# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### Inhomogeneous Timoshenko Beams with Uniform Cross-Sections  
Next, the ability of the proposed approach to accurately predict the frequencies of beams with uniform cross-sections made of inhomogeneous materials is evaluated. We assume that the beam's mass density and modulus of elasticity follow power law distributions of _Type 1_ as described in (3). Since natural frequencies for Timoshenko beams under these conditions are not documented in the literature, we will compare our results with those obtained using the Euler-Bernoulli beam theory in [14] for AFGM beams with a slenderness ratio of \(L/h=50\). The geometrical and material parameters are consistent with those detailed in [14] (see Tables 4-6 and Appendix B in [14]). The comparison of the first five dimensionless frequencies \(\overline{\omega}=\omega L^{2}\sqrt{\frac{\rho_{0}A}{E_{0}}}\) for the cantilever AFGM beam is presented in Table 3. Furthermore, to extend this study to beams, where shear deformation plays a significant role, in Table 3, we also provide the natural frequencies computed for beams with a slenderness ratio of 10. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{L}\), \(\mathbf{m}\) & \(\mathbf{b}\), \(\mathbf{m}\) & \(\mathbf{h}\), \(\mathbf{m}\) & \(\mathbf{E}\), \(\mathbf{Pa}\) & \(\mathbf{G}\), \(\mathbf{Pa}\) & \(\mathbf{\
u}\) & \(\mathbf{\kappa}\) & \(\mathbf{\rho}\), \(\mathbf{kg/m^{3}}\) \\ \hline
1 & 0.02 & 0.1 & \(2.1\times 10^{11}\) & \(0.875\times 10^{11}\) & 0.2 & 0.845 & 7860 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of uniform beams. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**S-S**} & \multicolumn{4}{c}{**C-C**} & \multicolumn{4}{c}{**C-F**} \\ \cline{2-9}  & **DTM** & [48] & \(\mathbf{\Delta}\) *, \% & **DTM** & [48] & \(\mathbf{\Delta}\), \% & **DTM** & [48] & \(\mathbf{\Delta}\), \% \\ \hline
1 & 230.7888 & 230.7890 & \(6.9\times 10^{-5}\) & 500.5870 & 500.5870 & \(3.6\times 10^{-6}\) & 82.88577 & 82.88600 & \(2.8\times 10^{-5}\) \\
2 & 884.4691 & 884.4690 & \(1.3\times 10^{-5}\) & 1288.763 & 1288.763 & \(1.7\times 10^{-5}\) & 498.1911 & 498.1910 & \(2.9\times 10^{-5}\) \\
3 & 1870.358 & 1870.358 & \(1.4\times 10^{-5}\) & 2339.160 & 2339.160 & \(7.0\times 10^{-6}\) & 1315.041 & 1315.041 & \(1.4\times 10^{-5}\) \\
4 & 3091.056 & 3091.056 & \(4.4\times 10^{-6}\) & 3565.013 & 3565.013 & \(9.4\times 10^{-6}\) & 2394.776 & 2394.776 & \(7.0\times 10^{-6}\) \\
5 & 4468.225 & 4468.225 & \(9.5\times 10^{-6}\) & 4910.439 & 4910.438 & \(2.3\times 10^{-5}\) & 3658.662 & 3658.662 & \(5.9\times 10^{-6}\) \\
6 & 5946.852 & 5946.679 & \(2.9\times 10^{-3}\) & 6338.321 & 6337.691 & \(9.9\times 10^{-3}\) & 5044.291 & 5044.277 & \(2.9\times 10^{-3}\) \\
7 & 7472.753 & 7489.650 & \(2.3\times 10^{-1}\) & 7822.471 & 7821.911 & \(7.2\times 10^{-3}\) & 6508.492 & 6509.954 & \(2.3\times 10^{-2}\) \\ \hline \hline \end{tabular} \({}^{\star}\)\(\Delta=\frac{|\omega_{\text{DSM}}-\omega_{\text{DTM}}|}{\omega_{\text{DSM}}}\times 100\%\).  
\end{table} 
Table 2: The natural frequencies of uniform S-S, C-C, and C-F homogeneous beams based on different methods (Hz). Upon reviewing Table 3, it becomes evident that both Euler-Bernoulli and Timoshenko beams with slenderness ratio of 50 demonstrate excellent agreement for the fundamental frequency and show slight deviations for higher frequencies. This is because the Euler-Bernoulli beam theory overestimates especially higher frequencies due to its neglect of shear effect. Specifically, when the slenderness ratio is 10, the fundamental frequencies of the beams show only satisfactory matching, while significant discrepancies are observed in higher-order frequencies, consistent with theoretical expectations. In the second example, we evaluate the accuracy of predicted natural frequencies for exponential AFGM beams. The current results are compared with those obtained in [11] using the Chebyshev collocation method.