## 3 Case Application
\begin{table}
\begin{tabular}{c c} \hline \hline
**Component Code** & **Fault Probability Function** \\ \hline NC & \(F_{\text{NC}}(t)=1-\exp\left[-\left(\frac{t}{891L3X}\right)^{0,9911}\right]\) \\ F & \(F_{\text{F}}(t)=1-\exp\left[-\left(\frac{t}{1208.971}\right)^{0,9681}\right]\) \\ M & \(F_{\text{M}}(t)=1-\exp\left[-\left(\frac{t}{128.342}\right)^{0,9083}\right]\) \\ E & \(F_{\text{E}}(t)=1-\exp\left[-\left(\frac{t}{4001.146}\right)^{1,1109}\right]\) \\ S & \(F_{\text{S}}(t)=1-\exp\left[-\left(\frac{t}{801.153}\right)^{1,0308}\right]\) \\ T & \(F_{\text{T}}(t)=1-\exp\left[-\left(\frac{t}{167.57}\right)^{0,8358}\right]\) \\ H & \(F_{\text{H}}(t)=1-\exp\left[-\left(\frac{t}{904.036}\right)^{1,1802}\right]\) \\ A & \(F_{\text{A}}(t)=1-\exp\left[-\left(\frac{t}{954.843}\right)^{0,8283}\right]\) \\ B & \(F_{\text{B}}(t)=1-\exp\left[-\left(\frac{t}{1349.36}\right)^{1,0421}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The fault probability function of system components in CNC lathe. Figure 4: Probability importance function curve of system components. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Propagation Path** & \begin{tabular}{c} _I(e\({}_{i\to j}\))_ \\ **(4000 h)** \\ \end{tabular}  & \begin{tabular}{c} \(\tau\)(\(e\)\({}_{i\to j}\)) \\ **(4000 h)** \\ \end{tabular}  & \ \begin{tabular}{c} _I(e\({}_{i\to j}\))_ \\ **(6000 h)** \\ \end{tabular}  & \begin{tabular}{c} \(\tau\)(\(e\)\({}_{i\to j}\)) \\ **(6000 h)** \\ \end{tabular}  \\ \hline \multicolumn{5}{c}{(E as a starting point)} \\ E\(\to\)NC & 0.918448361 & 0.000254995 & 0.796881103 & 0.000231419 \\ E\(\to\)H & 0.977647161 & 0.000271431 & 0.956705067 & 0.000277833 \\ H\(\to\)A & 0.956717540 & 0.000221941 & 0.848962399 & 0.000211872 \\ H\(\to\)F & 0.909115084 & 0.000210898 & 0.766856527 & 0.000191382 \\ NC\(\to\)M & 0.981947229 & 0.0001141439 & 0.955113065 & 0.000137078 \\ NC\(\to\)A & 0.995258634 & 0.000143395 & 0.982569130 & 0.000141019 \\ A\(\to\)B & 0.979554928 & 0.000098696 & 0.969648730 & 0.000091128 \\ A\(\to\)T & 0.996958535 & 0.000100450 & 0.994247429 & 0.000093440 \\ F\(\to\)S & 0.945228319 & 0.000078409 & 0.875293970 & 0.000071674 \\ \multicolumn{5}{c}{(NC as a starting point)} \\ NC\(\to\)A & 0.995526834 & 0.000143395 & 0.982569130 & 0.000141019 \\ NC\(\to\)M & 0.981947229 & 0.0001414141 & 0.955113065 & 0.000137078 \\ A\(\to\)B & 0.979554928 & 0.000098696 & 0.969648730 & 0.000091128 \\ A\(\to\)T & 0.96958535 & 0.000100450 & 0.994247429 & 0.000093440 \\ \multicolumn{5}{c}{(F as a starting point)} \\ F\(\to\)S & 0.945228319 & 0.000078409 & 0.875293970 & 0.000071674 \\ S\(\to\)T & 0.963840493 & 0.000169284 & 0.902043089 & 0.000160421 \\  & & & & \\ S\(\to\)T & 0.963840493 & 0.000169284 & 0.902043089 & 0.000160421 \\ S\(\to\)M & 0.958919595 & 0.000168419 & 0.898204379 & 0.000159738 \\  & & & & \\  & & & & \\ A\(\to\)T & 0.996958535 & 0.000100450 & 0.994247429 & 0.000093440 \\ A\(\to\)B & 0.979554928 & 0.000098696 & 0 The dynamic and real-time fault propagation influence value represents the importance of the components in the system. Through the ranking results in Table 6, it can be identified that the key components of the CNC lathe under current investigation at 4000 h and 6000 h are electrical systems, followed by hydraulic systems, which should be paid attention to in fault diagnosis and health maintenance.