# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 7 Conclusions
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\text{{}^{\text{{}^{\text{{}^{\text{\text{{}^{\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text{\text{ \text{ }}}}}}}}}}}}}}}}}}\) & \multicolumn{8}{c}{T\({}_{\text{top beam}}\)= 300 \({}^{\circ}\)K} & \multicolumn{8}{c}{T\({}_{\text{bottom beam}}\)= 300 \({}^{\circ}\)K} \\ \cline{2-13}  & \multicolumn{4}{c}{T\({}_{\text{bottom beam}}\) (\({}^{\circ}\)K)} & \multicolumn{8}{c}{T\({}_{\text{top beam}}\) (\({}^{\circ}\)K)} \\ \cline{2-13}  & 300 & 400 & 600 & 800 & 1000 & 300 & 400 & 600 & 800 & 1000 \\ \hline \multirow{8}{*}{\(\text{{}^{\text{{}^{\text{{}^{\text{{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{  \text{   \text{   \text{     }}}}}}}}}}}}}}}}\) & 1 & 5.6356 & 5.6120 & 5.5878 & 5.5590 & 5.5142 & 5.6356 & 6.0376 & 7.2624 & 9.8268 & 25.168 \\  & 2 & 9.2200 & 9.1812 & 9.1418 & 9.0944 & 9.0212 & 9.2200 & 9.8776 & 11.881 & 16.077 & 41.172 \\  & 3 & 19.493 & 19.411 & 19.327 & 19.228 & 19.072 & 19.493 & 20.884 & 25.120 & 33.990 & 87.022 \\  & 4 & 22.848 & 22.734 & 22.584 & 22.380 & 22.036 & 22.848 & 24.454 & 29.340 & 39.512 & 99.016 \\  & 5 & 23.324 & 23.244 & 23.194 & 23.160 & 23.132 & 23.324 & 25.006 & 30.144 & 40.928 & 105.02 \\  & 6 & 31.408 & 31.276 & 31.142 & 30.982 & 30.732 & 31.408 & 33.648 & 40.474 & 54.764 & 140.12 \\  & 7 & 43.854 & 43.666 & 43.472 & 43.228 & 42.830 & 43.854 & 46.978 & 56.498 & 76.408 & 140.61 \\  & 8 & 45.688 & 45.444 & 45.128 & 44.710 & 44.052 & 45.688 & 48.886 & 58.626 & 78.936 & 145.90 \\  & 9 & 46.250 & 46.110 & 46.040 & 45.998 & 45.96 & 46.25 & 49.606 & 59.832 & 81.262 & 159.43 \\  & 10 & 57.446 & 57.204 & 56.958 & 56.66 & 56.198 & 57.446 & 61.544 & 74.024 & 100.15 & 193.62 \\ \hline \multirow{8}{*}{\(\text{{}^{\text{{}^{\text{{\text{\text{\text{\text{\text{\texttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttext{    \texttexttexttexttexttexttexttexttexttexttext{    \texttexttexttexttexttexttexttexttexttexttexttext{   \texttexttexttexttexttexttexttext{   \texttexttexttexttexttexttexttext{  \texttexttexttext{  \texttexttexttexttexttexttext{  \texttexttexttext{ \texttexttexttexttext{  \texttexttexttexttext{  \texttexttext{  \texttexttexttexttexttext{  \texttexttexttexttext{  \texttexttexttexttext{  \texttexttexttexttextdistribution of elastic springs both in the normal and tangential direction. Each arm of the coupled specimen is made of a PMMA matrix reinforced with SWCNTs with a considerable and effective improvement in the vibrational response of the system. The equivalent temperature-dependent mechanical properties of the nanocomposite material have been obtained based on the well-known rule of mixture and approximate temperature-dependent formulation. Various distribution patterns of the reinforcement phase have been considered throughout the specimen, whose natural frequencies are determined by applying the GDQ technique as useful tool to discretize the governing differential equations of the problem, here determined according to a FSDT. A comparative analysis of the results against further theoretical predictions in the literature has confirmed the correctness and accuracy of the proposed procedure, where a large parametric investigation has evaluated the sensitivity of the vibrational response to different key input parameters, both in a geometric and mechanical sense--primarily, the curvature and the boundary conditions, along with the CNT distribution patterns, the interfacial stiffness, and the thermal dependence of the mechanical properties. It was observed that there is strong agreement between the obtained results and the responses from previous research (less than 5%). Some specific results can be pointed out:  
* The system frequency declines with an increased value of \(R/\hbar\). This holds true for all boundary conditions and various volume fractions of CNTs.