# Uncertainty Representation of Natural Frequency for
## 6 Numerical Examples
### Three-Layer Laminated Composite Cylindrical Shells  
The proposed method is applied to the accurate representation of the natural frequency of three-layer laminated composite cylindrical shells with each ply angle \([-\theta,0,\theta]\), where the range of \(\theta\) is from 0 to \(\pi/2\). Each layer of laminated composite cylindrical shells is uniformly distributed, that is the thickness of each layer is \(h/L=0.033\) m. When the cubic normal PDF of uncertain input variables is determined, the optimal Latin hypercube is needed for uniform sampling. To ensure the computational efficiency and the effectiveness of the proposed method, the six sampling points are first selected for each uncertain input variable to construct the initial Kriging model. Then, according to the method  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**NO.** & \(E_{1}(\text{GPa})\) & \(E_{2}(\text{GPa})\) & \(\mu_{1}\) & \(G(\text{GPa})\) \\ \hline \(1\) & 130.63 & 9.08 & 0.33 & 5.16 \\ \(2\) & 132.19 & 9.07 & 0.3 & 4.85 \\ \(3\) & 132 & 9.73 & 0.35 & 5 \\ \(4\) & 130.39 & 9.21 & 0.34 & 5.34 \\ \(5\) & 128.28 & 8.67 & 0.33 & 4.98 \\ \(6\) & 135.3 & 9.18 & 0.32 & 5.13 \\ \(7\) & 137.33 & 9.28 & 0.33 & 5.25 \\ \(8\) & 126.91 & 9.39 & 0.33 & 5.45 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Uncertain available data for laminated composite cylindrical shells.  
Figure 2: Cumulative probability density functions of interval variables (**a**) \(E_{1}\), (**b**) \(E_{2}\).  
described in the Section 5, the accurate Kriging model is constructed to accurately represent the frequency uncertainty for laminated composite cylindrical shells. Finally, the MCS method is adopted to replace the real calculation model and is compared with the initial Kriging and accurate Kriging methods. The mean and standard deviation of DNF for laminated composite cylindrical shells under different boundary conditions are shown in Figures 3 and 4.  
By calculating the error, the maximum relative mean error and standard deviation error of DNF for the laminated composite cylindrical shells with clamped-clamped boundary conditions are reduced to 0.05% and 6.85%, respectively. In a similar way, the maximum relative mean error and standard deviation error of DNF for the laminated composite cylindrical shells with supported-supported boundary conditions are reduced to 0.05% and 4.07%. Therefore, the DNF curves obtained by the accurate Kriging method are highly fitted with the curve obtained by the MCS method, which fully proves the effectiveness of the proposed method. At the moment, the calculation number of the DNF for the laminated composite cylindrical shells reduces from 10,000 to 1000, which exhibits the proposed method has high calculation efficiency. In general, the accurate Kriging model can be applied to the accurate representation of the uncertain natural frequency of laminated composite cylindrical shells with the other boundary conditions.  
Figure 4: The mean and standard deviation of DNF for the three-layer laminated composite cylindrical shells with supported–supported boundary conditions. (**a**) mean, (**b**) standard deviation.  
Figure 3: The mean and standard deviation of DNF for the three-layer laminated composite cylindrical shells with clamped–clamped boundary conditions. (**a**) mean, (**b**) standard deviation.