## 2 Basic Theory and Calculation Formula
\begin{array}{cccc}\widetilde{W}^{(4)}=\boldsymbol{\tau}\widetilde{W}(\xi_ {0})+\boldsymbol{\tau}\widetilde{W}^{\prime}(\xi_{0})+\boldsymbol{\tau} \widetilde{W}^{\prime\prime}(\xi_{0})+\boldsymbol{\tau}\widetilde{W}^{\prime \prime\prime}(\xi_{0})+\boldsymbol{\tau}\widetilde{W}^{\prime\prime\prime}( \xi_{0})+\boldsymbol{\Gamma}\widetilde{W}^{(4)}(\xi_{i})\\ =\left[\boldsymbol{\tau},\boldsymbol{\tau},\boldsymbol{\tau},\boldsymbol{I} \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}_{4}^{*} \boldsymbol{J}_{4}\right]_{(N+1)\times(N+5)}\left(\begin{array}{c} \widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}\\ \text{Here},\boldsymbol{J}_{4}^{*}=\left[\boldsymbol{\tau},\boldsymbol{\tau}, \boldsymbol{\tau},\boldsymbol{\tau}\right]_{(N+1)\times 4^{\prime}}\boldsymbol{J}_{4}=\left[\boldsymbol{I} \right]_{(N+1)\times(N+1)^{\prime}},\end{array}  \tag{31e}\]  
Equation (31) is substituted into Equation (23) and converted into matrix form as follows,  
\[ \begin{array}{cccc}\left[\boldsymbol{K}_{11}^{*},\boldsymbol{K}_{12} \right]_{(N+1)\times(N+5)}\left(\begin{array}{c}\widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}-\Omega^{2}[ \boldsymbol{M}_{11}^{*},\boldsymbol{M}_{12}]_{(N+1)\times(N+5)}\left(\begin{array} []{c}\widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}\\ -\Omega[\boldsymbol{M}_{13}^{*},\boldsymbol{M}_{14}]_{(N+1)\times(N+5)}\left( \begin{array}{c}\widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}=0\\ \text{where},\quad\boldsymbol{K}_{11}^{*}=\boldsymbol{J}_{4}^{*}+g_{222} \boldsymbol{J}_{2}^{*},\boldsymbol{K}_{12}=\boldsymbol{J}_{4}+g_{222} \boldsymbol{J}_{2},\boldsymbol{M}_{11}^{*}=q_{222}\boldsymbol{J}_{2}^{*}-q_{2 20}\boldsymbol{J}_{0}^{*},\\ \boldsymbol{M}_{12}=q_{222}\boldsymbol{J}_{2}-q_{220}J_{0},\boldsymbol{M}_{13}^{* }=-q_{221}\boldsymbol{I}_{1}^{*},\boldsymbol{M}_{14}=-q_{221}\boldsymbol{I}_{1} \end{array} .\end{array} \tag{32}\]  
Since Equation (32) contains \(\Omega^{2}\), which makes the eigenvalue solution of generalized algebraic equations nonlinear, a new \((N+1)\times 1\)-order column vector \(\widetilde{\boldsymbol{Z}}(\xi)\) is introduced to quasi-linearize Equation (32). \[\widetilde{\boldsymbol{Z}}(\xi)=\Omega[\boldsymbol{M}_{11}^{*},\boldsymbol{M}_{ 12}]_{(N+1)\times(N+5)}\left( \begin{array}{c}\widetilde{\boldsymbol{W}}^{*} \\ \widetilde{\boldsymbol{W}}^{(4)}\end{array} \right)_{(N+5)\times 1} \tag{33}\]  
By substituting Equation (33) into Equation (32), we have:  
\[ \begin{array}{cccc}\left[\boldsymbol{K}_{11}^{*},\boldsymbol{K}_{12}\right]_ {(N+1)\times(N+5)}\left(\begin{array}{c}\widetilde{\boldsymbol{W}}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}-\Omega\widetilde{\boldsymbol{Z}}_{(N+1)\times 1}- \Omega[\boldsymbol{M}_{13}^{*},\boldsymbol{M}_{14}]_{(N+1)\times(N+5)}\left( \begin{array}{c}\widetilde{\boldsymbol{W}}^{*}\\ \widetilde{W}^{(4)}\end{array}\right)_{(N+5)\times 1}=0\\ \end{array}  \tag{34}\]  
By combining Equations (33) and (34) with the boundary condition in Equation (24), we have:  
\[ \begin{array}{cccc}\left(\begin{matrix}0&\boldsymbol{K}_{b}^{*}&\boldsymbol{K}_ {b}\\ \boldsymbol{I}&0&0\\ 0&\boldsymbol{K}_{11}^{*}&\boldsymbol{K}_{12}\end{matrix}\right)\left(\begin{matrix} \widetilde{\boldsymbol{W}}^{*}\\ \widetilde{W}^{(4)}\end{matrix}\right)_{[2(N+1)+4]\times 1}-\Omega\left(\begin{matrix}0&0&0\\ \boldsymbol{0}&\boldsymbol{M}_{11}^{*}&\boldsymbol{M}_{12}\\ \boldsymbol{I}&\boldsymbol{M}_{13}^{*}&\boldsymbol{M}_{14}\end{matrix}\right) \left(\begin{matrix}\widetilde{\boldsymbol{Z}}\\ \widetilde{W}^{*}\\ \widetilde{W}^{(4)}\end{matrix}\right)_{[2(N+1)+4]\times 1}=\left(\begin{matrix}0\\ 0\end{matrix}\right)\end{array}  \tag{35}\]When the axially moving FGM beam is simply supported at both ends (SS), the vector form of the boundary condition is,  
\[K_{b}^{*}=