## 2 Basic Theory and Calculation Formula
The solution of the eigenvalue of ordinary differential equations is transformed into the solution of the eigenvalue of standard generalized algebraic equation sets by means of an integral matrix. Figure 2 shows the calculation model of an axially moving FGM beam by the interpolating matrix method (IMM). The normalized interval [0,1] is divided into \(N\) discrete elements, \(\xi_{0}=0\), \(\xi_{N}=1\). The length of a discrete element is \(\Delta L_{i}=\xi_{i}-\xi_{i-1}=1/N\), and the third derivative of an unknown function \(\widetilde{W}(\xi)\) in an ordinary differential Equation (26) is expressed by the function value of a discrete node on the beam length interval using the difference method. \[D_{2}\frac{\partial^{4}w}{\partial x^{4}}+I_{0}\bigg{(}\frac{\partial^{2}w}{ \partial t^{2}}+V^{2}\frac{\partial^{2}w}{\partial x^{2}}+2V\frac{\partial^{2 }w}{\partial t\partial x}\bigg{)}-I_{2}\frac{\partial^{4}w}{\partial t^{2} \partial x^{2}}+D_{1}\frac{\partial^{3}u}{\partial x^{3}}=0 \tag{13}\]0  
The unknown derivative \(\widetilde{W}^{(4)}(\xi)\) in the above equation is approximated by the interpolating function, i.e.,  
\[D_{2}\frac{\partial^{4}w}{\partial x^{4}}+I_{0}\bigg{(}\frac{\partial^{2}w}{ \partial t^{2}}+V^{2}\frac{\partial^{2}w}{\partial x^{2}}+2V\frac{\partial^{2 }w}{\partial t\partial x}\bigg{)}-I_{2}\frac{\partial^{4}w}{\partial t^{2} \partial x^{2}}+D_{1}\frac{\partial^{3}u}{\partial x^{3}}=0 \tag{13}\]1  
\[D_{2}\frac{\partial^{4}w}{\partial x^{4}}+I_{0}\bigg{(}\frac{\partial^{2}w}{ \partial t^{2}}+V^{2}\frac{\partial^{2}w}{\partial x^{2}}+2V\frac{\partial^{2 }w}{\partial t\partial x}\bigg{)}-I_{2}\frac{\partial^{4}w}{\partial t^{2} \partial x^{2}}+D_{1}\frac{\partial^{3}u}{\partial x^{3}}=0 \tag{13}\]2 Interpolation Matrix Method Calculation Model for Axially Moving FGM Beams. \[ \begin{array}{cccc}\text{Here},&J_{1}^{*}&=&\left[\boldsymbol{\tau},\boldsymbol{ \sigma},\boldsymbol{D}\boldsymbol{\sigma},\boldsymbol{D}^{2}\boldsymbol{\sigma} \right]_{(N+1)\times 4^{\prime}}\\ \boldsymbol{I}_{1}=\left[\boldsymbol{D}^{3}\right]_{(N+1)\times(N+1)}, \,\tilde{\boldsymbol{W}}^{\prime}=\left\{\widetilde{W}^{\prime}(\xi_{0}), \widetilde{W}^{\prime}(\xi_{1}),\widetilde{W}^{\prime}(\xi_{2}),\cdots, \widetilde{W}^{\prime}(\xi_{N})\right\}^{T}.\\ \widetilde{\boldsymbol{W}}=\sigma\widetilde{W}(\xi_{0})+\boldsymbol{D} \boldsymbol{\sigma}\widetilde{W}^{\prime}(\xi_{0})+\boldsymbol{D}^{2} \boldsymbol{\sigma}\widetilde{W}^{\prime\prime}(\xi_{0})+\boldsymbol{D}^{3} \boldsymbol{\sigma}\widetilde{W}^{\prime\prime\prime}(\xi_{0})+\boldsymbol{ D}^{4}\widetilde{W}^{(4)}(\xi_{i})\\ =\left[\boldsymbol{\sigma},\boldsymbol{D}\boldsymbol{\sigma},\boldsymbol{D}^{2 }\boldsymbol{\sigma},\boldsymbol{D}^{3}\boldsymbol{\sigma},\boldsymbol{D}^{4 }\right]_{(N+1)\times(N+5)}\left(\begin{array}{c}\widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}=\left[\boldsymbol{J}_{0}^{*} \boldsymbol{J}_{0}\right]_{(N+1)\times(N+5)}\left(\begin{array}{c} \widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}\\ \text{Here},\quad\boldsymbol{J}_{0}^{*}=\left[\boldsymbol{\sigma},\boldsymbol{D }\boldsymbol{\sigma},\boldsymbol{D}^{2}\boldsymbol{\sigma},\boldsymbol{D}^{3} \boldsymbol{\sigma}\right]_{(N+1)\times 4^{\prime}},\quad\boldsymbol{J}_{0}=\left[ \boldsymbol{D}^{4}\right]_{(N+1)\times(N+1)},\\ \widetilde{\boldsymbol{W}}=\left\{\widetilde{W}(\xi_{0}),\widetilde{W}(\xi_{1 }),\widetilde{W}(\xi_{2}),\cdots,\widetilde{W}(\xi_{N})\right\}^{T}.\\ \end{array}  \tag{31d}\]  
The highest derivative \(\widetilde{W}^{(4)}(\xi)\) of the ordinary differential equation (in Equation (23)) can be written directly in vector form as follows:  
\[