## 2 Basic Theory and Calculation Formula
\begin{array}{l}I_{0}=\int_{A}\rho(z)dA=\rho_{m}\int_{A}g(z)dA=\rho_{ m}A\alpha_{0}\\ I_{1}=\int_{A}\rho(z)zdA=\rho_{m}\int_{A}g(z)zdA=\rho_{m}S\alpha_{1}\\ I_{2}=\int_{A}\rho(z)z^{2}dA=\rho_{m}\int_{A}g(z)z^{2}dA=\rho_{m}I\alpha_{2} \end{array} \right. \tag{10}\]  
In Equation (10), the expressions for \(\alpha_{0}\), \(\alpha_{1}\), and \(\alpha_{2}\) are respectively,  
\[\alpha_{0}=\frac{1}{A}\int_{A}g(z)dA,\alpha_{1}=\frac{1}{S}\int_{A}g(z)zdA, \alpha_{2}=\frac{1}{I}\int_{A}g(z)z^{2}dA \tag{11}\]  
According to the Hamilton principle \(\delta\int_{I_{1}}^{I_{2}}(K-U)dt=0\), due to the arbitrariness of the variable term, its coefficient of the variable term is zero [18], thus,  
\[D_{1}\frac{\partial^{3}w}{\partial x^{3}}-I_{1}\frac{\partial^{3}w}{\partial t ^{2}\partial x}-D_{0}\frac{\partial^{2}u}{\partial x^{2}}=0 \tag{12}\]  
\[D_{2}\frac{\partial^{4}w}{\partial x^{4}}+I_{0}\bigg{(}\frac{\partial^{2}w}{ \partial t^{2}}+V^{2}\frac{\partial^{2}w}{\partial x^{2}}+2V\frac{\partial^{2 }w}{\partial t\partial x}\bigg{)}-I_{2}\frac{\partial^{4}w}{\partial t^{2} \partial x^{2}}+D_{1}\frac{\partial^{3}u}{\partial x^{3}}=0 \tag{13}\]  
By calculating the first-order partial derivatives of both sides of the equal sign in Equation (12) with regard to \(x\), the value \(\frac{\partial^{3}u}{\partial x^{3}}\) can be obtained as:  
\[D_{0}\frac{\partial^{3}u}{\partial x^{3}}=D_{1}\frac{\partial^{4}w}{\partial x ^{4}}-I_{1}\frac{\partial^{4}w}{\partial t^{2}\partial x^{2}} \tag{14}\]  
By substituting Equation (14) into Equation (13) for decoupling, the kinematic differential equation of axially moving FGM beam can be obtained,  
\[\left(D_{2}-\frac{D_{1}^{2}}{D_{0}}\right)\frac{\partial^{4}w}{\partial x^{4} }-\left(I_{2}-I_{1}\frac{D_{1}}{D_{0}}\right)\frac{\partial^{4}w}{\partial t^ {2}\partial x^{2}}+I_{0}\bigg{(}\frac{\partial^{2}w}{\partial t^{2}}+V^{2} \frac{\partial^{2}w}{\partial x^{2}}+2V\frac{\partial^{2}w}{\partial t \partial x}\bigg{)}=0 \tag{15}\]  
The boundary conditions of a simply supported beam (SS) at both ends are [24],  
\[w(0,t)=0,\;w(L,t)=0,\;w^{\prime\prime}(0,t)=0,\;w^{\prime\prime}(L,t)=0\] (16a) The boundary conditions of the clamp supported beam (CC) at both ends are, \[w(0,t)=0,\;w(L,t)=0,\;w^{\prime}(0,t)=0,\;w^{\prime}(L,t)=0 \tag{16b}\]  
The boundary conditions of a beam (CS) clamp supported at one end and simply supported at the other end are,  
\[w(0,t)=0,\;w(L,t)=0,\;w^{\prime}(0,t)=0,\;w^{\prime\prime}(L,t)=0 \tag{16c}\]Here, \((\cdots)^{\prime}=\mathrm{d}(\cdots)/\mathrm{d}x\), \((\cdots)^{\prime\prime}=\mathrm{d}^{2}(\cdots)/\mathrm{d}x^{2}\), the same as below. In order to facilitate the subsequent programming calculation, the following normalized variables are introduced. \[W\big{(}\xi,\widetilde{t}\big{)}=\frac{w(x,t)}{L},\,\xi=\frac{x}{L},\,\widetilde {t}=\frac{t}{L^{2}}\sqrt{\frac{E_{m}I}{\rho_{m}A}},\,v=VL\sqrt{\frac{E_{m}I}{ \rho_{m}A}} \tag{17}\]  
By substituting Equation (17) into the motion differential Equation (15) of an axially moving FGM beam, there is  
\[\phi_{1}\frac{\partial^{4}W\big{(}\xi,\widetilde{t}\big{)}}{\partial\xi^{4}}+ \frac{\partial^{2}W\big{(}\xi,\widetilde{t}\big{)}}{\partial\widetilde{t}^{2} }+v^{2}\frac{\partial^{2}W\big{(}\xi,\widetilde{t}\big{)}}{\partial\xi^{2}} +2v\frac{\partial^{2}W(\xi,v)}{\partial\xi\partial\widetilde{t}}-\phi_{2} \frac{\partial^{4}W(\xi,v)}{\partial\xi^{2}\partial\widetilde{t}^{2}}=0 \tag{18}\]  
where,  
\[D_{1}\frac{\partial^{3}w}{\partial x^{3}}-I_{1}\frac{\partial^{3}w}{\partial t ^{2}\partial x}-D_{0}\frac{\partial^{2}u}{\partial x^{2}}=0 \tag{12}\]0  
Here, \(\lambda\) is the length-height ratio of the FGM beam. If the axial motion velocity of the beam exceeds a certain critical value, the axially moving FGM beam may become unstable.