# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 3 Results and Discussions  
In this section, we present the findings of our study on the behavior of DWCNTs with different boundary conditions when subjected to an elastic foundation and a distributed longitudinal magnetic field. First, our results are validated through a comparison with the existing findings in the literature. The natural frequencies of DWCNTs are compared in Table 1. Table 2 presents the natural frequencies of simply supported DWCNTs with different stiffness coefficients. Finally, Table 3 shows the system's natural frequencies in the presence of the non-local parameter. As shown, the present work agrees with the corresponding results reported by [PERSON] et al. [54], [PERSON] et al. [55], [PERSON] et al. [56], and [PERSON] et al. [57]. The physical values defined in Table 4 are used for the configuration presented in this section, and the effects of different parameters on sensitivity are investigated in the following plots. The presented model is based on a DWCNT nanoresonator. Figure 2 shows how the magnetic field could affect the frequency shift, decreasing it for the cantilevered nanotube and increasing it for both the simply supported and bridged nanotubes. Increasing the value of the elastic foundation modulus can increase the frequency shift of the simply supported and cantilevered DWCNTs while decreasing the frequency shift of \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **[PERSON] et al. [55]** & **[PERSON] et al. [54]** & **Present Study** \\ \hline simply supported & 0.46830 & 0.46 & 0.47 \\ \hline cantilevered & 0.202 & 0.17 & 0.17 \\ \hline bridged & 1.0515 & 1.06 & 1.085 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Natural frequencies (\(\times 10^{12}\) Hz) of DWCNTs with different boundary conditions (\(d_{2}=1.4\) nm, \(d_{1}=0.7\) nm, \(\rho=2300\) kg/m\({}^{3}\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{\(\mathbf{k=10^{8}\) N/m\({}^{2}\)}} & \multicolumn{2}{c}{\(\mathbf{k=10^{9}\) N/m\({}^{2}\)}} \\ \cline{2-5}  & **[PERSON] et al. [56]** & **Present Study** & **[PERSON] et al. [56]** & **Present Study** \\ \hline
1 & 0.5113 & 0.4997 & 0.6665 & 0.6539 \\ \hline
2 & 1.9328 & 1.9127 & - & 1.9571 \\ \hline
3 & 4.1670 & 4.1817 & - & 4.2040 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: First three natural frequencies (\(\times 10^{12}\) Hz) of embedded hinged–hinged DWCNTs with different stiffness coefficients (\(k\)) (effective thickness : \(h=34\) nm, \(d_{1}=0.684\) nm, \(\rho=2300\) kg/m\({}^{3}\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{_c} \hline \hline \multicolumn{1}{c}{_c} \hline \multicolumn{1}{c}{_c} & \(\mathbf{e_{0}a\) (nm)} & **[PERSON] et al. [57]** & **Present Study** \\ \hline \multirow{4}{*}{0} & 0 & 1.6625 & 1.6625 \\ \cline{2-5}  & 0.5 & 1.6592 & 1.6592 \\ \cline{2-5}  & 1 & 1.6496 & 1.6496 \\ \hline \multirow{4}{*}{0.0694} & 0 & 4.4304 & 4.4304 \\ \cline{2-5}  & 0.5 & 4.4305 & 4.4305 \\ \cline{1-1} \cline{2-5}  & 1 & 4.4846 & 4.4846 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Natural frequencies (\(\times 10^{11}\) Hz) of simply supported DWCNTs with different non-local parameters when outer tube is fixed (\(w_{2}=0\))(\(d_{2}=1.4\) nm, \(d_{1}=0.7\) nm, \(\rho=2300\) kg/m\({}^{3}\)). the bridged DWCNT. Based on Equation (21), the magnetic field contributes to the overall stiffness of the system. Consequently, as the value of \(H_{x}\) increases, its influence on the total stiffness becomes more prominent compared to the other parameters. This is evident when \(H_{x}\) reaches approximately 1.5 \(\times\) 10\({}^{8}\) A/m. At this point, the effect of \(H_{x}\) dominates the total stiffness of both cases (with and without mass), and further increases in \(H_{x}\) result in a frequency shift of zero. This observation applies to both the simply supported and the bridged cases. The corresponding diagram converges for all three cases when the magnetic intensity parameter reaches a sufficiently high value.