## 7 Numerical Results
### Free vibrations of Laminated Composite Shells
This aspect has been noticed by the authors in their previous work [69], where the same considerations were made for the corresponding theories based on the ESL approach embedded with the Murakami's function. Thus, the shear correction factor should be neglected in those structural theories with the Murakami's function, if a sandwich structure is analyzed. Finally, the related first nine mode shapes can be found in Figure 6.  
Figure 5: First nine mode shapes for a CCFC laminated cylindrical surface with \((90/30/45/60/0)\) as lamination scheme. The next structure is a CFCF (clamped-free-clamped-free) doubly-curved panel of translation, obtained by sliding an elliptic profile (defined by the semi diameters \(\overline{a},\overline{b}\)) over a parabola (described \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(f\)**[Hz]** & \(\mathbf{ELD2^{1\times=1.2}_{RS}}\) & \(\mathbf{ELD2^{1\times=1}_{RS}}\) & \(\mathbf{ELD2^{2\times=1.2}}\) & \(\mathbf{ELD2^{2\times=1}}\) & \(\mathbf{ELD23}\) & \(\mathbf{ELD24}\) & \(\mathbf{3D-FEM}\) \\ \hline
1 & 357.779 & 365.750 & 361.564 & 369.297 & 369.263 & 368.569 & 368.755 \\
2 & 360.035 & 368.663 & 365.443 & 374.047 & 374.010 & 373.200 & 373.480 \\
3 & 375.519 & 385.066 & 379.308 & 389.515 & 389.469 & 388.520 & 388.730 \\
4 & 383.773 & 393.567 & 387.618 & 397.411 & 397.367 & 396.434 & 396.820 \\
5 & 417.052 & 430.082 & 421.774 & 434.854 & 434.791 & 433.342 & 433.661 \\
6 & 418.612 & 431.731 & 423.729 & 437.000 & 436.935 & 435.462 & 435.912 \\
7 & 461.401 & 474.554 & 470.283 & 483.093 & 483.035 & 481.662 & 481.188 \\
8 & 470.591 & 485.171 & 478.858 & 493.098 & 493.034 & 491.534 & 490.978 \\
9 & 475.427 & 492.995 & 481.536 & 499.146 & 499.050 & 496.646 & 497.065 \\
10 & 480.125 & 496.486 & 484.349 & 501.024 & 500.928 & 498.549 & 498.905 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: First 10 natural frequencies ([Hz]) for a CCCC sandwich panel of revolution with \((45/\text{Core}/-45)\) as lamination scheme for several higher-order ELW approaches embedded with the Murakami’s function. The Cheb-Gau-Lob grid distribution is used for the numerical solution assuming \(I_{N}=I_{M}=31\). Figure 6: First nine mode shapes for a CCCC sandwich panel of revolution with \((45/\text{Core}/-45)\) as lamination scheme. through the parameters \(a,b,d\)). It should be noticed that this surface is obtained by moving the elliptic curve along the parabolic curve, keeping the plane containing the ellipse always orthogonal to the parabola (Figure 4e). The position vector \(\mathbf{r}(\alpha_{1},\alpha_{2})\), which defines the doubly-curved translational surface in hand, can be written as a function of the curvilinear coordinates \(\alpha_{1},\alpha_{2}\)  
\[\mathbf{r}(\alpha_{1},\alpha_{2})=\left(R_{0}^{\alpha_{1}}(\alpha_{1})-x_{3} ^{\alpha_{2}}(\alpha_{2})\sin\alpha_{1}\right)\mathbf{e}_{1}-R_{0}^{\alpha_{2 }}(\alpha_{2})\mathbf{e}_{2}+\left(x_{3}^{\alpha_{1}}(\alpha_{1})+x_{3}^{ \alpha_{2}}(\alpha_{2})\cos\alpha_{1}\right)\mathbf{e}_{3} \tag{57}\]  
where the quantities expressed as a function of the coordinate \(\alpha_{1}\) are related to the parabola (curve which the other curve is sliding on), whereas the geometric parameters that depend on \(\alpha_{2}\) are linked to the ellipse (curve that is moving on the other one). As far as the parabolic branch is concerned, one gets  
\[R_{0}^{\alpha_{1}}(\alpha_{1})=\frac{k^{\alpha_{1}}\tan\alpha_{1}}{2} \tag{58}\]  
\[x_{3}^{\alpha_{1}}(\alpha_{1})=\frac{k^{\alpha_{1}}\tan^{2}\alpha_{1}}{4} \tag{59}\]  
where \(k_{\alpha_{1}}=\left(a^{2}-d^{2}\right)/b\) is the characteristic parameter of the curve at issue, assuming the following values \(a=2\,\mathrm{m}\), \(b=1\,\mathrm{m}\), \(c=-2\,\mathrm{m}\), \(d=0\,\mathrm{m}\). Further details concerning the geometric features of the parabolic arch, as well as the meaning of each parameter, can be found in the work [68].