## 3 Results
### Validation of the Rayleigh-Ritz Formulation  
In order to validate the Rayleigh-Ritz described in Section 2.1, literature data [14; 20; 21] on clamped plates have been considered. For convergence and validation studies, the comparison concerns six different laminates with unidirectional reinforcement plies. Furthermore, the Rayleigh-Ritz formulation has been validated on the experimental results of an isotropic aluminum plate loaded by a concentrated mass. In regard to the laminated composites, data and results are taken from [PERSON] et al. [14] and [PERSON] [21]. In these works, the dimensions of the plates were \(a=b=1270\) mm with thickness \(h=25.4\) mm. Hence, the plate was relatively thin with the aspect ratio, i.e., the ratio between the in-plane dimension and the thickness, equal to 50. The laminated composites were clamped at the extremities, and the first resonant frequency was acquired. The laminated composites were constituted by four layers with unidirectional reinforcement and present symmetric stacking sequences (\([0^{\circ},0^{\circ},0^{\circ},0^{\circ}]\), \([15^{\circ},-15^{\circ},-15^{\circ},15^{\circ}]\), \([30^{\circ},-30^{\circ},-30^{\circ},30^{\circ}]\), and \([45^{\circ},-45^{\circ},-45^{\circ},45^{\circ}]\)), an anti-symmetric stacking sequence (\([30^{\circ},-30^{\circ},30^{\circ},-30^{\circ}]\)), and a non-symmetric sequence (\([0^{\circ},30^{\circ},60^{\circ},90^{\circ}]\)). Material properties of the unidirectional plies are reported in Table 1 with reference to the axes of orthotropy. Firstly, the convergence study has been addressed. Results of the convergence study are reported in Table 2 in terms of frequency parameter \(\lambda\) for different numbers of terms \(n\) of the polynomials in Equation (11). Indeed, as the polynomial degree increases (i.e., \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(E_{11}\) [GPa] & \(E_{22}\) [GPa] & \(G_{12}\) [GPa] & \(\
u_{12}\) [-] & \(\rho\) [kg/m\({}^{3}\)] \\ \hline Unidirectional ply & 131.69 & 8.55 & 6.76 & 0.3 & 1610 \\ \hline \end{tabular}
\end{table} 
Table 1: Material properties at ply level. the number of terms \(n\) increases), the exact solution is better approximated through the Rayleigh-Ritz method. Table 2 shows that the convergence is reached even for \(n=36\). However, for a better approximation, in this paper, a degree of the polynomial equal to 7 has been assumed in the calculations, with the number of terms \(n\) equal to 49. Table 2 also reports a comparison of the results obtained with the formulation proposed in Section 2.1 with those obtained by [PERSON] et al. [14] and [PERSON] and [PERSON] [21] in their works. The discrepancy is very limited. The frequency parameter \(\lambda\) calculated through the proposed formulation is always smaller than those calculated in the literature. This is mainly due to the different shear deformation theories. Indeed, the present Rayleigh-Ritz formulation is based on the higher-order shear deformation theory. Instead, the first order shear deformation theory was adopted in the considered works. Indeed, the higher order of the presented formulation increases the compliance of the plates, resulting in lower values of the frequency parameter \(\lambda\)[18]. The Rayleigh-Ritz formulation has been also validated with respect to the experimental results reported by [PERSON] et al. [20], who analyzed the first resonant frequency of a clamped rectangular plate made of aluminum for different positions of a concentrated mass of 96 g. Particularly, the mass was moved along the axes of symmetry of the rectangle, and the resultant first resonant frequency was measured. The clamped plate had dimensions of \(a=600\) mm and \(b=300\) mm. Therefore, while moving the mass along the x direction, its y coordinate was equal to \(\frac{b}{2}=150\) mm. Instead, while moving the mass along the y direction, the x coordinate was 300 mm. Even though the Rayleigh-Ritz formulation of Section 2.1 is specifically developed for angle-ply laminates, an isotropic material can be considered as a particular case.