# 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
2.3 Calculation of the Fault Influence Degree of Edges in the Same Level Based on the Copula Function  
In this paper, it is assumed that a whole machine has \(n\) fault-related system components and its reliability function is \(R(t_{i}),\;i=1,2,\cdots,n\) and the joint reliability function of these system components is \(R(t_{1},t_{2},\cdots,t_{n})\). Based on the Copula theory and Sklar theorem [42; 43; 44], a Copula function can be uniquely determined. It is expressed as follows:  
\[C(R_{1}(x),R_{2}(x),\cdots,R_{n}(x))=\exp\left\{-\left[\sum_{i=1}^{n}(-lnR_{i} (t))^{1/\theta}\right]^{\theta}\right\}, \tag{10}\]  
where \(\theta\) is the parameter of the Copula function to characterize the degree of association between the system components (\(\theta\in(0,1]\)). Given that the calculation process of the Copula function parameters is relatively complex, this paper introduces a relatively intelligent artificial fish swarm algorithm [45] and uses MATLAB software to solve the parameters of the Copula function. The following relationship exists between the fault probability function and reliability function:  
\[F(t)=1-\exp\left[-\left(\frac{t}{\beta}\right)^{\gamma}\right]=1-R(t). \tag{11}\]  
As expressed in Equation (10), the degree of correlation between components obtained from the perspective of the reliability function is represented by \(\theta\) and as expressed in Equation (11), \(F(t)=1-R(t)\). Therefore, the fault influence degree of the edge at the same level in a machining centre is \(I\big{(}v_{i},v_{j}\big{)}=1-\theta\) and the greater the value is, the greater the fault influence degree of the edge will be. 2.4 Calculation of the Fault Influence Degree of Edges at Different Levels Based on the Coupling Degree Function  
In Reference [46], the effect function is used to describe the impact of a certain subsystem change on the whole machine, while the coupling degree model is used to characterize the coupling degree between subsystems. Thus, the importance of the node is taken as the effect function value and a model of the fault coupling degree is established to reflect the fault influence value between the components. The calculation equation is expressed in Equation (12):  
\[I\big{(}v_{i},v_{j}\big{)}=2\Bigg{[}\frac{I_{C}(v_{i})I_{C}(v_{j})}{\big{(}I_{C} (v_{i})+I_{C}(v_{j})\big{)}^{2}}\Bigg{]}^{1/2}, \tag{12}\]  
where \(I\big{(}v_{i},v_{j}\big{)}\in(0,1)\) and is the fault influence degree between \(v_{i}\) and \(v_{j}\). This value is used to characterize the effect of this edge on fault propagation. \(I_{C}(v_{i})\) and \(I_{C}(v_{j})\) are the critical importance values of \(v_{i}\) and \(v_{j}\), respectively. 2.2.5 Construction of the Fault Propagation Probability Model for Machining Centre System Components  
Based on the hierarchy model of a machining centre and the fault propagation mechanism, the state of a system component node depends on its own fault probability, which the influence of upstream component node is dependant on. The fault influence between the nodes reflects the possibility of a fault of the component node to cause another component fault to be associated with it. Therefore, by integrating the fault probability (\(F_{i}(t)\)) of the node and the fault influence degree (\(I\big{(}v_{i},v_{j}\big{)}\)) between the system component nodes, the value of the fault propagation probability between system components can be obtained. The equation is as follows:  
\[P\big{(}v_{i},v_{j}\big{)}(t)=F_{i}(t)I\big{(}v_{i},v_{j}\big{)} \tag{13}\]  
Evaluation of Fault Propagation Intensity of a Machining Centre Based on the Fault Propagation Model  
Given that the influence of the structural characteristics of the hierarchy model of fault propagation cannot be disregarded, the edge betweenness that represents the structural characteristics of the model is considered [47]. The larger the value is, the stronger its influence on the whole model will be.