# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### AFGM Timoshenko Beams with Curvilinearly Tapered Cross-Sections
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Material** & **E, GPa** & \(\rho\)**, **kg/m\({}^{3}\)** & \(v\)** & \(L\)**, **m** & \(b_{0}\)**, **m** & \(h_{0}\)**, **m** & \(l_{e}\)**, **m** & \(l_{f}\)**, **m** \\ \hline Aluminum & 70 & 2700 & 0.3 & & & & & & \\ Steel & 348.43 & 2370 & 0.24 & 1 & 0.2 & 0.2 & 0.04 & 0.1 \\ Si\({}_{3}\)N\({}_{4}\) & 201.04 & 8166 & 0.33 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The parameters of curvilinearly non-uniform AFGM beams. density (\(m=2\)) as described in (3). Detailed specifications of these beam configurations are provided in [23]. While benchmark results for the fundamental frequencies of these beams are available in [23], higher-order frequencies are absent in the existing literature. Therefore, the results obtained using the DTM are validated against finite element analysis for these beams, as discussed in Section 3.2. The comparison of first five dimensionless bending frequencies \(\overline{\omega}=\omega L^{2}\sqrt{\frac{\rho_{\omega}A_{0}}{\rho_{0}J_{0}}}\), depending on the truncation factor \(\delta_{1}\) for the two types of boundary conditions is shown in Table 11.  
It is observed from Table 11 that the differences between dimensionless fundamental frequencies obtained from the present analysis, published results, and FEM predictions are quite small. However, the higher frequencies show more significant deviations between the DTM and FEM solutions. Likely, these discrepancies between the two sets of data are attributed to methodological differences in formulation and solution approach. Overall, these findings confirm the reliability of the DTM approach within acceptable limits. As mode shapes play a crucial role in evaluating the accuracy of structural dynamics modeling, the first five bending mode shapes associated with the appropriate frequencies of AFGM beams with \(\delta_{1}=0.2\) for C-C and C-F boundary conditions, as listed in Table 11 and obtained using the DTM, are plotted in Figure 4a,b, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**AFGM**} & \multirow{2}{*}{**Model**} & \multirow{2}{*}{\(\gamma_{e}\)/\(\gamma_{c}\)} & \multicolumn{3}{c}{**0**} & \multicolumn{3}{c}{**1**} & \multicolumn{3}{c}{**5**} \\ \cline{3-13}  & & & **Mode** & [29] & **DTM** & \(\Delta^{*}\)**, \%** & [29] & **DTM** & \(\Delta\)**, \%** & [29] & **DTM** & \(\Delta\)**, \%** \\ \hline \multirow{3}{*}{_Type 2_} & \multirow{2}{*}{Plump} & 1 & 889.39 & 887.304 & 0.23 & 891.48 & 889.589 & 0.21 & 997.39 & 1002.94 & 0.56 \\  & & 2 & 1889.1 & 2034.36 & 7.69 & 1894.4 & 2041.25 & 7.75 & 2025.3 & 2203.40 & 8.79 \\ \cline{2-13}  & \multirow{2}{*}{Slender} & 1 & 800.93 & 779.141 & 2.72 & 805.63 & 783.893 & 2.70 & 920.25 & 899.469 & 2.26 \\  & & 2 & 1774.6 & 1728.28 & 2.61 & 1778.8 & 1732.06 & 2.63 & 1882.4 & 1838.13 & 2.35 \\ \hline \multirow{3}{*}{_Type 3_} & \multirow{2}{*}{Plump} & 1 & 2128.3 & 2133.71 & 0.25 & 1229.4 & 1226.23 & 0.26 & 947.74 & 947.074 & 0.07 \\  & & 2 & 4532.2 & 4911.19 & 8.36 & 2632.4 & 2837.54 & 7.79 & 2036.2 & 2199.87 & 8.04 \\ \cline{1-1} \cline{2-13}  & \multirow{2}{*}{Slender} & 1 & 1906.2 & 1865.29 & 2.15 & 1108.2 & 1078.24 & 2.70 & 851.24 & 828.860 & 2.63 \\  & & 2 & 4239.8 & 4151.87 & 2.07 & 2471.8 & 2406.75 & 2.63 & 1902.5 & 1856.66 & 2.41 \\ \hline \multirow{3}{*}{_Type 4_} & \multirow{3}{*}{Slender} & \multirow{3}{*}{\(\Delta=\frac{\left|\omega_{Ref}-\omega_{DTM}\right|}{\omega_{Ref}}\times 1 00\%\).} & \multicolumn{3}{c}{} & & & & & & & & \\ \cline{1-1} \cline{4-13}  & & & & & & & & & & & & & \\ \end{tabular}
\end{table} 
Table 10: Natural frequencies, Hz, of the C-C AFGM beams with curvilinear taper in height and constant width.