# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
## 5 Numerical Case: An MEMS Application  
The numerical case is a micro-engine consisting of orthogonal linear comb drive actuators mechanically connected to a rotating gear whose data comes from the related literature [4]. MEMS is widely used in industry, specifically in vibration monitoring [25] and micropumps [26]. Due to the working condition and environment, some industry systems with MEMS are always under external shocks and wear processes. Thus, the micro-engine experiences two competing failure processes: soft failure and hard failure. Soft failure is the wear between the gear and the pin joint. The wear volume is caused by the aging degradation process and instantaneous increments due to shock loads [4]. Additionally, different shock loads will lead to different increments. When the total wear volume exceeds the threshold \(H\), the system will fail because of soft failure. Furthermore, the shock loads may cause the fracture of springs, which is regarded as a hard failure. The models presented in this paper are used to study the reliability of the micro-engine. This widely used example is acknowledged to demonstrate the validity of methods or models. The necessary parameters for calculation are listed in Table 1. Wear degradation (soft failure) threshold H, maximum fracture strength \(D_{0}\), and degradation path parameters \(\varphi\) and \(\beta\) are obtained from Sandia's experimental results and other references. \(W_{i}\) and \(\lambda\) are normally distributed random variables according to references [27]. The degradation instantaneously increases variables \(B_{i}\) and \(C_{i}\), which are supposed to be normal distributions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Sources** \\ \hline H & 0.00125 & [PERSON] [27] \\ \hline \(D_{0}\) & 1.5 & [PERSON] [27] \\ \hline \(D_{1}\) & 1.2 & Assumption \\ \hline \(D_{2}\) & 1.0 & Assumption \\ \hline \(\varphi\) & 0 & [PERSON] [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter Values. The ratios calculated by Equation (2) are 0.1586, 0.3413, and 0.4331. General reliability and competing failure probabilities are calculated by Equations (10), (12) and (14), and modified values of system reliability and competing failure probabilities are calculated by Equations (15)-(17), which are illustrated in Figure 3. Figure 3A shows that the modified system reliability value is greater than the general reliability. The reason is that modified reliability eliminates the situation in which both failures occur together, which is impossible. Figure 3B,C show the competing failure and modified competing failure for soft and hard failure. Modified competing failure in Figure 3C indicates that the system certainly fails because of degradation when working circulation is large enough, which is more logical than general competing failure. Thus, the probabilities following are all modified values. To explore the sensitivity of parameters on reliability and competing failure probability, the results by different parameter values are illustrated. In Figure 4, the system reliability and probability of hard failure while there is no soft failure increases with \(H\) at a fixed time. It is easy to understand that the larger the hard threshold \(H\), the less possible for hard failure and system failure. Meanwhile, Figure 4C demonstrates that the competing failure probability 2 decreases with \(H\) because of the same reason. After \(t\) reaches about 1.5 \(\times\) 10\({}^{5}\), the system reliability decreases to 0, as Figure 4A shows. The wastage of the system is so serious after sufficient working circulation that the system will fail because of soft failure definitely. Figure 4B,C demonstrates the inference, too.