## 3 Establishment of the Dynamic Model of Three-Support Shafting
### Derivation of Motion Equation of the Three-Support Shafting  
The second kind of Lagrange equation is used to derive the motion differential equations of each element:  
\[\frac{d}{dt}\!\left(\frac{\partial T}{\partial\dot{q}_{j}}\right)-\left(\frac {\partial T}{\partial\dot{q}_{j}}\right)+\left(\frac{\partial V}{\partial\dot {q}_{j}}\right)=Q_{j}\ \ \ (j=1,2,\ldots N) \tag{3}\]  
Figure 7: Dynamic model of the rotor shaft system. where \(T\) is the kinetic energy of the system, \(V\) is the potential energy of the system, \(q_{j}\) is the generalized coordinate of the system, \(Q_{j}\) is the generalized force on the system and \(N\) is the number of degrees of freedom of the system. Suppose generalized coordinates are represented as  
\[q_{1}=y,\ q_{2}=z,\ q_{3}=\theta_{y},\ q_{4}=\theta_{z}\]  
If the mass, diameter and polar moment of inertia of a rigid disk are \(m\), \(J_{d}\) and \(J_{p}\), then the motion equation of the disk is  
\[\left\{ \begin{array}{l}[M_{d}]\Big{[}\ddot{u}_{1d}\Big{]}+\Omega[J_{d}] \Big{[}\dot{u}_{2d}\Big{]}=\{Q_{1d}\}\\ [M_{d}]\Big{[}\ddot{u}_{2d}\Big{]}+\Omega[J_{d}]\Big{[}\dot{u}_{1d}\Big{]}=\{Q _{2d}\}\end{array} \right. \tag{4}\]  
where \([M_{d}]\) is the mass matrix of the disk, \(\{Q_{1d}\}\) and \(\{Q_{2d}\}\) are the generalized forces of the vibration response along the y-axis and z-axis and, supposing \([G_{d}]=\Omega[J_{d}]\), then \([G_{d}]\) is the rotation matrix. Assuming that the disk has a small eccentricity (\(e_{y0}\), \(e_{z0}\)) and the influence of eccentricity on the moment of inertia is ignored, then the unbalanced force included in the generalized force can be approximately expressed as  
\[\left\{ \begin{array}{l}\{Q_{1d}^{u}\}=m\Omega^{2}\bigg{(}\left\{\begin{array} []{c}e_{y0}\\ 0\end{array}\right\}\cos\Omega t+\left\{\begin{array}{c}-e_{z0}\\ 0\end{array}\right\}\sin\Omega t\bigg{)}\\ \{Q_{2d}^{u}\}=m\Omega^{2}\bigg{(}\left\{\begin{array}{c}e_{z0}\\ 0\end{array}\right\}\cos\Omega t+\left\{\begin{array}{c}-e_{y0}\\ 0\end{array}\right\}\sin\Omega t\bigg{)}\end{array} \right. \tag{5}\]  
The equation of the motion of the elastic shaft segment element is as follows:  
\[\left\{ \begin{array}{l}[M_{x}]\Big{[}\ddot{u}_{1x}\Big{]}+\Omega[J_{x}] \Big{[}\dot{u}_{2x}\Big{]}+[K_{x}]\big{[}u_{1x}\big{]}=\{Q_{1x}\}\\ [M_{x}]\Big{[}\dot{u}_{2x}\Big{]}-\Omega[J_{x}]\Big{[}\dot{u}_{1x}\Big{]}+[K_{ x}]\big{[}u_{2x}\big{]}=\{Q_{2x}\}\end{array} \right. \tag{6}\]  
where \([M_{x}]=[M_{xT}]+[M_{xR}]\), \([M_{xT}]\) is the moving inertia matrix of the element, \([M_{xR}]\) is the rotational inertia matrix of the element, \([K_{x}]\) is the stiffness matrix, and \([Q_{1x}]\) and \([Q_{2x}]\) are generalized force vectors, including the forces and moments of inertia of disks or adjacent elements connected at the nodes. Assuming that the coordinates of the bearing support center are \(y_{b}\) and \(z_{b}\) and the node number corresponding to the journal center is \(s(j)\), then the coordinates of the journal center are \(y_{x}(j)\) and \(z_{x}(j)\). Assuming that the bearing support center coincides with the journal center, the motion equation of the bearing support is as follows:  
\[\left[ \begin{array}{cc}M_{b}&0\\ 0&M_{b}\end{array} \right]\left\{ \begin{array}{l}\ddot{y}_{x(j)}\\ \ddot{z}_{x(j)}\end{array} \right\}+\left[ \begin{array}{cc}c_{yy}&c_{yz}\\ c_{zy}&c_{zz}\end{array} \right]\left\{ \begin{array}{l}\dot{y}_{x(j)}\\ \dot{z}_{x(j)}\end{array} \right\}+\left[ \begin{array}{cc}k_{yy}&k_{yz}\\ k_{zy}&k_{zz}\end{array} \right]\left\{ \begin{array}{l}y_{x(j)}\\ z_{x(j)}\end{array} \right\}=\{0\} \tag{7}\]  
The generalized force acting on the journal node by the bearing support is  
\[\left\{ \begin{array}{l}Q_{1d}^{b}\\ Q_{2d}^{b}\end{array} \right\}=-\left[ \begin{array}{cc}M_{b}&0\\ 0&M_{b}\end{array} \right]\left\{ \begin{array}{l}\ddot{y}_{x(j)}\\ \ddot{z}_{x(j)}\end{array} \right\}-\left[ \begin{array}{cc}c_{yy}&c_{yz}\\ c_{zy}&c_{zz}\end{array} \right]\left\{ \begin{array}{l}\dot{y}_{x(j)}\\ \dot{z}_{x(j)}\end{array} \right\}-\left[