# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### Homogeneous Timoshenko Beams with Curvilinearly Tapered Cross-Sections
Once again, due to limited availability of data for Timoshenko beams in the existing literature, comparisons are made with results obtained using hypergeometric functions within the framework of the Euler-Bernoulli beam theory, as described in [3]. Table 8 displays the first four dimensionless frequencies \(\overline{\omega}=\omega(L/2)^{2}\sqrt{\frac{\rho A_{0}}{E_{0}}}\) of cantilever Euler-Bernoulli and Timoshenko beams with a slenderness ratio \(L/h_{0}=50\) for various truncation factors. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\delta_{0}\)} & \multirow{2}{*}{**BCs**} & \multicolumn{4}{c}{**C-C**} & \multicolumn{4}{c}{**S-S**} & \multicolumn{4}{c}{**F-C**} \\ \cline{3-14}  & & **Mode** & **[2]** & **DTM** & \(\Delta^{*}\)**, \%** & **[2]** & **DTM** & \(\Delta\)**, \%** & **[2]** & **DTM** & \(\Delta\)**, \%** \\ \hline \multirow{4}{*}{0.4} & 1 & 18.175 & 18.08114 & 0.52 & 8.0198 & 8.008940 & 0.14 & 4.2660 & 4.262320 & 0.09 \\  & 2 & 50.325 & 49.73045 & 1.18 & 32.561 & 32.39500 & 0.51 & 20.592 & 20.50043 & 0.44 \\  & 3 & 98.855 & 96.82087 & 2.06 & 73.098 & 72.27499 & 1.13 & 53.199 & 52.65124 & 1.03 \\ \hline \multirow{4}{*}{0.5} & 1 & 19.044 & 18.89066 & 0.81 & 8.4018 & 8.384520 & 0.21 & 4.0729 & 4.067820 & 0.12 \\  & 2 & 52.633 & 51.66934 & 1.83 & 33.904 & 33.63600 & 0.79 & 20.812 & 20.67111 & 0.68 \\  & 3 & 103.30 & 100.0365 & 3.16 & 76.179 & 74.85468 & 1.74 & 54.785 & 53.92227 & 1.57 \\ \hline \multirow{4}{*}{0.6} & 1 & 19.852 & 19.55547 & 1.49 & 8.7443 & 8.71463 & 0.34 & 3.9188 & 3.9112 & 0.19 \\  & 2 & 54.718 & 53.06449 & 3.02 & 35.148 & 34.68662 & 1.31 & 21.052 & 20.81738 & 1.11 \\  & 3 & 107.34 & 101.82207 & 5.14 & 79.021 & 76.76144 & 2.86 & 56.296 & 54.84588 & 2.58 \\ \hline \multirow{4}{*}{0.7} & 1 & 20.531 & 20.01665 & 2.51 & 9.0569 & 8.99939 & 0.63 & 3.7922 & 3.77915 & 0.34 \\  & 2 & 56.634 & 53.51562 & 5.51 & 36.315 & 35.43022 & 2.44 & 21.299 & 20.85956 & 2.06 \\  & 3 & 111.06 & 96.84761 & 12.80 & 81.676 & 77.40895 & 5.22 & 57.736 & 55.03705 & 4.67 \\ \hline \multirow{4}{*}{0.8} & 1 & 21.186 & 19.98305 & 5.68 & 9.3459 & 9.207100 & 1.49 & 3.6859 & 3.493240 & 5.23 \\  & 2 & 58.417 & 43.48449 & 25.6 & 37.419 & 35.37001 & 5.48 & 21.547 & 19.53236 & 9.35 \\ \cline{1-1}  & 3 & 114.54 & 69.00506 & 39.8 & 84.180 & 51.02468 & 39.4 & 59.112 & 63.26979 & 7.03 \\ \hline \hline \end{tabular} \({}^{*}\Delta=\frac{|\omega_{RF}-\omega_{DTM}|}{\omega_{RF}}\times 100\%\).  
\end{table} 
Table 7: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\delta_{0}\)} & \multirow{2}{*}{**0.15**} & \multirow{2}{*}{**0.2**} & \multirow{2}{*}{**0.25**} & \multirow{2}{*}{**0.3**} & \multirow{2}{*}{**0.3**} & \multirow{2}{*}{**0.35**} & \multirow{2}{*}{**0.2**} & \multirow{2}{*}{**0.2**} & \multirow{2}{*}{**0.2**} & \multirow{2}{*}{**0.3**} & \multirow{2}{*}{**0.35**} \\ \cline{2-14}  & & **[3]** & **DTM** & \(\Delta^{*}\)**, \%** & **[3]** & **DTM** & \(\Delta_{*}\)**, \%** & **[3]** & **DTM** & \(\Delta\)**, \%** & **[3]** & **DTM** & \(\Delta\)**, \%** \\ \hline
1 & 1.050 & 1.05057 & 0.05 & 1.469 & 1.46748 & 0.10 & 1.936 & 1.93539 & 0.03 & 2.465 & 2.46322 & 0.07  
From Table 8, one can see that the compared natural frequencies exhibit close agreement. Similar to the previous example, as the beam length decreases, discrepancies between the present and benchmarked results increase, especially for higher modes, due to the greater influence of shear deformation, as discussed earlier. Therefore, these examples thoroughly confirm the reliability of the proposed computational method for analyzing curvilinear tapered beams.