## 3 Numerical Scheme  
After the fundamental system of Equation (36), along with the proper boundary conditions of (38)-(40), is set, a numerical scheme has to be implemented in order to obtain the solution. In this paper, the fundamental system of equations is solved by the Generalized Differential Quadrature (GDQ) method. Although, only fundamental aspects of this techniques are presented in the current paper, the reader can find further information about this method in [137]. Using this approach, the \(n\)-th derivative at a generic point \(x_{i}\) of a sufficiently smooth one-dimensional function \(f(x)\) is obtained as a weighted linear sum of the function values at some chosen grid points  
\[\left.\frac{d^{n}f(x)}{dx^{n}}\right|_{x=x_{i}}\cong\sum_{j=1}^{I_{N}}\xi_{ij}^ {(n)}f(x_{j}) \tag{66}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Carbon Nanotubes** & \(\mathbf{k_{r}}\) [GPa] & \(\mathbf{l_{r}}\) [GPa] & \(\mathbf{m_{r}}\) [GPa] & \(\mathbf{n_{r}}\) [GPa] & \(\mathbf{p_{r}}\) [GPa] & **Refs.** \\ \hline SWCNT (5,5) & 536 & 184 & 132 & 2143 & 791 & [9; 12] \\ SWCNT (6,6) & 9.9 & 8.4 & 4.4 & 457.6 & 27 & [8] \\ SWCNT (10,10) & 271 & 88 & 17 & 1089 & 442 & [9] \\ SWCNT (15,15) & 181 & 58 & 5 & 726 & 301 & [9] \\ SWCNT (20,20) & 136 & 43 & 2 & 545 & 227 & [9; 12] \\ SWCNT (50,50) & 55 & 17 & 0.1 & 218 & 92 & [9; 12] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: [PERSON]’s elastic moduli for several Single-Walled Carbon Nanotubes. Figure 3: CNT reinforced composite layer: (**a**) shell element with inclusion model of CNT agglomeration; (**b**) effective CNT fiber and local co-ordinate system representation. for \(i=1,\,2,\ldots,\,I_{N}\), where quantities \(\varsigma_{ij}^{(n)}\) represent the weighting coefficients of the sum. To obtain the solution, weighting coefficients \(\varsigma_{ij}^{(n)}\) have to be evaluated at each grid point, as highlighted in the review paper by [PERSON] et al. [137]. Additionally, if adequate solutions are to be acquired, proper grid distribution is needed. In this paper, Chebyshev-Gauss-Lobatto grid distribution has been chosen to discretize the domain under consideration. If Expression (66) is extended on a two-dimensional problem, the points of the shell middle surface are placed according to the following Expressions along the principal co-ordinates \(\alpha_{1}\), \(\alpha_{2}\)  
\[\alpha_{1i}=\left(1-\cos\left(\frac{i-1}{I_{N}-1}\pi\right)\right)\frac{\left( \alpha_{1}^{1}-\alpha_{1}^{0}\right)}{2}+\alpha_{1}^{0} \tag{67}\]  
for \(i=1,2,\ldots,I_{N}\) with \(\alpha_{1}\in\left[\alpha_{1}^{0},\alpha_{1}^{1}\right]\), and  
\[\alpha_{2j}=\left(1-\cos\left(\frac{j-1}{I_{M}-1}\pi\right)\right)\frac{\left( \alpha_{2}^{1}-\alpha_{2}^{0}\right)}{2}+\alpha_{2}^{0} \tag{68}\]  
for \(j=1,2,\ldots,I_{M}\) with \(\alpha_{2}\in\left[\alpha_{2}^{0},\alpha_{2}^{1}\right]\), in which \(I_{N},I_{M}\) are the total number of nodes along \(\alpha_{1}\), \(\alpha_{2}\), respectively. To solve the fundamental system of equations the separation of variables is to be used. The generalized displacements can be expressed as  
\[\mathbf{u}^{(\tau)}(\alpha_{1},\alpha_{2},t)=\mathbf{U}^{(\tau)}(\alpha_{1}, \alpha_{2})\mathrm{e}^{\mathrm{i}\omega t} \tag{69}\]  
where \(\mathbf{U}^{(\tau)}=\left[ \begin{array}{cc}U_{1}^{(\tau)}(\alpha_{1},\alpha _{2})&U_{2}^{(\tau)}(\alpha_{1},\alpha_{2})&U_{3}^{(\tau)}(\alpha_{1},\alpha _{2})\end{array} \right]^{T}\) denotes the mode shape vector, whose components are the amplitude of the modes at issue, whereas \(\omega\) is the circular frequency of the system, which allows the natural frequency to be defined as \(f=\omega/2\pi\). If Expression (69) and its second-order temporal derivative are substituted in (36), the fundamental system of Equation (36) becomes  
\[\sum_{\eta=0}^{N+1}\left(\mathbf{L}^{(\tau\eta)}-\mathbf{L}_{f}^{(\tau\eta)} \right)\mathbf{U}^{(\eta)}+\omega^{2}\sum_{\eta=0}^{N+1}\left(\mathbf{M}^{( \tau\eta)}+\mathbf{M}_{f}^{(\tau\eta)}\right)\mathbf{U}^{(\eta)}=\mathbf{0} \tag{70}\]  
for \(\tau=0,1,2,\ldots,N,N+1\).