# 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
Figure 8: Reliability analysis results of the motor hanger under multiple failure modes. (**a**) CDF curve of the motor hanger; (**b**) Iterative curve of CDF. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(N_{\text{calls}}\) & \(P_{f}\)/\(10^{-3}\) & \(\beta\) & \(\delta_{\beta}\)/\% \\ \hline MC simulation & \(10^{6}\) & 3.2 & 2.7266 & — \\ Proposed method & 18 & 3.1 & 2.7370 & 0.3814 \\ MRGP + U & 22 & 3.1 & 2.7370 & 0.3814 \\ MRGP + H & 30 & 2.9 & 2.7589 & 1.1846 \\ MRGP + EFF & 19 & 3.0 & 2.7478 & 0.7775 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of example 4.4 with different methods. Reliability Analysis of A Bionic Robot Lower Extremity Exoskeleton (LEEX) under Multiple Failure Models  
LEEX is a piece of typical complex equipment, which is mainly composed of a power source, sensing system, control system, driving system, mechanical structure, etc. Due to the uncertainty of the environment, especially the road condition and random shocks, the LEEX will be continuously affected during operation, which will inevitably lead to failure. As the main executive part of the mechanical structure, the harmonic reducer plays a key role in driving LEEX. The harmonic reducer consists of the following four basic components: a wave generator, flexible gear, flexible bearing, and rigid gear. Its main failure modes are tooth surface contact fatigue failure, tooth root bending fatigue failure of flexible gear or rigid gear, and the transmission efficiency failure of harmonic reducer. Therefore, the reliability analysis of LEEX is performed in consideration of the three failure modes of the harmonic reducer. Figure 9 shows a three-dimensional model sketch of the LEEX and the composition of the harmonic reducer. For the tooth surface contact fatigue failure mode, the limit state function of LEEX can be expressed as follows:  
\[G_{1}(\mathbf{x})=\sigma_{Hlim}Z_{N}Z_{L}Z_{V}Z_{R}Z_{W}Z_{X}-\sigma_{H}\sqrt{KK_{A }K_{V}K_{H}} \tag{33}\]  
where \(\sigma_{H}\) is contact stress on the tooth surface, \(\sigma_{Hlim}\) is tooth surface contact fatigue limit, \(Z_{N}\) is life factor, \(Z_{R}\) is roughness coefficient of the tooth surface, \(Z_{V}\) is velocity coefficient, \(Z_{W}\) is working hardening factor, \(Z_{L}\) is lubricant coefficient, \(Z_{X}\) is size factor, \(K\) is calculation coefficient; \(K_{A}\) is usage factor, \(K_{V}\) is dynamic load factor, and \(K_{H}\) is load distribution factor along with the direction of tooth width. For the tooth root bending fatigue failure mode, the limit state function of LEEX can be expressed as follows:  
\[G_{2}(\mathbf{x})=\sigma_{Tiinj}-\sigma_{F}=\sigma_{Tiinj}Y_{ST}Y_{ST}Y_{ST}Y_{ST }Y_{X}-\frac{1.91\times 10^{7}k_{F}p}{dn_{n}nbm_{n}}Y_{x_{n}}Y_{y_{n}}Y_{x_{n}}Y_{y _{n}}K_{A}K_{V}K_{F}K_{F}K_{x} \tag{34}\]  
where \(\sigma_{Tiinj}\) is tooth surface bending fatigue strength, \(\sigma_{F}\) is bending fatigue stress, \(\sigma_{Tiinj}\) is tooth root bending fatigue limit, \(Y_{ST}\) is stress correction factor, \(Y_{NT}\) life factor, \(Y_{ST}\) is sensitivity coefficient of relative tooth root filter, \(Y_{RT}\) is coefficient of relative tooth root surface condition, \(m_{n}\) is gear normal modulus, \(Y_{Fa}\) is the tooth form factor, \(Y_{Sa}\) is stress correction factor, \(Y_{z}\) is contact ratio factor, \(Y_{g}\) is helix angle coefficient, \(K_{F\beta}\) is load distribution factor along with the direction of tooth width, and \(K_{Fa}\) load distribution factor between teeth. Figure 9: Model sketch of LEEX robot and harmonic reducer.