## 2 Theoretical Model  
### Dynamic Analysis of Rotor Vibration Response Based on Centroid Transformation  
As shown in Figure 1, the reference axis of the traditional stacking assembly method is set through the center \(O\) of the rotor bottom face and perpendicular to the bottom face. However, the center line of bearing at both ends of multistage rotor is taken as the actual rotation axis when doing vibration analysis of rotary parts rotor system. That is to say, the actual rotation axis \(OO_{HA}\) is the connection line of the center of bottom face and the center of top face which deviates from the reference axis. Figure 1: Multistage rotor centroid vector transformation model. The position vector of \(k_{th}\) rotor centroid deviation can be expressed as follows in a \(XOYZ\) coordinate system [26]:  
\[\mathbf{dp}_{0\cdot k}^{c}=\left[ \begin{array}{c}dx_{0\cdot k}^{c}\\ dy_{0\cdot k}^{c}\\ dz_{0\cdot k}^{c}\end{array} \right]=\sum_{i=1}^{k-1}\left(\prod_{j=1}^{i-1} \left(dR_{j}^{r}dR_{j}^{o}\right)dR_{i}^{r}dp_{i}^{o}\right)+\prod_{i=1}^{k-1} \left(dR_{i}^{r}dR_{i}^{o}\right)dR_{k}^{r}dp_{k}^{c} \tag{1}\]  
where, \(dx_{0\cdot k}^{c},dy_{0\cdot k},dz_{0\cdot k}^{c}\) are the accumulated centroid deviation of \(k_{th}\) stage rotor in \(X\), \(Y\), \(Z\) direction. \(\mathbf{dR}_{i}^{r}\) is the rotation matrix of \(i_{th}\) stage rotor and can be expressed as:  
\[\mathbf{dR}_{i}^{r}=\left[ \begin{array}{ccc}\cos\theta_{ri}&-\sin\theta_{ri}&0\\ \sin\theta_{ri}&\cos\theta_{ri}&0\\ 0&0&1\end{array} \right] \tag{2}\]  
where, \(\theta_{ri}\) is the rotation angle of the \(i_{th}\) stage rotor around the reference axis. \(\mathbf{dR}_{i}^{o}\) is the geometric eccentricity deviation matrix of \(i_{th}\) stage rotor and can be expressed as:  
\[\mathbf{dR}_{i}^{o}=\left[ \begin{array}{ccc}1&0&\sin\theta_{ii}\cos\theta_{ii}\\ 0&1&\sin\theta_{it}\sin\theta_{ii}\\ -\sin\theta_{ii}\cos\theta_{ii}&-\sin\theta_{it}\sin\theta_{ii}&1\end{array} \right] \tag{3}\]  
where, \(\theta_{ii}\) is the tilt angle of the assembly surface relative to the reference plane, and \(\theta_{ii}\) is the position of the tilt angle on the assembly surface. \(\mathbf{dp}_{i}^{o}\) is the geometric center position vector matrix of \(i_{th}\) stage rotor and can be expressed as:  
\[\mathbf{dp}_{i}^{o}=\left[ \begin{array}{c}dx_{i}^{o}\\ dy_{i}^{o}\\ dz_{i}^{o}\end{array} \right] \tag{4}\]  
where, \(dx_{i}^{o},dy_{i}^{o},dz_{i}^{o}\) are the position vectors of the \(i_{th}\) rotor top face center in \(XOYZ\) coordinate system. \(\mathbf{dp}_{i}^{c}\) is the position vector matrix of the mass centroid of the \(i_{th}\) stage rotor, and can be expressed as:  
\[\mathbf{dp}_{i}^{c}=\left[ \begin{array}{c}dx_{i}^{c}\\ dy_{i}^{o}\\ dz_{i}^{c}\end{array} \right] \tag{5}\]  
where, \(dx_{i}^{c},dy_{i}^{c},dz_{i}^{c}\) are the position vectors of the \(i_{th}\) rotor mass centroid in \(XOYZ\) coordinate system. The position vector of the measuring surface center of the \(k_{th}\) stage rotor after assembly in the coordinate system \(XOYZ\) can be expressed as:  
\[\mathbf{dp}_{0\cdot k}^{o}=\left[ \begin{array}{c}dx_{0\cdot k}^{o}\\ dy_{0\cdot k}^{o}\\ dz_{0\cdot k}^{o}\end{array} \right]=\sum_{i=1}^{k}\left(\prod_{j=2}^{i}\left( dR_{j-1}^{r}dR_{j-1}^{o}\right)dR_{i}^{r}d\mathbf{p}_{i}^{o}\right) \tag{6}\]  
The coordinate system \(X^{\prime}OY^{\prime}Z^{\prime}\) can be obtained through the rotation of the coordinate system \(XOYZ\). Set the rotation axis direction vector as \(I=(l_{x}l_{y}l_{z})^{T}\). The unit vector of \(I\) can be expressed as: \(w(w_{x},w_{y},w_{z})^{T}=\frac{I}{|I|}\). The rotation axis direction vector \(I\) and the rotation angle \(\theta\) can be obtained from Equations (6)-(8). The position vector of the rotor top face center in the coordinate system \(XOYZ\) can be expressed as \(\mathbf{P}\big{(}O_{nAx},O_{nAy},O_{nAz}\big{)}^{T}\).