## 5 Results and Discussion
### Free Vibration Analysis  
Modal analysis involves identifying the fundamental dynamic characteristics of the mast, including natural frequencies and mode shapes. These characteristics are then utilized to establish the system's dynamic or quasi-static behavior. Figure 9 shows the first eigenfrequencies and eigenmodes of the mast in the initial condition. Figure 8: Relationship between variables \(\alpha\) and \(\gamma\). Figure 9: _Cont._  
Table 7 shows the effective masses in each of the directions according to the Cartesian reference system shown in the figure above. With the determination of the first six eigenfrequencies, a total of 86.97% of the total mass is captured. It is observed that the first mode is by far the most important, so it will be the one to pay attention to in terms of structural design. The influence of mass redistribution on the determination of the modal analysis translated, in terms of obtaining eigenfrequencies and eigenmodes, is studied. The alternative is a beam of variable section resembling a truncated cone (tapered beam), of the following dimensions: a lower radius of 300 mm, an upper radius of 100 mm for the same initial height. In order to generalize the structural behavior and not to condition it to the same cross-section, a solid circular cross-section with the same value as the outer radius for the tubular section is considered, in order to compare its structural behavior with its counterpart in terms of constant cross-section. Figure 10 shows the eigenfrequencies and eigenmodes of the mast for the variable and solid cross-section case. Table 8 shows the comparison of the six eigenfrequencies between the constant solid section and the variable solid one. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode** & **Frequency (Hz)** & \begin{tabular}{c} **Effective Mass** \\ **\(x\)-Direction (\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Effective Mass** \\ **\(y\)-Direction (\%)** \\ \end{tabular} &
\begin{tabular}{c} **Effective Mass** \\ **\(z\)-Direction (\%)** \\ \end{tabular} \\ \hline
1 & 3.49 & 0 & 0 & 0 \\
2 & 3.49 & 61.35 & 0 & 0 \\
3 & 21.56 & 0 & 0 & 19.01 \\
4 & 21.56 & 19.01 & 0 & 0 \\
5 & 59.03 & 1.11 & 0 & 5.50 \\
6 & 59.03 & 5.50 & 0 & 1.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Effective mass in \(x\)-, \(y\)-, and \(z\)-direction of the mast with constant and tubular cross-section. Figure 9: Natural modes and frequencies of the mast. A difference of varying magnitude is observed between the eigenfrequencies of the solid circular cross-sections, the constant and the variable ones. In the first case, there is a difference of 56.19%, in the second one 22.18%, and in the third one 6.68%. A design tool based on regression models is established in terms of structural design, which allows us to establish the evaluation of the eigenfrequency of the constant or variable section mast from the knowledge of only one of them, with the advantages derived from this. Figure 11 shows graphically the percentage evolution of the differences between each of the eigenfrequencies previously calculated, for the constant and variable sections, with the comparison of different regression models. The potential regression is depicted by the blue line, while the orange line represents exponential regression. The purple line corresponds to a second-order polynomial regression, and lastly, the green line portrays logarithmic regression. Table 9 shows the numerical difference, in terms of the correlation coefficient, of the different regression models. Figure 10: Natural modes and frequencies of the mast with variable and solid cross-section. \begin{table}
\begin{tabular}{c c c} \hline
**Natural Frequency** & **Constant Section** & **Variable Section** \\ \hline \(f_{4}\) (Hz) & 15.80 & 20.31 \\ \(f_{5}\) (Hz) & 43.90 & 47.04 \\ \(f_{6}\) (Hz) & 43.900 & 47.034 \\ \hline \end{tabular}
\end{table} 
Table 8: _Cont._  
cross-section (constant tubular).