## 4 Applications
### Effect of Elastic Foundation  
Following example show the variation of the natural frequencies as a function of the parameters that define elastic foundations. A cylindrical surface with cycloidal profile, as shown in Figure 4b, is assumed, with the boundary conditions CFCF. The agglomeration parameters and the mass fraction  
Figure 7: Frequency variations of a CCCC square plate (Figure 3a) made of one lamina of constant thickness \(h=0.1\) m reinforced by CNTs distributed as in Figure 5a for the different parameters \(p^{(k)}\). The mass fraction was \(w_{r}=0.05\), and the agglomeration parameters \(\mu_{1}=\mu_{2}=1\). The Chebyshev-Gauss-Lobatto grid distribution was employed, with \(I_{N}=I_{M}=21\). The following mode shapes are considered: (**a**) 1 st mode; (**b**) 2 nd mode; (**c**) 3 rd mode; (**d**) 4 th mode; (**e**) 5 th mode; (**f**) 6 th mode; (**g**) 7 th mode; (**h**) 8 th mode. value of the CNTs are given in Tables 6 and 7. The applied layer schemes are shown in Figure 5b. In this example, the following theories were used: FSDT\({}_{RS}^{\kappa=5/6}\), FSDTZ\({}_{RS}^{\kappa=1}\), FSDTZ\({}_{RS}^{\kappa=5/6}\), EDZ\({}^{\kappa=5/6}\), EDZ\(2^{\kappa=1}\), EDZ\(2^{\kappa=5/6}\) and ED3. Tables 6 and 7 show the first seven natural frequencies for each theory for different values of the parameters of elastic foundation \(k_{3f}^{(-)}\), \(G_{f}^{(-)}\), respectively. The overall behavior of the variation of these elastic foundation parameters can be observed from the graphs depicted in Figures 8-11. In particular, Figures 8 and 10 show the variation of the first eight natural frequencies as a function of the foundation parameters \(k_{3f}^{(-)}\) and \(G_{f}^{(-)}\), respectively. In each graph, all the aforementioned theories have been considered. On the other hand, Figures 9 and 11 present seven different graphs (one for each structural theory), in which the variation of the first four natural frequencies are presented in terms of the foundation parameters \(k_{3f}^{(-)}\) and \(G_{f}^{(-)}\), respectively. Interesting behaviors can be observed. With reference to Figure 9, it can be noted that, initially, \(f_{1}\) and \(f_{2}\)--as well as \(f_{3}\) and \(f_{4}\)--are overlapped. Nevertheless, this feature changes when increasing the value of \(k_{3f}^{(-)}\). Then, a bifurcation point appears, and the frequencies \(f_{1}\) and \(f_{4}\) are not overlapped anymore; on the other hand, the curve related to \(f_{2}\) corresponds with that representing \(f_{3}\). Similar observations can be made with reference to Figure 11. In particular, \(f_{1}\) and \(f_{2}\) exhibit an intermediate behavior in which they are overlapped. Then, the bifurcation point occurs, following which \(f_{3}\) and \(f_{4}\) turn out to be overlapped. In general, all these frequencies seem to follow specific paths due to the regularity of the corresponding curves. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Elastic Foundation: \(\rho^{(-)}=1800\,kgm^{3}\), \(\quad h_{l}^{(-)}=0.15\,m_{r}\)**} & \(\mathbf{G_{f}^{(-)}=35\times 10^{7}\,Nm^{3}\), \(\quad k_{M}^{(-)}-variable\)} & \\ \hline \multirow{2}{*}{\(f\)**[Hz]**} & \multicolumn{3}{c}{\(\mathbf{k_{M}^{(-)}}\) [N/m\({}^{3}\)]**} \\ \cline{2-9}  & **0** & \(\mathbf{5\times 10^{7}}\) & \(\mathbf{10\times 10^{7}}\) & \(\mathbf{15\times 10^{7}}\) & \(\mathbf{20\times 10^{7}}\) & \(\mathbf{25\times 10^{7}}\) & \(\mathbf{30\times 10^{7}}\) & \(\mathbf{45\times 10^{7}}\) & \(\mathbf{60\times 10^{7}}\) & \(\mathbf{75\times 10^{7}}\) \\ \hline \multicolumn{9}{c}{FSDT\({}_{85}^{-5.6}\)} \\ \hline
1 & 184.846 & 193.075 & 198.086 & 202.754 & 207.129 & 203.898 & 201.873 & 200.118 & 199.564 & 199.304 \\
2 & 184.846 &