# Dynamic Analysis Method for Fault Propagation Behaviour of Machining Centres
## 1 Introduction
Due to the neglecting of the correlation of fault mode, a deviation in the calculation of fault rate occurs and the structural characteristics of fault trees are not considered, thus resulting in a wrong diagnosis. Given the shortcomings of the above diagnosis methods, some scholars use a Petri net, cellular automaton, and complex networks with topological characteristics such as regular networks, scale-free networks, small-world networks, and random networks to establish fault propagation models to study fault propagation [33; 34; 35; 36]. However, given the dependence of a model on the structure and the correlation of faults, a deviation occurs in the fault mechanism analysis and fault source location of a machining centre. In addition, the fault influence degree of system components obtained by traditional methods is mostly constant; in contrast, the fault influence degree of each system component will change over time. Thus, achieving the ideal effect in the fault diagnosis strategy is difficult. Therefore, this paper proposes a fault propagation intensity evaluation method that is used to dynamically describe the fault propagation process of machining centre system components. Compared with the existing methods, the DSM-based fault propagation hierarchy structure model of machining centres established in this paper can more clearly demonstrate the relationship between system components. The fault influence degree of system components obtained in this paper are also time-varying, thereby more in line with engineering practice. Moreover, for the calculation of the influence degree of the edge between components, most of the previous studies used the same method to calculate the fault influence degree of each level and did not consider the difference of fault influence between different levels that will inevitably lead to the deviation in the fault propagation analysis results. This paper considers the difference of fault influence degree between different levels and uses the Copula function and coupling function to calculate the fault influence degree of the edges between the same level and different levels, respectively. The structural characteristics of the model and fault mechanism of the system is considered synthetically, thus the description is more reasonable. We use the value of the fault propagation intensity as an index to study the fault propagation behaviour of a machining centre, to identify the critical fault propagation path of a machining centre, and to provide a theoretical basis and practical reference for later fault detection and maintenance. The remainder of this paper is organized as follows. Section 2 describes the method for evaluating the fault propagation intensity of machining centres on the basis of a fault propagation model. Section 3 introduces the dynamic analysis method of fault propagation behaviour of machining centres on the basis of the fault propagation intensity. Section 4 provides a case application of a machining centre to demonstrate the effectiveness of the proposed method. Section 5 engages in a discussion. Lastly, Section 6 presents our conclusions. Method for Evaluating Fault Propagation Intensity of Machining Centres on the Basis of the Fault Propagation Model  
Based on the basic working process and principle of machining centres, a machining centre is divided into modules and the relevant faults are identified in combination with the field fault data. Considering the fault correlation of components, a hierarchy structure model of machining centres' fault propagation on the basis of DSM is established. The Johnson method is applied to correct the component fault order and construct a time-dependent component fault probability model. On this basis, the importance of component nodes is calculated to reflect the fault influence degree of component nodes.