# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### Inhomogeneous Timoshenko Beams with Uniform Cross-Sections
The parameters of the AFGM beams are the same as those in [15] and are listed in Table 4.  
The first four dimensionless natural frequencies \(\overline{\omega}=\omega L^{2}\sqrt{\frac{\rho_{Zr}A}{E_{Zr}J}}\) of the AFGM beams under C-F, S-S, and C-C boundary conditions for various indices \(\beta\), compared with the solutions obtained using the differential quadrature element method in [15], are shown in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**BCs**} & \multirow{2}{*}{\(\beta\)} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**2**} & \multicolumn{2}{c}{**3**} \\ \cline{3-13}  & & **Mode** & [15] & **DTM** & \(\Delta\)**\({}^{*}\)**, \%** & [15] & **DTM** & \(\Delta\)**, \%** & [15] & **DTM** & \(\Delta\)**, \%** \\ \hline \multirow{4}{*}{C-F} & 1 & 3.89630 & 3.896310 & \(2.6\times 10^{-4}\) & 3.88280 & 3.883120 & \(8.2\times 10^{-3}\) & 3.79570 & 3.797640 & \(5.4\times 10^{-2}\) \\  & 2 & 15.0505 & 15.05049 & \(6.0\times 10^{-4}\) & 15.2590 & 15.25982 & \(6.0\times 10^{-3}\) & 15.2936 & 15.30040 & \(4.5\times 10^{-2}\) \\  & 3 & 30.9409 & 30.94087 & \(1.2\times 10^{-3}\) & 31.6180 & 31.61871 & \(3.2\times 10^{-3}\) & 31.9962 & 32.00717 & \(3.6\times 10^{-2}\) \\  & 4 & 46.3782 & 46.37814 & \(1.2\times 10^{-3}\) & 47.6349 & 47.63506 & \(1.6\times 10^{-3}\) & 48.33355 & 48.344272 & \(2.4\times 10^{-2}\) \\ \hline \multirow{4}{*}{S-S} & 1 & 7.84590 & 7.845920 & \(2.6\times 10^{-4}\) & 7.98730 & 7.988010 & \(1.0\times 10^{-2}\) & 8.06450 & 8.069630 & \(6.5\times 10^{-2}\) \\  & 2 & 23.9406 & 23.94060 & \(8.4\times 10^{-4}\) & 24.2521 & 24.25241 & \(2.1\times 10^{-3}\) & 24.3532 & 24.35745 & \(1.9\times 10^{-2}\) \\  & 3 & 41.6789 & 41.67887 & \(1.4\times 10^{-3}\) & 42.3024 & 42.30265 & \(2.0\times 10^{-3}\) & 42.5010 & 42.50566 & \(1.2\times 10^{-2}\) \\  & 4 & 53.7111 & 53.71108 & \(2.0\times 10^{-3}\) & 54.87717 & \(54.87722\) & \(2.2\times 10^{-3}\) & 55.3881 & 55.38793 & \(1.7\times 10^{-3}\) \\ \hline \multirow{4}{*}{C-C} & 1 & 12.9065 & 12.90653 & \(1.0\times 10^{-3}\) & 12.6873 & 12.68758 & \(3.8\times 10^{-3}\) & 12.6030 & 12.60572 & \(2.3\times 10^{-2}\) \\  & 2 & 26.7597 & 26.75967 & \(1.4\times 10^{-3}\) & 26.64146 & 26.64177 & \(1.8\times 10^{-3}\) & 26.5809 & 26.58381 & \(1.2\times 10^{-2}\) \\ \cline{1-1}  & 3 & 43.0417 & 43.04166 & \(1.5\times 10^{-3}\) & 43.3694 & 43.36938 & \(1.3\times 10^{-3}\) & 43.4958 & 43.49723 & \(4.7\times 10^{-3}\) \\ \cline{1-1}  & 4 & 58.5205 & 58.52006 & \(7.9\times 10^{-4}\) & 59.2779 & 59.27826 & \(2.0\times 10^{-3}\) & 59.5726 & 59.57154 & \(2.7\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Dimensionless natural frequencies of uniform power law AFGM beams. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**BCs**} & \multirow{2}{*}{\(\beta\)} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**2**} & \multicolumn{2}{c}{**3**} \\ \cline{3-13}  & & **Mode** & [15] & **DTM** & \(\Delta\)**\({}^{*}\)**, \%** & [15] & **DTM** & \(\Delta\)**, \%** & [15] & **DTM** & \(\Delta\)**, \%** & [15] & **DTM** & \(\Delta\)**, \%** \\ \hline \multirow{4}{*}{C-F} & 1 & 3.89630 & 3.896310 & \(2.6\times 10^{-4}\) & 3.88280  
Inspecting Table 6 reveals excellent agreement between the first four frequencies obtained in this study and those reported in the referenced research across all boundary conditions and power indices. It should be noted that throughout this series of examples, the proposed approach demonstrates high accuracy, efficiency, and rapid convergence. Only thirty terms in the series expansions (14) are sufficient to yield accurate results for the first several natural frequencies.