## 5 Example Analysis and Discussion
### Reliability Analysis of a Vehicle's Suspension Mechanics  
To exhibit the advantage of the US approach, a passive vehicle suspension problem is investigated in this subsection [43]. The simplified model is shown in Figure 4. The considered LSF is the road-holding ability of the vehicle, which can be expressed as follows:  
\[g(\xi)=\left(\frac{\pi AVm}{b_{0}kG^{2}}\right)\left(\left(\frac{c_{k}}{M+m}- \frac{c}{M}\right)^{2}+\frac{c^{2}}{Mm}+\frac{c_{k}k^{2}}{mM^{2}}\right)-1 \tag{17}\]  
where \(A=1.0\,\mathrm{cm}^{2}/\) cycle m, \(b_{0}=0.27\), \(V=10.0\,\mathrm{m}/\mathrm{s}\), \(M=3.2633\,\mathrm{kg}\cdot\mathrm{s}^{2}/\mathrm{cm}\), \(G=981\,\mathrm{cm}/\mathrm{s}^{2}\), and \(m=0.8158\,\mathrm{kg}\cdot\mathrm{s}^{2}/\mathrm{cm}\) are deterministic parameters. The considered input factors are the spring stiffness coefficient \(c\) (\(\mathrm{kg}/\mathrm{cm}\)), tire stiffness \(c_{k}\) (\(\mathrm{kg}/\mathrm{cm}\)), and shock absorber damping coefficient \(k\) (\(\mathrm{kg}/\mathrm{cm}\) s). In this paper, all input factors are regarded as uncertain variables \(\xi=(c,c_{k},k)^{\mathrm{T}}\). Distribution types and parameters of input factors are represented in Table 2.  
The LSF \(g(\xi)\) of this real engineering problem is not a strictly monotonic function for \(c\). Therefore, the URI cannot be calculated based on the equivalent analytical model. Hence, the US method is used to estimate the reliability of the vehicle suspension. The classical MCS method with a sample size of \(10^{7}\) is provided as a reference. Within the MCS method, all input factors are considered as random variables with normal probability distribution shown in Table 2. The corresponding reliability estimation results of the aforementioned two approaches are shown in Figure 5 under the assumption that the mean value \(\mu_{k}\) fluctuates within \(\mu_{k}\in[0,30]\).  
Figure 3: Reliability comparison for the latch lock mechanism.  
According to Figure 5, both URI and PRI increase with \(\mu_{k}\). It clearly shows that the structural reliability estimated by the US method is higher than the one calculated by the traditional MCS method when \(\mu_{k}\in[0,\,17.4819]\). In addition, the structural reliability calculated by the proposed method is lower than the one estimated by the MCS method when \(\mu_{k}\in[17.4819,\,30]\). Therefore, the proposed methods can help engineering designers estimate structural reliability when input factor information is insufficient. This applied  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Input Factors** & **Proposed Method** & **MCS Method** \\ \hline \(c\) (kg/cm) & \(\mathcal{N}(400.98,10)\) & \(N(400.98,10^{2})\) * \\ \(c_{k}(\text{kg/cm})\) & \(\mathcal{N}(1451.7,10)\) & \(N(1451.7,10^{2})\) \\ \(k\) (kg/cm s) & \(\mathcal{N}(\mu_{k},10)\) & \(N(\mu_{k},10^{2})\) \\ \hline \hline \end{tabular}  
* \(10^{2}\) is the variance of normal probability distribution.  
\end{table}
Table 2: Distribution parameters of input factors.  
Figure 4: Schematic view of a passive vehicle suspension model.  
Figure 5: Reliability comparison for the suspension mechanical model.  
engineering problem shows that the US method can be more widely employed than the equivalent analytical model.