# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 4 Case Studies
### Motor Hanger Reliability Analysis under Multiple Failure Models
For the transmission efficiency failure, the limit state function of LEEX can be expressed as follows:  
\[G_{3}(\pi)=\eta_{\rm pm}-\eta_{\rm min}=\frac{M_{2}}{\bigg{\{}\Big{[}\mu d_{1} \frac{7,194\times 10^{-11}E(K_{band}K_{band}K_{band})}{m^{2}(D+d)^{3}}\Big{]}+\mu d_{1} \frac{M_{2}\tan\left(\cos^{-1}\left(\frac{0.67Z_{2}}{0.57Z_{2}}\right)+\tan^{- 1}(f)\right)}{m\left(Z_{2}-2(d_{2}+c^{*})+2(3+0.61Z_{2})-\frac{c}{2}\right)} \bigg{\}}_{i}} \tag{35}\]  
where \(\eta_{\rm pm}\) is true transmission efficiency, \(\eta_{\rm min}\) is minimum transmission efficiency, \(M_{2}\) is output torque, \(\mu\) is dynamic resistance coefficient of bearing, \(E\) is the modulus of elasticity, \(K_{band}\) is correction coefficient of bending section, \(K_{h}\) is structure coefficient, \(d_{m}\) is flexspline diameter, \(m\) is the modulus, \(Z_{r}\) is the number of flexible teeth, \(h_{a}^{*}\) is addendum coefficient, \(c^{*}\) is top clearance coefficient, \(\delta\) is the thickness of flexible gear ring, and \(f\) is friction coefficient of the tooth surface. The failure probability of LEEX under multiple failure models based on Equations (33)-(35) can be given as follows:  
\[P_{f}=\Pr\biggl{(}\max_{\chi=1,2,3}G_{\chi}(x)\leq 0\biggr{)}=\Pr\{\max\left( \begin{array}{c}G_{1}(x)=\sigma_{Hlimj}-\sigma_{H}\leq 0\\ G_{2}(x)=\sigma_{Tlimj}-\sigma_{F}\leq 0\\ G_{3}(x)=\eta_{\rm pm}-\eta_{\rm min}\leq 0\end{array} \right)\} \tag{36}\]  
To verify the reliability of LEEX under uncertainty factors, five random variables with great influence on three failure modes are selected, and detailed information is presented in Table 4.  
As mentioned above, the LEEX reliability analysis under multiple failure modes using the proposed method was performed. The proposed RMQGS method is first used to build the initial MRGP model, and then the adaptive MRGP model is built by the update strategy and termination criterion. The LEEX reliability analysis is finally conducted, combined with the SS method, and compared with the MRGP + H, MRGP + H, and MRGP + EFF methods. Figure 10 illustrates the reliability analysis results of LEEX under multiple failure modes. It can be seen from the results that the reliability of LEEX under the combined effect of tooth surface contact fatigue failure, tooth root bending fatigue failure, and transmission efficiency failure is 0.9976, indicating that the LEEX has high reliability and can resist uncertain changes in the external environment. The solution accuracy of the proposed method is consistent with the MRGP + H and MRGP + EFF methods, but the computational efficiency is improved by 51.53% and 32.76%, respectively, as can be seen in Table 5. Although the number of calls and solution accuracy of the MRGP + U method are similar to those of the proposed method, the overall performance is still lower. The reason for the difference in analysis results is that the proposed method can achieve high-precision reliability analysis results with fewer updated samples, as shown in Figure 10b. The proposed method only needs to add 19 updated samples to reach the convergence condition, while the MRGP + H, MRGP + U, and MRGP + EFF methods require 111, 49, and 66, respectively. By comparing the total number of calls, it is found that the proportion of the updated samples in the four methods to the total number of calls is 24.36%, 68.94%, 49.50%, and 56.90%, respectively. This further proves that the initial MRGP built by the proposed method has high accuracy. Above, the analysis shows that the proposed method can achieve the efficient application of multi-failure engineering problems for complex structures under the condition of taking into account the solution efficiency and accuracy.