## 3 Performance Analysis of the Balancing Actuator
### Balancing Accuracy  
When the actuator is installed in the hollow rotor, the counterweight blocks of the two counterweight discs are in opposite directions and the compensation mass is zero. During the balancing process, through the step rotation of the two counterweight discs, the compensation mass is gradually produced. To analyze the balancing accuracy of the actuator, the step rotation procedure should be considered first.  
Assuming the number of magnets on the counterweight disc is \(N\), the center angle of each step rotation \(\Delta\varphi\) is equal to 360/N. When the compensation mass increases from 0 to the maximum value, there are three rotary methods for the two counterweight discs:  
(1) The two counterweight discs approach the symmetrical axis at the same time and rotate one step every time. During the procedure, the total steps is N/4 and the compensation vector \(U\) and the variation of compensation vector \(\Delta U\) can be expressed by:  
\[\left\{\begin{array}{l}U(j+1)=U_{AB}*\left(e^{(\varphi_{A}+j\Delta\varphi)i }+e^{(\varphi_{B}-j\Delta\varphi)i}\right)\\ \Delta U(j)=U(j+1)-U(j)\end{array}\right.\left(0\leq j\leq Z/4\right) \tag{4}\]  
where \(U_{\mathrm{AB}}\) is the balancing ability of each counterweight disc, \(\varphi_{A}\) is the initial phase of one disc and equal to 0 and \(\varphi_{B}\) is the initial phase of the other disc and equal to 180.  
(2) The two counterweight discs approach the symmetrical axis in turn and rotate one step every time. The total steps is N/2 and these two vectors are expressed by:  
\[\left\{\begin{array}{l}U(j+1)=U_{AB}*(e^{(\varphi_{A}+ceil(\frac{j}{2}) \Delta\varphi)i}+e^{(\varphi_{B}-floor(\frac{j}{2})\Delta\varphi)i})\\ U(j+1)=U_{AB}*(e^{(\varphi_{A}+\frac{j}{2}\Delta\varphi)i}+e^{(\varphi_{B}- \frac{j}{2}\Delta\varphi)i})\\ \Delta U(j)=U(j+1)-U(j)\end{array}\right.\qquad(j\,is\,even\,\,number)\qquad \qquad(0\leq j\leq Z/2) \tag{5}\]  
where ceil and floor are two integral functions.  
(3) One counterweight disc stays still while the other gradually approaches the opposite side. The total steps is also N/2 and these two vectors are expressed by:  
\[\left\{\begin{array}{l}U(j+1)=U_{AB}*(e^{(\varphi_{A}+j\Delta\varphi)i}+e^{ (\varphi_{B})i})\\ \Delta U(j)=U(j+1)-U(j)\end{array}\right.\ (0\leq j\leq Z/2), \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Value** & **Item** & **Value** \\ \hline Outer radius (mm) & 22.5 & Depth of the upper part (mm) & 6.5 \\ Ladder radius (mm) & 18 & Depth of the lower part (mm) & 4.25 \\ Inner radius (mm) & 13 & Density (g\(/\)mm\({}^{3}\)) & 0.0085 \\ Center angle (mm) & \(\pi\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of counterweight block.  
For the actuator in the balancing experiments, the balancing ability of each counterweight disc \(U_{AB}\) is 584 g\(\cdot\)mm and the number of magnets \(N\) is 36; the computational results of the three rotary methods are shown in Figure 8.  
Two conclusions can be drawn from the results: (1) the variation of compensation vector for the latter two rotary methods is a fixed value, that is, \(\Delta U\) for each step at any position is always 50.9 g\(\cdot\)mm; (2) for the first rotary method, \(\Delta U\) decreases from 101.4 g\(\cdot\)mm to 8.9 g\(\cdot\)mm, that is, when the phase difference between the two counterweight discs is larger than 90\({}^{0}\), the balancing accuracy of the first method is inferior to that of the latter two methods but when the phase difference is smaller than 90\({}^{0}\), the balancing accuracy of the first method is better. In the actual control process of the actuator, three methods should be combined and the variation range of the compensation vector is within [8.9, 50.9]. Therefore, the maximum value of \(\Delta U\) can be considered as the balancing accuracy of the actuator and is equal to 50.9 g\(\cdot\)mm.