## 4 Results
### Model Validation through Simulation
Table 1 shows a good approximation of the values obtained with our model, for the case \(m=0.2\) kg, \(a=0.01\) m. Computing the mean absolute proportional error of the four fundamental frequencies, we obtain \(e^{1}(0.01,0.2)=0.058\), \(e^{2}(0.01,0.2)=0.035\), \(e^{3}(0.01,0.2)=0.046\) and \(e^{4}(0.01,0.2)=0.058\), meaning 5.8% of error for the first frequency, 3.5% of error for the second frequency, 4.6% of error for the third frequency and 5.8% for the fourth frequency. After simulating for all the combinations of the considered parameters, the results shown in Table 2 were obtained. Finally, Table 3 shows the average errors for each combination of \(a\) and \(m\), considering the first four fundamental frequencies. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \multicolumn{2}{c}{\(\mathbf{l_{1}=0.2\) m}} & \multicolumn{2}{c}{\(\mathbf{l_{1}=0.3\) m}} & \multicolumn{2}{c}{\(\mathbf{l_{1}=0.4\) m}} & \multicolumn{2}{c}{\(\mathbf{l_{1}=0.5\) m}} \\ \hline  & **Model** & **FEM** & **Model** & **FEM** & **Model** & **FEM** & **Model** & **FEM** \\ \hline  & 45.5 & 48.6 & 36.9 & 39.0 & 29.2 & 30.9 & 26.5 & 28.2 \\ \(l_{5}=0.2\) m & 72.3 & 73.0 & 51.8 & 53.4 & 38.7 & 40.7 & 35.0 & 36.9 \\  & 75.7 & 77.2 & 71.4 & 74.9 & 66.7 & 71.2 & 65.6 & 70.3 \\  & 124.1 & 103.5 & 109.9 & 115.3 & 131.5 & 140.1 & 136.1 & 146.9 \\ \hline  & 42.0 & 44.9 & 35.3 & 37.4 & 28.6 & 30.3 & 26.1 & 27.7 \\ \(l_{5}=0.3\) m & 72.8 & 73.5 & 52.7 & 54.3 & 39.4 & 41.3 & 35.5 & 37.5 \\  & 75.5 & 76.9 & 69.6 & 72.7 & 63.6 & 67.7 & 62.2 & 66.6 \\  & 99.3 & 105.6 & 110.2 & 115.7 & 127.4 & 135.8 & 130.0 & 139.5 \\ \hline  & 39.3 & 42.0 & 34.0 & 36.1 & 28.1 & 29.8 & 25.8 & 27.4 \\ \(l_{5}=0.4\) m & 73.2 & 73.8 & 53.5 & 55.1 & 39.9 & 41.9 & 36.0 & 38.1 \\  & 75.2 & 76.4 & 66.8 & 69.6 & 59.6 & 63.4 & 58.0 & 61.9 \\  & 104.3 & 110.4 & 112.7 & 118.2 & 137.0 & 145.3 & 141.7 & 152.0 \\ \hline  & 38.4 & 41.0 & 33.5 & 35.6 & 27.9 & 29.6 & 25.7 & 27.3 \\ \(l_{5}=0.5\) m & 73.3 & 74.0 & 53.8 & 55.4 & 40.2 & 42.1 & 36.3 & 38.3 \\  & 75.1 & 76.2 & 65.7 & 68.3 & 58.0 & 61.6 & 56.3 & 60.1 \\  & 106.9 & 113.0 & 113.8 & 119.3 & 143.6 & 151.0 & 153.0 & 164.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Fundamental frequencies in Hertz obtained with our model and FEM, \(m=0.2\) kg, \(a=0.01\) m.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{2}{c}{\(\mathbf{m=0\) kg}} & \(\mathbf{m=0.2\) kg} & \(\mathbf{m=1\) kg} \\ \hline  & \(e^{1}\) & 0.064 & 0.058 & 0.059 \\ \(a=0.01\) m & \(e^{2}\) & 0.043 & 0.035 & 0.038 \\ \(e^{3}\) & 0.052 & 0.046 & 0.054 \\ \(e^{4}\) & 0.063 & 0.058 & 0.058 \\ \hline  & \(e^{From Table 3, it can be seen that the errors were around 5.5% for all the cases in which \(a=0.01\) m and \(a=0.03\) m, independently of the values for \(l_{1}\), \(l_{3}\) and m. The largest errors were obtained for \(a=0.1\) m with attached masses, i.e., with \(m=0.2\) kg and \(m=1\) kg. These results are coherent with the theory, since the Bernoulli-Euler formulation is valid for thin beams, i.e., when \(a\) is small. Moreover, in the case of \(a=0.01\) m and \(a=0.03\) m, the mode shapes are also well approximated. For instance, Figures 4-7 show the modes associated with the first four fundamental frequencies for the case \(a=0.01\) m, \(m=1\) kg, \(l