# Dynamic Analysis Method for Fault Propagation Behaviour of Machining Centres
## 4 Case Application of a Machining Centre
### Implementation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Any Two Nodes** & **Paths** \\ \hline D\(\rightarrow\)F & D\(\rightarrow\)NC\(\rightarrow\)F, D\(\rightarrow\)B\(\rightarrow\)F, D\(\rightarrow\)W\(\rightarrow\)F, \\ D\(\rightarrow\)NC\(\rightarrow\)B & D\(\rightarrow\)B, D\(\rightarrow\)NC\(\rightarrow\)B \\ NC\(\rightarrow\)F & NC\(\rightarrow\)F, NC\(\rightarrow\)B\(\rightarrow\)F \\ D\(\rightarrow\)T & D\(\rightarrow\)NC\(\rightarrow\)B\(\rightarrow\)T \\ D\(\rightarrow\)T & D\(\rightarrow\)T, D\(\rightarrow\)B\(\rightarrow\)T \\ R\(\rightarrow\)T & R\(\rightarrow\)T, R\(\rightarrow\)B\(\rightarrow\)T \\ R\(\rightarrow\)F & R\(\rightarrow\)F, R\(\rightarrow\)B\(\rightarrow\)F \\ H\(\rightarrow\)T & H\(\rightarrow\)T, H\(\rightarrow\)B\(\rightarrow\)T \\ D\(\rightarrow\)W & D\(\rightarrow\)W \\ Q\(\rightarrow\)B & Q\(\rightarrow\)B \\ D\(\rightarrow\)NC & D\(\rightarrow\)NC \\ H\(\rightarrow\)B & H\(\rightarrow\)B \\ NC\(\rightarrow\)B & NC\(\rightarrow\)B \\ NC\(\rightarrow\)T & NC\(\rightarrow\)B\(\rightarrow\)T \\ H\(\rightarrow\)F & H\(\rightarrow\)B\(\rightarrow\)F \\ R\(\rightarrow\)B & R\(\rightarrow\)B \\ B\(\rightarrow\)T & B\(\rightarrow\)T \\ W\(\rightarrow\)T & W\(\rightarrow\)T \\ K\(\rightarrow\)F & K\(\rightarrow\)F \\ B\(\rightarrow\)F & B\(\rightarrow\)F \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistical results of any two node paths. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Directed Edge** & \(L\left(v_{i}v_{j}\right)\) & **Directed Edge** & \(L\left(v_{i}v_{j}\right)\) \\ \hline \(E(v_{R},v_{B})\) & 1 & \(E(v_{Q},v_{B})\) & 1.5 \\ \(E(v_{D},v_{B})\) & 0.75 & \(E(v_{B},v_{T})\) & 3.5 \\ \(E(v_{NC},v_{B})\) & 2.75 & \(E(v_{B},v_{F})\) & 3.5 \\ \(E(v_{H},v_{B})\) & 1.5 & \(E(v_{R},v_{T})\) & 0 \\ \(E(v_{D},v_{NC})\) & 1.5 & \(E(v_{R},v_{F})\) & 0 \\ \(E(v_{D},v_{W})\) & 0.25 & \(E(v_{H},v_{T})\) & 0 \\ \(E(v_{W},v_{F})\) & 0.25 & \(E(v_{K},v_{F})\) & 0 \\ \(E(v_{NC},v_{F})\) & 0.25 & \(E(v_{Q},v_{T})\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The edge betweenness of the fault propagation hierarchy model of a machining centre. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Directed Edge** & \(In\left(v_{i}v_{j}\right)\) & **Directed Edge** & \(In\left(v_{i}v_{j}\right)\) \\ \hline \(E(v_{R},v_{B})\) & 0.1396 & \(E(v_{Q},v_{B})\) & 0.0775 \\ \(E(v_{D},v_{B})\) & 0.1400 & \(E(v_{B},v_{T})\) & 0.0314 \\ \(E(v_{NC},v_{B})\) & 0.1231 & \(E(v_{B},v_{F})\) & 0.0581 \\ \(E(v_{H},v_{B})\) & 0.0740 & \(E(v_{R},v_{T})\) & 0.0262 \\ \(E(v_{D},v_{NC})\) & 0.0521 & \(E(v_{R},v_{F})\) & 0.0267 \\ \(E(v_{D},v_{W})\) & 0.0486 & \(E(v_{H},v_{T})\) & 0.0368 \\ \(E(v_{W},v_{F})\) & 0.0686 & \(E(v_{K},v_{F})\) & 0.0477 \\ \(E(v_{NC},v_{F})\) & 0.0212 & \(E(v_{Q},v_{T})\) & 0.0283 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Fault propagation intensity of each directed edge of a machining centre fault propagation model at 1500 h.  
In reference to Figure 11 and Table 9, a hierarchy structure model of the fault propagation intensity of a machining centre can be obtained as shown in Figure 12.  
In reference to Equation (16) and Figure 11, the fault propagation probability values of each path in the fault propagation hierarchy model of a machining centre at 1500 h can be calculated as shown in Table 10.  
As expressed in Table 10, the fault propagation probability of each path is greater than the threshold value of \(10^{-8}\); thus, a fault propagation phenomenon exists in the model. Based on Table 9 and the hierarchy structure model of fault propagation in a machining centre, for the fault initiation layer there is only the electrical system component \(D\). Hence, the critical node of the fault initiation layer is \(D\). For the fault propagation layer at 1500 h, \(In(v_{D},v_{NC})=0.0740>In(v_{D},v_{B})=0.0521>In(v_{D},v_{W})=0.0368\), thus the critical node currently is \(N\)C. This indicates that the fault is more likely to be transferred from the electrical system to the numerical control system.