# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### AFGM Timoshenko Beams with Curvilinearly Double-Tapered Cross-Sections  
To the best of authors' knowledge, benchmark results for curvilinearly double-tapered AFGM beams are not available in the existing literature. Hence, the results of the present analysis are validated against axial plumb and slender beams with _Type 2_ and _Type 3_ material distributions, as considered in [29], which feature variable thickness and constant width. Further, the analysis will be extended to include beams with double-tapered cross-sectional variations. The geometrical and material parameters are identical to those used in the previous examples of curvilinear taper beams with exponential and power law gradations as presented in [29]. The first two dimensionless frequencies \(\overline{\omega}=\omega L\sqrt{\frac{\rho A}{\varepsilon_{Al}}}\) of axially graded _Type 2_ plump and slender C-C beams with variable thickness and constant width, as well beams with simultaneously variable thickness and width, depending on the exponential law parameter \(a\) and gradient index \(\gamma_{e}\) are provided in Table 12. Satisfactory agreement between the reference and present results for plump and slender AFGM beams with variable thickness is observed. The errors between the frequencies fall within the range observed for frequencies of similar beams with \(a=1\) and \(\gamma_{e}=1\) in Table 10. In particular, minor deviations are noted for the first frequency, while higher discrepancies are observed for the second frequency. Upon reviewing Table 12, it becomes evident that introducing variations in width influences the frequencies of the beams. Specifically, plump double-tapered beams show slightly lower frequencies compared to those with only variable thickness, whereas slender double-tapered beams exhibit a slight increase. However, the trends in frequency changes with increasing gradient index \(\gamma_{e}\) and parameter \(a\) remain consistent for both beams with  
Figure 5: Mode shapes of truncated AFGM beams with parabolic taper in height and constant width obtained from 3-D FEM analysis: (**a**) C-C boundary conditions; (**b**) C-F boundary conditions. only variable thickness and double-tapered beams. The frequencies increase with higher \(\gamma_{c}\) and \(a\). This increase is more pronounced as \(\gamma_{c}\) or \(a\) become larger. Analogously, axially graded _Type 3_ plump and slender C-C beams with variable thickness and constant width and with simultaneously variable thickness and width are examined. Table 13 presents the first two dimensionless frequencies \(\overline{\omega}=\omega L\sqrt{\frac{\overline{\rho}_{\omega}}{E_{ms}}}\) of these beams for different values of power law parameter \(b\) and gradient index \(\gamma_{c}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{\(\gamma_{c}\)} & \multirow{2}{*}{**29**} & \multirow{2}{*}{**DTM**} & \multirow{2}{*}{**DTM** * [29]} & \multirow{2}{*}{**DTM**} & \multirow{2}{*}{**29**} & \multirow{2}{*}{**DTM**} & \multirow{2}{*}{**MT** * [29]} &  
Table 13 shows that the agreement between the results obtained from the IGA analysis in [29] and the present study is satisfactory, with errors comparable to those observed for _Type 3_ plump and slender C-C beams with \(b=1\) and \(\gamma_{c}=1\) in Table 11. Additionally, Table 13 demonstrates that introducing variable width along with variable height has similar effects on the frequencies of both beams with variable thickness only and curvilinearly double-tapered beams, similar to what was observed for _Type 2_ plump and slender C-C beams, as mentioned earlier. On the other hand, the trends of the frequency changes with increasing \(b\) and \(\gamma_{c}\) are found to be opposite to those in _Type 2_ plump and slender C-C beams. In _Type 3_ plump and slender beams, the frequencies decrease with higher values of \(\gamma_{c}\) and \(b\). This decrease begins more rapidly and exhibits a faster decay as \(\gamma_{c}\) or \(b\) increase.