# Dynamic Analysis Method for Fault Propagation Behaviour of Machining Centres
## 4 Case Application of a Machining Centre
### Implementation
The system components of the first layer, which is the appearance layer, are classified as the fault absorption layer in the fault propagation model. The system components of the fourth layer, the root layer, are classified as the fault initiation layer and the remaining layers are classified as the fault propagation layers. Thus, the fault propagation hierarchy structure model of a machining centre can be obtained as shown in Figure 7.  
On the basis of obtaining the fault information of a machining centre, we calculated the fault probability function of each component according to the calculation method presented in Section 2.2.1. Considering the impact of the timing truncation test and the fault time truncation on the sequence of the fault data, we use the Johnson method to modify it. The parameters of the fault probability model are then estimated and the distribution hypothesis test is passed. Finally, the fault probability function of each component can be obtained as shown in Table 2 and the function curve is illustrated in Figure 8.  
Similarly, the fault probability function of a whole machining centre is \(F_{Z}(t)=1-\exp\left[-(t/620.984)^{0.951}\right]\). According to Table 2 and Equations (9) and (10), the probability importance and criticality importance of each system component node can be obtained at any time. The function curves of probability importance and criticality importance of each system component node are illustrated in Figures 9 and 10, respectively. Figure 6: The fault propagation DSM model of a machining centre. Figure 7: Hierarchy structure model of fault propagation in a machining centre. Figure 8: Fault probability function curve of machining centre system components. \begin{table}
\begin{tabular}{c c} \hline \hline
**Component Code** & **Fault Probability Function** \\ \hline T & \(F_{T}(t)=1-\exp\left[-(t/2791.105)^{1.015}\right]\) \\ B & \(F_{B}(t)=1-\exp\left[-(t/7832.664)^{0.641}\right]\) \\ R & \(F_{R}(t)=1-\exp\left[-(t/8161.512)^{0.791}\right]\) \\ H & \(F_{H}(t)=1-\exp\left[-(t/8905.062)^{0.656}\right]\) \\ NC & \(F_{\mathrm{NC}}(t)=1-\exp\left[-(t/7912.399)^{0.542}\right]\) \\ D & \(F_{\mathrm{D}}(t)=1-\exp\left[-(t/7935.696)^{0.771}\right]\) \\ Q & \(F_{\mathrm{O}}(t)=1-\exp\left[-(t/27503.400)^{0.524}\right]\) \\ F & \(F_{\mathrm{F}}(t)=1-\exp\left[-(t/2606.440)^{1.295}\right]\) \\ W & \(F_{\mathrm{W}}(t)=1-\exp\left[-(t/4145.720)^{0.910}\right]\) \\ K & \(F_{\mathrm{K}}(t)=1-\exp\left[-(t/5960.776)^{0.931}\right]\) \\ U & \(F_{\mathrm{U}}(t)=1-\exp\left[-(t/6869.766)^{0.773}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fault probability function of system components in a machining centre. Figure 9: Probability importance function curve of system component nodes. Given that the values of probability and the criticality importance of system components are varied, this paper takes t = 1500 h as an example to illustrate this concept. By substituting the fault probability values of system components obtained from Table 2 into Equations (9) and (10), the probability importance and criticality importance of each system component node at t = 1500 h can be obtained as shown in Table 3.  
On the basis of Equations (10) and (11), combined with Figure 7, the edge fault influence value of the same level at 1500 h can be calculated as shown in Table 4.  
In reference to Equation (12), Table 3, and Figure 7, the edge fault influence value between different levels in the fault propagation hierarchy structure model of a machining centre at 1500 h is calculated. The results are detailed in Table 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Directed Edge** & \(\mathbf{I}\Big{(}\mathbf{v}_{i}\mathbf{v}_{j}\Big{)}\) \\ \hline \(E(v_{R},v_{B})\) & 0.9870 \\ \(E(v_{H},v_{B})\) & 0.9876 \\ \(E(v_{Q},v_{B})\) & 0.9808 \\ \(E(v_{NC},v_{B})\) & 0.9914 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fault influence degree of the edge at the same level at 1500 h.  
Figure 10: Criticality importance function curve of system component nodes.