## 5 Examples
### The Influence of the Layer Height Uncertainty  
Four types of boundary conditions were considered, as shown in Figure 7. The first natural frequency \(\omega_{1}\) and the first nondimensional damping ratio \(\gamma_{1}\) were analyzed. The central parameter values are presented in Table 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Dynamic Characteristics**} & \multicolumn{4}{c}{**Mode**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline Natural frequency (present paper) [rad/s] & 404.875 & 1336.507 & 2860.876 & 4990.735 \\ Natural frequency [35] [rad/s] & 404.879 & 1336.560 & 2861.150 & 4991.590 \\ Nondimensional damping ratio (present paper) [-] & 0.1378 & 0.1343 & 0.1184 & 0.1042 \\ Nondimensional damping ratio [35] [-] & 0.1378 & 0.1343 & 0.1184 & 0.1042 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of natural frequencies of a sandwich beam.  
Figure 5: Scheme of the considered five-layered composite beam.  
Figure 6: Scheme of the sandwich beam considered in [35].  
Figure 7: Scheme of the considered five-layered composite beam (**a**) simply supported, (**b**) cantilever, (**c**) fixed-pinned and (**d**) fixed-fixed.  
In all cases, as shown in the graphs for variations up to 10%, the results from all four methods are very close and the lines in the graphs almost overlap. This applies to both the natural frequency and the nondimensional damping ratio for all considered boundary conditions. This confirms that even the simplest method of predicting a solution for uncertain design parameters (a first-order Taylor series expansion) is sufficient to achieve good results.  
Figure 11: Ranges of interval values of (**a**) natural frequency \(\omega_{1}\) and (**b**) nondimensional damping ratio \(\gamma_{1}\) for a fixed-fixed beam.  
Figure 9: Ranges of interval values of (**a**) natural frequency \(\omega_{1}\) and (**b**) nondimensional damping ratio \(\gamma_{1}\) for a cantilever beam.  
When large uncertainties are considered, in the case of natural frequency, the second-order Taylor series expansion with all components of the Hessian matrix provides results closest to the vertex method. However, it is relatively time consuming due to the number of combinations required. In the case of the nondimensional damping ratio, this method usually gives results closest to the vertex method, but the discrepancies are larger and, in some cases, a smaller range of results is obtained (i.e., the lower limit for simply supported or fixed-fixed beams) than using the vertex method. In this case, the solution does not take into account all cases that may occur.  
For this reason, it seems to be important to determine the same critical combination of design parameters as in the case of the vertex method. Therefore, the algorithm presented in Section 4 (Method V) was used. In Tables 5 and 6, the results for a simply supported beam with the uncertainty of design parameters equal to 10% and 30% are presented.  
Based on the results presented in Tables 5 and 6, it can be concluded that Method V yields the same critical combination of design parameters and the same results as the vertex method. In addition, it is more efficient than Method IV, as described in Section 5.4.