## 3 Results and Discussion
### Effects of the Concentrated Mass  
To analyze the effect of the taper ratio on the natural frequency of a cantilever beam with a concentrated mass, the effects of the concentrated mass and taper ratios on the natural frequencies of such beams were investigated for \(n=1\) and \(n=2\). The mass of the beam for \(c=0\) and \(1.884\) kg was considered the concentrated mass, and the ratio of the concentrated mass to the beam mass (\(\overline{m}=m_{p}/m_{b}\)) was added at arbitrary positions along the length of the beam from 0 to 1 at intervals of 0.1. The taper ratio ranged from 0 to 0.5, and the predicted results are shown in Figures 4 and 5. These figures show the effects of the concentrated mass on the natural frequencies of the single- and double-tapered beams under clamped free-end conditions. Figures 4a and 5a show the variations in the natural frequencies with respect to changes in the position of the entire considered concentrated mass, and Figures 4b and 5b show the results for \(\overline{m}=0\), 0.1, 0.5, and 1. As shown in Figures 4b and 5b, the larger the taper ratio and \(\overline{m}\), the greater the effect of the concentrated mass. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{8}{c}{**Natural Frequencies [Hz]**} \\ \cline{2-9}  & \multicolumn{2}{c}{**[PERSON] et al. [48]**} & \multicolumn{2}{c}{**Present**} & \\ \cline{2-9}  & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{\(n=1\)**} & \multicolumn{2}{c}{\(n=2\)**} & \\ \hline  & \multicolumn{2}{c}{**Experimental**} & \multicolumn{2}{c}{\(\underline{-}\)} & \multicolumn{2}{c}{\(\underline{-}\)} & \multicolumn{2}{c}{\(\underline{-}\)} & \multicolumn{2}{c}{\(\underline{-}\)} \\  & \multicolumn{2}{c}{**} & \multicolumn{2}{c}{\(\underline{-}\)**} & \multicolumn{2}{c}{\(\underline{-}\)} & \multicolumn{2}{c}{\(\underline{-}\)} \\ \hline
1 & 20.000 & 19.902 & 21.089 & 19.897 & 19.897 & 21.089 & 19.897 & 19.897 \\
2 & 124.500 & 124.543 & 132.063 & 124.431 & 124.431 & 132.063 & 124.431 & 124.431 \\
3 & 342.188 & 345.507 & 369.332 & 345.068 & 345.068 & 369.332 & 345.068 & 345.068 \\
4 & 664.375 & 664.317 & 722.465 & 663.545 & 663.546 & 722.465 & 663.545 & 663.546 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of results calculated between present and previous studies. In the case of the second natural frequency, the positional changes at the nodal point according to the changes in the size and location of the concentrated mass were not pronounced, whereas the third natural frequency showed a noticeable change in the location of the first nodal point. When analyzing the results of Figures 4b and 5b, the effect of the concentrated mass on the natural frequencies for \(n=1\) and \(n=2\) changed with a similar trend. As expected, in the case of the first natural frequency, the larger the taper ratio at all positions, the greater the effect of the concentrated mass. However, the second and third natural frequencies showed a greater effect for \(c=0\) than for \(c=0.5\) at some locations, and the location of the occurrence of this phenomenon depended on the size of the concentrated mass. Figure 4: Effects of the concentrated mass on the first three natural frequencies of single tapered beams with the cantilevered end condition: (**a**) results for entire concentrated masses, and (**b**) results for \(\overline{m}=0\), \(0.1\), \(0.5\), and \(1\).