# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 2 Theoretical Basis  
A schematic diagram of a simply supported DWCNT beam embedded in an elastic foundation, subjected to an external force (\(F_{ext}\)), and carrying an attached concentrated mass (\(m_{c}\)) along its length position (\(x=a\)) is shown in Figure 1. In this paper, the non local Euler-Bernoulli beam model is applied to study the free vibration of a DWCNT. The governing equations of a viscoelastic nano-beam under a longitudinal magnetic field based on non-local elasticity theory can be obtained as follows [43]:  
\[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w(x,t)}{ \partial t^{2}}-(e_{0}a)^{2}\bigg{(}\rho A\frac{\partial^{4}w(x,t)}{\partial x^ {2}\partial t^{2}}-\frac{\partial^{2}\ F_{ext}(x,t)}{\partial x^{2}}\bigg{)}= \ F_{ext}(x,t), \tag{1}\]  
where \(e_{0}\) is a constant for calibrating the model using experimental results and other validated models and \(a\) is the length of the internal characteristics. The external force (\(F_{ext}\)) is a combination of several factors acting on the nanotube, including the forces per unit length induced by the magnetic field \(\ (q_{mag})\), the Winkler foundation force \(\ (q_{elastic\ medium})\), and the force of the interaction between the inner and outer nanotubes \(\ (q_{vun})\), which can be expressed as follows:  
\[F_{ext}=q_{mag}+q_{elastic\ medium}+q_{vun}. \tag{2}\]  
_Maxwell's Relations_  
When applying a longitudinal magnetic field vector (\(\overline{H}(H_{x},\ 0,\ 0)\)), a Lorentz force acts on the DWCNT. For the present study, \(w=w(x,t)\); so, the Lorentz force in the z direction is written as follows:  
\[F_{z}=\eta H_{x}^{2}\bigg{(}\frac{\partial^{2}w(x,t)}{\partial x^{2}}\bigg{)}, \tag{3}\]  
where \(\eta\) is the magnetic field permeability and \(H_{x}\) denotes the magnetic intensity of the longitudinal magnetic field. Therefore, the pressure exerted on the nanotube due to a Lorentz magnetic field can be expressed as follows:  
\[q_{mag}=\int_{A}F_{z}dz=\eta AH_{x}^{2}\bigg{(}\frac{\partial^{2}w(x,t)}{ \partial x^{2}}\bigg{)}. \tag{4}\]  
As shown in Figure 1, the interaction between the inner and outer nanotubes is facilitated by van der Waals forces and can be obtained as follows:  
\[q_{vun}=c(w_{2}-w_{1}). \tag{5}\]  
The subscripts 1 and 2 denote the inner and outer nanotubes, respectively; furthermore, \(c\) is the van der Waals interaction coefficient, which is written as follows [28; 52]:  
\[c=\frac{320\times 2R_{1}}{0.16a^{2}},\ (a=0.142\ \mathrm{nm}), \tag{6}\]  
Figure 1: Schematic diagram of the DWCNT with attached mass under a longitudinal magnetic field. where \(R_{1}\) is the internal diameter of the nanotube. The Winkler foundation model describes the effect of the elastic medium on the outer nanotube. Therefore, for a DWCNT with transverse displacement in the \(z\) direction, we have  
\[q_{elastic\ medium}=kw_{2}, \tag{7}\]  
where \(k\) is the elastic foundation modulus. Substituting Equations (4), (5) and (7) into Equation (2) and substituting the result into Equation (1), one obtains the equation of motion for the inner nanotube:  
\[EI_{1}\frac{\partial^{4}w_{1}}{\partial x^{4}}-(e_{0}a)^{2}\bigg{(}\rho A_{1} \frac{\partial^{4}w_{1}}{\partial x^{2}\partial t^{2}}-\eta A_{1}H_{x}^{2} \frac{\partial^{4}w_{1}}{\partial x^{4}}+c\frac{\partial^{2}w_{1}}{\partial x ^{2}}-c\frac{\partial^{2}w_{2}}{\partial x^{2}}\bigg{)}+\rho A_{1}\frac{ \partial^{2}w_{1}}{\partial t^{2}}-\eta A_{1}H_{x}^{2}\frac{\partial^{2}w_{1} }{\partial x^{2}}=c(w_{2}-w_{1}). \tag{8}\]  
The equation for the outer nanotube is as follows:  
\[