## 3 Case Application
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Element Code and Name** & & & & & & & & \\ \hline Numerical control & 29.54 & 99.31 & 172.53 & 235.26 & 348.63 & 406.73 \\ system (NC) & 609.18 & 725.92 & 971.39 & 1058.25 & 1189.59 & 1363.59 \\  & 1564.23 & 1853.46 & 2111.67 & & & \\ Feed system (F) & 52.49 & 135.57 & 309.27 & 589.69 & 726.11 & 868.53 \\  & 1276.37 & 1486.23 & 2275.64 & & & \\ Electrical system (E) & 30.75 & 80.74 & 126.85 & 160.85 & 219.37 & 319.75 \\  & 333.39 & 435.97 & 581.55 & 683.86 & 689.83 & 750.35 \\  & 784.27 & 957.84 & 1015.73 & 1146.33 & 1268.24 & 1305.53 \\  & 1496.54 & 1596.24 & 1812.53 & 1856.96 & 2073.36 & 2546.35 \\  & 2943.71 & & & & & \\ Tool holder (M) & 58.63 & 72.63 & 116.78 & 348.36 & 420.75 & 551.71 \\  & 1029.27 & 1116.58 & 1531.33 & 1860.67 & 2755.26 & \\ Servo system (S) & 28.53 & 116.36 & 174.64 & 246.59 & 275.57 & 333.53 \\  & 406.36 & 449.63 & 592.73 & 707.59 & 896.47 & 1160.36 \\  & 1264.74 & 1408.49 & 1643.68 & 1785.24 & 1972.74 & 2465.46 \\ Main driving system (T) & 43.57 & 58.35 & 103.57 & 161.47 & 219.63 & 278.54 \\  & 420.47 & 565.97 & 841.977 & 1334.44 & 1667.38 & 1986.96 \\ Hydraulic system (H) & 67.45 & 143.53 & 246.52 & 297.68 & 347.53 & 406.86 \\  & 464.49 & 551.41 & 638.83 & 826.5 & 1015.58 & 1102.09 \\  & 1256.36 & 1580.56 & 1972.47 & 2436.37 & 243.67 \\ Auxiliary system (A) & 15.82 & 42.57 & 92.75 & 169.53 & 195.46 & 284.38 \\  & 363.36 & 498.75 & 652.85 & 797.3 & 975.35 & 1029.86 \\  & 1276.29 & 1421.96 & 1940.47 & 2812.5 & & \\ Base piece (B) & 89.75 & 278.93 & 568.27 & 758.53 & 1136.57 & 2264.43 \\ \hline \end{tabular}
\end{table} 
Table 1: System component code and failure time statistics table. \begin{table}
\begin{tabular}{c c c c} \hline
**Antecedent Component** & **Subsequent Component** & **Antecedent Component** & **Subsequent Component** \\ \hline NC & M & E & A \\ NC & T & E & B \\ NC & A & S & M \\ \hline NC & B & S & T \\ F & S & H & F \\ E & NC & H & M \\ E & F & H & A \\ E & M & H & B \\ E & T & A & T \\ E & H & A & B \\ \hline \end{tabular}
\end{table} 
Table 2: Statistical analysis of the related faults in CNC lathe components. The improved Floyd algorithm was used to iteratively calculate the number of steps and paths of fault propagation, and the corresponding indirect correlation matrix \(C^{u}\) and routing matrix \(C^{r}\) were obtained as Matrix (14) and (15). \[C^{u}=\begin{array}{ccccccccc}\text{NC}&\text{F}&\text{M}&\text{E}&\text{S}& \text{T}&\text{H}&\text{A}&\text{B}\\ \text{F}&\left[ \begin{array}{ccccccccc}0&0&1&0&0&2&0&1&2\\ 0&0&2&0&1&2&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 1&2&2&0&3&3&1&2&3\\ 0&0&1&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ \text{H}&0&1&3&0&2&2&0&1&2\\ 0&0&0&0&0&1&0&0&1\\ 0&0&0&0&0&0&0&0&0\end{array} \right]\\ \\ \text{B}&\begin{array}{ccccccccc}\text{NC}&\text{F}&\text{M}&\text{E}&\text{S}& \text{T}&\text{H}&\text{A}&\text{B}\\ \text{NC}&\text{F}&\left[ \begin{array}{ccccccccc}1&0&1&0&0&8&0&1&8\\ 0&2&5&0&2&5&0&0&0\\ 0&0&3&0&0&0&0&0\\ 4&7&1&4&2&8&4&1&8\\ 0&0&5&0&5&5&0&0&0\\ 0&0&0&0&6&0&0&0\\ 0&7&5&0&2&8&7&7&8\\ 0&0&0&0&0&8&0&8&8\\ 0&0&0&0&0&0&0&0&9\end{array} \right]\\ \\ C^{r1}= \begin{array}{ccccccccc}\text{NC}&\text{F}&\text{M}&\text{E}&\text{S}& \text{T}&\text{H}&\text{A}&\text{B}\\ \text{NC}&\text{F}&\left[\begin{array}{ccccccccc}1&0&1&0&0&8&0&1&8\\ 0&2&5&0&2&5&0&0&0\\ 0&0&3&0&0&0&0&0&0\\ 4&7&1&4&2&8&4&7&8\\ 0&0&5&0&5&6&0&0&0\\ 0&0&0&0&6&0&0&0\\ 0&7&5&0&2&8&7&7&8\\ 0&0&0&0&0&8&0&8&8\\ 0&0&0&0&0&0&0&0&9\end{array}\right]\\ \end{array}  \tag{15}\]  
Combined with an indirect correlation matrix and routing matrix, a fault propagation structure model starting from each component of CNC lathe was constructed, as shown in the Figure 3.  
Figure 3 clearly shows the extent to which each component of the CNC lathe is an initial fault source and may propagate the fault to other components associated with it. For example, in Figure 3a, when NC acts as the initial fault source, its failure may cause M or A to fail, and when A fails, the failure may be transmitted to T or B. This is further explained by an example.