## 2 Control Equations for Euler-Bernoulli Rotor
### Free Vibration Equation of the Beam Element
\begin{split}&\int_{x_{i}}^{x_{j}}\left(EI\frac{\partial^{4}w}{ \partial x^{4}}\right)\delta wdx=EI\frac{\partial^{3}w}{\partial x^{3}}(\delta w )_{x_{i}}^{x_{j}}-EI\frac{\partial^{2}w}{\partial x^{2}}\delta\bigg{(}\frac{ \partial w}{\partial x}\bigg{)}_{x_{i}}^{x_{j}}+\int_{x_{i}}^{x_{j}}EI\frac{ \partial^{2}w}{\partial x^{2}}\delta\bigg{(}\frac{\partial^{2}w}{\partial x^{2 }}\bigg{)}dx\\ &\int_{x_{i}}^{x_{j}}f_{\Omega}\frac{\partial^{2}\tilde{v}}{ \partial x^{2}}\delta wdx=J_{p}\Omega\frac{\partial\tilde{v}}{\partial x}( \delta w)_{x_{i}}^{x_{j}}-\int_{x_{i}}^{x_{j}}J_{p}\Omega\frac{\partial\tilde{v }}{\partial x}\delta\bigg{(}\frac{\partial w}{\partial x}\bigg{)}dx\\ &\int_{x_{i}}^{x_{j}}J_{d}\frac{\partial^{2}\tilde{w}}{\partial x ^{2}}\delta vdx=J_{d}\frac{\partial\tilde{w}}{\partial x}(\delta w)_{x_{i}}^{x _{j}}-\int_{x_{i}}^{x_{j}}J_{d}\frac{\partial\tilde{w}}{\partial x}\delta \bigg{(}\frac{\partial w}{\partial x}\bigg{)}dx\\ &\int_{x_{i}}^{x_{j}}\left(EI\frac{\partial^{4}w}{\partial x^{4}}+ m\tilde{w}+J_{p}\Omega\frac{\partial^{2}\tilde{v}}{\partial x^{2}}-J_{d} \frac{\partial^{2}\tilde{w}}{\partial x^{2}}\right)\delta wdx\\ &=\int_{x_{i}}^{x_{j}}\left[EI\frac{\partial^{2}w}{\partial x^{2 }}\delta\bigg{(}\frac{\partial^{2}w}{\partial x^{2}}\bigg{)}+m\tilde{w}\delta w -J_{p}\Omega\frac{\partial\tilde{v}}{\partial x}\delta\bigg{(}\frac{\partial w }{\partial x}\bigg{)}+\int_{x_{i}}^{x_{j}}J_{d}\frac{\partial\tilde{w}}{ \partial x}\delta\bigg{(}\frac{\partial w}{\partial x}\bigg{)}\right]dx\\ \end{split}  \tag{24}\]  
Hereby, the weak form of the beam element's control equation is rewritten as  
\[ \begin{split}&\int_{x_{i}}^{x_{j}}\left[EI\frac{\partial^{2}v}{ \partial x^{2}}\delta\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}\bigg{)}+m \tilde{w}\delta v+J_{p}\Omega\frac{\partial\tilde{w}}{\partial x}\delta\bigg{(} \frac{\partial w}{\partial x}\bigg{)}+J_{d}\frac{\partial\tilde{w}}{\partial x }\delta\bigg{(}\frac{\partial v}{\partial x}\bigg{)}dx\\ &+EI\frac{\partial^{2}w}{\partial x^{2}}\delta\bigg{(}\frac{ \partial^{2}w}{\partial x^{2}}\bigg{)}+m\tilde{w}\delta w-J_{p}\Omega\frac{ \partial\tilde{w}}{\partial x}\delta\bigg{(}\frac{\partial w}{\partial x} \bigg{)}+\int_{x_{i}}^{x_{j}}J_{d}\frac{\partial\tilde{w}}{\partial x}\delta \bigg{(}\frac{\partial w}{\partial x}\bigg{)}\right]dx\\ \end{split}  \tag{25}\]  
Considering a two-node beam element with eight degrees of freedom, the nodal displacement array is denoted as \(\{\delta^{\varepsilon}\}=\left\{v_{i},w_{i},v_{i^{\prime}}^{\prime},w_{i}^{ \prime},v_{j},w_{j},v_{j^{\prime}}^{\prime},w_{j}^{\prime}\right\}^{T}\). The Hermite interpolation expression is given by  
\[v=\{N_{1},N_{2},N_{3},N_{4}\}\left\{ \begin{array}{l}v_{i}\\ v_{i}^{\prime}\\ v_{j}\\ v_{j}^{\prime}\end{array} \right\},w=\{N_{1},N_{2},N_{3},N_{4}\}\left\{ \begin{array}{l}w_{i}\\ w_{i}^{\prime}\\ w_{j}\\ w_{j}^{\prime}\end{array} \right\} \tag{26}\]  
Alternatively,  
\[v=\{N_{p}\}\{\delta^{\varepsilon}\},w=\{N_{w}\}\{\delta^{\varepsilon}\},\left\{ \begin{array}{l}v\\ w\end{array} \right\}=\{N\}\{\delta^{\varepsilon}\} \tag{27}\]  
\[\{N\}=\left\{ \begin{array}{l}N_{p}\\ N_{w}\end{array} \right\}=\left\{ \begin{array}{l}N_{1}&0&N_{2}&0&N_{3}&0&N_{4} &0\\ 0&N_{1}&0&N_{2}&0&N_{3}&0&N_{4}\end{array} \right\} \tag{28}\]  
where the components of the shape function matrix are given by  
\[ \begin{split}& N_{1}=\left(1+2\frac{x-x_{i}}{x_{j}-x_{i}}\right) \left(\frac{x-x_{i}}{x_{i}-x_{j}}\right)^{2},N_{2}=\left(x-x_{i}\right)\left( \frac{x-x_{i}}{x_{i}-x_{j}}\right)^{2}\\ & N_{3}=\left(1+2\frac{x-x_{i}}{x_{i}-x_{j}}\right)\left(\frac{x-x_{i}}{x_{j}-x_ {i}}\right)^{2},N_{4}=\left(x-x_{j}\right)\left(\frac{x-x_{i}}{x_{j}-x_{i}} \right)^{2}\end{split}  \tag{29}\]  
Substituting Equation (27) into the weak form of the control equation, we obtain  
\[