## 3 Results and Discussion
### Effects of Concentrated Mass Attached to Elastic Supports  
To analyze the effects of the concentrated mass attached to the elastic supports under the same conditions, the rotational and translational springs at both ends used the same values of 22,026.5 (=exp(10)) for all cases. The Type B model shown in Figure 2 is considered, and the 12 cases are as follows: Cases A1-A4 are supported by \(K_{w1}\) and \(K_{\Phi 1}\) at \(\overline{x}\) = 0, and Cases A5-A8 are supported by \(K_{w1}\) and \(K_{\Phi 1}\) at \(\overline{x}\) = 0 and \(K_{w2}\) at \(\overline{x}\) = 1. In addition, Cases A9-A12 are beam models supported by \(K_{w1}\) and \(K_{\Phi 1}\) at \(\overline{x}\) = 0, and \(K_{w2}\) and \(K_{\Phi 2}\) at \(\overline{x}\) = 1. These values were listed in Table 5, and the concentrated mass \(\overline{m}\) is increased from 0 to 0.1 in intervals of 0.01, and the taper ratio is considered \(c\) = 0 and \(c\) = 0.5 for each case. The predicted results for the first three nondimensional natural frequencies of the single- and double-tapered beams are tabulated in Tables 6 and 7, respectively. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Approaches**} & & \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{6}{c}{**Nondimensional Natural Frequencies**} \\ \cline{3-8}  & & & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) \\  & & & \(m=0\) & \(m=0.2\) & \(m=0.4\) & \(m=0.6\) & \(m=0.8\) & \(m=1.0\) \\ \hline Exact [16] & BE & 1 & 3.5160 & 2.6127 & 2.1680 & 1.8925 & 1.7006 & 1.5573 \\  & & 2 & 22.0345 & 18.2078 & 17.1763 & 16.7007 & 16.4274 & 16.2501 \\  & & 3 & 61.6972 & 53.5586 & 52.0632 & 51.4451 & 51.1080 & 50.8958 \\ Present & \(n=1\) & \(r^{2}=0\) & 1 & 3.5160 & 2.6127 & 2.1679 & 1.8924 & 1.7005 & 1.5571 \\  & & 2 & 22.0343 & 18.2076 & 17.1761 & 16.7006 & 16.4273 & 16.2499 \\  & & 3 & 61.6961 & 53.5577 & 52.0624 & 51.4443 & 51.1071 & 50.8950 \\  & & \(r^{2}\
eq 0\) & 1 & 3.5159 & 2.6126 & 2.1679 & 1.8924 & 1.7005 & 1.5571 \\  & & 2 & 22.0296 & 18.2053 & 17.1744 & 16.6990 & 16.4258 & 16.2485 \\  & & 3 & 61.6651 & 53.5397 & 52.0463 & 51.4288 & 51.0921 & 50.8801 \\ \(n=2\) & \(r^{2}=0\) & 1 & 3.5160 & 2.6127 & 2.1679 & 1.8924 & 1.7005 & 1.5571 \\  & & 2 & 22.0343 & 18.2076 & 17.1761 & 16.7006 & 16.4273 & 16.2499 \\  & & 3 & 61.6961 & 53.5577 & 52.0624 & 51.4443 & 51.1071 & 50.8950 \\  & & \(r^{2}\
eq 0\) & 1 & 3.5159 & 2.6126 & 2.1679 & 1.8924 & 1.7005 & 1.5571 \\  & & 2 & 22.0296 & 18.2053 & 17.1744 & 16.6990 & 16.4258 & 16.2485 \\  & & 3 & 61.6651 & 53.5397 & 52.0463 & 51.4288 & 51.0921 & 50.8802 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparisons of the first three nondimensional natural frequencies of a clamped-free uniform beam with a concentrated mass at the free end. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(c\)} & \multicolumn{2}{c}{\(-\)} & \(K_{w1}\) & \(K_{w2}\) & \(K_{\Phi 2}\) & \(K_{\Phi 2}\) \\ \cline{3-8}  & & \(-\) & \(-\) & \(-\) & [N/m] & [N-m/rad] & [N/m] & [N-m/rad] \\ \cline{3-8}  & & \(x=0\) & \(x=1\) & & & & \\ \hline A1 & 0 & 0–0.1 & 0 & 22,026.5 & 22,026.5 & 0 & 0 \\ A2 & 0 & 0 & 0–0.1 & 22,026.5 & 22,026.5 & 0 & 0 \\ A3 & 0.5 & 0–0.1 & 0 & 22,026.5 & 22,026.5 & 0 & 0 \\ A4 & 0.5 & 0 & 0–0.1 & 22,026.5 & 22,026.5 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Values of \(\overline{m}\), \(K_{\Phi}\), and \(K_{w}\) attached to **Example 1**. The effects of the concentrated mass and taper ratio for Cases A5-A8 with \(n=1\) are illustrated in Figure 8. The concentrated mass attached at \(\overline{x}=0\) significantly affected the first natural frequency for \(c=0\) and \(c=0.5\), whereas the second and third natural frequencies were significantly affected when the concentrated mass was located at \(\overline{x}=1\). When comparing the taper ratios, the first natural frequency had a similar effect, regardless of the taper ratio, when the concentrated mass was located at \(\overline{x}=0\).