## 2 Control Equations for Euler-Bernoulli Rotor
### Free Vibration Equation of the Beam Element
\begin{split}& v=\{N_{p}\}\{\delta^{\varepsilon}\},w=\{N_{w}\}\{ \delta^{\varepsilon}\}\\ &\frac{\partial v}{\partial x}=\left\{\frac{\partial N_{w}}{\partial x} \right\}\{\delta^{\varepsilon}\},\frac{\partial w}{\partial x}=\left\{\frac{ \partial N_{w}}{\partial x}\right\}\{\delta^{\varepsilon}\},\frac{\partial^{2} v}{\partial x^{2}}=\left\{\frac{\partial^{2}N_{w}}{\partial x^{2}}\right\}\{ \delta^{\varepsilon}\},\frac{\partial^{2}w}{\partial x^{2}}=\left\{\frac{\partial^{2} N_{w}}{\partial x^{2}}\right\}\{\delta^{\varepsilon}\}\end{split}  \tag{30}\]\[\int_{x_{i}}^{x_{j}}\left[\begin{array}{l}EI\Bigg{(}\bigg{\{}\frac{ \partial^{2}N_{p}}{\partial x^{2}}\bigg{\}}\delta\{\delta^{\varepsilon}\} \bigg{)}^{T}\bigg{\{}\frac{\partial^{2}N_{p}}{\partial x^{2}}\bigg{\}}\{\delta^ {\varepsilon}\}+m(\{N_{v}\}\delta\{\delta^{\varepsilon}\})^{T}\{N_{v}\}\{ \tilde{\delta}^{\varepsilon}\}\\ +J_{p}\Omega\bigg{(}\bigg{\{}\frac{\partial N_{v}}{\partial x}\bigg{\}}\delta \{\delta^{\varepsilon}\}\bigg{)}^{T}\bigg{(}\bigg{\{}\frac{\partial N_{w}}{ \partial x}\bigg{\}}\bigg{\{}\tilde{\delta}^{\varepsilon}\}\bigg{)}+J_{d} \Big{(}\bigg{\{}\frac{\partial N_{v}}{\partial x}\bigg{\}}\delta\{\delta^{ \varepsilon}\}\bigg{)}^{T}\bigg{(}\bigg{\{}\frac{\partial N_{v}}{\partial x} \bigg{\}}\bigg{\{}\tilde{\delta}^{\varepsilon}\}\bigg{)}\\ +EI\bigg{(}\bigg{\{}\frac{\partial^{2}N_{w}}{\partial x^{2}}\bigg{\}}\delta \{\delta^{\varepsilon}\}\bigg{)}^{T}\bigg{(}\bigg{\{}\frac{\partial^{2}N_{w}} {\partial x^{2}}\bigg{\}}\{\delta^{\varepsilon}\}\bigg{)}+m(\{N_{w}\}\delta\{ \delta^{\varepsilon}\})^{T}\big{(}\{N_{w}\}\Big{\{}\tilde{\delta}^{\varepsilon }\}\big{)}\\ -J_{p}\Omega\bigg{(}\bigg{\{}\frac{\partial N_{w}}{\partial x}\bigg{\}}\delta \{\delta^{\varepsilon}\}\bigg{)}^{T}\bigg{(}\bigg{\{}\frac{\partial N_{v}}{ \partial x}\bigg{\}}\bigg{\{}\tilde{\delta}^{\varepsilon}\}\bigg{)}+J_{d} \Big{(}\bigg{\{}\frac{\partial N_{w}}{\partial x}\bigg{\}}\delta\{\delta^{ \varepsilon}\}\bigg{)}^{T}\bigg{(}\bigg{\{}\frac{\partial N_{w}}{\partial x} \bigg{\}}\bigg{\{}\tilde{\delta}^{\varepsilon}\}\bigg{)}\\ =\int_{x_{i}}^{x_{j}}\left[ \begin{array}{l}\delta\{\delta^{\varepsilon}\}^{ T}\Bigg{(}\bigg{\{}\frac{\partial^{2}N_{v}}{\partial x^{2}}\bigg{\}}^{T}EI \bigg{\{}\frac{\partial^{2}N_{v}}{\partial x^{2}}\bigg{\}}+\bigg{\{}\frac{ \partial^{2}N_{w}}{\partial x^{2}}\bigg{\}}^{T}EI\bigg{\{}\frac{\partial^{2}N _{w}}{\partial x^{2}}\bigg{\}}\bigg{)}\{\delta^{\varepsilon}\}\\ +\delta\{\delta^{\varepsilon}\}^{T}\bigg{(}\{N_{v}\}^{T}m\{N_{v}\}+\{N_{w}\}^{ T}m\{N_{w}\}+\bigg{\{}\frac{\partial N_{v}}{\partial x}\bigg{\}}^{T}J_{d} \bigg{\{}\frac{\partial N_{v}}{\partial x}\bigg{\}}+\bigg{\{}\frac{\partial N _{w}}{\partial x}\bigg{\}}^{T}J_{d}\bigg{\{}\frac{\partial N_{w}}{\partial x }\bigg{\}}\bigg{\}}\{\tilde{\delta}^{\varepsilon}\}\\ +\delta\{\delta^{\varepsilon}\}^{T}\bigg{(}\bigg{\{}\frac{\partial N_{v}}{ \partial x}\bigg{\}}^{T}I_{p}\bigg{\{}\frac{\partial N_{w}}{\partial x}\bigg{\}} -\bigg{\{}\frac{\partial N_{w}}{\partial x}\bigg{\}}^{T}I_{p}\bigg{\{}\frac{ \partial N_{v}}{\partial x}\bigg{\}}\bigg{)}\Omega\Big{\{}\tilde{\delta}^{ \varepsilon}\}\end{array} \right]dx\]  
Upon rearrangement, the motion equation for the beam element is derived as  
\[\{\textbf{K}^{\textbf{e}}\}\{\textbf{S}^{\textbf{e}}\}+\{\textbf{M}^{\textbf{e} }\}\Big{\{}\tilde{\textbf{S}}^{\textbf{e}}\}+\Omega\{\textbf{G}^{\textbf{e}} \}\Big{\{}\tilde{\textbf{S}}^{\textbf{e}}\}=0 \tag{33}\]  
Herein, \(\{\textbf{K}^{\textbf{e}}\}\), \(\{\textbf{M}^{\textbf{e}}\}\), and \(\{\textbf{G}^{\textbf{e}}\}\) represent the stiffness matrix, mass matrix, and gyroscopic matrix of the beam element, respectively.