# Size-Dependent Free Vibration of Axially Moving Nanobeams Using [PERSON]'s Two-Phase Integral Model
## 2 Model Equations  
We consider an axially moving nanobeam under an applied longitudinal tension \(P\), with a transport velocity \(\gamma(t)\) between two boundaries where the distance is \(L\). The beam has density \(\rho\) and cross-section \(A\). Here, we consider only the in-plane behavior of the beam, and the out-of-plane motion are not taken into account. The distance to the left boundary is measured by \(x\). The transverse displacement (in \(y\) direction) is denoted by \(w(x,t)\), while the longitudinal displacement is not considered for simplicity. By Euler beam theory, the kinetic energy \(T\) for the axially moving nanoscale beam can be expressed as  
\[ \begin{array}{rcl}T&=&\frac{1}{2}\int_{0}^{L}\rho V\cdot V \mathrm{d}x\\ &=&\frac{1}{2}A\rho\int_{0}^{L}\gamma(t)^{2}+(\frac{\partial w}{ \partial t}+\gamma(t)\frac{\partial w}{\partial x})^{2}\mathrm{d}x.\end{array}  \tag{1}\]  
We consider the Lagrange strain \(\epsilon_{x}\) for the nanoscale beam, which is used in calculations where large shape changes are expected. The expression of Lagrange strain which incorporates geometric non-linearity is given by  
\[\epsilon_{x}=\frac{1}{2}(\frac{\partial w}{\partial x})^{2}-y \frac{\partial^{2}w}{\partial x^{2}}. \tag{2}\]  
For homogenous isotropic materials, the nonlocal stress can be expressed as  
\[\sigma_{x}(x)=E(\xi_{1}\epsilon_{x}+\frac{\xi_{2}}{\tau}\int_{0 }^{L}e^{-\frac{|x-s|}{\tau}}\epsilon_{x}ds), \tag{3}\]  
where \(\xi_{1}>0,\xi_{2}>0\) and \(\xi_{1}+\xi_{2}=1\). \(\sigma_{x}(x)\) denotes the nonlocal stress, \(\tau\) is a size-dependent parameter and \(E\) is [PERSON]'s modulus. Equation (3) is the so-called two-phase nonlocal integral model: phase 1 (with volume fraction \(\xi_{1}\)) complies with classical (local) elasticity, while phase 2 (with volume fraction \(\xi_{2}\)) obeys nonlocal elasticity. It should be pointed out that the solution to two-phase model can not be reduced to the pure nonlocal model when the volume fraction tends to zero, and the pure nonlocal model has been proved to be ill-posed [19]. Thus, the model adopted herein has the advantages of well-posedness (as compared to the pure nonlocal model) and self-consistency (as compared to nonlocal differential model) [25]. With the above constitutive relation, the bending moment can be expressed as  
\[M(x,t)=-\int_{A}y\sigma_{x}(x)dA=\frac{EI}{2}(\xi_{1}\frac{ \partial^{2}w}{\partial x^{2}}+\frac{\xi_{2}}{2\tau}\int_{0}^{L}e^{-\frac{|x- s|}{\tau}}\frac{\partial^{2}w}{\partial s^{2}}ds), \tag{4}\]  
where  
\[I=\int_{A}y^{2}\mathrm{d}A,\quad\int_{A}y\mathrm{d}A=0 \tag{5}\]  
are used. The potential energy as a result of the bending moment, deformation and the external tension \(P\) is given by  
\[\phi(t)=\frac{1}{2}\int_{0}^{L}M(x,t)\frac{\partial^{2}w}{\partial x^{2}}dx+ \frac{1}{2}\int_{0}^{L}\int_{A}(P+\sigma_{x})\epsilon_{x}dAdx. \tag{6}\]Substituting Equations (2) and (3) into Equation (6), after some manipulations, we have  
\[ \begin{array}{rcl}\phi(t)&=&\int_{0}^{L}\bigg{(}\frac{EA}{ \delta}(\frac{\partial w}{\partial x})^{2}\big{(}\frac{\xi_{2}}{2\tau}\int_{0}^{ L}e^{-\frac{|x-i|}{\tau}}(\frac{\partial w}{\partial x})^{2}ds+\xi_{1}(\frac{ \partial w}{\partial x})^{2}\big{)}\\ &+&EI\frac{\partial^{2}w}{\partial x^{2}}(\frac{\xi_{2}}{2\tau}\int_{0}^{L}e^{- \frac{|x-i|}{\tau}}\frac{\partial^{2}w}{\partial s^{2}}ds+\xi_{1}\frac{ \partial^{2}w}{\partial x^{2}}\bigg{)}\bigg{)}dx+\frac{1}{2}\int_{0}^{L}P\frac{ \partial w^{2}}{\partial x}dx.\end{array}  \tag{7}\]  
By Hamilton's Principle, the governing equation for the nonlinear vibration can be obtained as  
\[