## 2 Reliability Modeling of NLG EmergencyExtension
### Reliability Method and Mixture of Models  
The reliability models of the emergency extension mechanism include the mechanism starting model, mechanism continuous movement model, mechanism movement precision model, and static strength model [23]. These reliability models can be established based on the fault tree analysis of the emergency extension mechanism and the mechanism reliability analysis theory [24]. For reliability analysis, the four reliability models actually have similar procedures. In this study, the starting reliability model is considered as a typical example to explain the reliability method. To start the mechanism from stationary to motion state, it is ensured that the driving torque \(M_{d}\) is larger than the resistance torque \(M_{r}\), i.e.,  
\[M=M_{d}-M_{r}>0 \tag{1}\]  
The starting failure probability \(P_{f}\) is defined as the probability that driving torque is less than resistance torque, i.e.,  
\[P_{f}=\{M_{d}-M_{r}<0\} \tag{2}\]  
With regard to the distribution characteristics of driving torque and resistance torque, the starting reliability index of the mechanism is expressed as  
\[\beta=\frac{\overline{M}_{d}-\overline{M}_{r}}{\sqrt{C_{d}^{2}\overline{M}_{ d}{}^{2}+C_{r}^{2}\overline{M}_{r}{}^{2}}} \tag{3}\]  
where \(C_{d}\) is the variable coefficient of driving torque; \(C_{r}\) is the variable coefficient of resistance torque; and \(\overline{M}_{d}\) and \(\overline{M}_{r}\) denote the mean values of driving torque \(M_{d}\) and resistance torque \(M_{r}\).  
In terms of the standard normal cumulative distribution function \(\Phi(\bullet)\)[25; 26], the starting reliability \(R\) is  
\[R=\Phi(\beta) \tag{4}\]  
The starting failure probability \(P_{f}\) is  
\[P_{f}=1-R=\Phi(-\beta) \tag{5}\]  
Figure 2: Fault tree of emergency extension.  
In this case, the first-order second-moment method is used to calculate the starting reliability. The rest of the reliability models are not repeatedly respected. The reliability model, safety boundary equation, reliability index, and parameter distributions of the bottom event [23; 27; 28; 29] are shown in Table 2.  
In terms of the above fault tree analysis, the relationship between bottom events and top events is \"or\" for emergency extension. All the bottom events are minimal cut sets. The occurrence probability \(P_{T}\) of the top event is equal to the sum of the probability of occurrence for all the minimal cut-sets. In this case, the failure probability of emergency extension is  
\[P_{T}=\sum_{i=1}^{5}P(X_{i}) \tag{6}\]  
where \(X_{i}\) (\(i=1,\,2,\,\ldots,\,5\)) is \(i\)th bottom event in the fault tree; and \(P(.)\) indicates the failure probability.  
First, the failure probability of each bottom event with the reliability model is obtained by the related parameters, such as geometric sizes, material performance parameters, lowering speed, and so forth. With respect to Equation (6), the failure probability of the top event is obtained in the fault tree of emergency extension.  
As for the bottom event \(X_{1}\) (excess passive torque), the reliability models involve starting reliability \(M\) and continuous movement reliability \(M_{\omega}\). In the unlocking stage of emergency extension for the strut rotation angle in [0\({}^{\circ}\), 2.66\({}^{\circ}\)], the starting reliability model is applied to compute the failure probability of the starting driving mechanism, while after unlocking (i.e., the strut rotation angle of landing gear excess 2.66\({}^{\circ}\)), the continuous movement reliability model is employed to evaluate the failure probability of lowering the driving mechanism. In this paper, the maximum failure probability in the whole motion process is regarded as the failure probability of the bottom event \(X_{1}\).