# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 5 Validation and Convergence Study  
A numerical algorithm has been developed to compute the natural frequencies of coupled curved CNT beams, where the accuracy of results has been checked against frequency predictions in the literature [13], based on a finite element method (FEM) for different boundary conditions. The geometry of the S-S curved beam is defined as R = 0.75 m, A = 4 m\({}^{2}\), I = 0.01 m\({}^{4}\), and R/h = 4, whereas the C-C example features the geometrical properties R = 0.6366 m, A = 1 m\({}^{2}\), I = 0.0016 m\({}^{4}\), and R/h = 6. The mechanical properties of the structure are set as E = 70 GPa, \(\upsilon\) = 0.4166, \(\rho\) = 2777 kg/m\({}^{3}\), and \(\kappa\) = 0.85, in line with Ref. [13]. Table 2 summarizes the dimensionless frequency for the curved beam obtained in this research compared to results from Ref. [13]. Based on the results, the frequencies computed with our approach are very close to predictions from Ref. [13], thus confirming the accuracy of the proposed formulation. At the same time, [PERSON] and [PERSON] [16] applied the finite element approach to the vibration of coupled arched beams, connected elastically by a layer of springs at the interface, for different boundary conditions. Table 3 validates our method for coupled arched beams with the results from Ref. [16]. In this example, the geometry of the beams is defined as L\({}_{i}\) = 10, R\({}_{1}\) = R\({}_{2}\) = 7.5, and h\({}_{1}\) = h\({}_{2}\) = b\({}_{1}\) = b\({}_{2}\) = 0.1. Material properties are E = 200 GPa, \(\upsilon\) = 0.34, \(\rho\) = 7850 kg/m\({}^{3}\), and \(\kappa\) = 0.87, and mid-layer stiffness is 16,000. Based on Table 3, it is worth observing the very good correspondence among our results and the literature [16] for all boundary conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**S-S**} & \multicolumn{2}{c}{**C-C**} \\ \cline{2-5}  & **This Study** & **Ref. [13]** & **This Study** & **Ref. [13]** \\ \hline
1 & 29.2757 & 29.2850 & 36.7045 & 36.7160 \\
2 & 33.2995 & 33.3210 & 42.2656 & 42.2780 \\
3 & 67.1136 & 67.2020 & 82.2368 & 82.3610 \\
4 & 79.9578 & 80.0490 & 84.4943 & 84.5650 \\
5 & 107.835 & 108.169 & 122.311 & 122.722 \\ \hline \end{tabular}
\end{table} 
Table 2: Dimensionless frequency of the homogenous curved beam. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**C-C/C-C**} & \multicolumn{2}{c}{**C-S/C-S**} & \multicolumn{2}{c}{**S-S/S-S**} & \multicolumn{2}{c}{**C-F/C-F**} \\ \cline{2-9}  & **This Study** & **Ref. [16]** & **This Study** & **Ref. [16]** & **This Study** & **Ref. [16]** \\ \hline
1 & 10.931 & 10.932 & 8.6624 & 8.6624 & 6.7019 & 6.7023 & 2.9378 & 2.9378 \\
2 & 11.342 & 11.342 & 9.1684 & 9.1684 & 7.3345 & 7.3349 & 3.5271 & 3.5271 \\
3 & 20.620 & 20.621 & 18.031 & 18.032 & 15.545 & 15.546 & 4.5757 & 4.5756 \\
4 & 20.860 & 20.861 & 18.304 & 18.304 & 15.860 & 15.861 & 11.003 & 11.003 \\
5 & 37.380 & 37.382 & 33.306 & 33.306 & 29.528 & 29.530 & 11.423 & 11.424 \\ \hline \end{tabular}
\end{table} 
Table 3: Natural frequencies of coupled homogenous curved beams. In accordance to [PERSON] [30], we studied the free vibration of FG-CNTRC curved beams on nonlinear elastic foundations, and a convergence study was done to obtain the optimum value of grid points (N) in GDQM. Based on results, it was found that nine grid points is sufficient to ensure the accuracy of the frequency results for CNT beams. Table 4 also indicates the first three dimensionless frequency \(\left(\omega\frac{L^{2}}{h}\sqrt{\frac{\rho_{0}}{E_{0}}}\right)\) for the CNT curved beam, while comparing successfully them with our results, under the following assumptions for the properties of CNTs and size-dependent coefficient  
\[