# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 3 Results and Discussions
As can be seen, larger masses cause greater increases in the frequency shift in the presence of a magnetic field versus its absence for both the simply supported and the bridged nanotubes. However, the frequency shift decreases with increasing magnetic intensity for the cantilevered resonator DWCNTs. The disparity in the behavior of the different systems under the influence of a magnetic field arises from the nature of the magnetic field's effect on each system's total stiffness. When a cantilevered system is subjected to a longitudinal magnetic field, the total stiffness decreases, resulting in a stiffer system. Moreover, if we focus solely on the scenario in  
Figure 2: The effect of the magnetic intensity on the frequency shift for various values of the elastic foundation modulus for (**a**) cantilevered, (**b**) simply supported, and (**c**) bridged nanotubes (\(\mu=1\,\)nm). which Hx is zero, the pattern observed in the diagram aligns with the results presented by [PERSON] et al. [23]. Conversely, in the case of the simply supported and bridged systems, the magnetic field makes the systems harder. Consequently, the frequencies of the systems decrease. Then, the frequency shifts decrease based on Equation (24). Therefore, in the presence of a magnetic field, one can potentially obtain higher sensitivity using simply supported and bridged DWCNTs. The observation that a stiffer nanosystem is more capable of sensing mass aligns with the output shown in Figure 4, where smaller nanosystems with higher stiffness are shown to be more effective in capturing small masses. Finally, Figure 6 shows the effect of the non-local parameter on the frequency shift for the three nanosystems (cantilevered, simply supported, and bridged nanotubes). The non-local parameter decreases the frequency shift for both the simply supported and the bridged nanotubes, while it increases the frequency shift for the cantilevered nanotubes. The effect of non-local parameters is more significant for a bridged nanotube than for the cantilevered and simply supported ones. Here, a connection between the effect of the non-local parameter on the system stiffness and the frequency shift can be made: if the system is stiffer, the frequency shift will be more significant than for a softer system. The effect of the non-local parameter is in agreement with the results published in the literature [61]. Figure 3: The effect of the magnetic intensity on the frequency shift for various values of the aspect ratio for (**a**) cantilevered, (**b**) simply supported, and (**c**) bridged nanotubes (\(\mu\) = 1 nm). _This Study's Limitations and Potential Future Research Areas_  
As mentioned above, the current study focuses on the possibility of increasing the mass sensitivity of a DWCNT nano-mass sensor using a longitudinal magnetic field. The mass sensitivity of the current model is on the order of \(10^{-24}\). In comparison to the sensitivity values of some of the other models in the literature, this value is bigger than some and smaller than others. For example, [PERSON] et al. [62] introduced a sensitivity on the order of \(10^{-18}\) for an SWCNT model with a diameter of 2.7 nm and a length of 55 nm. In other works, carried out by [PERSON] et al. [63] for SWCNT models with a diameter of 1.1 nm and lengths of 4.1, 5.6, and 8 nm, the sensitivity was on the order of \(10^{-21}\). [PERSON] et al. [64], using a 10 nm \(\times\) 10 nm single-layer graphene sheet [65], increased the sensitivity to \(10^{-27}\). These differences were due to the use of different control parameters, such as geometry, environmental control parameters, temperature, and so on, as mentioned in the introduction. Some of these control parameters are among the limitations of the current model and should be considered when the results of the current model are compared to those in the literature. For example, in the current model, we did not consider the ambient temperature's effect on the nanoresonator's performance, which effectively adds external noise to the system.