## 2 Methods
### Assembly Datun for Unbalance Optimization  
The mathematical definition of unbalance is the product of the mass of a rotor and the eccentricity error of the center of mass and can also be expressed as the product of an unbalanced mass block and its effect radius. During the dynamic balance test of a rotor, journals of the front and back shafts of the rotor are respectively positioned on the support frames on left and right sides of a dynamic balancing machine. The axis of rotation should be approximately the central line between the journals of the front and back shafts, and the measured vector radius of the mass eccentricity should be perpendicular to such an axis of rotation (i.e., the orientation-varying axis in Figure 3). This axis of rotation will vary along with variations in the assembly orientations of rotors at different stages. For the optimization of the rotor unbalance, this axis of rotation should be used as the assembly datum. The unbalance of rotors at different stages after assembly can be obtained as follows:  
\[U_{n}=m_{n}r^{\prime}{}_{n}, \tag{6}\]  
where \(U_{n}\) is the unbalance of rotor \(n\) after assembly, that is a product of \(m_{n}\) (unbalanced mass) and \(r_{n}{}^{\prime}\) (i.e., the vector of the effect radius of rotor \(n\)). As for \(r_{n}{}^{\prime}\), it can be solved by the following steps:  
Figure 3: Assembly datum for unbalance optimization of four-stage rotor. Step-1: A linear equation expressing the orientation-varying axis can be written as follows:  
\[\frac{x-x_{0}}{e^{\prime}nx-x_{0}}=\frac{y-y_{0}}{e^{\prime}ny-y_{0}}=\frac{z-z_{ 0}}{e^{\prime}nz-z_{0}}=t_{n}, \tag{7}\]  
where (\(x_{0}\), \(y_{0}\), \(z_{0}\)) is a coordinate vector of the bottom center of the bottom rotor. Under the circumstance that the table axis serves as the datum for measurement of the rotor's geometric parameters, such a point is the origin of the coordinates by default. Step-2: As for the plane \(Q_{n}\) that is extended across an unbalanced mass point of rotor \(n\) and perpendicular to the orientation-varying axis, it is expressed in the equation below:  
\[\left\{ \begin{array}{l}\left(e^{\prime}nx-x_{0}\right)\left(x-r^{\prime}nx \right)+\left(e^{\prime}ny-y_{0}\right)\left(y-r^{\prime}ny\right)+\left(e^{ \prime}nz-z_{0}\right)\left(z-r^{\prime}nz\right)=0\\ \left(R^{\prime}n=\left[\begin{array}{cc}r^{\prime}nx&r^{\prime}ny&r^{ \prime}nz\end{array}\right],R_{n}=\left[\begin{array}{cc}r_{nx}&r_{ny}&r_{ nz}\end{array}\right]\right)\end{array} ,\right. \tag{8}\]  
where \(R_{n}{}^{\prime}\) refers to a coordinate vector of the unbalanced mass point of rotor \(n\) after assembly. \(R_{n}\) refers to the initial coordinate vector of the unbalanced mass point of rotor \(n\) before assembly, and it can be measured by the dynamic balancing machine or artificially added. By substituting \(R_{n}\) into Equation (5), \(R_{n}{}^{\prime}\) is solved. Step-3: Coordinates of an intersecting point between the orientation-varying axis and the plane \(Q_{n}\) are expressed as follows:  
First, Equations (7) and (8) form a simultaneous equation. Through such a simultaneous equation, parameter \(t_{n}\) in the linear equation expressing the orientation-varying axis can be obtained:  
\[t_{n}=\frac{\left(e^{\prime}nx-x_{0}\right)\left(r^{\prime}nx-x_{0}\right)+ \left(e^{\prime}ny-y_{0}\right)\left(r^{\prime}ny-y_{0}\right)+\left(e^{ \prime}nz-z_{0}\right)\left(r^{\prime}nz-z_{0}\right)}{\left(e^{\prime}nx-x_{ 0}\right)^{2}+\left(e^{\prime}ny-y_{0}\right)^{2}+\left(e^{\prime}nz-z_{0} \right)^{2}}. \tag{9}\]  
By substituting \(t_{n}\) into Equation (7), the coordinates of the intersection point \(Jn\) can be acquired. Step-4: The effect radius of the unbalanced mass relative to the orientation-varying axis, that is the modulus of, \(r_{n}{}^{\prime}\), is expressed in the following equation:  
\[r^{\prime}{}_{n}=\sqrt{\left(j_{nx}-r^{\prime}{}_{nx}\right)^{2}+\left(j_{ny }-r^{\prime}{}_{ny}\right)^{2}+\left(j_{nz}-r^{\prime}{}_{nz}\right)^{2}}\left( J_{n}=\left[ \begin{array}{cc}j_{nx}&j_{ny}&j_{nz}\end{array}