## 2 System Modelling and Problem Formulation
### Hub-Driven Electric Vehicle Modelling
#### 2.1.1 Unbalanced Electromagnetic Force Model  
The dynamic vibration in vehicle and UEF (calculated by the UEF model) can evoke a magnet gap deformation which results in magnetic force oscillation as shown in Figure 2a, where \(O_{s}\) and \(O_{r}\) are the geometric center of the stator and the rotor, respectively. In this paper, we are mainly concerned about the rotor eccentricity in the vertical direction. Hence, the uneven magnet gap of IWM is a function of angle \(a_{m}\) and its expression is  
\[\delta_{e}(a_{m})=\delta-e_{d}(z)\cos(a_{m}) \tag{1}\]  
where \(e_{d}\) (\(z\)) denotes relative displacement between the rotor and the stator. It can be provided by vertical vibration model of the EVs with IWMs. Rotor eccentricity leads to the change of the permeance in magnet gap, and it can be calculated by:  
\[ \begin{array}{ll}\lambda_{e}&=\frac{u_{0}}{\delta_{e}(a_{m})}=\frac{u_{0}}{ \delta-e_{d}(z)\cos(a_{m})}=\frac{u_{0}}{\delta}\cdot\frac{5}{\delta-e_{d}(z) \cos(a_{m})}\\ &=\lambda\cdot\frac{\delta}{\delta-e_{d}(z)\cos(a_{m})}=\lambda\xi_{\delta} \end{array}  \tag{2}\]  
where \(\xi_{\delta}\) is the correction coefficient for the permeance. In this study, the 27-slot/24-pole surface PMBDC motor is adopted. The analytical expressions of the radial and the tangential flux density (\(B_{\_}{pru}\)(\(r\), \(a_{m}\), \(t\)) and \(B_{\_}{ptu}\)(\(r\), \(a_{m}\), \(t\))) of the slotless field model in the middle of the air gap (\(r=R_{s}-g/2\)) produced by the permanent magnets (PMs) can be given by  
\[B_{\_}{pru}(r,a_{m},t)=\sum_{n=1,3,5\ldots}^{\infty}K_{B}(n)\cdot f_{\_}{Br}(r )\cdot\cos[np(a_{m}-\omega_{r}t-\theta_{0})] \tag{3}\]  
\[B_{\_}{ptu}(r,a_{m},t)=\sum_{n=1,3,5\ldots}^{\infty}K_{B}(n)\cdot f_{\_}{Br}(r )\cdot\sin[np(a_{m}-\omega_{r}t-\theta_{0})] \tag{4}\]  
Figure 1: Hub-driven electric vehicle model. Figure 2: (**a**) Magnet gap deformation of the in-wheel motor (IWM) (air-gap exaggerated) and (**b**) the winding distribution of the permanent magnet brushless direct contact (PMBDC) motor. where \(p\) denotes the rotor pole-pair number, \(\alpha_{m}\) denotes the angular position, and \(\omega_{r}\) denotes angular velocity of the rotor. The parameters of IWM are listed in Appendix A. (The details about the motor can be found in the literature [40,41]). The winding distribution of the PMBDC motor is shown in Figure 2b. The radial and the tangential flux density (\(B_{\_{an}}(r,a_{m},t)\) and \(B_{\_{an}}(r,a_{m},t)\)) produced by 3-phase winding of the IWM in the middle of the air gap can be calculated as  
\[ \begin{array}{rl}B_{\_{an}}(r,a_{m},t)=&\sum\limits_{\_}{=1}^{\infty}B_{\_}{ \_}{\textit{tr}}\Big{\{}I_{A}(t)\sum\limits_{\_}{i=1}^{9}s_{\_}{Ai}\cos\gamma \Big{[}a_{m}-\frac{2\pi}{Q_{\_}}(a_{\_}{Ai}-1)\Big{]}+I_{\_}{B}(t)\sum\limits_ {\_}{i=1}^{9}s_{\_}{Bi}\cos\gamma\Big{[}a_{m}-\frac{2\pi}{Q_{\_}}(a_{\_}{Bi}-1) \Big{]}\\ &\hskip 142.26378 pt+I_{\_}{C}(t)\sum\limits_{\_}{i=1}^{9}s_{\_}{C}\cos\gamma \Big{[}a_{m}-\frac{2\pi}{Q_{\_}}(a_{\_}{Ci}-1)\Big{]}\Big{\}}\\ \\ B_{\_{an}}(r,a_{\_}{m},t)=&\sum\limits_{\_}{=1}^{\infty}B_{\_}{\textit{tr}} \bigg{\{}I_{A}(t)\sum\limits_{\_}{i=1}^{9}s_{\_}{A1}\cos\gamma\Big{[}a_{m}- \frac{2\pi}{Q_{\_}}(a_{\_}{Ai}-1)\Big{]}+I_{\_}{B}(t)\sum\limits_{\_}{i=1}^{9} s_{\_}{Bi}\cos\gamma\Big{[}a_{\_}{m}-\frac{2\pi}{Q_{\_}}(a_{\_}{Bi}-1)\Big{]}\\ &\hskip 142.26378 pt+I_{\_}{C}(t)\sum\limits_{\_}{i=1}^{9}s_{\_}{C}\cos\gamma \Big{[}a_{\_}{m}-\frac{2\pi}{Q_{\_}}(a_{\_}{Ci}-1)\Big{]}\Big{\}}\end{array}  \tag{5}\]  
where \(I_{\_}{A}(t)\), \(I_{\_}{B}(t)\), and \(I_{\_}{C}(t)\) denote three phase winding current. The detailed derivation of Equations (5) and (6) are referred to in the literature [42,43]. The flux density can be decomposed as the superposition of stator and rotor magnetic fields in the linear case [34].