# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 5 Conclusions  
Free vibration analysis of axially functionally graded beams with curvilinear tapered cross-sections and various material distributions is performed using the semi-analytical differential transform method. Unlike the standard DTM procedure, which typically imposes boundary conditions on transformed images of unknown functions, the proposed DTM approach combines various boundary conditions subjected directly to the original continuous problem formulation. The non-uniform beams shaped by conic sections, are analytically represented using rational Bezier curves within the Grobner basis. The equations of motion for free vibration are derived using the principle of virtual work under the assumptions of Timoshenko beam theory. It is assumed that both the material properties of the beam and its geometric features vary continuously along the beam's axial direction. Comparisons are presented to validate the accuracy and assess the efficiency of the proposed computational technique. The computed examples encompass both documented cases and previously unexamined scenarios, ranging from uniform homogeneous and inhomogeneous beams to curvilinearly double-tapered axially functionally graded beams. In cases where benchmark results are unavailable in the literature, the DTM solutions are compared with finite element simulations using ABAQUS, incorporating material gradients into the 3-D C3D20 elements via a user-defined subroutine UMAT. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**AFGM**} & \multirow{2}{*}{**Model**} & \multirow{2}{*}{**BCs**} & \multicolumn{3}{c}{**C-C**} & \multicolumn{3}{c}{**C-S**} & \multicolumn{3}{c}{**C-F**} & \multicolumn{3}{c}{**S-S**} \\ \cline{3-16}  & & & **Mode** & **[**29**]** & **DTM** & **DTM \({}^{*}\)** & **[**29**]** & **DTM** & **DTM \({}^{*}\)** & **[**29**]** & **DTM** & **DTM \({}^{*}\)** & **[**29**]** & **DTM** & **DTM \({}^{*}\)** \\ \hline \multirow{4}{*}{_Type 2_} & Plump & 1 & 1.1001 & 1.09774 & 1.03149 & 0.8392 & 0.83336 & 0.80370 & 0.1669 & 0.16736 & 0.16262 & 0.6856 & 0.67765 & 0.66811 \\  & & 2 & 2.3377 & 2.51889 & 2.42384 & 1.5533 & 2.31624 & 2.25450 & 0.9638 & 1.13797 & 1.12997 & 1.8524 & 2.16694 & 2.13405 \\ \cline{2-16}  & & 1 & 0.9941 & 0.98513 & 1.06936 & 0.5764 & 0.56832 & 0.58619 & 0.1070 & 0.10628 & 0.09795 & 0.3468 & 0.34744 & 0.33893 \\  & & 2 & 2.1951 & 2.15107 & 2.27128 & 1.7610 & 1.74672 & 1.89129 & 0.7314 & 0.7207 & 0.72082 & 1.3943 & 1.4032 & 1.43892 \\ \hline \multirow{4}{*}{_Type 3_} & Plump & 1 & 0.6371 & 0.63543 & 0.59782 & 0.5215 & 0.51757 & 0.49565 & 0.1460 & 0.14617 & 0.13912 & 0.4070 & 0.40223 & 0.39643 \\  & & 2 & 0.8972 & 1.47041 & 1.41596 & 0.9537 & 1.36093 & 1.32281 & 0.6423 & 0.74588 & 0.73694 & 1.0946 & 1.27727 & 1.25659 \\ \cline{2-16}  & & 1 & 0.5743 & 0.55882 & 0.61731 & 0.3808 & 0.37381 & 0.39302 & 0.1008 & 0.09998 & 0.09508 & 0.2065 & 0.20703 & 0.2028 \\ \cline{2-16}  & & 2 & 1.2809 & 1.24838 & 1.31629 & 1.0547 & 1.0332 & 1.06555 & 0.4992 & 0.48937 & 0.49201 & 0.8293 & 0.82974 & 0.82545 \\ \hline \hline \end{tabular}  
* Curvilinear double-tapered beam.  
\end{table} 
Table 14: Dimensionless natural frequencies of curvilinear taper AFGM beams with various boundary conditions. Based on these results, it can be concluded that the proposed DTM formulation excels in modeling the free vibrations of beams with complex geometries and material distributions. The frequency analysis of Timoshenko beams, whether homogeneous or inhomogeneous with exponential and power law material distributions and with uniform cross-sections, reveals that the method achieves high accuracy, efficiency, and rapid convergence. Accurate results for the first few natural frequencies are achieved with only thirty terms in the series solution. For homogeneous truncated beams with a parabolic taper in height or double-parabolic taper, where the published literature lacks frequency data within Timoshenko beam theory, comparisons with Euler-Bernoulli beams indicate close agreement when the truncation factor is small.