# Dynamic Analysis Method for Fault Propagation Behaviour of Machining Centres
## 4 Case Application of a Machining Centre
### Implementation  
In this section, we take MDH series horizontal machining centres as the research object that are mainly used for processing rotary parts. We collected 108 on-site fault information details of 36 machining centres of this series during the course of one year. After fault analysis, we can determine whether each component fault is an independent fault or a related fault. If it is a related fault, the antecedent component that caused the component fault will be determined through fault analysis. For example, when the workpiece cannot be clamped, the tool magazine system (T) is the direct fault location but the root cause is that the workpiece cannot be clamped due to insufficient pressure in the hydraulic system (H). At this time, the faulty component is the tool magazine system (T) and its antecedent component is the hydraulic system (H). Considering the existence of this propagation fault, there is likely to be a directed arrow pointing from the hydraulic system (H) to the tool magazine system (T). Similarly, when the servo motor fails, the fault location is the feed system (F) but the root cause is the abnormality of the spindle system (B). Therefore, there is likely to be a directed arrow from the spindle system (B) to the feed system (F). In this manner, we can identify other related faults. The statistical analysis results of the related faults are shown in Table 1.  
As expressed in Table 1, combined with the knowledge of graph theory, the fault propagation relationship of a machining centre is modelled. We do not consider compo \begin{table}
\begin{tabular}{c c c} \hline
**Fault Component** & **Antecedent Component** & **Frequency** \\ \hline T & B & 3 \\ F & B & 2 \\ B & NC & 1 \\ F & NC & 1 \\ T & H & 1 \\ B & H & 1 \\ B & D & 1 \\ NC & D & 1 \\ W & D & 1 \\ F & R & 1 \\ T & R & 1 \\ B & R & 1 \\ F & W & 1 \\ F & K & 1 \\ B & Q & 2 \\ T & Q & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Statistical analysis of the related faults in machining centre system components. nents with uncorrelated faults such as the workbench (_U_). Thus, we can obtain the fault propagation digraph of a machining centre as shown in Figure 5.  
As expressed in the fault propagation digraph of a machining centre in Figure 5, the relationship matrix (_A_) and the design structure matrix (_M_) can be obtained as follows:  
\[ \begin{array}{ccccccccc}\text{B}&\text{T}&\text{F}&\text{NC}&\text{H}&\text{ D}&\text{Q}&\text{R}&\text{W}&\text{K}\\ \text{B}&\text{T}&\left[\begin{array}{cccccccccccc}0&1&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 1&0&1&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0&0&0\\ 1&1&1&0&0&0&0&0&0&0\\ 1&1&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ \end{array}\right]\\ \end{array} \]  
\[ \begin{array}{cccccccccccc}\text{B}&\text{T}&\text{F}&\text{NC}&\text{H}&\text{ D}&\text{Q}&\text{R}&\text{W}&\text{K}\\ \text{T}&\left[\begin{array}{cccccccccccc}1&1&1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 1&1&1&1&0&0&0&0&0&0\\ 1&1&1&0&1&0&0&0&0&0\\ 1&1&0&1&0&1&0&0&1&0\\ 1&1&1&0&0&0&1&0&0&0\\ 1&1&1&0&0&0&0&1&0&0\\ 0&0&1&0&0&0&0&0&1&0\\ 0&0&1&0&0&0&0&0&1\end{array}\right]\\ \end{array} \]  
The element 1 in the matrix \(M\) indicates that the fault of the system component \(i\) impacts \(j\). Contrarily, 0 indicates that no impact exists. Thus, on the basis of the modelling process of the DSM fault propagation hierarchy structure model, the fault propagation DSM model of a machining centre can be obtained as shown in Figure 6. The diagonal elements in Figure 6 are represented by black squares. Figure 5: Fault propagation digraph of a machining centre. On the basis of Figure 6, we can derive the result of module division after the DSM modeling processing: the first-layer system components are (T, F), the second-layer system components are (B, W, K), the third-layer system components are (NC, H, Q, R), and the fourth-layer system component is (D).