# Dynamic Analysis Method for Fault Propagation Behaviour of Machining Centres
## 1 Introduction
#### 2.1.1 General System Hierarchy Structure Modelling Process  
The system structure model describes the relationship structure between the components of the system with the concept of set. The model is a diagram that reflects the relationship between the components of the system. For machining centres, however, the relationship diagram is difficult to utilize for clearly reflecting the relationships between the components of the system. Through matrix and hierarchy processing, the related digraph of each system component is transformed into a hierarchy structure model that can clearly understand the structure of the system and relationship between the components of the system. Its general modelling process is shown in Figure 2.  
1.2 Construction of the Hierarchy Structure Model of Fault Propagation in Machining Centres on the Basis of DSM  
In accordance with the general modelling process of the hierarchy structure model, the machining centre is first divided into \(n\) system components on the basis of the working process and principles of the machining centre, and the fault correlation analysis of the collected fault data is conducted and aided by the relevant experience of the system structure function and the fault diagnosis manual. Furthermore, the fault time of each system component in the machining centre and the fault propagation relationship among the components is determined. The system component is expressed as a node set: \(V=\{v_{1},v_{2},\cdots,v_{n}\}\). The fault propagation relationship and connection relationship between system component nodes are represented by directed edge set \(E=\{\langle v_{1},v_{2}\rangle,\langle v_{2},v_{3}\rangle,\cdots,\langle v_{m },v_{n}\rangle\}\). Thus, a digraph (\(G=(V,E)\)) of fault propagation of a machining centre can be obtained. Then, according to the fault propagation digraph, the direct fault influence between the system components is expressed with a relation matrix (\(A\)) and the reachable matrix is obtained according to the relation matrix. In addition, the reachable matrix is decomposed to realize the construction of the fault propagation hierarchy structure model. To establish a clearer fault propagation hierarchy structure model, the design structure matrix (DSM) method is introduced [37]. In the form of binary values, '0' and '1' or '\(\times\)' and a blank space represent the relationship between the row and column elements in the  
Figure 2: Schemes follow the same formatting. design structure matrix (\(M\)). The design structure matrix can be regarded as a reachable matrix. The elements in the design structure matrix are defined as follows:  
\[m_{ij}=\left\{ \begin{array}{cc}1,v_{i}\text{ has at least one dependency on }v_{j}\\ 0,v_{i}\text{ has nothing to do with }v_{j}\end{array} \right.\left(i=1,2,\cdots,n;\ j=1,2, \cdots,n\right) \tag{1}\]  
The fault influence relationship between the system components of a machining centre can also be expressed as the dependency relationship between the elements in the design structure matrix. At this time, the elements are defined as follows:  
\[m_{ij}=\left\{ \begin{array}{cc}1,v_{i}\text{ has at least one effect on }v_{j}\\ 0,v_{i}\text{ has no effect on }v_{j}\end{array} \right.\left(i=1,2,\cdots,n;\ j=1,2, \cdots,n\right) \tag{2}\]  
The modelling principle of the hierarchy structure model is based on the design structure matrix (\(M\)) and according to the knowledge of matrix, row-column transformation is conducted. In addition, the design structure matrix is converted into the lower triangular matrix as much as possible to reduce the existence of positive and negative transfer relations in the matrix. To eliminate feedback information between modules, the risk of iteration is reduced [38].