## 2 System Modelling and Problem Formulation
### Characteristics of UEF and Its Influence on the Vehicle Performance  
In this paper, the rotor pole-pair number of PMBDC motor adopted is 12. Thus, rotational angular of the rotor is \(2\pi/12=0.52\) (rad) in one PWM duty cycle, as shown in Figure 2b. For above-mentioned mathematical formulas, it can be found that UMF is a function of rotor position \(\int\omega_{r}dt\), phase currents (\(I_{A}\), \(I_{B}\), \(I_{C}\)) and eccentricity \(e_{d}\) (\(z\)). Figure 4a shows the influence of the eccentricity and rotor position on UEF when the phase current is 20A. It can be observed that the mean value of UEF rises rapidly with the minor increase of the eccentricity. Similarly, Figure 4b presents the relationship between the rotor eccentricity, phase current and UMF when the rotor eccentricity is 0.1 mm. The mean value and amplitude value of the UEF rise as the phase current increases. By examining the influence of the eccentricity, rotor position and the phase current on the UEF, it can be noted that the sensitivity of the eccentricity is greater than the sensitivity of the rotor position and phase current. This means that the discussion of the eccentricity is of great significance.  
Figure 3: IWM-driven system without a speed reducer. (**a**) Main structure. (**b**) Quarter vehicle model.  
The parameters of the hub-driven EV are listed in the Appendix A. Based on the electric vehicle model driven by IWM established in Section 2.1, the performance of an electric vehicle with a passive suspension was investigated at a speed of 25 km/h on a B-class road (Figure 5). Figure 5a shows the error between the ideal and real vehicle speed; Figure 5b-f describe wheel speed (rotor speed), slip rate, A-phase current of IWM, driving torque \(T_{e}\) produced by PMBDC motor, and road speed excitation, respectively. It can be observed that all signals are reasonable, indicating that the model established is correct. Figure 5g-i describe the vertical component of the UEF, the eccentricity of rotor and the acceleration of the body with or without coupling effect, respectively. In Figure 5, the coupling effect is considered if there is \(F_{r}\), while it is not considered if there is no \(F_{r}\). It can be concluded that the coupling effect exacerbates the electromagnetic force oscillation, increases the eccentricity of the rotor, and aggravates the acceleration of sprung mass. The eccentricity of rotor provokes UEF, as shown in Figure 4a. Synchronously, the UEF intensifies the eccentricity, as shown in Figure 5e. This mutual promotion phenomenon aggravates the electromagnetic coupling effect, thus intensifying the wear of bearing and shortening the life of the motor. Furthermore, it aggravates the acceleration of sprung mass and provokes cacophony, deteriorating the vehicle comfort. Therefore, the coupling effect in IWM should be seriously considered when designing active suspension system for hub-driven EVs.  
Figure 4: The characteristics of unbalanced electromagnetic force (UEF): (**a**) under different rotor position and eccentricity; (**b**) under different rotor position and phase current.  
Figure 5: The response of the electric vehicle on stochastic uneven road: (**a**) vehicle speed; (**b**) wheel speed; (**c**) slip rate; (**d**) A-phase current; (**e**) driving torque; (**f**) road speed excitation; (**g**) vertical component of the UEF; (**h**) the rotor eccentricity; (**i**) the sprung mass acceleration.