## 3 Results
### Calculations of the Reference Frequencies  
Four different materials are here considered for the validation of the proposed methodology. In addition to the unidirectional ([0\({}^{\circ}\), 0\({}^{\circ}\), 0\({}^{\circ}\), 0\({}^{\circ}\)]), the symmetric ([30\({}^{\circ}\), \(-\)30\({}^{\circ}\), -30\({}^{\circ}\), 30\({}^{\circ}\)]), and the asymmetric materials ([30\({}^{\circ}\), \(-\)30\({}^{\circ}\), 30\({}^{\circ}\), \(-\)30\({}^{\circ}\)]) already presented in Section 3.1, a woven fabric laminate is considered. The woven laminate is also constituted by four layers, with the material properties reported in Table 3 with reference to the axes of orthotropy. These materials have been chosen to show the applicability of the proposed approach to the limited selection of laminates, whose stacked plies are made of the same material. The reference frequencies are here calculated through the Rayleigh-Ritz formulation. Dimensions of the rectangular region are \(a=150\) mm and \(b=100\) mm, as in the assumption of assessing local elastic properties, according to the methodology described in Section 2.2. The thicknesses of the four laminates are 1 mm for the unidirectional, 1.5 mm for both the symmetric and anti-symmetric laminates, and 2.5 mm for the woven fabric. With respect to the reference axes shown in Figure 1, the mass locations are \((x_{1},y_{1})=(50,\ 60)\) and \((x_{2},y_{2})=(100,\ 60)\), which are symmetric with respect to the vertical axis of symmetry of the rectangle. It is worth noting that in order to obtain a measurable discrepancy in the reduction of the first resonant frequency, the two locations must not be located far from the center of the of the plate. The mass is assumed equal to 1 g.  
For the unidirectional, the symmetric, and anti-symmetric plates, the reference values have been calculated considering the material orientated at 0\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), and 90\({}^{\circ}\) with respect to the clamped boundaries. For the woven fabric, the assumed rotations are 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\). Indeed, after 45\({}^{\circ}\), the frequency values are repeated as the longitudinal and transverse Young's moduli are equal. Results of the reference values of the first resonant frequencies are reported in Tables 4-7. Figure 5: Comparison between experimental data obtained by [PERSON] et al. [20] and the presented Rayleigh–Ritz formulation: variation of the frequency for the mass positioned along the x direction with the y coordinate equal to \(\frac{b}{2}=150\) mm (**a**) and along the y direction with the x coordinate equal to \(\frac{a}{2}=300\) mm (**b**). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(E_{11}\) [GPa] & \(E_{22}\) [GPa] & \(G_{12}\) [GPa] & \(\
u_{12}\) [-] & \(\rho\left[\mathbf{kg}/\mathbf{m}^{3}\right]\) \\ \hline Woven fabric & 59.0 & 59.0 & 3.4 & 0.04 & 1432.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties at the py level of the woven fabric. As expected, the presence of the concentrated mass reduces the value of the first resonant frequency. Furthermore, when the isolines are parallel to the clamped boundaries, as in the case of the unidirectional laminate with fibers oriented at \(0^{\circ}\) or at \(90^{\circ}\) with respect to the clamped boundaries, the first resonant frequency equally decreases for the two mass locations. This is particularly due to the modal shape combined with the choice of the mass positions, which are symmetric with respect to one of the axes of the rectangular plate. Instead, when the isolines are not parallel to the boundaries, the mass affects the first resonant frequency differently in the two locations. As a consequence, the discrepancy between the reductions of the first resonant frequency due to the two mass locations is a measure of the material anisotropy. Furthermore, in regard to the resultant first frequency, it is important to notice that its value is higher for the mass disposed in the first position.