# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 3 Results and Discussions
The effect of the aspect ratio (defined as the ratio of the outer tube diameter to the inner tube diameter) is shown in Figure 3. In all three cases, the aspect ratio has a consistent effect. Higher aspect ratio values result in more remarkable frequency shifts. In the absence of magnetic intensity (\(H_{x}=0\)), the results in this figure align with the findings of [PERSON] et al. [58]. According to their work, in a DWCNT with a greater length, a smaller frequency is observed. This correlation is based on Equation (24), where the frequency shift and frequency are directly related. Furthermore, from a physical perspective, this implies that a larger outer tube diameter in a system with a constant inner tube diameter is more effective in sensing nano-mass, as it exhibits a more significant frequency shift. Similarly, the same effect can be observed in a system with a fixed outer tube diameter where the inner tube has a smaller diameter. Regarding two systems with equal inner tube diameters but different outer tube diameters, Equation (23) reveals that the system with the larger outer tube diameter has a larger value. This larger value contributes to a smaller second part of Equation (24), resulting in a larger frequency shift. Therefore, the system with the larger outer tube diameter exhibits a greater frequency shift compared to the system with the smaller outer tube diameter. Furthermore, a tube with a higher aspect ratio exhibits greater stiffness than a tube with a smaller one. Consequently, this leads to increases in frequency and mass responsivity, enhancing sensitivity. This observation aligns with the experimental research findings presented by [PERSON] et al. [59]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Numerical Value** \\ \hline \(l\) & Length (nm) & 5.55 \\ \hline \(\rho\) & Density (kg/m\({}^{3}\)) & 2300 \\ \hline \(E\) & Young’s modulus (N/m\({}^{2}\)) & \(1\times 10^{12}\) \\ \hline \(d_{2}\) & Outside diameter (nm) & 1.4 \\ \hline \(d_{1}\) & Inside diameter (nm) & 0.7 \\ \hline \(h\) & Thickness (nm) & 0.34 \\ \hline \(A_{2}\) & Outside area (nm\({}^{2}\)) & 1.4954 \\ \hline \(A_{1}\) & Inside area (nm\({}^{2}\)) & 0.7477 \\ \hline \(I_{1},I_{2}\) & Moments of inertia (nm\({}^{4}\)) & 0.0566, 0.3879 \\ \hline \(H_{x}\) & Magnetic intensity (A/m) & \(1\times 10^{8}\) \\ \hline \(k\) & Elastic foundation modulus (N/m\({}^{2}\)) & \(5\times 10^{8}\) \\ \hline \(\zeta\) & Position of attached mass (nm) & - \\ \hline \(M\) & Attached mass (g) & - \\ \hline \(d_{2}/d_{1}\) & Aspect ratio & - \\ \hline \(\mu\) & Non-local parameter (nm) & 0–1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Geometrical and mechanical properties of the nanosensor. Furthermore, let us examine why increasing the magnetic intensity causes a more significant divergence among the three aspect ratio cases. It can be attributed to the varying effects of the magnetic field on different systems with different aspect ratios. This difference arises from the direct influence of the surface area on the magnetic force experienced by each system. As the magnetic intensity increases, the impact of the magnetic field becomes more pronounced and distinct for systems with varying aspect ratios. Figure 4 shows the effect of the length ratio (defined as the ratio of the length to the outer tube diameter) on the frequency shift for different boundary conditions. Shorter beams are more sensitive, due to the presence of the magnetic intensity for the bridged, cantilevered, and simply supported nanotubes. In the case of the absence of a magnetic field, this finding is in line with those reported in the literature [11, 60] and the experimental observations reported by [PERSON] et al. [37]. Figure 5 shows how the magnetic intensity can increase or decrease the range of the measurable mass in the three different boundary conditions.