## 3 Mechanism Reliability Modeling  
### Mechanism Reliability Calculation  
The maximum stress and damage expressions within the \(i\)th TP considering the clearance, local stiffness of the member and random loading effects are obtained above. For ease of calculation, the PDF of the maximum value of stress generated by the load \(L(N_{i})\) with the PDF of \(f(L(N_{i}))\) within the \(i\)th TP is expressed as \(\sigma(L(N_{i}),\varphi(N_{i}),E(N_{i}))\). Considering the structural damage due to random stresses, the initial strength is denoted by \(r_{0}\) and then the equivalent damage within the \(i\)th TP can be expressed as  
\[\int_{-\infty}^{\infty}f(L(N_{i}))d(L(N_{i}),\alpha,S_{m}(N_{i}),\varphi(N_{i }),E(N_{i}))dL(N_{i}) \tag{17}\]  
Figure 4: Stochastic stress analysis based on neural network models.  
The result from Equation (17) considers the combined effect of the distribution characteristics of the load \(L(N_{i})\) within the \(N_{i}\)th TP on the damage. The residual strength within the \(n1\)th TP can be expressed as  
\[r(n1)=r_{0}\bigg{(}1-\sum_{i=0}^{n1-1}\int_{-\infty}^{\infty}f(L(N_{i}))d(L(N_{ i}),\alpha,S_{m}(N_{i}),\varphi(N_{i}),E(N_{i}))dL(N_{i})\bigg{)} \tag{18}\]  
Equation (18) describes the effects of cumulative damage due to stress application. The difference between the residual strength and the initial strength is the value of the decrease in strength due to cumulative damage. Considering the maximum stress distribution within different TPs, the reliability within \(N\) TPs is calculated by  
\[\prod_{i=1}^{N}\int_{-\infty}^{r(i)}f_{max}(\sigma(L(N_{i}),\varphi(N_{i}),E(N _{i})))\mathrm{d}\sigma(L(N_{i}),\varphi(N_{i}),E(N_{i})) \tag{19}\]  
Considering the randomness and the distribution of the initial strength \(r_{0}\) with the PDF of \(f(r_{0})\), the reliability of the mechanism running \(N\) TPs is  
\[R(N)=\int_{-\infty}^{\infty}f(r_{0})\prod_{i=1}^{N}\int_{-\infty}^{r(i)}f_{max }(\sigma(L(N_{i}),\varphi(N_{i}),E(N_{i})))d\sigma(L(N_{i}),\varphi(N_{i}),E( N_{i}))df(r_{0}) \tag{20}\]  
Failure rate is the probability that a component or system that has worked to a moment where it has not yet failed will fail per unit of time after that moment. Hence, the failure rate of the mechanism can be calculated from the time-varying reliability as follows:  
\[\begin{split}\lambda(N)=\Big{\{}&\int_{-\infty}^{ \infty}f(r_{0})\prod_{i=1}^{N}f_{-\infty}^{r(i)}f_{max}(\sigma(L(N_{i}), \varphi(N_{i}),E(N_{i})))\,d\sigma(L(N_{i}),\varphi(N_{i}),E(N_{i}))df(r_{0}) -\\ &\int_{-\infty}^{\infty}f(r_{0})\prod_{i=1}^{N+1}\int_{-\infty}^{ r(i)}f_{max}(\sigma(L(N_{i}),\varphi(N_{i}),E(N_{i})))d\sigma(L(N_{i}), \varphi(N_{i}),E(N_{i}))df(r_{0})\Big{\}}\\ /&\int_{-\infty}^{\infty}f(r_{0})\prod_{i=1}^{N}\int _{-\infty}^{r(i)}f_{max}(\sigma(L(N_{i}),\varphi(N_{i}),E(N_{i})))\,d\sigma( L(N_{i}),\varphi(N_{i}),E(N_{i}))df(r_{0})\end{split} \tag{21}\]  
The proposed time-varying reliability models of the mechanism are able to take into account the effects of changes in clearance wear, local structural stiffness degradation, random loading and strength degradation on the time-varying reliability of the mechanism under the fatigue failure mode. The models not only take into account the nonlinear characteristics of the system caused by changes in clearance and local stiffness, but also consider analytically characterizing the effects of these factors on the reliability of the mechanisms over time. The proposed reliability models can be used to analyze the effects of system failure factors on the reliability and remaining life of the mechanism and can also be used for dynamic fault diagnosis of the mechanism systems.