## 4 FEM Verification and Analysis
### Validation Analysis of Electromagnetic Torque and Magnetic Levitation Force Models
#### 4.2.3 FEM Validation Analysis of Eccentric Magnetic Pull Force Model  
In order to verify the mathematical model of the eccentric magnetic pull force, the static eccentric displacement of the rotor was gradually increased along the \(y\)-axis within the range of 0 mm to 0.3 mm with a step size of 0.05 mm. Since the armature's reflected magnetic potential is generally much lower than that of the permanent magnets, the FEM analysis was performed under the condition of zero armature current.  
Figure 10 shows the dynamic waveform of the eccentric magnetic pull force with time (or rotor position angle), and Figure 11 shows the comparison between the FEM analysis results of the eccentric magnetic pull force and its model calculation results. Based on Figures 10 and 11, the following research results were obtained:  
(1) In the case of an unsaturated magnetic circuit, when the rotor experiences static eccentricity along the positive \(y\)-axis direction, the resultant eccentric magnetic pull force is directly proportional to the static eccentricity displacement.  
(2) The calculated values of the eccentric magnetic pulling force model are consistent with the FEM analysis values. For example, when the eccentric distance of the rotor is 0.1 mm, the model-calculated value of the eccentric magnetic pull force is 10.2035 N, while its FEM analysis value is 10.9014 N, and the error between them is about 6.4%. The main reason for the calculation error is that the influence of the harmonic magnetic potential was not considered in the process of deriving the radial electromagnetic force model based on the virtual displacement method. To obtain a more accurate model of eccentric magnetic pull force, it can be equipped with a calibration coefficient \(k_{ec}\) of about 1.072 (here, \(k_{ec}\) is equal to 1.068).  
Figure 10: Eccentric magnetic pull waveform at static eccentricity.  
Due to machining accuracy and other reasons, it is inevitable that the rotor will have different degrees of mass eccentricity, which will excite the rotor to produce periodic dynamic eccentric displacement. Under the conditions of different dynamic eccentric distances, based on the FEM analysis model of the six-phase DSW-12/2-BPMSM, Figure 12 shows the dynamic waveform of the eccentric magnetic pull force along the \(y\)-axis direction as a function of time (or rotor position angle). The waveform of the eccentric magnetic pull force along the \(x\)-axis direction has a 90-degree phase difference with that along the \(y\)-axis direction, which is omitted here.  
From Figure 12, it can be seen that the eccentric magnetic pull force changes dynamically with time (or rotor position angle) in a sinusoidal pattern under the dynamic eccentricity condition of the rotor. At the same eccentric distance, the amplitude of the dynamic eccentric magnetic pull force is consistent with the FEM analysis value and model calculation value of the static eccentric magnetic pulling force (refer to Figure 10). Therefore, the FEM experimental results of the dynamic eccentric magnetic pull force prove the validity of the eccentric magnetic pull force model from another side.  
Figure 11: Variation curve of eccentric magnetic tension with static eccentric displacement.  
Figure 12: Eccentric magnetic pull waveform during dynamic eccentricity.