## 3 Results and Discussion
### Effects of the Concentrated Mass
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(c\)} & \multicolumn{2}{c}{\(\widetilde{\omega}_{i}\)} & \multicolumn{1}{c}{\(\overline{x}=0\)} & \multicolumn{1}{c}{\(\overline{x}=0.1\)} & \multicolumn{1}{c}{\(\overline{x}=0.2\)} & \multicolumn{1}{c}{\(\overline{x}=0.3\)} & \multicolumn{1}{c}{\(\overline{x}=0.4\)} & \multicolumn{1}{c}{\(\overline{x}=0.5\)} & \multicolumn{1}{c}{\(\overline{x}=0.6\)} & \multicolumn{1}{c}{\(\overline{x}=0.7\)} & \multicolumn{1}{c}{\(\overline{x}=0.8\)} & \multicolumn{1}{c}{\(\overline{x}=0.9\)} & \multicolumn{1}{c}{\(\overline{x}=1\)} \\ \hline
0 & 1 & 3.516 & 3.516 & 3.513 & 3.503 & 3.479 & 3.437 & 3.375 & 3.293 & 3.196 & 3.086 & 2.968 \\  & 2 & 22.03 & 21.99 & 21.63 & 20.88 & 20.23 & 20.15 & 20.73 & 21.64 & 22.01 & 21.16 & 19.35 \\  & 3 & 61.66 & 61.00 & 57.43 & 56.13 & 59.00 & 61.66 & 59.49 & 57.51 & 60.04 & 61.21 & 55.49 \\
0.1 & 1 & 3.674 & 3.673 & 3.670 & 3.659 & 3.631 & 3.580 & 3.505 & 3.405 & 3.286 & 3.153 & 3.011 \\  & 2 & 21.55 & 21.51 & 21.16 & 20.38 & 19.68 & 19.54 & 20.13 & 21.12 & 21.52 & 20.54 & 18.54 \\  & 3 & 59.16 & 58.57 & 55.14 & 53.64 & 56.32 & 59.14 & 56.97 & 54.65 & 57.28 & 58.67 & 52.51 \\
0.2 & 1 & 3.855 & 3.855 & 3.851 & 3.838 & 3.805 & 3.743 & 3.650 & 3.527 & 3.380 & 3.217 & 3.045 \\  & 2 & 21.05 & 21.02 & 20.68 & 19.88 & 19.11 & 18.92 & 19.50 & 20.58 & 21.02 & 19.88 & 17.67 \\  & 3 & 56.61 & 56.08 & 52.81 & 51.12 & 53.57 & 56.56 & 54.42 & 51.71 & 54.40 & 56.07 & 49.45 \\
0.3 & 1 & 4.067 & 4.067 & 4.063 & 4.047 & 4.007 & 3.931 & 3.814 & 3.658 & 3.474 & 3.272 & 3.061 \\  & 2 & 20.55 & 20.52 & 20.19 & 19.37 & 18.54 & 18.28 & 18.85 & 20.02 & 20.51 & 19.17 & 16.73 \\  & 3 & 54.00 & 53.53 & 50.44 & 48.57 & 50.76 & 53.90 & 51.83 & 48.68 & 51.39 & 53.41 & 46.33 \\
0.4 & 1 & 4.319 & 4.318 & 4.314 & 4.295 & 4.246 & 4.150 & 4.000 & 3.801 & 3.567 & 3.313 & 3.051 \\  & 2 & 20.05 & 20.02 & 19.70 & 18.87 & 17.96 & 17.63 & 18.19 & 19.46 & 19.99 & 18.41 & 15.72 \\  & 3 & 51.32 & 50.92 & 48.03 & 45.99 & 47.89 & 51.14 & 49.22 & 45.57 & 48.21 & 50.67 & 43.16 \\
0.5 & 1 & 4.625 & 4.625 & 4.619 & 4.596 & 4.534 & 4.410 & 4.213 & 3.953 & 3.650 & 3.328 & 3.000 \\  & 2 & 19.55 & 19.52 & 19.21 & 18.37 & 17.39 & 16.98 & 17.52 & 18.89 & 19.46 & 17.58 & 14.65 \\  & 3 & 48.57 & 48.23 & 45.57 & 43.37 & 44.96 & 48.28 & 46.58 & 42.39 & 44.86 & 47.85 & 39.94 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effects of the point mass on the first three nondimensional natural frequencies of double tapered cantilever beams as a function of location when \(\overline{m}=0.1\). Figure 6: Comparison of natural frequencies for \(n=1\) and \(n=2\) when having \(\overline{m}=0.1\) at the arbitrary location. The accuracy of the predicted results for the concentrated mass was demonstrated by comparing the results with those of previous studies [16]. The values of the translational and rotational springs used to provide a clamped-end condition were \(K_{\text{W}}\) = 9.745 \(\times\) 10\({}^{9}\) N/m and \(K_{\Phi}\) = 1.319 \(\times\) 10\({}^{9}\) Nm/rad. The effect of concentrated mass on the first three nondimensional natural frequencies of a clamped-free uniform beam with a concentrated mass attached to the free end. When considering \(\overline{m}\) from 0 to 1.0 at intervals of 0.2, a taper ratio of \(c=10^{-4}\) was used for the uniform beam. The comparison results are presented in Table 4 and are in good agreement. BE and \(r^{2}\) = 0 shown in Table 4 are the results computed by classical beam theory, and \(r^{2}\
eq 0\) is the result calculated by Rayleigh beam theory for **Example 1**.