# Size-Dependent Free Vibration of Axially Moving Nanobeams Using [PERSON]'s Two-Phase Integral Model
## 2 Model Equations
\begin{array}{l}\rho A(\frac{\partial^{2}w}{\partial t^{2}}+2 \gamma(t)\frac{\partial^{2}w}{\partial t\partial x}+\frac{d\gamma(t)}{dt} \frac{\partial w}{\partial x}+\gamma^{2}(t)\frac{\partial^{2}w}{\partial x^{ 2}})-P\frac{\partial^{2}w}{\partial x^{2}}+EI_{5}\xi_{1}\xi_{s}\frac{\partial^ {4}w}{\partial x^{4}}\\ =\frac{3}{2}\xi_{1}EA\frac{\partial^{2}w}{\partial x^{2}}(\frac{ \partial w}{\partial x})^{2}+\frac{2 EI_{5}^{2}}{\tau^{2}}\frac{\partial^{2}w} {\partial x^{2}}+\frac{\xi_{2}}{4\tau}EA\frac{\partial}{\partial x}\bigg{(} \frac{\partial w}{\partial x}\int_{0}^{L}e^{-\frac{|x-i|}{\tau}}(\frac{ \partial w}{\partial s})^{2}ds\bigg{)}\\ -EI\frac{\xi_{2}}{\tau}\frac{\partial^{2}}{\partial x^{2}}\int_{0}^{L}e^{- \frac{|x-i|}{\tau}}\frac{\partial^{2}w}{\partial s^{2}}ds.\end{array}  \tag{8}\]  
If the following nondimensionlizations are introduced  
\[ \begin{array}{l}x=Lx^{*},\quad s=L\theta,\quad w=Lw^{*},\quad t=\sqrt{\frac {E}{\rho L^{2}}}t^{*},\quad\gamma=\sqrt{\frac{E}{\rho}}\gamma^{*},\\ \alpha=\frac{L}{\tau},\quad p^{*}=\frac{P}{AE},\quad k_{s}=\frac{I}{ AL^{2}},\end{array}  \tag{9}\]  
then model Equation (9) can be rewritten as  
\[ \begin{array}{l}\frac{\partial^{2}w}{\partial t^{2}}+2\gamma(t) \frac{\partial^{2}w}{\partial t\partial x}+\frac{d\gamma(t)}{dt}\frac{ \partial w}{\partial x}+(\gamma^{2}(t)-p)\frac{\partial^{2}w}{\partial x^{2} }+\xi_{1}k_{s}\frac{\partial^{4}w}{\partial x^{4}}\\ =\frac{3}{2}\xi_{1}\frac{\partial^{2}w}{\partial x^{2}}(\frac{ \partial w}{\partial x})^{2}+2k_{s}a^{2}\xi_{2}\frac{\partial^{2}w}{\partial x ^{2}}+\frac{1}{4}\xi_{2}\frac{\partial^{2}w}{\partial x^{2}}\int_{0}^{1}e^{-|x -\theta|}(\frac{\partial w}{\partial\theta})^{2}d\theta\\ +\frac{1}{4}a^{2}\xi_{2}\frac{\partial w}{\partial x}\frac{ \partial}{\partial x}\bigg{(}\int_{0}^{1}e^{-|x-\theta|}(\frac{\partial w}{ \partial\theta})^{2}d\theta\bigg{)}-k_{s}\xi_{2}a^{3}\int_{0}^{1}e^{-|x- \theta|a}\frac{\partial^{2}w}{\partial\theta^{2}}d\theta.\end{array}  \tag{10}\]  
In Equation (11), the superscript \"\(*\)\" are all removed for convenience. In this work, we consider the simply supported and fixed boundary conditions (for both ends) for Equation (11). For the simply supported boundary condition, it means that the displacement \(w(x,t)\) must be zero and the beam is free to rotate, which yields  
\[w=0,M(x,t)=0,\quad\text{at}\quad x=0,1. \tag{11}\]  
The above equations can be rewritten as  
\[w=0,\quad\xi_{1}w_{,xx}(x,t)-\frac{\xi_{2}}{2\alpha}\int_{0}^{1}e^{-|x-\theta |\alpha}w_{,ss}(\theta,t)d\theta=0,\quad x=0,1. \tag{12}\]  
For the fixed boundary condition, it implies that the displacement \(w(x,t)\) and the slope of the beam at the two ends must be zero. Then, the boundary condition is  
\[w=0,\quad w_{,x}=0,\quad\text{at}\quad x=0,1, \tag{13}\]  
where the subscript, \"\(x\)\" denotes the derivative along \(x\). It is to be noted that, when \(\xi_{1}=1\) model, Equation (11) reduces to  
\[\frac{\partial^{2}w}{\partial t^{2}}+2\gamma(t)\frac{\partial^{2}w}{\partial t \partial x}+\frac{d\gamma(t)}{dt}\frac{\partial w}{\partial x}+(\gamma^{2}(t) -p)\frac{\partial^{2}w}{\partial x^{2}}+k_{s}\frac{\partial^{4}w}{\partial x^{ 4}}-\frac{3}{2}\frac{\partial^{2}w}{\partial x^{2}}(\frac{\partial w}{\partial x })^{2}=0, \tag{14}\]  
In Figure 3, the effect of the geometric parameter \(k_{s}\) on the critical velocity of the axially moving beam is plotted. We see that the critical velocity increases with the growth of the parameter \(k_{s}\) for both the nonlocal beam and the classical beam. From Figure 3, we find that the critical velocity is almost identical for the first three modes. However, the critical velocity increases with the growth of the mode number \(k\). Furthermore, the critical velocity of the nonlocal beam is always much larger than that of the classical beam (Figure 3a). Figure 1: The critical velocity versus nonlocal parameter \(\alpha\) when \(p=0.04,k_{s}=0.004\) and \(\xi_{1}=0.5\). Black dotted line, red dotted line and blue dotted line denote the results for \(k=3,2,1\) mode, respectively.