## 2 Theoretical Model
### Dynamic Analysis of Rotor Vibration Response Based on Centroid Transformation
The position vector of the rotor top face center in the coordinate system \(X^{\prime}OY^{\prime}Z^{\prime}\) can be expressed as \(\mathbf{Q}(0,0,O_{nAz^{\prime}})^{T}\), where,\(\sqrt{\left(O_{nAx}\right)^{2}+\left(O_{nAy}\right)^{2}+\left(O_{nAz}\right)^{2}}\) according to the geometric relationship. The rotation axis direction vector \(I\) and rotation angle \(\theta\) are shown in Equations (7) and (8)  
\[\mathbf{I}=\mathbf{P}\times\mathbf{Q}=\left[ \begin{array}{c}O_{nAy}O_{nAz}^{\prime}\\ -O_{nAx}O_{nAz}^{\prime}\\ 0\end{array} \right] \tag{7}\]  
\[\theta=\arccos\left(\frac{\mathbf{P}\cdot\mathbf{Q}}{|\mathbf{P}||\mathbf{Q}|}\right) \tag{8}\]  
Set the mass centroid position vector of the \(k_{th}\) rotor as \((dx^{\prime c}_{0\cdot k},dy^{\prime c}_{0\cdot k},dz^{\prime c}_{0\cdot k})^{ \mathrm{T}}\), and it can be expressed in the \(X^{\prime}OY^{\prime}Z^{\prime}\) coordinate system:  
\[\left[ \begin{array}{c}dx^{\prime c}_{0\cdot k}\\ dy^{\prime}_{0\cdot k}\\ dz^{\prime c}_{0\cdot k}\end{array} \right]=\mathbf{A}\left[ \begin{array}{c}dx^{ c}_{0-k}\\ dy^{\prime}_{0-k}\\ dz^{\prime c}_{0-k}\end{array} \right] \tag{9}\]  
where, \(\mathbf{A}\) is the rotation matrix which can transfer coordinate system \(XOYZ\) to coordinate system \(OX^{\prime}Y^{\prime}Z^{\prime}\) and can be expressed as follows:  
\[\mathbf{A}=\left[ \begin{array}{ccc}\cos\theta+w_{x}^{2}(1-\cos\theta)&w_{x}w_{y} (1-\cos\theta)-w_{z}\sin\theta&w_{y}\sin\theta+w_{x}w_{z}(1-\cos\theta)\\ w_{z}\sin\theta+w_{x}w_{y}(1-\cos\theta)&\cos\theta+w_{y}^{2}(1-\cos\theta)& -w_{x}\sin\theta+w_{y}w_{z}(1-\cos\theta)\\ -w_{y}\sin\theta+w_{x}w_{z}(1-\cos\theta)&w_{x}\sin\theta+w_{y}w_{z}(1-\cos \theta)&\cos\theta+w_{z}^{2}(1-\cos\theta)\end{array} \right] \tag{10}\]  
where, \(\theta=\arccos\left(\frac{\mathbf{P}\cdot\mathbf{Q}}{|\mathbf{P}||\mathbf{Q}|}\right)\). \(Q^{\prime}\) is the unbalanced exciting force of rotor in the \(X^{\prime}OY^{\prime}Z^{\prime}\) coordinate system, and can be expressed as Equation (11) when the rotation speed of the rotor is \(\Omega\). \[\mathbf{Q}^{\prime}=m_{k}\Omega^{2}\left[ \begin{array}{c}dx^{\prime c}_{0\cdot k }\\ dy^{\prime c}_{0\cdot k}\\ 0\\ 0\end{array} \right]\cos\Omega t+m_{k}\Omega^{2}\left[ \begin{array}{c}-dy^{ \prime c}_{0\cdot k}\\ dx^{\prime c}_{0\cdot k}\\ 0\\ 0\end{array} \right]\sin\Omega t \tag{11}\]  
The unbalanced force of the system can be expressed as:  
\[\mathbf{Q}^{\prime}=\mathbf{Q}_{c}\cos\Omega t+\mathbf{Q}_{s}\sin\Omega t \tag{12}\]  
Then the steady solution of the motion differential equation of the rotor system can be assumed to be:  
\[\mathbf{q}=\mathbf{q}_{c}\cos\Omega t+\mathbf{q}_{s}\sin\Omega t \tag{13}\]  
The steady motion equation of rotor system is shown as follows:  
\[\mathbf{M}\ddot{\mathbf{q}}+(\mathbf{C}-\Omega\mathbf{G})\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}=\mathbf{Q} \tag{14}\]  
where \(\mathbf{M}\) is the mass assembly matrix which contains the translational and rotary effects of the shaft element and the disk element. Just like \(\mathbf{M}\), \(\mathbf{G}\) is the gyroscopic assembly matrix which considers the effects of the shaft element and the disk element. \(\mathbf{C}\) is the damping assembly matrix. \(\mathbf{K}\) is the stiffness assembly matrix which considers stiffness effects of the shaft element and the disk element. \(\mathbf{Q}\) is the excitation matrix. To study the influence of unbalanced exciting force caused by centroid offset on a rotor system, \(\mathbf{Q}\) is the unbalanced exciting force which is only considering the unbalanced force caused by multi-stage rotors stacking. We can get the following results by substituting Equation (13) into Equation (14):  
\[\left[ \begin{array}{c}\mathbf{q}_{c}\\ \mathbf{q}_{s}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{K}-M\Omega^{2}& \Omega(\mathbf{C}-\Omega\mathbf{G})\\ -\Omega(\mathbf{C}-\Omega\mathbf{G})&\mathbf{K}-M\Omega^{2}\end{array} \right]^{-1}\left[ \begin{array}{c}\mathbf{Q}_{c}\\ \mathbf{Q}_{s}\end{array}