Reliability Evaluation of CNC Machine Tools Considering Competing Failures of Fault Failure Data and Machining Accuracy Degradation Data

Traditional reliability evaluation of CNC machine tools usually considers a single failure mode of fault failure or degradation failure, or considers fault failure and degradation failure to be independent of each other. However, in the actual working conditions, fault failure and degradation failure are mutually affected, and the reliability evaluation of the competing failure models of CNC machine tools by considering the two failure modes comprehensively can get more accurate evaluation results. Therefore, this paper proposes a reliability evaluation method for CNC machine tools considering fault failure data competing with machining accuracy degradation data. A fault failure model of CNC machine tools is established based on a non-homogeneous Poisson process. The fault failure model is updated according to the different effects of each maintenance result of the failure on machining accuracy. By integrating multiple geometric errors of CNC machine tools through multi-body system theory, the amount of machining accuracy degradation is extracted. A machining accuracy degradation failure model is established using the Wiener process. Considering the correlation between fault failure and degradation failure, a competing failure model based on the Coupla function is developed for evaluating the reliability of CNC machine tools. Finally, the effectiveness of the proposed method is verified by example analysis.

The failure process when only degradation is considered in the operation of the machine tool is shown in Figure 1, and L denotes the accuracy failure threshold.   NHPP is defined as follows [24], if the counting process {X(t),t ≧ 0} is a homogeneous Poisson process with intensity function λ(t), then the following conditions are satisfied.
Reliability is an essential indicator of the reliability of 3.
Where λ(t) represents the intensity function, and F(t) represents the total number of faults. According to the different effects of fault repair results on machine tool accuracy, three forms of F + (t), F 0 (t) and F -(t) is used to represent the positive, no, and negative effects of fault repair results on machine tool accuracy. The positive impact is described as the starting point of machine accuracy error after the fault repair is significantly lower than before the fault repair; no impact is described as the starting point of machine accuracy error after the fault repair is equal to before the fault repair; the negative impact is described as the starting point of machine accuracy error after the fault repair is significantly higher than before the fault repair; formula (8) indicates the relationship between the three forms.
During a complete working time of a CNC machine, the probabilities of three different forms obey the following relations.
With i + , i 0 , i -, respectively, the number of three forms of failure in a fixed period, the respective probability of failure is expressed by the formula, as shown in Equation  and each part of the machine tool is treated as the corresponding "typical body", and the two branches are numbered according to the natural growth order, and the coordinate system of each unit is denoted by R [25], and the topology of the machine tool is shown in Figure 5.
Where L i (j) denotes the i-order low order body of body In the actual machining process, the actual position of the tool forming point can deviate from the preset ideal position due to error factors, resulting in spatial position errors [27]. The combined volumetric error E of the machine tool is expressed by Equation (14). The

Accuracy degradation modeling based on Wiener process
The accuracy degradation of CNC machine tools is due to the joint action of random homogeneously distributed small degradation amounts of various components inside the machine tools, and these small degradation amounts are proportional to time and can be considered to obey a normal distribution, so the accuracy degradation of CNC machine tools is generally described by the Wiener process, which has good computational and analytical properties [28].

18) When a one-dimensional Wiener process {X(t), t ≥ 0}
is used to describe the machine tool machining accuracy degradation process, the accuracy failure time T of the machine tool to be tested is defined as the length of time when the amount of performance degradation first reaches the failure threshold L, the lifetime T = inf(t > 0| X(t) ≥ L), and the random variable T obeys the inverse Gaussian distribution [29]. The distribution function FT(t) and the probability density function fT(t) of the accuracy lifetime of the machine tool to be measured can be derived as follows, respectively.
Where Φ(.) denotes the standard normal function.
The reliability function of the CNC machine based on the accuracy degradation process can be obtained by equation  (F(x),G(y )). Conversely if C is a Coupla function and F and G are two arbitrary probability distribution functions, then the H function must be a joint distribution function, and the marginal distributions are F and G [30].
The cumulative failure probabilities FF(t) and Fd(t) based on equation (11) and equation (21) are obtained for fault failure and degradation failure, respectively, as shown in equation (22). is the best choice [31]. In this paper, the Frank Coupla function is chosen to describe the correlation between the two, which is expressed by Equation (24).
Based on the above equation, the reliability function under the competitive failure of the CNC machine tool is obtained as shown in Equation (25). Δxl is the amount of performance degradation of the machine accuracy to be tested between t(l-1) and tl, Δtl is the interval between moments, and n is the number of accuracy testing time points. Let the unknown parameters be ξ= (λ1,λ2,β1,β2,µ,σ,θ). Then the corresponding likelihood function is obtained as shown in Equation (26).
The corresponding logarithmic function is obtained according to the likelihood function as shown in Equation Based on Equation (27) Table 1 and Table 2.
The fault data is truncated using a fixed number, and the last time point of each machine tool is faulted, and the degradation data is collected using a laser interferometer to collect the straightness and positioning accuracy of XYZ three axes within different time points, and the straightness and positioning accuracy are used as the geometric error source in the multi-body system, and the machining accuracy degradation amount ΔEi is calculated for each time point through the model in section 4.1, focusing on recording the time before and after the fault data The accuracy data within the moment, and the schematic diagram of the accuracy acquisition system is shown in Figure 5.
Based on the changes in the amount of machining accuracy degradation before and after analyzing the faulty data points as described in Section 3.2, the faulty data points for each machine were classified into three forms, and the parameters of the reliability function were estimated based on the faulty and degraded data. Table 3 shows the results of the model parameter estimation and indicates the goodness of fit by TMSE.

Conclusion
In this paper, the fault failure of CNC machine tools is

Conflicts of interest
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Code availability
Not applicable [20] Tao Y, Zhao Jun, S Feng (2020) A reliability as