# Reliability Modeling and Analysis of Multi-Degradation of Momentum Wheel Based on Copula Function
## 5 Analysis of Examples
### Copula-Based Multi-Degradation Reliability Calculation  
In this paper, the Frank Copula function is used to analyze the correlation between the residual quantity of lubricant and the edge distribution function of the current. The data obtained above are substituted into the formula, and the reliability function based on Copula function can be obtained and shown in Figure 3.  
The Frank Copula joint distribution curve and the two edge distribution curves in the comparison show that for the same research object momentum wheel, the lubricant-based momentum wheel life prediction curve conforms to the life degradation of mechanical components, while the current-based momentum wheel life prediction conforms to the life degradation of electronic components. The curve corresponds to the life degradation of the electronic device. However, there is a certain difference in the momentum wheel life predicted by the two. Therefore, it is necessary to establish a multi-degenerate joint distribution function for the momentum wheel. In this figure, the momentum wheel  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Time/Month** & **1** & **2** & **3** & **4** & **5** \\ \hline
0 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
4 & 0.077 & 0.083 & 0.182 & 0.100 & 0.182 \\
8 & 0.154 & 0.167 & 0.182 & 0.200 & 0.273 \\
12 & 0.385 & 0.250 & 0.364 & 0.400 & 0.546 \\
16 & 0.615 & 0.667 & 0.727 & 0.700 & 0.818 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Degradation data of momentum wheel current.  
Figure 2: Momentum wheel reliability curve based on current degradation.  
reliability curve based on the Frank Copula function is consistent with the lubricant-based momentum wheel life prediction curve before 55 months, and then is similar to the current-based momentum wheel life prediction curve. This situation shows that in the early stage of the operation of the momentum wheel, the life degradation has a lot to do with the lubricant. With the passage of time, the degradation of the momentum wheel life is more related to the degradation of the life of electronic components. Compared with single degradation reliability curve, the Copula-based multi-degradation reliability curve is more consistent with the actual operation of the momentum wheel.  
If ignoring the correlation between two performance degradations of the momentum wheel, \(R_{1}(t)\) and \(R_{2}(t)\) can be considered independent, and the reliability of the momentum wheel can be calculated by Equation (16).  
\[R^{\prime}(t)=R_{1}(t)\cdot R_{2}(t) \tag{16}\]  
The comparison between Couple-based reliability function and independent reliability function is shown in Figure 4. In this figure, the Couple-based reliability curve is consistent with the independent reliability curve before 65 months, but after 65 months, it is obvious that the reliability analysis results without the consideration of correlation between these 2 performance degradations are more conservative. This conclusion of the comparison fully conforms with the general rule of reliability research considering multiple failure correlation, which further confirms the importance of considering the multi-degradation correlation in reliability modeling and analysis of complex electromechanical coupling systems, for example, the momentum wheel.  
Figure 3: Momentum wheel reliability function curve based on frank copula function.