# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### AFGM Timoshenko Beams with Curvilinearly Tapered Cross-Sections  
This subsection presents the performance of the DTM for frequency analysis of inhomogeneous curvilinearly tapered beams of convex-up and concave-down shapes, referred to as plump and slender models in [29] and illustrated in Figure 1a,b, respectively. The material properties of the beams vary axially according to the laws specified in (4) for _Type 2_ and (5) for _Type 3_ distributions. Detailed geometric and material properties of the beams can be found in [29] and are summarized in Table 9.  
First, we analyze beams with varying thickness and constant width for both _Type 2_ and _Type 3_ material laws. The exponential law (4) with parameter \(a=1\) and gradient indices \(\gamma_{e}=0,\ 1,\ 5\) is applied to beams made of aluminum, whereas the power law (5) with parameter \(b=1\) and gradient indices \(\gamma_{c}=0,\ 1,\ 5\) is utilized for beams composed of a combination of steel (SUS304) and ceramic (Si\({}_{3}\)N\({}_{4}\)) phases, as noted in [29]. Table 10 presents the first two frequencies associated with the bending modes of the curvilinear taper beams with C-C boundary conditions. These frequencies are compared with those computed using NURBS in the 3-D finite element analysis conducted in [29]. The current results align satisfactory with the reference data, demonstrating differences of less than 0.6% for the fundamental frequency and less than 9% for the second frequency in the plump beams. For slender beams, the discrepancies are approximately 3% for both frequencies. These deviations arise from differences in modeling approaches. The current model uses a line beam structure with a first-order shear theory, whereas the IGA analysis in [29] employs a more detailed 3-D beam modeling approach. Next, we analyze the parabolic convex beams depicted in Figure 3b. These beams feature a parabolic taper in height and maintain a constant width. They are truncated at the left end with a fixed factor \(\delta_{0}=0.5\), while the right end is characterized by a variable truncation factor \(\delta_{1}\). The beam ends are subjected to C-C and C-F constraints. The material constants, given in Table 1, vary along the beam length according to the _Type 1_ material law with a linear distribution for [PERSON]'s modulus (\(n=1\)) and a parabolic distribution for mass \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**(3)**} & \multirow{2}{*}{**DTM**} & \multirow{2}{*}{\(\Delta\)**\({}^{*}\), \%**} & \multirow{2}{*}{**[3]**} & **DTM** & \multirow{2}{*}{\(\Delta\)**, \%**} & \multirow{2}{*}{**[3]**} & **DTM** & \multirow{2}{*}{\(\Delta\)**, \%**} & \multirow{2}{*}{**[3]**} & **DTM** & \multirow{2}{*}{\(\Delta\)**, \%**} & \multirow{2}{*}{**[3]**} & **DTM** & \multirow{2}{*}{\(\Delta\)**, \%**} & \multirow{2}{*}{**[3]**} & **DTM** & \multirow{2}{*}{\(\Delta\)**, \%**} \\ \hline \(\delta\) & & **0.65** & & & **0.7** & & & **0.75** & & & **0.8** & & & **0.85** \\ \hline
1 & 10.17 & 10.1532 & 0.16 & 12.72 & 12.6888 & 0.25 & 16.27 & 16.2322 & 0.23 & 21.60 & 21.5566 & 0.20 & 30.47 & 30.3768 & 0.31 \\
2 & 34.47 & 34.2610 & 0.61 & 41.78 & 41.5124 & 0.64 & 51.99 & 51.6294 & 0.69 & 67.26 & 66.6661 & 0.88 & 92.66 & 91.9427 & 0.77 \\
3 & 71.29 & 70.5089 & 1.10 & 85.72 & 84.7297 & 1.16 & 105.9 & 104.568 & 1.26 & 136.0 & 134.452 & 1.14 & 186.2 & 183.606 & 1.39 \\
4 & 120.5 & 118.443 & 1.71 & 144.4 & 141.823 & 1.78 & 177.8 & 174.436 & 1.89 & 227.8 & 222.612 & 2.28 & 310.9 & 304.354 & 2.11 \\ \hline \hline \end{tabular} \({}^{*}\Delta=\frac{\left|\omega_{Ref}-\omega_{DM}\right|}{\omega_{Ref}} \times 100\%\).  
\end{table} 
Table 8: _Cont._