Component Code** & **Dynamics Fault Rate Function** \\ \hline NC & \(\lambda_{\text{NC}}(t)=\frac{f(t)}{R(t)}=\frac{0.9911}{6914.379971}t^{-0.0089}\) \\ F & \(\lambda_{\text{F}}(t)=\frac{f(t)}{R(t)}=\frac{1}{1208.771897}t^{-0.0319}\) \\ M & \(\lambda_{\text{M}}(t)=\frac{f(t)}{R(t)}=\frac{0.9083}{11283.42080}t^{-0.0917}\) \\ E & \(\lambda_{\text{E}}(t)=\frac{f(t)}{R(t)}=\frac{1.1109}{4001.14671107}\) \\ S & \(\lambda_{\text{S}}(t)=\frac{f(t)}{R(t)}=\frac{1.0308}{802.155300}t^{0.008}\) \\ T & \(\lambda_{\text{T}}(t)=\frac{f(t)}{R(t)}=\frac{0.8838}{116.757888}t^{-0.1642}\) \\ H & \(\lambda_{\text{H}}(t)=\frac{1.1802}{R(t)}=\frac{0.1802}{49040.0466^{-1.082}}\) \\ A & \(\lambda_{\text{A}}(t)=\frac{f(t)}{R(t)}=\frac{0.8283}{9544.843928}t^{-0.1717}\) \\ B & \(\lambda_{\text{B}}(t)=\frac{f(t)}{R(t)}=\frac{1.0421}{13449.36010}t^{0.0421}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dynamics fault rate function of system components in CNC lathe. Similarly, the fault probability function of a whole CNC lathe is \(F_{Z}(t)=1-\exp\left[-\left(\frac{l}{892.821}\right)^{1,047}\right]\) and the fault probability function of system components can be obtained in Table 4.  
According to Equations (5) and (6) and Table 4, the probability importance and criticality importance of components at any time can be calculated, and the function curves of each component are illustrated as Figures 4 and 5, respectively. It can be seen from the figures that importance curves of CNC lathe components tend to be stable when they run to 4000 h. Therefore, 4000 h and 6000 h are taken as examples, and the fault influence degree and fault propagation rate of other associated components are calculated combined with the fault propagation structure diagram of the components. According to the value of the fault propagation rate solved in Table 5, combined with the third-order neighborhood component relationship of each component characterized in the fault propagation structure diagram, the improved ASP algorithm is used to calculate the fault propagation influence value of the components in the system at 4000 h and 6000 h, and the ranking is as shown in Table 6.