# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 6 Parametric Investigation
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(V_{CNT}^{*}\)} & \multicolumn{8}{c}{Modes} \\ \cline{2-11}  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \multicolumn{11}{c}{\(V_{CNT}^{*}\) in both beams is 0.12} \\ \hline
0.12 & 7.2572 & 13.217 & 23.18 & 23.678 & 23.952 & 35.684 & 46.194 & 46.352 & 48.478 & 61.03 \\ \hline \multicolumn{11}{c}{\(V_{CNT}^{*}\) at the top side of both beams is 0.12 and varies at the bottom beam} \\ \hline
0.17 & 8.0452 & 14.644 & 23.46 & 26.546 & 28.234 & 39.526 & 46.428 & 53.038 & 56.246 & 67.324 \\
0.28 & 9.3562 & 16.973 & 23.574 & 30.798 & 35.456 & 45.702 & 46.478 & 60.972 & 69.564 & 70.608 \\ \hline \multicolumn{11}{c}{\(V_{CNT}^{*}\) at the bottom side of both beams is 0.12 and varies at the top beam} \\ \hline
0.17 & 8.0452 & 14.644 & 23.46 & 26.546 & 28.234 & 39.526 & 46.428 & 53.038 & 56.246 & 67.324 \\
0.28 & 9.3562 & 16.973 & 23.574 & 30.798 & 35.456 & 45.702 & 46.478 & 60.972 & 69.564 & 70.608 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The first six dimensionless frequency of the system when the CNTs volume of one beam of system changed. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(V_{CNT}^{*}\)} & \multicolumn{8}{c}{Modes} \\ \cline{2-11}  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \multicolumn{11}{c}{Totally Matrix Beam} \\ \hline
0 & 1.1917 & 2.168 & 3.8106 & 3.8894 & 3.9268 & 5.848 & 7.5844 & 7.6196 & 7.9462 & 9.996 \\ \hline bottom & \multicolumn{8}{c}{Top Beam Made by pure Matrix} \\ \hline
0.12 & 3.8044 & 5.2178 & 7.6306 & 9.6256 & 11.439 & 15.219 & 16.913 & 19.022 & 19.156 & 20.510 \\
0.17 & 3.8066 & 6.2744 & 7.6400 & 11.311 & 11.675 & 15.224 & 19.022 & 19.173 & 19.501 & 21.104 \\
0.28 & 3.8082 & 7.5500 & 7.9988 & 11.418 & 14.532 & 15.225 & 19.022 & 19.201 & 20.186 & 22.138 \\ \hline top & \multicolumn{8}{c}{Bottom Beam Made by pure Matrix} \\ \hline
0.12 & 3.8044 & 5.2178 & 7.6306 & 9.6256 & 11.439 & 15.219 & 16.913 & 19.022 & 19.156 & 20.510 \\
0.17 & 3.8066 & 6.2744 & 7.6400 & 11.311 & 11.675 & 15.224 & 19.022 & 19.173 & 19.501 & 21.104 \\
0.28 & 3.8082 & 7.5500 & 7.9988 & 11.418 & 14.532 & 15.225 & 19.022 & 19.201 & 20.186 & 22.138 \\ \hline \end{tabular}
\end{table} 
Table 7: Dimensionless frequency for system which one of beams is made by matrix and another beam reinforced by different volumes of CNTs. under the same geometrical assumptions \(\text{R}_{1}=\text{R}_{2}=10\), \(\text{R}/\text{h}=10\), \(L_{1}=L_{2}=2\,R\,\sin\left(\frac{\theta}{2}\right)\), and \(k_{s}\) equal to 100. In this example, six values of \(\theta\) are employed, where the frequencies of the system are computed for three different types of boundary condition. Note that by increasing \(\theta\), the first frequency of the system decreases. For example, for C-C/C-C systems with a curvature factor of \(15^{\circ}\), the frequency value is about thirty-nine and it tends to the unit value for \(\theta=180^{\circ}\). In this section, we investigate the thermal-dependent vibration of the system. There are several studies in which the effect of temperature variation on the structural behavior has been investigated [79, 80]. Thus far, the mechanical properties of SWCNTs (10,10) and PMMA have been represented at room temperature (300 \({}^{\circ}\)K). For other environmental temperatures, we consider the following expressions to define the thermal-dependent mechanical properties for SWCNTs:  
\[