## 4 Results
### Model Validation through Simulation  
In order to validate the proposed model, fundamental frequencies and modes are computed with both our method and FEM. The goal of this comparison is to investigate the conditions under which the proposed model is valid. The choice for FEM as the ground-truth data source is due to the acceptance of the method for vibration analysis [30, 31, 34, 35, 36]. The analyzed structure is shown in Figure 1. The total lengths of the segments are 1 m, i.e.,  
\[l_{1}+l_{2}=1\text{ m}\]\[l_{5}+l_{6}=1\ \text{m}\]  
Moreover, we have the conditions \(l_{1}=l_{3}\) and \(l_{2}=l_{4}\). The cross-section of all beams is square and equal, denoting their side dimension as \(a\), in m. Moreover, equal masses are attached to the endpoints of \(l_{1}\), \(l_{3}\), and \(l_{5}\), as shown in Figure 1, denoting their value as m, in kg. Free endpoints are fixed, and displacements are restricted to the vertical direction. In order to compute the fundamental frequencies \(\omega^{k}\), the bisection method was used to find the values \(\omega^{k}\) at which the matrix \(F(\omega^{k})\), given by (22), is singular, as explained in the previous section. For the analysis with FEM, the well-known software Ansys was used. In detail, \"structural steel\" was selected as the material, with a density of \(\rho=7850\ \text{kg}/\text{m}^{3}\) and elastic coefficient \(E=200\ \text{GPa}\). The torsional elastic coefficient is \(G=85\ \text{N}/\text{m}^{2}\). Small high-density rigid bodies were attached (having a volume of \(20\ \text{mm}^{3}\)) to the endpoints of \(l_{1}\), \(l_{3}\), and \(l_{5}\), to simulate the attached masses. Moreover, frictional supports were added to the beams lateral faces to restrict displacements to the vertical direction, as shown in Figure 3. For meshing, fine mesh with an element size of \(5\ \text{mm}\) was used, using the element \"SOLID187\". Smaller elements were attempted, obtaining similar results. The \"Modal\" analysis was performed with the \"Mechanical APDL\", using \"Block Lanczos\" as the extraction method. Different simulations were performed for different values of \(l_{1}\), \(l_{5}\), \(a\), and \(m\), in order to investigate the influence of each of these parameters on the solution accuracy. The following metric was used to measure the errors for the computation of the fundamental frequencies:  
\[e^{k}(a,m)=\frac{1}{|L_{1}|\cdot|L_{5}|}\sum\limits_{l_{1}\in L_{1}}\sum \limits_{l_{5}\in L_{5}}abs\left(\frac{f^{k}_{Model}(l_{1},l_{5},a,m)-f^{k}_{ FEM}(l_{1},l_{5})}{f^{k}_{FEM}(l_{1},l_{5},a,m)}\right) \tag{30}\]  
where \(L_{1}\) and \(L_{5}\) are the set of values for \(l_{1}\) and \(l_{5}\), respectively, that were simulated, \(f^{k}_{Model}(l_{1},l_{5},a,m)\) and \(f^{k}_{FEM}(l_{1},l_{5},a,m)\) denote the \(k\)-th fundamental frequency estimated by our model and the FEM in Hertz, respectively, considering the specified values for \(l_{1}\),\(l_{5}\), \(a\) and \(m\). Then, \(e^{k}(a,m)\) represents the mean absolute proportional error of the \(k\)-th fundamental frequency. Several simulations were performed, covering all the combinations of values \(l_{1}\in\{0.2,0.3,0.4,0.5\}\), \(l_{5}\in\{0.2,0.3,0.4,0.5\}\), \(a\in\{0.01,0.03,0.1\}\) and \(m\in\{0,0.2,1\}\). However, for the case in which \(m=0\), the value of \(l_{5}\) is irrelevant. For instance, Table 1 shows the first four fundamental frequencies obtained for \(a=0.01\ \text{m}\), \(m=0.2\ \text{kg}\), and different values for \(l_{1}\) and \(l_{5}\), by using our model and FEM. Figure 3: Structure with restrictions and attached masses for \(FEM\). M1, M5 and M3 correspond to the position of mass \(1\), \(5\), and \(3\).