## 3 Results and Discussion
### Effects of the Combination of Axial Loading, Concentrated Mass, and Elastic Supports  
In this section, various environments that can cause variations in natural frequencies are addressed. Variations in the natural frequencies of tapered beams for combinations of multiple elastic supports, concentrated masses, and axial forces were investigated, and Types A, C, and D, as illustrated in Figure 2, were considered to analyze these effects. The values of \(K_{\Phi}\), \(K_{w}\), \(\overline{m}\), and \(F_{0}\) used to construct the various problems are presented in Table 10. Type A considers the effects of elastic supports, axial loading, and concentrated masses on the eigenpairs of single- and double-tapered beams. Types C and D consider the effects of the point mass and multiple elastic supports on the natural frequencies of such beams. Before analyzing the example beams, the computed results were compared with the natural frequencies obtained using a commercial finite element analysis program (ANSYS). The comparison results are presented in Table 11 and agree with each other. Because the comparative results demonstrated that the proposed method has excellent accuracy even for complex problems, the eigenvalues predicted by considering several cases that can be  
Figure 15: Relationships between compressive loadings and values of elastic springs at which the first natural frequency vanished. experienced in practical problems were analyzed. The results are summarized in Table 12. As various analyses were conducted, the variations in the mode shapes of these effects were investigated. As illustrated in Figure 16, the first three mode shapes for the three cases of Type A with \(c=0.5\) were compared with the mode shapes of a cantilever beam for \(n=1\) and \(n=2\). When the same elastic support or concentrated mass was attached to \(n=1\) rather than \(n=2\), the effect on the modal shapes was greater. Compared with the elastic support, the effect of the concentrated mass attached at \(\overline{x}=1\) appeared to be significantly greater. The effect of the axial load on the mode shape is greater for \(n=2\) than for \(n=1\). When analyzing the second and third modes, the nodal point was moved to the left by the effects \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(\epsilon\)} & \multirow{2}{*}{**Beam Type**} & \multirow{2}{*}{**Approaches**} & \multicolumn{6}{c}{**Natural Frequencies [Hz]**} \\ \cline{6-13}  & & & & & & & & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{3}\) \\ \hline A & 3 & 0.5 & \(n=1\) & ANSYS & 24.94 & 75.74 & 166.8 \\  & & & & & Present & 24.90 & 75.61 & 166.5 \\  & & & 0.5 & \(n=2\) & ANSYS & 28.46 & 81.38 & 174.8 \\  & & & & Present & 28.38 & 81.02 & 174.0 \\ C & 2 & 0.3 & \(n=1\) & ANSYS & 14.99 & 21.71 & 74.46 \\  & & & & Present & 14.99 & 21.71 & 74.32 \\  & & & 0.3 & \(n=2\) & ANSYS & 14.34 & 17.75 & 53.14 \\ D & 2 & 0 & \(n=1\) & ANSYS & 3.169 & 10.38 & 63.99 \\  & & & & Present & 3.169 & 10.38 & 63.77 \\  & & & 0 & \(n=2\) & ANSYS & 3.169 & 10.38 & 63.99 \\  & & & & Present & 3.169 & 10.38 & 63.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparisons of the results computed from the present method and ANSYS. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{6}{c}{**Nondimensional Natural Frequencies**} \\ \cline{3-14}  & & & & \multicolumn{6}{c}{\(n=1\)} & \multicolumn{6}{c}{\(n=2\)} \\ \cline{3-14}  & & & \(c=0\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) & \(c=\) \\  & & & **0.1** & **0.2** & **0.3**  
of the axial load and elastic support, whereas it was moved to the right by the effect of the concentrated mass. Finally, the effects of the elastic supports on the first three mode shapes of uniform beams, single- and double-tapered beams with \(c=0.5\), and elastically restrained cantilever beams were considered.