# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 3 Problem Formulation
### Mathematical Model
 \tag{19}\]  
Using Equation (19), the governing coupled equations for nonlocal AFG double nanobeam are given as  
\[\frac{\partial^{2}}{\partial\hat{x}^{2}}\!\left[E(x)I_{1}(x)\frac{ \partial^{2}\hat{w}_{1}}{\partial\hat{x}^{2}}\right]+\rho(x)A_{1}(x)\frac{ \partial^{2}\hat{w}_{1}}{\partial\hat{t}^{2}}-(\varepsilon_{0}a)^{2}\frac{ \partial^{2}}{\partial x^{2}}\!\left(\rho(x)A_{1}(x)\frac{\partial^{2}\hat{w} _{1}}{\partial\hat{t}^{2}}\right)+k(\hat{w}_{1}-\hat{w}_{2})-(\varepsilon_{0}a )^{2}-k\frac{\partial^{2}}{\partial\hat{t}^{2}}(\hat{w}_{1}-\hat{w}_{2})=0, \tag{20}\] \[ \begin{array}{c}\frac{\partial^{2}}{\partial x^{2}}\!\left(E(x)I_{2} (x)\frac{\partial^{2}\hat{w}_{2}}{\partial x^{2}}\right)+\rho(x)A_{2}(x)\frac{ \partial^{2}\hat{w}_{2}}{\partial\hat{t}^{2}}-(\varepsilon_{0}a)^{2}\frac{ \partial^{2}}{\partial x^{2}}\!\left(\rho(x)A_{2}(x)\frac{\partial^{2}\hat{w}_{ 2}}{\partial\hat{t}^{2}}\right)+k(\hat{w}_{2}-\hat{w}_{1})-\\ (\varepsilon_{0}a)^{2}k\frac{\partial^{2}}{\partial x^{2}}(\hat{w}_{2}-\hat{w} _{1})+k_{w}\hat{w}_{2}-k_{G}\frac{\partial^{2}\hat{w}_{2}}{\partial x^{2}}-k_{ w}(\varepsilon_{0}a)^{2}\frac{\partial^{2}\hat{w}_{2}}{\partial x^{2}}+k_{G}( \varepsilon_{0}a)^{2}\frac{\partial^{4}\hat{w}_{2}}{\partial x^{4}}=0,\end{array}  \tag{21}\]  
where \(k\), \(\hat{w}_{j}\), \(I_{j}\), and \(A_{j}\) (\(j=1\), \(2\)) are the stiffness of the elastic layer between beams, the transverse displacement, the area moment of inertia, and the cross sectional area for nanobeam-\(j\), respectively. It is assumed that both nanobeams have the same length \(L\), modulus of elasticity \(E(x)\), density \(\rho(x)\), \(I_{0}\), and \(A_{0}\), which are the area moment of inertia, and the cross sectional area at \(x=0\), respectively. In the present research, \(E(x)\) and \(I(x)\) are given as [36]:  
\[ \begin{array}{c}E(x)=E_{0}\!\left(1+\frac{x}{L}\right)\!,I_{j}(x)=I_{0}\! \left(1-c_{b_{j}}\frac{x}{L}\right)\!\left(1-c_{h_{j}}\frac{x}{L}\right)\!^{3}, \rho(\hat{x})=\rho_{0}\!\left(1+\frac{x}{L}+\left(\frac{x}{L}\right)^{2}\right) \!,\\ A(x)=A_{0}\!\left(1-c_{b_{j}}\frac{x}{L}\right)\!\left(1-c_{h_{j}}\frac{x}{L} \right)\!,\ j=\ 1,\ 2,\end{array}  \tag{22}\]  
where \(E_{0}\) and \(\rho_{0}\) are the modulus of elasticity and the density at \(x=0\), respectively, and \(c_{b_{j}}\) and \(c_{h_{j}}\) are the breadth and the height taper ratios of nanobeam-\(j\), respectively. For harmonic natural vibration, the transverse displacements are given as:  
\[\hat{w}_{j}(x,t)=w_{j}(x)e^{i\omega t},\ j\ =\ 1,\ 2, \tag{23}\]  
where \(\omega\) is the transverse natural frequency in rad/s. For convenience, we introduced the following dimensionless variables as  
\[ \begin{array}{c}X=\frac{x}{L},\ W1=\frac{w_{1}}{L},\ W2=\frac{w_{2}}{L},\ \mu=\frac{c_{0}a}{L},\ \Omega^{2}=\frac{\rho_{0}A_{0}L^{4}\omega^{2}}{E_{0}b_{0}},\ \tau=\frac{t}{L^{2}}\Big{(}\frac{Eql_{0}}{\rho_{0}A_{0}} \Big{)}^{1/2},\ K_{W}=\frac{k_{W}L^{4}}{E_{0}b_{0}},\\ K_{G}=\frac{k_{G}L^{2}}{E_{0}b_{0}},K=\frac{kt^{3}}{E_{0}b_{0}},\ E(X)=\frac{E(x)}{E_{0}},\ A(X)=\frac{A(x)}{A_{0}},\ I(X)=\frac{l(x)}{l_{0}},\ \rho(X)=\frac{\rho(x)}{\rho_{0}}.\end{array}  \tag{24}\]  
Substituting Equations (23) and (24) into Equations (20) and (21) yields:  
\[ \begin{array}{c}\frac{d^{2}}{dX^{2}}\Big{(}E(X)I_{1}(X)\frac{d^{2}W1}{dX^{2} }\Big{)}+K(W1-W2)-\mu^{2}K\frac{d}{dX^{2}}(W1-W2)=\\ \Omega^{2}\Big{(}\rho(X)A_{1}(X)W1-\mu^{2}\frac{d^{2}}{dx^{2}}(\rho(X)A_{1}(X) W1)\Big{)}.\end{array}  \tag{25}\]  
\[ \begin{array}{c}\frac{d^{2}}{dX^{2}}\Big{(}E(X)I_{2}(X)\frac{d^{2}W2}{dX^{2} }\Big{)}+K(W2-W1)-\mu^{2}K\frac{d}{dX^{2}}(W2-W1)+K_{W}W2-K_{G}\frac{d^{2}W2}{ dX^{2}}\\ -K_{W}\mu^{2}\frac{d^{2}W2}{dX^{2}}+K_{G}\mu^{2}\frac{d^{2}W2}{dX^{2}}=\Omega ^{2}\Big{(}\rho(X)A_{2}(X)W2-\mu^{2}\frac{d^{2}}{dx^{2}}(\rho(X)A_{2}(X)W2) \Big{)}.\end{array}  \tag{26}\]  
The dimensionless natural transverse frequencies \(\Omega\) are calculated by solving the eigenvalue problem resulted from the discretization of Equations (25) and (26).