# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 2 Theoretical Basis
\begin{split} EI_{2}\frac{\partial^{4}w_{2}}{\partial x^{4}}-(e _{0}a)^{2}\bigg{(}\rho A_{2}\frac{\partial^{4}w_{2}}{\partial x^{2}\partial t ^{2}}+M(\delta(x-\zeta)\frac{\partial^{4}w_{2}}{\partial x^{2}\partial t^{2}} )-\eta A_{2}H_{x}^{2}\frac{\partial^{4}w_{2}}{\partial x^{4}}-c\frac{ \partial^{2}w_{1}}{\partial x^{2}}\\ +c\frac{\partial^{2}w_{2}}{\partial x^{2}}\bigg{)}-\eta A_{2}H_{ x}^{2}\frac{\partial^{2}w_{2}}{\partial x^{2}}+[\rho A_{2}+M(\delta(x-\zeta))] \frac{\partial^{2}w_{2}}{\partial t^{2}}+kw_{2}\\ =c(w_{1}-w_{2}).\end{split}  \tag{9}\]  
Assuming the motion is harmonic, an approximate solution for Equations (8) and (9) is obtained using the Galerkin approximation. Thus, the transverse deformation of each point on the inner and outer nanotubes can be expressed as follows:  
\[w_{1}(x,t)=\sum_{m=1}^{N}\mathcal{O}_{1n}(x)e^{iw_{1}t},w_{2}(x,t)=\sum_{m=1} ^{N}\mathcal{O}_{2n}(x)e^{iw_{1}t}, \tag{10}\]  
where \(\omega_{n}\) denotes the natural frequency of the DWCNT and \(\mathcal{O}_{n}(x)\) are the eigenfunctions, which depend on the beam-end condition. For the \(n\)th vibration mode and simply supported beam conditions, \(\mathcal{O}_{1n}(x)=\mathcal{O}_{2n}(x)=\sin\frac{n\pi x}{l}\). Next, for the \(n\)th vibration mode of a cantilevered DWCNT,  
\[\mathcal{O}_{n}(x)=\bigg{(}\cosh{(z_{n}\frac{x}{l})}-\cos{(z_{n}\frac{x}{l})} \bigg{)}-\bigg{(}\sinh{(z_{n}\frac{x}{l})}-\sin{(z_{n}\frac{x}{l})}\bigg{)} \times\bigg{(}\frac{\sin{h}\,z_{n}-\sin{z_{n}}}{\cosh{z_{n}}+\cos{z_{n}}}\bigg{)}. \tag{11}\]  
Finally, for the \(n\)th vibration mode of a bridged DWCNT,  
\[\mathcal{O}_{n}(x)=\bigg{(}\cosh{z_{n}\frac{x}{l}}-\cos{z_{n}\frac{x}{l}} \bigg{)}-\bigg{(}\sinh{z_{n}\frac{x}{l}}-\sin{z_{n}\frac{x}{l}}\bigg{)}\times \bigg{(}\frac{\cosh{z_{n}}+\cos{z_{n}}}{\sin{h}\,z_{n}-\sin{z_{n}}}\bigg{)} \tag{12}\]  
where \(z_{n}\) are the roots of the following equation. The natural frequency of a cantilevered DWCNT is also related to the \(z_{n}\):  
\[\cosh(z_{n})\text{cos}(z_{n})=1. \tag{13}\]  
As mentioned in the introduction, the energy method is employed to determine the natural frequency of a cantilevered DWCNT and the frequency shift due to added mass. Therefore, the kinetic energy of the inner-walled DWCNT can be obtained as follows [20,53]:  
\[T_{1}=\frac{\omega^{2}}{2}\int_{0}^{l}\rho A_{1}\bigg{(}(\mathcal{O}_{1n}(x)) ^{2}-(e_{0}a)^{2}\bigg{[}\frac{\partial\mathcal{O}_{1n}(x)}{\partial x}\bigg{]} ^{2}\bigg{)}dx. \tag{14}\]  
The kinetic energy of the outer-walled DWCNT can be obtained as  
\[ \begin{split} T_{2}=\frac{\omega^{2}}{2}\bigg{(}\int_{0}^{l}\rho A _{2}\bigg{(}&(\mathcal{O}_{2n}(x))^{2}-(e_{0}a)^{2}\bigg{[}\frac{ \partial\mathcal{O}_{2n}(x)}{\partial x}\bigg{]}^{2}\bigg{)}dx+M\Big{(}( \mathcal{O}_{2n}(\zeta))^{2}\\ &-(e_{0}a)^{2}\bigg{[}\frac{\partial^{2}\mathcal{O}_{2n}(x)}{ \partial x^{2}}\bigg{]}^{2}\bigg{)}\bigg{)}.\end{split}  \tag{15}\]The total kinetic energy of a DWCNT is the sum of the kinetic energies of the inner and outer nanotubes. Therefore,  
\[T_{t}=T_{1}+T_{2}. \tag{16}\]  
According to the kinetic energy equivalence and the position of the mass particle in the middle of the cantilevered beam, substituting Equations (15) and (16) into (17), we obtain  
\[m_{eq}=\rho\left(l(A_{1}+A_{2})-(e_{0}a)^{2}\left(\int_{0}^{l}\left(A_{1}\left[ \frac{\partial\mathcal{D}_{1n}(x)}{\partial x}\right]^{2}+A_{2}\left[\frac{ \partial\mathcal{D}_{2n}(x)}{\partial x}\right]^{2}\right)dx\right)\right)+M \left((\mathcal{O}_{2n}(\zeta))^{2}-(e_{0}a)^{2}\left[\frac{\partial^{2} \mathcal{O}_{2n}(x)}{\partial x^{2}}\Big{|}_{\zeta}\right]^{2}\right). \tag{17}\]  
Next, the potential energy of the inner-walled DWCNT can be obtained as follows [20]:  
\[