## 3 Results and Discussion
### Numerical Results  
In our case, when measurements are made for the first natural frequency, which corresponds to the natural bending mode, the acceleration transducer of mass \(m_{0}\) was used, and the mathematical model is the one considered in Equation (17) representing mass-spring elements with a constant force \(F(t)=m_{0}g\), and \(m_{0}\) will be introduced as a concentrated mass at the top (free end) of the tested sample. For numerical simulation, the same dimensions as for the experimental model were considered: \(b=0.02\) m, \(h=0.004\) m, \(L_{1}=0.14\) m, \(L_{2}=0.18\) m, \(L_{3}=0.22\) m, \(L_{4}=0.26\) m.  
For the cantilever beam reinforced with hemp fibers for a cantilever beam density \(900\) kg/m\({}^{3}\) and the Young's modulus experimentally determined above \(E=1819\) MPa, the  
Figure 5: Plots of experimental data and their fit by analytical model. Figure 6: Comparison of free vibration time records and analytical fit by viscoelastic model with (**a**) hemp fibers; (**b**) coconut fibers. values \(f_{11}=10.13\) Hz, \(f_{21}=6.9\) Hz, \(f_{31}=5.07\) Hz, and \(f_{41}=3.92\) Hz were obtained by numerical simulation. In the case of the cantilever beam reinforced with coconut fibers for a cantilever beam with \(E=962\) MPa and density 750 kg/m\({}^{3}\), \(b=0.02\) m, \(h=0.004\) m, \(L_{1}=0.14\) m, \(L_{2}=0.18\) m, \(L_{3}=0.22\) m, and \(L_{4}=0.26\) m, the values \(f_{11}=7.36\) Hz, \(f_{21}=5.01\) Hz, \(f_{31}=3.69\) Hz, and \(f_{41}=2.85\) Hz were obtained by FEM. In Figure 7 are depicted the numerical results for the two considered composite beams with their fit using the analytical model \(f(L)=\frac{C}{L^{3/2}}\). For the composite beam made with hemp fibers, the value 0.52799 was obtained for the constant \(C\), and for the cantilever beam reinforced with coconut fibers, the constant \(C\) had the value 0.38352. Tables 1 and 2 provide a summary of the experimental and numerical results obtained in the case of cantilever beams reinforced with hemp and coir fibers, respectively. It can be observed that for the two considered composite beams, the damping ratio values are very small (\(0<\zeta<0.05\)) and, therefore, the damped and undamped natural frequencies of the system are very close. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Length [m]**} & **FEM** & **Measured** & \multirow{2}{*}{**C\({}_{\text{FEM}}\)**} & **Cexp** & \multirow{2}{*}{\(\zeta\)} \\  & **Frequency** & **Frequency** & **[m\({}^{3/2}\)s\({}^{-1}\)]** & **[m\({}^{3/2}\)s\({}^{-1}\)]** & \\ \hline
0.14 & 10.13 & 10.38 & & & 0.049 \\
0.18 & 6.9 & 6.96 & & & 0.035 \\
0.22 & 5.07 & 5.25 & & & 0.030 \\
0.26 & 3.92 & 4.03 & & & 0.029 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Compared results for hemp. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Length [m]**} & **FEM** & **Measured** & \multirow{2}{*}{**C\({}_{\text{FEM}}\)**} & **Cexp** & \multirow{2}{*}{\(\zeta\)} \\  & **Frequency** & **Frequency** & **[m\({}^{3/2}\)s\({}^{-1}\)]** & **[m\({}^{3/2}\)s\({}^{-1}\)]** & \\ \hline
0.14 & 7.36 & 7.08 & & & 0.041 \\
0.18 & 5.01 & 5.62 & & 0.38352 & 0.39494 & 0.039 \\
0.22 & 3.69 & 4.27 & & & 0.035 \\
0.26 & 2.85 & 3.42 & & & 0.033 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Compared results for coconut. Figure 7: Plots of numerical results (FEM) and their fit by analytical model. The comparison of experimental and numerical results in the case of the cantilever beam reinforced with hep fibers is shown in Figure 8, and in the case of coconut fibers in Figure 9.  
The average relative error between the values obtained using the analytical model for the experimental and numerical data is 2.9% in the case of hep fibers and 3% in the case of coconut fibers, consistent with the various specimen lengths. The differences between the values may come from slight dimensional and material variations, support and testing conditions, and instrument precision.