# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### Homogeneous Timoshenko Beams with Curvilinearly Tapered Cross-Sections  
In order to demonstrate the applicability and effectiveness of the DTM in solving the free vibration problem for axially curvilinear cross-sectional beams, we now consider various specific cases. Whenever possible, we compare the results obtained using DTM with those already documented in the literature. In cases, where comparative data are not available, we perform the frequency analysis using ABAQUS 6.14 software. The first problem addressed is a homogeneous beam with a double-parabolic taper, truncated at the left end. The shape of the parabolic taper beam is analytically defined by the weight \(w_{1}=1\), and the parameter \(l_{e}\) is equal to the truncation factor \(\delta_{0}\) and the triangle \(\mathbf{P}_{0}\mathbf{P}_{1}\mathbf{P}_{2}\) specified as shown in Figure 3a. Due to the lack of data for such Timoshenko beams in the published literature, the results are compared with those obtained using the Rayleigh-Ritz method within the Euler-Bernoulli beam theory in [2]. Table 7 presents the first three dimensionless frequencies \(\overline{\omega}=\omega(L-\delta_{0})^{2}\sqrt{\frac{\rho\,\delta_{0}}{\epsilon \delta_{0}}}\) for both the reference Euler-Bernoulli and present Timoshenko beams with a slenderness ratio \(L/h_{0}=50\) under three different boundary conditions and different truncation factors. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{\delta_{0}}\)} & **BCs** & \multicolumn{3}{c}{**C-C**} & \multicolumn{3}{c}{**S-S**} & \multicolumn{3}{c}{**F-C**} \\ \cline{3-10}  & **Mode** & **[**2**]** & **DTM** & \(\mathbf{\Delta^{*}}\)**, \%** & **[**2**]** & **DTM** & \(\mathbf{\Delta_{\text{}}}\)**, \%** & **[**2**]** & **DTM** & \(\mathbf{\Delta_{\text{}}}\)**, \%** \\ \hline \multirow{3}{*}{0.1} & 1 & 14.368 & 14.35795 & 0.07 & 6.3778 & 6.379480 & 0.03 & 5.3883 & 5.371650 & 0.31 \\  & 2 & 40.479 & 40.36953 & 0.27 & 27.310 & 27.28302 & 0.10 & 20.699 & 20.55328 & 0.70 \\  & 3 & 80.211 & 79.77238 & 0.55 & 61.023 & 60.84181 & 0.30 & 48.471 & 47.93399 & 1.11 \\ \hline \multirow{3}{*}{0.2} & 1 & 15.972 & 15.9348 & 0.23 & 7.0594 & 7.054580 & 0.07 & 4.8654 & 4.862540 & 0.06 \\  & 2 & 44.578 & 44.3374 & 0.54 & 29.395 & 29.32546 & 0.24 & 20.372 & 20.32497 & 0.23 \\  & 3 & 87.897 & 87.0621 & 0.95 & 65.800 & 65.45681 & 0.52 & 49.870 & 49.61366 & 0.51 \\ \hline \multirow{3}{*}{0.3} & 1 & 17.176 & 17.11698 & 0.34 & 7.5823 & 7.575210 & 0.09 & 4.5179 & 4.514900 & 0.07 \\  & 2 & 47.701 & 47.32308 & 0.79 & 31.082 & 30.97545 & 0.34 & 20.420 & 20.35704 & 0.31 \\ \cline{1-1}  & 3 & 93.828 & 92.52728 & 1.39 & 69.692 & 69.16329 & 0.76 & 51.543 & 51.17858 & 0.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Dimensionless natural frequencies of homogeneous beams with double parabolic taper. Figure 3: The geometries of double-parabolic taper beams truncated at the left end: (**a**) Parabola branches directed to the right; (**b**) Parabola branches directed downward. One can observe from Table 7 that the eigenfrequencies closely match for both types of beams when the truncation factor is small. However, as the truncation factor increases, discrepancies between them become more pronounced, especially for the third mode. This difference is anticipated because shorter beams necessitate accounting for shear deformation, which the simpler Euler-Bernoulli beam model does not address. The second benchmark example involves a beam with a parabolic-tapered height and constant width, truncated at the left end and tapered to a point at the right end (\(\delta_{1}=0\)), as illustrated in Figure 2(b). A detailed specification of this beam configuration is provided in [3]. The analytical presentation of this shape of the parabolic beam with Bezier curves is discussed in Section 2.1.