# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 3 Problem Formulation  
### Mathematical Model  
Figure 1 shows a tapered AFG double nano beam lying on an elastic foundation. According to [PERSON] theorem, the normal strain \(\epsilon_{xx}\), the bending strain \(\kappa\), and the bending moment \(M(x,\hat{t})\) are given as:  
\[\epsilon_{xx}=-z\frac{\partial^{2}\hat{w}}{\partial x^{2}}=z\kappa, \tag{6}\]  
\[\kappa=-\frac{\partial^{2}\hat{w}}{\partial x^{2}}, \tag{7}\]  
\[M(x,\hat{t})=\int_{A}\sigma_{xx}z\ dA, \tag{8}\]  
where \(\hat{w}\) is the transverse deflection, \(x\) is the longitudinal axis of the beam, \(z\) is the transverse axis taken from the mid-plane of the beam along its thickness, and \(E(x)\) is the modulus of elasticity, \(\sigma_{xx}\) is the normal stress, and \(A(x)\) is the beam's cross sectional area. The kinetic energy \(T\) may be written as:  
\[T=\frac{1}{2}\int_{0}^{L}\rho(x)A(x)\bigg{(}\frac{\partial^{2}\hat{w}}{ \partial t^{2}}\bigg{)}^{2}dx, \tag{9}\]  
where \(L\) is the length of the beam, \(\rho(x)\) is the density, and \(\hat{t}\) is the time. The potential energy is given as:  
\[U=\frac{1}{2}\int_{0}^{L}\sigma_{xx}\ \varepsilon_{xx}\ A(x)\ dx+\frac{1}{2}\int_{0 }^{L}k_{W}\hat{w}^{2}dx+\frac{1}{2}\int_{0}^{L}k_{G}\bigg{(}\frac{\partial\hat {w}}{\partial x}\bigg{)}^{2}dx, \tag{10}\]  
where \(k_{W}\) and \(k_{G}\) are the Winkler stiffness and the shear stiffness of the surrounding foundation, respectively. [PERSON]'s principle is applied to obtain the governing equation of motion as:  
\[\int_{t_{1}}^{t_{2}}(\delta T-\delta U)dt=0, \tag{11}\]  
where \(\delta\) is the variation operator. From Equation (9), the variation of the kinetic energy is given as:  
\[\delta T=\int_{0}^{L}\rho(x)A(x)\frac{\partial\hat{w}}{\partial t}\frac{ \partial\delta\hat{w}}{\partial t}dx. \tag{12}\]  
Figure 1: The tapered nonlocal axially functionally graded (AFG) Euler–Bernoulli double nanobeam resting on elastic foundation. Based on Equation (10), the variation of the potential energy is expressed as:  
\[\delta U=\int_{0}^{L}\sigma_{xx}\;\delta\varepsilon_{xx}A(x)\;dx\;+\int_{0}^{L}k_ {G}\frac{\partial\hat{w}}{\partial x}\;\delta\!\left(\frac{\partial\hat{w}}{ \partial x}\right)dx+\int_{0}^{L}k_{W}\hat{w}\;\delta\hat{w}\;dx\;. \tag{13}\]  
Using Equations (11)-(13), and after some manipulation and simplification, one obtains  
\[\int_{t_{1}}^{t_{2}}\int_{0}^{L}\!\!\left[\rho(x)A(x)\frac{\partial\hat{w}}{ \partial t}\frac{\partial\hat{w}}{\partial t}-M(x,\hat{f})\delta\kappa-k_{G} \frac{\partial\hat{w}}{\partial x}\;\delta\!\left(\frac{\partial\hat{w}}{ \partial x}\right)+k_{W}\hat{w}\;\delta\hat{w}\right]\!dxdt=0. \tag{14}\]  
Then, integrating by parts and since \(\delta\hat{w}\) is arbitrary, we get the following governing equation of motion as:  
\[\kappa=-\frac{\partial^{2}\hat{w}}{\partial x^{2}}, \tag{7}\]0  
In light of Equation (4), the normal stress for the Euler-Bernoulli nano beam is defined as:  
\[\kappa=-\frac{\partial^{2}\hat{w}}{\partial x^{2}}, \tag{7}\]1  
Multiplying Equation (16) by \(z\dd A\) and integrating over the area of the beam, we get  
\[\kappa=-\frac{\partial^{2}\hat{w}}{\partial x^{2}}, \tag{7}\]2  
Using Equations (15) and (17), the bending moment \(M(x,\hat{t})\) in nonlocal form is defined as  
\[\kappa=-\frac{\partial^{2}\hat{w}}{\partial x^{2}}, \tag{7}\]3  
In light of Equations (15) and (18), and after some manipulation and rearranging, one obtains  
\[ \begin{array}{c}\frac{\partial^{2}}{\partial x^{2}}\!\left[E(x)I(x)\frac{ \partial^{2}\hat{w}}{\partial x^{2}}\right]+k_{w}\hat{w}-k_{G}\frac{\partial^{ 2}\hat{w}}{\partial x^{2}}-(\varepsilon_{0}a)^{2}k_{w}\frac{\partial^{2}\hat {w}}{\partial x^{2}}+(\varepsilon_{0}a)^{2}k_{G}\frac{\partial^{4}\hat{w}}{ \partial x^{4}}+\rho(x)A(x)\frac{\partial^{2}\hat{w}}{\partial\hat{t}^{2}}-\\ \frac{\partial^{2}}{\partial x^{2}}\!\left(\rho(x)A(x)\frac{\partial^{2}\hat{ w}}{\partial\hat{t}^{2}}\right)=0.\end{array}