## 3 Performance Analysis of the Balancing Actuator  
### Balancing Ability  
According to the working principle of the actuator introduced in Section 2.2, the mass distribution of the actuator is mainly determined by the two counterweight discs with counterweight blocks. As shown in Figure 7, there are two symmetrical counterweight discs in the actuator and each one has a pair of counterweight blocks. Because the counterweight blocks are semicircular, they form an eccentric mass for each counterweight disc, that is, U\({}_{\text{A}}\) and U\({}_{\text{B}}\). When the actuator is pre-installed inside the rotor, the counterweight blocks of the two discs are in opposite directions, so U\({}_{\text{A}}\) and U\({}_{\text{B}}\) offset each other and the compensation mass is zero, shown in Figure 7b. During the balancing process, the angle between U\({}_{\text{A}}\) and U\({}_{\text{B}}\) decreases through the step rotation of the two counterweight discs and the compensation mass increases gradually. When the counterweight blocks of the two discs are in the same direction, U\({}_{\text{A}}\) and U\({}_{\text{B}}\) are superimposed and the compensation mass reaches maximal value. This value can be considered as the balancing ability of the actuator and is four times that of one counterweight block.  
Taking one counterweight block as an example, it is semicircular and its mass distribution is symmetrical to the \(y\)-axis. Therefore, its center of gravity must be on the \(y\)-axis. Assuming the coordinates of the center of gravity are \((0,y_{c})\), \(y_{c}\) can be obtained by:  
\[y_{c}=\frac{1}{A}\iint\limits_{D}ydS=\frac{1}{A}\iint\limits_{D}r\cos\theta \cdot rd\theta\cdot dr\ =\ \frac{1}{A}\int\limits_{-\frac{\theta}{2}}^{\frac{\theta}{2}}\cos\theta\cdot d \theta\int\limits_{r}^{R}r^{2}dr\ =\ \frac{4\sin\frac{\theta}{2}(R^{3}-r^{3})}{3 \theta(R^{2}-r^{2})} \tag{1}\]  
where \(R\) is the outer radius, \(r\) is the inner radius and \(\theta\) is the center angle of the counterweight block.  
The mass of the counterweight block is expressed by:  
\[m_{c}=\rho V=\frac{1}{2}\rho B\theta(R^{2}-r^{2}) \tag{2}\]  
where \(\rho\) is density, B is thickness and \(V\) is volume of the counterweight block.  
Because the designed counterweight block is a ladder structure, in the calculation process, it should be divided into two parts, upper and lower, with the ladder as the boundary. The specific parameters of the counterweight block are shown in Table 2. Assuming the center of gravity and mass  
Figure 7: Structure drawing of counterweight disc: (**a**) The distribution of two counterweight discs; (**b**) Initial state of two discs and blocks; (**c**) Specific view of counterweight disc; (**d**) Analytical diagram of counterweight block.  
of these two parts are \(y_{c1}\), \(y_{c2}\), \(m_{c1}\) and \(m_{c2}\), the balancing ability of the counterweight block \(U_{b}\) is written as:  
\[U_{b}=m_{c1}y_{c1}+m_{c2}y_{c2} \tag{3}\]  
Bringing these parameters into Equations (1)-(3), the balancing ability of one counterweight block is 292 g\(\cdot\)mm. Therefore, the balancing ability of the actuator is four times that of one block, that is, 1168 g\(\cdot\)mm. Here the counterweight block is made of brass; when the material is changed to tungsten-base alloy, the value of density can increase from 0.0085 to 0.017 and the balancing ability can be doubled to 2336 g\(\cdot\)mm.