## 4 Numerical Examples  
In this section, the proposed analytical models for reliability assessment will be validated, and then the effect of mechanism clearance and local stiffness variation together on the time-varying reliability of the mechanism will be analyzed. Finally, the effect of different strength dispersion on the time-varying reliability of a mechanism containing a clearance is considered. 1. The parameters of the mechanism system are shown in Table 1. \(\mu(\blacksquare)\) denotes the mean value of a random variable. In this section, the proposed reliability model is verified by the Monte Carlo simulation method. The obtained Monte Carlo simulation results and reliability calculation results are shown in Figure 6.  
The Monte Carlo simulation is based on the system working mechanism for simulation and does not depend on the proposed reliability models. As can be seen in Figure 6, the proposed time-varying reliability model of the mechanism is consistent with the Monte Carlo simulation results. Additionally, the reliability does not decrease linearly or in the form of an exponential function under the traditional assumption of constant failure rate. The rate of reliability decline is slower at the beginning of the operational duration, but as the strength continues to decrease, the effect of the mechanism clearance on the mechanism becomes more obvious, accelerating the rate of reliability decline and thus accelerating the failure of the mechanism system. Figure 5: Monte Carlo simulation flowchart. In practice, material parameters need to be obtained from random samples. For example, using \(x_{i}\) to denote the \(i\)th sample of a material parameter x and the total number of samples is \(N_{\text{s}}\), the standard deviation of the parameter \(\sigma(x)\) can be expressed as follows. \[\sigma(x)=\sqrt{\frac{1}{N_{\text{s}}}\sum_{i=1}^{N_{\text{s}}}(x_{i}-\mu(x))^{ 2}}\]  
where \(\mu(x)\) is the mean value of \(x\). 1. In order to consider the combined effect of the mechanism clearance and the local stiffness of the mechanism on the time-varying reliability of the mechanism, the results \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(l_{1}\) & \(1\) & m \\ \hline \(l_{2}\) & \(1\) & m \\ \hline \(l_{3}\) & \(0.1\) & m \\ \hline \(l_{4}\) & \(1\) & m \\ \hline Mean value of load \(\mu(L)\) & \(1000\) & N \\ \hline Standard deviation of working load \(\sigma(L)\) & \(120\) & N \\ \hline Modulus of elasticity \(E\) in flexible element & \(5\times 10^{9}\) & Pa \\ \hline Density \(\rho\) & \(7800\) & kg/m\({}^{3}\) \\ \hline Angular velocity & \(6.283\) & Rad/s \\ \hline Diameter of bearing & \(0.020\) & m \\ \hline Diameter of journal & \(0.018\) & m \\ \hline Mean value of initial strength \(\mu(\tau_{0})\) & \(1.3\) & MPa \\ \hline Standard deviation of initial strength \(\sigma(\tau_{0})\) & \(0.2\) & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanism system parameters. Figure 6: Reliability from the Monte Carlo simulation and the proposed model. of the time-varying reliability calculations for the mechanism with different values of clearance and local stiffness shown in Table 2 are shown in Figures 7 and 8. From Figures 7 and 8, it can be seen that the mechanism clearance and the local stiffness of the members have very important effects on the time-varying reliability of the mechanism. At the same local stiffness, as the clearance increases, the time-varying reliability of the mechanism system is lower and decreases faster. Due to the increase in clearance, the collision force of the system clearance is larger and the additional stress as well as the damage is also larger. Thus, it may not only cause a larger maximum stress, but also accelerate the system's strength degradation rate, resulting in a rapid decline in the time-varying reliability of the mechanism system. Increased clearance will significantly reduce the service life of the mechanism and affect the safety and reliability of the mechanism system.