# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 6 Parametric Investigation  
In this section, several examples have been solved to investigate the vibration properties of the system. For all examples, PMMA and SWCNTs (10,10) are used as matrix and reinforcement phase, respectively. More specifically, for CNTs we assume the following properties:  
\[ \begin{array}{l}E_{11}^{CNT}=600\text{ GPa, }\,E_{22}^{CNT}=10\text{ GPa, and }G_{12}^{CNT}=17.2\text{ GPa}\\ \rho_{CNT}=1400\,\frac{\text{kg}}{\text{cm}^{2}},\,\text{and }v_{CNT}=0.19\\ V_{CNT}^{*}=0.12,\,0.17,\,0.28\\ \end{array}  \tag{37}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{CNTs Pattern} & \multirow{2}{*}{\(V_{CNT}^{*}\)} & \multicolumn{5}{c}{This Study} & \multicolumn{5}{c}{Ref. [30]} \\ \cline{3-10}  & & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{3}\) & \(\omega_{4}\) & \(\omega_{5}\) & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{3}\) \\ \hline \multirow{3}{*}{UD} & 0.12 & 33.0910 & 53.0608 & 95.5477 & 137.195 & 178.955 & 31.6373 & 53.6617 & 94.5798 \\  & 0.17 & 39.7517 & 65.6635 & 119.925 & 173.818 & 227.912 & 38.0857 & 66.2890 & 118.645 \\  & 0.28 & 49.3183 & 75.6428 & 133.685 & 189.651 & 245.808 & 47.6137 & 77.3944 & 133.310 \\ \hline \multirow{3}{*}{FG-O} & 0.12 & 31.1475 & 43.3217 & 83.5019 & 124.939 & 167.510 & 29.8088 & 43.4827 & 79.4926 \\  & 0.17 & 37.3037 & 52.9235 & 103.529 & 156.791 & 211.891 & 35.8071 & 53.2657 & 99.2077 \\  & 0.28 & 46.4691 & 63.0713 & 119.963 & 177.582 & 236.519 & 45.0041 & 64.6479 & 117.180 \\ \hline \multirow{3}{*}{FG-X} & 0.12 & 34.4999 & 58.4039 & 101.363 & 142.768 & 184.100 & 33.1091 & 59.2156 & 101.272 \\  & 0.17 & 41.5231 & 72.9407 & 128.366 & 182.309 & 236.097 & 39.9296 & 73.3581 & 127.006 \\ \cline{1-1}  & 0.28 & 51.2700 & 83.6055 & 143.256 & 200.258 & 257.278 & 49.5523 & 83.0986 & 138.326 \\ \hline \end{tabular}
\end{table} 
Table 4: Dimensionless frequency of CNT curved beam with S-S boundary condition (L/R = 0.2 and L/h = 25). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\(V_{CNT}^{*}\)} & CNTs Pattern & \multicolumn{2}{c}{This Study} & \multicolumn{2}{c}{Ref. [30]} \\ \cline{3-6}  & & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{1}\) & \(\omega_{2}\) \\ \hline \multirow{2}{*}{0.17} & FG-O & 20.9321 & 47.1825 & 21.4803 & 48.6612 \\  & FG-X & 28.2739 & 64.3162 & 27.0170 & 64.3032 \\ \hline \end{tabular}
\end{table} 
Table 5: Dimensionless frequency of CNT curved beam with S-S boundary condition on elastic foundation. The PMMA instead features the following Young's modulus, density and Poisson's ratio:  
\[E_{Matrix}=2.5\,\text{GPa},\,\rho_{Matrix}=1190\,\frac{\text{kg}}{\text{cm}^{2}}, \,\text{and}\,v_{Matrix}=0.30 \tag{38}\]  
The following efficiency parameters are set as  
\[ \begin{array}{l}V_{\text{CNT}}^{\text{a}}=0.12\rightarrow\eta_{1}=1.2833, \,\eta_{2}=\eta_{3}=1.0556\\ V_{\text{CNT}}^{\text{a}}=0.17\rightarrow\eta_{1}=1.3414,\,\eta_{2}=\,\eta_{3 }=1.7101\\ V_{\text{CNT}}^{\text{a}}=0.28\rightarrow\eta_{1}=1.3238,\,\eta_{2}=\,\eta_{3 }=1.7380\end{array}  \tag{39}\]  
The stiffness of the interfacial springs is related to the mechanical and geometrical properties of the beams, as follows:  
\[K=k_{s}\times\frac{A_{BM}}{L^{2}} \tag{40}\]  
where \(K\) is the interfacial stiffness, \(k_{s}\) refers to the stiffness factor, and \(A_{BM}\) can be determined as  
\[A_{BM}=\int\frac{E}{2\left(1-\
u\right)}\,dz \tag{41}\]  
where \(E\) and \(\
u\) stand for the Young's modulus and [PERSON]'s ratio of the pure PMMA matrix beam which was made by matrix, respectively. For such a case, the dimensionless frequency of the structure is determined as  
\[\overline{\omega}=\omega\,R\,\sqrt{\frac{I_{BM}}{A_{BM}}} \tag{42}\]  
where the inertial quantity \(I_{BM}\) of the beam is obtained as \(\int\rho\,dz\). Figure 3 shows the effect of the rational radius-to-height quantity of the beam on the first dimensionless frequency of the coupled curved beams. To this end, four values of 10, 15, 20, and 30 are considered for \(R/h\), while setting the stiffness factor equal to 100.