## 4 System Reliability Modeling Considering Planetary Transmission Behavior Characteristics
### Fatigue Reliability Evaluation Model of Series System Considering Failure Dependence  
The fracture of any tooth in a planetary system will affect the transmission capacity of the whole system. Therefore, if each tooth in the system is regarded as a potential failure unit, then the system is a typical series system. In service process, the load borne by each tooth has significant dependence with the system input power, and the load dependence and general load randomness make the failure of each element not independent of each other, so the reliability of the series system cannot be simply considered as the reliability product of each unit. The fatigue reliability evaluation model of a series system is established based on conditional probability expectation algorithm and considering the failure dependence among unit parts. First, only under the action of the deterministic load is the failure of each part in the system independent of each other, so the conditional reliability of the series system is equal to the conditional reliability product of each part. Then considering load uncertainty effect at the system level, the probability that \(r\) parts in the system will not fail, that is, the fatigue reliability evaluation model of the series system, is  
\[R_{\mathrm{SYS}}(n)=\int_{0}^{\infty}h_{0}(\sigma)\prod_{i=1}^{r}\left[\int_{ n}^{\infty}f_{i}(N|(s_{i}\sigma+\mu_{i}))\mathrm{d}N\right]\mathrm{d}\sigma \tag{17}\]  
where \(h_{0}(\sigma)\) is the probability density function of standard normal distribution, and \(f_{i}(N|(s_{i}\sigma+\mu_{i}))\) is the life probability density function of the \(i\)th part under stress level \(s_{i}\sigma+\mu_{i}\). The model assumes that the load follows normal distribution, and realizes load normalization for each part through the mathematical relationship transformation between normal distribution and standard normal distribution, so as to consider the load uncertainty effect at system level. Structural Optimization of Reliability Model Considering Sequence Characteristics of Planetary System  
The kinematic equation of planetary transmission is deduced according to a periodic operation law in the planetary gear train, at the same time, the single tooth meshing times of various gears in the system within the same time interval are obtained, which will match sequence characteristic attributes for system fatigue reliability evaluation model. Assuming that the absolute angular velocities of the sun gear, planet gear, ring gear, and planet carrier are \(\omega_{\mathrm{S}}\), \(\omega_{\mathrm{P}}\), \(\omega_{\mathrm{R}}\), and \(\omega_{\mathrm{C}}\) respectively, then the mathematical relationship among them can be expressed as  
\[\left\{ \begin{array}{l}\omega_{\mathrm{S}}=i_{\mathrm{SR}}^{\mathrm{C}}\cdot \omega_{\mathrm{R}}+i_{\mathrm{SC}}^{\mathrm{R}}\cdot\omega_{\mathrm{C}}\\ \omega_{\mathrm{R}}=i_{\mathrm{RS}}^{\mathrm{C}}\cdot\omega_{\mathrm{S}}+i_{ \mathrm{BC}}^{\mathrm{R}}\cdot\omega_{\mathrm{C}}\\ \omega_{\mathrm{C}}=i_{\mathrm{CS}}^{\mathrm{C}}\cdot\omega_{\mathrm{S}}+i_{ \mathrm{CR}}^{\mathrm{R}}\cdot\omega_{\mathrm{R}}\\ \omega_{\mathrm{P}}=i_{\mathrm{PC}}^{\mathrm{R}}\cdot\omega_{\mathrm{C}}+i_{ \mathrm{PR}}^{\mathrm{R}}\cdot\omega_{\mathrm{R}}\\ \end{array} \right. \tag{18}\]where \(i_{\rm ab}^{\rm c}\) represents the ratio of the relative rotational speed of member a and member b, respectively, relative to member c, i.e., \(i_{\rm ab}^{\rm c}=(\omega_{\rm a}-\omega_{\rm c})/(\omega_{\rm b}-\omega_{\rm c})\). The kinematic equation of the planetary gear train can be derived from Equation (18)  
\[\left\{ \begin{array}{c}\omega_{\rm S}+p\cdot\omega_{\rm R}-(1+p)\cdot\omega_{ \rm C}=0\\ \omega_{\rm R}+\omega_{\rm S}/p-(1+p)\cdot\omega_{\rm C}/p=0\\ \omega_{\rm C}-\omega_{\rm S}/(1+p)-p\cdot\omega_{\rm R}/(1+p)=0\\ \omega_{\rm p}-(1+p)\cdot\omega_{\rm C}/(1-p)+2p\cdot\omega_{\rm R}/(1-p)=0 \end{array} \right.