## 4 Control Strategy of the Balancing Actuator  
### Basic Principle  
The initial unbalance vibration \(\overset{\rightarrow}{V}_{0}\) of the detected rotor system is set to \(\overset{\rightarrow}{V}_{0}=V_{0}\angle\alpha_{0}\) with the initial unbalance vector \(\overset{\rightarrow}{U}_{0}=U_{0}\angle\varphi_{0}\). Because the balancing ability of two counterweight discs is equal to \(U_{\text{AB}}\) and their initial phases are \(\varphi_{A0}\) and \(\varphi_{B0}\), respectively, the initial compensation vectors of two counterweight discs can be expressed by \(\overset{\rightarrow}{U}_{A0}=U_{AB}\angle\varphi_{A0}\) and \(\overset{\rightarrow}{U}_{B0}=U_{AB}\angle\varphi_{B0}\).  
After one trial step rotation, the phases of the two counterweight discs are changed to \(\varphi_{A1}\) and \(\varphi_{B1}\), respectively, their compensation vectors are expressed by \(\overset{\rightarrow}{U}_{A1}=U_{AB}\angle\varphi_{A1}\) and \(\overset{\rightarrow}{U}_{B1}=U_{AB}\angle\varphi_{B1}\), and the unbalance vibration is \(\overset{\rightarrow}{V}_{1}=V_{1}\angle\alpha_{1}\).  
Therefore, the influence coefficient vector of the detected rotor system can be written as:  
\[\overset{\rightarrow}{K}=k\angle\beta=\frac{\overset{\rightarrow}{V}_{1}- \overset{\rightarrow}{V}_{0}}{(\overset{\rightarrow}{U}_{0}+\overset{ \rightarrow}{U}_{A1}+\overset{\rightarrow}{U}_{B1})-(\overset{ \rightarrow}{U}_{0}+\overset{\rightarrow}{U}_{A0}+\overset{\rightarrow}{U }_{B0})}=\frac{\overset{\rightarrow}{V}_{1}-\overset{\rightarrow}{V}_{0}}{( \overset{\rightarrow}{U}_{A1}+\overset{\rightarrow}{U}_{B1})-(\overset{ \rightarrow}{U}_{A0}+\overset{\rightarrow}{U}_{B0})} \tag{8}\]  
and then, the initial unbalance vector is solved by:  
\[\overset{\rightarrow}{U}_{0}=U_{0}\angle\varphi_{0}=\frac{\overset{ \rightarrow}{V}_{0}}{K}=\frac{V_{0}}{K}\angle(\varphi_{0}-\beta) \tag{9}\]  
In order to eliminate the initial unbalance vector, a compensation vector that has equal magnitude but in the opposite position compared to the initial unbalance must be generated by the balancing actuator. The compensation vector can be expressed by:  
\[\overset{\rightarrow}{U}_{c}=U_{c}\angle\varphi_{c}=-\overset{ \rightarrow}{U}_{0}=U_{0}\angle(\varphi_{0}\pm 180^{0}) \tag{10}\]  
where the phase \(\varphi_{c}\) and \(\varphi_{0}\) must be within the range [0, 360].  
The relationship between the two compensation vectors of the counterweight discs and the resultant compensation vector during the balancing process are shown in Figure 11. In the triangle formed by these three vectors, the magnitudes of the two compensation vectors are equal, so it is an isosceles triangle and the angles between the two compensation vectors and the resultant compensation vector are the same. According to the cosine theorem, the included angle is obtained by:  
Figure 10: Analysis results of self-lock torque. (**a**) Variation of torque with rotating angle \(x_{1}\); (**b**) Variation of torque with air gap \(\delta_{3}\).  
\[\varphi=\arccos(\frac{U_{B}^{2}+U_{c}^{2}-U_{A}^{2}}{2U_{B}U_{c}})=\arccos(\frac{U_ {c}}{2U_{AB}}) \tag{11}\]  
To eliminate the initial unbalance vector, the two counterweight discs should step rotate to their phases and the phase values are given by  
\[\left\{\begin{array}{l}\varphi_{A}=\varphi_{c}+\varphi=\varphi_{0}\pm 180^{0}+ \arccos(\frac{U_{0}}{2U_{AB}})\\ \varphi_{B}=\varphi_{c}-\varphi=\varphi_{0}\pm 180^{0}-\arccos(\frac{U_{0}}{2U_{ AB}})\end{array}\right. \tag{12}\]  
Therefore, when the initial vibration of the detected rotor is achieved by the data collection unit, the new positions of the two counterweight discs can be solved by Equations (8)-(12).