## 2 Control Equations for Euler-Bernoulli Rotor
### Free Vibration Equation of the Beam Element  
The control equation for the Euler-Bernoulli beam under free vibration conditions is derived, assuming that the beam segment is an isotropic circular cross-section with a uniform cross-sectional area throughout; the material of the beam segment is an elastic material with an elastic modulus denoted as \(E\). The (generalized) displacement of the beam element's centerline nodes can be expressed in the following form:  
\[v,w \tag{1}\]  
\[\theta_{y}=-\frac{\partial w}{\partial x},\theta_{z}=\frac{\partial v}{ \partial x} \tag{2}\]  
The displacement field can be expressed as  
\[u_{x}=-y\frac{\partial v}{\partial x}+z\frac{\partial w}{\partial x} \tag{3}\]  
Consequently, the coordinate representation of the micro-element \(dA\) of the beam element in the moving coordinate system is given by  
\[ \begin{cases}\xi\\ \eta\\ \xi\end{cases} =R_{x}R_{y}R_{z}\left( \begin{cases}x\\ y\\ z\end{cases} - \begin{cases}0\\ v\\ w\end{cases} \right) \tag{4}\]  
\(R_{x}\), \(R_{y}\), and \(R_{z}\) represent the fundamental rotation matrices about the \(X\), \(Y\), and \(Z\) axes. Utilizing transformation, the coordinates of the micro-element \(dA\) in the inertial coordinate system are delineated as  
\[ \begin{cases}x\\ y\\ z\end{cases} =R_{z}^{T}R_{y}^{T}R_{x}^{T} \begin{pmatrix}\xi\\ \eta\\ \zeta\end{pmatrix} + \begin{cases}0\\ v\\ w\end{cases}  \tag{5}\]  
The stress and strain within the beam element are hereby illustrated:In the equation, \(v,w\) represent the displacements of the micro-element \(dA\) in the \(x\) and \(y\) directions, respectively, while \(E\) denotes the elastic modulus. \[\varepsilon_{xx}=-y\frac{\partial^{2}v}{\partial x^{2}}+z\frac{\partial^{2}w}{ \partial x^{2}} \tag{6}\]  
\[\sigma_{xx}=E\left(-y\frac{\partial^{2}v}{\partial x^{2}}+z\frac{\partial^{2} w}{\partial x^{2}}\right)^{2} \tag{7}\]  
Consequently, the elastic potential energy of the micro-element \(dA\) is expressed as  
\[dU=\int_{A}\frac{1}{2}\sigma_{xx}\varepsilon_{xx}dA=\frac{1}{2}\int_{A}E\left( -y\frac{\partial^{2}v}{\partial x^{2}}+z\frac{\partial^{2}w}{\partial x^{2}} \right)^{2}dA=\frac{1}{2}EI\left[\left(\frac{\partial^{2}v}{\partial x^{2}} \right)^{2}+\left(\frac{\partial^{2}w}{\partial x^{2}}\right)^{2}\right] \tag{8}\]  
\[U=\frac{1}{2}\int_{0}^{L}EI\left[\left(\frac{\partial^{2}v}{\partial x^{2}} \right)^{2}+\left(\frac{\partial^{2}w}{\partial x^{2}}\right)^{2}\right]dx \tag{9}\]  
\[ \begin{array}{l}\delta U=\int_{0}^{L}EI\left[\frac{\partial^{2}v}{\partial x ^{2}}\delta\left(\frac{\partial^{2}v}{\partial x^{2}}\right)+\frac{\partial^ {2}w}{\partial x^{2}}\delta\left(\frac{\partial^{2}w}{\partial x^{2}}\right) \right]dx\\ =\int_{0}^{L}EI\frac{\partial^{2}v}{\partial x^{2}}d\left(\frac{\partial\delta v }{\partial x}\right)+\int_{0}^{L}EI\frac{\partial^{2}w}{\partial x^{2}}d\left( \frac{\partial\delta w}{\partial x}\right)\\ =EI\frac{\partial^{2}v}{\partial x^{2}}\delta\left(\frac{\partial v}{\partial x }\right)-EI\frac{\partial^{2}v}{\partial x^{3}}(\delta v)+EI\frac{\partial^{ 2}w}{\partial x^{2}}\delta\left(\frac{\partial w}{\partial x}\right)-EI\frac {\partial^{3}w}{\partial x^{3}}(\delta w)\\ +\int_{0}^{L}EI\frac{\partial^{4}v}{\partial x^{4}}\delta vdx+\int_{0}^{L}EI \frac{\partial^{4}w}{\partial x^{4}}\delta wdx\end{array}  \tag{10}\]  
Examining the micro-element \(dA\), its coordinates in the moving coordinate system \(O\xi\eta\zeta\) are \((0,\eta,\zeta)\), and in the inertial coordinate system \(Oxyz\) are  
\[ \begin{cases}x\\ y\\ z\end{cases} =R_{z}^{T}R_{y}^{T}R_{x}^{T} \begin{pmatrix}0\\ \eta\\ \zeta\end{pmatrix} + \begin{cases}0\\ v\\ w\end{cases}  \tag{11}\]  
Since the moving coordinate system is rigidly attached to the plane, the velocity of the micro-element in the moving coordinate system is zero. Consequently, in the inertial coordinate system, the velocity of the micro-element is derived as  
\[ \begin{cases}\dot{x}\\ \dot{y}\\ \dot{z}\end{cases} =\frac{\partial\left(R_{x}^{T}R_{z}^{T}R_{y}^{T}\right)}{ \partial t} \begin{pmatrix}0\\ \eta\\ \zeta\end{pmatrix} +