# 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  
Importance refers to the contribution to the fault probability of the whole machine when a system component fails; it is a time-dependent function and can characterize both the structure of the system and a parameter of the reliability of the system components [39]. The importance of system components is quantitatively described and called the importance of system components. In this paper, it is regarded as the influence degree of a system component to reflect the influence degree of a whole machine fault caused by the fault of a certain system component. Calculation of Probability Importance of System Component Nodes  
In the late 1960s, Birnbaum put forward the Birnbaum probability importance for practical production [40]. The idea of probability importance is introduced by considering the contribution of reducing the fault probability of system components to reducing the fault probability of the whole system. When any system component of a machining centre fails, it may cause the whole machine to fail. Hence, the relationship between the system components of a machining centre can be regarded as the series relationship. According to this structural relationship, the fault probability function of the system components can be used to describe the fault probability function of a whole machining centre. The fault probability function is a function of time and their relationships are expressed in Equation (7):  
\[F_{Z}(t)=1-[(1-F_{1}(t))(1-F_{2}(t))\cdots(1-F_{i}(t))]. \tag{7}\]  
In addition, the equation for calculating the probability importance of system component nodes are shown in Equation (8):  
\[I_{P}(v_{i})=\frac{\partial F_{Z}(t)}{\partial F_{i}(t)}\ \ i=1,2,\cdots,n, \tag{8}\]  
where \(I_{P}(v_{i})\) is the probability importance of the system component node, \(F_{Z}(t)\) is the fault probability function of the whole machine, and \(F_{i}(t)\) is the fault probability function of the system component node. For any two-system component nodes, regarding \(v_{i}\) and \(v_{j}\), if \(I_{P}(v_{i})>I_{P}(v_{j})\) exists at some time then \(v_{i}\) is more important than \(v_{j}\). Calculation of the Criticality Importance of Nodes  
The probability importance expresses that the fault probability of the system component node changes by one unit, resulting in the change of the fault probability of the whole system. Given that the probability of the fault of each system component node is different, the difficulty of generating a unit change is also different. Therefore, the criticality importance is introduced to describe the properties that the probability importance cannot reflect. Criticality importance was first proposed by [PERSON] [41]. In the case of a given fault of a whole system, the conditional probability that the whole machine is in a certain state when the critical system component node \(v_{i}\) fails at time \(t\). The core idea is to improve the non-reliable system components rather than to further improve the reliable system components. Therefore, the criticality importance (\(I_{C}(v_{i})\)) of the system component node is defined as:  
\[I_{C}(v_{i})=\frac{F_{i}(t)}{F_{Z}(t)}\cdot\frac{\partial F_{Z}(t)}{\partial F _{i}(t)}=\frac{F_{i}(t)}{F_{Z}(t)}\cdot I_{p}(v_{i}). \tag{9}\]  
For a machining centre, when the system component node \(v_{i}\) fails and the fault is passed to the component node \(v_{j}\) at a certain value, \(v_{j}\) is affected by the fault and the component node \(v_{i}\) is also affected by the component node \(v_{j}\). There also exists a certain influence value on the directed edge between the two component nodes. This value represents the influence ability of the component node \(v_{i}\) fault to cause the fault of its connected component node \(v_{j}\). However, the fault influence degree of the edge of the same level differs from that of the different layers, hence it should be considered separately.