## 2 Basic Theory and Calculation Formula  
In a rectangular section beam, the elastic modulus \(E(z)\) and density \(\rho(z)\) of functionally graded materials change along the \(z\) direction of height. Let beam length be \(L\), rectangular section height be \(h\), width be \(b\), and axial velocity be \(V\). Then, a rectangular coordinate system is established, as shown in Figure 1. It is hypothesized that the FGM properties change along the \(z\) direction of height according to the arbitrary functions \(f(z)\) and \(g(z)\), with the concrete expression shown as follows:  
\[E(z) =E_{m}f(z) \tag{1a}\] \[\rho(z) =\rho_{m}g(z) \tag{1b}\]where, \(-h/2\leq z\leq h/2\), \(E_{m}\) and \(\rho_{m}\) are the elastic modulus and density of the reference material, respectively, and \(f(z)\) and \(g(z)\) are arbitrary functions with regard to coordinate \(z\). According to the Euler-Bernoulli beam theory, the displacement components \(\widetilde{u}(x,z,t)\) and \(\widetilde{w}(x,z,t)\) of any point on the beam in the directions \(x\) and \(z\) can be expressed as:  
\[\widetilde{u}(x,z,t)=u(x,t)-z\frac{\partial w(x,t)}{\partial x}+Vt \tag{2a}\] \[\widetilde{w}(x,z,t)=w(x,t) \tag{2b}\]  
where, \(u(x,t)\) and \(w(x,t)\) are the axial displacement and transverse displacement of any point on the beam axis, respectively, and \(Vt\) is the axial rigid body displacement generated by the axial motion velocity \(V\). \[\varepsilon_{x}=\frac{\partial u(x,t)}{\partial x}-z\frac{\partial^{2}w(x,t) }{\partial x^{2}} \tag{3a}\] \[\sigma_{x}=E(z)\varepsilon_{x}=E(z)\bigg{[}\frac{\partial u(x,t)}{\partial x }-z\frac{\partial^{2}w(x,t)}{\partial x^{2}}\bigg{]} \tag{3b}\]  
The strain energy \(U\) of an axially moving FGM beam is,  
\[U=\frac{1}{2}\int_{0}^{L}\int_{A}\sigma_{x}\varepsilon_{x}AdAx=\frac{1}{2} \int_{0}^{L}\bigg{\{}D_{0}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{2}-2 D_{1}\frac{\partial u}{\partial x}\frac{\partial^{2}w}{\partial x^{2}}+D_{2} \bigg{(}\frac{\partial^{2}w}{\partial x^{2}}\bigg{)}^{2}\bigg{\}}dx \tag{4}\]  
Here,  
\[\left\{ \begin{array}{l}D_{0}=\int_{A}E(z)dA=E_{m}\int_{A}f(z)dA=E_{m}A\gamma _{0}\\ D_{1}=\int_{A}E(z)zdA=E_{m}\int_{A}f(z)zdA=E_{m}S\gamma_{1}\\ D_{2}=\int_{A}E(z)z^{2}dA=E_{m}\int_{A}f(z)z^{2}dA=E_{m}I\gamma_{2}\end{array} \right. \tag{5}\]  
The expressions of \(\gamma_{0}\), \(\gamma_{1}\), and \(\gamma_{2}\) are respectively,  
\[\gamma_{0}=\frac{1}{A}\int_{A}f(z)dA,\gamma_{1}=\frac{1}{S}\int_{A}f(z)zdA, \gamma_{2}=\frac{1}{I}\int_{A}f(z)z^{2}dA \tag{6}\]  
In Equation (6), \(A=bh\), \(S=bh^{2}/2\), \(I=bh^{3}/12\), the same as below. In the transverse vibration of an axially moving FGM beam, the velocity of any point on the beam is,  
\[\widetilde{v}_{x}=\frac{\partial\widetilde{u}(x,z,t)}{\partial t}=\frac{ \partial u(x,t)}{\partial t}-z\frac{\partial w(x,t)}{\partial x\partial t}+V \tag{7}\]  
\[\widetilde{v}_{z}=\frac{\partial w(x,t)}{\partial t}+V\frac{\partial w(x,t)}{ \partial x} \tag{8}\]  
Figure 1: Schematic diagram of the functional gradient beam and coordinate system. The kinetic energy \(K\) of the axially moving FGM beam is,  
\[ \begin{array}{l}K=\frac{1}{2}\int_{0}^{L}\int_{A}\rho(z)\big{(}\tilde{v}_{x}^{2} +\tilde{v}_{z}^{2}\big{)}dAdx\\ =\frac{1}{2}\int_{0}^{L}\bigg{\{}I_{0}\bigg{[}\bigg{(}\frac{\partial u}{\partial t }\bigg{)}^{2}+V^{2}+2V\frac{\partial u}{\partial t}+\Big{(}\frac{\partial v}{ \partial t}\bigg{)}^{2}+V^{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}+2 V\frac{\partial w}{\partial t}\frac{\partial w}{\partial x}\bigg{]}\\ -2I_{1}\Big{(}V\frac{\partial^{2}w}{\partial t\partial x}+\frac{\partial u}{ \partial t}\frac{\partial^{2}w}{\partial t\partial x}\Big{)}+I_{2}\Big{(} \frac{\partial^{2}w}{\partial t\partial x}\Big{)}^{2}\bigg{\}}dx\\ \end{array}  \tag{9}\]  
Here, the expressions for \(I_{0}\), \(I_{1}\), and \(I_{2}\) are, respectively,  
\[\left\{