# Dynamic Analysis Method for Fault Propagation Behaviour of Machining Centres
## 1 Introduction
### Calculation of Fault Propagation Probability of Machining Centre System Components
#### 2.2.2 Determination of the Influence Degree between Components of Machining Centres
In this paper, the edge betweenness is defined as the proportion of the number of times that all paths pass through the edge \(E\big{<}v_{i},v_{j}\big{>}\) in the graph model. The edge betweenness of the directed edge \(E\big{<}v_{i},v_{j}\big{>}\) connecting \(v_{i}\) and \(v_{j}\) is denoted by \(  
Dynamic Analysis of the Fault Propagation Behaviour of a Machining Centre Based on the Fault Propagation Intensity  
To analyze the real-time fault propagation behaviour of a machining centre, the propagation range of the fault should be clarified to determine whether the propagation of the fault can be achieved. As expressed by the system and product safety manual [48], the occurrence of the fault propagation is within a certain interval and when the probability of fault propagation is greater than the threshold of \(10^{-8}\), the case of fault propagation may occur. Therefore, the probability of \(v_{i}\) fault to propagate to \(v_{j}\) can be obtained as follows:  
\[P_{E}\big{(}v_{i},v_{j}\big{)}\ =\ \sum_{b=1}^{w}P_{b}\big{(}v_{i},v_{j}\big{)}, \big{(}P_{b}\big{(}v_{i},v_{j}\big{)}\ =\ \prod P\big{(}v_{i},v_{j}\big{)}\big{)}, \tag{16}\]  
where \(b\) is one of the several paths in which a fault propagates from node \(v_{i}\) to \(v_{j}\). \(P_{b}\big{(}v_{i},v_{j}\big{)}\) is the fault propagation probability of the path. \(w\) is the number of paths from node \(v_{i}\) to \(v_{j}\). On the basis of Equation (16), it can be clarified whether the fault can propagate in the fault propagation model and then realize the determination of critical nodes and the fault propagation path. The process of the determination of critical nodes at each level is shown in Figure 4.  
As illustrated in Figure 4, to determine the system components of the fault initiation layer, the product of component fault probability and its own criticality importance at time \(t\) is calculated; in turn, the larger the value, the more critical it is. The critical component of the fault initiation layer is considered as the fault source. To determine the system components of the fault propagation layer, according to the hi  
Figure 4: Flow chart expressing how to determine critical nodes at each level. propagation intensity of a machining centre, the system components in the propagation layer connected with the fault source are determined at time \(t\) and the fault propagation intensity values of the system components connected with the fault source are calculated; in turn, the larger the value, the more critical it is. Similarly, the fault propagation intensity values of each system component connected with the critical nodes of the fault propagation layer in the fault absorption layer are calculated; in turn, the larger the value, the more critical it is and the critical nodes of the fault absorption layer are determined. The path composed of critical nodes is the critical fault propagation path, according to which fault prevention and maintenance can be conducted.