## 3 Performance Analysis of the Balancing Actuator
### Self-Lock Capacity  
As shown in Section 2.2.3, the normal condition of the balancing actuator is the self-lock condition and the torque generated by the permanent magnets and the tooth rings is considered as self-lock torque, which keeps the counterweight disc synchronously rotating with the balancing actuator. Therefore, the self-lock torque should be checked as to whether it can prevent changes in mass distribution without control commands. Because the actuator is an axisymmetric structure and the self-lock torque is only related to the counterweight disc with permanent magnets and the tooth rings, a finite element model of these three components is established based on Maxwell software, as shown in Figure 9.  
Figure 8: Diagram of balancing accuracy analysis.  
After importing the 3D models of these components from SolidWorks into Maxwell, the material properties of each component were defined separately and a computational area outside the component was created. The tetrahedron element was mainly adopted during the process of meshing and the grid of counterweight disc and magnets was encrypted compared to the tooth rings. In creating parameter scan settings, the rotating angle of the counterweight disc and all permanent magnets around the central axis was defined as variable \(x_{1}\). Since there were 36 permanent magnets on the counterweight disc and 18 iron teeth on the tooth rings, the change period of self-locking force was about 10 degrees. Therefore, the variable \(x_{1}\) was chosen from 0 to 20\({}^{\circ}\) in order to fully display the force change of the counterweight plate and the step length was 0.5\({}^{\circ}\). Before adaptive calculation, the parameters, such as maximum number of iterations and proportion of encryption cells per iteration, were respectively set and then the torque of the balancing disc and magnets relative to the central axis was calculated. To observe the distribution of magnetic field between magnets and iron teeth, a cross section was selected and the magnetic field between two pairs of magnets and iron teeth was displayed.  
According to the analysis results shown in Figure 10, two conclusions can be drawn: (1) the change period of self-lock torque is equal to the central angle between two adjacent magnets and in a step-by-step process, in accordance with the sinusoidal self-lock torque changes, the maximum torque value appears at 1/4 and 3/4 step positions; and (2) with the increase of air gap \(\delta_{3}\) between the counterweight disc and the iron teeth, the self-lock torque decreases rapidly and when the air gap is 1.0 mm, the self-lock torque is equal to 38.9 mN\(\cdot\)m.  
The relationship between the self-lock torque \(M_{\mathrm{T}}\) and the allowable acceleration \(\alpha\) of the actuator can be written as:  
\[\alpha_{n}\leq\frac{M_{\mathrm{Tmax}}-mgR}{k_{SF}J},t\geq\frac{2\pi n}{60 \alpha_{n}} \tag{7}\]  
where \(m\) and \(J\) are total mass and total rotary inertia of the counterweight disc respectively, \(R\) is the radius of the mass center, \(k_{SF}\) is the designed safety factor and n is the operation speed.  
From the inertia analysis of the counterweight disc and counterweight block [25], the total mass and total rotary inertia are 84.4 g and 32,000 g\(\cdot\)mm\({}^{2}\), respectively. The designed safety factor is set to 1.5, so the allowable acceleration of the actuator should be lower than 353 rad/s\({}^{2}\) according to Equation (7), that is, when the operation speed of the rotor system is 4200 rpm, the starting time of the spindle should be longer than 1.25 s. If the starting time is required to be shorter, the self-lock torque can be adjusted through the decrease of the air gap \(\delta_{3}\).  
Figure 9: Finite element analysis of self-lock torque.: (**a**) Finite element model; (**b**) Mesh generation; (**c**) Directional view of meshing; (**d**) Analysis of self-lock torque; (**e**)Select cross section to observe magnetic field distribution; (**f**) Local view of magnetic field.